Kuramoto oscillators in random networks

Abstract

By means of numerical analysis conducted with the aid of the computer, the collective synchronization of coupled phase oscillators in the Kuramoto model in the connected regime of random networks of various sizes is studied. The oscillators synchronize and achieve phase coherence, and this process is not significantly affected by the level of connectivity of the network. If the probability that two oscillators are coupled is around the network connectivity threshold synchronization still occurs, although in a more attenuated way. If the size of the network is sufficiently large the oscillators have a phase transition.

Figures (4)

• Figure 1: Geometric representation of oscillator dynamics in the Kuramoto model. Motion occurs on a circle of unit radius. The azimuthal angle $\theta$ represents the phase of the oscillator.
• Figure 2: Order parameter $R$ as a function of the coupling constant $K$ for random networks of size $N = 50, 100, 500, 1000$. For each value of $K$ the system of Eq.( \ref{['eq_4_kura_netw']}) is numerically integrated up to 5000 time iterations of amplitude 0.1 s. $R$ is calculated by averaging the last 1000 iterations. The figure shows $R(K)$ for different values of the probability $p$ that determines the structure of the network in the connected regime. For $p = p_c \simeq \hbox{ln}N/N$ the largest connected component absorbs all the nodes and has the minimum number of links. This is the configuration for which the network has the minimum level of connectivity. For $p = 1$ the maximum level of connectivity is achieved and every node is connected to all the others. The function $R(K)$ behaves roughly the same for $p > p_c$ while for $p = p_c$ it grows more slowly as $K$ increases. The figure shows in more detail the regions where the onset of synchronisation occurs and those where the oscillators are in phase coherence. For $p > p_c$ the value of the critical threshold of the coupling constant for which synchronization occurs is $K_c \sim 0.16 \div 0.19$, while for $p = p_c$ it is $K_c \sim 0.21$.
• Figure 3: The figure shows $R(K)$ when $p = p_c = \hbox{ln} N/N$ for all analyzed networks. As the size $N$ of the network increases the transition from $R \simeq 0$ to $R \simeq 1$ becomes abrupt and the oscillators have a phase transition. This is also evidenced by the inner figures which show that as $N$ increases the derivative $dR/dK$ has a larger maximum and the variation around it becomes smaller.
• Figure 4: The figure shows $R(N,K)$ for all values of $p$. For $K < K_c$, R decays towards a residual of $O(1/\sqrt{N})$. For $K > K_c$, R saturates towards a value $R_{\infty} < 1$ with fluctuations of $O(1/\sqrt{N})$. For $K = K_c$ the variation of $R(N,K_c)$ is minimal. The best estimate is $K_c \sim 0.16 \div 0.19$ for $p > p_c$, and $K_c \sim 0.21$ for $p = p_c$.