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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]
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.
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 ...
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} .
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 ...
[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:
The G(n, p) model was first introduced by Edgar Gilbert in a 1959 paper studying the connectivity threshold mentioned above. [3] The G(n, M) model was introduced by Erdős and Rényi in their 1959 paper. As with Gilbert, their first investigations were as to the connectivity of G(n, M), with the more detailed analysis following in 1960.