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Structure of networks
Published in Karthik Raman, An Introduction to Computational Systems Biology, 2021
This represents sets of nodes, where every pair of nodes have a path (not an edge) between them. A graph may have more than one connected component. In the case of directed graphs, there is the equivalent notion of a strongly connected component, where every pair of nodes are connected to each other in both directions. That is, for every pair of vertices {A,B} in a strongly connected component, there is a path from A to B, as well as B to A. The graph is weakly connected if the underlying undirected graph is connected. Connected components are maximal, i.e. they denote the maximal connected sub-graph of a graph or the maximal strongly connected sub-graph of a graph.
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.
Exploration strategies for balancing efficiency and comprehensibility in model checking with ant colony optimization
Published in Journal of Information and Telecommunication, 2022
Tsutomu Kumazawa, Munehiro Takimoto, Yasushi Kambayashi
Model checking is conducted in the following manner (Clarke et al., 2018). First, we construct a Büchi automaton that accepts the words conforming to the system model but contrary to the specification. When a specification is formulated with Linear Temporal Logic (Manna & Pnueli, 1992), its negation is transformed to the corresponding Büchi automaton (Clarke et al., 2018). The automaton constructed in this step is the intersection between the model and the automaton of the negated specification. Second, the emptiness of the Büchi automaton is examined with graph search algorithms. Classical model checking adopts exhaustive algorithms to find strongly connected components on directed graphs (Tarjan, 1972). If the Büchi automaton has an accepted word, the word is reported to the user as a counterexample. Otherwise, it concludes that the model satisfies the specification.
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.
Recent advances in distributed adaptive consensus control of uncertain nonlinear multi-agent systems
Published in Journal of Control and Decision, 2020
Wei Wang, Jiang Long, Changyun Wen, Jiangshuai Huang
A node is balanced if its in-degree equals its out-degree, i.e. . A directed graph is balanced if all its nodes are balanced. A direct path from agent i to agent j is a sequence of successive edges in the form . A digraph has a spanning tree, if there is an agent called root, such that there is a directed path from the root to each other agent in the graph. If there exists a directed path between any two distinct nodes in directed graph , the graph is said to be strongly connected.