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.

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The Structure and Function of Complex Networks

https://www.bebr.ufl.edu/sites/default/files/Centrality%20in%20Social%20Networks.pdf

Centrality in social networks conceptual clarification

A number of recent studies have focused on the statistical properties of networked systems such as social networks and the Worldwide Web. Researchers have concentrated particularly on a few properties that seem to be common to many networks: the small-world property, power-law degree distributions, and network transitivity. In this article, we highlight another property that is found in many networks, the property of community structure, in which network nodes are joined together in tightly knit groups, between which there are only looser connections. We propose a method for detecting such communities, built around the idea of using centrality indices to find community boundaries. We test our method on computer-generated and real-world graphs whose community structure is already known and find that the method detects this known structure with high sensitivity and reliability. We also apply the method to two networks whose community structure is not well known—a collaboration network and a food web—and find that it detects significant and informative community divisions in both cases.

/pdf/community-structure-in-social-and-biological-networks-1jamu4rcfv.pdf

Community structure in social and biological networks

We propose and study a set of algorithms for discovering community structure in networks-natural divisions of network nodes into densely connected subgroups. Our algorithms all share two definitive features: first, they involve iterative removal of edges from the network to split it into communities, the edges removed being identified using any one of a number of possible "betweenness" measures, and second, these measures are, crucially, recalculated after each removal. We also propose a measure for the strength of the community structure found by our algorithms, which gives us an objective metric for choosing the number of communities into which a network should be divided. We demonstrate that our algorithms are highly effective at discovering community structure in both computer-generated and real-world network data, and show how they can be used to shed light on the sometimes dauntingly complex structure of networked systems.

/pdf/finding-and-evaluating-community-structure-in-networks-1o9f6y8pyr.pdf

Finding and evaluating community structure in networks.

Recent developments in the quantitative analysis of complex networks, based largely on graph theory, have been rapidly translated to studies of brain network organization. The brain's structural and functional systems have features of complex networks--such as small-world topology, highly connected hubs and modularity--both at the whole-brain scale of human neuroimaging and at a cellular scale in non-human animals. In this article, we review studies investigating complex brain networks in diverse experimental modalities (including structural and functional MRI, diffusion tensor imaging, magnetoencephalography and electroencephalography in humans) and provide an accessible introduction to the basic principles of graph theory. We also highlight some of the technical challenges and key questions to be addressed by future developments in this rapidly moving field.

Complex brain networks: graph theoretical analysis of structural and functional systems

This is a note to introduce a new measure of a kind of structural centrality called pair-dependency. Pair-dependency explicates the centrality-related notion of the gatekeeper. Moreover, it turns out to be a fundamental structural property of communication networks that provides the basis for the derivation of two standard measures of structural centrality.

The gatekeeper, pair-dependency and structural centrality

The intuitive literature on segregation is reviewed and a segregation measure, S, is constructed to embody existing intuitions. S measures segregation over a network of symmetrical social relations. Distributional problems are explored and two kinds of applications are illustrated.

Segregation in Social Networks

Maintaining the duality of closeness and betweenness centrality

In a recent book I reviewed the development of social network analysis from its earliest beginnings until the late 1990s (Freeman, 2004). There I characterized social network analysis as an approach that involves four defining properties: (1) It involves the intuition that links among social actors are important. (2) It is based on the collection and analysis of data that record social relations that link actors. (3) It draws heavily on graphic imagery to reveal and display the patterning of those links. And (4) it develops mathematical and computational models to describe and explain those patterns.

The Development of Social Network Analysis—with an Emphasis on Recent Events

For more than 100 years, sociologists have been concerned with relatively small, cohesive social groups (Tonnies, [1887] 1940; Durkheim [1893] 1933; Spencer 1895-97; Cooley, 1909). The groups that concern sociologists are not simply categories—like redheads or people more than six feet tall. Instead they are social collectivities characterized by interaction and interpersonal ties. Concern with groups of this sort has been—and remains—at the very core of the field.

Linton C. Freeman

Papers

The gatekeeper, pair-dependency and structural centrality

Segregation in Social Networks

Maintaining the duality of closeness and betweenness centrality

The Development of Social Network Analysis—with an Emphasis on Recent Events

Finding Social Groups: A Meta-Analysis of the Southern Women Data