Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

17: Dynamical Networks II - Analysis of Network Topologies

  • Page ID
  • [ "article:topic-guide", "authorname:hsayama", "license:ccbyncsa", "showtoc:no" ]

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    • 17.1: Network Size, Density, and Percolation
      Networks can be analyzed in several different ways. One way is to analyze their structural features, such as size, density, topology, and statistical properties.
    • 17.2: Shortest Path Length
      Network analysis can measure and characterize various features of network topologies that go beyond size and density.  Many of the tools used here are actually borrowed from social network analysis developed and used in sociology [60].
    • 17.3: Centralities and Coreness
      The eccentricity of nodes discussed above can be used to detect which nodes are most central in a network. This can be useful because, for example, if you send out a message from one of the center nodes with minimal eccentricity, the message will reach every single node in the network in the shortest period of time.
    • 17.4: Clustering
      Eccentricity, centralities, and coreness introduced above all depend on the whole network topology (except for degree centrality). In this sense, they capture some macroscopic aspects of the network, even though we are calculating those metrics for each node. In contrast, there are other kinds of metrics that only capture local topological properties. This includes metrics of clustering, i.e., how densely connected the nodes are to each other in a localized area in a network.
    • 17.5: Degree Distribution
      Another local topological property that can be measured locally is, as we discussed already, the degree of a node.
    • 17.6: Assortativity
      Degrees are a metric measured on individual nodes. But when we focus on the edges, there are always two degrees associated with each edge, one for the node where the edge originates and the other for the node to where the edge points. So if we take the former for x and the latter for y from all the edges in the network, we can produce a scatter plot that visualizes a possible degree correlation between the nodes across the edges. Such correlations of node properties across edges can be generally
    • 17.7: Community Structure and Modularity
      The final topics of this chapter are the community structure and modularity of a network. These topics have been studied very actively in network science for the last several years. These are typical mesoscopic properties of a network; neither microscopic (e.g., degrees or clustering coefficients) nor macroscopic (e.g., density, characteristic path length) properties can tell us how a network is organized at spatial scales intermediate between those two extremes, and therefore, these concepts are