Discriminative Distance-Based Network Indices with Application to Link Prediction

TL;DR

This paper introduces a new discriminative distance measure for large networks that improves centrality indices, enables efficient estimation algorithms, and enhances link prediction accuracy in real-world graphs.

Contribution

It develops a novel discriminative distance measure, algorithms for its exact and approximate computation, and demonstrates its effectiveness in link prediction tasks.

Findings

Discriminative indices have higher discriminability than traditional indices.

Randomized algorithms accurately estimate average discriminative path length and eccentricity.

Real-world networks exhibit small average discriminative path lengths, often bounded by 2.

Abstract

In large networks, using the length of shortest paths as the distance measure has shortcomings. A well-studied shortcoming is that extending it to disconnected graphs and directed graphs is controversial. The second shortcoming is that a huge number of vertices may have exactly the same score. The third shortcoming is that in many applications, the distance between two vertices not only depends on the length of shortest paths, but also on the number of shortest paths. In this paper, first we develop a new distance measure between vertices of a graph that yields discriminative distance-based centrality indices. This measure is proportional to the length of shortest paths and inversely proportional to the number of shortest paths. We present algorithms for exact computation of the proposed discriminative indices. Second, we develop randomized algorithms that precisely estimate average discriminative path length and average discriminative eccentricity and show that they give $(\epsilon,\delta)$-approximations of these indices. Third, we perform extensive experiments over several real-world networks from different domains. In our experiments, we first show that compared to the traditional indices, discriminative indices have usually much more discriminability. Then, we show that our randomized algorithms can very precisely estimate average discriminative path length and average discriminative eccentricity, using only few samples. Then, we show that real-world networks have usually a tiny average discriminative path length, bounded by a constant (e.g., 2). Fourth, in order to better motivate the usefulness of our proposed distance measure, we present a novel link prediction method, that uses discriminative distance to decide which vertices are more likely to form a link in future, and show its superior performance compared to the well-known existing measures.

Tables

Table 1. Table 1. Specifications of the largest component of the real-world datasets.

Dataset	Link	# vertices	# edges	Discriminative diameter
dblp0305	http://www-kdd.isti.cnr.it/GERM/	109,045	233,962	2
dblp0507	http://www-kdd.isti.cnr.it/GERM/	135,116	290,364	2
dblp9202	http://www-kdd.isti.cnr.it/GERM/	129,074	277,082	2
facebook-uniform	http://odysseas.calit2.uci.edu/doku.php/public:online_social_networks	134,304	135,532	2
flickr	http://konect.uni-koblenz.de/networks/flickrEdges	73,342	2,619,711	5
gottron-reuters	http://konect.uni-koblenz.de/networks/gottron-reuters	38,677	978,461	5
petster-friendships	http://konect.uni-koblenz.de/networks/petster-friendships-cat	148,826	5,449,508	8
pics_ut	http://konect.uni-koblenz.de/networks/pics_ut	82,035	2,300,296	5
web-Stanford	http://snap.stanford.edu/data/web-Stanford.html	255,265	2,234,572	16
web-NotreDame	http://snap.stanford.edu/data/web-NotreDame.html	325,729	1,524,589	28
citeulike-ut	http://konect.uni-koblenz.de/networks/citeulike-ut	153,277	2,411,940	7
epinions	http://konect.uni-koblenz.de/networks/epinions	119,130	834,000	15
wordnet	http://konect.uni-koblenz.de/networks/wordnet-words	145,145	656,230	15

Table 2. Table 2. Comparison of discriminability of the centrality notions over different real-world networks. For each dataset, the most discriminative index is highlighted in bold.

Database	Discriminative closeness	Closeness	Betweenness	Length scaled betweenness	Linearly scaled betweenness	Katz
dblp0305	$2.7805$	$0.0201$	$0.0403$	$0.0403$	$0.0403$	$0.4447$
dblp0507	$2.7013$	$0.0155$	$0.0325$	$0.0325$	$0.0325$	$0.3804$
dblp9202	$3.2973$	$0.0147$	$0.0263$	$0.0263$	$0.0263$	$0.3091$
facebook-uniform	$5.6178$	$0.0446$	$0.0528$	$0.0528$	$0.0528$	$0.9039$
flickr	$92.7694$	$4.4435$	$84.8381$	$85.0835$	$85.0835$	$90.4365$
gottron-reuters	$88.9934$	$25.8810$	$74.3956$	$74.3956$	$74.3956$	$88.9753$
petster-friendships	$70.0764$	$39.2176$	$65.7049$	$65.7317$	$65.7317$	$70.0260$
pics_ut	$50.5113$	$36.3552$	$33.4028$	$33.5210$	$33.5210$	$42.6196$
web-Stanford	$97.3376$	$18.9258$	$26.6542$	$27.2861$	$27.2861$	$31.6122$
web-NotreDame	$29.9819$	$18.5230$	$18.1245$	$19.2402$	$19.2402$	$19.0277$
citeulike-ut	$45.2540$	$30.4546$	$28.6135$	$28.7185$	$28.7185$	$34.3032$
epinions	$70.0218$	$57.0679$	$42.1220$	$45.5745$	$45.5745$	$60.1922$
wordnet	$58.8907$	$51.8770$	$38.8838$	$40.5187$	$40.5187$	$52.8340$

Table 3. Table 3. Relative error of our randomized average discriminative path length estimation algorithm.

Database	Exact values		Approximate algorithm
	APL	ADPL	Sample size ($\%$)	Relative error ($\%$)
dblp0305	$1.99997$	$1.99995$	$10$	$0.0016$
			$1$	$0.0016$
			$0.1$	$0.0016$
dblp0507	$1.99997$	$1.99996$	$10$	$0.0008$
			$1$	$0.0008$
			$0.1$	$0.0008$
dblp9202	$1.99997$	$1.99996$	$10$	$0.0015$
			$1$	$0.0015$
			$0.1$	$0.0012$
facebook-uniform	$1.99998$	$1.99997$	$10$	$0.0097$
			$1$	$0.0099$
			$0.1$	$0.0102$
flickr	$2.3078$	$0.2787$	$10$	$2.5639$
			$1$	$0.2887$
			$0.1$	$2.7859$
gottron-reuters	$2.9555$	$0.6860$	$10$	$0.5827$
			$1$	$3.6259$
			$0.1$	$19.3603$
petster-friendships	$2.7028$	$0.2220$	$10$	$0.8221$
			$1$	$0.4389$
			$0.1$	$2.4948$
pics_ut	$3.6961$	$0.2953$	$10$	$0.4478$
			$1$	$1.8667$
			$0.1$	$2.9500$
web-Stanford	$6.8152$	$0.9509$	$10$	$0.6261$
			$1$	$1.4910$
			$0.1$	$2.7938$
web-NotreDame	$7.1731$	$1.5856$	$10$	$0.0618$
			$1$	$0.6948$
			$0.1$	$2.9328$
citeulike-ut	$3.9376$	$0.2361$	$10$	$0.0355$
			$1$	$1.6612$
			$0.1$	$2.3498$
epinions	$4.1814$	$0.9098$	$10$	$0.0240$
			$1$	$0.7403$
			$0.1$	$0.7527$
wordnet	$5.5320$	$1.1141$	$10$	$0.1610$
			$1$	$0.6710$
			$0.1$	$2.7228$

Table 4. Table 4. Relative error of our randomized average discriminative eccentricity estimation algorithm.

