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The idea of this approach is to pick a random pivot vertex and apply forward and backward reachability queries from this vertex. The two queries partition the vertex set into 4 subsets: vertices reached by both, either one, or none of the searches. One can show that a strongly connected component has to be contained in one of the subsets.
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 ...
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 ...
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]
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 ...
where is the (non-square) matrix having elements and is the so-called modularity matrix, which has elements B v w = A v w − k v k w 2 m . {\displaystyle B_{vw}=A_{vw}-{\frac {k_{v}k_{w}}{2m}}.} All rows and columns of the modularity matrix sum to zero, which means that the modularity of an undivided network is also always 0 {\displaystyle 0} .
In the context of network theory, a complex network is a graph (network) with non-trivial topological features—features that do not occur in simple networks such as lattices or random graphs but often occur in networks representing real systems.
[3] [4] [5] When speaking of closeness centrality, people usually refer to its normalized form which represents the average length of the shortest paths instead of their sum. It is generally given by the previous formula multiplied by N − 1 {\displaystyle N-1} , where N {\displaystyle N} is the number of nodes in the graph resulting in: