Math Insight

The stability of the asynchronous state as function of largest eigenvalue


One approach for determining a synchronizability index for the influence of a network on synchrony is by looking at the stability of the completely synchronous state. One can also look at the opposite end and look at the stability of the completely asynchronous, or incoherent, state.

Restrepo et al. analyzed the stability of the asynchronous state by looking at the following oscillator model 1,2 \begin{align*} \diff{\vc{x}_i}{t} = \vc{F}(\vc{x}_i,\eta_i) + S k(\vc{x}_i) \sum_{j} W_{ij} (\vc{H}(\vc{x}_j) - \langle \vc{H}(\vc{x})\rangle) + \xi_i(t), %\label{eq:model_for_async} \end{align*} where $\xi_i(t)$ is a noise term and the model is specified by the functions $\vc{F}$, $\vc{H}$ and $k$.

The is similar to the one used in the master stability function approach. Just like with the master stability function approach, they separate the influence of the network from the influence of the underlying dynamical system describing each node.

Restrepo et al.\ analyzed the emergence of synchrony from the completely asynchronous state, deriving a condition for the coupling strength at which the asynchronous state becomes unstable. Their key result is the coupling strength where this occurs depends on largest eigenvalue of the network adjacency matrix.

Summary of result of Restrepo, Ott, and Hunt, 20061,2

If linearize around the incoherent state, find the network structure influences the stability through $\lambda_{max}$, the largest eigenvalue of the adjacency matrix $A$.

The larger $\lambda_{max}$, the more likely the incoherent state is unstable.

The largest eigenvalue $\lambda_{\text{max}}$ of the adjacency matrix is a synchronizability index in the following sense. Given two networks with different large eigenvalues $\lambda_{\text{max}}$, synchrony will emerge at a lower coupling strength for the network with the largest $\lambda_{\text{max}}$. The precise coupling strengths where this occurs will depend on the dynamical model.

Both the master stability function approach and the Restrepo et al. analysis are in the same spirit of separating the effect of the network on synchrony from the underlying node dynamics. Both obtain their results by linearizing around a particular state (synchronous or asynchronous) to determine how network eigenvalues influence its stability.

Unfortunately, it's harder to analyze the effect of the connectivity on intermediate levels of synchrony. However, we can combine the two measures to get a fuller picture of the influence of network topology on synchrony.