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 βc(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 ≡ 0, corresponding to the network of science citations.

Information super-diffusion on structured networks

“…1a. Such a graph is grown by adjusting a single free parameter governing microscopic linking rules proposed in [9]. These rules have been suggested as an evolution model of the world-wide Web.…”

“…In the model [9,21] both the target node k and and the origin n of a link are selected by shifted preferential probabilities given by…”

“…(2) ensures an emergent scale-free structure of the graph after long evolution time. The global graph structure at t → ∞ is thus characterized by the power-law distributions of links [6,9,21], P (q in ) ∼ q −τ in in and P (q out ) ∼ q −τout out with τ in = 1 + γ −1 in and τ out = 1 + γ −1 out . Here we use the original one-parameter model [9] with α =α = 1/4 and M = 1.…”

“…In many networks a scale-free linking structure was found [6,7] which emerges from complex processes of self-organization with specific constraints. Maybe the most prominent example of such a scale-free network is the world-wide Web [8,9] and the Internet [10]. Transport processes on networks, such as traffic of information packets on the Internet or diffusion of signaling molecules in a biological cell, are physical processes which are closely related to the network geometry.…”

Relationship between the in-degree and out-degree of WWW

“…In addition, it is a small-world networks. Since the knowledge of the evolution mechanism is very important for the better understanding of the dynamics built upon WWW, many theoretical models have been constructed previously [28,29,30]. However, these models haven't considered the relationship between the in-degree distribution and the out-degree distribution.…”