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Mathematics LibreTexts

15.1: Network Models

 

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We 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 networks published in the late 1990s [56, 57], the science of complex networks, or network science for short, has been rapidly growing and producing novel perspectives, research questions, and analytical tools to study various kinds of systems in a number of disciplines, including biology, ecology, sociology, economics, political science, management science, engineering, medicine, and more [23, 24, 25]. 

The historical roots of network science can be sought in several disciplines. One is obviously discrete mathematics, especially graph theory, where mathematicians study various properties of abstract structures called graphs made of nodes (a.k.a. vertices—plural of 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 phase transitions) are studied using analytical means. A more applied root of network science is in the social sciences, especially social network analysis [58, 59, 60]. Yet another application-oriented root would be in dynamical systems, especially Boolean networks discussed in theoretical and systems biology [22, 61] and artificial neural networks discussed in computer science [20, 21]. In all of those investigations, the research foci were put on the connections and interactions among the components of a system, not just on each individual component.

Network models are different from other more traditional dynamical models in some fundamental aspects. First, the components of the system may not be connected uniformly and regularly, unlike cells in cellular automata that form regular homogeneous grids. This means that, in a single network, some components may be very well connected while others may not. Such non-homogeneous connectivity makes it more difficult to analyze the system’s properties mathematically (e.g., mean-field approximation may not apply to networks so easily). In the meantime, it also gives the model greater power to represent connections among system components more closely with reality. You can represent any network topology (i.e., shape of a network) by explicitly specifying in detail which components are connected to which other components, and how. This makes network modeling necessarily data-intensive. No matter whether the network is generated using some mathematical algorithm or reconstructed from real-world data, the created network model will contain a good amount of detailed information about how exactly the components are connected. We need to learn how to build, manage, and manipulate these pieces of information in an efficient way.

Second, the number of components may dynamically increase or decrease over time in certain dynamical network models. Such growth (or decay) of the system’s topology is a common assumption typically made in generative network models that explain self organizing processes of particular network topologies. Note, however, that such a dynamic change of the number of components in a system realizes a huge leap from the other more conventional dynamical systems models, including all the models we have discussed in the earlier chapters. This is because, when we consider states of the system components, having one more (or less) component means that the system’s phase space acquires one more (or less) dimensions! From a traditional dynamical systems point of view, it sounds almost illegal to change the dimensions of a system’s phase space over time, yet things like that do happen in many real-world complex systems. Networkmodels allow us to naturally describe such crazy processes.