{"title": "Confidence Sets for Network Structure", "book": "Advances in Neural Information Processing Systems", "page_first": 2097, "page_last": 2105, "abstract": "Latent variable models are frequently used to identify structure in dichotomous network data, in part because they give rise to a Bernoulli product likelihood that is both well understood and consistent with the notion of exchangeable random graphs.  In this article we propose conservative confidence sets that hold with respect to these underlying Bernoulli parameters as a function of any given partition of network nodes, enabling us to assess estimates of \\emph{residual} network structure, that is, structure that cannot be explained by known covariates and thus cannot be easily verified by manual inspection. We demonstrate the proposed methodology by analyzing student friendship networks from the National Longitudinal Survey of Adolescent Health that include race, gender, and school year as covariates.  We employ a stochastic expectation-maximization algorithm to fit a logistic regression model that includes these explanatory variables as well as a latent stochastic blockmodel component and additional node-specific effects.  Although maximum-likelihood estimates do not appear consistent in this context, we are able to evaluate confidence sets as a function of different blockmodel partitions, which enables us to qualitatively assess the significance of estimated residual network structure relative to a baseline, which models covariates but lacks block structure.", "full_text": "Con\ufb01dence Sets for Network Structure\n\nDavid S. Choi\n\nHarvard University\n\nCambridge, MA 02138\n\nPatrick Wolfe\n\nHarvard University\n\nCambridge, MA 02138\n\nSchool of Engineering and Applied Sciences\n\nSchool of Engineering and Applied Sciences\n\ndchoi@seas.harvard.edu\n\npatrick@seas.harvard.edu\n\nEdoardo M. Airoldi\nDepartment of Statistics\n\nHarvard University\n\nCambridge, MA 02138\n\nairoldi@fas.harvard.edu\n\nAbstract\n\nLatent variable models are frequently used to identify structure in dichotomous\nnetwork data, in part because they give rise to a Bernoulli product likelihood that\nis both well understood and consistent with the notion of exchangeable random\ngraphs. In this article we propose conservative con\ufb01dence sets that hold with re-\nspect to these underlying Bernoulli parameters as a function of any given partition\nof network nodes, enabling us to assess estimates of residual network structure,\nthat is, structure that cannot be explained by known covariates and thus cannot be\neasily veri\ufb01ed by manual inspection. We demonstrate the proposed methodology\nby analyzing student friendship networks from the National Longitudinal Survey\nof Adolescent Health that include race, gender, and school year as covariates. We\nemploy a stochastic expectation-maximization algorithm to \ufb01t a logistic regres-\nsion model that includes these explanatory variables as well as a latent stochastic\nblockmodel component and additional node-speci\ufb01c effects. Although maximum-\nlikelihood estimates do not appear consistent in this context, we are able to evalu-\nate con\ufb01dence sets as a function of different blockmodel partitions, which enables\nus to qualitatively assess the signi\ufb01cance of estimated residual network structure\nrelative to a baseline, which models covariates but lacks block structure.\n\nIntroduction\n\n1\nNetwork datasets comprising edge measurements Aij \u2208 {0, 1} of a binary, symmetric, and anti-\nre\ufb02exive relation on a set of n nodes, 1 \u2264 i < j \u2264 n, are fast becoming of paramount interest in the\nstatistical analysis and data mining literatures [1]. A common aim of many models for such data is\nto test for and explain the presence of network structure, primary examples being communities and\nblocks of nodes that are equivalent in some formal sense. Algorithmic formulations of this problem\ntake varied forms and span many literatures, touching on subjects such as statistical physics [2, 3],\ntheoretical computer science [4], economics [5], and social network analysis [6].