The emergence of informative higher scales in complex networks

TL;DR

This paper introduces a measure called effective information to analyze the informational content of network paths, demonstrating how higher-scale structures can emerge and increase certainty in various real-world and simulated networks.

Contribution

It derives a new measure for network information content and shows how macro-scale groupings can enhance effective information, revealing the emergence of informative higher scales.

Findings

Higher effective information indicates more meaningful node relationships.

Subgraph grouping into macro-nodes increases effective information.

Informative higher scales are prevalent across diverse network types.

Abstract

The connectivity of a network contains information about the relationships between nodes, which can denote interactions, associations, or dependencies. We show that this information can be analyzed by measuring the uncertainty (and certainty) contained in paths along nodes and links in a network. Specifically, we derive from first principles a measure known as effective information and describe its behavior in common network models. Networks with higher effective information contain more information in the relationships between nodes. We show how subgraphs of nodes can be grouped into macro-nodes, reducing the size of a network while increasing its effective information (a phenomenon known as causal emergence). We find that informative higher scales are common in simulated and real networks across biological, social, informational, and technological domains. These results show that the emergence of higher scales in networks can be directly assessed and that these higher scales offer a way to create certainty out of uncertainty.

Figures

Tables

Table 1. Table 1: Table of key terms . Quantities needed in order to calculate E ​ I 𝐸 𝐼 EI and create consistent macro-nodes.

Term	Description	Notation
Network size	the number of nodes in the network	$N$
Out-weight vector ($v_i$)	a vector of probabilities $w_{ij}$ that a random walker on node $v_i$ will transition to $v_j$	$W_i^{out}=\{w_{i1},w_{i2},,\dots w_{ij},\dots w_{iN}\}$
Effective information (network)	the total information in a causal structure, in bits	$EI=H\left(⟨W_i^{out}⟩\right)-⟨H\left(W_i^{out}\right)⟩$
Determinism ($v_i$)	how certain about next steps is a random walker on $v_i$	${\text{det}}_i={\log }_2\left(N\right)-H\left(W_i^{out}\right)$
Degeneracy (network)	how distributed the certainty is over the nodes of the network	$\text{degeneracy}={\log }_2\left(N\right)-H\left(⟨W_i^{out}⟩\right)$
Effect information ($v_i$)	the contribution of each node $v_i$ to the network’s $EI$	$E{I}_i={\text{D}}_{{}_{KL}}\left[W_i^{out}\right|\left|⟨W_i^{out}⟩\right]$
Micro-nodes in a macro-node	the set of micro-nodes grouped into a macro-node in the new network, $G_M$	$S=\{v_i,v_j,\dots \}$, of length $N_S$
Macro-node out-weights	the out-weights from macro-node, $\mu$, to its neighbors	$W_{\mu }^{out}=\sum _{i\in S}{W}_i^{out}\cdot \left(\frac{1}{N_S}\right)$
Macro-node out-weights given input weights	the out-weights from macro-node, $\mu$, to its neighbors, conditioned on in-weights to the micro-nodes, $v_i\in S$	$W_{\mu |j}^{out}=\sum _{i\in S}{W}_i^{out}\cdot \left(\frac{{\sum }_{j->i}{w}_{ji}}{{\sum }_{j->k\in S}{w}_{jk}}\right)$
Macro-node out-weights given the stationary distribution	the out-weights from macro-node, $\mu$, to its neighbors, conditioned on the stationary probabilities, ${\pi }_i$, of micro-nodes, $v_i\in S$	$W_{\mu |\pi }^{out}=\sum _{i\in S}{W}_i^{out}\cdot \left(\frac{{\pi }_i}{{\sum }_{k\in S}{\pi }_k}\right)$

Table 2. Table 2: Network datasets. Continued on the following page.

Network name	Domain	Source	Cat.	Description
HEP-th citations	Citation	Konect	Inf.	high-energy physics (HEP) citations - theory
HEP-ph citations	Citation	Konect	Inf.	HEP citations - phenomenology
Cora citations	Citation	Konect	Inf.	citations from the Cora database
DBLP citations	Citation	Konect	Inf.	database of scientific publications
Astro-ph coauthorships	Coauthorship	Konect	Inf.	coauthors on astronomy arXiv papers
HEP-th coauthorships	Coauthorship	Konect	Inf.	coauthors on HEP-theory arXiv papers
HEP-ph coauthorships	Coauthorship	Konect	Inf.	coauthors on HEP-phenomenology arXiv papers
Tarragona univ. emails	Communication	Konect	Soc.	emails from the University Rovira i Virgili
Dem. Nat. Comm. emails	Communication	Konect	Soc.	2016 Democratic National Committee email leak
Digg user-user replies	Communication	Konect	Soc.	reply network from the social news website Digg
UC Irvine messages	Communication	Konect	Soc.	messages between students at UC Irvine
Manufacturing emails	Communication	Konect	Soc.	internal emails between employees at a company
CAIDA autonomous systems	Computer	Konect	Tec.	autonomous systems network from CAIDA, 2007
Route views autonomous systems	Computer	Konect	Tec.	autonomous systems network
Internet autonomous systems	Computer	Konect	Tec.	connected IP routing
Haggle RFID contact	Human Contact	Konect	Soc.	human proximity, via carried wireless devices
Reality mining RFID	Human Contact	Konect	Soc.	RFID data from 100 MIT students’ interactions
California windsurfers	Human Contact	Konect	Soc.	contacts between windsurfers California, 1986
Train terrorists	Human Contact	Konect	Soc.	contacts between Madrid train bombing suspects
Hypertext conference	Human Contact	Konect	Soc.	face-to-face contacts at the ACM Hypertext 2009
Infectious conference	Human Contact	Konect	Soc.	face-to-face contacts at INFECTIOUS, 2009
Jazz musicians	Human Social	Konect	Soc.	collaboration network between Jazz musicians
Adolescent health	Human Social	Konect	Soc.	surveyed students list their best friends
Physicians	Human Social	Konect	Soc.	innovation spread network among 246 physicians
Resident hall	Human Social	Konect	Soc.	friendship ratings between students in a dorm
Sampson cloister	Human Social	Konect	Soc.	relations between monks in a monastery
Seventh graders	Human Social	Konect	Soc.	proximity ratings between seventh grade students
Taro gift-giving	Human Social	Konect	Soc.	gift-givings (taro) between households
Dutch college	Human Social	Konect	Soc.	friendship ratings between university freshmen
Highland tribes	Human Social	Konect	Soc.	tribes in the Gahuku-Gama alliance structure
Illinois school	Human Social	Konect	Soc.	friendships between boys at an Illinois highschool
Free online dict.	Hyperlink	Konect	Inf.	cross references in Free Online Dict. of Computing
Political blogs	Hyperlink	Konect	Inf.	hyperlinks between blogs, 2004 US election
Google internal	Hyperlink	Konect	Inf.	hyperlink network from pages within Google.com
Air traffic control	Infrastructure	Konect	Tec.	USA’s FAA, Preferred Routes Database
OpenFlights v1	Infrastructure	Konect	Tec.	flight network between airports, OpenFlights.org
OpenFlights v2	Infrastructure	Konect	Tec.	flight network between airports, OpenFlights.org
Contiguous U.S.	Infrastructure	Konect	Tec.	48 contiguous states and D.C. of the U.S.
European roads	Infrastructure	Konect	Tec.	international E-road network, mainly in Europe
Chicago roads	Infrastructure	Konect	Tec.	road transportation network of the Chicago region
West U.S. powergrid	Infrastructure	Konect	Tec.	power grid of the Western U.S.
U.S. Airports	Infrastructure	Konect	Tec.	flights between US airports in 2010
David Copperfield	Lexical	Konect	Inf.	network of common noun and adjective adjacencies
Edinburgh thesaurus	Lexical	Konect	Inf.	word association network, collected experimentally
King James Bible	Lexical	Konect	Inf.	co-occurrence between nouns in the Bible

Table 3. Table 2: Network datasets (continued).

