Spectral Network Principle for Frequency Synchronization in Repulsive Laser Networks

Network synchronization of lasers is critical for reaching high-power levels and for effective optical computing. Yet, the role of network topology for the frequency synchronization of lasers is not well understood. Here, we report our significant progress toward solving this critical problem for networks of heterogeneous laser model oscillators with repulsive coupling. We discover a general approximate principle for predicting the onset of frequency synchronization from the spectral knowledge of a complex matrix representing a combination of the signless Laplacian induced by repulsive coupling and a matrix associated with intrinsic frequency detuning. We show that the gap between the two smallest eigenvalues of the complex matrix generally controls the coupling threshold for frequency synchronization. In stark contrast with Laplacian networks, we demonstrate that local rings and all-to-all networks prevent frequency synchronization, whereas full bipartite networks have optimal synchronization properties. Beyond laser models, we show that, with a few exceptions, the spectral principle can be applied to repulsive Kuramoto networks. Our results may provide guidelines for optimal designs of scalable laser networks capable of achieving reliable synchronization

Dynamical and spectral properties of complex networks

Dynamical properties of complex networks are related to the spectral properties of the Laplacian matrix that describes the pattern of connectivity of the network. In particular we compute the synchronization time for different types of networks and different dynamics. We show that the main dependence of the synchronization time is on the smallest nonzero eigenvalue of the Laplacian matrix, in contrast to other proposals in terms of the spectrum of the adjacency matrix. Then, this topological property becomes the most relevant for the dynamics.Comment: 14 pages, 5 figures, to be published in New Journal of Physic

Assortativity Effects on Diffusion-like Processes in Scale-free Networks

We study the variation in epidemic thresholds in complex networks with different assortativity properties. We determine the thresholds by applying spectral analysis to the matrices associated to the graphs. In order to produce graphs with a specific assortativity we introduce a procedure to sample the space of all the possible networks with a given degree sequence. Our analysis shows that while disassortative networks have an higher epidemiological threshold, assortative networks have a slower diffusion time for diseases. We also used these networks for evaluating the effects of assortativity in a specific dynamic model of sandpile. We show that immunization procedures give different results according to the assortativity of the network considered

Spectral redemption: clustering sparse networks

Spectral algorithms are classic approaches to clustering and community detection in networks. However, for sparse networks the standard versions of these algorithms are suboptimal, in some cases completely failing to detect communities even when other algorithms such as belief propagation can do so. Here we introduce a new class of spectral algorithms based on a non-backtracking walk on the directed edges of the graph. The spectrum of this operator is much better-behaved than that of the adjacency matrix or other commonly used matrices, maintaining a strong separation between the bulk eigenvalues and the eigenvalues relevant to community structure even in the sparse case. We show that our algorithm is optimal for graphs generated by the stochastic block model, detecting communities all the way down to the theoretical limit. We also show the spectrum of the non-backtracking operator for some real-world networks, illustrating its advantages over traditional spectral clustering.Comment: 11 pages, 6 figures. Clarified to what extent our claims are rigorous, and to what extent they are conjectures; also added an interpretation of the eigenvectors of the 2n-dimensional version of the non-backtracking matri

Disease Localization in Multilayer Networks

We present a continuous formulation of epidemic spreading on multilayer networks using a tensorial representation, extending the models of monoplex networks to this context. We derive analytical expressions for the epidemic threshold of the SIS and SIR dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasi-stationary state method we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks, which are characterized analytically and numerically through the inverse participation ratio. Furthermore, when mapping the critical dynamics to an eigenvalue problem, we observe a characteristic transition in the eigenvalue spectra of the supra-contact tensor as a function of the ratio of two spreading rates: if the rate at which the disease spreads within a layer is comparable to the spreading rate across layers, the individual spectra of each layer merge with the coupling between layers. Finally, we verified the barrier effect, i.e., for three-layer configuration, when the layer with the largest eigenvalue is located at the center of the line, it can effectively act as a barrier to the disease. The formalism introduced here provides a unifying mathematical approach to disease contagion in multiplex systems opening new possibilities for the study of spreading processes.Comment: Revised version. 25 pages and 18 figure