The structure and function of complex networks

TL;DR

This survey analyzes the structure and dynamics of complex networks across social, information, technological, and biological systems. It connects empirical properties such as small-world behavior, high clustering, and heavy-tailed degree distributions to analytic models (Poisson and generalized random graphs, configuration models, exponential random graphs, and small-world/growth frameworks) and examines processes on networks (percolation, epidemics, and search). Key contributions include formal criteria for giant-component formation, insights into degree correlations and resilience, and the evaluation of growth mechanisms like preferential attachment and copying. The paper highlights gaps in modeling transitivity and community structure and outlines directions for future research to better understand how network topology shapes function and dynamics in real systems.

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.

Figures (15)

• Figure 1: A small example network with eight vertices and ten edges.
• Figure 2: Three examples of the kinds of networks that are the topic of this review. (a) A food web of predator-prey interactions between species in a freshwater lake Martinez91. Picture courtesy of Neo Martinez and Richard Williams. (b) The network of collaborations between scientists at a private research institution GN02. (c) A network of sexual contacts between individuals in the study by Potterat et al.Potterat02.
• Figure 3: Examples of various types of networks: (a) an undirected network with only a single type of vertex and a single type of edge; (b) a network with a number of discrete vertex and edge types; (c) a network with varying vertex and edge weights; (d) a directed network in which each edge has a direction.
• Figure 4: The two best studied information networks. Left: the citation network of academic papers in which the vertices are papers and the directed edges are citations of one paper by another. Since papers can only cite those that came before them (lower down in the figure) the graph is acyclic---it has no closed loops. Right: the World Wide Web, a network of text pages accessible over the Internet, in which the vertices are pages and the directed edges are hyperlinks. There are no constraints on the Web that forbid cycles and hence it is in general cyclic.
• Figure 5: Illustration of the definition of the clustering coefficient $C$, Eq. (\ref{['defsc1']}). This network has one triangle and eight connected triples, and therefore has a clustering coefficient of $3\times1/8=\frac{3}{8}$. The individual vertices have local clustering coefficients, Eq. (\ref{['defsci']}), of 1, 1, $\frac{1}{6}$, 0 and 0, for a mean value, Eq. (\ref{['defsc3']}), of $C=\frac{13}{30 }$.