Network name	Domain	Source	Cat.	Description
C. elegans metabolic	Metabolic	Konect	Bio.	metabolic network of the C. elegans roundworm
Human protein (Figeys)	Metabolic	Konect	Bio.	interactions network of proteins in Humans
PDZbase protein	Metabolic	Konect	Bio.	protein-protein interactions from PDZBase
Human protein (Stelzl)	Metabolic	Konect	Bio.	interactions network of proteins in Humans
Human protein (Vidal)	Metabolic	Konect	Bio.	proteome-scale map of Human protein interactions
Yeast protein	Metabolic	Konect	Bio.	protein interactions contained in yeast
Reactome humans	Metabolic	Konect	Bio.	protein interactions, from the Reactome project
Avogato	Social	Konect	Soc.	trust network for users of Advogato
Google+	Social	Konect	Soc.	Google+ user-user connections
Hamsterster	Social	Konect	Soc.	friendships between users of hamsterster.com
Twitter lists	Social	Konect	Soc.	Twitter user-user following network
Facebook NIPS	Social	Konect	Soc.	Facebook user-user friendship network
Linux dependency	Software	Konect	Inf.	Linux source code dependency network
J.D.K. dependency	Software	Konect	Inf.	software class dependencies, JDK 1.6.0.7
JUNG/javax dependency	Software	Konect	Inf.	software class dependencies, JUNG 2.0.1 & javax
Florida ecosystem - dry	Trophic	Konect	Bio.	food web in the Florida wetlands (dry season)
Florida ecosystem - wet	Trophic	Konect	Bio.	food web in the Florida wetlands (wet season)
Little Rock Lake ecosystem	Trophic	Konect	Bio.	food web of Little Rock Lake, Wisconsin
WHOIS protocol	Technological	NetworkRepository	Tec.	dataset of internet routing registries
PGP protocol	Technological	NetworkRepository	Tec.	trust protocol of private keys of internet users
Routers RF	Technological	NetworkRepository	Tec.	traceroute network between routers via Rocketfuel
Cat brain 1	Brain	NetworkRepository	Bio.	fiber tracts between brain regions of a cat
Drosophila medulla	Brain	NetworkRepository	Bio.	neuronal network from the medulla of a fly
Rhesus brain 1	Brain	NetworkRepository	Bio.	collation of tract tracing studies in primates
Rhesus brain 2	Brain	NetworkRepository	Bio.	inter-areal cortical networks from a primate
Macaque cerebral	Brain	NetworkRepository	Bio.	connections between cerebral cortex of a primate
Macaque interareal	Brain	NetworkRepository	Bio.	inter-areal cortical networks from a primate
Mouse Kasthuri	Brain	NetworkRepository	Bio.	neuronal network of a mouse
Mouse brain 1	Brain	NetworkRepository	Bio.	calcium imaging of neuronal networks in a mouse
Mouse retina 1	Brain	NetworkRepository	Bio.	electron microscopy of neurons in mouse retina
Mouse visual 1	Brain	NetworkRepository	Bio.	electron microscopy of visual cortex of a mouse
Mouse visual 2	Brain	NetworkRepository	Bio.	electron microscopy of visual cortex of a mouse
Power 1138BUS	Powergrid	NetworkRepository	Tec.	power system admittance, via Harwell-Boeing
Power 494BUS	Powergrid	NetworkRepository	Tec.	power system admittance, via Harwell-Boeing
Power 662BUS	Powergrid	NetworkRepository	Tec.	power system admittance, via Harwell-Boeing
Power 685BUS	Powergrid	NetworkRepository	Tec.	power system admittance, via Harwell-Boeing
U.S. power grid	Powergrid	NetworkRepository	Tec.	electricity / power transmission network in the U.S.
Power pcspwr 09	Powergrid	NetworkRepository	Tec.	BCSPWR09 powergrid data via Harwell-Boeing
Power pcspwr 10	Powergrid	NetworkRepository	Tec.	BCSPWR10 powergrid data via Harwell-Boeing
Power ERIS1176	Powergrid	NetworkRepository	Tec.	powergrid data via Erisman, 1973

Equations

$$EI=H(⟨W_i^{out}⟩)-⟨H(W_i^{out})⟩$$

$$={\log }_2(N)-⟨H(W_i^{out})⟩$$

$$={\log }_2(N)-H(⟨W_i^{out}⟩)$$

$$EI=\text{determinism}-\text{degeneracy}$$

$$\text{effectiveness}=\frac{EI}{{\log }_2(N)}$$

$$\text{inconsistency}=\sum _{t=0}^T{\text{D}}_{{}_{KL}}[P_M(t)||P_{M|m}(t)]$$

$$EI=\frac{1}{N}\sum _{i=1}^N{\text{D}}_{{}_{KL}}[W_i^{out}||⟨W_i^{out}⟩]$$

$$EI$$

$$\displaystyle=\dfrac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{N}w_{ij}\log_{2}\bigg{(}\dfrac{w_{ij}}{W_{j}}\bigg{)}$$

$$\displaystyle=\dfrac{1}{N}\sum_{i=1}^{N}\bigg{(}\sum_{j=1}^{N}w_{ij}\log_{2}(w_{ij})-\sum_{j=1}^{N}w_{ij}\log_{2}(W_{j})\bigg{)}$$

$$\displaystyle=\dfrac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{N}w_{ij}\log_{2}\big{(}w_{ij}\big{)}-\dfrac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{N}w_{ij}\log_{2}\big{(}W_{j}\big{)}$$

$$EI=\dfrac{1}{N}\sum_{i=1}^{N}-H(W_{i}^{out})-\sum_{j=1}^{N}W_{j}\log_{2}\big{(}W_{j}\big{)}$$

$$EI=H(⟨W_i^{out}⟩)-⟨H(W_i^{out})⟩$$

$$E{I}_{ER}$$

$$E{I}_i$$

$$E{I}_{ER}$$

$$E{I}_d$$

$$E{I}_i$$

$$E{I}_d$$

$$\displaystyle EI_{star}=\dfrac{1}{N}\cdot\Bigg{[}\sum_{i=1}^{N-1}$$

$$\displaystyle\text{D}_{{}_{KL}}\Big{[}W^{out}_{hub}||\langle W^{out}_{i}\rangle\Big{]}\Bigg{]}$$

$$\displaystyle EI_{star}=\dfrac{1}{N}\cdot\Bigg{[}\sum_{i=1}^{N-1}$$

$$\displaystyle\text{D}_{{}_{KL}}\Big{[}\dfrac{1}{N-1}||\dfrac{1}{N(N-1)}\Big{]}\Bigg{]}$$

$$E{I}_{star}$$

$$\displaystyle\hskip 78.24507pt\log_{2}\Bigg{(}\dfrac{\dfrac{1}{N-1}}{\dfrac{1}{N(N-1)}}\Bigg{)}\Bigg{]}$$

$$E{I}_{star}$$

Full text

I Introduction

Networks provide a powerful syntax for representing a wide range of systems, from the trivially simple to the highly complex Barabási (2016); Newman (2010); Amaral  and Ottino (2004). It is common to characterize networks based on structural properties like their degree distribution or clustering, and the study of such properties has been crucial for the growth of Network Science. Yet there remains a gap in our treatment of the information contained in the relationships between nodes in a network, particularly in networks that have both weighted connections and feedback, which are hallmarks of complex systems Koseska  and Bastiaens (2017); Rodrigues et al. (2016). As we will show, analyzing this information allows for modeling the network at the most appropriate, informative scale. This is especially critical for networks that describe interactions or dependencies between nodes such as contact networks in epidemiology Perra et al. (2012), neuronal and functional networks in the brain Bassett  and Sporns (2017), or interaction networks among cells, genes, or drugs Barabási et al. (2011), as these networks can often be analyzed at multiple different scales.

