Blind Inference of Centrality Rankings from Graph Signals

BLIND INFERENCE OF CENTRALITY RANKINGS FROM GRAPH SIGNALS

T. Mitchell Roddenberry, Santiago Segarra

Department of Electrical and Computer Engineering, Rice University

ABSTRACT eigenvector centrality, without observing the edges of the graph but given only a set of graph signals? We study the blind centrality ranking problem, where our goal is to As a motivating example, consider the observation of opinions infer the eigenvector centrality ranking of nodes solely from nodal of individuals (graph signals) in a social network. Intuitively, even observations, i.e., without information about the topology of the net- without having access to the social connections of this group of peo- work. We formalize these nodal observations as graph signals and ple, we might be able to determine the most central actors by tracing model them as the outputs of a network process on the underlying back the predominating opinions. In this paper, we analyze in which (unobserved) network. A simple spectral algorithm is proposed to cases this intuition holds. estimate the leading eigenvector of the associated adjacency matrix, thus serving as a proxy for the centrality ranking. A finite rate perfor- Related work. The standard paradigm for network inference from mance analysis of the algorithm is provided, where we find a lower nodal observations aims to infer the complete graph structure, also bound on the number of graph signals needed to correctly rank (with known as network topology inference. Network topology inference high probability) two nodes of interest. We then specialize our gen- has been studied from a statistical perspective where each node is eral analysis for the particular case of dense Erdos-Rényi˝ graphs, a random variable, and edges reflect the covariance structure of the where existing graph-theoretical results can be leveraged. Finally, ensemble of random variables [10–14]. Additionally, graph signal we illustrate the proposed algorithm via numerical experiments in processing methods have arisen recently, which infer network topol- synthetic and real-world networks, making special emphasis on how ogy by assuming the observed signals are the output of some under- the network features influence the performance. lying network process [15–19]. Our work differs from this line of research, in that we do not infer the network structure, but rather the Index Terms— graph signal processing, eigenvector centrality, centrality ranking of the nodes. This latter feature, being a coarse spectral methods. description of the network, can be inferred with less samples than those needed to recover the detailed graph structure. 1. INTRODUCTION In this same spirit, recent work has considered the inference of coarse network descriptors from graph signals, but have exclusively As the prevalence of complex, structured data has exploded in recent focused on community detection. More precisely, [20–22] provide years, so has the analysis of such data using graph-based represen- algorithms and statistical guarantees for community detection on a tations [1–3]. Indeed, abstracting relational structures as graphs has single graph from sampled graph signals, whereas [23, 24] analyze become an ever-increasing paradigm in science and engineering. In this problem for ensembles of graphs drawn from a latent random this context, network science aims to understand such structures, of- graph model. Lastly, in terms of eigenvector centrality estimation ten via analysis of their algebraic properties when represented as from partial data, [25] considers the case of missing edges in the matrices. graph of interest, but does not rely on node data as we propose here. In any network, the topology determines an influence structure In the direction of centrality inference from data, [26] is the most among the nodes or agents. Identifying the most important nodes in similar to our work. However, the authors are concerned with tem- a network helps in explaining the network’s dynamics, e.g., migra- poral data driven by consensus dynamics, which they use to infer an tion in biological networks [4], as well as in designing optimal ways ad hoc temporal centrality. Our approach does not depend on tempo- to externally influence the network, e.g., vulnerability to attacks [5]. ral structure, and infers the classical eigenvector centrality instead. arXiv:1910.10846v1 [cs.SI] 24 Oct 2019 Node centrality measures are tools designed to identify such impor- Contributions. The contributions of this paper are threefold. First, tant agents. However, node importance is a rather vague concept and we provide a simple spectral method to estimate the eigenvector cen- can be interpreted in various ways, giving rise to multiple coexisting trality ranking of the nodes. Second, and most importantly, we pro- centrality measures, some of the most common being closeness [6], vide theoretical guarantees for the proposed method. In particular, betweenness [7, 8] and eigenvector [9] centrality. In this latter mea- we determine the number of samples needed for a desired resolution sure – of special interest to this paper – the importance of a node is in the centrality ranking. Finally, we particularize our general theo- computed as a function of the importance of its neighbors. retical analysis to the case of Erdos-Rényi˝ graphs and showcase our Computation of the eigenvector centrality requires complete findings via illustrative numerical experiments. knowledge of the graph being studied. However, graphs are often difficult or infeasible to by fully observed, especially in large-scale settings. In these situations, we might rely on data supported on the 2. NOTATION AND BACKGROUND nodes (that we denominate graph signals) to infer network properties. In this paper, we seek to answer the question: Under what Graphs and eigenvector centrality. An undirected graph G con- conditions can one rank the nodes of a graph according to their sists of a set V of n := |V| nodes, and a set E ⊆ V × V of edges, corresponding to unordered pairs of elements in V. We commonly Emails: [email protected], [email protected]. index the nodes with the integers 1, 2, . . . , n. We then encode the n×n graph structure with the (symmetric) adjacency matrix A ∈ R , Algorithm 1 Blind centrality inference such that Aij = Aji = 1 for all (i, j) ∈ E, and Aij = 0 otherwise. N 1: INPUT: graph signals {yi}i=1 For a graph with adjacency matrix A, we consider the eigen- 2: Compute the sample covariance matrix vector centrality [9] given by the leading eigenvector of A. That Pn > is, if A has eigenvalue decomposition A = i=1 λivivi , where N N 1 X > λ1 ≥ λ2 ≥ · · · ≥ λn, the eigenvector centrality of node j is given Cb y := yiyi . (5) th N by the j entry of v1. For notational convenience, we denote this i=1 leading eigenvector by u, so that the centrality value of node j is u N given by j . 