\nOne popular modeling assumption for network data is to assume dyadic independence of the edge\nmeasurements when conditioned on a set of latent variables [7, 8, 9, 10]. The number of latent\nparameters in such models generally increases with the size of the graph, however, meaning that\ncomputationally intensive \ufb01tting algorithms may be required and standard consistency results may\nnot always hold. As a result, it can often be dif\ufb01cult to assess statistical signi\ufb01cance or quantify\nthe uncertainty associated with parameter estimates. This issue is evident in literatures focused\n\n1\n\n\fon community detection, where common practice is to examine whether algorithmically identi\ufb01ed\ncommunities agree with prior knowledge or intuition [11, 12]; this practice is less useful if additional\ncon\ufb01rmatory information is unavailable, or if detailed uncertainty quanti\ufb01cation is desired.\nCon\ufb01dence sets are a standard statistical tool for uncertainty quanti\ufb01cation, but they are not yet\nwell developed for network data. In this paper, we propose a family of con\ufb01dence sets for net-\nwork structure that apply under the assumption of a Bernoulli product likelihood. The form of\nthese sets stems from a stochastic blockmodel formulation which re\ufb02ects the notion of latent nodal\nclasses, and they provide a new tool for the analysis of estimated or algorithmically determined\nnetwork structure. We demonstrate usage of the con\ufb01dence sets by analyzing a sample of 26 ado-\nlescent friendship networks from the National Longitudinal Survey of Adolescent Health (available\nat http://www.cpc.unc.edu/addhealth), using a baseline model that only includes explanatory covari-\nates and heterogeneity in the nodal degrees. We employ these con\ufb01dence sets to validate departures\nfrom this baseline model taking the form of residual community structure. Though the con\ufb01dence\nsets we employ are conservative, we show that they are effective in identifying putative residual\nstructure in these friendship network data.\n\n2 Model Speci\ufb01cation and Inference\nWe represent network data via a sociomatrix A \u2208 {0, 1}N\u00d7N that re\ufb02ects the adjacency structure\nof a simple, undirected graph on N nodes. In keeping with the latent variable network analysis\nliterature, we assume entries {Aij} for i < j to be independent Bernoulli random variables with\nassociated success probabilities {Pij}i<j, and complete A as a symmetric matrix with zeros along\nits main diagonal. The corresponding data log-likelihood is given by\n\nL(A; P ) =\n\nAij log(Pij) + (1 \u2212 Aij) log(1 \u2212 Pij),\n\n(1)\n\n(cid:88)\n\ni<j\n\nwhere each Pij can itself be modeled as a function of latent as well as explanatory variables.\nGiven an instantiation of A and a latent variable model for the probabilities {Pij}i<j, it is natural\nto seek a quanti\ufb01cation of the uncertainty associated with estimates of these Bernoulli parameters.\nA standard approach in non-network settings is to posit a parametric model and then compute con\ufb01-\ndence intervals, for example by appealing to standard maximum-likelihood asymptotics. However,\nas mentioned earlier, the formulation of most latent variable network models dictates an increas-\ning number of parameters as the number of network nodes grows; this amount of expressive power\nappears necessary to capture many salient characteristics of network data. As a result, standard\nasymptotic results do not necessarily apply, leaving open questions for inference.\n\n2.1 A Logistic Regression Model for Network Structure\n\nTo illustrate the complexities that can arise in this inferential setting, we adopt a latent variable\nnetwork model with a standard \ufb02avor: a logistic regression model that simultaneously incorporates\naspects of blockmodels, additive effects, and explanatory variables (see [10] for a more general for-\nmulation). Speci\ufb01cally, we incorporate a K-class stochastic blockmodel component parameterized\nin terms of a symmetric matrix \u0398 \u2208 RK\u00d7K and a membership vector z \u2208 {1, . . . , K}N whose val-\nues denote the class of each node, with Pij depending on \u0398zizj . A vector of additional node-speci\ufb01c\nlatent variables \u03b1 is included to account for heterogeneity in the observed nodal degrees, along with\na vector of regression coef\ufb01cients \u03b2 corresponding to explanatory variables x(i, j). Thus we obtain\nthe log-odds parameterization\n\nwhere we further enforce the identi\ufb01ability constraint that(cid:80)\n\nPij\n1 \u2212 Pij\n\n= \u0398zizj + \u03b1i + \u03b1j + x(i, j)(cid:48)\u03b2,\n\nlog\n\ni \u03b1i = 0.