{"title": "Efficient Near-Optimal Testing of Community Changes in Balanced Stochastic Block Models", "book": "Advances in Neural Information Processing Systems", "page_first": 10364, "page_last": 10375, "abstract": "We propose and analyze the problems of \\textit{community goodness-of-fit and two-sample testing} for stochastic block models (SBM), where changes arise due to modification in community memberships of nodes. Motivated by practical applications, we consider the challenging sparse regime, where expected node degrees are constant, and the inter-community mean degree ($b$) scales proportionally to intra-community mean degree ($a$). Prior work has sharply characterized partial or full community recovery in terms of a ``signal-to-noise ratio'' ($\\mathrm{SNR}$) based on $a$ and $b$. For both problems, we propose computationally-efficient tests that can succeed far beyond the regime where recovery of community membership is even possible. Overall, for large changes, $s \\gg \\sqrt{n}$, we need only $\\mathrm{SNR}= O(1)$ whereas a na\\\"ive test based on community recovery with $O(s)$ errors requires $\\mathrm{SNR}= \\Theta(\\log n)$. Conversely, in the small change regime, $s \\ll \\sqrt{n}$, via an information theoretic lower bound, we show that, surprisingly, no algorithm can do better than the na\\\"ive algorithm that first estimates the community up to $O(s)$ errors and then detects changes. \nWe validate these phenomena numerically on SBMs and on real-world datasets as well as Markov Random Fields where we only observe node data rather than the existence of links.", "full_text": "Ef\ufb01cient Near-Optimal Testing of Community\nChanges in Balanced Stochastic Block Models\n\nAditya Gangrade\nBoston University\ngangrade@bu.edu\n\nBobak Nazer\n\nBoston University\n\nbobak@bu.edu\n\nPraveen Venkatesh\n\nCarnegie Mellon University\n\nvpraveen@cmu.edu\n\nVenkatesh Saligrama\n\nBoston University\n\nsrv@bu.edu\n\nAbstract\n\nWe propose and analyze the problems of community goodness-of-\ufb01t and two-\nsample testing for stochastic block models (SBM), where changes arise due to\nmodi\ufb01cation in community memberships of nodes. Motivated by practical applica-\ntions, we consider the challenging sparse regime, where expected node degrees are\nconstant, and the inter-community mean degree (b) scales proportionally to intra-\ncommunity mean degree (a). Prior work has sharply characterized partial or full\ncommunity recovery in terms of a \u201csignal-to-noise ratio\u201d (SNR) based on a and\nb. For both problems, we propose computationally-ef\ufb01cient tests that can succeed\nfar beyond the regime where recovery of community membership is even possible.\n\nOverall, for large changes, s \u226b \u221an, we need only SNR = O(1) whereas a na\u00efve\nConversely, in the small change regime, s \u226a \u221an, via an information theoretic\n\ntest based on community recovery with O(s) errors requires SNR = \u0398(log n).\n\nlower bound, we show that, surprisingly, no algorithm can do better than the na\u00efve\nalgorithm that \ufb01rst estimates the community up to O(s) errors and then detects\nchanges. We validate these phenomena numerically on SBMs and on real-world\ndatasets as well as Markov Random Fields where we only observe node data rather\nthan the existence of links.\n\nWhile community detection and recovery for the stochastic block model (SBM) [Abb18] and, more\ngenerally, inference of community structures underlying large-scale network data [GN02; New06;\nFor10] has received signi\ufb01cant interest across the machine learning, statistics and information theory\nliteratures, there has been limited work on the important problem of testing changes in community\nstructures. The general problem of testing changes in networks naturally arises in a number of\napplications such as discovering statistically signi\ufb01cant topological changes in gene regulatory net-\nworks [Zha+08] or differences in brain networks between healthy and diseased individuals [Bas+08].\nBuilding upon this perspective, we propose testing of differences in the underlying community struc-\nture of a network, which can encompass scenarios such as detecting structural changes over time in\nsocial networks [AG05; For10], determining whether a set of genes belong to different communities\nin disease and normal states [JTZ04], and deciding whether there are changes in functional modules,\nwhich represent communities, in protein-protein networks [CY06].\n\nTesting structural changes in networks is statistically challenging due to the fact that we may have rel-\natively few independent samples to evaluate combinatorially-many potential changes. In this paper,\nwe propose methods for goodness-of-\ufb01t (GoF) testing and two-sample testing (TST) for detecting\nchanges in community memberships under the SBM. The SBM naturally captures the community\nstructures commonly observed in large-scale networks, and serves as a baseline model for more com-\nplex networks. Speci\ufb01cally, there are n nodes partitioned into two equal-sized communities, and the\n\n33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada.