Mean-field Approximation for Stochastic Population Processes in Networks under Imperfect Information

This paper studies a general class of stochastic population processes in which agents interact with one another over a network. Agents update their behaviors in a random and decentralized manner based only on their current state and the states of their neighbors. It is well known that when the number of agents is large and the network is a complete graph (has all-to-all information access), the macroscopic behavior of the population converges to a differential equation called a {\it mean-field approximation}. When the network is not complete, it is unclear in general whether there exists a suitable mean-field approximation for the macroscopic behavior of the population. This paper provides general conditions on the network and policy dynamics for which a suitable mean-field approximation exists. First, we show that as long as the network is well-connected, the macroscopic behavior of the population concentrates around the {\it same} mean-field system as the complete-graph case. Next, we show that as long as the network is sufficiently dense, the macroscopic behavior of the population concentrates around a mean-field system that is, in general, {\it different} from the mean-field system obtained in the complete-graph case. Finally, we provide conditions under which the mean-field approximation is equivalent to the one obtained in the complete-graph case.Comment: 59 pages, 4 figure

MARGIN: Uncovering Deep Neural Networks using Graph Signal Analysis

Interpretability has emerged as a crucial aspect of machine learning, aimed at providing insights into the working of complex neural networks. However, existing solutions vary vastly based on the nature of the interpretability task, with each use case requiring substantial time and effort. This paper introduces MARGIN, a simple yet general approach to address a large set of interpretability tasks ranging from identifying prototypes to explaining image predictions. MARGIN exploits ideas rooted in graph signal analysis to determine influential nodes in a graph, which are defined as those nodes that maximally describe a function defined on the graph. By carefully defining task-specific graphs and functions, we demonstrate that MARGIN outperforms existing approaches in a number of disparate interpretability challenges.Comment: Technical Repor

Recovering the Graph Underlying Networked Dynamical Systems under Partial Observability: A Deep Learning Approach

We study the problem of graph structure identification, i.e., of recovering the graph of dependencies among time series. We model these time series data as components of the state of linear stochastic networked dynamical systems. We assume partial observability, where the state evolution of only a subset of nodes comprising the network is observed. We devise a new feature vector computed from the observed time series and prove that these features are linearly separable, i.e., there exists a hyperplane that separates the cluster of features associated with connected pairs of nodes from those associated with disconnected pairs. This renders the features amenable to train a variety of classifiers to perform causal inference. In particular, we use these features to train Convolutional Neural Networks (CNNs). The resulting causal inference mechanism outperforms state-of-the-art counterparts w.r.t. sample-complexity. The trained CNNs generalize well over structurally distinct networks (dense or sparse) and noise-level profiles. Remarkably, they also generalize well to real-world networks while trained over a synthetic network (realization of a random graph). Finally, the proposed method consistently reconstructs the graph in a pairwise manner, that is, by deciding if an edge or arrow is present or absent in each pair of nodes, from the corresponding time series of each pair. This fits the framework of large-scale systems, where observation or processing of all nodes in the network is prohibitive.Comment: Accepted at The 37th AAAI Conference on Artificial Intelligence (main track