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κPath CentralityDefinition
κpath node centrality
For each node v of a graph G = (V,E), the κpath node centrality C^{κ}(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. where s are all the possible source nodes, σ^{κ}_{s}(v) is the number of κpaths originating from s and passing through v and σ^{κ}_{s} is the overall number of κpaths originating from s. κpath edge centrality For each edge e of a graph G = (V,E), the κpath edge centrality L^{κ}(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. where s are all the possible source nodes, σ^{κ}_{s}(e) is the number of κpaths originating from s and traversing the edge e and, finally, σ^{κ}_{s} is the number of κpaths originating from s. In practical cases, the above Eq. for κpath edge centrality can not be feasible because it requires to count all the κpaths 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 κpath edge centrality. See Geodesic KPath Centrality SoftwareReferences
