Patrick N. McGraw

Clustering and the synchronization of oscillator networks.

By manipulating the clustering coefficient of a network without changing its degree distribution, we examine the effect of clustering on the synchronization of phase oscillators on networks with Poisson and scale-free degree distributions. For both types of networks, increased clustering hinders global synchronization as the network splits into dynamical clusters that oscillate at different frequencies. Surprisingly, in scale-free networks, clustering promotes the synchronization of the most connected nodes (hubs) even though it inhibits global synchronization. As a result, they show an additional, advanced transition instead of a single synchronization threshold. This cluster-enhanced synchronization of hubs may be relevant to the brain that is scale-free and highly clustered.

Topology and computational performance of attractor neural networks.

To explore the relation between network structure and function, we studied the computational performance of Hopfield-type attractor neural nets with regular lattice, random, small-world, and scale-free topologies. The random configuration is the most efficient for storage and retrieval of patterns by the network as a whole. However, in the scale-free case retrieval errors are not distributed uniformly among the nodes. The portion of a pattern encoded by the subset of highly connected nodes is more robust and efficiently recognized than the rest of the pattern. The scale-free network thus achieves a very strong partial recognition. The implications of these findings for brain function and social dynamics are suggestive.

Laplacian spectra as a diagnostic tool for network structure and dynamics.

We examine numerically the three-way relationships among structure, Laplacian spectra, and frequency synchronization dynamics on complex networks. We study the effects of clustering, degree distribution, and a particular type of coupling asymmetry (input normalization), all of which are known to have effects on the synchronizability of oscillator networks. We find that these topological factors produce marked signatures in the Laplacian eigenvalue distribution and in the localization properties of individual eigenvectors. Using a set of coordinates based on the Laplacian eigenvectors as a diagnostic tool for synchronization dynamics, we find that the process of frequency synchronization can be visualized as a series of quasi-independent transitions involving different normal modes. Particular features of the partially synchronized state can be understood in terms of the behavior of particular modes or groups of modes. For example, there are important partially synchronized states in which a set of low-lying modes remain unlocked while those in the main spectral peak are locked. We find therefore that spectra are correlated with dynamics in ways that go beyond results relating a single threshold to a single extremal eigenvalue.

Analysis of nonlinear synchronization dynamics of oscillator networks by Laplacian spectral methods.

We analyze the synchronization dynamics of phase oscillators far from the synchronization manifold, including the onset of synchronization on scale-free networks with low and high clustering coefficients. We use normal coordinates and corresponding time-averaged velocities derived from the Laplacian matrix, which reflects the networks topology. In terms of these coordinates, synchronization manifests itself as a contraction of the dynamics onto progressively lower-dimensional submanifolds of phase space spanned by Laplacian eigenvectors with lower eigenvalues. Differences between high and low clustering networks can be correlated with features of the Laplacian spectrum. For example, the inhibition of full synchoronization at high clustering is associated with a group of low-lying modes that fail to lock even at strong coupling, while the advanced partial synchronization at low coupling noted elsewhere is associated with high-eigenvalue modes.

Self-sustaining oscillations in complex networks of excitable elements.

Random networks of symmetrically coupled, excitable elements can self-organize into coherently oscillating states if the networks contain loops (indeed loops are abundant in random networks) and if the initial conditions are sufficiently random. In the oscillating state, signals propagate in a single direction and one or a few network loops are selected as driving loops in which the excitation periodically circulates. We analyze the mechanism, describe the oscillating states, identify the pacemaker loops, and explain key features of their distribution.