Synchronization in network geometries with finite spectral dimension

Recently there is a surge of interest in network geometry and topology. Here we show that the spectral dimension plays a fundamental role in establishing a clear relation between the topological and geometrical properties of a network and its dynamics. Specifically we explore the role of the spectral dimension in determining the synchronization properties of the Kuramoto model. We show that the synchronized phase can only be thermodynamically stable for spectral dimensions above four and that phase entrainment of the oscillators can only be found for spectral dimensions greater than two. We numerically test our analytical predictions on the recently introduced model of network geometry called complex network manifolds, which displays a tunable spectral dimension.

Figure 1

The cumulative distribution of eigenvalues ${\rho }_c\left(\lambda \right)$ for CNM of dimension $d=2,3,4,$ and 5, is shown in panels (a), (b), and (c) for the simplex, hypercube, and orthoplex CNM, respectively. Panel (d) represents the fitted spectral dimension of the CNM as a function of the dimension $d$ of its building blocks. Results are for $N=6400$ and the cumulative distribution of eigenevalue ${\rho }_c\left(\lambda \right)$ is averaged over 100 realizations of the network.

Figure 2

The probability distribution $P\left(Y\right)$, the cumulative distribution $P_c\left(Y\right)$ of the participation ratio $Y$, and the average value of the participation ratio $Y$ as a function of the corresponding eigenvalue $\lambda$ are shown for CNM formed by simplices [panels (a), (b), and (c)], hypercubes [panels (d), (e), and (f)], and orthoplexes [panels (g), (h), and (i)] networks with dimension $d=2,3,4,5$ of the polytopes.

Figure 3

Time series of the global order parameter calculated for different values of $\sigma =5$, 11 and 16, as indicated in the legend, and for CNM formed by simplices [panels (a), (d), (g), (j)], hypercubes [panels (b), (e), (h), (k)], and orthoplex [panels (c), (f), (i), (l)] of dimensions $d=2$ [panels (a), (b), (c)], $d=3$ [panels (d), (e), (f)], $d=4$ [panels (g), (h), (i)], and $d=5$ [panels (j), (k), (l)].

Figure 4

The average order parameter $\overline{R}$ and the standard deviation of the order parameter (${\text{std}}_R$) are plotted versus the coupling constant $\sigma$ for CNM formed by simplices [panels (a), (b), (c), (d)], hypercubes [panels (e), (f), (g), (h)], and orthoplexes [panels (i), (j), (k), (l)], and for dimension $d=2$ [panels (a), (e), (i)], $d=3$ [panels (b), (f), (j)], $d=4$ [panels (c), (g), (k)], and $d=5$ [panels (d), (h), (l)]. Results are shown for different network sizes $N=100,200,400,800,1600,$ and 3200 as indicated in the legend. Results are for $T=1000$ and have been averaged after equilibration for 20 realizations of the networks and internal frequencies.

Figure 5

Orbit diagrams of the system dynamics for CNM formed by simplices (left panels), hypercubes (center panels) and orthoplexes (right panel) for $d=2,3,4$ and 5 from top to bottom. The orbit diagrams are represented by the extremes (maxima and minima) $R^{*}$ taken by $R\left(t\right)$ for $t>0.8T$. Results are for $N=3200, T=1000$ and a given realization of the networks structure.