3: Compute the leading eigenvector ub of Cb y Often, we are not concerned with the precise centrality value of 4: OUTPUT: Centrality ranking induced by ub each node, but rather a rank ordering based on these values. In this regard, we define the centrality rank as notion that agents are positively influenced by the signals in their ri = {j ∈ N : uj ≥ ui} , (1) neighborhoods, e.g., in a social network the opinions of individuals so the most central node has r = 1 (in the absence of a tie) and the grow similar to those in their surroundings. least central node has r = n. Note that if two nodes have identical With this notation in place, we formally state our problem: centrality values they will have the same centrality rank. For any two N Problem 1. Given the observation of N signals {yi}i=1 generated nodes i, j ∈ N , two rankings r and r0 preserve the relative order of 0 0 as in (3), estimate the node centrality ranking induced by u, the i, j when ri ≥ rj if and only if ri ≥ rj . For convenience, we say leading eigenvector of the (unobserved) adjacency matrix A. that two vectors u and ub preserve the relative order of two nodes, where it is implicit that this refers to the ranking induced by each Our proposed method to solve Problem 1 is summarized in Al- N vector. gorithm 1, where we simply compute the sample covariance Cb y Graphs signals and graph filters. Graph signals are defined as of the observed outputs and use the ranking induced by its leading real-valued functions on the node set, i.e., x: V → R. Given an eigenvector as a proxy for the true centrality ranking. To see why indexing of the nodes, graph signals can be represented as vectors this simple spectral method is reasonable, notice that the (popula- n in R , such that xi = x(i).A graph filter H(A) of order T is a tion) covariance of the i.i.d. outputs yi is given by linear map between graph signals that can be expressed as a matrix > > 2 Cy = E[H (A) wiwi H (A) ] = H(A) , (4) polynomial in A of degree T , where we used the facts that wi is white and A is symmetric. Thus, T T X k X > from (4) it follows that A and Cy share the same set of eigenvectors. H (A) = γkA := H(λi)vivi , (2) Moreover, since γk ≥ 0 in (3), it must be the case that u is also the k=0 i=0 leading eigenvector of Cy. In this way, as N grows and the sample where H(λ) is the extension of the matrix polynomial H to scalars. covariance in (5) approaches Cy, it is evident that ub as computed Properly normalized and combined with a set of appropriately cho- in Algorithm 1 is a consistent estimator of the true centrality eigen- sen filter coefficients γk, the transformation H (A) can account for vector u. Hence, for a large enough sample size N, Algorithm 1 is a range of interesting dynamics including consensus [27], random guaranteed to return the true centrality ranking. However, the true walks, and diffusion [28]. practical value lies in the finite rate analysis, i.e., given a finite sample size N and two nodes of interest, when can we confidently state that we have recovered the true centrality ordering between these 3. BLIND CENTRALITY INFERENCE nodes? We answer this question next. Our goal is to infer the centrality rankings of the nodes without ever observing the graph structure, but rather exclusively relying on the 4. FINITE RATE ANALYSIS observation of graph signals. Naturally, these graph signals must be shaped by the underlying unknown graph for the described infer- We present our main result on the performance of Algorithm 1, ence task to be feasible. In particular, we model the observed graph where we state sufficient sampling conditions to preserve (with high N signals {yi}i=1 as being the output of graph filters excited by (un- probability) the relative centrality ordering of nodes. In stating this observed) white noise, i.e., result, we use the notation k · kp to denote the `p norm when the argument is a vector and the related induced norm when the argument T X k is a matrix. Also, we denominate the eigenvalues and eigenvectors y = H (A) w = γ A w , i i k i (3) of Cy by β1 ≥ ... ≥ βn ≥ 0 and z1, z2,..., zn, respectively. k=0 From our discussion following (4) it holds that z1 = u. > where [w] = 0, [ww ] = I, and γk ≥ 0 for all k. The ra- 2 N E E Theorem 1. Define µ := nkuk∞, E = Cy −Cb y , and κ = kCy − tionale behind the inputs wi being white is that we are interested > 2 β1uu k∞. Assume that kyik2 ≤ m for some constant m and in a situation where the original signal (e.g., the initial opinion of 2 β1 − κ = Ω µ kEk∞ . Consider any pair of nodes i, j where each individual in a social network) is uncorrelated among agents. |ui − uj | > α. If In this way, the correlation structure of the output is exclusively im- 2 posed by exchanges among neighboring agents as opposed to be- 4 t m log n N ≥ Cµ 2 , (6) ing driven by features that are exogenous to the graph. Regarding α β1(1 − κ/β1) the form of the filter H (A), we only require non-negative coefficients γk ≥ 0. Since successive powers of A aggregate information for some absolute constant C and t > 0, then u and ub preserve the −t2 in neighborhoods of increasing sizes, γk ≥ 0 simply imposes the relative order of i and j with probability at least 1 − n . 1 1 0.40 1 p = 0.1 A n = 50 B Erdos-Rényi˝ graph C D p = 0.3 n = 100 0.8 0.8 Barabási-Albert graph 0.8 p = 0.5 n = 500 0.30 1/√n 103 p = 0.7 n = 1000 0.6 p = 0.9 0.6 n = 2000 0.6 0.20

