Ramanujan Topologies for Decision Making in Sensor Networks

We consider the design of the topology of the communication graph G = (V, E) supporting distributed decision in sensor networks with N = |V | sensors. The numberM of links connecting the sensors, i.e., the number of edges |E| = M in the graph G, is fixed. We assume a simple binary decision test where the data may be spatially correlated. The global detector performs a threshold test on a weighted fusion of the local likelihood ratios, which can be computed in a distributed fashion using a consensus algorithm. The graph topology plays a central role in the convergence speed of the distributed detector. Exhaustive search over the class of possible communication networks is unrealistic. Our solution is constructive. We first reduce this topology design to a spectral graph optimization problem; specifically, to designing the topology that maximizes the ratio γ of the algebraic connectivity to the largest eigenvalue of the graph Laplacian. Borrowing results from spectral graph theory, we show that for the class of non-bipartite Ramanujan graphs γ ≥ γmin . The importance of this inequality is that γmin , asymptotically, is an upper bound onγ for most classes of graphs. The paper discusses the commonly used explicit constructions of Ramanujan graphs and their impact on the convergence speed of distributed consensus. In particular, it shows that these graphs perform much better even for finite values of N than highly structured networks, or small world type graphs, or Erdös-Reńyi random networks.