Quickest Search for Local Structures in Random Graphs

A network of agents that form a random graph is considered. Each agent represents an information source that generates a sequence of random variables (RVs). The RVs generated by an unknown subset of nodes are correlated according to a known kernel, while the remaining nodes generate independent and identically distributed random variables. To identify and localize the desired unknown subset of correlated nodes, this paper formalizes and delineates a quickest search process, which is the strategy that minimizes the average number of measurements. Despite its widespread applications, the problem of identifying subgraphs with such desired correlation structures is often investigated under the fixed sample-size settings, in which the data acquisition process and the inferential mechanisms are decoupled. Motivated by the significant advantages of sequential methods for agile inference, this paper analyzes this problem under a fully sequential setting. Specifically, it offers a framework that unifies the intertwined processes of information gathering and decision making, and through a constructive proof, it provides an optimal sequential data-gathering process as well as the attendant decision rules for the quickest search of interest.