\n\n(2)\n\n2.2 Likelihood-Based Inference\n\nExact maximization of the log-likelihood L(A; z, \u0398, \u03b1, \u03b2, x) is computationally demanding even for\nmoderately large K and N, owing to the total number of nodal partitions induced by z. Algorithm 1\ndetails a stochastic expectation-maximization (EM) algorithm to explore the likelihood space.\n\n2\n\n\fAlgorithm 1 Stochastic Expectation-Maximization Fitting of model (2)\n\n1. Set t = 0 and initialize (z(0), \u0398(0), \u03b1(0), \u03b2(0)).\n2. For iteration t, do:\n\nE-step Sample z(t) \u221d exp{L(z | A; \u0398(t), \u03b1(t), \u03b2(t), x)}\nM-step Set (\u0398(t), \u03b1(t), \u03b2(t)) = argmax\u0398,\u03b1,\u03b2 L(\u0398, \u03b1, \u03b2 | z(t); A, x)\n\n(e.g., via Gibbs sampling)\n\n(convex optimization)\n\n3. Set t \u2190 t + 1 and return to Step 2.\n\nWhen \u03b1 and \u03b2 are \ufb01xed to zero, model (2) reduces to a re-parameterization of the standard stochas-\ntic blockmodel. Consistency results for this model have been developed for a range of condi-\ntions [7, 13, 14, 15, 16]. However, it is not clear how uncertainty in z and \u0398 should be quanti\ufb01ed\nor even concisely expressed: in this vein, previous efforts to assess the robustness of \ufb01tted struc-\nture include [17], in which community partitions are analyzed under perturbations of the network,\nand [18], in which the behavior of local minima resulting from simulated annealing is examined; a\nlikelihood-based test is proposed in [19] to compare sets of community divisions.\nWithout the blockmodel components z and \u0398, the model of Eq. (2) reduces to a generalized lin-\near model whose likelihood can be maximized by standard methods. If \u03b1 is further constrained\nto equal 0, the model is \ufb01nite dimensional and standard asymptotic results for inference can be\napplied. Otherwise, the increasing dimensionality of \u03b1 brings consistency into question, and in\nfact certain negative results are known for a related model, known as the p1 exponential random\ngraph model [20]. Speci\ufb01cally, [21] reports that the maximum likelihood estimator for the p1 model\nexhibits bias with magnitude equal to its variance. Although estimation error does converge asymp-\ntotically to zero for the p1 model, it is not known how to generate general con\ufb01dence intervals or\nhypothesis tests; [22] prescribes reporting standard errors only as summary statistics, with no asso-\nciation to p-values. The predictions of [21] were replicated (reported below) when \ufb01tting simulated\ndata drawn from the model of Eq. (2) with parameters matched to observed characteristics of the\nAdolescent Health friendship networks.\nModel selection techniques, such as out-of-sample prediction, are sometimes used to validate sta-\ntistical network models. For example, [23] uses out-of-sample prediction to compare the stochastic\nblockmodel to other network models. We note that model selection techniques and the con\ufb01dence\nestimates presented here are complementary. To choose the best model for the data, a model selec-\ntion method should be used; however, if the parameter will be interpreted to draw conclusions about\nthe data, a con\ufb01dence estimate may be desired as well.\n\n2.3 Con\ufb01dence Sets for Network Structure\n\nInstead of quantifying the uncertainty associated with estimates of the model parameters\n(z, \u0398, \u03b1, \u03b2), we directly \ufb01nd con\ufb01dence sets for the Bernoulli likelihood parameters {Pij}i<j. To\nthis end, for any \ufb01xed K and class assignment z, de\ufb01ne symmetric matrices \u00af\u03a6, \u02c6\u03a6 in [0, 1]K\u00d7K\nelement-wise for 1 \u2264 a \u2264 b as\n\n\u00af\u03a6(z)\n\nab =\n\n1\nnab\n\nPij 1{zi = a, zj = b},\n\n\u02c6\u03a6(z)\n\nab =\n\n1\nnab\n\nAij 1{zi = a, zj = b},\n\nwith nab denoting the maximum number of possible edges between classes a and b (i.e., the cor-\nresponding number of Bernoulli trials). Thus \u00af\u03a6(z)\nab is the expected proportion of edges between (or\nwithin, if a = b) classes a and b, under class assignment z, and \u02c6\u03a6(z)\nab is its corresponding sample\nproportion estimator.