Parameterized Clique on Scale-Free Networks

Finding cliques in graphs is a classical problem which is in general NP-hard and parameterized intractable. However, in typical applications like social networks or protein-protein interaction networks, the considered graphs are scale-free, i.e., their degree sequence follows a power law. Their specific structure can be algorithmically exploited and makes it possible to solve clique much more efficiently. We prove that on inhomogeneous random graphs with n nodes and power law exponent γ, cliques of size k can be found in time O(n) for γ ≥ 3 and in time O(n exp(k)) for 2 < γ < 3.