Database	Exact values		Approximate algorithm
	AE ($\times 1000$)	ADE ($\times 1000$)	Sample size ($\%$)	Relative error ($\%$)
dblp0305	$0.0183$	$0.0183$	$10$	$0.0013$
			$1$	$0.0013$
			$0.1$	$0.0013$
dblp0507	$0.0148$	$0.0148$	$10$	$0.0007$
			$1$	$0.0007$
			$0.1$	$0.0007$
dblp9202	$0.0154$	$0.0154$	$10$	$0.0015$
			$1$	$0.0015$
			$0.1$	$0.0015$
facebook-uniform	$0.0148$	$0.0148$	$10$	$0.0096$
			$1$	$0.0096$
			$0.1$	$0.0096$
flickr	$0.0566$	$0.0323$	$10$	$0.2627$
			$1$	$1.1411$
			$0.1$	$1.6568$
gottron-reuters	$0.1159$	$0.0898$	$10$	$0.2327$
			$1$	$0.4596$
			$0.1$	$4.4685$
petster-friendships	$0.0432$	$0.0293$	$10$	$0.0749$
			$1$	$0.04465$
			$0.1$	$2.3459$
pics_ut	$0.0636$	$0.0450$	$10$	$0.0604$
			$1$	$0.7933$
			$0.1$	$0.8762$
web-Stanford	$0.4171$	$0.0314$	$10$	$0.3697$
			$1$	$1.0153$
			$0.1$	$2.4259$
web-NotreDame	$0.0852$	$0.0399$	$10$	$0.0235$
			$1$	$0.4150$
			$0.1$	$0.4150$
citeulike-ut	$0.0406$	$0.0262$	$10$	$0.3076$
			$1$	$0.2016$
			$0.1$	$3.2315$
epinions	$0.0894$	$0.0676$	$10$	$0.0780$
			$1$	$0.2170$
			$0.1$	$0.3784$
wordnet	$0.0780$	$0.0589$	$10$	$0.0724$
			$1$	$0.4879$
			$0.1$	$0.5011$

Table 5. Table 5. Specifications of the temporal real-world datasets used in our experiments for link prediction.

Dataset	Link	#vertices	#temporal edges	Time span
sx-stackoverflow	https://snap.stanford.edu/data/sx-stackoverflow.html	2,601,977	63,497,050	2774 days
sx-mathoverflow	http://snap.stanford.edu/data/sx-mathoverflow.html	24,818	506,550	2350 days
sx-superuser	https://snap.stanford.edu/data/sx-superuser.html	194,085	1,443,339	2773 days
sx-askubuntu	http://snap.stanford.edu/data/sx-askubuntu.html	159,316	964,437	2613 days
wiki-talk-temporal	https://snap.stanford.edu/data/wiki-talk-temporal.html	1,140,149	7,833,140	2320 days
CollegeMsg	http://snap.stanford.edu/data/CollegeMsg.html	1,899	20,296	193 days

Equations

$$C(v)=\frac{1}{n-1}\sum _{u\in V(G)\setminus \{v\}}d(v,u).$$

$$APL(G)=\frac{1}{n\times (n-1)}\sum _{v\in V(G)}\sum _{u\in V(G)\setminus \{v\}}d(v,u).$$

$$E(v)=\frac{1}{n-1}\underset{u\in V(G)\setminus \{v\}}{\max }d(v,u).$$

$$AE(G)=\frac{1}{n\times (n-1)}\sum _{v\in V(G)}\underset{u\in V(G)\setminus \{v\}}{\max }d(v,u).$$

$$DC(v)=\frac{1}{n-1}\sum _{u\in V(G)\setminus \{v\}}\frac{d(v,u)}{\sigma (v,u)}.$$

$$ADPL(G)=\frac{1}{n\times (n-1)}\sum _{v\in V(G)}\sum _{u\in V(G)\setminus \{v\}}\frac{d(v,u)}{\sigma (v,u)}.$$

$$DE(v)=\frac{1}{n-1}\underset{u\in V(G)\setminus \{v\}}{\max }\frac{d(v,u)}{\sigma (v,u)}.$$

$$ADE(G)=\frac{1}{n\times (n-1)}\sum _{v\in V(G)}\underset{u\in V(G)\setminus \{v\}}{\max }\frac{d(v,u)}{\sigma (v,u)}.$$

$$DD(G)=\underset{v\in V(G)}{\max }\underset{u\in V(G)\setminus \{v\}}{\max }\frac{d(v,u)}{\sigma (v,u)},$$

$$DR(G)=\underset{v\in V(G)}{\min }\underset{u\in V(G)\setminus \{v\}}{\max }\frac{d(v,u)}{\sigma (v,u)}.$$

$$HC(v)=\frac{1}{n-1}\sum _{u\in V(G)\setminus \{v\}}\frac{1}{d(v,u)}.$$

$$DHC(v)=\frac{1}{n-1}\sum _{u\in V(G)\setminus \{v\}}\frac{\sigma (v,u)}{d(v,u)},$$

$$\mathbf{E}\left[{\beta }_t\right]$$

$$\mathbf{P}\left[\left|ADPL(G)-\beta \right|>ϵ\right]\le 2\exp \left(-2\times T\times {\left(\frac{ϵ}{\Delta (G)}\right)}^2\right).$$

$$\mathbf{P}\left[\left|\mathbf{E}\left[\bar{X}\right]-\bar{X}\right|>ϵ\right]\le 2\exp \left(-2n{\left(\frac{ϵ}{b-a}\right)}^2\right).$$

$$\mathbf{P}\left[\left|ADPL(G)-\beta \right|>ϵ\right]\le 2\exp \left(-2\times T\times {\left(\frac{ϵ}{\log n}\right)}^2\right).$$

$$\frac{\#\text{distinct centrality scores}}{\#\text{vertices of }G}\times 100.$$

$$\frac{|\text{emexact score}-\text{emapproximate score}|}{\text{emexact score}}\times 100,$$

$$AUC(ind)=\frac{n_g+0.5n_e}{n_t},$$

$$Q(ind)=\sum _{\{u,v\}\in TE}\frac{rank(\{u,v\},L_{ind})}{|TE|},$$

Full text

Social network analysis, distance-based network indices, discriminative indices, closeness centrality, eccentricity, average path length, the tiny-world property, link prediction

††copyright: none††conference: ; ; ††ccs: Theory of computation Shortest paths

1. Introduction

An important category of network indices is based on the distance (the length of the shortest paths) between every two vertices in the network. It includes closeness centrality, average path length, vertex eccentricity, average graph eccentricity, etc. Indices in this category have many important applications in different areas. For example, in disease transmission networks, closeness centrality is used to measure vulnerability to disease and infectivity (Bell et al., 1999). In routing networks, vertex eccentricity is used to determine vertices that form the periphery of the network and have the largest worst-case response time to any other device (Magoni and Pansiot, 2001; Takes and Kosters, 2013). In biological networks, vertices with high eccentricity perceive changes in concentration of their neighbor enzymes or molecules (Pavlopoulos et al., 2011).

Using the length of shortest paths as the distance measure has shortcomings. A well-studied shortcoming is that extending it to disconnected graphs (and also directed graphs) is controversial (Wasserman et al., 1994; Cornwell, 2005; Marchiori and Latora, 2000; Opsahl et al., 2010). The other –less studied– shortcoming is that by using this measure, a huge number of vertices may find exactly the same closeness/eccentricity score. For instance, Shun (Shun, 2015) recently reported that around $30\%$ of the (connected) vertices of the Yahoo graph have the same non-zero eccentricity score. Our experiments, reported in Section 6.1, reveal that this happens in many real-world graphs. This restricts the applicability of distance-based indices such as closeness and eccentricity, as they cannot distinguish vertices. For example, when closeness or eccentricity are used for the facility location problem (Koschützki et al., 2004), they may not be able to distinguish one location among a set of candidate locations. Finally, in many cases, the distance between two vertices not only depends on the length of shortest paths, but also on the number of shortest paths between them. As a simple example, consider a network of locations where edges are roads connecting the locations. In a facility location problem, given two (or more) candidate locations, we want to choose the one which is more accessible from the rest of the network. Then, we may prefer the location which is slightly farther from the rest of the network but has more connections to the location which is closest to the rest of the network. In particular, if two locations have exactly the same distance from the other locations, the one connected to the rest of the network by more roads is preferred.

