gives a list of eccentricity centralities for the vertices in the graph g.
uses rules vw to specify the graph g.
- EccentricityCentrality will give high centralities to vertices that are at short maximum distances to every other reachable vertex.
- EccentricityCentrality for a graph g is given by , where is the maximum distance from vertex to all other vertices connected to .
- The eccentricity centrality for isolated vertices is taken to be zero.
- EccentricityCentrality works with undirected graphs, directed graphs, weighted graphs, multigraphs, and mixed graphs.
Background & Context
- EccentricityCentrality returns a list of non-negative machine numbers ("eccentricity centralities") that approximate particular centrality measures of the vertices of a graph. Eccentricity centrality is a measure of the centrality of a node in a network based on having a small maximum distance from a node to every other reachable node (i.e. the graph eccentricities). This measure has found applications in social networks, transportation, biology, and the social sciences.
- If is the maximum distance from vertex to all other vertices connected to , then the eccentricity centralities are given by . The eccentricity centrality for isolated vertices is taken to be zero. Eccentricity centralities lie between 0 and 1 inclusive.
- The eccentricity centrality of a vertex is the reciprocal of its VertexEccentricity. The full distance matrix of a graph can be computed using GraphDistanceMatrix.
Examplesopen allclose all
Basic Examples (2)
EccentricityCentrality works with undirected graphs:
Use rules to specify the graph:
EccentricityCentrality works with large graphs:
Rank vertices by their eccentricity:
Highlight the eccentricity centrality for CycleGraph:
Based on the infrastructure network of the Marshall Islands in eastern Micronesia, find a strategically located island:
It is different with the median of the graph:
A road network linking Chicago suburbs. Find the best location for hospitals and fire departments, to minimize the distance traveled by emergency vehicles:
For graphs with vertices, the largest sum in differences in eccentricity centrality between the most central vertex and all other vertices is the inverse of :
Measure how central the most central vertex is with respect to other vertices:
Properties & Relations (6)
EccentricityCentrality is the inverse of maximum distances to other reachable vertices:
The inverse of maximum distances to other reachable vertices:
Eccentricity centrality of a vertex is the reciprocal of the VertexEccentricity:
Eccentricity centrality is between 0 and 1:
Eccentricity centralities for an undirected graph are equivalent to centralities for each component:
Computing the centralities for each component yields the same result:
Use GraphCenter to find vertices with the highest eccentricity centrality:
Use VertexIndex to obtain the centrality for a specific vertex:
Wolfram Research (2012), EccentricityCentrality, Wolfram Language function, https://reference.wolfram.com/language/ref/EccentricityCentrality.html (updated 2015).
Wolfram Language. 2012. "EccentricityCentrality." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/EccentricityCentrality.html.
Wolfram Language. (2012). EccentricityCentrality. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/EccentricityCentrality.html