Global information from local observation

We observe a certain random process on a graph "locally", i.e., in the neighborhood of a node, and would like to derive information about "global" properties of the graph. For example, what can we know about a graph based on observing the returns of a random walk to a given node? Our main result concerns a graph embedded in an orientable surface with genus g, and a process, consisting of random excitations of edges and random balancing around nodes and faces. It is shown how to obtain the genus of the surface in polynomial time from local observations of the process restricted to a connected subgraph whose size is (essentially) O(g²).