SubgraphCentrality(​A,L0,SaveCoordinate​s)

Calculates the centrality (fraction of intercepted flows) of all subgraphs on L vertices of a graph.

This function calculates the centralities of all connected induced subgraphs on L vertices of a (weighted di)graph as per the article "Loop-centrality in economic and biological networks", to appear in the Proceedings of the Sixth International Conference on Complex Networks and Their Applications, also available at https://arxiv.org/abs/1707.00890
We recall that the centrality of cycle c or subgraph H is defined as the fraction of all networks flows intercepted by c (or H), that is passing through at least once by at least one vertex of c (or H). Equivalently, this fraction asymptotically determines the number of hikes on the entire graph, which are walks left-multiples of the prime c, in the context of the semi-commutative extension of number theory valid for walks on graphs ("Algebraic Combinatorics on Trace Monoids: Extending Number Theory to Walks on Graphs", SIAM J. Discrete Math., 31(2), 1428–1453).
To only compute the centralities of a few selected subgraphs or cycles, use the function FlowFraction on the File Exchange.
USAGE: SubgraphCentrality(A,L0,SaveCoordinates), A the adjacency matrix of the (weighted di)graph, L0 the size of the subgraphs for which the centrality is desired. Set SaveCoordinates to 0 if you do not wish to access the coordinates of the subgraphs, any other value will save the coordinates. All results are saved in a file 'CentralityResults.mat'.