\nIntuitively, \u00af\u03a6(z) measures assortativity by z; whenever the sociomatrix A is unstructured, elements\nof \u00af\u03a6(z) should be nearly uniform for any choice of partition z. When strong community structure is\npresent in A, however, these elements should instead be well separated for corresponding values of\nz. Thus, it is of interest to examine a con\ufb01dence set that relates \u02c6\u03a6(z)\nab to its expected value \u00af\u03a6(z) for\na range of partitions z. To this end, we may de\ufb01ne such a set by considering a weighted sum of the\n\n3\n\n(cid:88)\n\ni<j\n\n(cid:88)\n\ni<j\n\n\fElement of \u03b2\nIntercept\nGender\nRace\nGrade\n\n\u0398 = 0\n\u0398 = 0, \u03b1 = 0\n\u22120.001 (0.004)\n2.26 (0.070)\n0.003 (0.004) \u22120.005 (0.004)\n\u22120.03 (0.005)\n\u22120.001 (0.004)\n0.006 (0.003)\n0.04 (0.003)\n\nTable 1: Empirical bias (with standard errors) of ML-estimated components of \u03b2 under a baseline\nmodel, for the cases \u03b1 = 0 versus \u03b1 unconstrained. Note the change in estimated bias when \u03b1 is\nincluded in the model.\n\nform(cid:80)\nTheorem 1 ([14]) Let {Aij}i<j be comprised of(cid:0)N\n\nab ), where D(p||p(cid:48)) = p log(p/p(cid:48))+(1\u2212p) log[(1\u2212p)/(1\u2212p(cid:48))] denotes\nthe (nonnegative) Kullback\u2013Leibler divergence of a Bernoulli random variable with parameter p(cid:48)\nfrom that of one with parameter p. A con\ufb01dence set is then obtainable via direct application of the\nfollowing theorem.\n\n(cid:1) independent Bernoulli(Pij) trials, and let\n\na\u2264b nabD( \u02c6\u03a6(z)\n\nab || \u00af\u03a6(z)\n\nZ = {1, . . . , K}N . Then with probability at least 1 \u2212 \u03b4,\n\n2\n\n(cid:19)\n\n(cid:18) N\n\nK\n\n(cid:88)\n\na\u2264b\n\nsup\nz\u2208Z\n\nnabD( \u02c6\u03a6(z)\n\nab || \u00af\u03a6(z)\n\nab ) \u2264 N log K + (K 2 + K) log\n\n+ 1\n\n+ log\n\n1\n\u03b4\n\n.\n\n(3)\n\nBecause Eq. (3) holds uniformly over all class assignments, we may choose to apply it directly to the\nvalue of z obtained from Algorithm 1\u2014and because it does not assume any particular form of latent\nstructure, we are able to avoid the dif\ufb01culties associated with determining con\ufb01dence sets directly\nfor the parameters of latent variable models such as Eq. (2). However, it is important to note that\nthis generality comes at a price: In simulation studies undertaken in [14] as well as those detailed\nbelow, the bound of Eq. (3) is observed to be loose by a multiplicative factor ranging from 3 to 7 on\naverage.\n\n2.4 Estimator Consistency and Con\ufb01dence Sets\n\nRecalling our above discussion of estimator consistency for the related p1 model, we undertook\na small simulation study to investigate the consistency of maximum-likelihood (ML) estimation\nin a \u201cbaseline\u201d version of model (2) with K = 1 and the corresponding (scalar) value of \u0398\nset equal to zero. We compared estimates for the cases \u03b1 = 0 versus \u03b1 unconstrained for 500\ngraphs generated randomly from a model of the form speci\ufb01ed in Eq. (2) based on school 8 of\nthe Add-Health data set. The number of nodes N = 204 and covariates x(i, j) matched that of\nSchool 8 in the Adolescent Health friendship network dataset, and the regression coef\ufb01cient vector\n\u03b2 = (\u22122.6, 0.025, 0.9,\u22121.6)(cid:48), set to match the ML estimate of \u03b2 for School 8, \ufb01tted via logistic re-\ngression with \u0398 = 0, \u03b1 = 0. The covariates x(i, j) comprised of an intercept term, an indicator for\nwhether students i and j shared the same gender, an indicator for shared race, and their difference\nin school grade.\nThe inclusion of \u03b1 in the model of Eq. (2) appears to give rise to a loss of estimator consistency, as\nshown in Table 1 where the empirical bias of each component of \u03b2 is reported. This suggests, as\nwe alluded to above, that inferential conclusions based on parameter estimates from latent variable\nmodels should be interpreted with caution.\nTo explore the tightness of the con\ufb01dence sets given by the bound in Eq. (3), we \ufb01tted the full\nmodel speci\ufb01ed in Eq. (2) with K in the range 2\u20136 to 50 draws from a restricted version of the\nmodel corresponding to each of the 26 schools in our dataset.