{"title": "Modeling homophily and stochastic equivalence in symmetric relational data", "book": "Advances in Neural Information Processing Systems", "page_first": 657, "page_last": 664, "abstract": "This article discusses a latent variable model for inference and prediction of symmetric relational data. The model, based on the idea of the eigenvalue decomposition, represents the relationship between two nodes as the weighted inner-product of node-specific vectors of latent characteristics. This ``eigenmodel'' generalizes other popular latent variable models, such as latent class and distance models: It is shown mathematically that any latent class or distance model has a representation as an eigenmodel, but not vice-versa. The practical implications of this are examined in the context of three real datasets, for which the eigenmodel has as good or better out-of-sample predictive performance than the other two models.", "full_text": "Modeling homophily and stochastic equivalence in\n\nsymmetric relational data\n\nPeter D. Hoff\n\nDepartments of Statistics and Biostatistics\n\nUniversity of Washington\nSeattle, WA 98195-4322.\n\nhoff@stat.washington.edu\n\nAbstract\n\nThis article discusses a latent variable model for inference and prediction of sym-\nmetric relational data. The model, based on the idea of the eigenvalue decomposi-\ntion, represents the relationship between two nodes as the weighted inner-product\nof node-speci\ufb01c vectors of latent characteristics. This \u201ceigenmodel\u201d generalizes\nother popular latent variable models, such as latent class and distance models: It is\nshown mathematically that any latent class or distance model has a representation\nas an eigenmodel, but not vice-versa. The practical implications of this are exam-\nined in the context of three real datasets, for which the eigenmodel has as good or\nbetter out-of-sample predictive performance than the other two models.\n\n1 Introduction\nLet {yi,j : 1 \u2264 i < j \u2264 n} denote data measured on pairs of a set of n objects or nodes. The\nexamples considered in this article include friendships among people, associations among words\nand interactions among proteins. Such measurements are often represented by a sociomatrix Y ,\nwhich is a symmetric n \u00d7 n matrix with an unde\ufb01ned diagonal. One of the goals of relational data\nanalysis is to describe the variation among the entries of Y , as well as any potential covariation of\nY with observed explanatory variables X = {xi,j, 1 \u2264 i < j \u2264 n}.\nTo this end, a variety of statistical models have been developed that describe yi,j as some function\nof node-speci\ufb01c latent variables ui and uj and a linear predictor \u03b2T xi,j.\nIn such formulations,\n{u1, . . . , un} represent across-node variation in the yi,j\u2019s and \u03b2 represents covariation of the yi,j\u2019s\nwith the xi,j\u2019s. For example, Nowicki and Snijders [2001] present a model in which each node i\nis assumed to belong to an unobserved latent class ui, and a probability distribution describes the\nrelationships between each pair of classes (see Kemp et al. [2004] and Airoldi et al. [2005] for recent\nextensions of this approach). Such a model captures stochastic equivalence, a type of pattern often\nseen in network data in which the nodes can be divided into groups such that members of the same\ngroup have similar patterns of relationships.