Community structure: A comparative evaluation of community detection   methods

Vinh-Loc Dao; C\'ecile Bothorel; Philippe Lenca

arXiv:1812.06598·cs.SI·April 15, 2021

Community structure: A comparative evaluation of community detection methods

Vinh-Loc Dao, C\'ecile Bothorel, Philippe Lenca

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TL;DR

This paper provides a comprehensive empirical comparison of various community detection methods across multiple network types, analyzing their performance, community structures, and suitability for different applications.

Contribution

It offers a detailed evaluation of community detection algorithms, including their computational efficiency and community characteristics, aiding practitioners in selecting appropriate methods.

Findings

01

Different methods produce varying community size distributions.

02

Optimization schemes influence community detection outcomes.

03

Partitioning strategies differ significantly across methods.

Abstract

Discovering community structure in complex networks is a mature field since a tremendous number of community detection methods have been introduced in the literature. Nevertheless, it is still very challenging for practioners to determine which method would be suitable to get insights into the structural information of the networks they study. Many recent efforts have been devoted to investigating various quality scores of the community structure, but the problem of distinguishing between different types of communities is still open. In this paper, we propose a comparative, extensive and empirical study to investigate what types of communities many state-of-the-art and well-known community detection methods are producing. Specifically, we provide comprehensive analyses on computation time, community size distribution, a comparative evaluation of methods according to their optimisation…

Figures33

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Tables6

Table 1. Table 1 : Community detection methods involved in the study.

Approach	Publication	Ref. label	Time order	Implementation
Edge removal	[\citenameGirvan & Newman, 2002]	GN	$𝒪 (n m^{2})$	igraph¹¹1Published at http://igraph.org/
Edge removal	[\citenameRadicchi et al., 2004]	RCCLP	$𝒪 (m^{4} / n^{2})$	Authors²²2Published at http://homes.sice.indiana.edu/filiradi/resources.html
Modularity optimization	[\citenameClauset et al., 2004]	CNM	$𝒪 (m \log^{2} (n))$	igraph
	[\citenameBlondel et al., 2008]	Louvain	$𝒪 (n \log (n))$	Authors³³3Published at https://sourceforge.net/projects/louvain/
	[\citenameNewman, 2006a]	SN	$𝒪 (n m \log (n))$	igraph
Dynamic process	[\citenamePons & Latapy, 2005]	Walktrap	$𝒪 (n)$	igraph
	[\citenameRosvall & Bergstrom, 2007]	Infomod	NA	Authors⁴⁴4Published at http://www.tp.umu.se/~rosvall/code.html
	[\citenameRosvall et al., 2009]	Infomap	$𝒪 (m)$	Authors⁵⁵5Published at http://www.mapequation.org//igraph
Statistical inference	[\citenameLancichinetti et al., 2011]	Oslom	$𝒪 (n^{2})$	Authors⁶⁶6Published at http://www.oslom.org/
Statistical inference	[\citenameRiolo et al., 2017]	(DC)SBM	Parametric	Authors⁷⁷7Published at http://www-personal.umich.edu/~mejn/
Other methods	[\citenameReichardt & Bornholdt, 2006]	RB	$𝒪 (n^{2} l o g (n))$	igraph
	[\citenameRaghavan et al., 2007]	LPA	$𝒪 (m)$	igraph
	[\citenameXie & Szymanski, 2012]	SLPA	$𝒪 (m)$	Authors⁸⁸8Published at https://sites.google.com/site/communitydetectionslpa/
	[\citenameMeo et al., 2014]	Conclude	$𝒪 (n + m)$	Authors⁹⁹9Published at http://www.emilio.ferrara.name/code/conclude/

Table 2. Table 3 : Ranking of analyzed methods according to the amount of time consumed, to identify community structures on networks of the dataset.

Method label	Rank by average	Rank by median	Scalability
RCCLP-3	9	8	Low
CNM	5	3	Medium
Louvain	1	2	High
SN	3	5	High
Walktrap	4	4	High
Infomod	12	9	Low
Infomap	6	7	Medium
Oslom	11	14	Low
DCSBM	8	12	Low
RB	10	13	Low
LPA	2	1	High
SLPA	7	6	Medium
Concude	13	11	Low

Table 3. Table 4 : Ranking of analyzed methods according to their number of detected communities. A method is considered to over-fit if it detects asymptotically more than n 𝑛 \sqrt{n} clusters. The group numbers exhibit the estimated similarity based on fitting quality.

Method label	Size wrt. k-planted model	Fitting
GN	Bigger	Over-fit
RCCLP-3	Bigger	Over-fit
RCCLP-4	Bigger	Over-fit
CNM	Close	Over-fit
Louvain	Close	Under-fit
SN	Smaller	Under-fit
Walktrap	Bigger	Over-fit
Infomod	Close	Under-fit
Infomap	Bigger	Over-fit
Oslom	Smaller	Under-fit
SBM	Smaller	Under-fit
DCSBM	Smaller	Under-fit
RB	Smaller	Under-fit
LPA	Bigger	Over-fit
SLPA	Bigger	Over-fit
Concude	Bigger	Over-fit

Table 4. Table 5 : Extract of the smallest Kolmogorov-Smirnov distances between community size distributions

Method	Method	KS-Distance	p-value
SLPA	LPA	0,007	0,017
Conclude	LPA	0,067	0,075
DCSBM	RB	0,070	0,000
RCCLP-4	Walktrap	0,080	0,000
GN	RCCLP-3	0,098	0,000
Conclude	SLPA	0,118	0,000
Infomod	SBM	0,118	0,000
Infomod	SN	0,126	0,000
DCSBM	Infomod	0,127	0,000
Louvain	SN	0,127	0,000

Table 5. Table 6 : Contingency table of P 1 subscript 𝑃 1 P_{1} and P 2 subscript 𝑃 2 P_{2} on the same graph provides information about the similarity between the two partitions.

Partition $P_{2}$
	$c_{1}^{(2)}$	$c_{2}^{(2)}$	$\dots$	$c_{S}^{(2)}$	$\sum$
$c_{1}^{(1)}$	$n_{11}$	$n_{12}$	$\dots$	$n_{1 S}$	$n_{1, \cdot}$
$c_{2}^{(1)}$	$n_{21}$	$n_{22}$	$\dots$	$n_{2 S}$	$n_{2, \cdot}$
$⋮$	$⋮$	$⋮$	$⋱$	$⋮$	$⋮$
$c_{R}^{(1)}$	$n_{R 1}$	$n_{R 2}$	$\dots$	$n_{R S}$	$n_{R, \cdot}$
$\sum$	$n_{\cdot 1}$	$n_{\cdot 2}$	$\dots$	$n_{\cdot S}$	$n$

Table 6. Table 7 : Some popular validation metrics for comparing community partitions

Label	Range	Measure
RI	$[0, 1]$	Fraction of commonly grouped and separated vertices in two partitions.
ARI	$[0, 1]$	Rand index with a chance correction, less sensitive to differences of community sizes.
NMI	$[0, 1]$	Information theoretic approach, indicates how much information in one partition will help to guess the other.
AMI	$[0, 1]$	Similar to NMI, with a chance correction to set a constant baseline for random partitions.
VI	$[0, \log (n)]$	Shared information distance measures the amount of mutual information. The higher the value, the less resemblance between the two partitions.
NVI	$[0, 1]$	Normalized version of shared information distance with chance correction.

Equations44

S_{G} (A, B) = \frac{1}{2} i = 1 \sum r j = 1 \sum s min {\frac{n ^{a} ( x _{i}^{a} )}{N ^{a}}, \frac{n ^{b} ( x _{j}^{b} )}{N ^{b}}} δ (x_{i}^{a}, x_{j}^{b}),

S_{G} (A, B) = \frac{1}{2} i = 1 \sum r j = 1 \sum s min {\frac{n ^{a} ( x _{i}^{a} )}{N ^{a}}, \frac{n ^{b} ( x _{j}^{b} )}{N ^{b}}} δ (x_{i}^{a}, x_{j}^{b}),

f (x_{0}) = \frac{1}{hn} i \sum K (\frac{x _{i} - x _{0}}{h}),

f (x_{0}) = \frac{1}{hn} i \sum K (\frac{x _{i} - x _{0}}{h}),

S_{G} (A, B) = \int min {f^{(a)} (x), f^{(b)} (x)} d x,

S_{G} (A, B) = \int min {f^{(a)} (x), f^{(b)} (x)} d x,

f^{(u)} (x) = \frac{1}{h N ^{u}} i \sum N^{u} [n^{u} (x_{i}^{u}) K (\frac{x _{i}^{u} - x}{h})],

f^{(u)} (x) = \frac{1}{h N ^{u}} i \sum N^{u} [n^{u} (x_{i}^{u}) K (\frac{x _{i}^{u} - x}{h})],

Q_{N G} (P) = \frac{1}{m} c \in P \sum [m_{c} - \frac{( 2 m _{c} + l _{c} ) ^{2}}{4 m}]

Q_{N G} (P) = \frac{1}{m} c \in P \sum [m_{c} - \frac{( 2 m _{c} + l _{c} ) ^{2}}{4 m}]

Q_{E R} (P) = \frac{1}{m} c \in P \sum [m_{c} - \frac{m n _{c} ( n _{c} - 1 )}{n ( n - 1 )}]

Q_{E R} (P) = \frac{1}{m} c \in P \sum [m_{c} - \frac{m n _{c} ( n _{c} - 1 )}{n ( n - 1 )}]

Q_{D} (P) = c \in P \sum \frac{1}{n _{c}} (i \in c \sum k_{i c}^{in t} - i \in c \sum k_{i c}^{e x t})

Q_{D} (P) = c \in P \sum \frac{1}{n _{c}} (i \in c \sum k_{i c}^{in t} - i \in c \sum k_{i c}^{e x t})

Q_{Z} (P) = [c \in P \sum \frac{m _{c}}{m} - c \in P \sum (\frac{D _{c}}{2 m})^{2}] [c \in P \sum (\frac{D _{c}}{2 m})^{2} (1 - c \in P \sum (\frac{D _{c}}{2 m})^{2})]^{\frac{- 1}{2}}

Q_{Z} (P) = [c \in P \sum \frac{m _{c}}{m} - c \in P \sum (\frac{D _{c}}{2 m})^{2}] [c \in P \sum (\frac{D _{c}}{2 m})^{2} (1 - c \in P \sum (\frac{D _{c}}{2 m})^{2})]^{\frac{- 1}{2}}

S (P) = - lo g k = m^{in t} \sum m i n (m, M^{in t}) \frac{( k M ^{in t} ) ( m - k M - M ^{in t} )}{( m M )} .

S (P) = - lo g k = m^{in t} \sum m i n (m, M^{in t}) \frac{( k M ^{in t} ) ( m - k M - M ^{in t} )}{( m M )} .

S (P) \approx m D (q ∣∣ ⟨ q ⟩),

S (P) \approx m D (q ∣∣ ⟨ q ⟩),

D (q ∣∣ ⟨ q ⟩) = p lo g \frac{p}{⟨ q ⟩} + (1 - q) lo g \frac{1 - q}{1 - ⟨ q ⟩} .

D (q ∣∣ ⟨ q ⟩) = p lo g \frac{p}{⟨ q ⟩} + (1 - q) lo g \frac{1 - q}{1 - ⟨ q ⟩} .

Z (P) = C \in P \sum (2 n _{C}) D (p_{C} ∣∣ p) .

Z (P) = C \in P \sum (2 n _{C}) D (p_{C} ∣∣ p) .