\nIn the same manner described\nabove, each simulated graph shared the same size and covariates as its corresponding school in\nthe dataset, with \u03b2 \ufb01xed to its ML-\ufb01tted value with \u0398 = 0, \u03b1 = 0. The empirical divergence term\nab ) under the approximate ML partition determined via Algorithm 1 was then\ntabulated for each of these 1300 \ufb01ts, and compared to its 95% con\ufb01dence upper bound given by\nEq. (3). The empirical divergences are reported in the histogram of Fig. 1 as a fraction of the up-\nper bound. It may be seen from Fig. 1 that the largest divergence observed was less than 41% of\nits corresponding bound, with 95% of all divergences less than 22% of their corresponding bound.\n\na\u2264b nabD( \u02c6\u03a6(z)\n\nab || \u00af\u03a6(z)\n\n(cid:80)\n\n4\n\n\fFigure 1: Divergence terms (cid:80)\n\nab || \u00af\u03a6(z)\n\nab ) as fractions of 95% con\ufb01dence set values,\nshown for approximate maximum-likelihood \ufb01ts to 1300 randomly graphs matched to the 26-school\nfriendship network dataset.\n\na\u2264b nabD( \u02c6\u03a6(z)\n\nThis analysis provides an indication of how in\ufb02ated the con\ufb01dence set sizes are expected to be in\npractice; while conservative in nature, they seem usable for practical situations.\n\n3 Analysis of Adolescent Health Networks\n\nThe National Longitudinal Study of Adolescent Health (Add Health) is a study of adolescents in the\nUnited States. To date, four waves of surveys have been collected over the course of \ufb01fteen years.\nMany statistical studies have been performed using the data to explore a variety of social and health\nissues1. For example, [24, 25] discusses effects of racial diversity on community formation across\nschools. Here we examine the schools individually to \ufb01nd residual block structure not explained by\ngender, race, or grade. Since we will be unable to verify such blocks by checking against explanatory\nvariables, we rely on the con\ufb01dence sets developed above to assess signi\ufb01cance of the discovered\nblock structure.\nOur approach is as follows. As discussed in Section 2.3, Eq. (3) enables us to calculate con\ufb01dence\nsets with respect to Bernoulli parameters {Pij} for any class membership vector z in terms of the\ncorresponding sample proportion matrices \u02c6\u03a6(z). Then, by comparing values of \u02c6\u03a6(z) to a baseline\nmodel obtained by \ufb01tting K = 1, \u0398 = 0 (thus removing the stochastic block component from\nEq. (2)), we may evaluate whether or not the observed sample counts are consistent with the structure\npredicted by the baseline model. This procedure provides a kind of notional p-value to qualitatively\nassess signi\ufb01cance of the residual structure induced by any choice of z.\n\n3.1 Model Checking\n\nWe \ufb01rst \ufb01t model (2) with \u0398 = 0 and \u03b1 = 0, since it reduces to a logistic regression with explanatory\nvariables x(i, j), for which standard asymptotic results apply. The parameter \ufb01ts were examined and\nan analysis of deviance was conducted. The \ufb01ts were observed to be well behaved in this setting;\nestimates of \u03b2 and their corresponding standard errors indicate a clustering effect by grade that is\nstronger than that of either shared gender or race. An analysis of deviance, where each variable was\nwithheld from the model, resulted in similar conclusions: Average deviances across the 26 schools\nwere \u221269, \u2212238, and \u22123760 for gender, race, and grade respectively, with p-values below 0.05\nindicating signi\ufb01cance in all but 3, 7, and 0 of the schools for each of the respective covariates; these\nschools had small numbers of students, with a maximum N of 108.\nWhen \u03b1 was re-introduced into the model of Eq. (2), its components were observed to correlate\nhighly with the sequence of observed nodal degrees in the data, as expected. (Recall that consistency\nresults are not known for this model, so that p-values cannot be associated with deviances or standard\nerrors; however, in our simulations the maximum-likelihood estimates showed moderate errors, as\ndiscussed in Section 2.4.) For two of the schools, the resulting model was degenerate, whereas for\nthe remaining schools the \u03b1-degree correlation had a range of 0.78\u20130.94 and a median value of 0.89.