Kuramoto Lab: Finite-Size Effects

Module 2: Synchronization and Collective Behaviour

The formulas in Critical Coupling hold in the limit $N \to \infty$. Your simulations use finite $N$. This page covers the last two steps of the workflow: interpretation and limitations.

Two finite-$N$ effects

Noise floor. For incoherent phases, $\mathbb{E}[r^2] = 1/N$ (The Order Parameter). Below $K_c$ you measure $r_\infty$ of order $N^{-1/2}$, not zero.
Sampling of frequencies. With $N$ random draws, the empirical frequency distribution differs from $g$. The transition point of one sample moves around $K_c$, and the sharp corner of the $N \to \infty$ curve is rounded off.

Simulations and physical arguments suggest that the fluctuations of $r(t)$ are of order $N^{-1/2}$, except very close to $K_c$ where they may be amplified. As of 2000 there were no rigorous convergence results for finite $N$ (Strogatz 2000, sec. 5.1). Both effects shrink as $N$ grows. Figure 1 shows them with a normal frequency density, $\sigma = 1$, so $K_c \approx 1.60$.

Figure 1: Time-averaged order parameter against $K$ for three population sizes (normal frequencies, $\sigma = 1$, one random sample each, RK4 with $\Delta t = 0.1$ up to $t = 80$, average over $40 \le t \le 80$). Error bars show the standard deviation of $r(t)$ in that window. Dotted lines mark $K_c$ and the noise floors $N^{-1/2}$.

Interpretation

Below $K_c$, each curve sits near its own floor $N^{-1/2}$: about $0.14$, $0.07$ and $0.03$.
Near $K_c$ the fluctuations of $r(t)$ are largest. The transition is sharpest for the largest $N$.
The $N = 50$ sample rises before $K_c$. Its 50 frequencies have a sample standard deviation of about $0.88$, smaller than $\sigma = 1$, so this particular population is easier to synchronize. A different seed would move its curve.
Well above $K_c$, the three curves agree. Most oscillators are locked and sampling details matter little.

Limitations

One sample of frequencies per $N$ is not enough to separate the two finite-size effects. Average over several seeds if you need error bars on $r_\infty$ itself, not just on $r(t)$.
Close to $K_c$, $r(t)$ settles slowly. A fixed time window can stop too early.
The mean-field model has all-to-all coupling. Real oscillators often interact through a network, which changes the threshold. Module 5 returns to this with the swing equation for power grids.

Averaging over samples

This part is optional.

Exercise

Repeat Figure 1 with 10 seeds for each $N$ and plot the mean of $r_\infty$ over seeds. Then plot $r_\infty\sqrt{N}$ against $K$. Below $K_c$ the curves for different $N$ should collapse onto a single curve. Where does the collapse fail?

Solution

At $K = 0$ all three curves start at $r_\infty\sqrt{N} \approx 0.89$, the mean of $\sqrt{N}\,r$ for independent uniform phases ($\sqrt{\pi}/2$). Up to $K \approx 1$ they stay together and rise slowly, to about $1.6$: coupling below threshold enhances the fluctuations but keeps them of order $N^{-1/2}$. Close to $K_c$ the curves separate: $r_\infty$ no longer scales like $N^{-1/2}$, and the largest population departs first. The run takes about 15 seconds.

import numpy as np
import matplotlib.pyplot as plt
from amlab.synchronization.kuramoto_sweep import critical_coupling, order_parameter_sweep

sigma = 1.0
kc = critical_coupling(1 / (np.sqrt(2 * np.pi) * sigma))
k_values = np.linspace(0, 4, 17)
fig, axs = plt.subplots(1, 2, figsize=(9, 4))
for n in [50, 200, 1000]:
    r_seeds = []
    for seed in range(10):
        rng = np.random.default_rng(seed)
        omega = rng.normal(0, sigma, n)
        theta0 = rng.uniform(0, 2 * np.pi, n)
        r_mean, _ = order_parameter_sweep(k_values, omega, theta0, t_end=80, dt=0.1, t_average=40)
        r_seeds.append(r_mean)
    r_avg = np.mean(r_seeds, axis=0)
    axs[0].plot(k_values, r_avg, "o-", label=f"N = {n}")
    axs[1].plot(k_values, r_avg * np.sqrt(n), "o-", label=f"N = {n}")
for ax in axs:
    ax.axvline(kc, color="black", linestyle=":")
    ax.set_xlabel("Coupling strength K")
    ax.legend()
axs[0].set_ylabel(r"mean of $r_\infty$ over seeds")
axs[1].set_ylabel(r"$r_\infty \sqrt{N}$")
axs[1].set_xlim(0, kc)
axs[1].set_ylim(0, 5)
plt.tight_layout()
plt.show()