I_{G} (A, B, Q) = \frac{N \sum q _{i}^{a} q _{i}^{b} - \sum q _{i}^{a} \sum q _{i}^{b}}{[ N \sum ( q _{i}^{a} ) ^{2} - ( \sum q _{i}^{a} ) ^{2} ] ^{1/2} [ N \sum ( q _{i}^{b} ) ^{2} - ( \sum q _{i}^{b} ) ^{2} ] ^{1/2}},

I_{G} (A, B, Q) = \frac{N \sum q _{i}^{a} q _{i}^{b} - \sum q _{i}^{a} \sum q _{i}^{b}}{[ N \sum ( q _{i}^{a} ) ^{2} - ( \sum q _{i}^{a} ) ^{2} ] ^{1/2} [ N \sum ( q _{i}^{b} ) ^{2} - ( \sum q _{i}^{b} ) ^{2} ] ^{1/2}},

R I (P_{1}, P_{2}) = \frac{( 2 n ) + 2 \sum _{i} \sum _{j} ( 2 n _{ij} ) - [ \sum _{i} ( 2 n _{i \cdot} ) + \sum _{j} ( 2 n _{\cdot j} ) ]}{( 2 n )}

R I (P_{1}, P_{2}) = \frac{( 2 n ) + 2 \sum _{i} \sum _{j} ( 2 n _{ij} ) - [ \sum _{i} ( 2 n _{i \cdot} ) + \sum _{j} ( 2 n _{\cdot j} ) ]}{( 2 n )}

A d j u s t_in d e x = \frac{I n d e x - E x p ec t e d _ I n d e x}{M a x _ I n d e x - E x p ec t e d _ I n d e x}

A d j u s t_in d e x = \frac{I n d e x - E x p ec t e d _ I n d e x}{M a x _ I n d e x - E x p ec t e d _ I n d e x}

A R I (P_{1}, P_{2}) = \frac{\sum _{ij} ( 2 n _{ij} ) - [ \sum _{i} ( 2 n _{i} ) \sum _{j} ( 2 n _{j} ) ] / ( 2 n )}{\frac{1}{2} [ \sum _{i} ( 2 n _{i} ) + \sum _{j} ( 2 n _{j} ) ] - [ \sum _{i} ( 2 n _{i} ) \sum _{j} ( 2 n _{j} ) ] / ( 2 n )}

A R I (P_{1}, P_{2}) = \frac{\sum _{ij} ( 2 n _{ij} ) - [ \sum _{i} ( 2 n _{i} ) \sum _{j} ( 2 n _{j} ) ] / ( 2 n )}{\frac{1}{2} [ \sum _{i} ( 2 n _{i} ) + \sum _{j} ( 2 n _{j} ) ] - [ \sum _{i} ( 2 n _{i} ) \sum _{j} ( 2 n _{j} ) ] / ( 2 n )}

I (P_{1}, P_{2}) = ij \sum p (c_{i}^{(1)}, c_{j}^{(2)}) lo g \frac{p ( c _{i}^{(1)} , c _{j}^{(2)} )}{p ( c _{i}^{(1)} ) p ( c _{j}^{(2)} )} = ij \sum \frac{n _{ij}}{n} lo g \frac{n _{ij} n}{n _{i \cdot} n _{\cdot j}}

I (P_{1}, P_{2}) = ij \sum p (c_{i}^{(1)}, c_{j}^{(2)}) lo g \frac{p ( c _{i}^{(1)} , c _{j}^{(2)} )}{p ( c _{i}^{(1)} ) p ( c _{j}^{(2)} )} = ij \sum \frac{n _{ij}}{n} lo g \frac{n _{ij} n}{n _{i \cdot} n _{\cdot j}}

H (P) = - k \sum \frac{n _{k}}{n} lo g \frac{n _{k}}{n}

H (P) = - k \sum \frac{n _{k}}{n} lo g \frac{n _{k}}{n}

N M I (P_{1}, P_{2}) = \frac{2 I ( P _{1} , P _{2} )}{H ( P _{1} ) + H ( P _{2} )} = \frac{- 2 \sum _{ij} n _{ij} lo g ( \frac{n _{ij} n}{n _{i} n _{j}} )}{\sum _{i} n _{i} lo g ( \frac{n _{i}}{n} ) + \sum _{j} n _{j} lo g ( \frac{n _{j}}{n} )}

N M I (P_{1}, P_{2}) = \frac{2 I ( P _{1} , P _{2} )}{H ( P _{1} ) + H ( P _{2} )} = \frac{- 2 \sum _{ij} n _{ij} lo g ( \frac{n _{ij} n}{n _{i} n _{j}} )}{\sum _{i} n _{i} lo g ( \frac{n _{i}}{n} ) + \sum _{j} n _{j} lo g ( \frac{n _{j}}{n} )}

A M I (P_{1}, P_{2}) = \frac{I ( P _{1} , P _{2} ) - E { I ( M ) ∣ n _{i \cdot} n _{\cdot j} }}{\frac{1}{2} ( H ( P _{1} ) + H ( P _{2} )) - E { I ( M ) ∣ n _{i \cdot} n _{\cdot j} }},

A M I (P_{1}, P_{2}) = \frac{I ( P _{1} , P _{2} ) - E { I ( M ) ∣ n _{i \cdot} n _{\cdot j} }}{\frac{1}{2} ( H ( P _{1} ) + H ( P _{2} )) - E { I ( M ) ∣ n _{i \cdot} n _{\cdot j} }},

V I (P_{1}, P_{2}) = H (P_{1}) + H (P_{2}) - 2 I (P_{1}, P_{2})

V I (P_{1}, P_{2}) = H (P_{1}) + H (P_{2}) - 2 I (P_{1}, P_{2})

N V I (P_{1}, P_{2}) = \frac{H ( P _{1} ) + H ( P _{2} ) - 2 I ( P _{1} , P _{2} )}{H ( P _{1} ) + H ( P _{2} ) - 2 E { I ( M ) ∣ n _{i \cdot} n _{\cdot j} }} .

N V I (P_{1}, P_{2}) = \frac{H ( P _{1} ) + H ( P _{2} ) - 2 I ( P _{1} , P _{2} )}{H ( P _{1} ) + H ( P _{2} ) - 2 E { I ( M ) ∣ n _{i \cdot} n _{\cdot j} }} .

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Nov 2019

\pagerangeCommunity Structure: A Comparative Evaluation of Community Detection Methods–References

Community Structure: A Comparative Evaluation of Community Detection Methods

VINH LOC DAO

CECILE BOTHOREL and PHILIPPE LENCA

IMT Atlantique

Lab-STICC CNRS UMR 6285

F-29238

Brest

France

{vinh.dao, cecile.bothorel, philippe.lenca}@imt-atlantique.fr

(2019)

Abstract

Discovering community structure in complex networks is a mature field since a tremendous number of community detection methods have been introduced in the literature. Nevertheless, it is still very challenging for practioners to determine which method would be suitable to get insights into the structural information of the networks they study. Many recent efforts have been devoted to investigating various quality scores of the community structure, but the problem of distinguishing between different types of communities is still open. In this paper, we propose a comparative, extensive and empirical study to investigate what types of communities many state-of-the-art and well-known community detection methods are producing. Specifically, we provide comprehensive analyses on computation time, community size distribution, a comparative evaluation of methods according to their optimisation schemes as well as a comparison of their partioning strategy through validation metrics. We process our analyses on a very large corpus of hundreds of networks from five different network categories and propose ways to classify community detection methods, helping a potential user to navigate the complex landscape of community detection.

Keywords: community detection, community structure, comparative analysis, empirical analysis, computation time, community size, structural quality function, validation metric, decision-making assistance for practioners

1 Introduction
2 Community detection methods and dataset
2.1 Community detection methods
2.1.1 Edge removal based methods
2.1.2 Modularity optimization methods
2.1.3 Dynamic process based methods
2.1.4 Statistical inference based methods
2.1.5 Other methods
2.2 Experimental dataset
3 Preliminary analysis of community detection methods
3.1 Computation time performance
3.2 Analysis on community size distribution
3.2.1 Edge removal approach: GN, RCCLP-3 and RCCLP-4
3.2.2 Modularity optimization approach: CNM, Louvain and SN
3.2.3 Dynamic process approach: Infomap, Infomod and Walktrap
3.2.4 Statistical inference approach: SBM, DCSBM and Oslom
3.2.5 RB, LPA, SLPA and Conclude methods
3.2.6 Summary
3.3 Similarity based on community size distribution
3.3.1 Experimental results
4 Quality profiling of community detection methods
4.1 Community quality metrics
4.1.1 Newman-Girvan Modularity
4.1.2 Erdős-Rényi Modularity
4.1.3 Modularity Density (D-value or D-modularity)
4.1.4 Z-modularity
4.1.5 Surprise
4.1.6 Significance
4.2 Co-performance index
5 Partitioning strategy comparison
5.1 Validation metrics
5.1.1 Rand Index (RI)
5.1.2 Adjusted Rand Index (ARI)
5.1.3 Normalized Mutual Information (NMI)
5.1.4 Adjust Mutual Information (AMI)
5.1.5 Normalized Variation of Information (NVI)
5.2 Empirical results
6 Related work
7 Conclusion

1 Introduction

In network science, community detection (sometimes called graph clustering) is one of the fundamental challenges to discovering the structure of networks on the mesoscopic level. However, it is an ill-defined problem. According to Arifin et al. [\citenameArifin et al., 2017], “it does not have clear goals, solution paths or expected solution”. There is no universal definition or closed form formula of what kind of objects one should be looking for [\citenameFortunato & Hric, 2016], and consequently there is no golden standard to assess the quality of a community structure and the performance of a community detection algorithm.

The most frequently found definition of community in the network science literature is derived from the mechanism of connection preference. It implies that a community is a group of nodes (a subgraph) in a graph where there must be more edges (denser) connecting them together than edges connecting the community with the rest of the graph [\citenameRadicchi et al., 2004, \citenameFortunato, 2010]. Newman defines a community as a “group of vertices with a higher-than-average density of edges connecting them” [\citenameNewman, 2006b]. Depending on the context, a community may be called a cluster, a module, a class or a modular group. This is the most basic definition that sets the fundamental requirement for most of its derivative definitions. Many different variations of community could be found in [\citenameWasserman, 1994], for instance, LS-set, which is a set of nodes in a network such that each of its proper subsets has more ties to its complement within the set than outside; or k-core, which is a subgraph in which each node is adjacent to at least a minimum number $k$ of the other nodes in the subgraph. However, in recent developments of community detection algorithms, there is no consensus of the quantity of edges in reality that could be considered as “many”. Communities are just algorithmically defined, i.e. they are final products of the algorithm without any precise a priori definition [\citenameFortunato, 2010].

In practice, there are even more constraints, which are sometimes not explicitly expressed, than which appeared in the announcement of the problem. If one only looks for a partition of a graph that maximizes the number of internal edges and minimizes the number of external edges, then the graph itself can be considered as a big community with no external connection. Another solution is to leave the node having the smallest degree of a graph in one community, and all other nodes in another community. This solution could also maximize the ratio between external and internal edges. However, these monotonous solutions do not seem to seduce most (if not all) analysts considering using a community detection algorithm to detect communities. In fact, it is preferable to cluster a network into groups of relatively similar size [\citenameNewman, 2010]. This means that somehow, the relative size of communities is significant but this notion of size has not been explicitly announced. Besides, there are many other criteria that could be mentioned such as community complete mutuality, reachability, vertex degree distribution and the comparison of internal versus external cohesion [\citenameWasserman, 1994, \citenameFortunato, 2010]. There exists a subtle compromise between adding into a community new vertices—along with their edges— and conserving the common property that defines the group. In fact, different community detection methods have different ways to define what defines the groups and how to consider those constraints or not. They produce different community structures. The following are the main reasons that could lead to these disagreements between detection methods:

•

Different algorithms may implement different meanings of the notion of community.

•

When two algorithms define the same concept of community, it may also mathematically and algorithmically be formalized in different ways (same objective but different objective functions) and hence leads us to different results.

•

Even when two algorithms have exactly the same objective function, the algorithmic mechanism they employ to find communities also decides what they are going to find, especially in heuristic searching approaches.

•

Initial configuration is also another important factor that affects the final result of an algorithm, many community detection methods are not deterministic.

•

Each method may include a consideration between obtaining an optimized solution and providing reasonable performances (in terms of calculation time, memory consumption, etc.). This trade-off may be considered differently across the methods.

•

Some algorithms are variable according to input data and will prove more or less efficient on some kinds of inputs than on others.

•

Variations due to implementation factors could also impact the final result of an algorithm.

•

Finally, in some algorithms, there are tie-break situations where they have to choose randomly without any factor related to their final objectives. It may also significantly affect the result that one would get if the tie-break problems have been resolved in a different way.

