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Network topology & synchronisation

Raising the coupling makes oscillators lock — but how they are wired decides how easily. A hub pulls a whole population into step almost at once; a ring passes order slowly, neighbour to neighbour; a dense graph locks at a whisper of coupling because every edge helps. Same oscillators, same equation — only the wiring diagram changes. Choose one below and watch.

From mean field to a coupling matrix

The mean-field model couples every oscillator to the population average. The networked model replaces that single average with a coupling matrix $K_{ij}$: oscillator $i$ feels only the neighbours $j$ that the wiring connects it to, each with its own strength. Set $K_{ij}$ to a constant on a fully connected graph and you recover mean field; make it sparse and structure takes over.

$$\dot\theta_i \;=\; \omega_i \;+\; \sum_{j=1}^{N} K_{ij}\,\sin(\theta_j-\theta_i),\qquad R \;=\; \Big|\tfrac{1}{N}\textstyle\sum_j e^{i\theta_j}\Big|$$

Here each present edge carries the same per-edge coupling $K$, so denser graphs feel far more total pull at the same slider value — the all-to-all graph locks at a fraction of the coupling a ring needs. That is the honest lesson of the panel: connectivity, not just strength, sets the synchronisation.

Live — networked Kuramoto, RK4 on a coupling matrix
order parameter R: — edges: — state: —

Nodes sit on a circle and are coloured by phase; lines are the couplings. Start with the ring: order spreads slowly and $R$ climbs only gradually, if at all. Switch to the star and one hub drags the whole population into near-lock. All-to-all snaps to synchrony at tiny per-edge coupling. The frequency spread is the opposing force — widen it and every topology needs more help to hold together.

Deeper: the same kernel, a matrix instead of a mean
This is the library's networked Kuramoto kernel: the derivative sums $K_{ij}\sin(\theta_j-\theta_i)$ over the coupling matrix rather than folding it into a single population average, and the trajectory is advanced by the same fixed-step RK4 and read out by the same order parameter as the mean-field panel. It runs against the same physics-validated Python reference; verified here across ring, star and all-to-all coupling matrices, the largest divergence in $R$ was under $2\times10^{-15}$ — floating-point roundoff, well inside the reference's tolerance.

Why topology is the point

Real coupled systems are rarely all-to-all. Qubit lattices have a fixed connectivity graph; sensor arrays and oscillator hardware wire nearest neighbours; a control signal reaches the network through a few drive lines. The networked kernel is how the platform maps a synchronisation problem onto the actual connectivity of the target — and how it studies which wiring makes a target easy or hard to lock and steer.

TopologyCharacterSynchronises
Ring / nearest-neighbourlocal coupling, long path across the graphslowly, needs strong coupling
Ring + long chordslocal plus a few shortcuts (small-world)much faster than a plain ring
Starone hub couples to allfast — the hub sets the pace
All-to-allevery pair coupled (mean-field limit)at the lowest per-edge coupling
The mean-field model The science Algorithms Back to overview

Evidence boundary: a classical networked simulation running live from the library's exact kernel — a baseline and a teaching tool, not a hardware result and not a quantum-advantage claim. Topologies and initial phases are generated on the page; the integrator is what is checked against the Python reference, to better than $2\times10^{-15}$.