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