\nAn alternative approach to representing across-node variation is based on the idea of homophily, in\nwhich the relationships between nodes with similar characteristics are stronger than the relationships\nbetween nodes having different characteristics. Homophily provides an explanation to data patterns\noften seen in social networks, such as transitivity (\u201ca friend of a friend is a friend\u201d), balance (\u201cthe\nenemy of my friend is an enemy\u201d) and the existence of cohesive subgroups of nodes. In order to\nrepresent such patterns, Hoff et al. [2002] present a model in which the conditional mean of yi,j is a\nfunction of \u03b20xi,j \u2212 |ui \u2212 uj|, where {u1, . . . , un} are vectors of unobserved, latent characteristics\nin a Euclidean space. In the context of binary relational data, such a model predicts the existence\nof more transitive triples, or \u201ctriangles,\u201d than would be seen under a random allocation of edges\namong pairs of nodes. An important assumption of this model is that two nodes with a strong\n\n1\n\n\fFigure 1: Networks exhibiting homophily (left panel) and stochastic equivalence (right panel).\n\nrelationship between them are also similar to each other in terms of how they relate to other nodes:\nA strong relationship between i and j suggests |ui \u2212 uj| is small, but this further implies that\n|ui \u2212 uk| \u2248 |uj \u2212 uk|, and so nodes i and j are assumed to have similar relationships to other nodes.\nThe latent class model of Nowicki and Snijders [2001] and the latent distance model of Hoff et al.\n[2002] are able to identify, respectively, classes of nodes with similar roles, and the locational prop-\nerties of the nodes. These two items are perhaps the two primary features of interest in social network\nand relational data analysis. For example, discussion of these concepts makes up more than half of\nthe 734 pages of main text in Wasserman and Faust [1994]. However, a model that can represent\none feature may not be able to represent the other: Consider the two graphs in Figure 1. The graph\non the left displays a large degree of transitivity, and can be well-represented by the latent distance\nmodel with a set of vectors {u1, . . . , un} in two-dimensional space, in which the probability of an\nedge between i and j is decreasing in |ui \u2212 uj|. In contrast, representation of the graph by a latent\nclass model would require a large number of classes, none of which would be particularly cohesive\nor distinguishable from the others. The second panel of Figure 1 displays a network involving three\nclasses of stochastically equivalent nodes, two of which (say A and B) have only across-class ties,\nand one (C) that has both within- and across-class ties. This graph is well-represented by a latent\nclass model in which edges occur with high probability between pairs having one member in each\nof A and B or in B and C, and among pairs having both members in C (in models of stochastic\nequivalence, nodes within each class are not differentiated). In contrast, representation of this type\nof graph with a latent distance model would require the dimension of the latent characteristics to be\non the order of the class membership sizes.\nMany real networks exhibit combinations of structural equivalence and homophily in varying de-\ngrees. In these situations, use of either the latent class or distance model would only be representing\npart of the network structure. The goal of this paper is to show that a simple statistical model based\non the eigenvalue decomposition can generalize the latent class and distance models: Just as any\nsymmetric matrix can be approximated with a subset of its largest eigenvalues and corresponding\neigenvectors, the variation in a sociomatrix can be represented by modeling yi,j as a function of\ni \u039buj, where {u1, . . . , un} are node-speci\ufb01c factors and \u039b is a diagonal matrix. In this\n\u03b20xi,j + uT\narticle, we show mathematically and by example how this eigenmodel can represent both stochastic\nequivalence and homophily in symmetric relational data, and thus is more general than the other two\nlatent variable models.