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

A family of new measures of point and graph centrality based on early intuitions of Bavelas (1948) is introduced. These measures define centrality in terms of the degree to which a point falls on the shortest path between others and there fore has a potential for control of communication. They may be used to index centrality in any large or small network of symmetrical relations, whether connected or unconnected.

A Set of Measures of Centrality Based on Betweenness

The process of centrifugally casting an article such as a tire or the like from a curable or hardenable liquid polymeric material, which process includes the steps of selecting a mold and placing a core within the mold which core is hollow and/or is readily deformable under pressure but which has sufficient memory to resume its original position when the pressure is removed. The article being formed is formed between the core and the mold. The space between the core and the mold is filled with the curable liquid material of which the article is to be formed and the hollow core is also filled with a liquid material. The liquid material in the hollow core, the material from which the core is constructed and the curable liquid material all have about the same specific gravity. The mold and core are rotated to centrifugally cast the article which is formed of the curable liquid material. The deformable core permits easy removal from the completed article and the matching of the specific gravities as aforementioned keeps the deformable core from distorting during the centrifugal casting operation. The order of introducing the liquid material into the core, introducing the curable liquid into the space between the mold and core and rotation of the mold and core can be varied within the limits as set forth in the following description. Variations in the structure for accomplishing the principle of matching specific gravities are illustrated and described.

The Development of Social Network Analysis: A Study in the Sociology of Science

http://www.analytictech.com/borgatti/papers/borgatti%20-%20Centrality%20in%20valued%20graphs%20-%20A%20measure%20of%20betweenness%20based%20on%20network%20flow.pdf

Centrality in valued graphs: A measure of betweenness based on network flow

The use of visual images is common in many branches of science. And reviewers often suggest that such images are important for progress in the various fields (Koestler, 1964; Arnheim, 1970; Taylor, 1971; Tukey, 1972; Klovdahl, 1981; Tufte, 1983; Belien and Leenders). The historian Alfred Crosby (1997) has gone much further. He has proposed that visualization is one of only two factors that are responsible for the explosive development of all of modern science. The other is measurement.

/pdf/visualizing-social-networks-1c7v81t9q3.pdf

Linton C. Freeman

Papers

Centrality in social networks conceptual clarification

A Set of Measures of Centrality Based on Betweenness

The Development of Social Network Analysis: A Study in the Sociology of Science

Centrality in valued graphs: A measure of betweenness based on network flow

Visualizing Social Networks.