\n\n\fnetwork is observed as a random n \u00d7 n adjacency matrix, representing the instantaneous pairwise\ninteractions among individuals in the population. Both intra- and inter-community interactions are\nallowed. Members within the same community interact with uniform probability a/n, while mem-\nbers belonging to different communities with a smaller probability b/n. We restrict attention to the\ncommonly-considered and practically-relevant setting of a/b = \u0398(1).\nFor our testing problems, we assume that the network samples are aligned on n \u226b 1 vertices,\nand that the latent communities are either the same, or they differ in at least some s \u226a n nodes.\nWe pose the GoF problem as: Decide whether or not the observed random incidence matrix is an\ninstantiation of a given community structure. For the TST problem, we ask: Given two random\nincidence matrices, decide whether or not their latent community structure is identical.\n\nSparse vs. Dense Graphs. We focus on scenarios where the observed random incidence matrices\nare sparse with average node degree bounded by a constant independent of the network size. Within\nthis context we develop minimax optimal methods for GoF and TST in this context. We are moti-\nvated by both practical and theoretical concerns. Practically, as observed in [Chu10], realistic graphs\nsuch as social networks are sparse (friendships do not grow with network size); in temporal settings,\nat any given time, only a small subset of interactions are observed; and in other cases ascertaining\nthe presence or absence of each edge in the network being observed is an expensive process, and it\nmakes sense to understand the fundamental limits for when testing is even possible.\nFrom a theoretical standpoint, the sparse setting is challenging due to signal-to-noise ratio (SNR)\nconstraints that do not arise in the dense case. Recovery of the latent community with up to s errors\nis possible iff \u039b & log(n/s) [CRV15; ZZ16; FC19], where \u039b is a SNR parameter that, in the setting\na/b = \u0398(1), scales linearly with the mean degree. In particular, for \u039b of constant order, recovery\nwith sublinear distortion fails. The question of whether testing is possible when recovery fails is\nmathematically intriguing. Further, this is the only theoretically interesting setting. Indeed, if test-\ning for s changes requires a graph dense enough to allow recovery with \u223c s errors, then one might\nas well recover these communities and compare them.\n\nOur Contributions. We show that optimal tests exhibit a surprising two-phase behavior:\n\nthat succeed with \u039b = O(1) - far below the SNR threshold for recovery. For GoF, this requirement\n\n1. For s \u226b \u221an, or \u2018large changes,\u2019 we propose computationally-ef\ufb01cient schema for GoF and TST\nis even weaker - we only need \u039b & n/s2, which vanishes with n since s \u226b \u221an. Further, we match\n2. In contrast, we show via an information-theoretic lower bound that for s \u226a \u221an, or \u2018small\n\nthese bounds up to constants with information-theoretic lower bounds.\n\nchanges,\u2019 both testing problems require \u039b = \u2126(log(n)) for reliable testing. This means that the\nna\u00efve strategy of recovering communities and comparing them is tight up to constants in this regime.\nWe complement the above theoretical study by three experiments: the \ufb01rst implements the above\ntests on synthetic SBMs, and the second on the political blogs dataset - a popular real world dataset\nfor community detection [AG05]. Both of these experiments show excellent agreement with the\ntheoretical predictions. The third experiment casts a wider net, and instead studies the related prob-\nlem of testing the underlying community structure of a Gaussian Markov Random Field that has\nprecision matrix I + \u03b3G for G drawn from an SBM. This experiment explores the more realistic set-\nting where instead of receiving a graph, we obtain observations at each node of a hidden graph, and\nwish to reason about the underlying structure. Remarkably, a simple adaptation of our procedure for\nSBMs shows excellent performance for this problem. This indicates that our observations are not\nrestricted to raw SBMs, but may signal a more general phenomenon that merits exploration.\n\nRelated Work. For work on recovery communities we refer to the survey [Abb18]. However, we\nexplicitly point out the papers [CRV15; ZZ16; FC19], which provide various schemes and necessary\nconditions that show that the partial recovery problem with distortion s can be solved with vanishing\nerror probability if and only if \u039b & log(n/s). We further point out the lower bounds of [MNS15;\nDAM17], which assert that if \u039b < 2, then asymptotically, the best possible distortion for partial\nrecovery (or weak recovery, as it is referred to in this constant SNR regime) is n/2 \u2212 o(n). Note\nthat reporting a uniformly random community achieves distortion of s = n/2 \u2212 O(\u221an).\nOurs is the \ufb01rst work to study GoF and TST where both hypothesized models are SBMs. Neverthe-\nless, both GoF and TST in the context of network data as well as SBMs have been studied. Below\nwe highlight the key differences in modeling assumptions and the ensuing technical implications,\nwhich renders much of the prior work inapplicable to our setting.