These observations motivate us to develop a new distance measure between vertices of a graph that yields more discriminative centrality notions. Furthermore, it considers both shortest path length and the number of shortest paths. In this paper, our key contributions are as follows.

• •

We propose new distance-based network indices, including discriminative closeness, discriminative path length, discriminative vertex eccentricity and average discriminative graph eccentricity. These indices are proportional to the length of shortest paths and inversely proportional to the number of shortest paths. Our empirical evaluation of these notions reveals an interesting property of real-world networks. While real-world graphs have the small-world property which means they have a small average path length bounded by the logarithm of the number of their vertices, they usually have a considerably smaller average discriminative path length, bounded by a constant (e.g., $2$). We refer to this property as the tiny-world phenomena.

• •

We present algorithms for exact computation of the proposed discriminative indices. We then develop a randomized algorithm that precisely estimate average discriminative path length (and average discriminative eccentricity) and show that it can give an $(\epsilon,\delta)$-approximation, where $\epsilon\in\mathbb{R}^{+}$ and $\delta\in(0,1)$.

• •

We perform extensive experiments over several real-world networks from different domains. First, we examine discriminability of our proposed indices and show that compared to the traditional indices, they are usually much more discriminative111Note that having a total ordering of the vertices is not always desirable and by discriminative indices, we do not aim to do so. Instead, we want to have a partial ordering over a huge number of vertices that using traditional distance-based measures, find exactly the same value. . Second, we evaluate the empirical efficiency of our simple randomized algorithm for estimating average discriminative path length and show that it can very precisely estimate average discriminative path length, using only few samples. Third, we show that our simple randomized algorithm for estimating average discriminative eccentricity can generate high quality results, using only few samples. This has analogy to the case of average eccentricity where a simple randomized algorithm significantly outperforms more advanced techniques (Shun, 2015).

• •

In order to better motivate the usefulness of our proposed distance measure in real-world applications, we present a novel link prediction method, that uses discriminative distance to indicate which vertices are more likely to form a link in future. By running extensive experiments over several real-world datasets, we show the superior performance of our method, compared to the well-known existing methods.

The rest of this paper is organized as follows. In Section 2, preliminaries and necessary definitions related to distance-based indices are introduced. A brief overview on related work is given in Section 3. In Section 4, we introduce our discriminative distance-based indices and discuss their extensions and properties. We present exact and approximate algorithms for computing discriminative indices in Section 5. In Section 6, we empirically evaluate discriminability of our indices and the efficiency and accuracy of our randomized algorithms. In Section 7, we present our link prediction method and show its superior performance. Finally, the paper is concluded in Section 8.

2. Preliminaries

In this section, we present definitions and notations widely used in the paper. We assume that the reader is familiar with basic concepts in graph theory. Throughout the paper, $G$ refers to a graph (network). For simplicity, we assume that $G$ is a connected, undirected and loop-free graph without multi-edges. Throughout the paper, we assume that $G$ is an unweighted graph, unless it is explicitly mentioned that $G$ is weighted. $V(G)$ and $E(G)$ refer to the set of vertices and the set of edges of $G$, respectively. We use $n$ and $m$ to refer to $|V(G)|$ and $|E(G)|$, respectively. We denote the set of neighbors of a vertex $v$ by $\mathcal{N}(v)$.

A shortest path (also called a geodesic path) between two vertices $v,u\in V(G)$ is a path whose length is minimum, among all paths between $v$ and $u$. For two vertices $v,u\in V(G)$, we use $d(v,u)$, to denote the length (the number of edges) of a shortest path connecting $v$ and $u$. We denote by $\sigma(v,u)$ the number of shortest paths between $v$ and $u$. By definition, $d(v,v)=0$ and $\sigma(v,v)=0$. We use $deg(v)$ to denote the degree of vertex $v$. The diameter of $G$, denoted by $\Delta(G)$, is defined as $\max_{v,u\in V(G)}d(v,u)$. The radius of $G$ is defined as $\min_{v\in V(G)}\max_{u\in V(G)\setminus\{v\}}d(v,u)$.

Closeness centrality of a vertex $v\in V(G)$ is defined as (Kang et al., 2011):222 The more common definition of closeness centrality is as follows (Bergamini et al., 2016): $C(v)=\frac{n-1}{\sum_{u\in V(G)\setminus\{v\}}d(v,u)}$. In this paper, due to consistency with the definitions of the other distance-based indices, we use the definition presented in Equation 1. Note that this change has no effect on the results presented in the paper and they are still valid for the more common definition of closeness.

[TABLE]

Average path length of graph $G$ is defined as (Newman, 2003):

[TABLE]

Eccentricity of a vertex $v\in V(G)$ is defined as (Chartrand et al., 1996; Husfeldt, 2017):333Again, while the common definition of eccentricity does not have the normalization factor $\frac{1}{n-1}$, here in order to have consistent definitions for all the distance-based indices, we add it to Equation 3.

[TABLE]

Average eccentricity of graph $G$ is defined as (Chartrand et al., 1996; Husfeldt, 2017):

[TABLE]

Center of a graph is defined as the set of vertices that have the minimum eccentricity. Periphery of a graph is defined as the set of vertices that have the maximum eccentricity.

3. Related work

The widely used distance-based indices are closeness centrality, average path length, eccentricity and average eccentricity defined in Section 2. In all these indices, it is required to compute the distance between every pair of vertices. The best algorithm in theory for solving all-pairs shortest paths is based on matrix multiplication (Williams, 2012) and its time complexity is $O(n^{2.3727})$. However, in practice breadth first search (for unweighted graphs) and Dijkstra’s algorithm (for weighted graphs with positive weights) are more efficient. Their time complexities for all vertices are $O(nm)$ and $O(nm+n^{2}\log n)$, respectively. In the following, we briefly review exact and inexact algorithms proposed for computing closeness and eccentricity.

3.1. Closeness centrality and average path length

Eppstein and Wang (Eppstein and Wang, 2004) presented a uniform sampling algorithm that with high probability approximates the inverse closeness centrality of all vertices in a weighted graph $G$ within an additive error $\epsilon\Delta(G)$. Their algorithm requires $O(\frac{\log n}{\epsilon^{2}})$ samples and spends $O(n\log n+m)$ time to process each one. Brandes and Pich (Brandes and Pich, 2007) extended this sampler by considering different non-uniform ways of sampling. Cohen et.al. (Cohen et al., 2014) combined the sampling method with the pivoting approach (Cohen, 2000) and (Ullman and Yannakakis, 1991), where pivoting is used for the vertices that are far from the given vertex. Olsen et.al. (Olsen et al., 2014) suggested storing and re-using the intermediate results that are common among different vertices. Okamoto et.al. (Okamoto et al., 2008) presented an algorithm for ranking top $k$ highest closeness centrality vertices that runs in $O((k+n^{\frac{2}{3}}\log^{\frac{1}{3}}n)(n\log n+m))$ time. There are several extensions of closeness centrality for specific networks. Kang et.al. (Kang et al., 2011) defined closeness centrality of a vertex $v$ as the (approximate) average distance from $v$ to all other vertices in the graph and proposed algorithms to compute it in MapReduce. Tarkowski et.al. (Tarkowski et al., 2016) developed a game-theoretic extension of closeness centrality to networks with community structure.

