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An example graph, with 6 vertices, diameter 3, connectivity 1, and algebraic connectivity 0.722 The algebraic connectivity (also known as Fiedler value or Fiedler eigenvalue after Miroslav Fiedler) of a graph G is the second-smallest eigenvalue (counting multiple eigenvalues separately) of the Laplacian matrix of G. [1]
NetworkX is suitable for operation on large real-world graphs: e.g., graphs in excess of 10 million nodes and 100 million edges. [ clarification needed ] [ 19 ] Due to its dependence on a pure-Python "dictionary of dictionary" data structure, NetworkX is a reasonably efficient, very scalable , highly portable framework for network and social ...
The connectivity and edge-connectivity of G can then be computed as the minimum values of κ(u, v) and λ(u, v), respectively. In computational complexity theory , SL is the class of problems log-space reducible to the problem of determining whether two vertices in a graph are connected, which was proved to be equal to L by Omer Reingold in ...
Several algorithms based on depth-first search compute strongly connected components in linear time.. Kosaraju's algorithm uses two passes of depth-first search. The first, in the original graph, is used to choose the order in which the outer loop of the second depth-first search tests vertices for having been visited already and recursively explores them if not.
The degree of a node in a network (sometimes referred to incorrectly as the connectivity) is the number of connections or edges the node has to other nodes. If a network is directed, meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of ...
All rows and columns of the modularity matrix sum to zero, which means that the modularity of an undivided network is also always . For networks divided into just two communities, one can alternatively define s v = ± 1 {\displaystyle s_{v}=\pm 1} to indicate the community to which node v {\displaystyle v} belongs, which then leads to
Given a simple graph with vertices , …,, its Laplacian matrix is defined element-wise as [1],:= { = , or equivalently by the matrix =, where D is the degree matrix and A is the adjacency matrix of the graph.
Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity. [3]