Due to the many reasons stated above, choosing the community detection method that corresponds well to a particular scenario or to an expectation of quality is not straightforward. This paper is not the first effort as a guide for choosing community detection methods. For the same purpose, readers can also refer to [\citenameFortunato & Hric, 2016], [\citenameGhasemian et al., 2018], [\citenameJebabli et al., 2015], and [\citenameJebabli et al., 2018]. The difference with existing work is that we provide an exhaustive empirical evaluation on many state-of-the-art methods, using a large network dataset (more than 100 different real networks, of which only a few are synthetic) and using popular quality scoring functions of structural characteristics of communities. Hence, we are able to disclose a performance review of many different methods based solely on empirical experiments and measurements that help network practitioners to reveal the functionality of each method, and behaviours that they could exhibit in real world networks, independently of theoretical mechanisms. The objective of this work does not differ from existing work in the literature. It does not aim to bring readers an innovative factor in the community detection problem nor discuss every related challenging point of the community detection problem. However it provides an evaluation as well as an approach to examine community detection algorithms. This evaluation is very useful for network practitioners to obtain a quick notion of community structure quality before actually applying a method to their concrete dataset. These notions of quality are expected to guide users to appropriate choices of methods according to their quality criteria.

The paper is organized as follows: Section 2 introduces some popular and state-of-the-art community detection methods that will be analyzed in this paper as well as the benchmarking dataset employed in our experiments. Then, Section 3 presents analyses of the most essential aspects of community detection performance including computation time, community size distribution as well as numbers of detected communities by each method. In Section 4, we address different structural quality aspects of community structures. This is followed by a comparative evaluation using many popular clustering validation metrics in Section 5, which are widely used in the context of community detection. The Sections 3 to 5 are independent. They bring complementary material and criteria to describe the quality of community structures and compare the different methods from different perspectives. Each section is detailed, making this article a little long, but they end with a summary that allows readers to get an overview and provide them with the necessary key lessons to be learned. Finally, we present some close results of related work that can be found in the literature in Section 6 and conclude our study with some discussions and recommendations in Section 7.

2 Community detection methods and dataset

2.1 Community detection methods

We present, in this section, some popular community detection methods that have been widely used and discussed in the literature. Note that in recent years, there are a large number of innovative methods which have been proposed to solve either generic or specific cases. However, an empirical and exhaustive analysis of all methods would be impractical, even unrealizable. To the best of our knowledge, we are trying to include the most important and representative methods among the community detection approaches.

There are many possible theoretical taxonomies for community detection methods, depending on the final objective of each categorization. For instance, one could classify methods according to differences in searching mechanisms, objective functions, assumptions about the structure to be found, expected qualities, hypothesis models, or even the theoretical model employed, etc. Moreover, what makes the problem trickier is that many methods are not just some simple algorithms to resolve a specific problem, but instead are combinations of many different approaches in order to leverage as much algorithmic power as possible from each one. There is not a consensus on how different methods are similar and how they can be classified into different families whose functionality can be resumed in some simple words. Porter et al. use centrality based, local techniques, modularity optimization, spectral clustering to describe communities in networks [\citenamePorter et al., 2009]. In [\citenameFortunato, 2010, \citenameFortunato & Hric, 2016], the authors group community detection methods into traditional data clustering methods, divisive algorithms, modularity-based methods, spectral algorithms, dynamic algorithms and methods based on statistical inference. Coscia et al. classify community discovering into feature distance based, internal density, bridge detection, diffusion process, closeness based, structural pattern based, link clustering, meta clustering [\citenameCoscia et al., 2011]. In a context of Social Media, Papadopulos et al. compare methods in substructure detection, vertex clustering, community quality optimization, divisive and model-based approaches [\citenamePapadopoulos et al., 2011]. Bohlin et al. aggregate different approaches into three principle classes representing different network models: null models, block models and flow models [\citenameBohlin et al., 2014]. Schaub et al. classify methods into four perspectives: cut based, clustering internal density based, stochastic equivalent based and dynamic based showing four different facets of community structure [\citenameSchaub et al., 2017]. Finally, Ghasemian et al. adopt an experimental classification [\citenameGhasemian et al., 2018] as well as group community detection methods in distinct families based on their outputs on many real-world networks using a validation metric (a topic we will address in Section 5.1).

In the following sections, we choose to classify community detection methods according to different theoretical approaches including edge removal, modularity optimization, spectral partitioning, dynamic process and statistical inference. Although every theoretical taxonomy may be questionable, this categorization is expected to support the empirical analyses in the next sessions to verify whether theoretical and conceptual closeness could engender quality closeness in practice.

2.1.1 Edge removal based methods

Edge betweenness (GN) [\citenameNewman & Girvan, 2004] detects communities by removing edges progressively according to their betweenness centrality scores. This method is based on the intuition that dense areas of a graph are loosely connected by a few edges that are located in the shortest paths between pairs of nodes. Removing these edges would reveal densely connected communities.

Edge clustering coefficient (RCCLP) [\citenameRadicchi et al., 2004] suggests replacing the edge betweenness centrality of Girvan-Newman’s method by an edge clustering coefficient, which requires less computation time and hence reduces the complexity. In this paper, we analyze two configurations of this method corresponding to triangular ( $g=3$ denoted by RCCLP-3) and quadrangular ( $g=4$ denoted by RCCLP-4) versions.

2.1.2 Modularity optimization methods

Greedy optimization (CNM) [\citenameClauset et al., 2004] greedily maximizes the modularity function $Q$ by aggregating iteratively connected communities which induces a maximum increase or minimum decrease in modularity $\Delta Q$ .

Louvain method [\citenameBlondel et al., 2008] adopts a two-step agglomerative process similar to the greedy optimization method. However, in each iteration of the first step, it allows nodes to move between communities until no additional gain in modularity can be obtained with these local switches. Then, a new graph whose vertices are the communities resulting from the first step is built and the process is repeated on the new graph to reduce computation time, leading to a hierarchical clustering.

Spectral method (SN) [\citenameNewman, 2006a] reformulates the idea of maximizing modularity as a spectral partitioning problem by constructing a modularity matrix and using its leading eigenvector to spectrally partition networks into sub-networks. The community structure is identified using the eigenvectors of this matrix. Eigenvectors are used to project each node into low-dimensional node vectors, and the community structure is identified through clustering the node vectors (e.g. with the k-means clustering method).

2.1.3 Dynamic process based methods

Walktrap [\citenamePons & Latapy, 2005] defines a pairwise dynamic distance between nodes and then applies traditional hierarchical clustering to detect community structure. The distance between two nodes is defined in terms of a random walk process. The basic idea is that if two nodes are in the same community, they tend to ”see” other nodes in the same way, i.e. the probability of getting to a third node through a random walk should not be very different for these two nodes, with higher distance for nodes belonging to other communities.

Infomod [\citenameRosvall & Bergstrom, 2007] uses an information theoretic model where a signaler tries to send the structure of a network over a limited capacity transmission channel to a receiver. The network must be encoded in community structure in a way that minimizes the transferred information and the information loss.

Infomap [\citenameRosvall et al., 2009] represents networks by a two-level structure description. Analogically, each node in a network is encrypted by a unique codeword composed of two parts: a prefix representing the community to which it belongs and a suffix representing the local code. Detecting community structure becomes equivalent to searching for the coding rule to minimize the average code length describing random walks on the network.

2.1.4 Statistical inference based methods

Stochastic Block Model (SBM) [\citenameRiolo et al., 2017]. The Stochastic Block Model was introduced by Holland et al. [\citenameHolland et al., 1983]. Here we use the implementation by Riolo et al. which uses a Monte Carlo sampling scheme to maximize a Bayesian posterior probability distribution over possible divisions of the network into communities. This probability implies an expected network model to be fitted from the observed network data. In this block model variant, the authors employ a new prior on the number of communities based on a queueing-type mechanism to calculate posterior probability. We analyze, in the following sections, both traditional $SBM$ and the degree-corrected version DCSBM, which had been proven to perform better in practice.

Order statistics local optimization (OSLOM) [\citenameLancichinetti et al., 2011] measures the statistical significance of a community by calculating the probability of finding a similar one in a null model. Following this concept, nodes are gradually aggregated into communities to find significant communities. Then nodes are considered to be swapped between communities in order to increase significance level.

2.1.5 Other methods

Spin glass model (RB) [\citenameReichardt & Bornholdt, 2006] is a method relying on an analogy between the statistical mechanics of complex networks and physical spin glass models. It finds communities by fitting the ground state of a spin glass model. Instead of favoring only intra-community edges and penalizing inter-community edges like the traditional modularity measure, this model also favors inter-community non edges and penalizes intra-community non-edges.

Label propagation (LPA) [\citenameRaghavan et al., 2007] exploits the topology of networks to infer community structure. It is closely related to the context of message passing paradigms or epidemic spreading. The principled idea of this method is based on the concept that nodes should belong to the community of most of their neighbors. Hence, they gradually update their memberships according to their incident nodes.

Speaker-listener label propagation (SLPA) by Xie and Szymanski [\citenameXie & Szymanski, 2012] modifies the propagation mechanism above by a new label update strategy. Also, instead of keeping only hard membership information, each node is equipped by a memory to contain the labels that it receives. Then, in the update phase, nodes transmit the membership to their neighbors according to the membership frequency in the memories.

Mixing global and local information (Conclude) [\citenameMeo et al., 2014] combines a dynamic distance with a modularity optimization process to identify community structure. Firstly, the authors define a new pairwise proximity function using random and non backtracking walks of finite length to determine distances between vertices. Then, the multi-level modularity optimization strategy of Louvain method [\citenameBlondel et al., 2008] is combined with the defined distance to find community structure.

Table 1 summarizes the methods presented previously, grouped by different approaches. Since community detection is receiving more and more attention in the network science community, there is a huge volume of work that has been published in recent years to evaluate different methods including both theoretical and empirical approaches. However, there is no formal and quantitative definition of community that is explicitly implemented inside algorithms. Therefore it is challenging to distinguish the topological differences of community structures using different methods, even when the associated concepts are quite theoretically discernible. Additionally, it is still not clear whether proximity in the assumption of community concept will engender a structural similarity of communities that could be detected. Our comparative analysis in the next sections will try to address these questions in more detail.

2.2 Experimental dataset

In this section, we describe some statistical properties of networks that will be included in the following analysis. Available biological networks that have been published and analyzed widely are relatively small in comparison to the other networks of the other families. Besides, due to the complexity of the analysis process, we limit the domains of interest to 5 categories which are commonly researched and where numerous networks are available. We introduce a 6th category with various types of networks, each of which is under-represented, such as ecological networks for example. In this study, we consider 108 different networks, which is relatively large in comparison to many other studies. Many notable related works where some of these networks are also employed could be mentioned for quick reference: Orman et al. use 6 networks to evaluate the structure of communities discovered by several detection techniques [\citenameOrman et al., 2012]; Lancichinetti et al. use 15 networks to characterize structural communities [\citenameLancichinetti et al., 2010]; Hric et al. use 16 networks to reveal differences between structural communities and ground truth [\citenameHric et al., 2014]; Leskovec et al. use over 100 networks to analyze network community profile [\citenameLeskovec et al., 2008] and 230 networks to evaluate the goodness of ground-truth communities in social networks. Within this number, 225 samples of the Ning online social networking platform networks101010https://www.ning.com/ are aggregated [\citenameYang & Leskovec, 2013]. Let us mention, finally, the related work by Ghasemian et al. which has introduced the large CommunityFitNet corpus111111https://github.com/AGhasemian/CommunityFitNet containing 572 real-world networks. Table 2 summarizes the composition of networks that have been analyzed in this section.

Some notable structural characteristics of networks in the dataset are illustrated in Figure 1. It is noticeable that apart from biological networks which are relatively small, the other classes cover quite a wide range of numbers of nodes, edges, mean degree, clustering coefficient and edge density. Since real world networks are relatively sparse, the number of edges increases linearly according to the number of nodes and consequently, the edge densities decrease linearly by the number of nodes (since the number of possible connections increases quadratically by the number of nodes in a community). This sparsity property can easily be seen in Figure 1(a,d). Specifically, the number of edges increases linearly according to the number of nodes with equivalent rates among different network categories as can be deduced from the gradients of the linear estimates. From Figure 1(b), it can be seen that the average degree of the networks in the dataset varies principally between $1$ and $100$ edges per node except for 2 communication networks. Also, the majority of networks have an average degree from 10 to 20 connections. From a global point of view, networks in the dataset have a quite strong modular quality since most of them have relatively high clustering coefficients as shown in Figure 1(c).