0.4 0.4 102 0.4 Centrality

0.10 Sample size 0.2 0.2 0.2 Spearman Correlation Spearman Correlation 0.00 1 0 1 2 3 0 1 0 1 2 10 10 10 10 10− 10 10 10 0 200 400 0 100 200 300 400 Sample size Normalized sample size N log n/n Node index Window region

Fig. 1: Blind centrality inference on synthetic graphs. Each plot is the average over 10 runs. (A) Spearman correlation for ﬁxed graph size n = 500 with varying edge probability p. (B) Spearman correlation for edge probability p = 4 log n/n with varying graph size n, plotted against the normalized sample size N log n/n. (C) Comparison of eigenvector centrality for Erdos-Rényi˝ graph with p = log n/n and Barabási-Albert graph with m = m0 = 3, where both graphs are of size n = 500. Notice that both sets of parameters yield graphs of similar density. (D) Blind centrality inference for Barabási-Albert graphs with m = m0 = 4 and n = 500. For each sample size, the Spearman correlation is evaluated for a windowed region (width = 100) with respect to the true ordered eigenvector centrality.

Proof. (sketch) First, we bound the inﬁnity-norm of the matrix E. matrices, which is characterized by the Davis-Kahan sin Θ Theo- By [29, Corollary 5.5.2] and the equivalence of norms, under our rem [31]. For the ranking problem, we are concerned with element- 2 assumptions, the following holds with probability at least 1 − n−t : wise perturbations of the leading eigenvectors, requiring the use of more modern statistical results [30]. 2 4 t 4β1 If N ≥ C0µ 0 2 m log n α (β1 − κ) (7) 4.1. Graph spectra and eigenvector delocalization √ √ then kEk∞ ≤ nkEk2 ≤ nβ1 The spectrum of the covariance is determined by the spectrum of (the adjacency matrix of) the graph itself and the frequency response 0 α (β1−κ) of the ﬁlter. To illustrate this, we consider an Erdos-Rényi˝ graph of where = 2 , and C0 is an absolute constant. It follows 2β1µ n nodes, where each edge exists independently with probability p. u from [30, Theorem 2.1], up to the sign of b, In the dense regime, where pn ≥ C log n, C ≥ 1, it is well-known

