This article studies the problem of reconstructing the topology of a network of interacting agents via observations of the state-evolution of the agents. We focus on the large-scale network setting with the additional constraint of @math observations, where only a small fraction of the agents can be feasibly observed. The goal is to infer the underlying subnetwork of interactions and we refer to this problem as @math @math . In order to study the large-scale setting, we adopt a proper stochastic formulation where the unobserved part of the network is modeled as an Erd " o s-R 'enyi random graph, while the observable subnetwork is left arbitrary. The main result of this work is establishing that, under this setting, local tomography is actually possible with high probability, provided that certain conditions on the network model are met (such as stability and symmetry of the network combination matrix). Remarkably, such conclusion is established under the @math - @math @math , where the cardinality of the observable subnetwork is fixed, while the size of the overall network can increase without bound.
We propose a graphical model for representing networks of stochastic processes, the minimal generative model graph. It is based on reduced factorizations of the joint distribution over time. We show that under appropriate conditions, it is unique and consistent with another type of graphical model, the directed information graph, which is based on a generalization of Granger causality. We demonstrate how directed information quantifies Granger causality in a particular sequential prediction setting. We also develop efficient methods to estimate the topological structure from data that obviate estimating the joint statistics. One algorithm assumes upper bounds on the degrees and uses the minimal dimension statistics necessary. In the event that the upper bounds are not valid, the resulting graph is nonetheless an optimal approximation in terms of Kullback-Leibler (KL) divergence. Another algorithm uses near-minimal dimension statistics when no bounds are known, but the distribution satisfies a certain criterion. Analogous to how structure learning algorithms for undirected graphical models use mutual information estimates, these algorithms use directed information estimates. We characterize the sample-complexity of two plug-in directed information estimators and obtain confidence intervals. For the setting when point estimates are unreliable, we propose an algorithm that uses confidence intervals to identify the best approximation that is robust to estimation error. Last, we demonstrate the effectiveness of the proposed algorithms through the analysis of both synthetic data and real data from the Twitter network. In the latter case, we identify which news sources influence users in the network by merely analyzing tweet times. Many tools from the field of graph signal processing exploit knowledge of the underlying graph's structure (e.g., as encoded in the Laplacian matrix) to process signals on the graph. Therefore, in the case when no graph is available, graph signal processing tools cannot be used anymore. Researchers have proposed approaches to infer a graph topology from observations of signals on its vertices. Since the problem is ill-posed, these approaches make assumptions, such as smoothness of the signals on the graph, or sparsity priors. In this paper, we propose a characterization of the space of valid graphs, in the sense that they can explain stationary signals. To simplify the exposition in this paper, we focus here on the case where signals were i.i.d. at some point back in time and were observed after diffusion on a graph. We show that the set of graphs verifying this assumption has a strong connection with the eigenvectors of the covariance matrix, and forms a convex set. Along with a theoretical study in which these eigenvectors are assumed to be known, we consider the practical case when the observations are noisy, and experimentally observe how fast the set of valid graphs converges to the set obtained when the exact eigenvectors are known, as the number of observations grows. To illustrate how this characterization can be used for graph recovery, we present two methods for selecting a particular point in this set under chosen criteria, namely graph simplicity and sparsity. Additionally, we introduce a measure to evaluate how much a graph is adapted to signals under a stationarity assumption. Finally, we evaluate how state-of-the-art methods relate to this framework through experiments on a dataset of temperatures.  Many applications collect a large number of time series, for example, the financial data of companies quoted in a stock exchange, the health care data of all patients that visit the emergency room of a hospital, or the temperature sequences continuously measured by weather stations across the US. These data are often referred to as un structured. The first task in its analytics is to derive a low dimensional representation, a graph or discrete manifold, that describes well the inter relations among the time series and their intra relations across time. This paper presents a computationally tractable algorithm for estimating this graph that structures the data. The resulting graph is directed and weighted, possibly capturing causal relations, not just reciprocal correlations as in many existing approaches in the literature. A convergence analysis is carried out. The algorithm is demonstrated on random graph datasets and real network time series datasets, and its performance is compared to that of related methods. The adjacency matrices estimated with the new method are close to the true graph in the simulated data and consistent with prior physical knowledge in the real dataset tested. We introduce a measure of causality that captures the functional dependencies in dynamical systems and subsequently, define a new type of graphical model, functional dependency graph, to encode such dependencies. We study the relationship between this type of graphical model and other graphical models such as directed information graphs and linear dynamical graphs that have been proposed to capture causal influences in dynamical systems. We show that functional dependency graphs are a generalization of these previously introduced graphical models and learn the functional dependencies in a larger class of models. We also establish sufficient conditions under which the functional dependency graph defined through our measure is equivalent to the directed information graphs. Some simulation results on linear and nonlinear dynamics are provided.