Exploring the interplay of excitatory and inhibitory interactions in the Kuramoto model on circle topologies.

In the field of collective dynamics, the Kuramoto model serves as a benchmark for the investigation of synchronization phenomena. While mean-field approaches and complex networks have been widely studied, the simple topology of a circle is still relatively unexplored, especially in the context of excitatory and inhibitory interactions. In this work, we focus on the dynamics of the Kuramoto model on a circle with positive and negative connections paying attention to the existence of new attractors different from the synchronized state. Using analytical and computational methods, we find that even for identical oscillators, the introduction of inhibitory interactions modifies the structure of the attractors of the system. Our results extend the current understanding of synchronization in simple topologies and open new avenues for the study of collective dynamics in physical systems.

Cavity approach for modeling and fitting polymer stretching.

The mechanical properties of molecules are today captured by single molecule manipulation experiments, so that polymer features are tested at a nanometric scale. Yet devising mathematical models to get further insight beyond the commonly studied force-elongation relation is typically hard. Here we draw from techniques developed in the context of disordered systems to solve models for single and double-stranded DNA stretching in the limit of a long polymeric chain. Since we directly derive the marginals for the molecule local orientation, our approach allows us to readily calculate the experimental elongation as well as other observables at wish. As an example, we evaluate the correlation length as a function of the stretching force. Furthermore, we are able to fit successfully our solution to real experimental data. Although the model is admittedly phenomenological, our findings are very sound. For single-stranded DNA our solution yields the correct (monomer) scale and yet, more importantly, the right persistence length of the molecule. In the double-stranded case, our model reproduces the well-known overstretching transition and correctly captures the ratio between native DNA and overstretched DNA. Also in this case the model yields a persistence length in good agreement with consensus, and it gives interesting insights into the bending stiffness of the native and overstretched molecule, respectively.

Entropies of complex networks with hierarchically constrained topologies.

The entropy of a hierarchical network topology in an ensemble of sparse random networks, with "hidden variables" associated with its nodes, is the log-likelihood that a given network topology is present in the chosen ensemble. We obtain a general formula for this entropy, which has a clear interpretation in some simple limiting cases. The results provide keys with which to solve the general problem of "fitting" a given network with an appropriate ensemble of random networks.

Synchronization reveals topological scales in complex networks.

We study the relationship between topological scales and dynamic time scales in complex networks. The analysis is based on the full dynamics towards synchronization of a system of coupled oscillators. In the synchronization process, modular structures corresponding to well-defined communities of nodes emerge in different time scales, ordered in a hierarchical way. The analysis also provides a useful connection between synchronization dynamics, complex networks topology, and spectral graph analysis.

Application of the microcanonical multifractal formalism to monofractal systems.

The design of appropriate multifractal analysis algorithms, able to correctly characterize the scaling properties of multifractal systems from experimental, discretized data, is a major challenge in the study of such scale invariant systems. In the recent years, a growing interest for the application of the microcanonical formalism has taken place, as it allows a precise localization of the fractal components as well as a statistical characterization of the system. In this paper, we deal with the specific problems arising when systems that are strictly monofractal are analyzed using some standard microcanonical multifractal methods. We discuss the adaptations of these methods needed to give an appropriate treatment of monofractal systems.