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RandomWalk Closeness CentralityDefinition
Random walk closeness centrality is a measure of centrality in a network, which describes the average speed with which randomly walking processes reach a node from other nodes of the network.
Consider a weighted network – either directed or undirected – with n nodes denoted by j=1, …, n; and a random walk process on this network with a transition matrix M. The element of M describes the probability of the random walker that has reached node i, proceeds directly to node j. These probabilities are defined in the following way. where is the (i,j)th element of the weighting matrix A of the network. When there is no edge between two nodes, the corresponding element of the A matrix is zero. The random walk closeness centrality of a node i is the inverse of the average mean first passage time to that node: Mean first passage timeThe mean first passage time from node i to node j is the expected number of steps it takes for the process to reach node j from node i for the first time: where P(i,j,r) denotes the probability that it takes exactly r steps to reach j from i for the first time. To calculate these probabilities of reaching a node for the first time in r steps, it is useful to regard the target node as an absorbing one, and introduce a transformation of M by deleting its jth row and column and denoting it by . As the probability of a process starting at i and being in k after r1 steps is simply given by the (i,k)th element of , P(i,j,r) can be expressed as Substituting this into the expression for mean first passage time yields Using the formula for the summation of geometric series for matrices yields where I is the n1 dimensional identity matrix. For computational convenience, this expression can be vectorized as where is the vector for first passage times for a walk ending at node j, and e is an n1 dimensional vector of ones. Mean first passage time is not symmetric, even for undirected graphs.
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