We provide a framework for modeling social network formation through conditional multinomial logit models from discrete choice and random utility theory, in which each new edge is viewed as a “choice” made by a node to connect to another node, based on (generic) features of the other nodes available to make a connection. This perspective on network formation unifies existing models such as preferential attachment, triadic closure, and node fitness, which are all special cases, and thereby provides a flexible means for conceptualizing, estimating, and comparing models. The lens of discrete choice theory also provides several new tools for analyzing social network formation; for example, the significance of node features can be evaluated in a statistically rigorous manner, and mixtures of existing models can be estimated by adapting known expectation-maximization algorithms. We demonstrate the flexibility of our framework through examples that analyze a number of synthetic and real-world datasets. For example, we provide rigorous methods for estimating preferential attachment models and show how to separate the effects of preferential attachment and triadic closure. Non-parametric estimates of the importance of degree show a highly linear trend, and we expose the importance of looking carefully at nodes with degree zero. Examining the formation of a large citation graph, we find evidence for an increased role of degree when accounting for age.
Estimating the parameters that drive edge formation is different from identifying the factors that could have lead to the observed graph. There is vast literature on pursuing the latter question by estimating a logit model with maximum likelihood, called exponential random graph models (ERGMs) @cite_46 @cite_42 @cite_54 . However, these models do not consider individual edge events, are hard to estimate, and have known pathologies @cite_2 @cite_68 . Should check out TERGMs again, temporal ERGMs, and put that work in it's place here. I think Carter Butts is in that literature.
@cite_54 @cite_42 @cite_2 @cite_46 @cite_68
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We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph G from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit ‘single-letter’ characterization of the per-vertex mutual information between the vertex labels and the graph, when the graph average degree diverges. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and -in particular- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent of the vertex labels. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing. We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph @math from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit single-letter' characterization of the per-vertex mutual information between the vertex labels and the graph. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and --in particular-- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing. We consider the popular and well-studied push model, which is used to spread information in a given network with n vertices. Initially, some vertex owns a rumour and passes it to one of its neighbours, which is chosen randomly. In each of the succeeding rounds, every vertex that knows the rumour informs a random neighbour. It has been shown on various network topologies that this algorithm succeeds in spreading the rumour within O(log n) rounds. However, many studies are quite coarse and involve huge constants that do not allow for a direct comparison between different network topologies. In this paper, we analyse the push model on several important families of graphs, and obtain tight runtime estimates. We first show that, for any almost-regular graph on n vertices with small spectral expansion, rumour spreading completes after log2n + log n+o(log n) rounds with high probability. This is the first result that exhibits a general graph class for which rumour spreading is essentially as fast as on complete graphs. Moreover, for the random graph G(n,p) with p=c log n n, where c > 1, we determine the runtime of rumour spreading to be log2n + γ (c)log n with high probability, where γ(c) = clog(c (c−1)). In particular, this shows that the assumption of almost regularity in our first result is necessary. Finally, for a hypercube on n=2d vertices, the runtime is with high probability at least (1+β) ⋅ (log2n + log n), where β > 0. This reveals that the push model on hypercubes is slower than on complete graphs, and thus shows that the assumption of small spectral expansion in our first result is also necessary. In addition, our results combined with the upper bound of O(log n) for the hypercube (see [11]) imply that the push model is faster on hypercubes than on a random graph G(n, clog n n), where c is sufficiently close to 1. The “classical” random graph models, in particular G(n,p), are “homogeneous,” in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power-law degree distributions. Thus there has been a lot of recent interest in defining and studying “inhomogeneous” random graph models. One of the most studied properties of these new models is their “robustness”, or, equivalently, the “phase transition” as an edge density parameter is varied. For G(n,p), p = c n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogeneous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this property already, and others can be approximated by models with independence. Here we introduce a very general model of an inhomogeneous random graph with (conditional) independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p = c n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real-world graphs. Our model includes as special cases many models previously studied. We show that, under one very weak assumption (that the expected number of edges is “what it should be”), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is “stable”: for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n). © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 31, 3–122, 2007 The growing availability of network data and of scientific interest in distributed systems has led to the rapid development of statistical models of network structure. Typically, however, these are models for the entire network, while the data consists only of a sampled sub-network. Parameters for the whole network, which is what is of interest, are estimated by applying the model to the sub-network. This assumes that the model is consistent under sampling, or, in terms of the theory of stochastic processes, that it defines a projective family. Focusing on the popular class of exponential random graph models (ERGMs), we show that this apparently trivial condition is in fact violated by many popular and scientifically appealing models, and that satisfying it drastically limits ERGM’s expressive power. These results are actually special cases of more general results about exponential families of dependent random variables, which we also prove. Using such results, we offer easily checked conditions for the consistency of maximum likelihood estimation in ERGMs, and discuss some possible constructive responses. This paper considers the inverse problem with observed variables Y = BGX ⊕Z, where BG is the incidence matrix of a graph G, X is the vector of unknown vertex variables with a uniform prior, and Z is a noise vector with Bernoulli(e) i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery of X is possible if and only the graph G is connected, with a sharp threshold at the edge probability log(n) n for Erdős-Renyi random graphs. The first goal of this paper is to determine how the edge probability p needs to scale to allow exact recovery in the presence of noise. Defining the degree (oversampling) rate of the graph by α = np log(n), it is shown that exact recovery is possible if and only if α > 2 (1−2e)+o(1 (1−2e)). In other words, 2 (1−2e) is the information theoretic threshold for exact recovery at lowSNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. Full version available in [1]. A statistical approach to a posteriori blockmodeling for graphs is proposed. The model assumes that the vertices of the graph are partitioned into two unknown blocks and that the probability of an edge between two vertices depends only on the blocks to which they belong. Statistical procedures are derived for estimating the probabilities of edges and for predicting the block structure from observations of the edge pattern only. ML estimators can be computed using the EM algorithm, but this strategy is practical only for small graphs. A Bayesian estimator, based on Gibbs sampling, is proposed. This estimator is practical also for large graphs. When ML estimators are used, the block structure can be predicted based on predictive likelihood. When Gibbs sampling is used, the block structure can be predicted from posterior predictive probabilities. A side result is that when the number of vertices tends to infinity while the probabilities remain constant, the block structure can be recovered correctly with probability tending to 1. (Less) We describe some of the capabilities of the ergm package and the statistical theory underlying it. This package contains tools for accomplishing three important, and inter-related, tasks involving exponential-family random graph models (ERGMs): estimation, simulation, and goodness of fit. More precisely, ergm has the capability of approximating a maximum likelihood estimator for an ERGM given a network data set; simulating new network data sets from a fitted ERGM using Markov chain Monte Carlo; and assessing how well a fitted ERGM does at capturing characteristics of a particular network data set. Random graph models with limited choice have been studied extensively with the goal of understanding the mechanism of the emergence of the giant component. One of the standard models are the Achlioptas random graph processes on a fixed set of (n ) vertices. Here at each step, one chooses two edges uniformly at random and then decides which one to add to the existing configuration according to some criterion. An important class of such rules are the bounded-size rules where for a fixed (K 1 ), all components of size greater than (K ) are treated equally. While a great deal of work has gone into analyzing the subcritical and supercritical regimes, the nature of the critical scaling window, the size and complexity (deviation from trees) of the components in the critical regime and nature of the merging dynamics has not been well understood. In this work we study such questions for general bounded-size rules. Our first main contribution is the construction of an extension of Aldous’s standard multiplicative coalescent process which describes the asymptotic evolution of the vector of sizes and surplus of all components. We show that this process, referred to as the standard augmented multiplicative coalescent (AMC) is ‘nearly’ Feller with a suitable topology on the state space. Our second main result proves the convergence of suitably scaled component size and surplus vector, for any bounded-size rule, to the standard AMC. This result is new even for the classical Erdős–Renyi setting. The key ingredients here are a precise analysis of the asymptotic behavior of various susceptibility functions near criticality and certain bounds from (The barely subcritical regime. Arxiv preprint, 2012) on the size of the largest component in the barely subcritical regime. Exponential-family random graph models (ERGMs) provide a principled and flexible way to model and simulate features common in social networks, such as propensities for homophily, mutuality, and friend-of-a-friend triad closure, through choice of model terms (sufficient statistics). However, those