In this paper we consider the problem of learning undirected graphical models from data generated according to the Glauber dynamics. The Glauber dynamics is a Markov chain that sequentially updates individual nodes (variables) in a graphical model and it is frequently used to sample from the stationary distribution (to which it converges given sufficient time). Additionally, the Glauber dynamics is a natural dynamical model in a variety of settings. This work deviates from the standard formulation of graphical model learning in the literature, where one assumes access to i.i.d. samples from the distribution. Much of the research on graphical model learning has been directed towards finding algorithms with low computational cost. As the main result of this work, we establish that the problem of reconstructing binary pairwise graphical models is computationally tractable when we observe the Glauber dynamics. Specifically, we show that a binary pairwise graphical model on @math nodes with maximum degree @math can be learned in time @math , for a function @math , using nearly the information-theoretic minimum number of samples.
Several works have studied the problem of learning the graph underlying a random process for various processes. These include learning from epidemic cascades @cite_1 @cite_0 @cite_34 and learning from delay measurements @cite_38 . Another line of research asks to find the source of infection of an epidemic by observing the current state, where the graph is known @cite_36 @cite_42 .
@cite_38 @cite_36 @cite_42 @cite_1 @cite_0 @cite_34
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We consider the problem of finding the graph on which an epidemic spreads, given only the times when each node gets infected. While this is a problem of central importance in several contexts -- offline and online social networks, e-commerce, epidemiology -- there has been very little work, analytical or empirical, on finding the graph. Clearly, it is impossible to do so from just one epidemic; our interest is in learning the graph from a small number of independent epidemics. For the classic and popular "independent cascade" epidemics, we analytically establish sufficient conditions on the number of epidemics for both the global maximum-likelihood (ML) estimator, and a natural greedy algorithm to succeed with high probability. Both results are based on a key observation: the global graph learning problem decouples into n local problems -- one for each node. For a node of degree d, we show that its neighborhood can be reliably found once it has been infected O(d2 log n) times (for ML on general graphs) or O(d log n) times (for greedy on trees). We also provide a corresponding information-theoretic lower bound of Ω(d log n); thus our bounds are essentially tight. Furthermore, if we are given side-information in the form of a super-graph of the actual graph (as is often the case), then the number of epidemic samples required -- in all cases -- becomes independent of the network size n. The paper is concerned with new methodology for statistical inference for final outcome infectious disease data using certain structured population stochastic epidemic models. A major obstacle to inference for such models is that the likelihood is both analytically and numerically intractable. The approach that is taken here is to impute missing information in the form of a random graph that describes the potential infectious contacts between individuals. This level of imputation overcomes various constraints of existing methodologies and yields more detailed information about the spread of disease. The methods are illustrated with both real and test data. Copyright 2005 Royal Statistical Society. We study the problem of estimating the origin of an epidemic outbreak -- given a contact network and a snapshot of epidemic spread at a certain time, determine the infection source. Finding the source is important in different contexts of computer or social networks. We assume that the epidemic spread follows the most commonly used susceptible-infected-recovered model. We introduce an inference algorithm based on dynamic message-passing equations, and we show that it leads to significant improvement of performance compared to existing approaches. Importantly, this algorithm remains efficient in the case where one knows the state of only a fraction of nodes. We consider the contact process on a random graph with fixed degree distribution given by a power law. We follow the work of Chatterjee and Durrett (2009), who showed that for arbitrarily small infection parameter @math , the survival time of the process is larger than a stretched exponential function of the number of vertices, @math . We obtain sharp bounds for the typical density of infected sites in the graph, as @math is kept fixed and @math tends to infinity. We exhibit three different regimes for this density, depending on the tail of the degree law.