Detection of graph structures via communications over a multiaccess Boolean channel

In this paper, we propose a novel model to study the efficiency of detecting latent connection relationships, represented by a given set of graphs, among N users. A subset of active nodes transmit following a common codebook over a multiple access Boolean channel. To maximize the error exponent of the structure detection, we formulate an optimization problem whose objective is to max-minimize the pairwise Chernoff information, and the constraint is a probability simplex due to the users´ multiple dependency relationships, which are further shown to have close relationship to the internal connectivity of graphs. Case studies are provided to show certain inherent properties of the optimal solution. In addition, we present a particular case with two equally weighted complementary Paley graphs of prime square order, whose optimal solution for the codebook is proved and the resulting exponent is shown to be O(1/N). The case study demonstrates how the fundamental graph discrepancy property affects the solution to the problem.