On the occasion of SIAM Review’s 60th volume, the author of the journal’s most popular article offers insight as to why the paper continues to spark so much interest.
Many systems of scientific significance can be represented as graphs or networks — sets of nodes or vertices joined in pairs by edges. Examples include the internet; the World Wide Web; social, professional, and personal networks; road, rail, and airline routes; metabolic networks; food webs; and the power grid. In each of these cases, the network’s structure can substantially impact system behavior. For instance, the flow of data traffic online depends strongly on the internet’s network topology. How long will it take for data to travel from one part of the network to another? Are there bottlenecks or weak points in the structure? Would certain changes improve the system’s performance or stability? The answers to these questions depend on the specific shape that the network takes.
Mathematicians have long studied graphs and networks in the context of graph theory, a branch of discrete mathematics that has yielded many beautiful formal results about network structure. Recent research, however, differs from traditional graph theory in its focus on the structure and properties of empirical networks as they appear in the real world. As a new area of applied research—sometimes dubbed “complex networks” or “network science”—this field has been driven in part by the increasing ubiquity of detailed data sets describing network structures across a range of different areas of science and technology, as well as the widespread availability of inexpensive and powerful computers with which to analyze them. It focuses on the development of novel mathematical theory and methods to analyze, quantify, and understand real-world networks.
A scientific collaboration network. The nodes represent scientists and the links represent collaborations among them. Image credit: Mark Newman.
Beginning in the 1990s, research on complex networks quickly identified a number of central issues important to the understanding of network phenomena that are still relevant today. These issues include the following: the construction and solution of formal models of network structure, such as random graphs and models of network growth; metrics that quantify specific structural features of networks, like path lengths, correlations, clustering coefficients, subgraph densities, and centrality measures; methods for quantifying large-scale structure in networks, particularly community structure; spectral graph theory and random matrix methods; networks’ resilience to failure or attack; and processes taking place on networks, such as the spread of diseases in human populations or the flow of information across the internet.
By 2003, the field’s focus had coalesced to the point where a survey of the mathematical developments was needed, and SIAM Review invited me to contribute an article. The timing was ideal, coinciding with rapidly-increasing interest in networks across the mathematical sciences, and the article received a record number of citations in the years following its publication.
The field of network science has since grown to encompass thousands of researchers, with new papers appearing every day. Even after 15 years, the topics covered in the original review garner a significant amount of research attention, and the paper continues to be highly cited. But many new developments have emerged as well, including the study of dynamic networks (those that change over time), the development of new algorithmic and analytic methods for network data (including statistical inference and spectral methods), the study of multilayer and multiplex networks (those with multiple different types of edges), and theories of dynamical systems and processes occurring on networks (such as flow processes, synchronization, and cascading dynamics).
The field remains extremely active, with a number of new journals devoted to network topics and numerous conferences attracting large numbers of researchers, including the SIAM Workshop on Network Science, held each year in conjunction with the SIAM Annual Meeting. This year’s workshop will take place on July 12 and 13 in Portland, Ore.
Table 1. The 10 most downloaded SIAM Review articles.
Table 1 displays the 10 most-downloaded SIAM Review articles, all of which are freely accessible through the end of the year. This information is also available online.