The Structure and Function of Complex Networks

Abstract

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.

Heritability of Resting State EEG Functional Connectivity Patterns

“…L indicates the average number of steps required to go from a node to all others, taking the shortest route. In addition, the value of +Ý was assigned to the path length involving unconnected nodes and the harmonic mean was used to average the values obtained for path length across different nodes (Newman, 2003). For each participant, the average CC and L over all epochs and electrodes were calculated.…”

Network structure and the biology of populations

“…The oldest and best-studied network model is the ErdosRenyi random graph [3], in which n nodes (or vertices) are connected by links (or edges) placed randomly between pairs of nodes. Less interesting is a uniform network, in which each node is linked to exactly m others.…”

“…Two other interesting statistical properties of large networks are the network 'diameter' (d) and the 'clustering coefficient' (C) [3,[7][8][9]. The former is calculated by first finding the shortest path (smallest number of links) between each pair of nodes; d is then the maximum such shortest path (some biological authors take d to be the overall average shortest path).…”

“…The asymptotic result of this process is a SF distribution, P(i)wi Kg , with gZ(2Kp)/(1Kp). As p ranges from 0 to 1, g goes from 2 to N. More restrictively, if we assume that each link connects a new node to the existing network, we have pZ0.5 and, thus, gZ3 (reviewed in [3,7]). …”

“…The mathematics of networks or 'graphs' are reviewed extensively and in detail by Bollobas [7] and Newman [3]. Bollobas focuses on the mathematical technicalities, whereas Newman integrates the mathematics with applications in a readable way.…”