Here we introduce information-theoretic measures that capture the information contained in the connectivity of a network, which can be used to identify when these networks possess informative higher scales. To do so, we focus on the out-weight vector, $W^{out}_{i}$, of each node, $v_{i}$, in a network. This vector consists of weights $w_{ij}$ between $v_{i}$ and its neighbors, $v_{j}$, and $w_{ij}=0$ if there is no edge from $v_{i}$ to $v_{j}$. For each $W^{out}_{i}$ we assume $\sum_{j}w_{ij}=1$, which means $w_{ij}$ can be interpreted as the probability $p_{ij}$ that a random walker on $v_{i}$ will transition to $v_{j}$ in the next time step, where a random walker might represent the passing of a signal, an interaction, or a state-transition Masuda et al. (2017). The information contained in a network’s connectivity can be characterized by the uncertainty among its nodes’ out-weights and in-weights. The total information in the relationships between nodes is a function of this uncertainty and can be derived from two properties.

The first is the uncertainty of a node’s outputs, which is the Shannon entropy Shannon (1948) of its out-weights, $H(W^{out}_{i})$. The average of this entropy, $\langle H(W^{out}_{i})\rangle$, across all nodes is the amount of noise present in the network’s relationships. Only if $\langle H(W^{out}_{i})\rangle=0$ is the network is deterministic.

The second property is how weight is distributed across the whole network, $\langle W^{out}_{i}\rangle$. This vector is composed of elements that are the sum of the in-weights $w_{ji}$ to each node $v_{i}$ from each of its incoming neighbors, $v_{j}$ (then normalized by total weight of the network). Its entropy, $H(\langle W^{out}_{i}\rangle)$, reflects how certainty is distributed across the network. If all nodes link only to the same node, then $H(\langle W^{out}_{i}\rangle)=0$, and the network is totally degenerate since all nodes lead to the same node.

The effective information ($EI$) of a network is the difference between these two quantities:

[TABLE]

The entropy of the distribution of out-weights in the network forms an upper bound of the amount of unique information in the network’s relationships, from which the information lost due to the uncertainty of those relationships is subtracted. Networks with high $EI$ contain more certainty in the relationships between nodes in the network (since the links represent less uncertain dependencies, unique associations, or deterministic transitions), whereas networks with low $EI$ contain less certainty. Note that $EI$ can be interpreted simply as a structural property of random walkers on a network and their behavior, similar to other common network measures Masuda et al. (2017).

Here, we use this measure to develop a general classification of networks (key terms can be found in Supplementary Materials, SM V.1). Furthermore, we show how the connectivity and different growth rules of a network have a deep relationship to that network’s $EI$. This also provides a principled means of quantifying the amount of information among the micro-, meso-, and macroscale dependencies in a network. We introduce a formalism for finding and assessing the most informative scale of a network: the scale that minimizes the uncertainty in the relationships between nodes. For some networks, a macroscale description of the network can be more informative in this manner, demonstrating a phenomenon known as causal emergence Hoel et al. (2013); Hoel (2018), which here we generalize to complex networks. This provides a rigorous means of identifying when networks possess an informative higher scale.

II Results

II.1 Effective information quantifies a network’s dependencies

This work expands to networks previous research on using effective information to measure the amount of information in the causal relationships between the mechanisms or states of a system. Originally, $EI$ was introduced to capture the causal influence between two subsets of neurons as a step in the calculation of integrated information in the brain Tononi  and Sporns (2003). Later, a system-wide version of $EI$ was shown to capture fundamental causal properties in Boolean networks of logic gates, particularly their determinism and degeneracy Hoel et al. (2013).

Our current derivation from first principles of an $EI$ for networks is equivalent to this system-wide definition (SM V.2), which was based originally on interventions upon system states. For example, if a system in a particular state, A, always transitions to state B, the causal relationship between A and B can be represented by a node-link diagram wherein the two nodes—A and B—are connected by a directed arrow, indicating that B depends on A. This might be a node pair in a “causal diagram” (often represented as a directed acyclic graph, or a DAG) such as those used in Pearl (2000, 1995) to represent interventions and causal relationships. In such a case, the information in the causal relationship between A and B can be assessed by intervening to randomize A ($do(A=H^{max})$) and measuring the effects on B. The $EI$ would be the mutual information between A and B under such randomization: $I(do(A=H^{max}),B)$ Hoel (2017).

To expand this framework to networks in general, we relax this intervention requirement by assuming that the elements in $W^{out}_{i}$ sum to $1$. In this case, an “intervention” can be interpreted as dropping a random walker on the network. For example, if the network represents a DAG or Markov chain, then dropping a random walker on a node $v_{i}$ would be equivalent to $do(v_{i})$. The entropy of the transitions of the random walkers and how those transitions are distributed defines the $EI$ of a network. In this generalized formulation, only in networks where the nodes and edges actually represent dynamics, interactions, or couplings does $EI$ indicate information about causation. In the case where edges represent correlations, or where what nodes or edges represent is undefined, $EI$ is merely a structural property of the information contained in the behavior of hypothetical random walkers (however, this situation is no different from other analysis methods that rely on random walkers).

Here we describe how this generalized structural $EI$ property behaves in common network models, asking basic questions about the relationship between a network’s $EI$ and its size, density, and structure. These inquiries allow for the exhaustive classification and quantification of the information contained in the connectivity of real networks. It is intuitive that the $EI$ of a network will increase as the network grows in size. In general, adding more nodes should increase the entropy, which should in turn increase the amount of information. However, in cases of randomness rather than structure, $EI$ should reflect this randomness. We found this is indeed the case.

Fig. 1a shows the relationship between a network’s $EI$ and its size under several parameterizations of Erdős-Rényi (ER) random graphs Erdős  and Rényi (1959); Bollobás (1984). As the size of an ER network increases (while keeping constant the probability that any two nodes will be connected, $p$), its $EI$ converges to a value of $-\log_{2}(p)$. That is, in random networks $EI$ is dominated solely by the probability that any two nodes are connected, a key finding which demonstrates that, after a certain point, a random network structure does not contain more information as its size increases. This shift occurs in ER networks at approximately $\langle k\rangle=\log_{2}(N)$, which is also the point at which we can expect all nodes to be in a giant component Barabási (2016). This finding illustrates that network connectivity must be non-random to increase the amount of information in the relationships between nodes (see SM V.3.1 for derivation). Note that if a network is maximally dense (i.e. a fully connected network, with self-loops), $EI=0.0$. However, we expect such dense low-$EI$ structures to be uncommon, since network structures found in nature and society tend to be sparse Del Genio et al. (2011).

We report another key relationship between a network’s connectivity and its $EI$ in Fig. 1b. We again compare the $EI$ of a network to its size, focusing on networks grown under different parameterizations of a preferential attachment model Krapivsky et al. (2000); Barabási et al. (1999). Under a preferential attachment growth model, a new node is added to the network at each time step, contributing $m$ new edges to the network; these $m$ edges connect to nodes already in the network, $v_{j}$, with a probability proportional to $k_{j}^{\alpha}$. Here $k_{j}$ is the degree of node $v_{j}$ and $\alpha$ tunes the amount of preferential attachment. A value of $\alpha=0.0$ corresponds to each node having an equal chance of receiving a new node’s link (i.e., no preferential attachment). The classic Barabási-Albert network corresponds to linear preferential attachment, $\alpha=1.0$ Barabási et al. (1999). Superlinear preferential attachment, $\alpha>1.0$, creates networks that have less and less $EI$, eventually resembling star-like structures (see SM V.3.2 for derivation). As shown in Fig. 1b, only in cases of sublinear preferential attachment, $\alpha<1.0$, does the network’s $EI$ continue to increase with its size. When $\alpha=0.0$—creating a random tree—the network’s $EI$ increases logarithmically as its size increases.

