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Subgraph CentralityDefinition
Accounts for the participation of a node in all sub graphs of the network.
the number of closed walks of length k starting and ending node v in the network is given by the local spectral moments μ_{k}(v).
The subgraph centrality is just the diagonal entry of the exponential of the adjacency matrix. Notice that this is not the matrix in which you rise every entry to the exponential. In Matlab there is a function that compute this. For A being the adjacency matrix the function is expm(A) and the subgraph centrality of the nodes is diag(expm(A)). According to eigenvectors and eigenvalues, if the adjacency matrix is symmetric. Let phi_j(p) be the p_{th} entry of the j_{th} eigenvactor associated with eigenvalue lambda_j of the matrix A. The subgraph centrality is then: SC(p) = SUM (phi_j(p)^2*exp(lambda_j) Subgraph centrality (SC) of a node is a weighted sum of the numbers of all closed walks of different lengths in the network starting and ending at the node. These closed walks are related to partial subgraphs of a network, e.g., a closed walk with four nodes can ‘‘go through’’ different subgraphs on four nodes, such as along the same edge AB twice (as described above: from node A to node B along edge AB, then back to A along the same edge and then again from A to B and back to A along the same edge), or along a 4node cycle ABCD that includes edge AB (along the ‘‘square’’ from node A to node B to node C to node D and back to A; this is regardless of whether edges CA and DB that ‘‘go along the diagonal of the square’’ exist) etc. The above mentioned sum is weighted so that the contribution of the closed walks decreases as the length of the walks increases, i.e., shorter walks (smaller subgraphs) have higher weight [MILENKOVIĆ, T., 2011]. Estrada index Let G=(V,E) be a simple undirected graph with n nodes and let λ1≤λ2≤⋯λn be a nonincreasing ordering of the eigenvalues of its adjacency matrix A. The Estrada index is: [A HAGBERG, D. S., 2008, ESTRADA, E. 2000] Communicability centrality, also called subgraph centrality, of a node n is the sum of closed walks of all lengths starting and ending at node n. [A HAGBERG, 2008] Requirements
Require undirected and loop free network.
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