ERGMs modeling the more complex features have, to date, been limited to binary data: presence or absence of ties. Thus, analysis of valued networks, such as those where counts, measurements, or ranks are observed, has necessitated dichotomizing them, losing information and introducing biases. In this work, we generalize ERGMs to valued networks. Focusing on modeling counts, we formulate an ERGM for networks whose ties are counts and discuss issues that arise when moving beyond the binary case. We introduce model terms that generalize and model common social network features for such data and apply these methods to a network dataset whose values are counts of interactions. In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs (G n ). Let λn be the largest eigenvalue of the adjacency matrix of G n , and let G n (p n ) be the random subgraph of G n obtained by keeping each edge independently with probability p n . We show that the appearance of a giant component in G n (p n ) has a sharp threshold at p n = 1 λ n . In fact, we prove much more: if (G n ) converges to an irreducible limit, then the density of the largest component of G n (c n) tends to the survival probability of a multi-type branching process defined in terms of this limit. Here the notions of convergence and limit are those of Borgs, Chayes, Lovasz, S6s and Vesztergombi. In addition to using basic properties of convergence, we make heavy use of the methods of Bollobas, Janson and Riordan, who used multi-type branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs. We introduce stochastic time-dependency in evolving graphs: starting from an arbitrary initial edge probability distribution, at every time step, every edge changes its state (existing or not) according to a two-state Markovian process with probabilities p (edge birth-rate) and q (edge death-rate). If an edge exists at time t then, at time t+1, it dies with probability q. If instead the edge does not exist at time t, then it will come into existence at time t+1 with probability p. Such evolving graph model is a wide generalization of time-independent dynamic random graphs [6] and will be called edge-Markovian dynamic graphs. We investigate the speed of information dissemination in such dynamic graphs. We provide nearly tight bounds (which in fact turn out to be tight for a wide range of probabilities p and q) on the completion time of the flooding mechanism aiming to broadcast a piece of information from a source node to all nodes. In particular, we provide: i) A tight characterization of the class of edge-Markovian dynamic graphs where flooding time is constant and, thus, it does not asymptotically depend on the initial probability distribution. ii) A tight characterization of the class of edge-Markovian dynamic graphs where flooding time does not asymptotically depend on the edge death-rate q. Consider the random graph models G(n, #edges = M) and G(n, Prob(edge) = p) with M = M(n) = (1 + )n-1 3)n 2 and p = p(n) = (1 +An-1 3) n . For 1 > -1 define an i-component of a random graph as a component which has exactly I more edges than vertices. Call an i-component with I > 1 a complex component. For both models, we show that when A is constant, the expected number of complex components is bounded, almost surely (a.s.) each of these components (if any exist) has size of order n2 3, and the maximum value of I is bounded in probability. We prove that a.s. the largest suspended tree in each complex component has size of order n2 3, and deletion of all suspended trees results in a "smoothed" graph of size of order n 3, with the maximum vertex degree 3. The total number of branching vertices, i.e., of degree 3, is bounded in probability. Thus, each complex component is almost surely topologically equivalent to a 3-regular multigraph of a uniformly bounded size. Lengths of the shortest cycle and of the shortest path between two branching vertices of a smoothed graph are each of order n 3. We find a relatively simple integral formula for the limit distribution of the numbers of complex components, which implies, in particular, that all values of the "complexity spectrum" have positive limiting probabilities. We also answer questions raised by Erdos and Renyi back in 1960. It is proven that there exists p(A), the limiting planarity probability, with 0 < p(A) < 1, p(-oo) = 1, p(oo) = 0. In particular, G(n, M) (G(n, p), resp.) is almost surely nonplanar iff (M n 2)n-213 -* 00 ((np I)n-1 3) -+ oo, resp.). Abstract Exponential random graph models are extremely difficult models to handle from a statistical viewpoint, since their normalising constant, which depends on model parameters, is available only in very trivial cases. We show how inference can be carried out in a Bayesian framework using a MCMC algorithm, which circumvents the need to calculate the normalising constants. We use a population MCMC approach which accelerates convergence and improves mixing of the Markov chain. This approach improves performance with respect to the Monte Carlo maximum likelihood method of Geyer and Thompson (1992) . The percolation phase transition and the mechanism of the emergence of the giant component through the critical scaling window for random graph models, has been a topic of great interest in many dierent communities ranging from statistical physics, combinatorics, computer science, social networks and probability theory. The last few years have witnessed an explosion of models which couple random aggregation rules, that specify how one adds edges to existing congurations, and choice, wherein one selects from a ‘limited’ set of edges at random to use in the conguration. These models exhibit fascinating new phenomenon, ranging from delaying or speeding up the emergence of the giant component, to percolation", where the diameter of the scaling window is several orders of magnitude smaller than that for standard