\nWith regards to GoF, [AV14; VA15] study the problem of detecting if a graph is an unstructured\n\n2\n\n\fErd\u02ddos-R\u00e9nyi (ER) graph, or if it has a planted dense subgraph, providing detailed characterizations\nof the feasiblity regions and statistical phase transitions in this setting. While this work is aligned\nwith ours in the techniques used, the modeled setting and problem there are different (ER vs. planted\ndense subgraph), and TST is not explored. Particularly, the dense subgraph model and the SBM are\nqualitatively different, and conclusions from one cannot be transferred to the other directly.\nA number of papers, including [Lei16; BS16; Ban+16; GL17] study various techniques and regimes\nof determining if a graph is a SBM or an unstructured ER graph, and if the former, the number of\ncommunities in the model. Of these, [GL17] approach the problem by counting small motifs in\nthe graphs, [Ban+16] propose a simple scan and [Lei16; BS16] propose testing of the number of\ncommunities on the basis of the top singular values of the graph.\n[Tan+17] study TST of the model parameters in random dot product graphs, and propose the dis-\ntance between aligned spectral embeddings of the two graphs as a statistic to do so. They use this to\ntest equality against various transformations of the underlying models, and in particular for SBMs,\ntest if the connectivity probabilities (a/n, b/n) are identical or not for two graphs with latent com-\nmunities that are randomly drawn. [LL18] adapt these tests by considering the same distance, but\nweighted by the corresponding singular values of one of the graphs, and use this to study two-sample\ntesting of equality of the latent communities in the graphs - as in this paper.\nIn contrast to the low-rank structure assumptions in the above work, [Gho+17a; Gho+17b; GL18]\nstudy two-sample testing of inhomogeneous ER graphs (i.e., ER graphs where each edge may have\na distinct probability of existing). Within this setting, they provide a number of statistics based both\non estimates of the Frobenius and operator norms of the differences of the expected graph adjacency\nmatrices, as well as those based on motifs such as triangles, and explore the limits of these tests.\nA fundamental drawback of these approaches, in our context, is their reliance on singular values,\nspectral norms and Frobenius norms. Singular embeddings are particularly sensitive to noise, and\nstable embeddings require signi\ufb01cant edge density (particularly when a sublinear number of alter-\nations to the communities are to be tested). Indeed, in this context, we note that, in contrast to our\nlow SNR, sparse setting, [LL18] require both a degree of n1/2\u2212\u01eb and an SNR of log(n) correspond-\ning to a high SNR, high edge-density regime, where full community recovery is possible.\nSimilarly, Frobenius and Spectral norms based tests of [GL18; Gho+17a] are not stable enough to\ntest a sublinear number of changes in a low SNR regime. Functionally, this can be seen by the fact\nthat the square-Frobenius norm of the difference of two graphs is equal to the number of edges that\nappear in one graph but not the other, and for sparse graphs, most edges appear in only one of the\ntwo graphs. Similarly, arguments about spectral norms rely on concentration of the same for ER\ngraphs, but the best known concentration radius [LLV17] is far too large to allow testing of small\ndifferences in sparse graphs. Indeed, for any of the statistics of [GL18] to have power in our setting,\nthe results of the paper require that the expected degree diverges with n, and that \u039b & n/s, which is\nexponentially above the SNR required to recover communities up to distortion s/2.\n\n1 De\ufb01nitions\n\nThe Stochastic Block Model. A vector x \u2208 {\u00b11}n is said to be a balanced community vector (or\nG on n nodes such that all edges are drawn mutually independently given x, and\n\npartition) ifP xi = 0. The stochastic block model is de\ufb01ned as a random, simple, undirected graph\n\nP ({i, j} \u2208 G|x) =\n\na + b\n\n2n\n\n+\n\na \u2212 b\n2n\n\nxixj.\n\nNote that we treat x as a deterministic but unknown quantity, and thus, P (\u00b7|x) is a slight abuse of\nnotation. The parameters (a, b) may vary with n, and we focus on the setting a, b = O(log n), with\nemphasis on O(1)1, and a/b = \u0398(1). For technical convenience, we require that a + b < n/4.\n\nThe signal-to-noise ratio (SNR) of an SBM is the quantity \u039b :=\n\nrecovery problem, as described in earlier discussions.\n\n(a \u2212 b)2\na + b\n\n, which characterises the\n\n1While our main interest is in the constant degree regime, we also show that testing for small changes is\nimpossible in this setting (e.g Thm 1), and instead logarithmic degrees are needed. Thus, to present our results\ncompletely, we must allow a, b to vary at least in the range [\u2126(1), O(log(n))], or, more succinctly O(log n).\nLarge scales are not of interest since exact recovery is possible at the logarithmic scale.