3 Preliminary analysis of community detection methods

3.1 Computation time performance

Since computation time is a crucial factor to be considered in the selection of an algorithm, it is worth analyzing experimental performances to see how different community detection methods accomplish their task in real-world networks. We tested the official implementations of community detection methods introduced in Table 1 on the dataset collection presented in Table 2. The implementations are provided officially either by their authors or by popular network analysis libraries which can be easily accessed by a large public.

We ran the implementations stated above to identify community structures on all the networks in the dataset. We measured the time needed for each implementation to compute each partition on each network. The default parameters configured by the implementations remained unchanged during the test. The calculations were executed on a server equipped by an Intel Xeon CPU E5-2650 with 32 cores of 2.60 GHz and a memory capacity of approximately 100 GBytes. However, due to the high complexity of some methods, only processes that finish within a practical time limit (less than 4 hours) are taken into account. However, for a reference purpose, we let some longer computations continue. For example Conclude method took approximately 9 days to identify community structures on a network of 300 thousand vertices and 1 million edges; GN method did not finish its calculation for networks of more than 4 thousand nodes and 40 thousand edges within 2 days. Consequently, the experiments that theoretically require too much time are neglected in the test. It is also worth noting that the calculations of communities on large-scale networks are also restrained by limited memory. Thus, calculations that should be finished within 4 hours but required too much memory cannot be shown here either. We repeat the calculations 5 times for each pair graph/method to reduce the fluctuation impact. Eliminating all the cases that do not satisfy our requirements, the final successful rate (number of partitions identified over the number of possible tests) ended at around $44.72\%$ , mainly because of time/memory overflow.

In the following figures (from Figure 2 to Figure 7) that illustrate the analyses of experimental time consumption, some conventions are commonly used. Points in the figures correspond to separated executions that have been measured. The solid lines with the same colors are estimated relations between computation time and network size (number of vertices and number of edges), using a local regression model [\citenameCleveland, 1979]. The dark colored backgrounds around the regression curves represent $95\%$ confidence intervals of the model parameters. Besides, for reference purposes, we also show theoretical execution time (i.e. in the worst case, in terms of number of calculations needed) for each algorithm. In our estimates, we plot this theoretical execution time by assigning $n=m$ , as from the analysis of structural characteristics of the dataset shown above in Figure 1(a), most networks are sparse, i.e. the number of edges ( $m$ ) increases linearly with the number of nodes ( $n$ ). For simplicity of illustration, we grouped our results according to our classification (Table 1).

The first group of methods consists of centrality detection techniques to identify community structure. As we can see in Figure 2, the GN method cannot be accomplished in our test for networks of more than 4 thousand nodes or 30 thousand edges. The outcome is quite reasonable since the theoretical estimation for this method is $\mathcal{O}(nm^{2})$ , which grows quickly with the network size. It should be remembered that the primary goal of the RCCLP method is to reduce the time complexity of the GN method. We can easily observe that this objective is achieved since the RCCLP-3 reduces by an order of around $10^{3}$ times for graphs of 3 hundred nodes. RCCLP-3 can function well with graphs up to millions of edges. However, when we used the same test with RCCLP-4, the method rarely reached its terminus for large graphs as well as small graphs. As we can see in the figure, there are few dots at either side. The reason is that there are not many (they may even be absent) 4-step close paths on real world networks. As it is not very probable that such structures exist in small graphs, finding them in large graphs also requires a huge amount of time, RCCLP-4 shows a poor performance in our tests. Therefore, this configuration of the method is not recommended, as versions with $g>4$ would logically perform poorly. It is also worth noticing that RCCLP-3 and RCCLP-4 are extremely memory consuming and are not suitable for limited resource devices. Finally, theoretical and practical time seem to find a consensus as the increments of time according to network size are quite consistent in the three cases.

The next group includes methods using modularity optimization processes whose experimental measures are shown in Figure 3. Practically, the three methods in this family require a reasonable time for calculating community structures. The longest experiment took less than 2 hours for a graph of 1 million edges. The Louvain method is the fastest in this group; its computation increases approximately in linear time. It took only 9 seconds for the largest graph. Among the three methods, the optimization using the spectral approach is the most expensive. However, all three methods perform better than the methods in the edge removal group previously stated. The experimental results also verify theoretical estimates concerning the complexity of these methods.

Similarly to the two previous groups, the computation time needed by methods in the dynamic process group is illustrated in Figure 4. In terms of time consumption, this group performs better with respect to the first group, but generally worse than the modularity optimization group (except for the Walktrap method for small and average size graphs). Among these three methods, Infomod generally requires more calculation time than the others. At the same time, Walktrap and Infomap work asymptotically equally well with a slightly better rendition for Walktrap in small and average size graphs.

The same analyses for methods in the two final groups are shown in Figure 5 and Figure 6. We can easily see that DCSBM and Oslom have practically identical performances in terms of time consumption, DCSBM being slightly better. In the last group, the results are quite different between different methods. The label propagation method LPA shows a clear distinctive curve indicating its out-performance compared with the other methods. Besides, SLPA works quite well, but not as fast as LPA although it employs some additional techniques to reduce the number of necessary calculations [\citenameXie & Szymanski, 2012]. This difference in performance is due to the more complicated mechanism that SLPA uses in comparison to LPA. It is due to the fact that SLPA manages dedicated memories for all the nodes to stock the membership information and update it regularly. Therefore, despite the improvement of 5 to 10 times in the label update strategy, the global performance cannot surpass that of LPA method. In terms of scalability, LPA and SLPA seem to exhibit the same behavior which is almost linear for small and medium graphs but increases in large graphs. The spin glass model RB manifests a better than expected presentation with an undeviating linear augmentation. The only unexpected behavior is spotted in Conclude method, as when the size of input graphs exceeds some thousands, the required time has been inflated by a factor of $n$ , making it very demanding for large graphs.

Finally, we aggregate all the results into a common illustration as shown in Figure 7. At the same time, for more convenient observation, we remove all the points corresponding to each experiment and keep only the regression curves, which are the execution time estimates as a function of the number of vertices on the left side and similarly on the right side for the number of edges. At first sight, it is easy to see that except for GN, the necessary execution time for all other methods is limited in a range that increases polynomially with the network size, which accurately reflects theoretical estimates. This range is upper-bounded by Conclude/Oslom and lower-bounded by LPA, corresponding to the worst and the best tested method(s) respectively. Another important fact which can be deduced from this figure is that, for most real world networks in the range up to 1 million edges, choosing a fast detection method could economize an order of $10^{3}$ times to $10^{5}$ times the calculation effort. This is an important element to be considered where time consumption is a serious problem.

We demonstrate in Table 3 the ranking of these methods according to our tests for reference purposes. GN and RCCLP-4 are not involved in this ranking since they failed to accomplish their tasks in large graphs, which also means they are the most time-consuming methods within the methods that we analyzed. We show both the rankings according to the average and the median of time. Since the average-time ranking is heavily affected by the measurements on large graphs, the methods that were successful in discovering communities on very large graphs are ranked lower than methods that were not able to do so. In these cases, the median ranking is more accurate and reflects therefore the relative performance on small and medium graphs. For large graphs, we recommend using the average ranking.

3.2 Analysis on community size distribution

After these performance considerations, we focus on the nature of the results produced, i.e. the communities themselves. The number of latent communities that should be induced from a given network is one of the major questions in community detection context [\citenameFortunato & Hric, 2016], [\citenameRiolo et al., 2017]. It is equivalent to the subject of the expected number of clusters in a classical clustering problem. Observing the number of communities reveals useful information about the mesoscopic structure of a network. The variation of the number of communities in a network involves different levels of resolutions. An analogous way to describe the concept of resolution is the distance from an object that we prefer in order to contemplate it. The closer you get to an object, the greater the detail of its micro-structures that can be perceived, while, at the same time, information about the global organization tends to be less clear. Although several multi-resolution approaches [\citenameLambiotte, 2010, \citenamePons & Latapy, 2011] incorporate resolution parameters into their solutions providing more flexible mechanisms and different modular scales of networks, it is not always obvious to regulate these parameters appropriately without ad-hoc cases. The inclusion of multi-resolution parameters, of course, widens the possibility of understanding networks, but at the expense of automation convenience, that is sometimes required in clustering problems.

In this section, we compare, once again, the previously mentioned methods but this time according to their resolution abilities. We use the same dataset collection and again we keep all default configurations of the implementations unchanged to ensure the consistency of future results. From the previous analyses, some modifications will be applied to our testing process as follows:

From the observation of the network size distribution in Figure 1(a), as well as the previous computation time analyses, the linear relation between number of vertices and number of edges of networks in our corpus becomes clear. As a consequence, it will be redundant to address the relation of dependent variables in respect of these two latter predictors. Therefore, only analyses according to the number of vertices will be rendered. 2. 2.

In the case of the community detection problem, showing only the numbers of communities discovered would not always be enough. If we assume that community size in an arbitrary network exhibiting a negative power law distribution (as shown for example in [\citenameClauset et al., 2004]), it means that the number of communities depends heavily on the number of tiny communities. Therefore, we propose to observe the distribution of community size to discern the differences between methods which could not be recognized by seeing solely the number of blocks. 3. 3.

Due to the huge number of calculations required and a limited hardware resource, discovering processes in the last section were interrupted unless they could be finished in a few hours. Here, some more efforts have been made, when a method is supposed to be finished in a reasonable amount of time (a few days).

For a given network in the dataset, we applied all of the methods presented, to identify the set of communities predicted by each one and then measured their volumes. Similarly, to the previous section, for simplicity of observation, we group methods by different families depending on their approaches. We illustrate the results in Figures 8 to 12 by using some conventions as follows:

A figure (denoted a) at the top contains three following sub-figures:

(a)

The central figure (a1) shows a scatter plot of the distribution of community size. The solid lines in the figure represent the estimated average community size using a local regression model [\citenameCleveland, 1979]. Dark colored backgrounds around the lines are $95\%$ confidence intervals of the estimates.

(b)

The top figure (a2) exhibits marginal density distributions of communities found in each range of network sizes. They are rendered by a Gaussian kernel estimator.

(c)

The right hand figure(a3) illustrates another type of marginal density distribution of communities, as a function of their sizes. They are also rendered by a Gaussian kernel estimator.

(d)

The axes of marginal figures are the same as the axes of the corresponding central figure. We simply omit them for ease of representation.

A figure on the bottom (denoted b) presents the number of communities as a function of network size (i.e. their number of vertices) as well as the estimated relation between these variables using the regression model stated above. Dark colored backgrounds around the lines show $95\%$ confidence intervals of the estimate relations.

3.2.1 Edge removal approach: GN, RCCLP-3 and RCCLP-4

From Figure 8, we can notice again that the GN method is only able to function on small and medium networks, due to its high complexity, which is quite obvious from theoretical analysis. RCCLP-3 and RCCLP-4 can detect up to the largest networks in our corpus. By observing the right marginal density distribution, all of these methods identify a surprisingly high number of singleton communities. The average number of singleton communities is around $24\%$ but the number can reach $60\%$ in some cases. The reason for this aberrant phenomenon is that in some dense, small networks, there exist too many high and equivalent central vertices and edges. The separating mechanism employed here keeps removing central nodes or edges until a large number of vertices are isolated, creating singletons or very small communities. Since GN only works on small graphs, it is highly impacted by this phenomenon in our experiment. Besides, from a global observation, we can see in the top figure that the majority of communities detected by these methods are very small for the same reason. From Figure 8(a), we can see that a large number of communities have less than 10 vertices, even in very large networks. This makes the number of communities increase rapidly, as illustrated in Figure 8(b). Bear in mind that the distributions of community size have right-skewed shapes, meaning that the majority of communities are small, and most are found under the lines of average community size. Therefore, the three methods of this family have very high resolutions. Notwithstanding, this result needs to be interpreted with caution, for the following reasons:

The density function in Figure 8(a1) reveals that the successful rates for discovering community structures of the three methods are fundamentally very different. In fact, due to the high complexity of time and memory, many networks are not successfully resolved, which significantly degrades the comparison quality. 2. 2.

