Statistical and spectral properties of generators.

<p>Box plots of the distributions of persistences (panel ), births (panel ) and lengths (panel ) for the 1d cycles ( generators) of real networks (<i>black</i>), reshuffled (<i>white</i>) and randomized (<i>gray</i>). The gray and green shaded areas identify the two network classes described in the main text: class I is significantly different from the random expectations, with shorter, less persistent cycles that appear across the entire filtration; class II networks are not significantly different from the random versions, with long cycles and late birth times in the filtration. The characteristics of class I networks imply a stratification of cycles that betrays the presence of large, non-local organisation in the network structure, which is not present in class II networks. For comparison, an example of RGG network (600 nodes in the unitary disk, linking distance 0.01), known to have higher order degree correlations, had edge weights set according to , with (linearly correlated weight RGG) and (random weight RGG). In both cases, the distributions of cycles’ properties resemble closely those of class I networks. Panel finally reports the distribution of adjacency spectral gaps and (left plot) and the Laplacian eigenratio (right plot). All the quantities show significant () differences between the two classes, implying that the homological structure affect the dynamical properties of networks, e.g. the synchronizability threshold.