\nThe next section motivates the use of latent variables models for relational data, and shows mathe-\nmatically that the eigenmodel generalizes the latent class and distance models in the sense that it can\ncompactly represent the same network features as these other models but not vice-versa. Section 3\ncompares the out-of-sample predictive performance of these three models on three different datasets:\na social network of 12th graders; a relational dataset on word association counts from the \ufb01rst chap-\nter of Genesis; and a dataset on protein-protein interactions. The \ufb01rst two networks exhibit latent\nhomophily and stochastic equivalence respectively, whereas the third shows both to some degree.\n\n2\n\nllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll\fIn support of the theoretical results of Section 2, the latent distance and class models perform well\nfor the \ufb01rst and second datasets respectively, whereas the eigenmodel performs well for all three.\nSection 4 summarizes the results and discusses some extensions.\n\n2 Latent variable modeling of relational data\n\n2.1 Justi\ufb01cation of latent variable modeling\n\nThe use of probabilistic latent variable models for the representation of relational data can be moti-\nvated in a natural way: For undirected data without covariate information, symmetry suggests that\nany probability model we consider should treat the nodes as being exchangeable, so that\n\nPr({yi,j : 1 \u2264 i < j \u2264 n} \u2208 A) = Pr({y\u03c0i,\u03c0j : 1 \u2264 i < j \u2264 n} \u2208 A)\n\nfor any permutation \u03c0 of the integers {1, . . . , n} and any set of sociomatrices A. Results of Hoover\n[1982] and Aldous [1985, chap. 14] show that if a model satis\ufb01es the above exchangeability condi-\ntion for each integer n, then it can be written as a latent variable model of the form\n\nyi,j = h(\u00b5, ui, uj, \u0001i,j)\n\n(1)\nfor i.i.d. latent variables {u1, . . . , un}, i.i.d. pair-speci\ufb01c effects {\u0001i,j : 1 \u2264 i < j \u2264 n} and some\nfunction h that is symmetric in its second and third arguments. This result is very general - it says\nthat any statistical model for a sociomatrix in which the nodes are exchangeable can be written as a\nlatent variable model.\nDifference choices of h lead to different models for y. A general probit model for binary network\ndata can be put in the form of (1) as follows:\n\n{\u0001i,j : 1 \u2264 i < j \u2264 n} \u223c i.i.d. normal(0, 1)\n\n{u1, . . . , un} \u223c i.i.d. f(u|\u03c8)\n\nyi,j = h(\u00b5, ui, uj, \u0001i,j) = \u03b4(0,\u221e)(\u00b5 + \u03b1(ui, uj) + \u0001i,j),\n\nwhere \u00b5 and \u03c8 are parameters to be estimated, and \u03b1 is a symmetric function, also potentially\ninvolving parameters to be estimated. Covariation between Y and an array of predictor variables\nX can be represented by adding a linear predictor \u03b2T xi,j to \u00b5. Finally, integrating over \u0001i,j we\nobtain Pr(yi,j = 1|xi,j, ui, uj) = \u03a6[\u00b5 + \u03b2T xi,j + \u03b1(ui, uj)]. Since the \u0001i,j\u2019s can be assumed to be\nindependent, the conditional probability of Y given X and {u1, . . . , un} can be expressed as\n\nPr(yi,j = 1|xi,j, ui, uj) \u2261 \u03b8i,j = \u03a6[\u00b5 + \u03b2T xi,j + \u03b1(ui, uj)]\n\n(2)\n\nPr(Y |X, u1, . . . , un) = Y\n\ni,j (1 \u2212 \u03b8i,j)yi,j\n\u03b8yi,j\n\ni 0 or \u03bbk < 0. In this way,\nthe model can represent both positive or negative homophily in varying degrees, and stochastically\nequivalent nodes (nodes with the same or similar latent vectors) may or may not have strong rela-\ntionships with one another.\nWe now show that the eigenmodel generalizes the latent class and distance models: Let Sn be the\nset of n \u00d7 n sociomatrices, and let\n\nCK = {C \u2208 Sn : ci,j = mui,uj , ui \u2208 {1, . . . , K}, M a K \u00d7 K symmetric matrix};\nDK = {D \u2208 Sn : di,j = \u2212|ui \u2212 uj|, ui \u2208 RK};\nEK = {E \u2208 Sn : ei,j = uT\n\ni \u039buj, ui \u2208 RK, \u039b a K \u00d7 K diagonal matrix}.\n\nIn other words, CK is the set of possible values of {\u03b1(ui, uj), 1 \u2264 i < j \u2264 n} under a K-\ndimensional latent class model, and similarly for DK and EK.