As a consequence of the first reason, there is a high fluctuation in the dependent variables which makes the confidence intervals quite large. A deeper investigation of the quality on small and medium networks could partially palliate this problem.

Despite the previously mentioned issues, within this class of methods there is a large consensus as regards the discovery of the highest number of communities.

3.2.2 Modularity optimization approach: CNM, Louvain and SN

In this second group, our measurements are more complete since all three methods successfully resolved large networks. From Figure 9(a2), it can be seen that there is a regularity between the distributions of communities over the whole range of networks, except for the range of very large networks. Actually, in this range, the behavior is very different in the three methods. While CNM determines a very large number of medium and small communities, Louvain identifies fewer small communities and more medium and large communities. On the other hand, SN only proposes a partition of two giant communities. For instance, if we take the $Amazon$ network [\citenameLeskovec & Krevl, 2014], while CNM detected $1480$ clusters, the number is $249$ for Louvain, and only two for SN. The same phenomenon is also identified for another example, the $DBLP$ network [\citenameLeskovec & Krevl, 2014]. The corresponding numbers are $3077$ , $275$ and $2$ in the same order for the three methods. This fact can also be remarked in smaller networks as can be seen in Figure 9(b). However the gap between the number of communities reduces gradually from the right to the left of the figure. But in general, the order remains unaltered as experienced in our observations, i.e. the average number of communities detected by CNM is larger than that of Louvain which is in turn larger than that of SN. Consequently, the order of community sizes is inversed, since the sizes of graphs are fixed, as can be seen in Figure 9(a1). Another fact can be extracted from Figure 9(a3) concerning the diversity of community size: while CNM and SN consistently move towards small and medium communities respectively, Louvain on the other hand tends to propose both small and medium size communities.

3.2.3 Dynamic process approach: Infomap, Infomod and Walktrap

At first glance, we can see a clear separation within the three methods. While Infomap and Walktrap display quite a comparable evolution of average community size, depicted by Figure 10(a1), as well as marginal distribution, as depicted by Figure 10(a2-a3), Infomod is driven distinctly apart. After close examination, we notice that in Infomod, there is a relatively uniform partitioning of communities which is upper-bounded by the largest community containing $6948$ vertices. Unlike many other methods including Infomap and Walktrap, the number of medium and large communities discovered by Infomod does not outnumber the number of small communities, as stipulated by heavy-tailed distributions. As a consequence, the total number of communities observed remains low and increases at a slow, constant pace.

Infomap and Walktrap tend to keep their average community size limited to around $10$ to $30$ , over the whole range of networks. This phenomenon keeps them apart from the resolution limit issue. In both methods, the most popular community size can be found around $10$ nodes or smaller. Our more specific experiment on the median community size shows almost similar results for Infomap while the number decreases slightly for Walktrap. Above these values, the number of communities decreases significantly. The biggest difference between these two methods can be easily observed at the spurious region on the marginal distribution of Figure 10(a3). In fact, unlike Infomap, which produces moderately small communities, Walktrap identifies a huge number of isolated nodes (around $10\%$ according to the statistics), similarly to RCCLP-3 and RCCLP-4, as indicated in the previeus section. This problem may be due to the agglomerative hierarchical clustering employed by Walktrap to detect communities, which engenders orphaned peripheral vertices. This behavior has been identified in particular by Newman and Girvan, cf. Figure 3 in [\citenameNewman & Girvan, 2004]. This problem, however, is quite simple to palliate since these peripheral vertices could be assigned to their closest neighbor’s community. By removing this issue, we obtained quite a similar result for Infomap and Walktrap.

In terms of average number of communities, Infomap and Walktrap show practically the same behavior. The evolutions coincide across almost all networks with small confidence intervals, particularly in the mid-range networks. For medium and large networks, as seen in Figure 10(b), it is very likely that Infomod identifies a much smaller number of communities. In fact, more than $75\%$ of Infomod’s partitions have fewer communities than those of the other two methods.

3.2.4 Statistical inference approach: SBM, DCSBM and Oslom

In the case of statistical inference, we see quite a similar phenomenon ro that previously experienced in the dynamic approach. Specifically, the distributions of community size of the two implementations SBM and DCSBM nearly coincide with a slightly higher average community size for the former. In fact, in this Bayesian block model it is necessary that the prior distribution of number of blocks be given. In the implementation that has been employed, the authors initialize the community discovering process by assigning nodes randomly to groups, according to a queuing-type mechanism and then use a Monte Carlo sampling process to maximize the posteriori probability. However, the calculation becomes extremely time-consuming when the maximum number of communities is too large [\citenameRiolo et al., 2017]. Hence, by default, the maximum number of communities is configured at $25$ , as proposed by the authors, which leads to an underestimation of medium and large graphs, as shown in Figure 11(b), and also noticed by the authors. One can see the impact of this regulation as the number of communities approaches asymptotically $25$ , independently with the network size on the right hand side of the figure. Discovering community structures using this method in large networks (more than $25^{2}$ nodes, for example) would have to be done recursively to avoid resolution limits. In other words, one could apply community detection again on very large communities (larger than the square root of the number of nodes).

By observing the distribution of community size in Figure 11(a1), it is understandable that the average block size of SBM and DCSBM increases linearly according to the number of vertices. As the number of communities remains constant, the average community size must increase proportionately. Furthermore, Figure 11(a3) also reveals that community sizes are well distributed around their mean values, which makes the marginal distribution quite symmetric for both SBM and DCSBM. There is almost no particular inclination towards small communities, as acknowledged in some previous methods.

For the case of Oslom, the separation is quite clear. It uncovers many more communities, making their sizes very small. Figure 11(a1) shows that the majority of Oslom communities are found under the average values of the associated partitions of SBM and DCSBM. Our demonstrations show that there is indeed a significant difference in the partitioning strategies of these methods.

3.2.5 RB, LPA, SLPA and Conclude methods

In the last group, we discover that there is a remarkable coincidence in all distributions of the three methods LPA, SLPA and Conclude. In fact, the difference between them is almost indistinguishable on the marginal measures. There is only a small discrepancy in the number of detected communities in very large networks, as can be seen in Figure 12(a2), such that LPA detected slightly more communities than SLPA and Conclude. From Figure 12(a3), one can see that the majority of communities are quite small in these three methods. Similarly to CNM, Infomap or Walktrap, the majority of communities are small, i.e. have less than 10 nodes.

In the three methods, one can see that the variation of the data is significantly large, which also produces a large variation in our estimates. Since the associated prediction intervals for the estimates are likely to be larger, predictions related to community size distribution are not expected to be accurate.

On the other hand, $RB$ method shows a solid consistency with far fewer variations in our examination. Average community size increases regularly and the number of communities becomes saturated from medium size networks. The behavior of $RB$ method closely resembles that of DCSBM observed in Figure 10. Consequently, it is supposed to suffer the resolution limit for large networks. Nevertheless, since $RB$ is provided with a resolution tune parameter, the method may avoid this effect if the parameter is correctly chosen.

3.2.6 Summary

For the final step in this section, in the same manner as the previously presented time computational analysis, we aggregate, for all methods, the estimates of average community size and the number of detected communities, as a function of number of vertices in the network in Figures 13(a) and 13(b), respectively. One can see that there exist several partitioning strategies hidden in these methods. If we use the preference of a theoretical number of recoverable communities in a $k$ -planted partition model [\citenameAmes, 2013], being $O(\sqrt{n})$ , the methods studied could be considered to over-fit (create more than $k$ clusters) or under-fit (create less than $k$ clusters) as presented in Table 4, in the third column.

With an overview of the second and third columns of Table 4, methods belonging to the same theoretical class, which shares a common assumption about the definition of community, have a tendency to show the same fitting quality. This has also been identified by [\citenameGhasemian et al., 2018]. However, although being useful to help practitioners to presume the expected number of clusters which a method would detect with respect to the theoretical experience, it is still very challenging to decide which method to use, since the reference is based on a hypothesis about an underlying model. This also means that if the hypothesis about the partition model changes (another model than $k$ -planted model), the expected number of communities will be diversified, and hence the indicated fitting quality preference becomes disproved. As a consequence, in the next section, we propose a novel technique to estimate the similarity of community detection methods based on community size distributions.

3.3 Similarity based on community size distribution

A very naive but efficient approach to evaluate the similarity of two methods is to inquire into the “closeness” of the two corresponding community size distributions [\citenameDao et al., 2018b]. Two methods could be supposed to be similar, if their corresponding density distributions expose a large intersection area, as shown in Figure 14(a). From this notice, we can define our new similarity function as follows:

First, we denote two 2-tuples $(\mathcal{A},n^{a})$ and $(\mathcal{B},n^{b})$ being the multisets representing all communities detected on a set of networks $\mathcal{G}=\{G\}$ by method $A$ and method $B$ respectively, where $\mathcal{A}=\{x_{1}^{a},x_{2}^{a},...,x_{r}^{a}\}$ and $\mathcal{B}=\{x_{1}^{b},x_{2}^{b},...,x_{s}^{b}\}$ being the ascending ordered sets of sizes of communities: $1\leq x_{1}^{a}<x_{2}^{a}<...<x_{r}^{a}$ and $1\leq x_{1}^{b}<x_{2}^{b}<...<x_{s}^{b}$ . The multiplicity functions $n^{a}:\mathcal{A}\rightarrow\mathbb{N}_{\geq 1}$ and $n^{b}:\mathcal{B}\rightarrow\mathbb{N}_{\geq 1}$ measure the number of communities of sizes $x_{i}^{a}$ and $x_{i}^{b}$ respectively. Let $N^{a}=\sum_{i=1}^{r}n^{a}(x_{i}^{a})$ and $N^{b}=\sum_{i=1}^{s}n^{b}(x_{i}^{b})$ being the total number of communities of all sizes detected by each method, we define a similarity function describing the closeness of $A$ and $B$ on $\mathcal{G}$ as:

[TABLE]

where $\delta(x_{i}^{a},x_{j}^{b})=1$ if $x_{i}^{a}=x_{j}^{b}$ and [math] otherwise. Equation (1) is simply the common fraction of same-size communities detected on $\mathcal{G}$ by both $A$ and $B$ : $0\leq S_{\mathcal{G}}(A,B)\leq 1$ . This definition seems to be intuitive but does not work well in practice. As illustrated in Figure 14(b), when the sizes interlace with each other, a low score will be produced, although the level of similarity in this case is that of Figure 14(a). Choosing an appropriate binning interval would mitigate the problem. This solution is quite inflexible, however.A straightforward alternative can be envisioned by using a kernel density estimator to uncover the probability density function as shown by the solid lines in Figure 14(b). In this way, we approximate the common fraction of same-size communities of Equation (1) by the overlapping area of two corresponding continuous distributions. The premise behind this estimation is that two similar methods do not always produce a large portion of exact same-size communities but rather a large portion of comparable-size ones. Hence, we consider the following estimator to take into account local information of community size $x_{0}$ :

[TABLE]

where $h$ is the bandwidth controlling the neighborhood interval around $x_{0}$ and $K$ is the kernel function controlling the weight given to the observations $\{x_{i}\}$ , chosen as Gaussian in our analysis. One would wonder why we use a Gaussian, whereas Power law may be a better fit, as some papers (eg. [\citenameClauset et al., 2004]) show that community size in many real world networks exhibits a power law distribution. We would like to recall that expecting such groundtruth-like partitions does not mean that community detection methods impose this property into their mechanism and produce Power law distribution. For example, on the Zachary network, although we try to discover two equivalent-size groups, community detection methods identify two to four similar-size communities. Some methods indeed fit a power law, but as a matter of fact, many others have a tendency to partition network nodes into quite balanced groups (spectral methods, modularity optimization methods, SBM, etc.) as seen in the previous section. This led us to use a Gaussian kernel, as it fits well in most of our cases on our large dataset and our panel of methods. In a less generic and exhaustive context, when the resulting community size varies significantly, following a Power law distribution, we, of course, recommend using the appropriate law.

Using our estimator, we rewrite the similarity function defined in Equation (1) as follows:

[TABLE]

where

[TABLE]

with $u\in\{a,b\}$ . In the estimations of this paper, the bandwidth $h$ is selected based on the normal reference rule [\citenameSilverman, 1986] to minimize the mean integrated squared error.

