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KatzCentrality

KatzCentrality[g,α]

gives a list of Katz centralities for the vertices in the graph g and weight α.

KatzCentrality[g,α,β]

gives a list of Katz centralities using weight α and initial centralities β.

KatzCentrality[{vw,…},…]

uses rules vw to specify the graph g.

Details and Options

KatzCentrality gives a list of centralities that satisfy , where is the adjacency matrix of g.
If β is a scalar, it is taken to mean {β,β,…}.
KatzCentrality[g,α] is equivalent to KatzCentrality[g,α,1].
The option WorkingPrecision->p can be used to control the precision used in internal computations.
KatzCentrality works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

Examples

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Basic Examples (2)

Compute Katz centralities:

Highlight:

Rank vertices by centrality; higher value means more influence:

Scope (8)

KatzCentrality works with undirected graphs:

Directed graphs:

Multigraphs:

Mixed graphs:

Use rules to specify the graph:

Use weights:

Nondefault initial centralities:

KatzCentrality works with large graphs:

Options (3)

WorkingPrecision (3)

By default, KatzCentrality finds centralities using machine-precision computations:

Specify a higher working precision:

Infinite working precision corresponds to exact computation:

Applications (6)

Rank vertices of a graph by their importance in their reachable neighborhood:

Highlight the Katz centrality for CycleGraph:

CompleteKaryTree:

Simulate a citation network:

Find the top five most important papers and highlight them:

Predict a partition of the Zachary karate club in case of a conflict between influential members:

Show the partition:

Find the most common ancestor in a family tree:

Find descendants at the bottom of the tree:

In a trust network among employees, select employees who could efficiently spread the corporate culture:

Employees who are less likely to influence others:

Properties & Relations (4)

The centrality vector satisfies the equation :

EigenvectorCentrality is a special case of KatzCentrality:

Take and with the largest eigenvalue of the adjacency matrix:

As , all vertices get the same centrality:

Use VertexIndex to obtain the centrality of a specific vertex:

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