Related papers: Spectral dimension reduction of complex dynamical …
Despite the knowledge that social, economical, and ecological networks are often of a small-world nature with inter-nodal distance growing even slower than logarithmically with system size, we often assume theoretical systems to be outside…
Networks are fundamental for our understanding of complex systems. Interactions between individual nodes in networks generate network motifs - small recurrent patterns that can be considered the network's building-block components,…
Dynamic networks are used in a wide range of fields, including social network analysis, recommender systems, and epidemiology. Representing complex networks as structures changing over time allow network models to leverage not only…
We analyse a collection of empirical networks in a wide spectrum of disciplines and show that strong non-normality is ubiquitous in network science. Dynamical processes evolving on non-normal networks exhibit a peculiar behaviour, as…
In this paper, we address the problem of dynamic network embedding, that is, representing the nodes of a dynamic network as evolving vectors within a low-dimensional space. While the field of static network embedding is wide and…
One of the hallmarks of real networks is their ability to perform increasingly complex tasks as their topology evolves. To explain this, it has been observed that as a network grows certain subsets of the network begin to specialize the…
To cope with the complexity of large networks, a number of dimensionality reduction techniques for graphs have been developed. However, the extent to which information is lost or preserved when these techniques are employed has not yet been…
The eigenvalues and eigenvectors of the connectivity matrix of complex networks contain information about its topology and its collective behavior. In particular, the spectral density $\rho(\lambda)$ of this matrix reveals important network…
Universality is one of the key concepts in understanding critical phenomena. However, for interacting inhomogeneous systems described by complex networks a clear understanding of the relevant parameters for universality is still missing.…
A network provides powerful means of representing complex relationships between entities by abstracting entities as vertices, and relationships as edges connecting vertices in a graph. Beyond the presence or absence of relationships, a…
Epidemic spreading processes in the real world can interact with each other in a cooperative, competitive, or asymmetric way, requiring a description based on coevolution dynamics. Rich phenomena such as discontinuous outbreak transitions…
Networks have been studied mainly using statistical methods. Here I collect some dynamical systems tools which are useful to study both the dynamics on networks and their evolution. They include decomposition of differential dynamics,…
We propose a conceptually novel method of reconstructing the topology of dynamical networks. By examining the correlation between the variable of one node and the derivative of another node, we derive a simple matrix equation yielding the…
Real-world complex systems such as ecological communities and neuron networks are essential parts of our everyday lives. These systems are composed of units which interact through intricate networks. The ability to predict sudden changes in…
Centrality is an important notion in network analysis and is used to measure the degree to which network structure contributes to the importance of a node in a network. While many different centrality measures exist, most of them apply to…
Network filtering is an important form of dimension reduction to isolate the core constituents of large and interconnected complex systems. We introduce a new technique to filter large dimensional networks arising out of dynamical behavior…
The ability to achieve coordinated behavior --engineered or emergent-- on networked systems has attracted widespread interest over several fields. This has led to remarkable advances on the development of a theoretical understanding of the…
We present a general framework to study stability of the synchronous solution for a hypernetwork of coupled dynamical systems. We are able to reduce the dimensionality of the problem by using simultaneous block-diagonalization of matrices.…
Determining the effect of structural perturbations on the eigenvalue spectra of networks is an important problem because the spectra characterize not only their topological structures, but also their dynamical behavior, such as…
In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic)…