Topologically protected synchronization in networks

TL;DR

The paper introduces topologically equivalent (TE) groups in Kuramoto networks and proves that a TE group forming a fully connected subgraph can synchronize with the rest of the network acting as a noninfluential background, provided the topological condition k^{(OUT)} ≤ k^{(IN)} holds. It derives rigorous bounds on phase differences using a bounding ODE \dot{Θ} ≤ Ω + J B Θ, showing that all eigenvalues of B must be negative (the TC) for protection of synchronization, and explicitly solves the cases N' = 2, 3, 4, 5 FC to obtain precise TC inequalities. The work generalizes to arbitrary N' TE nodes and discusses other topologies (disconnected unions, regular polygons), identifying how TC depends on internal and external degrees and illustrating the phenomenon with simulations that reveal TE groups behaving as independent pacemakers. This mechanism offers a robust, topology-driven route to localized synchronization in heterogeneous oscillator networks, with potential implications for biological and engineered systems.

Abstract

In a graph, we say that two nodes are topologically equivalent if their sets of first neighbors, excluding the two nodes, coincide. We prove that nonlinearly coupled heterogeneous oscillators located on a group of topologically equivalent nodes can get easily synchronized when the group forms a fully connected subgraph (or combinations thereof), regardless of the status of all the other oscillators. More generally, any change occurring in the remainder of the graph will not alter the synchronization status of the group. Typically, the group can synchronize when $k^{(\mathrm{OUT})}\leq k^{(\mathrm{IN})}$, $k^{(\mathrm{IN})}$ and $k^{(\mathrm{OUT})}$ being the common internal and outgoing degree of each node in the group, respectively. Simulations confirm our rigorous analysis and suggest that groups of topologically equivalent nodes act as independent pacemakers.

Figures (7)

• Figure 1: Graph associated to model (\ref{['Kuramoto']}) with $N=22$ nodes and $L=46$ links (15 red-ticker and 31 black-thin). The graph contains five groups of TE nodes: one FC with $N'=3$ (nodes 1, 2 and 3), two FC with $N'=2$ (nodes 7 and 6, and nodes 15 and 16, respectively); one with $N'=4$ forming a square (nodes 9, 10, 11, and 12); and one FC with $N'=4$ (nodes 19, 20, 21, 22). In each group, the internal links are drawn as ticker (red). In the four FC groups we have $k^{(\mathrm{OUT})}\leq k^{(\mathrm{IN})}$ while in the other group we have $k^{(\mathrm{OUT})}< k^{(\mathrm{IN})}$, hence, in each group, the TC for synchronization are satisfied.
• Figure 2: Numerical solution of model (\ref{['Kuramoto']}) with $J=1$ and $N=22$ oscillators on the top of the graph of Fig. \ref{['figIllustrative']}. Plots correspond to the variables $\theta_{i+1}-\theta_i$, for $i=1,\ldots,N-1$. In each group of TE nodes, the initial conditions, as well as the natural frequencies, are chosen very close to each other as to guarantee the strict sufficient conditions explained in the text. It is evident that for several indices $i$, $\theta_{i+1}-\theta_i$ drifts away, while there exists a group of indices where $\theta_{i+1}-\theta_i$ remains bounded. As can be better checked by the Insets, the bounded variables includes all the five groups of TE nodes and a few others which, however, are manifestly less synchronized than the TE nodes. Note in particular that, unlike the latter, in each group, as analytically predicted, the sign of $\theta_{i+1}-\theta_i$ does not change over time. See Fig. \ref{['fig2_SM']} and SM for further visualization.
• Figure 3: As in Fig. \ref{['figIllustrative2']} but without strict conditions. Note that $\theta_{i+1}-\theta_i$ becomes almost constant and with constant sign whenever $i$ belongs to a group of TE nodes (unlike the case with strict conditions, a change of sign can occur at short times as for the plot of $\theta_{21}-\theta_{20}$). Inset: rotating numbers for the same system. Asymptotically, there emerge six sets of converging lines. Each set is indicated by different colors as follows (top to bottom): black (nodes 1, 13, 19, 20, 21, 22); red (nodes 2, 3, 4, 5, 6, 7); green (nodes 8, 9, 10, 11, 12); blue (node 17); magenta (node 18); cyan (nodes 14, 15, 16).
• Figure 4: Particular of Fig. 2. It shows in full detail all the ten phase differences $\theta_{i+1}-\theta_i$ associated to the ten links writable in the form $(i,i+1)$ inside the groups of TE nodes of the graph of Fig. 1. It demonstrates the consistency of the general behavior $\theta_{i+1}-\theta_i$ with the bounds established for example by Eq. (15), as well as by Eqs. (S11)-(S12) of SM: After an initial exponential drop, $\theta_{i+1}-\theta_i$ in each group reaches a smooth regime with a bounded value that is proportional to the spread of the group's frequencies and inversely proportional to the coupling $J$.
• Figure S1: All possible groups of $N'$ TE nodes for $N'=2$ to $5$ (top to bottom). Here we focus only on the concept of topological equivalence within the group, while the topological equivalence with respect to the neighbors of the group --- indicated by the surrounding black corona --- is assumed.