Dynamics of directed graphs: the world-wide Web

Abstract

We introduce and simulate a growth model of the world-wide Web based on the dynamics of outgoing links that is motivated by the conduct of the agents in the real Web to update outgoing links (re)directing them towards constantly changing selected nodes. Emergent statistical correlation between the distributions of outgoing and incoming links is a key feature of the dynamics of the Web. The growth phase is characterized by temporal fractal structures which are manifested in the hierarchical organization of links. We obtain quantitative agreement with the recent empirical data in the real Web for the distributions of in- and out-links and for the size of connected component. In a fully grown network of $N$ nodes we study the structure of connected clusters of nodes that are accessible along outgoing links from a randomly selected node. The distributions of size and depth of the connected clusters with a giant component exhibit supercritical behavior. By decreasing the control parameter---average fraction $β$ of updated and added links per time step---towards $β_c(N) < 10% $ the Web can resume a critical structure with no giant component in it. We find a different universality class when the updates of links are not allowed, i.e., for $β\equiv 0$, corresponding to the network of science citations.