\nEK generalizes CK: Let C \u2208 CK and let \u02dcC be a completion of C obtained by setting ci,i = mui,ui.\nThere are at most K unique rows of \u02dcC and so \u02dcC is of rank K at most. Since the set EK contains\nall sociomatrices that can be completed as a rank-K matrix, we have CK \u2286 EK. Since EK includes\nmatrices with n unique rows, CK \u2282 EK unless K \u2265 n in which case the two sets are equal.\nEK+1 weakly generalizes DK: Let D \u2208 DK. Such a (negative) distance matrix will generally be\nof full rank, in which case it cannot be represented exactly by an E \u2208 EK for K < n. However,\nwhat is critical from a modeling perspective is whether or not the order of the entries of each D can\nbe matched by the order of the entries of an E. This is because the probit and ordered probit model\nwe are considering include threshold variables {\u00b5y : y \u2208 Y} which can be adjusted to accommodate\nmonotone transformations of \u03b1(ui, uj). With this in mind, note that the matrix of squared distances\namong a set of K-dimensional vectors {z1, . . . , zn} is a monotonic transformation of the distances,\nnzn] \u2212 2ZZ T ) and so is\nnzn]T 1T + 1[z0\nis of rank K + 2 or less (as D2 = [z0\ni zi) \u2208 RK+1 for each i \u2208 {1, . . . , n}, we have\n\nizj +p(r2 \u2212 |ui|2)(r2 \u2212 |uj|2). For large r this is approximately r2\u2212|zi\u2212 zj|2/2, which\n\nin EK+2. Furthermore, letting ui = (zi,pr2 \u2212 zT\n\nu0\niuj = z0\nis an increasing function of the negative distance di,j. For large enough r the numerical order of the\nentries of this E \u2208 EK+1 is the same as that of D \u2208 DK.\nDK does not weakly generalize E1: Consider E \u2208 E1 generated by \u039b = 1, u1 = 1 and ui =\nr < 1 for i > 1. Then r = e1,i1 = e1,i2 > ei1,i2 = r2 for all i1, i2 6= 1. For which K is such an\nordering of the elements of D \u2208 DK possible? If K = 1 then such an ordering is possible only if\nn = 3. For K = 2 such an ordering is possible for n \u2264 6. This is because the kissing number in\nR2, or the number of non-overlapping spheres of unit radius that can simultaneously touch a central\nsphere of unit radius, is 6. If we put node 1 at the center of the central sphere, and 6 nodes at the\ncenters of the 6 kissing spheres, then we have d1,i1 = d1,i2 = di1,i2 for all i1, i2 6= 1. We can only\nhave d1,i1 = d1,i2 > di1,i2 if we remove one of the non-central spheres to allow for more room\nbetween those remaining, leaving one central sphere plus \ufb01ve kissing spheres for a total of n = 6.\nIncreasing n increases the necessary dimension of the Euclidean space, and so for any K there are\nn and E \u2208 E1 that have entry orderings that cannot be matched by those of any D \u2208 DK.\n\n1z1, . . . , z0\n\n1z1, . . . , z0\n\n4\n\n\fA less general positive semi-de\ufb01nite version of the eigenmodel has been studied by Hoff [2005],\nin which \u039b was taken to be the identity matrix. Such a model can weakly generalize a distance\nmodel, but cannot generalize a latent class model, as the eigenvalues of a latent class model could\nbe negative.\n\n3 Model comparison on three different datasets\n\n3.1 Parameter estimation\n\nBayesian parameter estimation for the three models under consideration can be achieved via Markov\nchain Monte Carlo (MCMC) algorithms, in which posterior distributions for the unknown quantities\nare approximated with empirical distributions of samples from a Markov chain. For these algo-\nrithms, it is useful to formulate the probit models described in Section 2.1 in terms of an additional\nlatent variable zi,j \u223c normal[\u03b20xi,j + \u03b1(ui, uj)], for which yi,j = y if \u00b5y < zi,j < \u00b5y+1. Using\nconjugate prior distributions where possible, the MCMC algorithms proceed by generating a new\nstate \u03c6(s+1) = {Z (s+1), \u00b5(s+1), \u03b2(s+1), u(s+1)\n\n} from a current state \u03c6(s) as follows:\n1. For each {i, j}, sample zi,j from its (constrained normal) full conditional distribution.