\n\n3\n\n\fNote that the partitions x and \u2212x induce the same distribution. Accordingly, the distortion between\npartitions x and y is d(x, y) := min(dH(x, y), dH(x,\u2212y)), where dH is the Hamming distance.\nMinimax Testing Problems. We formally de\ufb01ne two minimax hypothesis testing problems.\n\nGoodness-of-Fit. We are given a balanced partition x0 and a parameter s. We receive a graph\nG \u223c P (G|x), where x is an unknown balanced partition that is either exactly equal to x0 or differs\nin at least s places. Our goal is to solve the hypothesis test:\n\nH0 : d(x, x0) = 0\n\nvs.\n\nH1 : d(x, x0) \u2265 s\n\nWe measure the minimax risk of this problem by\n\nRGoF(n, s, a, b) := inf\n\u03c6\n\nsup\n\nx0 nP (FA) + sup\n\nx\n\nP (MD(x))o\n\nwhere \u03c6(G) outputs either 0 or 1, P (FA) := P (\u03c6(G) = 1 | x0), P (MD(x)) := P (\u03c6(G) = 0 | x),\nare respectively the false alarm and missed detection probabilities, and the second supremum is over\nall x such that d(x, x0) \u2265 s.\nTwo-Sample Testing. We are given a parameter s and two independent graphs G \u223c P (G|x), H \u223c\nP (H|y), where x and y are unknown balanced communities satisfying d(x, y) \u2208 {0} \u222a [s : n/2].\nThe goal is to solve the following (composite null) testing problem:\n\n(1)\n\n(2)\n\nH0 : d(x, y) = 0\n\nvs.\n\nH1 : d(x, y) \u2265 s,\n\nwith the measure of risk\n\nRTST(n, s, a, b) := inf\n\u03c6\n\nsup\nx,y\n\nP(cid:0)\u03c6(G, H) 6= 1{x = y}| x, y(cid:1),\n\nwhere \u03c6(G, H) outputs either 0 or 1 and the supremum is over balanced x, y such that d(x, y) \u2208\n{0} \u222a [s : n/2].\nAs we vary n and (s, a, b) with n as some functions (sn, an, bn), the above de\ufb01ne a sequence\nof hypothesis tests. We say that the GoF problem can be solved reliably for such a sequence if\nRGoF(n, sn, an, bn) \u2192 0 as n \u0580 \u221e, and similarly for TST. Below, we will target O(1/n) bounds.\nFor conciseness, we will suppress the dependence of risks on (n, s, a, b), writing just RGoF/RTST.\n\nOn balance: The strict balance assumption above can be relaxed to only requiring that both com-\nmunities are of size linear in n, at the cost of weakening some of the constants left implicit in the\ntheorem statements. While the majority of the analysis in the paper will assume exact balance, we\nbrie\ufb02y discuss unbalanced but linearly sized communities whilst detailing the proofs. Note that\nsince the communities are no longer balanced, the differences between x and y can be \u2018one-sided\u2019\ni.e., more nodes can move from, say, + to \u2212, than in the other direction. We do not require any\ncontrol on these other than the total number of changes is at least s.\n\nOn constants: We use C and c, and their modi\ufb01cations, as unspeci\ufb01ed constants that may change\nfrom line to line. While these can be explicitly bounded, we do not expect them to be tight.\n\n2 Community Goodness-of-Fit\n\nWe begin by stating our main results regarding the community goodness-of-\ufb01t problem.\nTheorem 1. Community goodness-of-\ufb01t testing is possible with risk RGoF \u2264 \u03b4 if s\u039b \u2265 C log(2/\u03b4)\nand \u039b \u2265 C\nConversely, in order to attain RGoF \u2264 \u03b4 \u2264 0.25, we must have that s\u039b \u2265 C \u2032 log(1/\u03b4) and \u039b \u2265\n\nn\ns2 log(2/\u03b4) for some constant C > 0.\n\nC \u2032 log(cid:16)1 +\n\nn\n\ns2(cid:17) for some constant C \u2032 > 0.\n\nThese bounds reveal the following behavior in terms of large and small changes:\n\n\u2022 For large changes (s \u2265 n1/2+c for some c > 0), since n/s2 \u2264 1 and log(1 + x) \u2265 x/2 for\nx \u2264 1, the second converse bound behaves as \u039b \u2265 Cn/s2, matching the suf\ufb01cient condition up\n\u2022 For small changes (s \u2264 n1/2\u2212c for some c > 0), since n/s2 \u223c n2c, the second converse bound\ninstead behaves as \u039b & log n. In this regime, community recovery up to s/2 errors requires\n\nto a constant.\n\n4\n\n\fconstants.\n\n\u039b \u2265 C log 2n/s = \u02dcC log n. Thus, estimating x from G and comparing it to x0 is optimal up to\n\u2022 The above indicate a phase transition in the GoF testing problem at \u03c3 := logn(s) = 1/2. Consider\nthe thermodynamic limit of n \u0580 \u221e. For \u03c3 < 1/2, the problem is \u2018hard\u2019 in that the SNR \u039b is\nrequired to diverge to \u221e, while for \u03c3 > 1/2, the SNR can tend to zero.\n\nProof Sketch for the Achievability. Let us begin with an intuitive development of the test. Since we\nstart with a partition x0 in hand to test, it is natural to look at the edges across and within the cut\nde\ufb01ned by x0. We thus de\ufb01ne the number of edgess across and within this cut:\n\nN x0\na (G) := |{(i, j) \u2208 G : x0,i 6= x0,j}| =\nN x0\nw (G) := |{(i, j) \u2208 G : x0,i = x0,j}| =\n\n1\n4\n1\n4\n\nxT\n0 (D(G) \u2212 G)x0\nxT\n0 (D(G) + G)x0\n\n(3)\n\nwhere the latter expressions treat G as an adjacency matrix and D(G) = diag(degree(i))i\u2208[1:n]. 2 In\n\nthe null case, these are respectively Bin(n2/4, b/n) and Bin(2(cid:0)n/2\n\nin the alternate case some s/2 \u00b7 (n \u2212 s)/2 of each behave like edges of the opposite polarity (i.e. as\nb/n instead of a/n and vice versa), leading to a excess/de\ufb01cit of edges of this type. Note that while\nthe \u2018average signal strength\u2019, i.e., the amount by which edges are over- or underrepresented is the\nsame in both cases (\u223c s|a \u2212 b|), the group with the larger null parameter suffers greater \ufb02uctuations.