The maximum possible $EI$ in a network of $N$ nodes is $\log_{2}(N)$. This can be seen in the case of a directed ring network where each node has one incoming link and one outgoing link, each with a weight of $1.0$, so each node has one node uniquely connecting to it. In such a network, each node contributes zero uncertainty, since ${\langle H(W^{out}_{i})\rangle}=0.0$, and ${H}(\langle W^{out}_{i}\rangle)=\log_{2}(N)$, and therefore its $EI$ is always $\log_{2}(N)$. In general, the $EI$ of undirected lattices is fixed entirely by its size and the dimension of the ring lattice (i.e. $d=1$ is an undirected ring, $d=2$ is a taurus, etc. Watts  and Strogatz (1998)), so for such lattices $EI=\log_{2}(N)-\log_{2}(2d)$ (see SM V.3.2 for derivation).

The picture that emerges is that $EI$ is inextricably linked a network’s connectivity and growth (even network motifs, as shown in SM V.4) and therefore to the fundamentals of Network Science. Random networks have a fixed amount of $EI$, and scale-freeness ($\alpha=1.0$) represents the critical bound for the growth of $EI$. In general, dense networks and star-like networks have less $EI$. The next section explores how $EI$’s components explain these associations.

II.2 Determinism and degeneracy

Determinism and degeneracy are the two fundamental components of $EI$ Hoel et al. (2013). They are based on a network’s connectivity (see Figure 2a for a visual explanation), specifically the degree of overlapping weight in the networks. Determinism and degeneracy are derived from the uncertainty over outputs and uncertainty in how those outputs are distributed, respectively:

[TABLE]

In a maximally deterministic network wherein all nodes have a single output, $w_{ij}=1.0$, the determinism is $\log_{2}(N)$ because $\langle H(W^{out}_{i})\rangle=0.0$. Conceptually, this means that a random walker will move deterministically starting from any node. Degeneracy is the amount of information in the connectivity lost via an overlap in input weights (e.g., if multiple nodes output to the same node). In a perfectly non-degenerate system where all nodes have equal input weights, the degeneracy is zero since ${H}(\langle W^{out}_{i}\rangle)=\log_{2}(N)$. Together, determinism and degeneracy can be used to define $EI$:

[TABLE]

These two quantities provide clear explanations for why different networks have the $EI$ they do. For example, as the size of an Erdős-Rényi random network increases, its degeneracy approaches zero, which means the $EI$ of a random network is driven only by the determinism of the network, which is in turn the negative log of the probability of connection, p. Similarly, in d-dimensional ring lattice networks, the degeneracy term is always zero, which means the $EI$ of a ring lattice structure also reduces to the determinism of that structure. Ring networks with an average degree $\langle k\rangle$ will have a higher $EI$ than ER networks with the same average degree because ring networks will have a higher determinism value. In the case of star networks, the degeneracy term alone governs the decay of the $EI$ such that hub-and-spoke-like structures quickly become uninformative in terms of cause and effect (see SM V.3 for derivations concerning these cases). In general, this means that canonical networks can be characterized by their ratio of determinism to degeneracy (see Fig. 2b).

II.3 Effective information in real networks

So far, we have been agnostic as to the origin of the network under analysis. As described previously, to measure the $EI$ of a network one can create each $W^{out}_{i}$ by normalizing each node’s out-weight vector to sum to $1.0$. Regardless of what the relationships between the nodes represent, the network’s determinism reflects how targeted the out-weights of the nodes are (networks with more targeted links possess higher $EI$), while the degeneracy captures the overlap of the targeting of nodes. High $EI$ reflects the greater specificity in the connectivity, whereas low $EI$ indicates a lack of specificity (as in Fig. 2a). This generalizes our results to multiple types of representations, although the origin of the normalized network should be kept in mind when interpreting the value of the measure.

Since the $EI$ of a network will change depending on the network’s size, we use a normalized form of $EI$ known as effectiveness in order to compare the $EI$ of real networks. Effectiveness ranges from $0.0$ to $1.0$ and is defined as:

[TABLE]

As the determinism and degeneracy of a network increase to their minimum and maximum possible values, respectively, the effectiveness of that network will trend to $0.0$. Regardless of its size, a network wherein each node has a deterministic output to a unique target has an effectiveness of $1.0$.

In Fig. 3, we examine the effectiveness of 84 different networks corresponding to data from real systems. These networks were selected primarily from the Konect Network Database Kunegis (2013), which was used because its networks are publicly available, range in size from dozens to tens of thousands of nodes, often have a reasonable interpretation as being based on interactions between nodes, and they are diverse, ranging from social networks, to power networks, to metabolic networks. We defined four categories of interest: biological, social, informational, and technological. We selected our networks by using all the available networks (under 40,000 nodes due to computational constraints) in the domains corresponding to each category within the Konect database, and where it was appropriate, the Network Repository as well Rossi  and Ahmed (2015). See the Materials & Methods section and SM Table 2 for a full description of this selection process.

Lower effectiveness values correspond to structures that either have high degeneracy (as in right column, Fig. 2a), low determinism (as in left column, Fig. 2a), or a combination of both. In the networks we measured, biological networks on average have lower effectiveness values, whereas technological networks on average have the highest effectiveness. This finding aligns intuitively with what we know about the relationship between $EI$ and network structure, and it also supports long-standing hypotheses about the role of redundancy, degeneracy, and noise in biological systems Edelman  and Gally (2001); Tononi et al. (1999). On the other hand, technological networks like power grids, autonomous systems, or airline networks on average are associated with higher effectiveness values. One explanation for this difference is that efficiency in human-made technological networks tends to create sparser, non-degenerate networks with higher effectiveness on average, wherein the nodes relationships are more specific in their targeting.

Perhaps it might be surprising to find that evolved networks have such low effectiveness. But, as we will show, a low effectiveness can actually indicate that there is informative higher-scale (macroscale) connectivity in the system. That is, a low effectiveness can reflect the fact that biological systems often contain higher-scale structure, which we demonstrate in the following section.

II.4 Causal emergence in complex networks

This new global network measure, $EI$, offers a principled way to answer an important question: what is the scale that best captures the connectivity of a complex system? The resolution to this question is important because science analyzes the structure of different systems at different spatiotemporal scales, often preferring to intervene and observe systems at levels far above that of the microscale Hoel (2018). This is likely because relationships at the microscale can be extremely noisy and therefore uninformative, and coarse-graining can minimize this noise Hoel et al. (2013). Indeed, this noise minimization is actually grounded in Claude Shannon’s noisy-channel coding theorem Shannon (1948), wherein dimension reductions can operate like codes that use more of a channel’s capacity Hoel (2017). Higher-level causal relationships often perform error-correction on the lower-level relationships, thus generating extra effective information at those higher scales. Measuring this difference provides a principled means of deciding when higher scales are more informative (emergence) or when higher scales are extraneous, epiphenomenal, or lossy (reduction).

Bringing these issues to network science, we can now ask: what representation will minimize the uncertainty present in a network? We do this by examining causal emergence, which is is when a dimensionally-reduced network contains more informative connectivity, in the form of a higher $EI$ than the original network. Note that, as discussed, $EI$ can be interpreted solely as a general structural property of networks. Therefore, while we still call this phenomenon “causal emergence” because it has the same mathematical formalization as previous work in Boolean networks and Markov chains Hoel et al. (2013); Hoel (2017, 2018), here we focus on how it can be used to identify the informative higher scales of networks regardless of what those networks represent.

Notably, the phenomenon can be measured by recasting networks at higher scales and observing how the $EI$ changes, a process which identifies whether the network’s higher scales add information above and beyond lower scales.

