arXiv subjects

Hildeberto Jardón Kojakhmetov

Publications and source records attributed to Hildeberto Jardón Kojakhmetov.

3 recordsLinked to original sources

arXiv2025

Higher-order Laplacian dynamics on hypergraphs with cooperative and antagonistic interactions

Laplacian dynamics on a signless graph characterize a class of linear interactions, where pairwise cooperative interactions between all agents lead to the convergence to a common state. On a structurally balanced signed graph, the agents converge to values of the same magnitude but opposite signs (bipartite consensus), as illustrated by the well-known Altafini model. These interactions have been modeled using traditional graphs, where the relationships between agents are always pairwise. In comparison, higher-order networks (such as hypergraphs), offer the possibility to capture more complex, group-wise interactions among agents. This raises a natural question: can collective behavior be analyzed by using hypergraphs? The answer is affirmative. In this paper, higher-order Laplacian dynamics on signless hypergraphs are first introduced and various collective convergence behaviors are investigated, in the framework of homogeneous and non-homogeneous polynomial systems. Furthermore, by employing gauge transformations and leveraging tensor similarities, we extend these dynamics to signed hypergraphs, drawing parallels to the Altafini model. Moreover, we explore non-polynomial interaction functions within this framework. The theoretical results are demonstrated through several numerical examples.

eess.SY↗

arXiv2024

Nonlinear Diffusion on Networks: Perturbations and Consensus Dynamics

In this paper, we study a class of equations representing nonlinear diffusion on networks. A particular instance of our model can be seen as a network equivalent of the porous-medium equation. We are interested in studying perturbations of such a system and describing the consensus dynamics. The nonlinearity of the equations gives rise to potentially intricate structures of equilibria that can intersect the consensus space, creating singularities. For the unperturbed case, we characterise the sets of equilibria by exploiting the symmetries under group transformations of the nonlinear vector field. Under small perturbations, we obtain a slow-fast system. Thus, we analyse the slow-fast dynamics near the singularities on the consensus space. The analysis at this stage is carried out for complete networks, allowing a detailed characterisation of the system. We provide a linear approximation of the intersecting branches of equilibria at the singular points; as a consequence, we show that, generically, the singularities on the consensus space turn out to be transcritical. We prove under local assumptions the existence of canard solutions. For generic graph structures, assuming more strict conditions on the perturbation, we prove the existence of a maximal canard, which coincides with the consensus subspace. In addition, we validate by numerical simulations the principal findings of our main theory, extending the study to non-complete graphs. Moreover, we show how the delayed loss of stability associated to the canards induces transient spatio-temporal patterns.

math.DS↗

arXiv2024

On the Eigenvalues of Graphs with Mixed Algebraic Structure

We study some spectral properties of a matrix that is constructed as a combination of a Laplacian and an adjacency matrix of simple graphs. The matrix considered depends on a positive parameter, as such we consider the implications in different regimes of such a parameter, perturbative and beyond. Our main goal is to relate spectral properties to the graph's configuration, or to basic properties of the Laplacian and adjacency matrices. We explain the connections with dynamic networks and their stability properties, which lead us to state a conjecture for the signature.

math.DS↗