Introduction to the Modeling and Analysis of Complex Systems

Chapter 15Basics of Networks15.1 Network ModelsWe are now moving into one of the most recent developments of complex systems science:networks. Stimulated by two seminal papers on small-world and scale-free networkspublished in the late 1990s [56, 57], the science of complex networks, or networkscience for short, has been rapidly growing and producing novel perspectives, researchquestions, and analytical tools to study various kinds of systems in a number of disciplines,including biology, ecology, sociology, economics, political science, managementscience, engineering, medicine, and more [23, 24, 25].The historical roots of network science can be sought in several disciplines. One is obviouslydiscrete mathematics, especially graph theory, where mathematicians study variousproperties of abstract structures called graphs made of nodes (a.k.a. vertices—pluralof vertex) and edges (a.k.a. links, ties). Another theoretical root is statistical physics,where properties of collective systems made of a large number of entities (such as phasetransitions) are studied using analytical means. A more applied root of network scienceis in the social sciences, especially social network analysis [58, 59, 60]. Yet anotherapplication-oriented root would be in dynamical systems, especially Boolean networksdiscussed in theoretical and systems biology [22, 61] and artificial neural networks discussedin computer science [20, 21]. In all of those investigations, the research foci wereput on the connections and interactions among the components of a system, not just oneach individual component.Network models are different from other more traditional dynamical models in somefundamental aspects. First, the components of the system may not be connected uniformlyand regularly, unlike cells in cellular automata that form regular homogeneous295