The communities of a social network are sets of vertices with more connections inside the set than outside. We theoretically demonstrate that two commonly observed properties of social networks, heavy-tailed degree distributions and large clustering coefficients, imply the existence of vertex neighborhoods (also known as egonets) that are themselves good communities. We evaluate these neighborhood communities on a range of graphs. What we find is that the neighborhood communities often exhibit conductance scores that are as good as the Fiedler cut. Also, the conductance of neighborhood communities shows similar behavior as the network community profile computed with a personalized PageRank community detection method. The latter requires sweeping over a great many starting vertices, which can be expensive. By using a small and easy-to-compute set of neighborhood communities as seeds for these PageRank communities, however, we find communities that precisely capture the behavior of the network community profile when seeded everywhere in the graph, and at a significant reduction in total work.
Much of the modern work on networks rests on surprising empirical observations about the structure of real world connections. For instance, information networks were found to have a power-law in the degree distribution @cite_3 @cite_29 . These same networks were also found to have considerable local structure in the form of large clustering coefficients @cite_12 , but retained a small global diameter. Our theory shows that a third potential observation -- the existence of vertex neighborhood with low conductance -- is in fact implied by these other two properties. We formally show that heavy tailed degree distributions and high clustering coefficients imply the existence of large dense cores.
@cite_29 @cite_12 @cite_3
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Despite the apparent randomness of the Internet, we discover some surprisingly simple power-laws of the Internet topology. These power-laws hold for three snapshots of the Internet, between November 1997 and December 1998, despite a 45 growth of its size during that period. We show that our power-laws fit the real data very well resulting in correlation coefficients of 96 or higher.Our observations provide a novel perspective of the structure of the Internet. The power-laws describe concisely skewed distributions of graph properties such as the node outdegree. In addition, these power-laws can be used to estimate important parameters such as the average neighborhood size, and facilitate the design and the performance analysis of protocols. Furthermore, we can use them to generate and select realistic topologies for simulation purposes. Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.