Using equations (3) and (4) to estimate the similarity between pairs of detection methods on a large dataset will help us to discover different behaviors of community detection methods. Since the accuracy of the estimator depends on the networks of the dataset that we analyze, the result will obviously have to be relativized. However, our large and representative corpus helps to reduce the dependency impact.

3.3.1 Experimental results

From the communities identified in the previous section, we proceed to measure the volumes of communities detected by each method to determine the elements of the corresponding 2-tuples. Finally, we use the similarity function defined by Equation (3) to estimate the closeness between each pair of methods. Due to the huge number of experiments, only processes having a reasonable theoretical estimated time and memory consumption are maintained (less than a few days and at most 30 to 40 GBytes of memory). The outcome distributions are illustrated in Figure 15.

As we can see, there is a clear difference in the densities of community size, showing that these methods have various partitioning strategies. Knowing that methods belonging to the same theoretical group (as shown in Table 1) are placed next to each other, we can see some agreements between the theoretical families, with practical outcomes as follows:

Edge removal: GN and RCCLP-3 have very similar distributions where a large number of communities are very small. This is due to the fact that in some highly local centralized networks, having star-like structures [\citenameDao et al., 2018a], they have a tendency to remove edges connecting hub and peripheral nodes and to create singletons (single node community). This phenomenon is less distinguishable on RCCLP-4, since there are far fewer quadrangular than triangular connections in networks.
Modularity optimization: Modularity is known to suffer from resolution limit phenomenon [\citenameFortunato & Barthelemy, 2006], which often aggregates small communities in large scale networks. We can see from Figure 15 that Louvain and SN found very large communities, as predicted. At the same time, there are also a comparable number of small communities which are found on small graphs. However, the behavior is a little different on CNM method, which is an agglomerative clustering algorithm based on modularity optimization.
Dynamic process: Methods in this family show very discernible distributions although all are based on dynamic processes. In fact, they make different assumptions about community structure and searching mechanisms. Therefore, belonging to the same theoretical family does not lead to a similarity in practical results.
Statistical inference: the Bayesian SBM and DCSBM use the Monte Carlo sampling process, which is very time-consuming, in order to sweep the solution space. This makes the method unfeasible, if the maximum number of clusters is not limited. Indeed, in the default version, the maximum number of communities is limited to 25, meaning that the $(DC)SBM$ methods find very large communities in large networks. On the other hand, the Oslom method uses an agglomerative discovery mechanism and identifies globally smaller communities.
Other methods: In this group, LPA, $SPLA$ (both based on label propagation) and Conclude display almost identical distributions. $RB$ method, being based on a very close concept with modularity (with a tuning parameter), exhibits a similarity with modularity optimization based methods.

Quantitatively, applying the estimator presented in Equation (4), to compute pairwise similarities between the methods, leads us to the results demonstrated in Figure 16.

A well-known method for comparing two distributions is the Kolmogorov-Smirnov test (KS). Under the null hypothesis, the two distributions are identical. The test generates the cumulative distributions of the distributions, reports the maximum difference between them and calculates a p-value. If the Kolmogorov-Smirnov statistic (distance) is small or the $p$ value is high (above the significance level, e.g. $0.05$ ), the assumption that the distributions of the two samples are identical cannot be rejected. Conversely, we can reject the null hypothesis, if the p-value is low. In our case, $p-value<0.05$ for all the pairs, which means that the distributions do not fit the same model, except for a single pair $(Conclude,LPA)$ with a value $p=0.07$ . A deeper exploration shows that these two methods do not fit a normal law, but they fit better a power law distribution than other heavy-tailed distributions (with a non-significant difference, however). In practice, the KS distance is useful, if one is mostly interested in how similar two datasets are. As shown on the Table 5, the smallest distances occur with couples (SLPA, LPA), (Conclude, LPA), (DCSBM, RB), (RCCLP-4, Walktrap), (GN, RCCLP-3), (Infomod, SN), (Louvain, SN), which is consistent with our results.

Thus, according to the community size criterion, the methods can be classified into different classes of partitioning strategy. Similar results are also discovered by [\citenameGhasemian et al., 2018], as they found that methodological similar methods usually lead to a similar outcome. The separations are clearly shaped, showing that the distinction is very clear between groups. Therefore, we choose to characterize these methods by 3 (possibly 4) principle groups as follows:

Group 1 - $RB$ , DCSBM, SBM, Infomod, SN, Louvain: Methods in this group discover communities whose sizes vary in wide range of spectra, from very small to very large communities. The characterized community size distribution is quite flat, meaning all sizes are nearly equally considered. 2. 2.

Group 2 - GN and RCCLP-3: These two methods identify a huge number of very small communities including singletons regardless of network size. As a consequence, there are few variations in community volume. 3. 3.

Group 3 - the others: These methods produce communities whose sizes approach bell-shaped distribution. The strategy can be translated as: neither left nor right, i.e. communities that are neither too small and nor too big.

This characterization could help us to identify appropriate groups of community detection methods, according to different community size fitting strategies. Also, it helps to avoid brute-force attempts when a method does not succeed in proposing desired partitions by proposing substitute solutions. Moreover, by combining with the previous time-computation analysis in Section 3.1, one could also choose a group of methods corresponding to size distinction criteria and then select the fastest method that leads to a desired outcome.

Community distribution (or number of communities) is just one possible quality dimension, albeit possibly one of the most intuitive and important pieces of information when choosing a clustering method. In the next part, we demonstrate some techniques that can be used to define other similarity aspects. We show that these notions of similarity can be combined to accentuate the distinction between different community detection methods.

4 Quality profiling of community detection methods

4.1 Community quality metrics

A popular way to evaluate the structure of communities is to design quality metrics, in order to measure different expected characteristics from subgraphs that we want to obtain. In practice, metrics using network generative models are sometimes preferable, as they reflect different assumptions about the underlying mechanisms that create community structure. One of the most widely used metrics, that quantifies the quality of community structure, is the modularity function. The idea here is to reveal how the quality of an identified community structure is different from that which would be expected. Although some unexpected phenomena, known as resolution limit [\citenameFortunato & Barthelemy, 2006, \citenameTraag et al., 2011] have been exposed, when the scale of community size is too small, modularity remains the standard measure of quality.

The advantage of this approach is that one can “embed” the assumption of community structure inside quality functions. Hence, they provide better performance in some cases. However, community structure is quite an open question, such that, according to different mechanisms that render the structure of networks, there will be models that are more suitable than others.

We present some quality metrics (also called “community scoring functions” or less often “goodness metrics”) to evaluate community structure. Many of them are initially or gradually employed as objective functions in some community detection methods, since they expose good performance in searching processes. Firstly, we recall some notations that will be used to describe structural characteristics of communities. A graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ consisting of $n=|\mathcal{V}|$ vertices and $m=|\mathcal{E}|$ edges can be represented by an associated adjacency matrix $A$ . Given a community $C$ of $n_{C}$ vertices being a subgraph of $\mathcal{G}$ in an arbitrary partition $P$ , a function $f(C)$ or $f(P)$ quantifies a structural goodness feature of community $C$ or the whole partition $P$ , according to a particular expectation of community structure. Let $m_{c}$ be the number of edges inside community $C$ , $m_{C}=|(i,j)\in\mathcal{E}:i\in C,j\in C|$ , $l_{C}$ be the number of edges that connect $C$ to other vertices outside of $C$ , $l_{C}=|(i,j)\in\mathcal{E}:i\in C,j\not\in C|$ . Any vertex $i$ belonging to community $C$ has an internal degree $k_{iC}^{int}$ and an external degree $k_{iC}^{ext}$ satisfying $k_{iC}^{int}+k_{iC}^{ext}=k_{i}$ , where $k_{i}$ is the total degree of vertex $i$ . The internal and external degrees can be expressed via the adjacency matrix $A$ as: $k_{iC}^{int}=\sum_{j\in C}A_{ij}$ and $k_{iC}^{ext}=\sum_{j\notin C}A_{ij}$ . If vertex $i$ in community $C$ has $k_{iC}^{ext}>0$ and $k_{iC}^{int}\geq 0$ , $i$ is called boundary vertex since $i$ has neighbor(s) outside of $C$ . Otherwise, if $k_{iC}^{ext}=0$ and $k_{iC}^{ext}>0$ , $i$ is called internal vertex which only has connections with other vertices in the same community. In this paper, we employ the following functions to evaluate community structure:

4.1.1 Newman-Girvan Modularity

The standard version of modularity [\citenameNewman & Girvan, 2004] reflects the difference for the fraction of intra community edges of a partition, with the expected number of such edges, if distributed according to a null model. In the standard version of modularity, the null model preserves the expected degree sequence of the graph under consideration. In other words, the modularity compares the real network structure with a corresponding one where nodes are connected without any preference about their neighbors. There are several ways to mathematically express modularity. In order to compare standard modularity with other variants, it is convenient to consider modularity as a sum of contributions from pairs of vertices of the same community:

[TABLE]

Despite some existing issues such as the widely discussed resolution limits [\citenameFortunato & Barthelemy, 2006] or the indistinguishability structures of two partitions having similar modularity scores, the Newman-Girvan modularity is still widely used in the community as a proof of quality.

4.1.2 Erdős-Rényi Modularity

The Newman-Girvan modularity has attracted much attention in the research literature. Many alternative derivations have been proposed to adapt to different contexts. Some use different null models to quantify the modular structure of partitions. For example, one could assume that vertices in a network are connected randomly with a constant probability, $p$ , as formulated in the Erdős-Rényi (ER) model [\citenameErdős & Rényi, 1959]. The connection probability is calculated as $p=\frac{2m}{n(n-1)}$ , being the number of presented edges over the total number of edges that could be established. The expected number of edges in a community of size $n_{c}$ becomes $\langle m_{c}\rangle=p{n_{c}\choose 2}$ . This null model leads us to the ER Modularity:

[TABLE]

4.1.3 Modularity Density (D-value or D-modularity)

Standard modularity is found to be impacted by resolution limits [\citenameFortunato & Barthelemy, 2006], i.e., it is claimed that the sizes of detected modules depend on the size of the whole network, such that optimizing standard modularity can not identify communities having a small number of vertices. The expected number of intra-community edges is highly sensitive to the total number of edges in the whole network [\citenameRosvall & Bergstrom, 2007], as can be observed in the second term of Equation (5). Modularity density [\citenameLi et al., 2008] is one of several propositions envisioned to palliate this issue. The idea of this metric is to include the information about community size in the expected density of a community to avoid the negligence of small and dense communities. For each community $C$ in partition $P$ , it uses the average modularity degree calculated by $d(C)=d^{int}(C)-d^{ext}(C)$ where $d^{int}(C)$ and $d^{ext}(C)$ are the average internal and external degrees of $C$ respectively to evaluate the fitness of $C$ in its network. Finally, the modularity density can be calculated as follows:

[TABLE]

4.1.4 Z-modularity

Z-modularity is another variant of the standard modularity that is proposed to avoid the resolution limit [\citenameMiyauchi & Kawase, 2016]. The founding idea of this version is based on the observation that the difference between the fraction of edges inside communities and the expected number of such edges in a null model should not be considered as the only contribution to the final quality of community structure. Specifically, the authors recommend that the statistical rareness of a community should also be taken into consideration. In other words, the rarer the structure of a commnunity, the greater is its contribution to the final modularity. Therefore, the variance of the probability distribution of the fraction of the number of edges within each community is included in the quality function through a standardization using Z-score. Following the null model of the standard modularity, the probability that an edge in placed inside community $C$ is $p=(\frac{D_{C}}{2m})^{2}$ , where $D_{C}=2m_{C}+l_{C}$ is the total degree of community $C$ . The number of edges in each community follows a binomial distribution with the probability $p$ and its normalized value approaches a normal distribution when the number of edges is sufficiently large. The statistical rarity of partition $P$ in terms of the fraction of the number of intra-community edges using Z-score is thus translated into Z-modularity as follows:

[TABLE]

4.1.5 Surprise

This statistical approach proposes a quality metric assuming that edges between vertices emerge randomly according to a hyper-geometric distribution [\citenameAldecoa & Marín, 2011]. Specifically, for a graph of $n$ vertices and $m$ edges, there are $M={n\choose 2}$ possible ways of drawing $m$ edges. For a particular partition, there are $M^{int}=\sum_{C\in P}{n_{c}\choose 2}$ possible ways of drawing an intra-community edge. Surprise metric computes the (minus logarithm of) probability of observing at least $m^{int}=\sum_{C\in P}\frac{k_{C}^{int}}{2}$ intra-community edges within $m$ draws without replacement from the population of $M$ possible choices in which there are precisely $M^{int}$ possible intra-community edges. This probability is formalized as follows:

[TABLE]

However, this formulation is not easy to work with in large-scale networks due to numerical computational problems. Hence, [\citenameTraag et al., 2015] provides an asymptotic approximation for the metric, which is a good alternative. By assuming that the relative number of intra-community edges $q=\frac{m^{int}}{m}$ and the relative number of expected intra-community edges $\langle q\rangle=\frac{M^{int}}{M}$ remains fixed, Surprise metric is approximated at:

[TABLE]

where $D(q||\langle q\rangle)$ is the Kullback–Leibler divergence [\citenameKullback & Leibler, 1951]:

[TABLE]

According to the Surprise metric, the higher the score of a partition, the less likely it is that it resulted from a random realization, and thus the better the quality of the community structure will be.