3.2. Eccentricity and average eccentricity

Dankelmann et.al. (Peter Dankelmann, 2004) showed that the average eccentricity of a graph is at least $\frac{9n}{4(deg_{m}+1)}+O(1)$, where $deg_{m}$ is the minimum degree of the graph. Roditty and Williams (Roditty and Vassilevska Williams, 2013) developed an algorithm that gives an estimation $\hat{E}(v)$ of the eccentricity of vertex $v$ in an undirected and unweighted graph, such that $\hat{E}(v)$ is bounded as follows: $\frac{2}{3}E(v)\leq\hat{E}(v)\leq\frac{3}{2}E(v)$. Time complexity of this algorithm is $O(m\sqrt{n\log n})$. Takes and Kosters (Takes and Kosters, 2013) presented an exact eccentricity computation algorithm, based on lower and upper bounds on the eccentricity of each vertex of the graph. They also presented a pruning technique and showed that it can significantly improve upon the standard algorithms. Chechik et.al. (Chechik et al., 2014) introduced an $O(\left(m\log m\right)^{3/2})$ time algorithm that gives an estimate $\hat{E}(v)$ of the eccentricity of vertex $v$ in an undirected and weighted graph, such that $\frac{3}{5}E(v)\leq\hat{E}(v)\leq E(v)$. Shun (Shun, 2015) compared shared-memory parallel implementations of several average eccentricity approximation algorithms. He showed that in practice a two-pass simple algorithm significantly outperforms more advanced algorithms such as (Roditty and Vassilevska Williams, 2013) and (Chechik et al., 2014).

4. Discriminative distance-based indices

In this section, we present the family of discriminative distance-based indices.

4.1. Indices

The first index is discriminative closeness centrality. Similar to closeness centrality, discriminative closeness is based on the length of shortest paths between different vertices in the graph. However, unlike closeness centrality, discriminative closeness centrality considers the number of shortest paths, too. For a vertex $v\in V(G)$, discriminative closeness of $v$, denoted with $DC(v)$, is formally defined as follows:

[TABLE]

If in the definition of average path length, closeness centrality is replaced by discriminative closeness centrality defined in Equation 5, we get average discriminative path length of $G$, defined as follows:

[TABLE]

In a similar way, discriminative eccentricity of a vertex $v\in V(G)$, denoted by $DE(v)$, is defined as follows:

[TABLE]

Finally, average discriminative eccentricity of $G$ is defined as follows:

[TABLE]

All these notions are based on replacing distance by discriminative distance, defined as follows. For $v,u\in V(G)$, discriminative distance between $v$ and $u$, denoted with $dd(v,u)$, is defined as $\frac{d(v,u)}{\sigma(v,u)}$. We define discriminative diameter and discriminative radius of $G$ respectively as follows:

[TABLE]

[TABLE]

Finally, we define discriminative center of a graph as the set of vertices that have the minimum discriminative eccentricity; and discriminative periphery of a graph as the set of vertices that have the maximum discriminative eccentricity.

Generalizations

We can consider two types of generalizations of Equations 5-10. In the first generalization, in the denominator of the equations, instead of using the number of shortest paths, we may use the number of a restricted class of shortest paths, e.g., vertex disjoint shortest paths, edge disjoint shortest paths etc. In the second generalization, instead of directly using distances and the number of shortest paths, we may introduce and use functions $f$ and $g$, defined respectively on the length and the number of shortest paths. Then, by changing the definitions of $f$ and $g$, we can switch among different distance-based notions. For example, for any two vertices $v,u\in V(G)$, if $f(d(v,u))$ and $g(\sigma(v,u))$ are respectively defined as $d(v,u)$ and $1$, we will have the traditional distance-based indices introduced in Section 2. If $f(d(v,u))$ and $g(\sigma(v,u))$ are respectively defined as $\frac{1}{d(v,u)}$ and $1$, we will have harmonic closeness centrality (Marchiori and Latora, 2000) defined as follows:

[TABLE]

Then, someone may define discriminative harmonic closeness centrality of vertex $v$ as:

[TABLE]

where $f(d(v,u))$ and $g(\sigma(v,u))$ are respectively defined as $\frac{1}{d(v,u)}$ and $\frac{1}{\sigma(v,u)}$.

Connection to the other indices

Path-based indices such as betweenness centrality (Chehreghani, 2014b; Chehreghani et al., 2018) (and its generalizations such as group betweenness centrality (Veremyev et al., 2017) and co-betweenness centrality (Chehreghani, 2014a)) consider the number of shortest paths that pass over a vertex. However, betweenness centrality does not consider the shortest path length and it is used as an indicator of the amount of control that a vertex has over shortest paths in the network. Some variations of betweenness centrality, such as length-scaled betweenness centrality and linearly scaled betweenness centrality (Brandes, 2008), are more similar to our proposed notions. However, they still measure the amount of control that a vertex has over shortest paths, but give a weight (which is a function of distance) to the contribution of each shortest path. In our proposed notions, the number of shortest paths passing over a vertex does not always contribute to the centrality of the vertex. Indices such as Katz centrality (Katz, 1953) and personalized PageRank (Haveliwala, 2002) consider both the length and the number of paths between two vertices. However, there are important differences, too. For example, Katz centrality is proportional to both the length and the number of paths. Furthermore, it considers all paths. This makes it inappropriate for the applications where the concept of shortest paths is essential. This index is mainly used in the analysis of directed acyclic graphs. If in the Katz index of two vertices $v$ and $u$, denoted by $K(v,u)$, the paths are limited to shortest paths (and the bias constant $\beta$ is set to [math]), we can express it using our generalization of discriminative indices. If this limitation is applied, $K(v,u)$ will be defined as $\alpha^{d(v,u)}\sigma(v,u)$, where $\alpha$ is the attenuation factor (Katz, 1953). Then, this index can be seen as a special case of our generalized discriminative index, where $f(d(v,u))$ is defined as $\alpha^{d(v,u)}$ and $g(\sigma(v,u))$ is defined as $\frac{1}{\sigma(v,u)}$.

The other index that may have some connection to our discriminative indices is clustering coefficient (Watts and Strogatz, 1998). Both clustering coefficient and discriminative indices are sensitive to the local density of the vertices, however, they have different goals. While clustering coefficient aims to directly reflect the local density, discriminative indices aim to take into account the density of different regions of the graph, when computing distances.

Disconnected or directed graphs

When the graph is disconnected or directed, it is possible that there is no (shortest) path between vertices $v$ and $u$. In this case, $d(v,u)=\infty$ and $\sigma(v,u)=0$, hence, $\frac{d(v,u)}{\sigma(v,u)}$ is undefined. For closeness centrality, when $d(v,u)=\infty$, a first solution is to define $d(v,u)$ as $n$. The rationale is that in this case $d(v,u)$ is a number greater than any shortest path length. We can use a similar technique for discriminative distance: when there is no path from $v$ to $u$, we define $d(v,u)$ as $n$ and $\sigma(v,u)$ as $1$. This discriminative distance will be greater than the discriminative distance between any two vertices $v^{\prime}$ and $u^{\prime}$ that are connected by a path from $v^{\prime}$ to $u^{\prime}$. The second solution suggested for closeness centrality is to use harmonic centrality (Marchiori and Latora, 2000). As stated in Equation 11, this can be applied to discriminative closeness, too. When $d(v,u)=\infty$, Equation 11 yields $\frac{0}{\infty}$, which is conventional to define as [math].

A property

A nice property of shortest path length is that for vertices $v,u,w\in V(G)$ such that $w$ is on a shortest path between $v$ and $u$, the following holds: $d(v,u)=d(v,w)+d(w,u)$. This property is useful in e.g., designing efficient distance computation algorithms. This property does not hold for discriminative distance as $dd(v,u)$ can be less than or equal to $dd(v,w)+dd(w,u)$. An example is presented in Figure 1. However, we believe this is not a serious problem. The reason is that more than shortest path length that satisfies the above mentioned property, discriminative distance is based on the number of shortest paths, which satisfies the following property: $\sigma_{w}(v,u)=\sigma(v,w)\times\sigma(w,u)$, where $\sigma_{w}(v,u)$ is the number of shortest paths between $v$ and $u$ that pass over $w$. As we will discuss in Section 5, these two properties can help us to design efficient algorithms for computing discriminative distance-based indices.

