Dynamics of directed graphs: the World-Wide Web

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 constan tly changing selected nodes. Emergent statistical correlation between the d istributions of outgoing and incoming links is a key feature of the dynamic s of the Web. The growth phase is characterized by temporal fractal structu res which are manifested in the hierarchical organization of links. We obta in quantitative agreement with the recent empirical data in the real Web fo r the distributions of in- and out-links and for the size of the connected component. In a fully grown network of N nodes, we study the structure of c onnected clusters of nodes that are accessible along outgoing links from a randomly selected node. The distributions of size and depth of the connecte d clusters with a giant component exhibit supercritical behavior. By decrea sing the control parameter - average fraction beta of updated and added lin ks per time step - towards beta (c)(N) < 10% the Web can resume a critical structure with no giant component in it. We find a different universality c lass when the updates of links are not allowed, i.e., for <beta> equivalent to 0, corresponding to the network of science citations. (C) 2001 Elsevier Science B.V. All rights reserved.