On perturbation theory and an algorithm for maximal clique enumeration in uncertain and noisy graphs

The maximal clique enumeration (MCE) problem can be used to find very tightly-coupled collections of objects inside a network or graph of relationships. However, when such networks are based on noisy or uncertain data, the solutions to the MCE problem for several closely related graphs may be necessary to accurately define the collections. Thus, we propose an algorithm that efficiently solves the MCE problem on altered, or perturbed, graphs. The algorithm utilizes the enumeration of a baseline graph and identifies only those maximal cliques that the perturbation adds and/or removes. We detail the algorithm and the underlying theory required to guarantee correctness. Further, we report average runtime speedups of 7 and 9 for our algorithm over traditional enumeration techniques in the cases of adding and removing edges, respectively, from graphs constructed from protein interaction data.