Topological Measure Locating the Effective Crossover between Segregation and Integration in a Modular Network

We introduce an easily computable topological measure which locates the effective crossover between segregation and integration in a modular network. Segregation corresponds to the degree of network modularity, while integration is expressed in terms of the algebraic connectivity of an associated hypergraph. The rigorous treatment of the simplified case of cliques of equal size that are gradually rewired until they become completely merged, allows us to show that this topological crossover can be made to coincide with a dynamical crossover from cluster to global synchronization of a system of coupled phase oscillators. The dynamical crossover is signaled by a peak in the product of the measures of intracluster and global synchronization, which we propose as a dynamical measure of complexity. This quantity is much easier to compute than the entropy (of the average frequencies of the oscillators), and displays a behavior which closely mimics that of the dynamical complexity index based on the latter. The proposed topological measure simultaneously provides information on the dynamical behavior, sheds light on the interplay between modularity and total integration, and shows how this affects the capability of the network to perform both local and distributed dynamical tasks.

Figure 1

(a)–(c) Graph representations of (a) a network consisting of two cliques of size $N_C=16$, and (b–c) the resulting networks after rewiring. (b) $j^{\max }=60$ (${\ell }_{\mathrm{out}}/2L=0.5$ and $Q=0$), and (c) $j=30$ (${\ell }_{\mathrm{out}}/2L=0.25$ and $Q=0.25$). Intracommunity (intercommunity) links are light gray (black colored). Blue (red) nodes are phase oscillators whose natural frequency is randomly distributed around 0.8 (0.2) within a range $\pm 0.1$. (d)–(f) Time evolution of the instantaneous frequencies of the $N=32$ oscillators after solving Eq. (4) with the networks of the left panels and $d_{\mathrm{in}}=d_{\mathrm{in}}^{*}=0.14$. (d) $d_{\mathrm{out}}=0$, (e) $d_{\mathrm{out}}=d_{\mathrm{out}}^{*}=1.10$, and (f) $d_{\mathrm{out}}=1.40$. Dashed line in (d) marks the mean frequency between the two clusters. (g)–(h) $R$ (squares) and ${\sigma }_{\omega }$ (circles) (see text for definition) for: (g) the two clique network of (a) as a function of $d_{\mathrm{in}}$, (h) the 0-modularity network of (b) as a function of $d_{\mathrm{out}}$ and for $d_{\mathrm{in}}=0.14$. The vertical dashed line in (g) marks the critical coupling for cluster synchronization ($d_{\mathrm{in}}^{*}=0.14$), while in (h) it corresponds to $d_{\mathrm{out}}=1.4$, well above the onset of global synchronization ($d_{\mathrm{out}}^{*}=1.10$). Panel (i) reports the $S$ and $I$ indices (see text for definitions) vs the ratio ${\ell }_{\mathrm{out}}/(2L)$, after solving Eq. (4) with $d_{\mathrm{in}}=0.14$ and $d_{\mathrm{out}}=1.40$. Oblique dashed line is the corresponding modularity index $Q$.

Figure 2

(a) Complexity index $E$ (circles) and $SI$ index (triangles) (see text for definitions) corresponding to the data reported in Fig. 1i. (b) $SI$ curves as a function of the mixing ratio ${\ell }_{\mathrm{out}}/(2L)$ for $C=2$ and for increasing values of $d_{\mathrm{out}}$ (in the legend). (c) Maxima of the $SI$ curves in (b) vs $d_{\mathrm{out}}$ for $C=2$, 3, and 4. Horizontal lines mark the maxima provided by Eq. (3): $C=2$ (red dotted), $C=3$ (green dash dotted), and $C=4$ (blue dashed). Each line intersects the corresponding $SI$ maxima curve at a particular $d_{\mathrm{out}}$, that coincides with the threshold of complete synchronization, as shown by the corresponding vertical lines in (d).