Why our proposed indices are more discriminative

It is very difficult to theoretically prove that for general graphs, our proposed indices are always more discriminative than the existing distance-based indices. However, we can still provide arguments explaining why in practice our proposed indices are more discriminative. We here focus on unweighted graphs. Let $n$ be the number of vertices. While the maximum possible shortest path length in a graph is $n-1$, for real-world networks it is much smaller and it is $\log n$. This means the range of all possible values of distance is narrow and as a result, a vertex may have the same distance from many other vertices. This yields that many vertices may have the same closeness/eccentricity scores. However, the range of all possible values of the number of shortest paths between two vertices is much wider and it varies between $1$ and an exponential function of $n$. For example, on the one hand, in the graph of Figure 2, there are $2^{(n/2)-1}$ shortest paths between $s$ and $t$ and on the other hand, in a tree there is only one shortest path between any two vertices. This wider range yields that when the number of shortest paths is involved, the probability that two vertices have exactly the same score drastically decreases.

4.2. Intuitions

In several cases, the distance between two vertices in the graph not only depends on their shortest path length, but also (inversely) on the number of shortest paths they have. In the following, we discuss some of them.

Time and reliability of traveling

A key issue in transportation and logistics (Polus, 1979; Bonsall et al., 1997) and in vehicular social networks (VSNs) (Ning et al., 2017c) is to estimate the traveling time and route reliability between two points $A$ and $B$. The time and reliability of traveling from $A$ to $B$ depend on the structure of the road network and also on the stochastic factors such as weather, traffic incidents. One of the factors that depends on the network structure is the number of ways that someone can travel from $A$ to $B$. Having several ways to travel from $A$ to $B$, on the one hand, increases the reliability of traveling. On the other hand, it deceases traffic between $A$ and $B$ and as a result, the traveling time. Therefore, taking into account both the length and the number of shortest paths between $A$ and $B$ (in other words, defining the distance between $A$ and $B$ in terms of both the length and the number of shortest paths) can help to better estimate the time and reliability of traveling between $A$ and $B$.

Spread of infections

It it known that infection and contact rates in a network depend on the community structure of the network and the spread of infections inside a community is faster (Ball et al., 1997; Pellis et al., 2011; Liu and Hu, 2005). Consider vertices $v_{1}$, $v_{2}$ and $v_{3}$ such that $d(v_{1},v_{2})=d(v_{1},v_{3})$, $v_{1}$ and $v_{2}$ are in the same community but $v_{1}$ and $v_{3}$ are not. An infection from $v_{1}$ usually spreads to $v_{2}$ faster than $v_{3}$, or the probability that (after some time steps) $v_{2}$ becomes infected by $v_{1}$ is higher than the probability that $v_{3}$ becomes infected. Here, to describe the distances between the vertices, our discriminative distance measure is a better notion than the shortest path length. Vertices $v_{1}$ and $v_{2}$ that are inside a community are heavily connected and as a result, they usually have many shortest paths between themselves. In contrast, $v_{1}$ and $v_{3}$ do not belong to the same community and are not heavily connected, hence, they usually have less shortest paths between themselves. This means $dd(v_{1},v_{2})$ is smaller than $dd(v_{1},v_{3})$, which is consistent with the infection rate.

5. Algorithms

In this section, we discuss how discriminative indices can be computed. First in Section 5.1, we present the exact algorithms and then in Section 5.2, we present the approximate algorithms.

5.1. Exact Algorithms

In this section, we present the DCC444DCC is an abbreviation for Discriminative Closeness Calculator. algorithm for computing discriminative closeness centrality of all vertices of the network and show how it can be revised to compute the other discriminative indices.

Algorithm 1 shows the high level pseudo code of the algorithm. DCC is an iterative algorithm where at each iteration, discriminative closeness of a vertex $v$ is computed. This is done by calling the ShortestPathDAG method for $v$. Inside ShortestPathDAG, the distances and the number of shortest paths between $v$ and all other vertices in the graph are computed. If $G$ is unweighted, this is done by a breadth-first search starting from $v$. Otherwise, if $G$ is weighted with positive weights, this is done using Dijkstra’s algorithm (Dijkstra, 1959). A detailed description of ShortestPathDAG can be found in several graph theory books, including (Diestel, 2010), hence, we here ignore it.

Computing the other indices

Algorithm 1 can be revised to compute average discriminative path length of $G$, discriminative eccentricity of vertices of $G$ and average discriminative eccentricity of $G$.

• •

$ADPL(G)$. After Line 12 of Algorithm 1 (where the $I[v]$ values are already computed), $ADPL(G)$ can be computed as $\frac{\sum_{v\in V(G)}I[v]}{n}$.

• •

$DE(v)$. If Line 10 of Algorithm 1 is replaced by the following lines:

  if $\frac{1}{n-1}\times\frac{D[u]}{N[u]}>I[v]$ then

     $I[v]\leftarrow\frac{1}{n-1}\times\frac{D[u]}{N[u]}$.

  end if

then, the algorithm will compute discriminative eccentricity of the vertices of $G$ and will store them in $I$.

• •

$ADE(G)$. After computing discriminative eccentricity of all vertices of $G$ and storing them in $I$, $ADE(G)$ can be computed as $\frac{\sum_{v\in V(G)}I[v]}{n}$.

In a similar way, Algorithm 1 can be revised to compute discriminative diameter and discriminative radius of $G$.

Complexity analysis

For unweighted graphs, each iteration of the loop in Lines 7-12 of method DCC takes $O(m)$ time. For weighted graphs with positive weights, using a Fibonnaci heap, it takes $O(m+n\log n)$ time (Diestel, 2010). This means discriminative closeness centrality and discriminative eccentricity of a given vertex can be computed respectively in $O(m)$ time and $O(m+n\log n)$ time for unweighted and weighted graphs with positive weights. However, computing average discriminative path length and/or average eccentricity of the graph requires respectively $O(nm)$ time and $O(nm+n^{2}\log n)$ time for unweighted graphs and weighted graphs with positive weights. Space complexity of each iteration (and the whole algorithm), for both unweighted graphs and weighted graphs with positive weights, is $O(n+m)$ (Diestel, 2010). Note that these complexities are the same as complexities of computing traditional distance-based indices. The reason is that in addition to computing the distances between $v$ and all other vertices of the graph, ShortestPathDAG can also compute the number of shortest paths, without having any increase in the complexity (Diestel, 2010).

5.2. Randomized Algorithms

While discriminative closeness and discriminative eccentricity of a given vertex can be computed efficiently, the algorithms of computing average discriminative path length and average discriminative eccentricity of the graph are expensive in practice, even for mid-size networks. This motivates us to present randomized algorithms for ADPL and ADE that can be performed much faster, at the expense of having approximate results.

Algorithm 2 shows the high level pseudo code of the RandomADPL algorithm, proposed to estimate average discriminative path length. The inputs of the algorithm are the graph $G$ and the number of samples (iterations) $T$. In each iteration $t$, the algorithm first chooses a vertex $v$ uniformly at random and calls the ShortestPathDAG method for $v$ and $G$, to compute distances and the number of shortest paths between $v$ and any other vertex in $G$. Then, it estimates average discriminative path length of $G$ at iteration $t$ as $\frac{1}{n-1}\times\sum_{u\in V(G)\setminus\{v\}}\frac{d(v,u)}{\sigma(v,u)}$ and stores it in $\beta_{t}$. The average of all $\beta_{t}$ values computed during different iterations gives the final estimation $\beta$ of average discriminative path length. Clearly, for unweighted graphs, time complexity of Algorithm 2 is $O(T\times m)$ and for weighted graphs with positive weights, it is $O(T\times m+T\times n\log n)$. In a way similar to Algorithm 1, Algorithm 2 can be modified to estimate discriminative eccentricity of graph $G$, where the details are omitted.