\n2. For each y \u2208 Y, sample \u00b5y from its (normal) full conditional distribution.\n3. Sample \u03b2 from its (multivariate normal) full conditional distribution.\n4. Sample u1, . . . , un and their associated parameters:\n\n, . . . , u(s+1)\n\nn\n\n1\n\n\u2022 For the latent distance model, propose and accept or reject new values of the ui\u2019s with\nthe Metropolis algorithm, and then sample the population variances of the ui\u2019s from\ntheir (inverse-gamma) full conditional distributions.\n\u2022 For the latent class model, update each class variable ui from its (multinomial) con-\nditional distribution given current values of Z,{uj : j 6= i} and the variance of the\nelements of M (but marginally over M to improve mixing). Then sample the elements\nof M from their (normal) full conditional distributions and the variance of the entries\nof M from its (inverse-gamma) full conditional distribution.\n\u2022 For the latent vector model, sample each ui from its (multivariate normal) full con-\nditional distribution, sample the mean of the ui\u2019s from their (normal) full conditional\ndistributions, and then sample \u039b from its (multivariate normal) full conditional distri-\nbution.\n\nTo facilitate comparison across models, we used prior distributions in which the level of prior vari-\nability in \u03b1(ui, uj) was similar across the three different models (further details and code to imple-\nment these algorithms are available at my website).\n\n3.2 Cross validation\n\nTo compare the performance of these three different models we evaluated their out-of-sample pre-\ndictive performance under a range of dimensions (K \u2208 {3, 5, 10}) and on three different datasets\nexhibiting varying combinations of homophily and stochastic equivalence. For each combination of\ndataset, dimension and model we performed a \ufb01ve-fold cross validation experiment as follows:\n\n(cid:1) data values into 5 sets of roughly equal size, letting si,j be the set\n\n1. Randomly divide the(cid:0)n\n\nto which pair {i, j} is assigned.\n\n2\n\n2. For each s \u2208 {1, . . . , 5}:\n\n(a) Obtain posterior distributions of the model parameter conditional on {yi,j : si,j 6= s},\n(b) For pairs {k, l} in set s, let \u02c6yk,l = E[yk,l|{yi,j : si,j 6= s}], the posterior predictive\n\nthe data on pairs not in set s.\n\nmean of yk,l obtained using data not in set s.\n\nThis procedure generates a sociomatrix \u02c6Y , in which each entry \u02c6yi,j represents a predicted value\nobtained from using a subset of the data that does not include yi,j. Thus \u02c6Y is a sociomatrix of\nout-of-sample predictions of the observed data Y .\n\n5\n\n\fTable 1: Cross validation results and area under the ROC curves.\n\nK\n\n3\n5\n10\n\ndist\n0.82\n0.81\n0.76\n\nAdd health\n\nclass\n0.64\n0.70\n0.69\n\neigen\n0.75\n0.78\n0.80\n\nGenesis\nclass\n0.82\n0.82\n0.82\n\neigen\n0.82\n0.82\n0.82\n\ndist\n0.62\n0.66\n0.74\n\nProtein interaction\neigen\ndist\n0.83\n0.88\n0.90\n0.84\n0.85\n0.90\n\nclass\n0.79\n0.84\n0.86\n\nFigure 2: Social network data and unscaled ROC curves for the K = 3 models.\n\n3.3 Adolescent Health social network\n\nThe \ufb01rst dataset records friendship ties among 247 12th-graders, obtained from the National Longi-\ntudinal Study of Adolescent Health (www.cpc.unc.edu/projects/addhealth). For these data,\nyi,j = 1 or 0 depending on whether or not there is a close friendship tie between student i and j\n(as reported by either i or j). These data are represented as an undirected graph in the \ufb01rst panel of\nFigure 2. Like many social networks, these data exhibit a good deal of transitivity. It is therefore not\nsurprising that the best performing models considered (in terms of area under the ROC curve, given\nin Table 1) are the distance models, with the eigenmodels close behind. In contrast, the latent class\nmodels perform poorly, and the results suggest that increasing K for this model would not improve\nits performance.