random graph models. While an intense study of such models has ensued, understanding the actual emergence of the giant component and merging dynamics in the critical scaling window has remained impenetrable to a rigorous analysis. In this work we take an important step in the analysis of such models by studying one of the standard examples of such processes, namely the Bohman-Frieze model, and provide rst results on the asymptotic dynamics, through the critical scaling window, that lead to the emergence of the giant component for such models. We identify the scaling window and show that through this window, the component sizes properly rescaled converge to the standard multiplicative coalescent. Proofs hinge on a careful analysis of certain innite-type branching processes with types taking values in the space of RCLL paths, and stochastic analytic techniques to estimate susceptibility functions of the components all the way through the scaling window where these functions explode. Previous approaches for analyzing random graphs at criticality have relied largely on classical breadth-rst search techniques that exploit asymptotic connections with Brownian excursions. For dynamic random graph models evolving via general Markovian rules, such approaches fail and we develop a quite dierent set of tools that can potentially be used for the study of critical dynamics for all bounded size rules. We propose a random graph model which is a special case of sparse random graphs with given degree sequences. This model involves only a small number of parameters, called logsize and log-log growth rate. These parameters capture some universal characteristics of massive graphs. Furthermore, from these parameters, various properties of the graph can be derived. For example, for certain ranges of the parameters, we will compute the expected distribution of the sizes of the connected components which almost surely occur with high probability. We will illustrate the consistency of our model with the behavior of some massive graphs derived from data in telecommunications. We will also discuss the threshold function, the giant component, and the evolution of random graphs in this model. Let G = G(d) be a random graph with a given degree sequence d, such as a random r-regular graph where r â¥ 3 is fixed and n = |G| â ∞. We study the percolation phase transition on such graphs G, i.e., the emergence as p increases of a unique giant component in the random subgraph G[p] obtained by keeping edges independently with probability p. More generally, we study the emergence of a giant component in G(d) itself as d varies. We show that a single method can be used to prove very precise results below, inside and above the 'scaling window' of the phase transition, matching many of the known results for the much simpler model G(n, p). This method is a natural extension of that used by Bollobas and the author to study G(n, p), itself based on work of Aldous and of Nachmias and Peres; the calculations are significantly more involved in the present setting. This article provides an introductory summary to the formulation and application of exponential random graph models for social networks. The possible ties among nodes of a network are regarded as random variables, and assumptions about dependencies among these random tie variables determine the general form of the exponential random graph model for the network. Examples of different dependence assumptions and their associated models are given, including Bernoulli, dyad-independent and Markov random graph models. The incorporation of actor attributes in social selection models is also reviewed. Newer, more complex dependence assumptions are briefly outlined. Estimation procedures are discussed, including new methods for Monte Carlo maximum likelihood estimation. We foreshadow the discussion taken up in other papers in this special edition: that the homogeneous Markov random graph models of Frank and Strauss [Frank, O., Strauss, D., 1986. Markov graphs. Journal of the American Statistical Association 81, 832–842] are not appropriate for many observed networks, whereas the new model specifications of [Snijders, T.A.B., Pattison, P., Robins, G.L., Handock, M. New specifications for exponential random graph Let P be a Poisson process of intensity one in a square S n of area n. We construct a random geometric graph G n , k by joining each point of P to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that G n , k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that G n , k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With G n , k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover S n tends to 0 as n → ∞ while, if k > 0.9967 log n, then the probability that the discs cover S n tends to 1 as n → ∞. Learning about the structure of hidden or covert networks is a major challenge in epidemiology, sociology, and intelligence analysis. Vertices in hidden networks usually cannot be enumerated or sampled in a systematic way; they can only be revealed by tracing links emanating from already-observed vertices. Observers sometimes cannot follow links directly, and instead must rely on passive observation of a dynamic process to reveal vertices and edges. This paper outlines a framework for estimating network structures from partial observation of information diffusion through the network. Diffusion is modeled by a continuous-time Markov epidemic model. Edges are revealed by transmission events and new vertices are uncovered when information is transmitted to them. The approach is a generalization of tools developed to reconstruct drug-user networks from respondent-driven sampling studies in epidemiology. The likelihood of the diffusion process can be interpreted as an exponential random graph model. A Bayesian method for probabilistic reconstruction of the transmission-induced subgraph is described.