EigenvectorCentrality

How well-connected each vertex of a Graph is to OTHER well-connected vertices.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageEigenvectorCentrality✓↗

EigenvectorCentrality(g)the dominant (Perron) eigenvector of g's undirected adjacency matrix, normalized so the entries SUM to 1, in VertexList(g) order.

Typed signatures

Type	Package	Applies
(value) -> list<number>	combinatorics	always

Domain: Collections

Details

• Edge direction is ignored -- built from the underlying (undirected) adjacency matrix, same convention as IsConnectedGraph.
• Normalized so the entries SUM to 1 -- kernel-verified against Wolfram 15: PathGraph({1,2,3,4}) -> {0.190983, 0.309017, 0.309017, 0.190983}, StarGraph(4) -> {0.366025, 0.211325, 0.211325, 0.211325}.
• Computed by power iteration on (adjacency + identity), not the bare adjacency matrix -- a bipartite graph's spectrum is symmetric about 0, and the +1 shift is what keeps the iteration from oscillating between the two equal-magnitude extreme eigenvalues.

Examples

Basic #

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In$\mathrm{EigenvectorCentrality}(\mathrm{Graph}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(3, 4)\bigr\rbrack))$Out$\bigl\lbrack0.190\,983\,005\,625\,053, 0.309\,016\,994\,374\,947, 0.309\,016\,994\,374\,947, 0.190\,983\,005\,625\,053\bigr\rbrack$

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In$\mathrm{EigenvectorCentrality}(\mathrm{StarGraph}(4))$Out$\bigl\lbrack0.366\,025\,403\,784\,439, 0.211\,324\,865\,405\,187, 0.211\,324\,865\,405\,187, 0.211\,324\,865\,405\,187\bigr\rbrack$

Applications #

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In$\mathrm{Reverse}(\mathrm{SortBy}(\bigl\lbrack\mathrm{VertexList}(g), \mathrm{EigenvectorCentrality}(g)\bigr\rbrack^T, \mathrm{Last}))$Out$\bigl\lbrack\mathrm{EigenvectorCentrality}(g), \mathrm{VertexList}(g)\bigr\rbrack$

Properties #

Rescales centrality by the edge count of a spanning forest, $|V| - c$, cancelling the $c$ against $|V| - 1$ symbolically

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In$\frac{(\mathrm{VertexCount}(h)-1)\mathrm{EigenvectorCentrality}(h)}{\mathrm{VertexCount}(g)-\mathrm{Length}(\mathrm{ConnectedComponents}(g))}$Out$\frac{(\mathrm{VertexCount}(h)-1)\mathrm{EigenvectorCentrality}(h)}{\mathrm{VertexCount}(g)-1}$

About

Examples	4
Checked against	Wolfram Language 3 agree1 unevaluated
Crosswalk	Wolfram Language

See also: ClosenessCentrality, VertexDegree