Contagion processes are strongly linked to the network structures on which they propagate, and learning these structures is essential for understanding and intervention on complex network processes such as epidemics and (mis)information propagation. However, using contagion data to infer network structure is a challenging inverse problem. In particular, it is imperative to have appropriate measures of uncertainty in network structure estimates, however these are largely ignored in most machine-learning approaches. We present a probabilistic framework that uses samples from the distribution of networks that are compatible with the dynamics observed to produce network and uncertainty estimates. We demonstrate the method using the well known independent cascade model to sample from the distribution of networks P(G) conditioned on the observation of a set of infections C. We evaluate the accuracy of the method by using the marginal probabilities of each edge in the distribution, and show the bene ts of quantifying uncertainty to improve estimates and understanding, particularly with small amounts of data.
@cite_38: Can evolving networks be inferred and modeled without directly observing their nodes and edges? In many applications, the edges of a dynamic network might not be observed, but one can observe the dynamics of stochastic cascading processes (e.g., information diffusion, virus propagation) occurring over the unobserved network. While there have been efforts to infer networks based on such data, providing a generative probabilistic model that is able to identify the underlying time-varying network remains an open question. Here we consider the problem of inferring generative dynamic network models based on network cascade diffusion data. We propose a novel framework for providing a non-parametric dynamic network model---based on a mixture of coupled hierarchical Dirichlet processes---based on data capturing cascade node infection times. Our approach allows us to infer the evolving community structure in networks and to obtain an explicit predictive distribution over the edges of the underlying network---including those that were not involved in transmission of any cascade, or are likely to appear in the future. We show the effectiveness of our approach using extensive experiments on synthetic as well as real-world networks.
@cite_37: We present a scalable nonparametric Bayesian method to perform network reconstruction from observed functional behavior, that at the same time infers the communities present in the network. We show that the joint reconstruction with community detection has a synergistic effect, where the edge correlations used to inform the existence of communities are inherently also used to improve the accuracy of the reconstruction, which in turn can better inform the uncovering of communities. We illustrate the use of our method with observations arising from epidemic models and the Ising model, both on synthetic and empirical networks, as well as on data containing only functional information.
@cite_33: Networks play a central role in modern data analysis, enabling us to reason about systems by studying the relationships between their parts. Most often in network analysis, the edges are given. However, in many systems it is difficult or impossible to measure the network directly. Examples of latent networks include economic interactions linking financial instruments and patterns of reciprocity in gang violence. In these cases, we are limited to noisy observations of events associated with each node. To enable analysis of these implicit networks, we develop a probabilistic model that combines mutually-exciting point processes with random graph models. We show how the Poisson superposition principle enables an elegant auxiliary variable formulation and a fully-Bayesian, parallel inference algorithm. We evaluate this new model empirically on several datasets.
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@cite_15: The analysis of network connections, diffusion processes and cascades requires evaluating properties of the diffusion network. Properties of interest often involve variables that are not explicitly observed in real world diffusions. Connection strengths in the network and diffusion paths of infections over the network are examples of such hidden variables. These hidden variables therefore need to be estimated for these properties to be evaluated. In this paper, we propose and study this novel problem in a Bayesian framework by capturing the posterior distribution of these hidden variables given the observed cascades, and computing the expectation of these properties under this posterior distribution. We identify and characterize interesting network diffusion properties whose expectations can be computed exactly and efficiently, either wholly or in part. For properties that are not nice' in this sense, we propose a Gibbs Sampling framework for Monte Carlo integration. In detailed experiments using various network diffusion properties over multiple synthetic and real datasets, we demonstrate that the proposed approach is significantly more accurate than a frequentist plug-in baseline. We also propose a map-reduce implementation of our framework and demonstrate that this can analyze cascades with millions of infections in minutes.
@cite_16: 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.
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@cite_11: In real-world and online social networks, individuals receive and transmit information in real time. Cascading information transmissions (e.g. phone calls, text messages, social media posts) may be understood as a realization of a diffusion process operating on the network, and its branching path can be represented by a directed tree. The process only traverses and thus reveals a limited portion of the edges. The network reconstruction inference problem is to infer the unrevealed connections. Most existing approaches derive a likelihood and attempt to find the network topology maximizing the likelihood, a problem that is highly intractable. In this paper, we focus on the network reconstruction problem for a broad class of real-world diffusion processes, exemplified by a network diffusion scheme called respondent-driven sampling (RDS). We prove that under realistic and general models of network diffusion, the posterior distribution of an observed RDS realization is a Bayesian log-submodular model.We then propose VINE (Variational Inference for Network rEconstruction), a novel, accurate, and computationally efficient variational inference algorithm, for the network reconstruction problem under this model. Crucially, we do not assume any particular probabilistic model for the underlying network. VINE recovers any connected graph with high accuracy as shown by our experimental results on real-life networks.
Embar propose a Bayesian framework for estimating properties of the network, edge strengths as well as cascade properties, propagation trees @cite_15 . Other Bayesian approaches of network inference aim to infer the underlying network when the exact infection time is unknown in both standard and online algorithms @cite_33 @cite_8 @cite_1 . Additionally, some approaches based on MLE use Bayesian techniques at intermediate steps to improve results @cite_11 . Only recently have Bayesian techniques have been introduced to quantify the uncertainty of network structure inference from observation of dynamics @cite_16 @cite_42 @cite_40 @cite_13 . Ghalebi highlight the need for general probabilistic frameworks for inference problems and propose the algorithm DYFERENCE that samples edge and node probabilities in an online algorithm for dynamic network inference @cite_38 . Peixoto @cite_37 recently proposed the most closely related work in an algorithm designed to jointly reconstruct network structure and community labels assuming a stochastic block model structure and highlights the benefits of recovering the full posterior distribution of networks.