  Let $N$ local decision makers in a sensor network communicate with their
neighbors to reach a decision \emph{consensus}. Communication is local, among
neighboring sensors only, through noiseless or noisy links. We study the design
of the network topology that optimizes the rate of convergence of the iterative
decision consensus algorithm. We reformulate the topology design problem as a
spectral graph design problem, namely, maximizing the eigenratio~$\gamma$ of
two eigenvalues of the graph Laplacian~$L$, a matrix that is naturally
associated with the interconnectivity pattern of the network. This
reformulation avoids costly Monte Carlo simulations and leads to the class of
non-bipartite Ramanujan graphs for which we find a lower bound on~$\gamma$. For
Ramanujan topologies and noiseless links, the local probability of error
converges much faster to the overall global probability of error than for
structured graphs, random graphs, or graphs exhibiting small-world
characteristics. With noisy links, we determine the optimal number of
iterations before calling a decision. Finally, we introduce a new class of
random graphs that are easy to construct, can be designed with arbitrary number
of sensors, and whose spectral and convergence properties make them practically
equivalent to Ramanujan topologies.
