Fig. 3
Effective degrees of random graphs. a A 2D spatial model in which individuals are randomly placed in the unit square and then randomly paired according to their distance from each other. b A shifted-linear preferential attachment network44,45, with isothermal edge weights obtained by minimizing \({\sum }_{i,j}{w}_{ij}^{2}\) subject to \({\sum }_{j}{w}_{ij}=1\) for all \(i\). c, d For each graph generated by these two models, the effective degree, \(\tilde{\kappa }\) (black line) is plotted against the arithmetic mean topological degree, \(\bar{k}\) (green dots); the arithmetic mean Simpson degree, \({\kappa }_{{\rm{A}}}\) (blue dots); and the harmonic mean Simpson degree, \({\kappa }_{{\rm{H}}}\) (magenta dots). Vertical gray bars show the quantile bounds (6) for each graph. In almost all cases, \({\kappa }_{{\rm{H}}}\) provides the best estimate for \(\tilde{\kappa }\); \({\kappa }_{{\rm{H}}}\) performs better only when \(\tilde{\kappa }\) is very small. Note that the arithmetic mean topological degree, \(\bar{k}\), is significantly larger than the other degree measures in almost all cases. See Methods for further details and parameter values
