  Identifying the nodes of small sub-graphs with no a priori information is a
hard problem. In this work, we want to find each node of a sparse sub-graph
embedded in both dynamic and static background graphs, of larger average
degree. We show that exploiting the summability over several background
realizations of the Estrada-Benzi communicability and the Krylov approximation
of the matrix exponential, it is possible to recover the sub-graph with a fast
algorithm with computational complexity O(N n). Relaxing the problem to
complete sub-graphs, the same performance is obtained with a single background.
The worst case complexity for the single background is O(n log(n)).