In the rest of this section, we provide an error bound for our estimation of average discriminative path length. First in Proposition 5.1, we prove that in Algorithm 2 the expected value of $\beta$ is $ADPL(G)$. Then in Proposition 5.2, we provide an error bound for $\beta$.

Proposition 5.1.

In Algorithm 2, expected value of $\beta_{t}$’s ($1\leq t\leq T$) and $\beta$ is $ADPL(G)$.

Proof.

We have:

[TABLE]

where $\frac{1}{n}$ comes from the uniform distribution used to choose vertices of $G$. Then, we have: $\mathbf{E}\left[\beta\right]=\frac{\sum_{t=1}^{T}\mathbf{E}\left[\beta_{t}\right]}{T}=\frac{T\times\mathbf{E}\left[\beta_{t}\right]}{T}=ADPL(G).$ ∎

Proposition 5.2.

In Algorithm 2, let $G$ be a connected and undirected graph. For a given $\epsilon\in\mathbb{R}^{+}$, we have:

[TABLE]

Proof.

The proof is done using Hoeffding’s inequality (Hoeffding, 1963). Let $X_{1},\ldots,X_{n}$ be independent random variables bounded by the interval $[a,b]$, i.e., $a\leq X_{i}\leq b$ ($1\leq i\leq n$). Let also $\bar{X}=\frac{1}{n}\left(X_{1}+\ldots+X_{n}\right)$. Hoeffding (Hoeffding, 1963) showed that:

[TABLE]

On the one hand, for any two distinct vertices $v,u\in V(G)$, we have: $d(v,u)\leq\Delta(G)$ and $\sigma(v,u)\geq 1$. Therefore, $\frac{d(v,u)}{\sigma(v,u)}\leq\Delta(G)$ and as a result, $\beta_{t}\leq\Delta(G)$ ($1\leq t\leq T$). On the other hand, for any two distinct vertices $v,u\in V(G)$, we have: $\frac{d(v,u)}{\sigma(v,u)}>0$. Therefore, $\beta_{t}>0$, for $1\leq t\leq T$. Note that in Algorithm 2 vertices $u$ are chosen independently and therefore, variables $\beta_{t}$ are independent. Hence, we can use Hoeffding’s inequality, where $X_{i}$’s are $\beta_{t}$’s, $\bar{X}$ is $\beta$, $n$ is $T$, $a$ is [math] and $b$ is $\Delta(G)$. Putting these values into Inequality 13 yields Inequality 12. ∎

Real-world networks have a small diameter, bounded by the logarithm of the number of vertices in the network (Watts and Strogatz, 1998). This, along with Inequality 12, yields555Note that in Inequality 14, both $\beta$ and $\epsilon$ are in $\mathbb{R}^{+}$ and since $\beta$ and its expected value are not bounded by $(0,1)$ and they are considerably larger than 0 (and they can be larger than 1), $\epsilon$ is usually set to a value much larger than 0 (and even larger than 1, such as $\log n$).:

[TABLE]

Inequality 14 says that for given values $\epsilon\in\mathbb{R}^{+}$ and $\delta\in(0,1)$, if $T$ is chosen such that $T\geq\frac{\ln\left(\frac{2}{\delta}\right)\left(\log n\right)^{2}}{2{\epsilon}^{2}},$ Algorithm 2 estimates average discriminative path length of $G$ within an additive error $\epsilon$ with a probability at least $\delta$. Our extensive experiments reported in Table 1 of Section 6 (the rightmost column) show that many real-world networks have a very small discriminative diameter, much smaller than the logarithm of the number of vertices they have. So, we may assume that their discriminative diameter is bounded by a constant $c$. For such networks, using only $\frac{c^{2}\times\ln\left(\frac{2}{\delta}\right)}{2{\epsilon}^{2}}$ samples, Algorithm 2 can estimate average discriminative path length within an additive error $\epsilon$ with a probability at least $\delta$.

6. Experimental results

We perform extensive experiments on real-world networks to assess the quantitative and qualitative behavior of our proposed algorithms. The programs are compiled by the GNU C++ compiler 5.4.0 using optimization level 3. We do our tests over (largest connected components of) several real-world datasets from different domains, including the dblp0305, dblp0507 and dblp9202 co-authorship networks (Berlingerio et al., 2009), the facebook-uniform social network (Gjoka et al., 2010), the flickr network (McAuley and Leskovec, 2012), the gottron-reuters network (Lewis et al., 2004), the petster-friendships network (Kunegis, 2013), the pics_ut network (Kunegis, 2013), the web-Stanford network (Leskovec et al., 2009), the web-NotreDame network (Albert et al., 1999), the citeulike-ut network (Emamy and Cameron, 2007), the epinions network (Massa and Avesani, 2005) and the wordnet network (Fellbaum, 1998). All the networks are treated as undirected graphs. When a graph is disconnected, we consider only its largest component. Table 1 summarizes specifications of the largest components of our real-world networks.

6.1. Empirical Evaluation of Discriminability

We measure discriminability of a centrality notion in terms of its power in assigning distinguished values to the vertices. Hence, for each centrality notion and over each network $G$, we define discriminability as:

[TABLE]

Among different distance-based notions studied in this paper, we investigate discriminability of discriminative closeness centrality. The reason is that on the one hand, notions such as average discriminative path length and average discriminative eccentricity are graph characteristics, rather than vertex properties. Hence, it does not make sense to measure their discriminability. On the other hand, closeness centrality is a much more common distance-based notion to rank vertices than the other distance-based notions such eccentricity. We compare discriminative closeness centrality against closeness centrality, as well as a number of centrality notions that are not based on distance, including betweenness centrality (Chehreghani, 2014b), length-scaled betweenness centrality (Brandes, 2008) linearly scaled betweenness centrality (Brandes, 2008) and Katz centrality (Katz, 1953)666To compute betweenness centrality and its variations, we use boost graph library (http://www.boost.org/doc/libs/1_66_0/libs/graph/doc/index.html) and to compute Katz centrality, we use NetworKit (https://networkit.iti.kit.edu/), where all these algorithms are implemented in C++.. Katz centrality has three parameters to adjust: the damping factor $\alpha$, the bias constant $\beta$ and the convergence tolerance $tol$. Similar to e.g., (Zhan et al., 2017), we set $\beta$ to $1$. In order to guarantee convergence, $\alpha$ must be less than the inverse of the largest eigenvalue of the graph. Here we set it to $0.001$, which is one of the values used in the experiments of (Zhan et al., 2017). We set $tol$ to $1e-10$.

Table 2 reports the discriminability results. In the table, a ’higher percentage’ means a ’higher discriminability’ of the centrality notion. The followings can be seen in the table. First, discriminative closeness centrality is always more discriminative than the other indices. Second, over datasets such as dblp0305, dblp0507, dblp9202, facebook-uniform and web-Stanford, discriminability of discriminative closeness centrality is significantly larger than discriminability of the other indices. In fact, when over a network discriminability of the other indices is very low, discriminative closeness centrality becomes significantly more discriminative than them. However, when the other indices are discriminative enough, the difference between discriminability of the indices is less considerable. Third, Katz centrality is usually more discriminative than closeness centrality, betweenness centrality, length scaled betweenness centrality and linearly scaled betweenness centrality. The only exception is web-NotreDame, where length scaled betweenness centrality and linearly scaled betweenness centrality are more discriminative than Katz centrality. However, unlike the other indices, Katz centrality uses all paths, which makes it improper for the applications where the concept of shortest paths is essential. Fourth, while in most cases betweenness centrality and its variations are more discriminative than closeness centrality, in a few cases, e.g., over pics_ut, epinions and wordnet, closeness centrality is more discriminative than betweenness centrality and its variations. Fifth, while length scaled and linearly scaled betweenness centrality always show the same discriminability, they slightly improve discriminability of betweenness centrality. However, this improvement is not considerable.