II.5 Network macroscales

First we must introduce how to recast a network, $G$, at a higher scale. This is represented by a new network, $G_{M}$. Within $G_{M}$, a micro-node is a node that was present in the original $G$, whereas a macro-node is defined as a node, $\mu$, that represents a subgraph, $S_{i}$, from the original $G$ (replacing the subgraph within the network). Since the original network has been dimensionally reduced by grouping nodes together, $G_{M}$ will always have fewer nodes than $G$.

A macro-node $\mu$ is defined by some $W^{out}_{\mu}$, derived from the edge weights of the various nodes within the subgraph it represents. One can think of a macro-node as being a summary statistic of the underlying subgraph’s behavior, a statistic that takes the form of a single node. Ultimately there are many ways of representing a subgraph, that is, building a macro-node, and some ways are more consistent than others in capturing the subgraph’s behavior, depending on the connectivity. We highlight here that macroscales of networks should in general be consistent with their underlying microscales in terms of their dynamics. While this has never been assessed within networks or systems generally, there has been previous research that has asked whether the macroscales of structural equation models are consistent with the effect of all possible interventions Rubenstein et al. (2017).

Here, to decide whether or not a macro-node is an consistent summary of its underlying subgraph, we formalize consistency as measure of whether random walkers behave identically on $G$ and $G_{M}$. We do this because random walks are often used to represent dynamics on networks Masuda et al. (2017), and therefore many important analyses and algorithms—such as PageRank for determining a node’s centrality Page et al. (1998) or InfoMap for community discovery Rosvall  and Bergstrom (2008)—are based on random walks.

Specifically, we define the inconsistency of a macroscale as the Kullback-Leibler divergence Cover  and Thomas (2012) between the expected distribution of random walkers on $G$ vs. $G_{M}$, given some identical initial distribution on each. The expected distribution over $G$ at some future time, $t$, is $P_{m}(t)$, while the distribution over $G_{M}$ at some future time $t$ is $P_{M}(t)$. To compare the two, the distribution $P_{m}(t)$ is summed over the same nodes in the macroscale $G_{M}$, resulting in the distribution $P_{M|m}(t)$ (the microscale given the macroscale). We can then define the macroscale inconsistency over some series of time steps $T$ as:

[TABLE]

This consistency measure addresses the extent to which a random dynamical process on the microscale topology will be recapitulated on a dimensionally-reduced topology (for how this is applied in our analysis, see Materials & Methods).

What constitutes a consistent macroscale depends on the connectivity of the subgraph that gets grouped into a macro-node, as shown in Fig. 4. The $W^{out}_{\mu}$ can be constructed based on the collective $W^{out}$ of the subgraph (shown in Fig. 4a). For instance, in some cases one could just coarse-grain a subgraph by using its average $W^{out}$ as the $W^{out}_{\mu}$ of some new macro-node $\mu$ (as in Fig. 4b). However, it may be that the subgraph has dependencies not captured by such a coarse-grain. Indeed, this is similar to the recent discovery that when constructing networks from data it is often necessary to explicitly model higher-order dependencies by using higher-order nodes so that the dynamics of random walks to stay true to the original data Xu et al. (2016). We therefore introduce higher-order macro-nodes (HOMs), which draw on similar techniques to consistently represent subgraphs as single nodes Xu et al. (2016).

Different subgraph connectivities require different types of HOMs to consistently represent them. For instance, HOMs can be based on the input weights to the macro-node, which take the form $\mu|j$. In these cases the $W^{out}_{\mu|j}$ is a weighted average of each node’s $W^{out}$ in the subgraph, where the weight is based on the input weight to each node in the subgraph (Fig. 4c). Another type of HOM that generally leads to consistent macro-nodes over time is when the $W^{out}_{\mu}$ is based on the stationary output from the subgraph to the rest of the network, which we represent as $\mu|\pi$ (Fig. 4d). These types of HOMs may have minor inconsistencies given some initial state, but will almost always trend toward perfect consistency as the network approaches its stationary dynamics (outlined in Section IV).

Subgraphs with complex internal dynamics can require a more complex type of HOM in order to preserve the macro-node’s consistency. For instance, in cases where subgraphs have a delay between their inputs and outputs, this can be represented by a combination of $\mu|j$ and $\mu|\pi$, which when combined captures that delay (Fig. 4e). In these cases the macro-node $\mu$ has two components, one of which acts as a buffer over a timestep. This means that macro-nodes can possess memory even when constructed from networks that are at the microscale memoryless, and in fact this type of HOM is sometimes necessary to consistently capture the microscale dynamics.

We present these types of macro-nodes not as an exhaustive list of all possible HOMs, but rather as examples of how to construct higher scales in a network by representing subgraphs as nodes, and also sometimes using higher-order dependencies to ensure those nodes are consistent. This approach offers a complete generalization of previous work on coarse-grains Hoel et al. (2013) and also black boxes Ashby (1957); Hoel (2017); Marshall et al. (2018), while simultaneously solving the previously unresolved issue of macroscale consistency by using higher-order dependencies. The types of macro-nodes formed by subgraphs also provides substantive information about the network, such as whether the macroscale of a network possesses memory or path-dependency.

II.6 Causal emergence reveals the scale of networks

A network has an informative macroscale when a recast network, $G_{M}$ (a macroscale), has more $EI$ than the original network, $G$ (the microscale). In general, networks with lower effectiveness (low $EI$ given their size) have a higher potential for such emergence, since they can be recast to reduce their uncertainty. Searching across groupings allows the identification or approximation of a macroscale that maximizes the $EI$.

Checking all possible groupings is computationally intractable for all but the smallest networks. Therefore, in order to find macro-nodes which increase the $EI$, we use a greedy algorithm that groups nodes together and checks if the $EI$ increases. By choosing a node and then pairing it iteratively with its surrounding nodes we can grow macro-nodes until pairings no longer increase the $EI$, and then move on to a new node (see the Materials & Methods section for details on this algorithm).

By generating undirected preferential attachment networks and varying the degree of preferential attachment, $\alpha$, we observe a crucial relationship between preferential attachment and causal emergence. One of the central results in network science has been the identification of “scale-free” networks Barabási et al. (1999). Our results show that networks that are not “scale-free” can be further separated into micro-, meso-, and macroscales depending on their connectivity. This scale can be identified based on their degree of causal emergence (Fig. 5a). In cases of sublinear preferential attachment ($\alpha<1.0$) networks lack higher scales. Linear preferential attachment ($\alpha=1.0$) produces networks that are scale-free, which is the zone of preferential attachment right before the network develops higher scales. Such higher scales only exist in cases of superlinear preferential attachment ($\alpha>1.0$). And past $\alpha>3.0$ the network begins to converge to a macroscale where almost all the nodes are grouped into a single macro-node. The greatest amount of causal emergence is found in mesoscale networks, which is when $\alpha$ is between 1.5 and 3.0, when networks possess a rich array of macro-nodes. Note that the increase in $EI$ following macro-scale groupings for $\alpha>1.0$ shown in Fig. 5a resembles the decrease in $EI$ with higher $\alpha$ that we observe in Fig. 1b. This is because after $\alpha>1.0$ the decreasing $EI$ of the microscale leaves room for improvement of the $EI$ at the macroscale, following a grouping of nodes.

Correspondingly the size of $G_{M}$ decreases as $\alpha$ increases and the network develops an informative higher scale, which can be seen in the ratio of macroscale network size, $N_{M}$, to the original network size, $N$ (Fig. 5b). As discussed previously, networks generated with higher values for $\alpha$ will be more and more star-like. Star-like networks have higher degeneracy and thus less $EI$, and because of this, we expect that there are more opportunities to increase the network’s $EI$ through grouping nodes into macro-nodes. Indeed, the ideal grouping of a star network is when $N_{M}=2$ and $EI=1$ bit. This result is similar to recent advances in spectral coarse-graining that also observe that the ideal coarse-graining of a star network is to collapse it into a two-node network, grouping all the spokes into a single macro-node Laurence et al. (2019), which is what happens to star networks that are recast as macroscales.