\nThus, we base our test only on edges of smaller bias. This reduces the SNR by at most a factor of 4.\n\n2 (cid:1), a/n) random variables, while\n\nWe now de\ufb01ne the test. C1 below is the constant implicit in Lemma 3 in Appendix A.1.\n\n\u2022 If a > b, we use the test N x0\n\na (G)\n\n\u2022 If b > a, we use the test N x0\n\nw (G)\n\nH1\n\u2277\nH0\nH1\n\u2277\nH0\n\nbn\n4\nan\n4 \u2212\n\n+ C1 max(cid:16)pnb log(1/\u03b4), log(1/\u03b4)(cid:17) .\n+ C1 max(cid:16)pna log(1/\u03b4), log(1/\u03b4)(cid:17) .\n\na\n2\n\nThe risks of these tests can be controlled by separating the null and alternate ranges using Bernstein\u2019s\ninequality. Indeed, the threshold above is just the the expectation plus the concentration radius of the\nstatistic under the null distribution. Let us brie\ufb02y develop the statistic\u2019s behaviour in the alternate -\n\ncontinue to behave like Bern(b/n) bits, while the remaining s(n\u2212 s)/2 edges behave as Bern(a/n)\nbits. Thus, the expectation of N x0\n2n \u2265 s(a\u2212b)/4.\n\n2(cid:1) of the edges in N x0\nconsidering only the case a > b, we \ufb01nd that under the alternate,(cid:0)n\u2212s\nNext, Bernstein\u2019s inequality controls the \ufb02uctuations at scalepmax(nb, s(a \u2212 b)) log(2/\u03b4). The\n\nconclusion is straightforward to draw from here, and the proof is carried out in Appendix A.13.\n\na is increased by an amount greater than s(n\u2212s) a\u2212b\n\n2 (cid:1) +(cid:0)s\n\na\n\nProof Sketch for the Converse. The proof is relegated to Appendix A.2, and we discuss the strategy\nhere. The converse proof follows Le Cam\u2019s method, which lower bounds the minimax risk by the\nBayes risk for conveniently chosen priors - which can be expressed using the TV distance.\nTo show \u039b & log(1 + n/s2), we pick the null x0 to be any balanced community, and choose the\nuniform prior on communities that are exactly s-far from x0 (in fact, we only use a subset of these in\norder to facilitate easier computations). This is an obvious choice for this setting - we are interested\nin balanced communities that are at least s far, and choosing a large number of them allows for a\ngreater \u2018confusion\u2019 in the testing problem due to a richer alternate hypothesis. The bound follows\nby invoking inequalities between TV and \u03c72 divergences and a lengthy calculation due to the com-\nbinatorial objects involved.\nTo show s\u039b & \u2212 log(\u03b4), we again pick the null to be any balanced community, and pick the alternate\nto be an s-far singleton. We then proceed to control dTV by the Hellinger divergence.\n\n3 Two-Sample Testing\n\nWe again begin with the main results on community two-sample testing problem.\n\n2Note that D(G) \u2212 G is the Laplacian of the graph.\n3The same also describes the extention of the claims to linearly sized communities\n\n5\n\n\f1\n\n2 +\u03b3 . There exist constants C, C \u2032 such that if C \u2032 \u2264\nTheorem 2. Assume, for some \u03b3 > 0, s \u2265 n\na, b \u2264 (n/2)1/3, then two-sample testing of s changes with RTST \u2264 4/n is possible if the SNR\nsatis\ufb01es \u039b \u2265 C.\nConversely, for n \u2265 200, there exist constants c, c\u2032 such that if s < ( 1\n2 \u2212 c\u2032)n, then two-sample\ntesting of s changes cannot be carried out with RTST \u2264 1/4 unless \u039b \u2265 c.\nLarge Changes. The above theorem makes an achievability claim for the setting of large changes.\nNotice that in this regime the stated upper and lower bounds match up to constants. Speci\ufb01cally, if\n2 +\u03b3 < s < ( 1\n2 \u2212 c\u2032)n, two-sample testing can be solved iff \u039b & 1. Further, the condition a, b & 1\nn\nis also tight, as it follows from a/b = \u0398(1), and the necessary condition \u039b & 1, since \u039b \u2264 a + b.\nThis leaves the condition max(a, b) \u2264 (n/2)1/3, which we suspect is an artifact of the proof\ntechnique and conjecture that, even for our proposed test, it can be removed. In any case, observe\nthat this condition is irrelevant in the setting a, b = O(log n) considered in this paper. Further, if\na/b is bounded away from 1, then TST is directly possible when a, b = \u2126(log n) by recovering the\ncommunities and comparing them, demonstrating that this condition is not present in general.\n\n1\n\n1\n\n2 \u2212\u03b3 for some \u03b3 > 0 - the na\u00efve scheme of\nSmall Changes. We claim that for small changes - s < n\nrecovering the communities and comparing them is minimax. To see this, note that that GoF testing\nis reducible to TST - given a TST scheme of a known risk, one may construct a GoF tester of that\nrisk by feeding the TST algorithm the observed graph and a graph drawn from P (\u00b7|x0). Thus, the\nlower bounds of Theorem 1 apply to TST, and for a/b = \u0398(1), we \ufb01nd that it is necessary that\ns\u039b = \u03c9(1) and that \u039b & log(1 + n/s2) to attain vanishing RTST. For small s, the latter lower\nbound is \u2126(log n), the claim follows since recovery with up to s errors is possible if \u039b & log n.