Figure 3 compares running times of computing different indices. Since betweenness centrality, length scaled betweenness centrality and linearly scaled betweenness centrality follow exactly the same procedure and differ only in the way of aggregating the computed scores, we report only one time for all of them. As can be seen in the figure, since Katz centrality does not find shortest paths and at each vertex, simply follows all its neighbors, it is computed much faster than the other indices. Closeness centrality is computed faster than discriminative closeness and discriminative closeness is computed faster betweenness centrality and its variations. Note that the algorithms of computing closeness centrality and discriminative closeness centrality have the same time complexity. However, to compute discriminative closeness centrality, compared to closeness centrality we require to perform extra operations (e.g., counting the number of shortest paths). This makes it in practice slower than closeness centrality.

6.2. Empirical Evaluation of Randomized Algorithms

Table 3 presents the results of the empirical evaluation of our proposed randomized algorithm for estimating average discriminative path length. When estimating average discriminative path length or average discriminative eccentricity, we define relative error of the approximation algorithm as:

[TABLE]

where exact score and approximate score are respectively the values computed by the exact and approximate algorithms. Sample sizes are expressed in terms of the percentages of the number of vertices of the graph. We examine the algorithm for three sample sizes: $10\%$ of the number of vertices, $1\%$ of the number of vertices and $0.1\%$ of the number of vertices. As can be seen in the table, only a very small sample size, e.g., $0.1\%$ of the number of vertices, is sufficient to have an accurate estimation of average discriminative path length. Over all the datasets, except gottron-reuters, this sample size gives a relative error less than $3\%$. In particular, relative error in the datasets dblp0305, dblp0507, dblp9202 and facebook-uniform is very low. This is consistent with our analysis presented in Section 5.2 and is due to very small discriminative diameter of these networks.

Table 3 also compares average discriminative path length of the networks with their average path length. For all the datasets, except dblp0305, dblp0507, dblp9202 and facebook-uniform, average discriminative path length is considerably smaller than average path length. It may seem surprising that despite very high discriminability of discriminative closeness compared to closeness over dblp0305, dblp0507, dblp9202 and facebook-uniform, the differences between average discriminative path length and average path length are tiny. The reason is that over these datasets, on the one hand, between a huge number of pairs of vertices there is only one shortest path; a few pairs have two shortest paths and only a very tiny percentage of pairs have three or more shortest paths. Therefore, those pairs that have more than one shortest paths do not have a considerable contribution to $ADPL$, hence, $ADPL$ and $APL$ find very close values. However, on the other hand, for each vertex $v$ there are a different number of vertices to which $v$ is connected by two or more shortest paths. This is sufficient to distinguish its discriminative closeness from the other vertices.

Table 4 reports the results of the empirical evaluation of our randomized algorithm for estimating average discriminative eccentricity. Similar to the case of average discriminative path length, we test the algorithm for three different sample sizes and our experiments show that only a small sample size, e.g., $0.1\%$ of the number of vertices, can yield a very accurate estimation of average discriminative eccentricity. In our experiments, for the sample size $0.1\%$, relative error is always less than $5\%$. This high accuracy is due to very small discriminative diameter of the networks. Similar to the case of average eccentricity where a simple randomized algorithm significantly outperforms advanced techniques (Shun, 2015), our simple algorithms show very good efficiency and accuracy for estimating average discriminative path length and average discriminative eccentricity. Table 4 also shows that similar to $ADPL$, while the datasets dblp0305, dblp0507, dblp9202 and facebook-uniform have (almost) the same values for $AE$ and $ADE$, over the rest of the datasets $ADE$ is less than $AE$.

6.3. The Tiny-World Property

It is well-known that in real-world networks, average path length is proportional to the logarithm of the number of vertices in the graph and it is considerably smaller than the largest distance that two vertices may have in a graph (Watts and Strogatz, 1998). Our extensive experiments presented in Table 3 reveal that in real-world networks average discriminative path length is much more smaller than the largest discriminative distance777Both the largest distance and the largest discriminative distance that two vertices may have in a graph are equal to the number of vertices in the graph minus 1. that two vertices may have in a graph and it is bounded by a constant (i.e., $2$). This also implies that average discriminative path length of a network is usually considerably smaller than its average path length.

This property means that in real-world networks, not only most vertices can be reached from every other vertex by a small number of steps, but also there are many different ways to do so. We call this property the tiny-world property. A consequence of this property is that removing several vertices from a real-world network does not have a considerable effect on its average path length. Note that this property does not contradict the high discriminability of discriminative closeness; while this property implies that vertices in average have a tiny discriminative distance from each other, the high discriminability of discriminative closeness implies that the discriminative closeness scores of different vertices are less identical than their closeness scores. In Table 3, it can be seen that networks such as flickr, petster-friendships, pics_ut and citeulike-ut have an average discriminative path length considerably smaller than the others. This is due to the high density of these networks, which yields that any two vertices may have a shorter distance or more shortest paths.

7. Link prediction

In order to better motivate the applicability and usefulness of our proposed distance measure, in this section we present a novel link prediction method, which is based on our new distance measure. We empirically evaluate our method and show that it outperforms the well-known existing link prediction methods.

In the link prediction problem studied in this paper, we are given an unweighted and undirected graph $G$ in which each edge $e=\{u,v\}$ has a timestamp. For a time $t$, let $G[t]$ denote the subgraph of $G$ consisting of all edges with a timestamp less than or equal to $t$. Then the link prediction task is defined as follows. Given network $G[t]$ and a time $t^{\prime}>t$, (partially) sort the list of all pairs of vertices that are not connected in $G[t]$, according to their probability (likelihood) of being connected during the interval $(t,t^{\prime}]$. We refer to the intervals $[0,t]$ and $(t,t^{\prime}]$ as the training interval and the test interval, respectively.

To generate this (decreasingly) sorted list, existing methods during the training interval compute a similarity matrix $S$ whose entry $S_{uv}$ is the score (probability/likelihood) of having an edge between vertices $u$ and $v$. Generally, $S$ is symmetric, i.e., $S_{uv}=S_{vu}$. The pairs of the vertices that are at the top of the ordered list are most likely to be connected during the test interval (Martínez et al., 2016). To compute $S_{u,v}$, several methods have been proposed in the literature, including the number of common neighbors (Newman, 2001), negative of shortest path length (Liben-Nowell and Kleinberg, 2007) and its variations (Lebedev et al., 2017), the Jaccard’s coefficient (Salton and McGill, 1986), the preferential attachment index (Barabasi et al., 2002), hitting time (Liben-Nowell and Kleinberg, 2007), SimRank (Jeh and Widom, 2002), Katz index (Katz, 1953), the Adamic/Adar index (Adamic and Adar, 2001) and resource allocation based on common neighbor interactions (Zhang et al., 2014). In the literature, there are also many algorithms that exploit a classification algorithm, with these indices as the features, and try to predict whether a pair of unconnected vertices will be connected during the test interval or not (Lu and Zhou, 2011; Hasan et al., 2006; Martínez et al., 2016).

In this section, we propose a new method, called LIDIN888LIDIN is an abbreviation for LInk prediction based on DIstance and the Nmber of shortest paths., for sorting the list of pairs of unconnected vertices, which is a combination of shortest path length and discriminative distance. For two pairs of unconnected vertices $\{u_{1},v_{1}\}$ and $\{u_{2},v_{2}\}$, using LIDIN we say vertices $u_{2}$ and $v_{2}$ are more likely to form a link during the test interval than vertices $u_{1}$ and $v_{1}$ if:

• •

$d(u_{1},v_{1})>d(u_{2},v_{2})$, or

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$d(u_{1},v_{1})=d(u_{2},v_{2})$ and $dd(u_{1},v_{1})>dd(u_{2},v_{2})$.