\n\n3.4 Word neighbors in Genesis\n\nThe second dataset we consider is derived from word and punctuation counts in the \ufb01rst chapter\nof the King James version of Genesis (www.gutenberg.org/dirs/etext05/bib0110.txt).\nThere are 158 unique words and punctuation marks in this chapter, and for our example we take\nyi,j to be the number of times that word i and word j appear next to each other (a model extension,\nappropriate for an asymmetric version of this dataset, is discussed in the next section). These data\ncan be viewed as a graph with weighted edges, the unweighted version of which is shown in the\n\ufb01rst panel of Figure 3. The lack of a clear spatial representation of these data is not unexpected,\nas text data such as these do not have groups of words with strong within-group connections, nor\ndo they display much homophily: a given noun may appear quite frequently next to two different\nverbs, but these verbs will not appear next to each other. A better description of these data might be\nthat there are classes of words, and connections occur between words of different classes. The cross\nvalidation results support this claim, in that the latent class model performs much better than the\ndistance model on these data, as seen in the second panel of Figure 3 and in Table 1. As discussed in\nthe previous section, the eigenmodel generalizes the latent class model and performs equally well.\n\n6\n\nlllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll0500015000250000100200300400false positivestrue positivesdistanceclassvector\fFigure 3: Relational text data from Genesis and unscaled ROC curves for the K = 3 models.\n\nWe note that parameter estimates for these data were obtained using the ordered probit versions of\nthe models (as the data are not binary), but the out-of-sample predictive performance was evaluated\nbased on each model\u2019s ability to predict a non-zero relationship.\n\n3.5 Protein-protein interaction data\n\nOur last example is the protein-protein interaction data of Butland et al. [2005], in which yi,j = 1\nif proteins i and j bind and yi,j = 0 otherwise. We analyze the large connected component of this\ngraph, which includes 230 proteins and is displayed in the \ufb01rst panel of 4. This graph indicates\npatterns of both stochastic equivalence and homophily: Some nodes could be described as \u201chubs\u201d,\nconnecting to many other nodes which in turn do not connect to each other. Such structure is better\nrepresented by a latent class model than a distance model. However, most nodes connecting to hubs\ngenerally connect to only one hub, which is a feature that is hard to represent with a small number\nof latent classes. To represent this structure well, we would need two latent classes per hub, one for\nthe hub itself and one for the nodes connecting to the hub. Furthermore, the core of the network\n(the nodes with more than two connections) displays a good degree of homophily in the form of\ntransitive triads, a feature which is easiest to represent with a distance model. The eigenmodel is\nable to capture both of these data features and performs better than the other two models in terms of\nout-of-sample predictive performance. In fact, the K = 3 eigenmodel performs better than the other\ntwo models for any value of K considered.\n\n4 Discussion\n\nLatent distance and latent class models provide concise, easily interpreted descriptions of social\nnetworks and relational data. However, neither of these models will provide a complete picture of\nrelational data that exhibit degrees of both homophily and stochastic equivalence. In contrast, we\nhave shown that a latent eigenmodel is able to represent datasets with either or both of these data\npatterns. This is due to the fact that the eigenmodel provides an unrestricted low-rank approximation\nto the sociomatrix, and is therefore able to represent a wide array of patterns in the data.