Related papers: Order in the chaos with examples from graph theory
We analyze the problem of discovering long cycles inside a graph. We propose and test two algorithms for this task. The first one is based on recent advances in statistical mechanics and relies on a message passing procedure. The second…
Ramsey theory looks for regularities in large objects. Model theory studies algebraic structures as models of theories. The structural Ramsey theory combines these two fields and is concerned with Ramsey-type questions about certain…
This paper is a direct continuation of an earlier work, where we studied Erd\"os-R\'enyi random graphs perturbed by an interaction Hamiltonian favouring the formation of short cycles. Here, we generalize these results. We keep the same…
Understanding which system structure can sustain stable dynamics is a fundamental step in the design and analysis of large scale dynamical systems. Towards this goal, we investigate here the structural stability of systems with a random…
Turing patterns, arising from the interplay between competing species of diffusive particles, has long been an important concept for describing non-equilibrium self-organization in nature, and has been extensively investigated in many…
Given a system $\mathcal{G} =\{G_1,G_2,\dots,G_m\}$ of graphs/digraphs/hypergraphs on the common vertex set $V$ of size $n$, an $m$-edge graph/digraph/hypergraph $H$ on $V$ is transversal in $\mathcal{G}$ if there exists a bijection…
The theory of pattern formation in reaction-diffusion systems is extended to the case of a directed network. Due to the structure of the network Laplacian of the scrutinised system, the dispersion relation has both real and imaginary parts,…
For real application and theoretical investigation of ordinary hypergraphs and non-ordinary hypergraphs, researchers need to establish standard rules and feasible operating methods. We propose a visualization tool for investigating…
In this partly expository paper we discuss and describe some of our old and recent results on partial orders on the set (m,n)-graphs (i.e. graphs with n vertices and m edges) and some operations on graphs that are monotone with respect to…
There are many examples of dualities between topological spaces and algebras in the literature. Particularly, many of those examples come from the algebraic counterpart of a logical system, e.g, boolean and heyting algebras, MV-algebras,…
We introduce a modification of the Tur\'an density of ordered graphs and investigate this graph parameter.
For any fixed integer $R \geq 2$ we characterise the typical structure of undirected graphs with vertices $1, ..., n$ and maximum degree $R$, as $n$ tends to infinity. The information is used to prove that such graphs satisfy a labelled…
This study examines second-order dynamical systems incorporating Tikhonov regularization. It focuses on how nonlinearities induce bifurcations and chaotic dynamics. By using Lyapunov functions, bifurcation theory, and numerical simulations,…
This is an investigation of the role of shuffling and concatenating in the theory of graph drawing. A simple syntactic description of these and related operations is proved complete in the context of finite partial orders, as general as…
Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is shown that sequences of such graphs have…
Existence theory in economics is usually in real domains such as the findings of chaotic trajectories in models of economic growth, tatonnement, or overlapping generations models. Computational examples, however, sometimes converge rapidly…
A chaotic system under periodic forcing can develop a periodically visited strange attractor. We discuss simple models in which the phenomenon, quite easy to see in numerical simulations, can be completely studied analytically.
In the past two decades, significant advances have been made in understanding the structural and functional properties of biological networks, via graph-theoretic analysis. In general, most graph-theoretic studies are conducted in the…
In recent years, attempts to generalize lattice gauge theories to model topological order have been carried out through the so called $2$-gauge theories. These have opened the door to interesting new models and new topological phases which…
We construct tree-decompositions of graphs that distinguish all their k-blocks and tangles of order k, for any fixed integer k. We describe a family of algorithms to construct such decompositions, seeking to maximize their diversity subject…