\nEf\ufb01ciency. Finally, we point out that the above bounds can be attained with computationally ef\ufb01cient\ntests. Further, for large changes, the test can be made agnostic to knowledge of (a, b). Instead, it\n\ncan be done by simply counting the number of edges in the graphs.\n\nonly requires one to be able to estimate n(a+b) to within an additive error of eO(pn(a + b)), which\n\nProof Sketch of the Achievability. We describe the proposed test, and sketch its risk analysis below,\ncompleting the same in Appendix B.1. Recall the de\ufb01nition of N z\n\na from (3) in \u00a72, and let\n\nw, N z\n\nT \u02c6x(G) := N \u02c6x\n\nw(G) \u2212 N \u02c6x\n\na (G).\n\n(4)\n\nWe show that the routine \u2018TwoSampleTester\u2019 below attains a risk smaller than 4/n. In words, the test\ncomputes a partition \u02c6x for the graph G by using about half the edges in the graph. This is represented\nin the \u2018PartialRecovery\u2019 step below, for which any such method may be used - concretely, that of\n[CRV15]. Next, we compute the statistic T \u02c6x above for both the remaining part of the \ufb01rst graph, and\nfor the second graph. Notice that unlike the GoF\nstatistic, which was only Na, T \u02c6x takes the difference\nof Na and Nw. This is necessary because the partition\n\u02c6x derived from partial recovery cannot be very well\ncorrelated with the true partition x. This means the\nreduced \ufb02uctuations from only considering one part\ndoes not apply, and we instead use the whole cut.\n\nAlgorithm 1: TwoSampleTester(G, H, \u03b4)\n1: G1 \u2190 subsampling of edges of G at\n\nrate 1/2 uniformly at random.\n\n2: eG \u2190 G \u2212 G1.\n3: bx \u2190 PartialRecovery(G1).\n4: Compute T bx(eG), T bx(H).\n5: T \u2190 |2T bx(eG) \u2212 T bx(H)|.\n\n6: Return T\n\nH1\n\u2277\n\nSince the edges within communities, and across com-\nmunities in the graph are (separately) exchangable,\nthe errors made in \u02c6x distribute uniformly over the two\ncommunities4. This allows us to explicitly control the\nbehaviour of T as de\ufb01ned in the test provided \u02c6x is non-\ntrivially correlatd with x - i.e., given that it makes\n< (1/2 \u2212 c)n errors for some c > 0. The condition \u039b & 1 in the theorem arises from this.\nrecovered community \u02c6x. This is handled in the analysis by introducing an independent copy of G,\n\nA complication in this strategy is that the remaining graph eG in the scheme is not independent of the\ncalled G\u2032, and arguing that T \u02c6x(eG) \u2248 1/2T \u02c6x(G\u2032). This step is the origin of the nuisance condition\n\nH0pCn(a + b) log(6n).\n\nmax(a, b) . n1/3 in the theorem.\n\n4For a proof: since x, \u2212x induce the same law, and since the communities are balanced, for every real-\nerrors in the community +, \u2212 respectively, there is a realization of\n, e+ errors. Further, within community exchangability implies that errors\n\nization of G such that \u02c6x makes e+, e\nequal probability where it makes e\ndistribute uniformly.\n\n\u2212\n\n\u2212\n\n6\n\n\fLastly, we point out that while the above exploits the exact balance by using the description of the\nerror distribution it enables, one can derive the same results (but with weakened constants) even\nwithout this assumption, so long as both communities are at of size linear in n. In this case, one can-\nnot rely on the errors distributing uniformly over the nodes, but the within-community uniformity\nof errors, which follows due to within community exchangability, can be exploited in a similar way.\nWe describe this extension in Appendix B.1.1.\n\nProof Sketch of the Converse. The necessary condition is shown via Le Cam\u2019s method, but with\nthe twist that the null model is chosen to be a two-step procedure - one that draws a balanced com-\nmunity uniformly at random, and then generates a graph according to it, while the alternate models\nare drawn uniformly from the balanced communities that are at least s-far from the chosen null.\nThis allows a comparison to the unstructured Erd\u02ddos-R\u00e9nyi graph on n vertices with mean degree\n(a + b)/2. Bounds can then be drawn in from the study of the so-called distinguishability problem\n[Ban+16], and we invoke results from [WX18] to show that total variation distance between the null\nand alternate distributions is small when \u039b is a small enough constant, allowing us to conclude using\nNeyman-Pearson. See Appendix B.3 for a detailed argument.\n\n4 Experiments\n\nWe perform three different sets of numerical experiments. We \ufb01rst run our tests on SBMs with\n1000 nodes. Next, we demonstrate that our tests perform similarly for a real dataset, speci\ufb01cally\nthe Political Blogs dataset [AG05]. Finally, we examine SBM-supported Gaussian Markov Random\nFields (GMRFs) as an example of a \u201cnode observation\u201d model, where the SBM-generated edges\nform the precision matrix for the Gaussian vector consisting of the random variables assigned to\neach node. In particular, we need to determine if the underlying community of the graph has changed\nwithout explicitly observing (or recovering) the edges of the graph. For the sake of brevity, precise\ndetails of the experiments are moved to Appendix C.