A signature of power law network dynamics

Can one hear the 'sound' of a growing network? We address the problem of recognizing the topology of evolving biological or social networks. Starting from percolation theory, we analytically prove a linear inverse relationship between two simple graph parameters—the logarithm of the average cluster size and logarithm of the ratio of the edges of the graph to the theoretically maximum number of edges for that graph—that holds for all growing power law graphs. The result establishes a novel property of evolving power-law networks in the asymptotic limit of network size. Numerical simulations as well as fitting to real-world citation co-authorship networks demonstrate that the result holds for networks of finite sizes, and provides a convenient measure of the extent to which an evolving family of networks belongs to the same power-law class. 3 Mark Kac posed the question " Can one hear the shape of a drum? " (1), asking whether one could infer the geometric shape of a drum from its vibrational modes. Here we explore a similar question about the evolution of complex networks. The structure and properties of complex networks (2-4), such as the Internet (5, 6), citation networks (7), biological interaction network such as metabolic (8), protein-interaction networks (9), and disease-gene network (10) have been investigated intensively over the past decade. Many of these real world networks belong to the power-law family where the number of nodes with neighbors, , scales as a power law of the form where is the scaling coefficient (11). Several mechanisms by which power law networks arise through evolutionary mechanisms were proposed by considering the structure of the real-world network as a target and by constructing models of network growth that produce such a network given an arbitrary seed graph as the initial condition (12-15). An important problem that arises in evaluating models of network evolution and in testing these against real data is the need to definitively characterize experimentally determined networks as belonging to the power-law family (16). Here we consider a related problem—given a time-series of network structure data, can one deduce whether the collection of networks belongs to the same power-law class? In percolation theory (17), the vertices of a lattice are " occupied " or connected with a certain probability , and the emergence of connected clusters are studied as a function of. Of interest is the existence of a path that would allow a liquid, as …