摘要：Network motifs are small patterns that occur in a network significantly more often than expected. They have gathered a lot of interest, as they may describe functional dependencies of complex networks and yield insights into their basic structure [Milo et al., 2002]. Therefore, a large amount of work went into the development of methods for network motif detection in complex networks [Kashtan et al., 2004; Schreiber and Schwöbbermeyer, 2005; Chen et al., 2006; Wernicke, 2006; Grochow and Kellis, 2007; Alon et al., 2008; Omidi et al., 2009]. The underlying problem of motif detection is to count how often a copy of a pattern graph H occurs in a target graph G. This problem is #W[1]-hard when parameterized by the size of H [Flum and Grohe, 2004] and cannot be solved in time f( H )n^o( H ) under #ETH [Chen et al., 2005]. Preferential attachment graphs [Barabási and Albert, 1999] are a very popular random graph model designed to mimic complex networks. They are constructed by a random process that iteratively adds vertices and attaches them preferentially to vertices that already have high degree. Preferential attachment has been empirically observed in real growing networks [Newman, 2001; Jeong et al., 2003]. We show that one can count subgraph copies of a graph H in the preferential attachment graph G^n_m (with n vertices and nm edges, where m is usually a small constant) in expected time f( H ) m^O( H ^6) log(n)^O( H ^12) n. This means the motif counting problem can be solved in expected quasilinear FPT time on preferential attachment graphs with respect to the parameters H and m. In particular, for fixed H and m the expected run time is O(n^(1+epsilon)) for every epsilon>0. Our results are obtained using new concentration bounds for degrees in preferential attachment graphs. Assume the (total) degree of a set of vertices at a time t of the random process is d. We show that if d is sufficiently large then the degree of the same set at a later time n is likely to be in the interval (1 +/- epsilon)d sqrt(n/t) (for epsilon > 0) for all n >= t. More specifically, the probability that this interval is left is exponentially small in d.
关键词：random graphs; motif counting; average case analysis; preferential attachment graphs
