Hamiltonian mean field model: Effect of network structure on synchronization dynamics

The Hamiltonian mean field model of coupled inertial Hamiltonian rotors is a prototype for conservative dynamics in systems with long-range interactions. We consider the case where the interactions between the rotors are governed by a network described by a weighted adjacency matrix. By studying the linear stability of the incoherent state, we find that the transition to synchrony begins when the coupling constant $K$ is inversely proportional to the largest eigenvalue of the adjacency matrix. We derive a closed system of equations for a set of local order parameters to study the effect of network heterogeneity on the synchronization of the rotors. When $K$ is just beyond the transition to synchronization, we find that the degree of synchronization is highly dependent on the network's heterogeneity, but that for large $K$ the degree of synchronization is robust to changes in the degree distribution. Our results are illustrated with numerical simulations on Erdös-Renyi networks and networks with power-law degree distributions.

Figure 1

Time average of the order parameter $R$, i.e., ${\langle R\rangle }_t$ as a function of $K$ [(a) and (c)] and $\lambda K$ [(b) and (d)] for a variety of network structures having $N={10}^4$. (a) and (b) show the results for simulated Erdös-Renyi networks with varying $q$. (c) and (d) show the results for simulated scale-free networks with varying $\alpha$. From (b) and (d), we see that plotting ${\langle R\rangle }_t$ against $\lambda K$ for any network causes the transitions to line up at the value $2{\sigma }_0^{2}N$ (black dashed lines), in agreement with (15).

Figure 2

Time-averaged global order parameter $R$ as a function of $K$ and $\lambda K$ for scale-free networks with varying edge-degree correlation $\rho$. The three networks have distinct $\lambda$ [the inset of (b)], and (a) shows they have distinct $K_c$. (b) plots ${\langle R\rangle }_t$ against $\lambda K$ showing that the three curves nearly coincide as they did in Fig. 1. The dashed black line is at $2{\sigma }_0^{2}N$.

Figure 3

Time-averaged global order parameter for a fixed Erdös-Renyi network as a function of (a) $K$ and (b) $K/{\sigma }_0^2$. The initial momentum distribution is a Gaussian with mean $\mathrm{\Omega }=6$ and varying standard deviation ${\sigma }_0$, as shown in the inset of (b). The dashed black line is at $2N/\lambda$.

Figure 4

Time-averaged global order parameter as a function of (a) network size, $N$, and (b) edge probability, $q$ for Erdös-Renyi networks with $K=\frac{1}{2}{K}_c$. The dashed lines have a slope of $-\frac{1}{2}$ and an arbitrary intercept, corresponding to the theoretical estimate ${\langle R\rangle }_t\sim {\left(qN\right)}^{-1/2}$.

Figure 5

Growth of the order parameter $R$ (log scale) as a function of time for an incoherent initial state. The solid lines show simulated data for an Erdös-Renyi network with $N=2.5{\left(10\right)}^4$ and the dashed lines show the slopes of the theoretical result (14). The value of $K/K_c$ for each curve is shown.

Figure 6

Time-averaged global order parameter, ${\langle R\rangle }_t$ as a function of the coupling strength $K$ for (a) Erdös-Renyi ($q=0.01$) and (b) scale-free ($\alpha =2.5, d_{\min }=33$) networks with $N={10}^4$ and $\parallel d\parallel =100$. The circles (blue) are the result of numerical integration for an initial Gaussian distribution $g_0\left(p\right)$ with mean $P_0=6$ and variance ${\sigma }_0^2=0.12$ (for Erdös-Renyi) and ${\sigma }_0^2=1$ (for scale-free) and random phases uniformly distributed in $[0,2\pi )$. The solid curves (black) are the numerical solutions to (34). The dashed curves (red) show the second-order approximation (39), valid near $K=K_c$ (see Sec. 4a). The dashed black lines show the theoretical $K_c$ given by (15).

Figure 7

(a) Time-averaged oscillator frequencies and (b) fluctuation amplitudes for oscillators in an Erdös-Renyi network ($q=0.01, \parallel d\parallel =100, N={10}^4$), with $K=600=3K_c$. The initial momenta had a Gaussian distribution with mean $P_0=6$, and variance ${\sigma }_0^2=1$. Time averages were taken over an interval $[2000,10000]$. (b) shows the time average of the fluctuating terms in (32) relative to the synchronized terms.

Figure 8

Histogram of ${\psi }_n\left(t\right)-{\psi }_1\left(t\right)$, for the network of Fig. 7, but with $N={10}^3$ oscillators. Here the $\psi$ are sampled each unit of time over a total of 2000 time units, so the total number of events is $2{\left(10\right)}^6$.

Figure 9

Single oscillator phase space distribution for an Erdös-Renyi network ($N=100, \parallel d\parallel =10$) with $K=1000=50K_c$ with an initial Gaussian distribution of momenta with mean $P_0=6$ and variance ${\sigma }_0^2=1$ and uniform initial phases. (a) shows the theoretical Boltzmann distribution (24) for node $n=1$, with $\beta =0.965$ and $r_1=0.036$. (b) shows the numerical distribution of $[{\overline{p}}_1\left(t\right),{\overline{\theta }}_1\left(t\right)]$, averaged over the time interval $\left[2000,10000\right]$. The histograms in both panels use bins of $\mathrm{\Delta }\theta =0.0628$ and $\mathrm{\Delta }p=0.0718$.