Showing papers by "Duncan J. Watts published in 2000"

Network Robustness and Fragility: Percolation on Random Graphs

Abstract: Recent work on the Internet, social networks, and the power grid has addressed the resilience of these networks to either random or targeted deletion of network nodes or links. Such deletions include, for example, the failure of Internet routers or power transmission lines. Percolation models on random graphs provide a simple representation of this process but have typically been limited to graphs with Poisson degree distribution at their vertices. Such graphs are quite unlike real-world networks, which often possess power-law or other highly skewed degree distributions. In this paper we study percolation on graphs with completely general degree distribution, giving exact solutions for a variety of cases, including site percolation, bond percolation, and models in which occupation probabilities depend on vertex degree. We discuss the application of our theory to the understanding of network resilience.

Mean-field solution of the small-world network model.

Abstract: The small-world network model is a simple model of the structure of social networks, which possesses characteristics of both regular lattices and random graphs. The model consists of a one-dimensional lattice with a low density of shortcuts added between randomly selected pairs of points. These shortcuts greatly reduce the typical path length between any two points on the lattice. We present a mean-field solution for the average path length and for the distribution of path lengths in the model. This solution is exact in the limit of large system size and either a large or small number of shortcuts.

A Simple Model of Fads and Cascading Failures

Abstract: The origin of large but rare cascades that are triggered by small initial shocks is a problem that manifests itself in social and natural phenomena as diverse as cultural fads and innovations [1-3], social movements [4, 5], and cascading failures in large infrastructure networks [6-8]. Here we present a possible explanation of such cascades in terms of a network of interacting agents whose decisions are determined by the actions of their neighbors according to a simple threshold rule. We identify conditions under which the network is susceptible to very rare, but very large cascades and explain why such cascades may be difficult to anticipate in practice.

Random Graphs with Arbitrary Degree Distribution and Their Applications

Abstract: Recent work on the structure of social networks and the internet has focussed attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions which have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact expressions for the position of the phase transition at which a giant component first forms, the mean component size, the size of the giant component if there is one, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. We also apply our theory to some real-world graphs, including the world-wide web and collaboration graphs of scientists and business-people. We demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network which is not captured by the random graph.