Our results offer a principled and general approach to such community detection by asking when there is an informational gain from replacing a subgraph with a single node. Therefore we can define causal communities as being when a cluster of nodes, or some subgraph, forms a viable macro-node (note that this assumes the connections in the network actually represent possible causal interactions, but it also merely a topological property). Fundamentally causal communities represent noise at the microscale. The closer a subgraph is to complete noise, the greater the gain in $EI$ by replacing it with a macro-node (see SM V.7). Minimizing the noise in a given network also identifies the optimal scale to represent that network. However, there must be some structure that can be revealed by noise minimization in the first place. In cases of random networks that form a single large component which lacks any such structure, causal emergence does not occur (as shown in SM V.7).

II.7 Causal emergence in real networks

The presence and informativeness of macroscales should vary across real networks, dependent on connectivity. Here we investigate the disposition toward causal emergence of real networks across different domains. We draw from the same set of networks that are analyzed in Fig. 3, the selection process and details of which is outlined in the Materials & Methods section. The network sizes span up to 40,000 nodes, thus making it unfeasible to find the the best macroscales for each of them. Therefore, we focus specifically on the two categories that previously showed the greatest divergence in terms of the $EI$: biological and technological. Since we are interested in the general question of whether biological or technological networks show a greater disposition or propensity for causal emergence, we approximate causal emergence by calculating the causal emergence of sampled subgraphs of growing sizes. Each sample is found using a “snowball sampling” procedure, wherein a node is chosen randomly and then a weakly connected subgraph of a specified size is found around it Heckathorn  and Cameron (2017). This subgraph is then analyzed using the previously described greedy algorithmic approach to find macro-nodes that maximized the $EI$ in each network. Each available network is sampled 20 times for each size taken from it. In Fig. 6, we show how the causal emergence of these real networks differentiates as we increase the sampled subgraph size, in a sequence of 50, 100, 150, and finally 200 nodes per sample. Networks of these sizes previously provided ample evidence of causal emergence in simulated networks, as in Fig. 5a. Comparing the two categories of real networks, we observe a significantly greater propensity for causal emergence in biological networks, and that this is more articulated the larger the samples are. Note that constructing a random null model of these networks (e.g., a configuration model) would tend to create networks with minimal or negligible causal emergence, as is the case for ER networks (Fig. 13 in SM V.7).

That subsets of biological systems show a high disposition toward causal emergence is consistent, and even explanatory, of many long-standing hypotheses surrounding the existence of noise and degeneracy in biological systems Tononi et al. (1998). It also explains the difficulty of understanding how the causal structure of biological systems function, since they are cryptic by containing certainty at one level and uncertainty at another.

III Discussion

We have shown that the information in the relationships between nodes in a network is a function of the uncertainty intrinsic to their connectivity, as well as how that uncertainty is distributed. To capture this information we adapted a measure, effective information ($EI$), for use in networks, and analyzed what it reveals about common network structures that have been studied by network scientists for decades. For example, the $EI$ of an ER random network tends to $-\log_{2}(p)$, and whether the $EI$ of a preferential attachment network grows or shrinks as new nodes are added is a function of whether its degree of preferential attachment, $\alpha$, is greater or less than 1.0. In networks where the mechanisms or transitions are unknown, but the structure is known, $EI$ captures the degree of unique targeting in the network. In real networks, we showed that the $EI$ of biological networks tends to be much lower than technological networks.

We also illustrated that what has been called “causal emergence” can occur in networks. This is the gain in $EI$ that occurs when a network, $G$, is recast as a new network, $G_{M}$. Finding this sort of informative higher scale means balancing the minimization of uncertainty while simultaneously maximizing the number of nodes in the network. These methods may be useful in improving scientific experimental design, the compression and search of big data, model choice, and even machine learning. Importantly, not every recast network, $G_{M}$, will have a higher $EI$ than the $G$ that it represents, that is, these same techniques can identify cases of reduction. Ultimately, this is because comparing the $EI$ of different network representations provides a ground for comparing the effectiveness of any two network representations of the same complex system. These techniques allow for the formal identification of the scale of a network. Scale-free networks can be thought of as possessing a fractal pattern of connectivity Barabási et al. (2001), and our results show that the scale of a network is the breaking of that fractal in one direction or the other.

The study of higher-order structures in networks is an increasingly rich area of research Rosvall  and Bergstrom (2008); Benson et al. (2016); Scholtes (2017); Yin et al. (2018); Lambiotte et al. (2019), often focusing on constructing networks that better capture the data they represent. Here we introduce a formal and generalized way to recast networks at a higher scales while preserving random walk dynamics. In many cases, a macroscale of a network can be just as consistent in terms of random walk dynamics and also possess greater $EI$. Some macro-nodes in a macroscale may be of different types with different higher-order properties. In other words, we show how to turn a lower-order network into a higher-order network. One noteworthy and related aspect of our work is demonstrating how a system that is memoryless at the microscale can actually possess memory at the macroscale, indicating that whether a system has memory is a function of scale.

While some Stanley et al. (2018) have previously recast subgraphs as individual nodes as we do here, they have not done so in ways that are based on noise minimization and maximizing consistency, focusing instead on gains to algorithmic speed via compression. Explicitly creating macro-nodes to minimize noise brings the dependencies of the network into focus. This means that causal emergence in networks has a direct relationship to community detection, a vast sub-discipline that treats dense subgraphs within a network as representing shared properties, membership, or functions Fortunato  and Hric (2016); Radicchi et al. (2004). However, the relationship between causal emergence and traditional community detection is not as direct as it may seem. For one, causal emergence is high in networks with high degeneracy (i.e. networks with high-degree hubs, as we show in Fig. 5a). Community detection algorithms do not typically select for such structural properties, instead focusing on dense subgraphs that connect more highly within the subgraph than outside Fortunato  and Hric (2016). In SI Fig. 12, we show a landscape of stochastic block model networks and their associated values for causal emergence. Indeed in networks that would have high modularity Newman  and Girvan (2004) (e.g. two disconnected cliques), we do observe causal emergence, but only when the two disconnected cliques are of different sizes. This distinction is key and situates networks that display causal emergence in a meaningful place in the study of complex networks. In light of this, macro-nodes offer a sort of community detection where the micro-nodes that make up a macro-node are a community, and ultimately can be replaced by a macro-node that summarizes their behavior while reducing the subgraph’s noise. Under this interpretation, community structure is characterized by noise rather than shared memberships.

IV Materials and Methods

IV.1 Selection of real networks

Networks were chosen to represent the four categories of interest: social, informational, biological, and technological (see SM Fig. 10, where we detail the same information as in Fig. 3, but also include the source of the network data in addition to the effectiveness value of each network). We used all the available networks under 40,000 nodes (due to computational constraints) within all the domains in the Konect database that reflected our categories of interest. For our social category we used the domains Human Contact, Human Social, Social, and Communication. For our information category we used the domains Citations, Co-authorship, Hyperlinks, Lexical, and Software. For our biological category we used the domains Trophic and Metabolic. Due to overlaps between the Konect database and the Network Repository Rossi  and Ahmed (2015) in these domains, and the paucity of other biological data in the Konect database, we also included the Brains domain and the Ecology domain from the Network Repository to increase our sample size (again, all networks within these domains under 40,000 nodes were included). For our technological category, we used the domains Computer and Infrastructure from the Konect database. Again due to overlap between the Konect database and the Network Repository, we also included the Technological and Power Networks domains from the Network Repository. For a full table of the networks used in this study, along with their source and categorization, see Table 2.

