Parameterized clique on inhomogeneous random graphs

Finding cliques in graphs is a classical problem which is in general NP-hard and parameterized intractable. In typical applications like social networks or biological networks, however, 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 nn nodes and power law exponent  ββ, cliques of size kk can be found in time O(n)O(n) for β≥3β≥3 and in time O(nek4)O(nek4) for 2<β<32<β<3.