Distributed computation of classic and exponential closeness on tree graphs

Closeness centrality is a basic centrality measure that characterizes how centrally located a node is, within a network, based on its distances to all other nodes. In this paper, we address the distributed computation of two variants of this measure, known as classic closeness and exponential closeness, which differ in how the distances are taken into account. For each variant, we construct continuous- and discrete-time distributed algorithms, with which nodes in an undirected and unweighted tree graph can cooperatively determine their own closeness by talking only to neighbors, executing simple homogeneous update rules, and consuming minimal physical memories. We show that each algorithm is a networked dynamical system whose affine state equation has a unique equilibrium point that is always exponentially or finite-time stable, and whose output equation at the equilibrium point always yields the unknown closeness, thereby solving the problem.