Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

/pdf/the-structure-and-function-of-complex-networks-34deompc9l.pdf

The Structure and Function of Complex Networks

http://www.maths.qmul.ac.uk/%7Elatora/report_06.pdf

Complex networks: Structure and dynamics

Given a snapshot of a social network, can we infer which new interactions among its members are likely to occur in the near future? We formalize this question as the link-prediction problem, and we develop approaches to link prediction based on measures for analyzing the “proximity” of nodes in a network. Experiments on large coauthorship networks suggest that information about future interactions can be extracted from network topology alone, and that fairly subtle measures for detecting node proximity can outperform more direct measures. © 2007 Wiley Periodicals, Inc.

/pdf/the-link-prediction-problem-for-social-networks-1n3vz3x8vn.pdf

The link-prediction problem for social networks

Recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that 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 apply our theory to some real-world graphs, including the worldwide web and collaboration graphs of scientists and Fortune 1000 company directors. 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 that is not captured by the random graph.

Random graphs with arbitrary degree distributions and their applications.

The structure and dynamics of multilayer networks

Euclidean and Non-Euclidean Geometries, Development and History.

Small Worlds: The Dynamics of Networks between Order and Randomness.

The notion of Erd} os number has oated around the mathematical research community for more than thirty years, as a way to quantify the common knowledge that mathematical and scientiic research has become a very collaborative process in the twentieth century, not an activity engaged in solely by isolated individuals. In this paper we explore some (fairly short) collaboration paths that one can follow from Paul Erd} os to researchers inside and outside of mathematics. In particular, we nd that all the Fields Medalists up through 1998 have Erd} os numbers less than 6, and that over 60 Nobel Prize winners in physics, chemistry, economics, and medicine have Erd} os numbers less than 9. 1 An outstanding component of the collaboration graph The collaboration graph C has as vertices all researchers (dead or alive) from all academic disciplines, with an edge joining vertices u and v if u and v have jointly published a research paper or book (with possibly more co-authors). As is the case for any simple (undirected) graph, in C we have a notion of distance between two vertices u and v: d(u; v) is the number of edges in the shortest path between u and v, if such a path exits, 1 otherwise (it is understood that d(u; u) = 0). In this paper we are concerned with the collaboration subgraph centered at Paul Erd} os (1913{1996). For a researcher v, the number d(Paul Erd} os, v) is called

Jerrold W. Grossman

Papers

Euclidean and Non-Euclidean Geometries, Development and History.

Small Worlds: The Dynamics of Networks between Order and Randomness.

Famous trails to Paul Erdős

On weakly connected domination in graphs

Fractional arboricity, strength, and principal partitions in graphs and matroids