\n\n4.1 SBM Experiments\n\nWe perform experiments implementing our GoF and TST strategies as well as the na\u00efve scheme of\nreconstructing communities and comparing. Recovery is performed by regularised spectral cluster-\ning, for which a detailed description is given in Appendix C.1. The graphs are drawn on n = 1000\nnodes for a range of (s, \u039b) pairs and the high and low risk regimes are plotted in Figure 1. First,\n\nnote that for \u2018large changes,\u2019 s \u2265pn log(10) \u2248 50, our GoF and TST tests can succeed for lower\nSNR values. In contrast, for \u2018small changes,\u2019 s < \u221an \u2248 30, the na\u00efve test is more powerful in the\n\nhigh SNR regime. Additionally, both tests fail for TST unless the SNR is larger than a constant, as\npredicted by our lower bound in Theorem 2.\n\nRisk against (s, \u039b) for GoF tests\n\nProposed Scheme\nNa\u00a8\u0131ve Scheme\nBoth low risk\nBoth fail\n\n250\n\n200\n\n150\n\n100\n\n50\n\ns\n\n,\n\nn\no\ni\nt\nr\no\nt\ns\ni\nD\n\n250\n\n200\n\n150\n\n100\n\n50\n\ns\n\n,\n\nn\no\ni\nt\nr\no\nt\ns\ni\nD\n\nRisk Against (s, \u039b) for TST\n\nProposed Scheme\nNa\u00a8\u0131ve Scheme\nBoth low risk\nBoth fail\n\n1\n\n2\n\n3\n\n4\n\n5\n\n6\n\n7\n\n8\n\n9\n\n10\n\n1\n\n2\n\n3\n\n4\n\n5\n\n6\n\n7\n\n8\n\n9\n\n10\n\n\u039b/\u039b0\n\n\u039b/\u039b0\n\nFigure 1: Risks of the proposed tests from sections 2 and 3 for GoF and TST respectively, and the\nperformance of the na\u00efve scheme, on synthetic SBMs with n = 1000, a/b = 3. Both schemes attain\nhigh risk (> 1\u2212 \u03b4) in the grey region, intermediate risk in the white, and the colours indicate which\nof the schema attain low risk (< \u03b4), where \u03b4 = 0.01 for GoF and \u03b4 = 0.1 for TST.\n\n7\n\n\f4.2 Political Blogs Dataset [AG05]\n\nThe political blogs dataset [AG05] is canonical in the study of community detection, and consists of\nn = 1222 nodes. Here, we vary the effective SNR by randomly subsampling the edges of the graphs\nat rate \u03c1. See Appendix C.2 for further details. In this dataset, the ground truth partition xTrue is\navailable, which in turn yields accurate estimates of the connectivity probabilities (a, b). For this\ngraph a/b \u2248 10. Further, spectral clustering alone incurs \u2248 50 errors in this graph, which is larger\nthan \u221a1222 \u2248 35. As a consequence, the behaviour in the \u2018small changes\u2019 regime where the test\nrelies on recovery - is not well illustrated in the following.\n\nPolitical blogs: Risk against (s, \u03c1) for GoF\n\n250\n\nPolitical blogs: Risk against (s, \u03c1) for TST\n\n250\n\ns\n\n,\n\nn\no\ni\nt\nr\no\nt\ns\ni\nD\n\n200\n\n150\n\n100\n\n50\n\n0\n0.0\n\nProposed Scheme\nNa\u00a8\u0131ve Scheme\nBoth low risk\nBoth fail\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\nSparsity factor, \u03c1\n\ns\n\n,\n\nn\no\ni\nt\nr\no\nt\ns\ni\nD\n\n200\n\n150\n\n100\n\n50\n\n0\n0.0\n\nProposed Scheme\nNa\u00a8\u0131ve Scheme\nBoth low risk\nBoth fail\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\nSparsity factor, \u03c1\n\nFigure 2: Risks of the tests applied to the Political Blogs graphs - colour scheme is retained from\nFig. 1. The X-axis plots the sparsi\ufb01cation factor, which serves as a proxy for SNR. Features similar\nto Fig. 1 can be seen. The GoF plot improves since a/b is bigger, while the TST plot suffers since\nthe political blogs graph is not completely described as a 2-community SBM [Lei16].\n\nGoodness-of-Fit. We determine the size of the test by running the GoF procedures against xTrue.\nTo determine power, we construct a partition y by relabelling a random set of nodes of size s, and\nrunning the GoF procedures against y with the same graph.\nTwo-Sample Testing. We compare the political blogs graph G against two other graphs drawn from\nSBMs. Size is detemined by drawing G\u2032 according to an SBM of community xTrue and running\nthe TST procedure, and power is determined by drawing a y as above, generating H according\nto an SBM of community y, and running the TST procedure. Note that this experiment is thus\nsemi-synthetic.\n\n4.3 Gaussian Markov Random Fields (GMRFs)\n\nFrequently instead of simply receiving a graph, one receives i.i.d. samples from a graph-structured\ndistribution, and it is of interest to be able to cluster nodes with respect to the latent graph. For\nexample, in large-scale calcium imaging, it is possible to simultaneously record the activity pattern\nof thousands of neurons, but not their underlying synaptic connectivity [Pne+16]. Here, we explore\nthe behavior of our tests for GMRFs where the underlying graph structure is randomly drawn from\nan SBM and and we only observe the nodes.\n\nA heuristic reason for why our methods might succeed in such a situation arises from the local tree-\nlike property of sparse random graphs (see, e.g. [DM10]). For graphs with mean degree d \u226a n,\ntypical nodes do not lie in cycles shorter than \u223c log n\n2 log d . In MRFs, this tree-like property induces\ncorrelation decay: the correlation between two nodes decays geometrically up to graph-distance\n2 log d . Thus, the covariance matrix closely approximates \u03c31G +Pk\n\u223c log n\ni=2(\u03c31G)i + \u03c3011T for some\n\u03c30 \u226a \u03c31, small k, and G, the adjacency matrix of the graph. Since the local structure of the graph is\nso expressed, both clustering and testing applied directly to the covariance matrix should be viable.