4.1.6 Significance

This metric uses a similar approach to the Surprise metric. It also estimates how likely a partition of dense communities is to appear in a random graph, but in a different way [\citenameTraag et al., 2015]. While Surprise uses global quantities $q$ and $1-\langle q\rangle$ , Significance compares each community density $p_{C}=\frac{m_{C}}{{n_{C}\choose 2}}$ to the average graph density $p=\frac{m}{M}$ . The asymptotic form of Significance can be written as:

[TABLE]

Similarly, $D(x||y)$ is the Kullback–Leibler divergence defined in Equation (11). Generally, if the number of communities is relatively large or the graph is relatively dense, Significance is more discriminative than Surprise. On the other hand, in the case where $\langle q\rangle>p$ , Surprise can be better than Significance [\citenameTraag et al., 2015].

4.2 Co-performance index

We propose a new comparative approach using a matrix called community detection co-performance matrix. The idea is that, given an expected quality function, one could investigate whether there exists a correlation in the efficiency of enhancing (or aggravating) its scores between different methods. The co-performance matrices reveal how understanding the performance of a method in optimizing a quality would allow us to predict the performance of other methods on the same quality. Therefore, an exhaustive analysis of co-performance matrices on many qualities allows us to profile the characteristics of community detection methods in a comparative way. The index could be calculated as follows:

Let methods $A$ and $B$ divide a graph $G_{i}=(V_{i},E_{i})$ of dataset $\mathcal{G}=\{G_{i}|i=1..N\}$ into $\alpha$ and $\beta$ communities described by partitions $P_{G_{i}}^{a}=\{C^{a}_{1G_{i}},C^{a}_{2G_{i}},...,C^{a}_{\alpha G_{i}}\}\in\mathcal{P}_{G_{i}}$ and $P_{G_{i}}^{b}=\{C^{b}_{1G_{i}},C^{b}_{2G_{i}},...,C^{b}_{\beta G_{i}}\}\in\mathcal{P}_{G_{i}}$ respectively. We consider, solely, hard clustering methods, meaning $C_{uG_{i}}^{a}\cap C_{vG_{i}}^{a}=\varnothing:1\leq u<v\leq\alpha$ and $C_{uG_{i}}^{b}\cap C_{vG_{i}}^{b}=\varnothing:1\leq u<v\leq\beta$ . A function $Q:\mathcal{P}_{G_{i}}\rightarrow\mathbb{R}$ quantifies a quality of a partition of graph $G_{i}$ according to a particular goodness aspect (or model).

We define a co-performance index of two methods $A$ and $B$ on $\mathcal{G}$ by their mutual capacity in discovering community structures showing a particular quality $Q$ . In other words, each couple of methods should be assigned a high index according to a quality $Q$ , if knowing the performance of one method significantly reveals the information about the performance of the other. A straightforward solution for defining the index is using the Pearson correlation. In other words, it reflects the covariance of a quality score on two sets of partitions detected by two methods, indicating whether two methods agree or disagree about a quality criterion in the context of a network dataset.

Denoting $q^{a}_{i}=Q(P_{G_{i}}^{a})$ and $q^{b}_{i}=Q(P_{G_{i}}^{b})$ , the co-performance index can be calculated as follows:

[TABLE]

where $0\leq I_{\mathcal{G}}(A,B,Q)\leq 1$ . With a Pearson correlation, a high positive (resp. negative) score implies that two methods often find a strong consensus (resp. disagreement) in discovering communities with regards to a particular quality. Given a co-performance index, knowing the quality scores of one method could provide predictive information about the outcomes of the other method on the same dataset. This information could, in fact, be very useful in a context where alternative solutions must be deployed while maintaining an assumed quality, as expected. We present in the following section the mutual performance of the presented detection methods by the previously presented quality functions.

Figure 18 illustrates the co-performance matrices, according to six different quality goodness criteria. Again, similarly to the previous section, goodness functions with a close concept are placed together. For instance, NG modularity and ER modularity are both based on null models whose concept uses an expected fraction of intra-community edges. While the hypothesis of NG version is to keep the expected degree sequence of the graph in question, the ER version redistributes edges randomly with a constant average degree for every node. $D$ -modularity and $Z$ -modularity attempt to penalize large communities by including community sizes and significance level, respectively. One can notice a very slight similarity in the experimental results of the co-performance indexes between different quality functions. Also, it seems that the assumption about the quality of community structure has an impact on the co-performance outcome.

As shown in Figure 18, there is a class of methods (Louvain, GN, CNM, RB, Infomod, Infomap, Walktrap, Oslom, LPA, SLPA, Conclude) in which all methods show very consistent results, except for the case of $D$ -modularity121212In fact, density modularity is somewhat apart from other traditional ways of defining modularity, as it is not defined based on a null model but solely on edge density. The term $D$ -modularity is misused in this sense. Furthermore, there is also a strong relation between SBM and DCSBM. For the other methods (RCCLP and SN), no clear tendency could be observed from this experiment. The similarity of a large number of methods in many quality functions implies that, globally, if a method performs well on a given network, there is a signal that the others (from the same group) could also obtain good results. In other words, if the community structure in a network is clear, most methods will be able to detect it with more or less accuracy and conversely. However, as the co-performance indexes also vary significantly ( $0.2$ to $0.3$ ) inside each group, there will be always a remarkable difference if one goes from one method to another.

Within the case of density modularity shown in Figure 18(c), we discover that the sizes of detected communities have a great impact on co-performance. Since density is a measure that penalizes large size communities heavily, especially in sparse networks, $D$ -modularity gives very small values for giant communities and very high values for small ones. Concretely, the methods SBM, DCSBM, Infomod, RB, SN discover very large communities (as shown in Section 3.3) and their co-performances in terms of $D$ -modularity are very weak, showing that internal densities of communities detected by these methods are not linearly correlated. The reason is that the corresponding densities fluctuate unpredictably around zero. Similarly, GN and RCCLP-3 found many tiny communities making the density either very high or zero (if the internal degree is equal to the external degree), consequently the co-performance index can not show significant information. On the other hand, we notice a consistency between the similarity of community size and the co-performance when methods identify medium-sized communities. Specifically, we find high co-performance indexes between CNM, Conclude, Oslom, Walktrap, LPA, SLPA, Infomap methods in most of the cases of the six quality fitness functions. This finding exposes a global agreement with our categorization, determined by community size distributions.

The co-performance matrices also disclose interesting information about quality functions. As we can see in Figure 18(a,b,d), the matrices imply a similarity between NG modularity, ER modularity and $Z$ -modularity in the assumptions of quality. In the same way, Surprise and Significance are quite close in practice, as illustrated in 18(d,e). This experiment gives more proof about the closeness between the theoretical assumptions of community structure and the practical outcome. Moreover, although being based on different aspects of goodness, the performance of many methods tends to reach agreement on the modular structure of networks in general. This is to say, methods in the same group identify roughly and globally comparable results, although there are always significant differences. In order to strengthen and validate our conclusion, we are interested in using other popular approaches in the literature to compare these community detection algorithms, which will be presented in the next section.

5 Partitioning strategy comparison

This section is dedicated to using conventional clustering validation metrics from the literature to verify the previous similarity analyses. We employ some popular metrics in the traditional clustering context (and also widely used in community detection context), which measure directly the similarity of partitions using their corresponding contingency tables. These metrics do not take into consideration the structural information of community structures, but only use the common numbers of nodes that are shared by pairs of communities in two partitions.

5.1 Validation metrics

The consensus of two partitions $P_{1}=\{c_{1}^{(1)},c_{2}^{(1)},...,c_{R}^{(1)}\}$ and $P_{2}=\{c_{1}^{(2)},c_{2}^{(2)},...,c_{S}^{(2)}\}$ can be more practically observed using a contingency table (sometimes called confusion matrix or association matrix), whose elements $n_{ij}=|c_{i}^{(1)}\cap c_{j}^{(2)}|$ correspond to the number of common vertices between the $i$ -th community of $P_{1}$ and the $j$ -th community of $P_{2}$ as shown in Table 6.

In the evaluation of community structure using a validation metric, some of the following validation metrics are often used in the context of community detection, to define the matching coefficient between two arbitrary partitions of a network.

5.1.1 Rand Index (RI)

The rand index is a pair-counting based measure, defined as the ratio of the number of vertex pairs correctly classified (either in the same community or in different communities) by the total number of pairs [\citenameRand, 1971]. The RI penalizes both false positive and false negative decisions of the clusterings. When the false positive has to be neglected, we can refer to the Jaccard index [\citenameKuncheva & Hadjitodorov, 2004]. The rand index value of two partitions can be calculated by the following:

[TABLE]

The value varies between $1$ (meaning the two partitions are identical) and [math] (indicating that the two partitions do not agree on any pair of vertices). However, this value is only observed in the scenario when one partition consists of one community and the other consists of $n$ community of $1$ vertex, which has little practical value. Another shortcoming of the rand index is that its expected value for two randomly chosen partitions does not take a constant value, which is normally expected for a good matching index [\citenameVinh et al., 2010]. Therefore, a modified version of RI has been suggested, taking into consideration the expected value of randomness [\citenameHubert & Arabie, 1985], which is introduced in the following.

5.1.2 Adjusted Rand Index (ARI)

The corrected version of rand index takes the form:

[TABLE]

It quickly becomes a replacement recommended for measuring agreement between two partitions in the analysis of clusterings. Its values ranges from $-1$ to $1$ indicating completely different and identical partitions, respectively. It is known to be less sensitive to the difference of the number of communities between two partitions. An ARI value of [math] indicates that the similarity is equal to the expected value from randomly chosen partitions. It can be calculated as:

[TABLE]

5.1.3 Normalized Mutual Information (NMI)

Information theoretic based metrics constitute another approach for validating community structure with a given reference partition. Using the same notations as previously presented, the Mutual Information (MI) between two partitions quantifying the mutual dependence is calculated as:

[TABLE]

It measures how much knowing a partitioning of vertices in one way would reduce the uncertainty about the other way. In order words, it could be considered as an indicator of information closeness expressed by the joint distribution between two variables. Therefore, the mutual information can be used as a similarity metric between two partitions. However, it needs to be normalized to reflect a consistency between different measures. The normalization uses the entropy of each partition defined by:

[TABLE]

Several variants of normalization can be considered, for instance taking the average, the root or the maximum of entropy of the two partitions as the denominator [\citenameAna & Jain, 2003]. In this document, we use the average version which is widely used in the context of community analysis [\citenameDanon et al., 2005, \citenameChakraborty et al., 2017]. The closed form of NMI is hence defined from Equation (17) and (18), as follows:

[TABLE]

Likewise, the NMI similarity between two partitions varies between [math], corresponding to independent relation, and $1$ when two partitions are identical. NMI does not follow triangle inequality.