\n\n1For a bibliography, see http://www.cpc.unc.edu/projects/addhealth/pubs.\n\n5\n\n\f(a) K = 2\n\n(b) K = 4\n\n(c) K = 6\n\nFigure 2: Student counts resulting from a stochastic blockmodel \ufb01t for K \u2208 {2, . . . , 6}, arranged by\nlatent block and school year (grade) for School 6. The inferred block structure approximately aligns\nwith the grade covariate (which was not included in this model).\n\nEstimates of \u03b2 did not undergo qualitative signi\ufb01cant changes from their earlier values when the\nrestriction \u03b1 = 0 was lifted.\nA \u201cpure\u201d stochastic blockmodel (\u03b1 = 0, \u03b2 = 0) was \ufb01tted to our data over the range K \u2208 {2, . . . , 6},\nto observe if the resulting block structure replicates that of any of the known covariates. Figure 2\nshows counts of students by latent class (under the approximate maximum-likelihood estimate of\nz) and grade for School 6; it can be seen that the recovered grouping of students by latent class is\nclosely aligned with the grade covariate, particularly for grades 7\u201310.\n\n3.2 Residual Block Structure\n\nWe now report on the assessment of residual block structure in the Adolescent Health friendship\nnetwork data. Recalling that the con\ufb01dence sets obtained with Eq. (3) hold uniformly for all par-\ntitions of equal size, independently of how they are chosen, we therefore may freely modify the\n\ufb01tting procedure of Algorithm 1 to obtain partitions that exhibit the greatest degree of structure.\nBearing in mind the high observed \u03b1-degree correlation as discussed above, we replaced the latent\nvariable vector \u03b1 in the model of Eq. (2) with a more parsimonious categorical covariate determined\nby grouping the observed network degrees according to the ranges 0\u20133, 4\u20137, and 8\u2013\u221e. We also\nexpanded the covariates by giving each race and grade pairing its own indicator function. These\nmodi\ufb01cations would be inappropriate for the baseline model, as dyadic independence conditioned\non the covariates would be lost, and standard errors for \u03b2 would be larger; however, the changes\nwere useful for improving the output of Algorithm 1 without invalidating Eq. (3).\nFig. 3 depicts partitions for which the observed \u02c6\u03a6(z), \ufb01tted for various K > 1 using the modi\ufb01ed\nversion of Algorithm 1 detailed above, is \u201cfar\u201d from its nominal value under the baseline model \ufb01tted\nwith K = 1, in the sense that the corresponding 95% Bonferroni-corrected con\ufb01dence set bound is\nexceeded. We observe that in each partition, the number of apparently visible communities exceeds\nK, and they are comprised of small numbers of students. This effect is due to the intersection of\ngrade and z-induced clustering.\nWe take as our de\ufb01nition of nominal value the quantity \u00af\u03a6(z) computed under the base-\nline model, which we denote by \u03a6(z).\nTable 2 lists normalized divergence terms\nab ), Bonferroni-corrected 95% con\ufb01dence bounds, and measures of\nalignment between the corresponding partitions z and the explanatory variables. The alignment\nwith the covariates are small, as measured by the Jacaard similarity coef\ufb01cient and ratio of within-\nclass to total variance2, signifying the residual quality of the partitions, while the relatively large\ndivergence terms signify that the Bonferroni-corrected con\ufb01dence set bounds for each school have\nbeen met or exceeded.\n\n(cid:1)\u22121(cid:80)\n\na\u2264b nabD( \u02c6\u03a6(z)\n\nab ||\u03a6(z)\n\n(cid:0)N\n\n2\n\nB|, were A, B \u2282 (cid:0)N\n\n2The alignment scores are de\ufb01ned as follows. The Jacaard similarity coef\ufb01cient is de\ufb01ned as |A \u2229 B|/|A \u222a\n\n(cid:1) are the student pairings sharing the same latent class or the same covariate value,\n\nrespectively. See [12] for further network-related discussion. Variance ratio denotes the within-class degree\nvariance divided by the total variance, averaged over all classes.