Proposal for a Correction to the Temporal Correlation Coefficient Calculation for Temporal Networks

Measuring the topological overlap of two graphs becomes important when assessing the changes between temporally adjacent graphs in a time-evolving network. Current methods depend on the fraction of nodes that have persisting edges. This breaks down when there are nodes with no edges, persisting or otherwise. The following outlines a proposed correction to ensure that correlation metrics have the expected behavior. [1] defines the topological overlap in the neighborhood of i between two consecutive time steps [t m , t m+1 ] as: C i (t m , t m+1) = j a ij (t m)a ij (t m+1) [ j a ij (t m)][ j a ij (t m+1)] (1) where a ij represents an entry in the unweighted adjacency matrix of the graph, so summing over a ij gives the interactions between i and every other node. Then the average topological overlap of the neighborhood of node i as the average of C i (t m , t m+1) over all possible subsequent temporal snapshots: C i = 1 M − 1 M −1 m=1 C i (t m , t m+1) (2) The average topological overlap of all C i can be said to represent the temporal clustering of all of the edges in the network, which we will call the temporal clustering coefficient (TCC). C = 1 N N i=1 C i (3) Identically, the order of summations in this formulation can be reversed (simply to make the following more obvious), to give the average topological overlap of the graph at t m with the subsequent graph at t m+1 :