IV.2 Creating consistent macro-nodes

Previously we outlined methods for creating consistent macro-nodes of different types. Here we explore their implementation, which requires deciding which macroscales are consistent. Inconsistency is measured as the Kullback-Leibler divergence between the expected distribution of random walkers on both the microscale ($G$) and the macroscale ($G_{M}$), given an initial distribution, as in Eq. 6.

To measure the inconsistency we use an initial maximum entropy distribution on the shared nodes between $G$ and $G_{M}$. That is, only the set of nodes that are left ungrouped in $G_{M}$. Similarly, we only analyze the expected distribution over that same set of micro-nodes. Since such distributions are only over a portion of the network, to normalize each distribution to 1.0 we include a single probability that represents all the non-shared nodes between $G$ and $G_{M}$ (representing when a random walker is on a macro-node).

We focus on the shared nodes between $G$ and $G_{M}$ for the inconsistency measure because: a) it is easy to calculate which is necessary during an algorithmic search, b) except for unusual circumstances the inconsistency over the shared nodes still reflects the network as a whole, and c) even in cases of the most extreme macroscales (such as when $\alpha>4$ in Fig. 5), there are still nodes shared between $G$ and $G_{M}$.

Here we examine our methods of using higher-order dependencies in order to demonstrate that this creates consistent macro-nodes. We use 1000 simulated preferential attachment networks, which were chosen as a uniform random sample between parameters $\alpha=1.0$ and $2.0$, $n=25$ to $35$, and with either $m=1$ or $2$. These networks were then grouped via the algorithm described in the following section. All macro-nodes were of the $\mu|\pi$ type and their inconsistency was checked over 1000 timesteps. These macro-nodes generally have consistent dynamics, either because they start that way or because they trend to that over time, and of the 1000 networks, only 4 had any divergence greater than 0 after 1000 timesteps. In Fig. 11 in SM V.6, we show 15 of these simulated networks, along with their parameters, number of macro-nodes, and consistencies. Note that even in the cases with early nonzero inconsistency, this is always very low in absolute terms of bits, and of the randomly chosen 15 none do not trend toward consistency over time. In our observations most macro-nodes converge before 500 timesteps, so therefore, in analyzing the real world networks using the $\mu|\pi$ macro-node we check all macro-nodes for consistency and only reject those that are inconsistent at 500 timesteps. More details about the algorithmic approach to finding causal emergence can be found in the following section.

IV.3 Greedy algorithm for causal emergence

The greedy algorithm used for finding causal emergence in networks is structured as follows: for each node, $v_{i}$, in the shuffled node list of the original network, collect a list of neighboring nodes, $\{v_{j}\}\in B_{i}$, where $B_{i}$ is the Markov blanket of $v_{i}$ (in graphical models, the Markov blanket, $B_{i}$, of a node, $v_{i}$, corresponds to the “parents”, the “children”, and the “parents of the children” of $v_{i}$ Friston (2013)). This means that $\{v_{j}\}\in B_{i}$ consists of nodes with outgoing edges leading into $v_{i}$, nodes that the outgoing edges from $v_{i}$ lead into, and nodes that have outgoing edges leading into the out-neighbors of $v_{i}$. For each node in $\{v_{j}\}$, the algorithm calculates the $EI$ of a macroscale network after $v_{i}$ and $v_{j}$ are combined into a macro-node, $v_{M}$, according to one of the macro-node types in Fig. 4. If the resulting network has a higher $EI$ value, the algorithm stores this structural change and, if necessary, supplements the queue of nodes, $\{v_{j}\}$, with any new neighboring nodes from $v_{j}$’s Markov blanket that were not already in $\{v_{j}\}$. If a node, $v_{j}$, has already been combined into a macro-node via a grouping with a previous node, $v_{i}$, then it will not be included in new queues, $\{v_{j}^{\prime}\}$, of later nodes to check. The algorithm iteratively combines such pairs of nodes until every node, $v_{j}$, in every node, $v_{i}$’s Markov blanket is tested.

Additional information

Author contributions: B.K. and E.H. conceived the project. B.K. and E.H. wrote the article. B.K. performed the analyses. Acknowledgements: The authors thank Conor Heins, Harrison Hartle, and Alessandro Vespignani for their insights about notation and formalism of effective information. Funding: This research was supported by the Allen Discovery Center program through The Paul G. Allen Frontiers Group (12171). This publication was made possible through the support of a grant from Templeton World Charity Foundation, Inc. (TWCFG0273). The opinions expressed in this publication are those of the author(s) and do not necessarily reflect the views of Templeton World Charity Foundation, Inc. This work was also supported in part by the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. Competing interests: The authors declare no competing interests. Data and software availability: All data used in this work was retrieved from the Konect Database Kunegis (2013) and also the Network Repository Rossi  and Ahmed (2015), which are publicly available. Software for calculating $EI$ in networks and for finding causal emergence in networks is available by request or at https://github.com/jkbren/einet.

V Supplementary Materials

V.1 Table of key terms

A table of key terms can be found in Table 1.

V.2 Effective information calculation

Mathematically, $EI$ has been expressed in a number of previous ways. The first was as the mutual information between two subsets of a system (while injecting noise into one), originally proposed as a step in the calculation of integrated information between neuron-like elements Tononi (2004); Balduzzi  and Tononi (2009). More recently, it was pointed out that in general an intervention distribution, $I_{D}$, defined as a probability distribution over the $do(x)$ operator (as in Pearl (2000)), creates some resultant effect distribution, $E_{D}$. Then the $EI$ is the mutual information, $I(I_{D};E_{D})$, between the two, when the interventions are done like a randomized trial to reveal the dependencies (i.e., at maximum entropy Fisher (1936); Hoel (2017)).

In order to calculate the total information contained in the causal relationships of a system, $EI$ is applied to the system as a whole Hoel et al. (2013). There, $EI$ was defined over the set of all states of a system and its state transitions. Because the adjacency matrix of a network can be cast as a transition matrix (as in Fig. 7a), the $EI$ of a network can be expressed as:

[TABLE]

where $EI$ is the average of the effect information, $EI_{i}$, of each node (see Table 1 and Fig. 7b). This is equivalent to our derivation of $EI$ from first principles in Equation 1, since:

[TABLE]

Note that for a given node, $v_{i}$, the term in the first summation in Equation 8 above, \displaystyle\sum_{j=1}^{N}w_{ij}\log_{2}\big{(}w_{ij}\big{)}, is equivalent to the negative entropy of the out-weights from $v_{i}$, $-H(W_{i}^{out})$. Also note that $W_{j}$, the jth element in the $\langle W^{out}_{i}\rangle$ vector, is the normalized sum of the incoming weights to $v_{j}$ from its neighbors, $v_{i}$, such that $W_{j}=\dfrac{1}{N}\displaystyle\sum_{i=1}^{N}w_{ij}$. We substitute these two terms into Equation 8 above such that:

[TABLE]

This is equivalent to the formulation of $EI$ from Equation 1, since $H(\langle W^{out}_{i}\rangle)=-\displaystyle\sum_{j=1}^{N}W_{j}\log_{2}(W_{j})$:

[TABLE]

In the derivations of SM V.3 we adopt the relative entropy formulation of $EI$ from Equation 7 for ease of derivation. For a visual intuition behind the calculations involved in this formulation of $EI$, see how the network in Fig. 7a is used to calculate its $EI$ (Fig. 7b), by calculating the mean effect information, $EI_{i}$, of nodes in the network.

V.3 Deriving the effective information of common network structures

Here we inspect the $EI$ for iconic graphical structures, and in doing so, we see several interesting relationships between a network structure and its $EI$. First, however, we will introduce key terminology and assumptions.

