Learning Erd\H{o}s-R\'enyi Random Graphs via Edge Detecting Queries

In this paper, we consider the problem of learning an unknown graph via queries on groups of nodes, with the result indicating whether or not at least one edge is present among those nodes. We establish such bounds for a variety of algorithms inspired by the group testing problem, with explicit constant factors indicating a near-optimal number of tests, and in some cases asymptotic optimality including constant factors. I. INTRODUCTION Graphs are a ubiquitous tool in modern statistics and machine learning for depicting interactions, relations, and physical connections in networks, such as social networks, biological networks, sensor networks, and so on. Often, the graph is not known a priori, and must be learned via queries to the network. In this paper, we consider the problem of graph learning via edge detecting queries, where each query contains a subset of the nodes, and the binary outcome indicates whether or not there is at least one edge among these nodes. See Section IA for previous work on this problem. An application of this problem highlighted in previous works such as [15] is that of learning which chemicals react with each other, using tests that are able to detect whether any reaction occurs. Another potential application is learning connectivity in large wireless networks: Each node is given a unique identifier, and in response to a query, each node sends feedback to a central unit if both itself and one or more of its neigbors are included in that query.