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κ-Path Centrality


Definition

κ-path node centrality
For each node v of a graph G = (V,E), the κ-path node centrality Cκ(v) of v is defined as the sum, over all possible source nodes s, of the frequency with which a message originated from s goes through v, assuming that the message traversals are only along random simple paths of at most κ edges.
κ-Path Centrality
where s are all the possible source nodes, σκs(v) is the number of κ-paths originating from s and passing through v and σκs is the overall number of κ-paths originating from s.

κ-path edge centrality
For each edge e of a graph G = (V,E), the κ-path edge centrality Lκ(e) of e is defined as the sum, over all possible source nodes s, of the frequency with which a message originated from s traverses e, assuming that the message traversals are only along random simple paths of at most κ edges.
κ-Path Centrality
where s are all the possible source nodes, σκs(e) is the number of κ-paths originating from s and traversing the edge e and, finally, σκs is the number of κ-paths originating from s.
In practical cases, the above Eq. for κ-path edge centrality can not be feasible because it requires to count all the κ-paths originating from all the source nodes s and such a number can be exponential in the number of nodes of G. To this purpose, we need to design some algorithms capable of efficiently approximating the value of κ-path edge centrality.

See Geodesic K-Path Centrality

Software

References

  • DE MEO, P., FERRARA, E., FIUMARA, G. & RICCIARDELLO, A. 2012. A novel measure of edge centrality in social networks. Knowledge-Based Systems, 30, 136-150. DOI: 10.1016/j.knosys.2012.01.007 Publisher web site Endnote RIS file


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