Let $\langle k\rangle$ be the average degree of a network, $G$, and each node, $v_{i}$, has degree, $k_{i}$. In directed graphs each $v_{i}$ has an in-degree, $k^{in}_{i}$, and an out-degree, $k^{out}_{i}$. These correspond to the number of edges leading in to $v_{i}$ and edges going out from $v_{i}$. The total number of edges in $G$ is represented by $E$. In undirected Erdős-Rényi (ER) networks, the total number of edges is given by $E=p\frac{N(N-1)}{2}$, where $p$ represents the probability that any two nodes, $v_{i}$ and $v_{j}$, will be connected. In the following subsections, we derive the $EI$ of several prototypical network structures, from random graphs to ring lattices to star networks. Note that for the following derivations we proceed from the relative entropy formalism from SM V.2, and note that therefore $N$ is the number of nodes with the output, $N=N_{out}$.

V.3.1 Derivation: effective information of ER networks

In Erdős-Rényi networks, $EI$ does not depend on the number of nodes in the network, $N$. Instead, the network’s $EI$ reaches its maximum at $-\log_{2}(p)$. This is because in ER networks, each node is expected to connect to $\langle k\rangle=pN$ neighboring nodes, such that every value in $W^{out}_{i}=\frac{1}{\langle k\rangle}$ and every value in $\langle W^{out}_{i}\rangle=\frac{\langle k\rangle}{N\langle k\rangle}=\frac{1}{N}$, which can be represented as:

[TABLE]

Each node in an ER network is expected to be identical to all other nodes in the network, and calculating the expected effect information, $EI_{i}$, is equivalent to calculating the network’s $EI$. As such, we observe:

[TABLE]

V.3.2 Derivation: effective information of ring-lattice and star networks

Here, we compare two classes of networks with the same average degree—ring lattice networks and star, or hub-and-spoke, graphs (see Fig. 8). In each network, we assume an average degree $\langle k\rangle=2d$, with $d$ being the dimension. The $EI$ of star network, $EI_{star}$, approaches $0.0$ as $N$ gets larger, while the $EI$ of ring lattices approaches $\log_{2}(N)-\log_{2}(2d)$. These derivations are shown below, first for the d-dimensional ring lattice, $EI_{d}$.

As every node in a ring lattice is connected to its $2d$ neighbors, each element of $\langle W^{out}_{i}\rangle$ is $\frac{1}{2d}$ and each element of $W^{in}$ is $\frac{2d}{2d\times N}=\frac{1}{N}$.

[TABLE]

Each node in a d-dimensional ring lattice is expected to be identical, so calculating the expected effect information, $EI_{i}$, is equivalent to calculating the network’s $EI$. As such, we observe:

[TABLE]

Note: the $EI$ of ring lattice networks reduces to simply the determinism of the network. The $EI$ of ring lattice networks scale logarithmically with the size of the network, which is contrasted by the behavior of $EI$ in star networks. Star networks have a hub-and-spoke structure, where $N-1$ nodes of degree $k_{spoke}=1$ are connected a hub node, which itself has degree $k_{hub}=N-1$. For star networks, $EI$ approaches $0.0$ as the number of nodes increases. This derivation is shown below.

[TABLE]

Every spoke has an out-weight vector $W^{out}_{i}$ with $N-1$ elements of $w_{ij}=0.0$ and one with $w_{ij}=1.0$. The single hub, however, has $N-1$ elements of $w_{ij}=\frac{1}{N-1}$ with a single $w_{ij}=0.0$. Similarly, $\langle W^{out}_{i}\rangle$ consists of $N-1$ elements with values $\frac{1}{N(N-1)}$.

[TABLE]

Using the same techniques as above, this equation reduces to:

[TABLE]

V.4 Network motifs as causal relationships

It is important to understand why certain motifs have more $EI$ while others have less. In Fig. 9, we show the $EI$ in 13 directed three-node network motifs. The connectivity of each motif drastically influences the $EI$. Motif 09—the directed cycle—is the motif with the highest $EI$. Intuitively, this fits with our definition of $EI$: the amount of certainty in the network (notably, each link in Motif 09, if taken to represent a causal relationship, is both necessary and sufficient). A random walker in this system has zero entropy (even if the direction of its path were reversed), whereas every other three-node motif does not contain that degree of certainty. Second, we see that Motif 04—a system with a “sink” node—has no $EI$, suggesting that a causal structure with that architecture is not informative, since all causes lead to the same effect. Similarly, because there are no outputs from two nodes in Motif 01, we see an $EI$ value of zero.

V.5 Table of network data

In Table 2, we report the name, domain, source, category, and description of each of the 84 networks used in our comparison of $EI$ in real networks. These networks were selected primarily from the Konect database Kunegis (2013), with supplemental datasets added from NetworkRepository Rossi  and Ahmed (2015) when the Konect database lacked a sufficient number of datasets in a given category, since the two databases already significantly overlapped. In many cases, the interactions among nodes in these networks (i.e., their edges) can reasonably be interpreted as causal, directed influence, or dependencies such that the behavior of a node, $v_{i}$, at a given time can be thought to impact the behavior of its neighbors, $v_{j}$. By instituting relatively minimal requirements for selecting the above networks, we are able to assess the $EI$ in a variety of complex systems across different domains. However, while we can measure the $EI$ of any given network, the further interpretation of this $EI$ depends also on what the nodes and edges of a network represent. In a case where the nodes represent states of a system, such as a Markov process, then the $EI$ directly captures the information in the causal structure. In the case where the nodes represent merely dependencies or influence, $EI$ can still be informative as a metric to compare different networks. In a network specifically composed of non-causal correlations, then $EI$ is merely a structural property of the network’s connectivity.

V.6 Examples of consistent macro-nodes

In Fig. 11 we display 15 different parameterizations of small networks grown under degree-based preferential attachment. Each plot shows to the inconsistency of the mapping from the microscale to the macroscale, in bits, which corresponds to the KL divergence of the distribution of random walkers on microscale nodes and the same distribution at the macroscale. Each of these networks are consistent after 1000 timesteps, with eight showing full consistency from the start. These 15 example networks also show the range of causal emergence values that is found in networks.

V.7 Emergent subgraphs

What sort of subgraph connectivity leads to causal emergence? To explore this we take two independent subgraphs, and couple them together while varying their size, moving from clique-like to bipartite connectivity. We then check to see if grouping those clusters into macro-nodes leads to causal emergence (Fig. 12). Specifically, we simulate many small unweighted, undirected networks ($N=100$) from a stochastic block model with two clusters, and we vary the probability of within-cluster edges (from 0.0 to 1.0) as well as the size-asymmetry of the two clusters (illustrated around the border of Fig. 12). In each simulation, we group the microscale network into two macro-nodes, each corresponding to one cluster. What we observe is a causal emergence landscape with several important characteristics (Fig. 12). First, in these networks we observe causal emergence when the fraction of within-cluster connections is either very high or very low (right and left sides of the heatmap in Fig. 12). These are the conditions in which there is a large amount of uncertainty, or noise, in that subgraph. Not only that, however, causal emergence is most likely when there is a size asymmetry between the two clusters, suggesting that macroscales that maximize a network’s $EI$ often do so by creating a more evenly distributed $\langle W^{out}_{i}\rangle$. In general, however, the space of subgraphs leans toward causal reduction (a loss of $EI$ after grouping), which fits with the success of reduction historically and explains why researchers and modelers should generally be biased toward reduction.

In cases of complete noise, with no asymmetries or differences between intra- or inter-connectivity between subgraphs, we should expect causal emergence to be impossible. Indeed, this is what we see for many parameterizations of Erdős-Rényi networks of various sizes (Fig. 13). This result follows from insights in Fig. 1a, where the $EI$ of ER networks converges to a fixed value of $-\log_{2}(p)$ as the size of the network increases. Here, we observe some causal emergence in ER networks but only when the networks are very small. Importantly, the amount of causal emergence is also very small, especially relative to the causal emergence in networks with preferential attachment. This further suggests that causal emergence moves the existent structure of the network into focus by examining the network at a certain scale, rather than creating that structure from nothing.