Giant component

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.

Abstract: In this paper, we consider the evolution of structure within large online social networks. We present a series of measurements of two such networks, together comprising in excess of five million people and ten million friendship links, annotated with metadata capturing the time of every event in the life of the network. Our measurements expose a surprising segmentation of these networks into three regions: singletons who do not participate in the network; isolated communities which overwhelmingly display star structure; and a giant component anchored by a well-connected core region which persists even in the absence of stars.We present a simple model of network growth which captures these aspects of component structure. The model follows our experimental results, characterizing users as either passive members of the network; inviters who encourage offline friends and acquaintances to migrate online; and linkers who fully participate in the social evolution of the network.

Abstract: In this paper, we consider the evolution of structure within large online social networks. We present a series of measurements of two such networks, together comprising in excess of five million people and ten million friendship links, annotated with metadata capturing the time of every event in the life of the network. Our measurements expose a surprising segmentation of these networks into three regions: singletons who do not participate in the network; isolated communities which overwhelmingly display star structure; and a giant component anchored by a well-connected core region which persists even in the absence of stars.We present a simple model of network growth which captures these aspects of component structure. The model follows our experimental results, characterizing users as either passive members of the network; inviters who encourage offline friends and acquaintances to migrate online; and linkers who fully participate in the social evolution of the network.

Abstract: In this paper we study the small-world network model of Watts and Strogatz, which mimics some aspects of the structure of networks of social interactions. We argue that there is one nontrivial length-scale in the model, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero, giving a normal critical point in this limit. This length-scale governs the crossover from large- to small-world behavior in the model, as well as the number of vertices in a neighborhood of given radius on the network. We derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex-vertex distance on the network, and, using series expansion and Pade approximants, find an approximate analytic form for the scaling function. We calculate the effective dimension of small-world graphs and show that this dimension varies as a function of the length-scale on which it is measured, in a manner reminiscent of multifractals. We also study the problem of site percolation on small-world networks as a simple model of disease propagation, and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms (in epidemiological terms, the point at which an epidemic occurs). The typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph. All our analytic results are confirmed by extensive numerical simulations of the model.