Exit question

Exercise

You run a simulation with $N = 400$ and $K = 1.0$ (normal frequencies, $\sigma = 1$) and find $r_\infty \approx 0.05$. A classmate says this shows weak partial synchronization. Is that right?

Solution

No. $K = 1.0$ is below $K_c \approx 1.60$, so the $N \to \infty$ prediction is $r_\infty = 0$. The measured value is consistent with the incoherent noise floor $N^{-1/2} = 0.05$. To claim partial synchrony you would need $r_\infty$ well above $N^{-1/2}$ and a value that does not decrease as you increase $N$.

The graded Kuramoto Assignment is released in this session.

References

Strogatz, Steven H. 2000. “From Kuramoto to Crawford: Exploring the Onset of Synchronization in Populations of Coupled Oscillators.” Physica D: Nonlinear Phenomena 143: 1–20.

--- title: "Kuramoto Lab: Finite-Size Effects" subtitle: "Module 2: Synchronization and Collective Behaviour" format: html --- The formulas in [Critical Coupling](critical-coupling.qmd) hold in the limit $N \to \infty$. Your simulations use finite $N$. This page covers the last two steps of the workflow: interpretation and limitations. ## Two finite-$N$ effects - **Noise floor.** For incoherent phases, $\mathbb{E}[r^2] = 1/N$ ([The Order Parameter](order-parameter.qmd)). Below $K_c$ you measure $r_\infty$ of order $N^{-1/2}$, not zero. - **Sampling of frequencies.** With $N$ random draws, the empirical frequency distribution differs from $g$. The transition point of one sample moves around $K_c$, and the sharp corner of the $N \to \infty$ curve is rounded off. Simulations and physical arguments suggest that the fluctuations of $r(t)$ are of order $N^{-1/2}$, except very close to $K_c$ where they may be amplified. As of 2000 there were no rigorous convergence results for finite $N$ [@strogatz2000kuramoto, Section 5.1]. Both effects shrink as $N$ grows. @fig-finite-size shows them with a normal frequency density, $\sigma = 1$, so $K_c \approx 1.60$. ```{python} #| label: fig-finite-size #| fig-cap: "Time-averaged order parameter against $K$ for three population sizes (normal frequencies, $\\sigma = 1$, one random sample each, RK4 with $\\Delta t = 0.1$ up to $t = 80$, average over $40 \\le t \\le 80$). Error bars show the standard deviation of $r(t)$ in that window. Dotted lines mark $K_c$ and the noise floors $N^{-1/2}$." #| fig-width: 7 #| fig-height: 4 #| echo: false import numpy as np import matplotlib.pyplot as plt from amlab.synchronization.kuramoto_sweep import critical_coupling, order_parameter_sweep sigma = 1.0 kc = critical_coupling(1 / (np.sqrt(2 * np.pi) * sigma)) k_values = np.linspace(0, 4, 17) fig, ax = plt.subplots(figsize=(7, 4)) for idx, n in enumerate([50, 200, 1000]): rng = np.random.default_rng(1) omega = rng.normal(0, sigma, n) theta0 = rng.uniform(0, 2 * np.pi, n) r_mean, r_std = order_parameter_sweep( k_values, omega, theta0, t_end=80, dt=0.1, t_average=40 ) ax.errorbar( k_values + 0.03 * (idx - 1), r_mean, yerr=r_std, fmt="o-", markersize=4, capsize=2, label=f"N = {n}", ) ax.axhline(1 / np.sqrt(n), color=f"C{idx}", linestyle=":", linewidth=0.8) ax.axvline(kc, color="black", linestyle=":", linewidth=1) ax.text(kc + 0.05, 0.02, r"$K_c$") ax.set_xlabel("Coupling strength K") ax.set_ylabel(r"$r_\infty$") ax.set_ylim(0, 1) ax.legend() plt.tight_layout() plt.show() ``` ## Interpretation - Below $K_c$, each curve sits near its own floor $N^{-1/2}$: about $0.14$, $0.07$ and $0.03$. - Near $K_c$ the fluctuations