Community structure in directed networks

We consider the problem of finding communities or modules in directed networks. The most common approach to this problem in the previous literature has been simply to ignore edge direction and apply methods developed for community discovery in undirected networks, but this approach discards potentially useful information contained in the edge directions. Here we show how the widely used benefit function known as modularity can be generalized in a principled fashion to incorporate the information contained in edge directions. This in turn allows us to find communities by maximizing the modularity over possible divisions of a network, which we do using an algorithm based on the eigenvectors of the corresponding modularity matrix. This method is shown to give demonstrably better results than previous methods on a variety of test networks, both real and computer-generated.Comment: 5 pages, 3 figure

Clustering and preferential attachment in growing networks

We study empirically the time evolution of scientific collaboration networks in physics and biology. In these networks, two scientists are considered connected if they have coauthored one or more papers together. We show that the probability of scientists collaborating increases with the number of other collaborators they have in common, and that the probability of a particular scientist acquiring new collaborators increases with the number of his or her past collaborators. These results provide experimental evidence in favor of previously conjectured mechanisms for clustering and power-law degree distributions in networks.Comment: 13 pages, 2 figure

Characterizing the structure of small-world networks

We give exact relations which are valid for small-world networks (SWN's) with a general `degree distribution', i.e the distribution of nearest-neighbor connections. For the original SWN model, we illustrate how these exact relations can be used to obtain approximations for the corresponding basic probability distribution. In the limit of large system sizes and small disorder, we use numerical studies to obtain a functional fit for this distribution. Finally, we obtain the scaling properties for the mean-square displacement of a random walker, which are determined by the scaling behavior of the underlying SWN

Degree Correlations in Random Geometric Graphs

Spatially embedded networks are important in several disciplines. The prototypical spatial net- work we assume is the Random Geometric Graph of which many properties are known. Here we present new results for the two-point degree correlation function in terms of the clustering coefficient of the graphs for two-dimensional space in particular, with extensions to arbitrary finite dimension

Mean-field solution of the small-world network model

The small-world network model is a simple model of the structure of social networks, which simultaneously 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 large or small number of shortcuts.Comment: 14 pages, 2 postscript figure

Community Structure in the United States House of Representatives

We investigate the networks of committee and subcommittee assignments in the United States House of Representatives from the 101st--108th Congresses, with the committees connected by ``interlocks'' or common membership. We examine the community structure in these networks using several methods, revealing strong links between certain committees as well as an intrinsic hierarchical structure in the House as a whole. We identify structural changes, including additional hierarchical levels and higher modularity, resulting from the 1994 election, in which the Republican party earned majority status in the House for the first time in more than forty years. We also combine our network approach with analysis of roll call votes using singular value decomposition to uncover correlations between the political and organizational structure of House committees.Comment: 44 pages, 13 figures (some with multiple parts and most in color), 9 tables, to appear in Physica A; new figures and revised discussion (including extra introductory material) for this versio

Dynamics of Epidemics

This article examines how diseases on random networks spread in time. The disease is described by a probability distribution function for the number of infected and recovered individuals, and the probability distribution is described by a generating function. The time development of the disease is obtained by iterating the generating function. In cases where the disease can expand to an epidemic, the probability distribution function is the sum of two parts; one which is static at long times, and another whose mean grows exponentially. The time development of the mean number of infected individuals is obtained analytically. When epidemics occur, the probability distributions are very broad, and the uncertainty in the number of infected individuals at any given time is typically larger than the mean number of infected individuals.Comment: 4 pages and 3 figure

Small-World Networks: Links with long-tailed distributions

Small-world networks (SWN), obtained by randomly adding to a regular structure additional links (AL), are of current interest. In this article we explore (based on physical models) a new variant of SWN, in which the probability of realizing an AL depends on the chemical distance between the connected sites. We assume a power-law probability distribution and study random walkers on the network, focussing especially on their probability of being at the origin. We connect the results to L\'evy Flights, which follow from a mean field variant of our model.Comment: 11 pages, 4 figures, to appear in Phys.Rev.