\n\n2\n\n6\n\n\fSchool\n10\n18\n21\n22\n26\n29\n38\n55\n56\n66\n67\n72\n78\n80\n\nStudents\n678\n284\n377\n614\n551\n569\n521\n336\n446\n644\n456\n352\n432\n594\n\nEdges K\n6\n2795\n5\n1189\n1531\n6\n5\n2450\n3\n2066\n6\n2534\n5\n1925\n803\n4\n6\n1394\n6\n2865\n3\n926\n1398\n4\n6\n1334\n1745\n4\n\nDiv. (Bound) Gender\n0.14\n0.17\n0.15\n0.18\n0.25\n0.15\n0.17\n0.20\n0.15\n0.15\n0.25\n0.21\n0.15\n0.20\n\n0.0064 (0.0062)\n0.0150 (0.0150)\n0.0140 (0.0120)\n0.0064 (0.0061)\n0.0049 (0.0045)\n0.0091 (0.0075)\n0.0073 (0.0073)\n0.0100 (0.0100)\n0.0120 (0.0099)\n0.0069 (0.0066)\n0.0055 (0.0055)\n0.0099 (0.0095)\n0.0100 (0.0100)\n0.0054 (0.0053)\n\nJaccard coef\ufb01cient or Variance ratio\nRace Grade Degree\n0.93\n0.16\n0.88\n0.19\n0.16\n0.95\n0.99\n0.14\n0.99\n0.21\n0.88\n0.16\n0.86\n0.18\n0.18\n0.97\n0.98\n0.14\n0.91\n0.16\n1.00\n0.23\n0.21\n0.96\n0.98\n0.12\n0.19\n0.99\n\n0.097\n0.14\n0.12\n0.11\n0.13\n0.10\n0.17\n0.21\n0.15\n0.099\n0.25\n0.12\n0.15\n0.15\n\nTable 2: Block structure assessments corresponding to Fig. 3. Small Jacaard coef\ufb01cient values (for\ngender, race, and grade) and variance ratios approaching 1 for degree indicate a lack of alignment\nwith covariates and hence the identi\ufb01cation of residual structure in the corresponding partition.\n\nWe note that the usage of covariate information was necessary to detect small student groups; with-\nout the incorporation of grade effects, we would require a much larger value of K for Algorithm 1 to\ndetect the observed network structure (a concern noted by [23] in the absence of covariates), which\nin turn would in\ufb02ate the con\ufb01dence set, leading to an inability to validate the observed structure\nfrom that predicted by a baseline model.\n\n4 Concluding Remarks\n\nIn this article we have developed con\ufb01dence sets for assessing inferred network structure, by lever-\naging our result derived in [14]. We explored the use of these con\ufb01dence sets with an application to\nthe analysis of Adolescent Health survey data comprising friendship networks from 26 schools.\nOur methodology can be summarized as follows. In lieu of a parametric model, we assume dyadic\nindependence with Bernoulli parameters {Pji}. We introduced a baseline model (K = 1) that in-\ncorporates degree and covariate effects, without block structure. Algorithm 1 was then used to \ufb01nd\nhighly assortative partitions of students which are also far from partitions induced by the explana-\ntory covariates in the baseline model. Differences in assortativity were quanti\ufb01ed by an empirical\ndivergence statistic, which was compared to an upper bound computed from Eq. (3) to check for sig-\nni\ufb01cance and to generate con\ufb01dence sets for {Pij}. While the upper bound in Eq. (3) is known to be\nloose, simulation results in Figure 1 suggest that the slack is moderate, leading to useful con\ufb01dence\nsets in practice.\nIn our procedure, we cannot quantify the uncertainty associated with the estimated baseline model,\nsince the parameter estimates lack consistency. As a result, we cannot conduct a formal hypothesis\ntest for \u0398 = 0. However, for a baseline model where the MLE is known to be consistent, we con-\njecture that such a hypothesis test should be possible by incorporating the con\ufb01dence set associated\nwith the MLE.\nDespite concerns regarding estimator consistency in this and other latent variable models, we were\nable to show that the notion of con\ufb01dence sets may instead be used to provide a (conservative)\nmeasure of residual block structure. We note that many open questions remain, and are hopeful\nthat this analysis may help to shed light on some important current issues facing practitioners and\ntheorists alike in statistical network analysis.\n\n7\n\n\f(a) School 10, K = 6\n\n(b) School 18, K = 5\n\n(c) School 21, K = 6\n\n(d) School 22, K = 5\n\n(e) School 26, K = 3\n\n(f) School 29, K = 6\n\n(g) School 38, K = 5\n\n(h) School 55, K = 4\n\n(i) School 56, K = 6\n\n(j) School 66, K = 6\n\n(k) School 67, K = 3\n\n(l) School 72, K = 4\n\n(m) School 78, K = 6\n\n(n) School 80, K = 4\n\nFigure 3: Adjacency matrices for schools exhibiting residual block structure as described in Sec-\ntion 3.2, with nodes ordered by grade (solid lines) and corresponding latent classes (dotted lines).\n\n8\n\n\fReferences\n\n[1] A. Goldenberg, A. X. Zheng, S. E. Fienberg, and E. M. Airoldi, \u201cA survey of statistical\nnetwork models\u201d, Foundation and Trends in Machine Learning, vol. 2, pp. 1\u2013117, Feb. 2010.