of $r(t)$ are largest. The transition is sharpest for the largest $N$. - The $N = 50$ sample rises before $K_c$. Its 50 frequencies have a sample standard deviation of about $0.88$, smaller than $\sigma = 1$, so this particular population is easier to synchronize. A different seed would move its curve. - Well above $K_c$, the three curves agree. Most oscillators are locked and sampling details matter little. ## Limitations - One sample of frequencies per $N$ is not enough to separate the two finite-size effects. Average over several seeds if you need error bars on $r_\infty$ itself, not just on $r(t)$. - Close to $K_c$, $r(t)$ settles slowly. A fixed time window can stop too early. - The mean-field model has all-to-all coupling. Real oscillators often interact through a network, which changes the threshold. Module 5 returns to this with the [swing equation](../networks/swing-equation.qmd) for power grids. ## Averaging over samples This part is optional. ::: {.callout-note} ## Exercise Repeat @fig-finite-size with 10 seeds for each $N$ and plot the mean of $r_\infty$ over seeds. Then plot $r_\infty\sqrt{N}$ against $K$. Below $K_c$ the curves for different $N$ should collapse onto a single curve. Where does the collapse fail? ::: ::: {.callout-tip collapse="true"} ## Solution At $K = 0$ all three curves start at $r_\infty\sqrt{N} \approx 0.89$, the mean of $\sqrt{N}\,r$ for independent uniform phases ($\sqrt{\pi}/2$). Up to $K \approx 1$ they stay together and rise slowly, to about $1.6$: coupling below threshold enhances the fluctuations but keeps them of order $N^{-1/2}$. Close to $K_c$ the curves separate: $r_\infty$ no longer scales like $N^{-1/2}$, and the largest population departs first. The run takes about 15 seconds. ```python import numpy as np import matplotlib.pyplot as plt from amlab.synchronization.kuramoto_sweep import critical_coupling, order_parameter_sweep sigma = 1.0 kc = critical_coupling(1 / (np.sqrt(2 * np.pi) * sigma)) k_values = np.linspace(0, 4, 17) fig, axs = plt.subplots(1, 2, figsize=(9, 4)) for n in [50, 200, 1000]: r_seeds = [] for seed in range(10): rng = np.random.default_rng(seed) omega = rng.normal(0, sigma, n) theta0 = rng.uniform(0, 2 * np.pi, n) r_mean, _ = order_parameter_sweep(k_values, omega, theta0, t_end=80, dt=0.1, t_average=40) r_seeds.append(r_mean) r_avg = np.mean(r_seeds, axis=0) axs[0].plot(k_values, r_avg, "o-", label=f"N = {n}") axs[1].plot(k_values, r_avg * np.sqrt(n), "o-", label=f"N = {n}") for ax in axs: ax.axvline(kc, color="black", linestyle=":") ax.set_xlabel("Coupling strength K") ax.legend() axs[0].set_ylabel(r"mean of $r_\infty$ over seeds") axs[1].set_ylabel(r"$r_\infty \sqrt{N}$") axs[1].set_xlim(0, kc) axs[1].set_ylim(0, 5) plt.tight_layout() plt.show() ``` ::: ## Exit question ::: {.callout-note} ## Exercise You run a simulation with $N = 400$ and $K = 1.0$ (normal frequencies, $\sigma = 1$) and find $r_\infty \approx 0.05$. A classmate says this shows weak partial synchronization. Is that right? ::: ::: {.callout-tip collapse="true"} ## Solution No. $K = 1.0$ is below $K_c \approx 1.60$, so the $N \to \infty$ prediction is $r_\infty = 0$. The measured value is consistent with the incoherent noise floor $N^{-1/2} = 0.05$. To claim partial synchrony you would need $r_\infty$ well above $N^{-1/2}$ and a value that does not decrease as you increase $N$. ::: The graded [Kuramoto Assignment](assignment-kuramoto.qmd) is released in this session.