5.1.4 Adjust Mutual Information (AMI)

Similarly to the Rand Index, Mutual Information is also subject to the effect of randomness, i.e., there is not a constant baseline value between random partitions of a graph. This issue raises many difficulties in the comparison mechanism, since it is expected that a comparative index should preserve the relativity between different clusterings and enhance intuitiveness about the mutual agreement. For that reason, traditional Mutual Information is normalized with a supplementary correction for chance. This normalization method has recently received attention for comparing graph partitions. It is calculated as follows [\citenameVinh et al., 2010]:

[TABLE]

where $I(P_{1},P_{2})$ and $H(P)$ are introduced in equations (17) and (18) respectively. $E\{I(M)|n_{i\bm{\cdot}}n_{\bm{\cdot}j}\}$ is the expected mutual information value of all feasible contingency tables constructed from the actual table $M$ with the same marginals $n_{i\bm{\cdot}}$ , $n_{\bm{\cdot}j}$ .

5.1.5 Normalized Variation of Information (NVI)

Another popular metric that is often used in the context of comparing community partition similarity is the Variation of Information (VI) [\citenameMeilă, 2003], which is defined as:

[TABLE]

The VI metric can be interpreted as an index of shared information distance between two partitions. Its lower bound is [math] and it occurs when the two partitions are identical, whereas the upper bound $\log(n)$ happens when they are completely different. It is also preferable to use a normalized version, chance-corrected, to avoid the effect of randomness. Similarly to the construction of the Adjusted Mutual Information, with the same notation, the Normalized Variation of Information is calculated as follows:

[TABLE]

However, it turns out that $NVI$ discloses the same information with $AMI$ , since from Equation (20) and (22), one has $NVI(P_{1},P_{2})=1-AMI(P_{1},P_{2})$ . Consequently, calculating $VI$ and $NVI$ is unnecessary. We have chosen the $RI$ , $ARI$ , $NMI$ , and $AMI$ metrics in our experiment. A summary of these validation metrics are shown in Table 7.

Validation metrics are often used in the context of community structure evaluation to measure the difference between the partition identified by a method with an expected partition of the network under consideration (ground-truth). The more similar the discovered partition is to the ground-truth, the better the performance of the method. However, in this section, validation metrics are exploited as a tool to compare community structures of different detection methods. They estimate the practical proximity of different algorithms through detected partitions, which constitutes a supplementary source of information for evaluating their performance in a comparative approach.

5.2 Empirical results

Once again, the experimental process is the same as those of the previous sections. From the partitions detected by the methods on the dataset, we calculate pairwise scores, quantified by each validation metric on each network. Figure 19 illustrates pairwise average scores of the 4 metrics over the networks of the dataset131313Where the corresponding methods are able to finish using a reasonable amount of time and memory as mentioned in the previous experiments..

Again, by observing the dendrograms in Figure 19, one can see that all 4 metrics classify methods into two principle groups in a similar way to the co-performance matrices exposed in the previous section. The group of methods CNM, Conclude, Oslom, Walktrap, LPA, SLPA, Infomap mentioned in the last section also show very strong similarities in this experiment. LPA and SLPA, especially, being based on label propagation mechanism show nearly identical results in many cases. In addition, one could discern another group including RB, CNM, GN and Louvain (modularity based), which shows high consistency in general. Additionally, even with weaker scores, SBM and DCSBM are often found in the same group, with RCCLP-3 and RCCLP-4 too. Globally, it seems that methods with a close theoretical approach tend to provide more similar results, which is also revealed in the previous sections.

Another fact that could be extracted from this experiment is that $RI$ should not be used as a validation metric for evaluating detection performance. Since its average values vary generally over a small range ( $0.9$ to $1.0$ ), it is more difficult to see the difference between partitions. On the other hand, $NMI$ and $AMI$ show very close results in our experiment (between $0.5$ and $1.0$ ), meaning that structural communities detected by different methods are quite comparable, as concluded in the previous section. Finally, $ARI$ seems to magnify the differences between methods. However, there is no major difference in the similarity evaluation in comparison with the other metrics.

6 Related work

Orman et al. published a comparative evaluation of eight community detection algorithms [\citenameOrman et al., 2012], most of which are also studied in this paper. Different validation metrics are used to evaluate the agreement between the discovered partitions and reference community structures (ground-truths). As in our work, they find that these metrics ( $RI$ , $ARI$ , $NMI$ ) “agree with each other with small differences when considering the way they rank algorithms”, as illustrated in Section 5. Furthermore, the authors also focus on analyzing many topological aspects of community structure including transitivity, density, community size, etc. These topological qualities are then used to inspect community structures detected by different algorithms. The analyses allow the authors to conclude that these two approaches (topological metrics and validation metrics), used to evaluate community structures, are “complementary and needed to perform a relevant and complete analysis of community detection results”. They also note that the “traditional approach ( $RI$ , $ARI$ , $NMI$ ) is much faster and easier to apply”, and hence they advise using these metrics first. However, in practice, ground-truths are not usually available141414In the context where a new algorithm is invented, one normally uses networks whose community structures are well known in order to validate the proposed method. In reality, since community detection is often employed to discover structures of new networks, it is therefore not likely that reference community structures always exist.. Therefore, from the above observations, our analyses in this paper could be an important support, dispensing additional information about the closeness between methods, both in terms of the topological aspect and the partition-based aspect.

Agreste et al. evaluate different community detection algorithms in an empirical and comparative approach, especially for the context of web data analytic [\citenameAgreste et al., 2017]. The authors find that “time complexity is a crucial factor in the selection of a community detection algorithm” and find that the label propagation method (LPA) “has outstanding performance in scalability on both artificial and real graphs”, which is also in global agreement with our analysis in Section 3.1 providing predictions about required time for each method, according to network size. They also conclude that “Infomap algorithm showcased the best trade-off between accuracy and computational performance” based on $NMI$ score. Such a conclusion is valid in some specific cases where the expected ground-truth community structure is well understood. Otherwise, some additional analyses should be done to determine whether ground-truth information aligns the final objective of community detection algorithms [\citenamePeel et al., 2017]. Indeed, metadata information of nodes are often used in practice as a ground-truth community structure, whereas it has been found that metadata communities are sometimes very sparse [\citenameDao et al., 2017]. One should be cautious as regards generalizing. Algorithms may have difficulty agreeing with some non-aligned metadata-based reference, but one should not assume this level of performance on other datasets.

Ghasemian et al. present, in a recent publication, an evaluation of overfitting and underfitting of several community detection models [\citenameGhasemian et al., 2018]. The authors study the number of communities detected in practice by many methods, and the maximum number of detectable clusters, according to a theoretical model. Some conclusions are drawn about fitting qualities of methods in comparison to theoretical estimates. This study helps to choose an appropriate method according to a fitting quality. Community detection methods are also grouped in distinct families, based on their outputs on many real-world networks (similarly to our analysis in Section 5.1) using $AMI$ metric. The authors also find that “what an algorithm finds in a network depends strongly on the assumptions it makes about what to look for”, which is aligned with our results through several analyses.

Jebabli et al. also propose a framework to assess the performance of community detection algorithms, based on the topological characteristics of the resulting community-graphs [\citenameJebabli et al., 2018]. In their paper, the authors restrict their attention to networks with overlapping community structures, whereas our comparative evaluation concerns hard clustering. However, they discuss how their framework can be applied to networks with non-overlapping community structure. They present and evaluate an efficient alternative methodology, as compared to the classical quality and clustering measures. Similarly to our work, algorithms are compared in a decision-making scheme. They provide different rankings of the algorithms, according to different topological properties. They also introduce a Multiple-Criteria Decision-Making strategy in order to find the best compromise between these different rankings. A single ranking is produced. Such a decision making strategy is therefore a very interesting approach. Converting recommendations into a decision process is one of our perspective studies.

7 Conclusion

It is quite challenging to recommend which method is best for which scenario. It is at least as demanding as defining all possible scenarios. We carried out several experiments, demonstrating different aspects of community structure quality, which can be combined together in a flexible way to assist network analysts to find appropriate methods, according to the context. The following questions, for example, could be asked sequentially during decision making processes:

What is the size of the network under consideration and what is the acceptable computation time for the task of community detection? 2. 2.

What are the expectations about the number of communities as well as the community size distribution? 3. 3.

Is there any fitness function that should be optimized? 4. 4.

In circumstances where the targeted method cannot be deployed, is there an alternative solution?

The experiments and results in this paper could help to identify suitable method(s) quickly, if one is able to answer the above questions.

The consideration of computation time is indeed crucial in the process of choosing a community detection method for a problem at hand. Even if the theoretical estimate of time complexity is important and reveals the scalability of a community detection method, computation time is worth being studied in practice. Our estimates provide detailed information of practical time required by many popular community detection methods, according to network size. In particular the most scalable methods that we tested (Louvain, LPA and SN) approximately reduce $10^{4}$ times the required computation time, compared with most of the other methods. This is not only significant but also crucial for a large network. Given a network size, our results help in filtering non suitable methods.

In addition, the expected number of communities to be obtained is another important criterion in choosing a community detection method. Depending on the context, one could prefer different granularity levels. Our study shows that there are globally three main strategies that community detection methods use to decompose a network. Specifically, some identify communities whose sizes vary regularly over a wide range of values, from very small to very large communities. Some others divide networks into a huge number of very small communities and very few large communities. And finally, the latter distribute nodes into similar medium-size communities (around 10 members). Therefore, knowing how a network should be broken down is very useful in order to end up with an appropriate community discovery method.

In cases where (advanced) network analysts can determine a targeted objective function, designing new algorithms (or employing existing algorithms) that optimize the function would be the most effective. Since improving an objective function usually means expending more computation time, a compromise between getting higher fitness scores and using less time needs to be considered. However, finding a good method to optimize an objective function, satisfying a time constraint condition, is not straightforward and needs much investigation. Our approach, as presented in the co-performance analysis, provides network practitioners with a quick view of how different methods perform in improving some widely-used quality functions. This predictive information about the effect of using alternative methods in achieving good fitness scores would suggest multiple solutions for network analysts to reach a certain objective function. This scenario is specifically useful when the desired method is too expensive, in terms of computation time. Therefore, a combination of our empirical analyses about scalability and/or community size distribution with the co-performance index can identify eligible alternatives for specific cases.

Finally, we find that using some validation metrics to estimate the similarity between community detection methods could also provide interesting information that could help the decision process of network analysts. In situations where one knows exactly (or has notions of) what should be found (ground-truth information), studying the way nodes are allocated to communities is important, as it provides useful information about how a method is able to reach the desired clusters. However, this scenario does not often occur, since community detection is in general used to discover the structures of networks when no a-priori information is available. When performing several methods is possible, validation metrics are used to compare their results and then identify different types of partitions. From our empirical study, we noticed significant differences in the way that nodes are distributed in communities. Methods such as SBM or RCCLP-4, especially, seem to detect partitions which are very discernible from those of the others. Hence, the use of these methods needs to be examined, and we shall recommend their use, along with other methods, as they may bring totally different, and probably complementary insights to the data.

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TL;DR

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Community Structure: A Comparative Evaluation of Community Detection Methods

Abstract

Contents

1 Introduction

2 Community detection methods and dataset

2.1 Community detection methods

2.1.1 Edge removal based methods

2.1.2 Modularity optimization methods

2.1.3 Dynamic process based methods

2.1.4 Statistical inference based methods

2.1.5 Other methods

2.2 Experimental dataset

3 Preliminary analysis of community detection methods

3.1 Computation time performance

3.2 Analysis on community size distribution

3.2.1 Edge removal approach: GN, RCCLP-3 and RCCLP-4

3.2.2 Modularity optimization approach: CNM, Louvain and SN

3.2.3 Dynamic process approach: Infomap, Infomod and Walktrap

3.2.4 Statistical inference approach: SBM, DCSBM and Oslom

3.2.5 RB, LPA, SLPA and Conclude methods

3.2.6 Summary

3.3 Similarity based on community size distribution

3.3.1 Experimental results

4 Quality profiling of community detection methods

4.1 Community quality metrics

4.1.1 Newman-Girvan Modularity

4.1.2 Erdős-Rényi Modularity

4.1.3 Modularity Density (D-value or D-modularity)

4.1.4 Z-modularity

4.1.5 Surprise

4.1.6 Significance

4.2 Co-performance index

5 Partitioning strategy comparison

5.1 Validation metrics

5.1.1 Rand Index (RI)

5.1.2 Adjusted Rand Index (ARI)

5.1.3 Normalized Mutual Information (NMI)

5.1.4 Adjust Mutual Information (AMI)

5.1.5 Normalized Variation of Information (NVI)

5.2 Empirical results

6 Related work

7 Conclusion