\n[2] R. Albert and A. L. Barabasi, \u201cStatistical mechanics of complex networks\u201d, Reviews of\n\nModern Physics, vol. 74, no. 47, Jan. 2002.\n\n[3] M. E. J. Newman, \u201cThe structure and function of complex networks\u201d, SIAM Review, vol. 45,\n\npp. 167\u2013256, June 2003.\n\n[4] C. Cooper and A. M. Frieze, \u201cA general model of web graphs\u201d, Random Structures and\n\nAlgorithms, vol. 22, no. 3, pp. 311\u2013335, Mar. 2003.\n\n[5] M. O. Jackson, Social and Economic Networks, Princeton University Press, 2008.\n[6] S. Wasserman and K. Faust, Social Network Analysis: Methods and Applications, Cambridge\n\nUniversity Press, Cambridge, U.K., 1994.\n\n[7] T. A. B. Snijders and K. Nowicki, \u201cEstimation and prediction for stochastic blockmodels for\n\ngraphs with latent block structure\u201d, J. Classif., vol. 14, pp. 75\u2013100, Jan. 1997.\n\n[8] M. S. Handcock, A. E. Raftery, and J. M. Tantrum, \u201cModel-based clustering for social net-\n\nworks\u201d, J. R. Stat. Soc. A, vol. 170, pp. 301\u2013354, Mar. 2007.\n\n[9] E. M. Airoldi, D. M. Blei, S. E. Fienberg, and E. P. Xing, \u201cMixed membership stochastic\n\nblockmodels\u201d, J. Mach. Learn. Res., vol. 9, pp. 1981\u20132014, June 2008.\n\n[10] P. D. Hoff, \u201cMultiplicative latent factor models for description and prediction of social net-\n\nworks\u201d, Computational Math. Organization Theory, vol. 15, pp. 261\u2013272, Dec. 2009.\n\n[11] M. E. J. Newman, \u201cModularity and community structure in networks\u201d, Proc. Natl Acad. Sci.\n\nU.S.A., vol. 103, pp. 8577\u20138582, June 2006.\n\n[12] A. L. Traud, E. D. Kelsic, P. J. Mucha, and M. A. Porter, \u201cComparing community structure\n\nto characteristics in online collegiate social networks\u201d, SIAM Rev., 2011, to appear.\n\n[13] P. J. Bickel and A. Chen, \u201cA nonparametric view of network models and Newman-Girvan\nand other modularities\u201d, Proc. Natl Acad. Sci. U.S.A., vol. 106, pp. 21068\u201321073, Dec. 2009.\n[14] D.S. Choi, P.J. Wolfe, and E.M. Airoldi, \u201cStochastic blockmodels with growing numbers of\n\nclasses\u201d, Biometrika, 2011, to appear.\n\n[15] K. Rohe, S. Chatterjee, and B. Yu, \u201cSpectral clustering and the high-dimensional stochastic\n\nblockmodel\u201d, Ann. Stat., 2011, to appear.\n\n[16] A. Celisse, J.J. Daudin, and L. Pierre, \u201cConsistency of maximum-likelihood and variational\n\nestimators in the stochastic block model\u201d, Arxiv preprint 1105.3288, 2011.\n\n[17] B. Karrer, E. Levina, and MEJ Newman, \u201cRobustness of community structure in networks\u201d,\n\nPhys. Rev. E, vol. 77, pp. 46119\u201346128, Apr. 2008.\n\n[18] C.P. Massen and J.P.K. Doye, \u201cThermodynamics of community structure\u201d, Arxiv preprint\n\ncond-mat/0610077, 2006.\n\n[19] J. Copic, M. O. Jackson, and A. Kirman, \u201cIdentifying community structures from network\n\ndata via maximum likelihood methods\u201d, B.E. J. Theoretical Economics, vol. 9, Sept. 2009.\n\n[20] P.W. Holland and S. Leinhardt, \u201cAn exponential family of probability distributions for di-\n\nrected graphs\u201d, J. Am. Stat. Assoc., vol. 76, pp. 33\u201350, Mar. 1981.\n\n[21] SJ Haberman, \u201cComment on Holland and Leinhardt\u201d, J. Am. Stat. Assoc., vol. 76, pp. 60\u201362,\n\nMar. 1981.\n\n[22] S. Wasserman and S.O.L. Weaver, \u201cStatistical analysis of binary relational data: parameter\n\nestimation\u201d, J. Math. Psychol., vol. 29, pp. 406\u2013427, Dec. 1985.\n\n[23] P. D. Hoff, \u201cModeling homophily and stochastic equivalence in symmetric relational data\u201d,\n\nin Adv. in Neural Information Processing Systems, pp. 657\u2013664. MIT Press, 2008.\n\n[24] S.M. Goodreau, J.A. Kitts, and M. Morris, \u201cBirds of a feather, or friend of a friend? using\nexponential random graph models to investigate adolescent social networks\u201d, Demography,\nvol. 46, pp. 103\u2013125, Feb. 2009.\n\n[25] M.C. Gonz\u00b4alez, H.J. Herrmann, J. Kert\u00b4esz, and T. Vicsek, \u201cCommunity structure and ethnic\npreferences in school friendship networks\u201d, Physica A., vol. 379, no. 1, pp. 307\u2013316, 2007.\n\n9\n\n\f", "award": [], "sourceid": 1176, "authors": [{"given_name": "David", "family_name": "Choi", "institution": null}, {"given_name": "Patrick", "family_name": "Wolfe", "institution": null}, {"given_name": "Edo", "family_name": "Airoldi", "institution": null}]}