The rationale behind LIDIN is that when comparing a pair of vertices $u_{1},v_{1}$ with another pair $u_{2},v_{2}$, if $d(u_{1},v_{1})=d(u_{2},v_{2})$ but $u_{2}$ and $v_{2}$ are connected to each other by more shortest paths than $u_{1}$ and $v_{1}$, then they are more likely to form a link during the test interval. As a special case, for a fixed $k$, consider the list $L(k)$ consisting of all pairs of unconnected vertices $u$ and $v$ such that $d(u,v)=k$. A network may have many such pairs. It is known that compared to the pairs of unconnected vertices that have distance $k+1$, members of $L(k)$ are more likely to form a link during the test interval (Pasta et al., 2014; Chehreghani and Chehreghani, 2016). However, the question remaining open is what elements of $L(k)$ are more likely to be connected than the other members? Using LIDIN, we argue that by increasing the number of shortest paths between the two vertices, the probability of forming a link increases, too.

In order to empirically evaluate this argument, we perform tests over several temporal real-world networks, including sx-stackoverflow (Paranjape et al., 2017), sx-mathoverflow (Paranjape et al., 2017), sx-superuser (Paranjape et al., 2017), sx-askubuntu (Paranjape et al., 2017), wiki-talk-temporal (Paranjape et al., 2017; Leskovec et al., 2010) and CollegeMsg (Panzarasa et al., 2009). Table 5 summarizes the specifications of the used temporal real-world datasets. We consider all these networks as simple and undirected graphs, where multi-edges and self-loops are ignored. Since the networks sx-stackoverflow, sx-superuser, sx-askubuntu and wiki-talk-temporal are too large to load their unconnected pairs of vertices in the memory, after sorting their edges based on timestamp, we only consider the subgraphs generated by their first 300,000 edges.

We compare LIDIN with negative of shortest path length (Liben-Nowell and Kleinberg, 2007) and the Adamic/Adar index (Adamic and Adar, 2001), denoted respectively by -SPL and AA. We choose -SPL because LIDIN is inherently an improvement of -SPL, furthermore, the experiments reported in (Hasan et al., 2006) show that this index outperforms the other topological (global) indices studied in that paper. We choose AA because the experiments reported in (Liben-Nowell and Kleinberg, 2007) show that among $11$ studied indices, the Adamic/Adar index has the best relative performance ratio versus random predictions, the best relative performance ratio versus negative of shortest path length predictor and the best relative performance ratio versus common neighbors predictor999We can use these indices either as the features of a classification algorithm (like e.g., (Hasan et al., 2006)), or as the criteria of sorting the list of unconnected pairs of vertices (like e.g., (Liben-Nowell and Kleinberg, 2007)). Here, since we want to omit the effect of the classification algorithm and study only the effect of our new notion, we follow the second option..

For the graph formed during training interval, we sort (increasingly when LIDIN is used and decreasingly when -SPL and AA are used) the list $L$ of all pairs of unconnected vertices, based on each of the indices101010 When any of these indices is used, there might exist two or more pairs that are not sorted by the index. In this case, these pairs are sorted according to the identifiers of the end-points of the edges.. Then, during the test interval, for each edge that connects a pair in $L$, we examine its rank in $L$. In order to evaluate the accuracy of a link prediction method, we use two measures area under the ROC curve ($AUC$) and ranking error ($Q$). In $AUC$, we measure the probability that a randomly chosen pair of vertices that find a link during the test interval have a higher score than a randomly chosen pair of vertices that do not find a link during the test interval. Formally, $AUC$ of a method $ind$ is defined as (Zhang et al., 2016):

[TABLE]

where $n_{t}$ is the number of times that we randomly choose two pairs of vertices; one from those that form a link during the test interval and the other from those that do not; $n_{g}$ is the number of times that the one that forms a link gets a higher score than the other, and $n_{e}$ is the number of times that the scores of the two chosen pairs are equal. A higher value of $AUC$ implies a better link prediction method. In our experiments, we set $n_{t}$ to the number of edges in the test interval divided by $10$. However, similar results can be seen for other values of $n_{t}$.

We define the ranking error of a method $ind$ as:

[TABLE]

where $L_{ind}$ is the list $L$ sorted according to $ind$, $TE$ contains those edges of the test interval that connect a pair in $L$, and $rank(\{u,v\},L_{ind})$ returns the rank of $\{u,v\}$ in $L_{ind}$. For two given indices $ind1$ and $ind2$, $Q(ind1)<Q(ind2)$ means that compared to $ind2$, $ind1$ gives more priority (i.e., a better rank) to the pairs that form a link during the test interval, hence, $ind1$ is a better method than $ind2$.

We sort the edges of each network according to their timestamps and form the training and test intervals based on the timestamps, i.e., for some given value $\tau$, training interval contains those edges that have a timestamp at most $\tau$ and test interval contains those edges that have a timestamp larger than $\tau$. A factor that may affect the empirical behavior of the indices is the value of $\tau$. Therefore and to examine this, we consider 4 different settings for each network, and choose the values of $\tau$ in such a way that training interval includes $60\%$, $70\%$, $80\%$ and $90\%$ of the edges. In each case, the rest of the edges which are between a pair of vertices unconnected in the training interval, belong to the test interval.

Figure 4 compares $AUC$ of different methods. Over all the datasets and in all the settings, LIDIN has the highest $AUC$, therefore, it has the best performance. Figure 5 reports the $Q$ of the studied methods over different datasets. As can be seen in the figure, in all the cases, LIDIN has the lowest $Q$ and hence, the best performance. These tests empirically verify our above mentioned argument that among all the pairs of unconnected vertices in $L(k)$, those that have a smaller discriminative distance (and hence, are closer!), are more likely to form a link. While in most cases -SPL outperforms AA, over sx-mathoverflow and for all values of ratio, AA has a higher $AUC$ and a lower $Q$ than -SPL. The superior performance of our proposed link prediction method suggests that the inverse of discriminative distance might be useful in determining similarity between vertices of a network. This means, for example, more than using the fixed criteria for determining the similarity of objects in a Social Internet of Vehicle (SIoV) (Ning et al., 2017a) or friendship of User Equipments (Ning et al., 2017b), someone may also use the inverse of discriminative distance.

8. Conclusion and future work

In this paper, we proposed a new distance measure between vertices of a graph, which is proportional to the length of shortest paths and inversely proportional to the number of shortest paths. We presented exact and randomized algorithms for computation of the proposed discriminative indices and analyzed them. Then, by performing extensive experiments over several real-world networks, we first showed that compared to the traditional indices, discriminative indices have usually much more discriminability. We then showed that our randomized algorithms can very precisely estimate average discriminative path length and average discriminative eccentricity, using only a few samples. In the end, we presented a novel link prediction method, that uses discriminative distance to decide which vertices are more likely to form a link in future, and showed its superior performance compared to the well-known existing measures.

The current work can be extended in several directions. An interesting direction is to investigate distribution of discriminative closeness and discriminative vertex eccentricity in large networks. In particular, it is useful to see whether there exist correlations among discriminative indices on the one hand and other centrality indices such as betweenness and degree on the other hand. The other direction for future work is to develop efficient randomized algorithms for estimating discriminative closeness and discriminative eccentricity of one vertex or a set of vertices and discriminative diameter of the graph. For example, it is interesting to develop algorithms similar to (Bergamini et al., 2016) that estimate $k$ highest discriminative closeness scores in the graph. The other extension of the current work is the empirical evaluation of the generalizations of the discriminative indices presented in Section 4. Finally, another extension of the current work is to study discriminative indices of different network models (Chehreghani and Abdessalem, 2017).

Acknowledgement

This work has been supported by the ANR project IDOLE.