\n\nWe report experimentation on the GMRF (see, e.g. [WJ08, Ch. 3]), which comprises random vectors\n\n\u03b6 \u223c N (0, \u0398\u22121), where the non-zero entries of the precision matrix \u0398 encode the conditional depen-\ndence structure of \u03b6. Following standard parametrisations [WWR10], we set \u0398 = I + \u03b3G, where\nG \u223c P (G|x) is an adjacency matrix from an SBM with latent parameter x, and \u03b3 is a scalar. Below,\nwe \ufb01x the SBM parameters a, b and the level \u03b3, and explore risks against s and sample size t.\n\n8\n\n\fFollowing the above heuristic, we na\u00efvely adapt community recovery and testing to this setting, by\nreplacing all instances of the graph adjacency matrix in previous settings with the sample covariance\nmatrix. Figure 3 presents our simulations of the risk of this test when n = 1000, and (a, b) \u2248\n(12.3 log n, 1.23 log n), at \u039b \u2248 9 log(n) (for details see Appx. C.3). This large SNR is chosen\nso that community recovery would be easy if the graph was recovered;5 this emphasizes the role\nof the sample size, t. Importantly, in this implementation, the threshold for rejecting the null has\nbeen \ufb01t using data (unlike in the previous sections). This is since we lack a rigorous theoretical\nunderstanding of this problem, and have not analytically derived expressions for the thresholds. As\na result, these plots should be treated as speculative research intended to underscore the presence of\ninteresting testing effects in this scenario, and to encourage future work along these lines.\n\nRisk Against (s, t) for GoF in GMRFs\n\nProposed Scheme\nSpec. Clustering\nBoth low risk\nBoth fail\n\n250\n\n200\n\n150\n\n100\n\n50\n\ns\n\n,\n\nn\no\ni\nt\nr\no\nt\ns\ni\nD\n\nRisk Against (s, t) for TST in GMRFs\n\nProposed Scheme\nSpec. Clustering\nBoth low risk\nBoth fail\n\n250\n\n200\n\n150\n\n100\n\n50\n\ns\n\n,\n\nn\no\ni\nt\nr\no\nt\ns\ni\nD\n\n1000\n\n2000\n\n3000\n\n4000\n\n5000\n\n1000\n\n2000\n\n3000\n\n4000\n\n5000\n\nSample size, t\n\nSample size, t\n\nFigure 3: Risks for adaptation of our tests to GMRFs - colour scheme is retained from Fig. 1. The\nplots show structural similarity to Fig. 1, but with two differences - In GoF, we don\u2019t \ufb01nd a high\nrisk region at the sample sizes considered, and the proposed scheme always outperforms the Na\u00efve\nscheme based on spectral clustering.\n\n5 Directions for Future Work\n\nThe development of the recovery problem for SBMs suggests a number of directions for further work\non the testing problems considered above. For instance, one may investigate the exact constants in\nthe testing threshold that the above work suggests, or one may study the testing problem for SBMs\nwith k > 2 communities, which is a practically relevant setting since many real-world networks are\nsigni\ufb01cantly better described as k-SBMs than as 2-SBMs. In the latter vein, testing problems such\nas the above may be studied in richer random graph models, such as degree corrected SBMs, or\ngeometric block models. Additionally, testing of strongly imbalanced communities, where one of\nthe communities has size sublinear in n is conceptually unexplored and of interest.\n\nOne open problem that draws from the above exposition is if there exists an algorithm for TST in\nthe 2 community setting that does not pass through a partial recovery step and yet works for sparse\ngraphs. We expect that such a method would be necessary for determining exact testing thresh-\nolds (for large changes), since the recovery step neccessarily requires some subsampling, which\nreduces the effective SNR available for testing. In addition, this would be conceptually pleasant,\nand would eliminate the dissonance in the above work where showing testing guarantees requires\npassing through recovery guarantees. Such a scheme would also more generally allow study of the\ntesting problem for situations where partial recovery is ill understood.\n\nFinally, we mention that more work is needed on the practical investigation of the effectiveness of the\nabove methods - while the experiments we have run validate the theory, the real-world applicability\nof the methods above require deeper experimentation. A signi\ufb01cant lacuna for this line is the lack\nof a good real-world dataset for the testing of commumity changes.\n\n5Note, however, we expect graph recovery to be impossible at these sample sizes. 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In: The Annals of Statistics 44.5 (2016), pp. 2252\u20132280.\n\n12\n\n\f", "award": [], "sourceid": 5460, "authors": [{"given_name": "Aditya", "family_name": "Gangrade", "institution": "Boston University"}, {"given_name": "Praveen", "family_name": "Venkatesh", "institution": "Carnegie Mellon University"}, {"given_name": "Bobak", "family_name": "Nazer", "institution": "Boston University"}, {"given_name": "Venkatesh", "family_name": "Saligrama", "institution": "Boston University"}]}