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Network Science
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
A strongly connected component is a subset of vertices in which each vertex can be reached from every other vertex. Typically, the network has one giant strongly connected component (GSCC), which contains a significant fraction of the entire network, as well as a number of smaller strongly connected components. The network studied here has a giant component of size ~54% of the total network.
On the Use of Graph Theory to Interpret the Output Results from a Monte-Carlo Depletion Code
Published in Nuclear Science and Engineering, 2021
One of the first basic questions one could ask when dealing with a graph of any sort has to do with the number of separate pieces it contains. A graph can indeed be constituted by different pieces or connected components that do not interact with one another. For undirected graphs (i.e., no notion of direction is defined on the edges), this has a straightforward meaning, and the number of components simply reflects the fact that the object is made up of separate independent pieces that can be analyzed separately. The question is more subtle when working with directed graphs, for which connectedness can bear two separate meanings, embodied in the notions of strong and weak components of a graph. Weakly connected components of a directed graph correspond to the intuitive notion of separate pieces that the graph is made of. But, when the graph is directed, a situation might arise where regions (i.e., subsets of vertices) can be accessed by following a path along the edges of the graph but cannot be left because there is no directed way out of the subset of vertices. In this latter situation, the subset of vertices is said to form a strongly connected component of the graph.
Stabilisability of discrete-time multi-agent systems under fixed and switching topologies
Published in International Journal of Systems Science, 2019
Let be a weighted directed graph with n vertices, where is the set of vertices, is the set of edges, is the adjacency matrix with if , and otherwise. In this paper, we assume that the graph is simple, i.e. there are no repeated edges or self-loops. Denote as a diagonal matrix with . Then the Laplacian matrix of the graph can be defined as . A directed path is a sequence of distinct vertices with for . A weak path is the same sequence with either or . A directed graph is called strongly connected if there are directed paths between any two distinct vertices. Similarly, a directed graph is called weakly connected if there is a weak path between any two vertices. A subgraph of is an induced subgraph if and . A strongly connected component (SCC) of is an induced subgraph that is maximal and subject to being strongly connected. A SCC of is an independent strongly connected component (iSCC) if for any and . The union graph of a collection of digraphs , , with the same node set , is denoted by , where , , and .