2 that as the graph becomes sufficiently large, the leading eigenvalue C1µ β1 √ converges to pn, and λ2(A) = O( pn) [32]. For simplicity, con- ku − ubk∞ ≤ , (8) β1 − κ sider the case where the graph filter is equal to the adjacency matrix: H(A) = A. So, the covariance matrix is the square of the adjacency for some constant C1. Furthermore, by applying the substitutions 2 2 0 matrix, and thus has leading eigenvalue β1 ≈ p n . α (β1−κ) 2 0 = 2 , C = 4C0C1 , and α = C1α in Equations 7 and 8, κ 2β1µ In order to bound in (6) for this specific graph type we apply we get the equivalence of norms to yield 2 √ 4 t m log n > 3/2 κ = kCy − β1uu k∞ ≤ nβ2 = O(n p). (10) If N ≥ Cµ 2 α β1(1 − κ/β1) (9) α We emphasize that the graph filter has significant influence on κ; a then ku − uk∞ ≤ . b 2 filter that strongly decreases the ratio β2/β1 will improve the performance of Algorithm 1. For instance, an ‘ideal filter’ that annihilates This completes the proof, as an element-wise perturbation of mag- the lower spectrum of the adjacency matrix will yield optimal per- nitude at most α/2 is guaranteed to preserve the ordering of nodes formance, since the ratio κ/β1 would be zero in (6). whose centralities differ by more than α. 2 We proceed to analyze the value µ = nkuk∞. This measures concentrated the eigenvector centrality is in the most central Theorem 1 characterizes the sampling requirements of Algo- node, i.e., µ reflects the localization of the leading eigenvector. We rithm 1 in terms of a desired resolution α, i.e., we determine the consider the result of [33] which shows that in the dense regime, 2 −1+g(n) g(n) samples required to correctly order two nodes that differ by at least kuk∞ ≤ n , for some g(n) = o(1). Thus, µ ≤ n , so the α in their centrality values. As α decreases, the sampling require- eigenvector centrality becomes increasingly delocalized as n → ∞. ment increases by a factor of 1/α2, reflecting the difficulty of dif- Moreover, the largest possible gap between any nodes i, j is attained ferentiating nodes whose centralities are very close together.√ The when a node has eigenvector centrality 0. Thus, the largest mean- y m ingful α in the context of Theorem 1 is less than ng(n)/2−1/2, since assumption of being bounded in a Euclidean ball of radius √ √ √ −1+g(n) g(n) g(n)/2−1/2 holds for bounded inputs to a graph filter. Considering the example kuk∞ ≤ n = n / n = n . So, to pre- of opinions in a social network, it is reasonable to assume that the serve the order of the most central and least central nodes in this measured graph signal is bounded, i.e. there is a limit to the extremes scenario, it can be shown that in opinion dynamics. no(1)+1 The result in Theorem 1 differs from existing work in blind N ≥ Ct2m (11) network inference [20, 21], where the performance of similar ap- log n proaches for community detection are justified in terms of the align- samples are sufficient, under the same conditions as Theorem 1. This ment of principal subspaces between the true and sample covariance quantity tends to infinity with n, reflecting the increasing difficulty √ 1,000 close to 1/ n, Barabási-Albert graphs are characterized by a highly localized centrality structure. 800 Due to this unevenness in the distribution of the eigenvector centrality, we evaluate the Spearman correlation over a sliding win- 600 dow, i.e., for groups of nodes with consecutive ground-truth centrality rankings. This highlights the varying sensitivity to perturbation 400 based on differences in centrality, as described in Theorem 1. As shown in Fig. 1D, performance over every region of the graph (where “region” refers to nodes of similar centrality) improves as 200 the number of samples increases, with the most central nodes being properly ranked with fewer samples than the rest. In particular, for the top 100 nodes (nodes 400-500), we achieve a Spearman corre- Fig. 2: Karate club network, with the president (P) and instructor lation of 0.84 with as few as ≈ 600 samples, while the next lower (I) labelled. Nodes are scaled proportionally to their eigenvector block (nodes 300-400) needs 5000 samples to achieve a Spearman centrality, and colored according to the number of samples sufficient correlation of 0.81. This illustrates the highly centralized nature of for proper ranking. the Barabási-Albert model, reflected in the power-law distribution of node degrees [34]. The most central nodes in the graph are more of the ranking problem as the eigenvector centrality becomes in- separated from other nodes on average, compared to the nodes with creasingly delocalized. We further assess empirically the tightness lower eigenvector