Related papers: Graphical notation reveals topological stability c…
We investigate the connection between the dynamics of synchronization and the modularity on complex networks. Simulating the Kuramoto's model in complex networks we determine patterns of meta-stability and calculate the modularity of the…
Adaptive Kuramoto models admit a variety of nontrivial phase-locked configurations, including antipodal and rotating-wave states. A central open question is whether the observed persistence of such configurations can be attributed to…
Collective stable chaos consists of the persistence of disordered patterns in dynamical spatiotemporal systems possessing a negative maximum Lyapunov exponent. We analyze the role of the topology of connectivity on the emergence and…
Topological landscape is introduced for networks with functions defined on the nodes. By extending the notion of gradient flows to the network setting, critical nodes of different indices are defined. This leads to a concise and…
Complex networks exhibit a wide range of collective dynamic phenomena, including synchronization, diffusion, relaxation, and coordination processes. Their asymptotic dynamics is generically characterized by the local Jacobian, graph…
Given a holomorphic family of Bridgeland stability conditions over a surface, we define a notion of spectral network which is an object in a Fukaya category of the surface with coefficients in a triangulated DG-category. These spectral…
In the present study we consider a random network of Kuramoto oscillators with inertia in order to mimic and investigate the dynamics emerging in high-voltage power grids. The corresponding natural frequencies are assumed to be bimodally…
Kuramoto networks constitute a paradigmatic model for the investigation of collective behavior in networked systems. Despite many advances in recent years, many open questions remain on the solutions for systems composed of coupled Kuramoto…
The recent discovery of universal principles underlying many complex networks occurring across a wide range of length scales in the biological world has spurred physicists in trying to understand such features using techniques from…
We propose a novel method for topological analysis of unweighted graphs which is based on \textit{persistent homology}. The proposed method maps the input graph to a complete weighted graph where the weighting function maps each edge to a…
We discuss a link between graph theory and geometry that arises when considering graph dynamical systems with odd interactions. The equilibrium set in such systems is not a collection of isolated points, but rather a union of manifolds,…
The complexity of biological, social and engineering networks makes it desirable to find natural partitions into communities that can act as simplified descriptions and provide insight into the structure and function of the overall system.…
It is basic question in biology and other fields to identify the char- acteristic properties that on one hand are shared by structures from a particular realm, like gene regulation, protein-protein interaction or neu- ral networks or…
Many real-world systems of coupled agents exhibit directed interactions, meaning that the influence of an agent on another is not reciprocal. Furthermore, interactions usually do not have identical amplitude and/or sign. To describe…
The topological structure of the power grid plays a key role in the reliable delivery of electricity and price settlement in the electricity market. Incorporation of new energy sources and loads into the grid over time has led to its…
Networks are universally considered as complex structures of interactions of large multi-component systems. In order to determine the role that each node has inside a complex network, several centrality measures have been developed. Such…
Using the diagrammatic method, we derive a set of self-consistent equations that describe eigenvalue distributions of large correlated asymmetric random matrices. The matrix elements can have different variances and be correlated with each…
In this paper, we study combinatorial properties of stable curves. To the dual graph of any nodal curve, it is naturally associated a group, which is the group of components of the N\'eron model of the generalized Jacobian of the curve. We…
In many large systems, such as those encountered in biology or economics, the dynamics are nonlinear and are only known very coarsely. It is often the case, however, that the signs (excitation or inhibition) of individual interactions are…
This paper addresses analytical aspects of deterministic, continuous-time dynamical systems defined on networks. The goal is to model and analyze certain phenomena which must be framed beyond the context of networked dynamical systems,…