\nThe concept behind the eigenmodel is the familiar eigenvalue decomposition of a symmetric ma-\ntrix. The analogue for directed networks or rectangular matrix data would be a model based on the\nsingular value decomposition, in which data yi,j could be modeled as depending on uT\ni Dvj, where\nui and vj represent vectors of latent row and column effects respectively. Statistical inference using\nthe singular value decomposition for Gaussian data is straightforward. A model-based version of\n\n7\n\n,;:.aaboveabundantlyafterairallalsoandappearbebearingbeastbeginningbeholdblessedbringbroughtcalledcattlecreatedcreaturecreepethcreepingdarknessdaydaysdeepdividedivideddominiondryeartheveningeveryfacefemalefifthfillfinishedfirmamentfirstfishflyforformforthfourthfowlfromfruitfruitfulgatheredgatheringgivegivengodgoodgrassgreatgreatergreenhadhathhaveheheavenheavensherbhimhishostiimageinisititselfkindlandlesserletlifelightlightslikenesslivingmademakemalemanmaymeatmidstmorningmovedmovethmovingmultiplynightofoneopenouroverownplacereplenishrulesaidsawsayingseaseasseasonssecondseedsetshallsignssixthsospiritstarssubduethatthetheirthemtherethingthirdthustotogethertreetwounderuntouponusveryvoidwaswaterswerewhaleswhereinwhichwhosewingedwithoutyearsyieldingyou040008000120000100200300400false positivestrue positivesdistanceclassvector\fFigure 4: Protein-protein interaction data and unscaled ROC curves for the K = 3 models.\n\nthe approach for binary and other non-Gaussian relational datasets could be implemented using the\nordered probit model discussed in this paper.\n\nAcknowledgment\n\nThis work was partially funded by NSF grant number 0631531.\n\nReferences\nEdoardo Airoldi, David Blei, Eric Xing, and Stephen Fienberg. A latent mixed membership model\nIn LinkKDD \u201905: Proceedings of the 3rd international workshop on Link\nfor relational data.\ndiscovery, pages 82\u201389, New York, NY, USA, 2005. ACM Press. ISBN 1-59593-215-1. doi:\nhttp://doi.acm.org/10.1145/1134271.1134283.\n\nDavid J. Aldous. Exchangeability and related topics. In \u00b4Ecole d\u2019\u00b4et\u00b4e de probabilit\u00b4es de Saint-Flour,\n\nXIII\u20141983, volume 1117 of Lecture Notes in Math., pages 1\u2013198. Springer, Berlin, 1985.\n\nG. Butland, J. M. Peregrin-Alvarez, J. Li, W. Yang, X. Yang, V. Canadien, A. Starostine, D. Richards,\nB. Beattie, N. Krogan, M. Davey, J. Parkinson, J. Greenblatt, and A. Emili. Interaction network\ncontaining conserved and essential protein complexes in escherichia coli. Nature, 433:531\u2013537,\n2005.\n\nPeter D. Hoff. Bilinear mixed-effects models for dyadic data. J. Amer. Statist. Assoc., 100(469):\n\n286\u2013295, 2005. ISSN 0162-1459.\n\nPeter D. Hoff, Adrian E. Raftery, and Mark S. Handcock. Latent space approaches to social network\n\nanalysis. J. Amer. Statist. Assoc., 97(460):1090\u20131098, 2002. ISSN 0162-1459.\n\nD. N. Hoover. Row-column exchangeability and a generalized model for probability. In Exchange-\nability in probability and statistics (Rome, 1981), pages 281\u2013291. North-Holland, Amsterdam,\n1982.\n\nCharles Kemp, Thomas L. Grif\ufb01ths, and Joshua B. Tenenbaum. Discovering latent classes in rela-\n\ntional data. AI Memo 2004-019, Massachusetts Institute of Technology, 2004.\n\nKrzysztof Nowicki and Tom A. B. Snijders. Estimation and prediction for stochastic blockstructures.\n\nJ. Amer. Statist. Assoc., 96(455):1077\u20131087, 2001. ISSN 0162-1459.\n\nStanley Wasserman and Katherine Faust. Social Network Analysis: Methods and Applications.\n\nCambridge University Press, Cambridge, 1994.\n\n8\n\nllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll0500015000250000100300500700false positivestrue positivesdistanceclassvector\f", "award": [], "sourceid": 587, "authors": [{"given_name": "Peter", "family_name": "Hoff", "institution": null}]}