Fig. 2.

Simpson degree and remeeting time. The Simpson degree ${\kappa }_i={\left({\sum }_j{w}_{ij}^2\right)}^{-1}$ quantifies the effective number (or diversity) of neighbors of a vertex $i$, taking their edge weights into account. a If the edge weights to neighbors are nonuniform, the Simpson degree ${\kappa }_i$ is less than the topological degree $k_i$. Here, ${\kappa }_i=4$, which is less than the topological degree, $k_i=5$. b If each neighbor has equal edge weight $1∕k$, the Simpson degree is equal to the topological degree, $k$. c The remeeting time ${\tau }_i$ is the expected time for two independent random walks from $i$ to meet each other. The effective degree $\tilde{\kappa }$ of a graph is the weighted harmonic average of the Simpson degrees, with weights given by the remeeting times