Introduction to the Modeling and Analysis of Complex Systems

296 CHAPTER 15. BASICS OF NETWORKSgrids. This means that, in a single network, some components may be very well connectedwhile others may not. Such non-homogeneous connectivity makes it more difficultto analyze the system’s properties mathematically (e.g., mean-field approximation maynot apply to networks so easily). In the meantime, it also gives the model greater powerto represent connections among system components more closely with reality. You canrepresent any network topology (i.e., shape of a network) by explicitly specifying in detailwhich components are connected to which other components, and how. This makes networkmodeling necessarily data-intensive. No matter whether the network is generatedusing some mathematical algorithm or reconstructed from real-world data, the creatednetwork model will contain a good amount of detailed information about how exactly thecomponents are connected. We need to learn how to build, manage, and manipulatethese pieces of information in an efficient way.Second, the number of components may dynamically increase or decrease over timein certain dynamical network models. Such growth (or decay) of the system’s topologyis a common assumption typically made in generative network models that explain selforganizingprocesses of particular network topologies. Note, however, that such a dynamicchange of the number of components in a system realizes a huge leap from theother more conventional dynamical systems models, including all the models we have discussedin the earlier chapters. This is because, when we consider states of the systemcomponents, having one more (or less) component means that the system’s phase spaceacquires one more (or less) dimensions! From a traditional dynamical systems point ofview, it sounds almost illegal to change the dimensions of a system’s phase space overtime, yet things like that do happen in many real-world complex systems. Network modelsallow us to naturally describe such crazy processes.15.2 Terminologies of Graph TheoryBefore moving on to actual dynamical network modeling, we need to cover some basicsof graph theory, especially the definitions of technical terms used in this field. Let’s beginwith something we have already discussed above:A network (or graph) consists of a set of nodes (or vertices, actors) and a set of edges(or links, ties) that connect those nodes.As indicated above, different disciplines use different terminologies to talk about networks;mathematicians use “graph/vertex/edge,” physicists use “network/node/edge,” computer