Abstract-We consider the problem of detecting a random walk on a graph, based on observations of the graph nodes. When visited by the walk, each node of the graph observes a signal of elevated mean, which we assume can be different across different nodes. Outside of the path of the walk, and also in its absence, nodes measure only noise. Assuming the NeymanPearson setting, our goal then is to characterize detection performance by computing the error exponent for the probability of a miss, under a constraint on the probability of false alarm. Since the exact computation of the error exponent is known to be difficult, equivalent to the computation of the Lyapunov exponent, we approximate its value by finding a tractable lower bound. The bound reveals an interesting detectability condition: the walk is detectable whenever the entropy of the walk is smaller than one half of the expected signal-to-noise ratio. We derive the bound by extending the notion of Markov types to GaussMarkov types. These are sequences of the state-observation pairs with a given number of node-to-node transition counts and the same average signal values across nodes, computed from the measurements made during the times the random walk visited each node's respective location. The lower bound has an intuitive interpretation: among all Gauss-Markov types that are asymptotically feasible in the absence of the walk, the bound finds the most typical one under the presence of the walk. Finally, we show by a sequence of judicious problem reformulations that computing the bound reduces to solving a convex optimization problem, which is a result of in its interest own right. Index Terms-Random walk, hypothesis testing, error exponent, large deviations principle, threshold effect, Gauss-Markov type, convex analysis, Lyapunov exponent.
In paper REF , the authors discuss the issue of detecting a random walk on a graph based on observations of the graph nodes.
Detecting Random Walks on Graphs With Heterogeneous Sensors
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Computer Science Mathematics Physics