centrality (see Figure 1C), making the ranking of bound (11) in our numerical experiments. task easier for these regions of the graph. In terms of Theorem 1, the sampling requirements here are low due to a large tolerance α. Zachary’s Karate Club. Finally, we show the application of 5. NUMERICAL EXPERIMENTS Algorithm 1 to the karate club network [35, 37]. To evaluate its performance, for an inferred ranking r, we consider node i to be cor- We demonstrate the performance of our algorithm on Erdos-Rényi˝ b rectly ranked with respect to the true centrality ranking r if |ri − and Barabási-Albert graph models, as well as the karate club net- b ri| ≤ 1. Moreover, we consider a number of samples N to be suf- work [2, 34, 35]. The graph filter used is of the form ficient to rank node i if for all sample counts N 0 such that 1000 ≥ N 0 ≥ N, node i is ranked correctly with probability greater than 4 k X A 0.95. If this is not attained, we saturate the required number of sam- H (A) = . (12) λ1(A) ples at 1000. k=0 The results of this experiment are shown in Fig. 2. It is clear that We evaluate our results with the absolute value of the Spearman the nodes with higher eigenvector centrality require fewer samples correlation, a measure of monotonicity between two ranking func- to be properly ranked, as they are more separated from other nodes. tions [36]. More precisely, the most central nodes are the Instructor (I) with Erdos-Rényi˝ graphs. We consider an Erdos-Rényi˝ random eigenvector centrality 0.36, and the President (P) with eigenvector graph model with graph size n and edge probability p. We first show centrality 0.37, requiring 80 and 10 samples respectively for proper how the edge probability influences the performance of Algorithm 1 ranking. The least central nodes, found on the periphery of the graph, in the ranking task. For fixed n = 500, we vary p from 0.1 to 0.9, have very similar centralities, and are difficult to distinguish. For in- and compute the Spearman correlation of the inferred eigenvector stance, 5 nodes in this network have centrality close to 0.101, mak- centrality against the true eigenvector centrality for an increasing ing the sampling requirements for distinguishing them from other number of samples N. similarly-ranked nodes very high due to small tolerance α. As pictured in Fig. 1A, the performance of the algorithm varies inversely with the parameter p. This is expected, as dense graphs 6. CONCLUSIONS AND FUTURE WORK have more delocalized eigenvectors, and thus have small average distances between centralities. Then, by Theorem 1, the ranking We have considered the problem of ranking the nodes of an unob- problem becomes increasingly difficult due to small values of α. served graph based on their eigenvector centrality, given only a set Looking past the edge probability, we consider the relationship of graph signals regularized by the graph structure. Using tools from between the number of samples N and the graph size n in the dense matrix perturbation theory, we characterize the `∞ perturbation of Erdos-Rényi˝ graph regime. For each n, we set p = 4 log n/n. We the leading eigenvector of the empirical covariance matrix in terms consider the algorithm performance against the number of samples of the graph filter’s spectrum and number of samples taken. We taken, normalized by log n/n, by the derivation in (11). Fig. 1B then present an analysis of dense Erdos-Rényi˝ graphs in this regime, shows that for each graph size n, the performance closely matches showing the interplay between the ranking problem and eigenvector over the normalized sample size N log n/n, as expected. Barring a delocalization. These theoretical results are then demonstrated on slight deviation for small n, due to the unknown no(1) term in (11) Erdos-Rényi˝ and Barabási-Albert random graphs, highlighting the unaccounted for by the normalized sample size, this demonstrates influence of the graph structure on the performance of our approach. the tightness of the sampling requirements for dense Erdos-Rényi˝ Finally, we demonstrate the sampling requirements for approximate graphs and, for this setting, confirms the practical validity of Theo- ranking on the karate club network. rem 1. Future research avenues include the consideration of additional Barabási-Albert graphs. To illustrate our algorithm on a more (spectrum-based) centrality measures, such as Katz and PageRank realistic type of graph, we consider a Barabási-Albert graph model centralities, as well as the relaxation of some of the assumptions in of size n = 500 with parameter m = m0 = 3 [34]. 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