arXiv subjects

J. A. Mendez-Bermudez

Publications and source records attributed to J. A. Mendez-Bermudez.

At least 19 recordsLinked to original sources

arXiv2026

Fuzzy Erdos-Renyi random graphs: Spectral statistics

Within a random-matrix-theory (RMT) approach, we perform a detailed numerical study of the spectral statistics of fuzzy Erdos-Renyi (ER) random graphs. Specifically, we consider the membership values of the vertices and edges of a fuzzy ER graph model as random variables. In this way, the corresponding adjacency matrices become sparse matrices with random entries. Then, we apply standard RMT measures to characterize the eigenvalues and eigenvectors of ensembles of Hermitian and non-Hermitian adjacency matrices of both undirected and directed fuzzy graphs. We demonstrate that the average degree of the ER model serves as the scaling parameter of the spectral properties of the corresponding fuzzy setups.

cond-mat.dis-nn↗

arXiv2024

Dissipative fractional standard maps: Riemann-Liouville and Caputo

In this study, given the inherent nature of dissipation in realistic dynamical systems, we explore the effects of dissipation within the context of fractional dynamics. Specifically, we consider the dissipative versions of two well known fractional maps: the Riemann-Liouville (RL) and the Caputo (C) fractional standard maps (fSMs). Both fSMs are two-dimensional nonlinear maps with memory given in action-angle variables $(I_n,θ_n)$; $n$ being the discrete iteration time of the maps. In the dissipative versions these fSMs are parameterized by the strength of nonlinearity $K$, the fractional order of the derivative $α\in(1,2]$, and the dissipation strength $γ\in(0,1]$. In this work we focus on the average action $\left< I_n \right>$ and the average squared action $\left< I_n^2 \right>$ when~$K\gg1$, i.e. along strongly chaotic orbits. We first demonstrate, for $|I_0|>K$, that dissipation produces the exponential decay of the average action $\left< I_n \right> \approx I_0\exp(-γn)$ in both dissipative fSMs. Then, we show that while $\left< I_n^2 \right>_{RL-fSM}$ barely depends on $α$ (effects are visible only when $α\to 1$), any $α< 2$ strongly influences the behavior of $\left< I_n^2 \right>_{C-fSM}$. We also derive an analytical expression able to describe $\left< I_n^2 \right>_{RL-fSM}(K,α,γ)$.

nlin.CD↗

arXiv2024

Non-Hermitian diluted banded random matrices: Scaling of eigenfunction and spectral properties

Here we introduce the non-Hermitian diluted banded random matrix (nHdBRM) ensemble as the set of $N\times N$ real non-symmetric matrices whose entries are independent Gaussian random variables with zero mean and variance one if $|i-j|<b$ and zero otherwise, moreover off-diagonal matrix elements within the bandwidth $b$ are randomly set to zero such that the sparsity $α$ is defined as the fraction of the $N(b-1)/2$ independent non-vanishing off-diagonal matrix elements. By means of a detailed numerical study we demonstrate that the eigenfunction and spectral properties of the nHdBRM ensemble scale with the parameter $x=γ[(bα)^2/N]^δ$, where $γ,δ\sim 1$. Moreover, the normalized localization length $β$ of the eigenfunctions follows a simple scaling law: $β= x/(1 + x)$. For comparison purposes, we also report eigenfunction and spectral properties of the Hermitian diluted banded random matrix ensemble.

cond-mat.dis-nn↗

arXiv2024

Singular-value statistics of directed random graphs

Singular-value statistics (SVS) has been recently presented as a random matrix theory tool able to properly characterize non-Hermitian random matrix ensembles [PRX Quantum {\bf 4}, 040312 (2023)]. Here, we perform a numerical study of the SVS of the non-Hermitian adjacency matrices $\mathbf{A}$ of directed random graphs, where $\mathbf{A}$ are members of diluted real Ginibre ensembles. We consider two models of directed random graphs: Erdös-Rényi graphs and random regular graphs. Specifically, we focus on the ratio $r$ between nearest neighbor singular values and the minimum singular value $λ_{min}$. We show that $\langle r \rangle$ (where $\langle \cdot \rangle$ represents ensemble average) can effectively characterize the transition between mostly isolated vertices to almost complete graphs, while the probability density function of $λ_{min}$ can clearly distinguish between different graph models.

stat.AP↗

arXiv2024

Tunable subdiffusion in the Caputo fractional standard map

The Caputo fractional standard map (C-fSM) is a two-dimensional nonlinear map with memory given in action-angle variables $(I,θ)$. It is parameterized by $K$ and $α\in(1,2]$ which control the strength of nonlinearity and the fractional order of the Caputo derivative, respectively. In this work we perform a scaling study of the average squared action $\left< I^2 \right>$ along strongly chaotic orbits, i.e. when $K\gg1$. We numerically prove that $\left< I^2 \right>\propto n^μ$ with $0\leμ(α)\le1$, for large enough discrete times $n$. That is, we demonstrate that the C-fSM displays subdiffusion for $1<α<2$. Specifically, we show that diffusion is suppressed for $α\to1$ since $μ(1)=0$, while standard diffusion is recovered for $α=2$ where $μ(2)=1$. We describe our numerical results with a phenomenological analytical estimation. We also contrast the C-fSM with the Riemann-Liouville fSM and Chirikov's standard map.

nlin.CD↗

arXiv2024

Scaling properties of the action in the Riemann-Liouville fractional standard map

The Riemann-Liouville fractional standard map (RL-fSM) is a two-dimensional nonlinear map with memory given in action-angle variables $(I,θ)$. The RL-fSM is parameterized by $K$ and $α\in(1,2]$ which control the strength of nonlinearity and the fractional order of the Riemann-Liouville derivative, respectively. In this work, we present a scaling study of the average squared action $\left< I^2 \right>$ of the RL-fSM along strongly chaotic orbits, i.e. for $K\gg1$. We observe two scenarios depending on the initial action $I_0$, $I_0\ll K$ or $I_0\gg K$. However, we can show that $\left< I^2 \right>/I_0^2$ is a universal function of the scaled discrete time $nK^2/I_0^2$ ($n$ being the $n$th iteration of the RL-fSM). In addition, we note that $\left< I^2 \right>$ is independent of $α$ for $K\gg1$. Analytical estimations support our numerical results.

nlin.CD↗

arXiv2023

Multiplicative topological indices: Analytical properties and application to random networks

We make use of multiplicative degree-based topological indices $X_Π(G)$ to perform a detailed analytical and statistical study of random networks $G=(V(G),E(G))$. We consider two classes of indices: $X_Π(G) = \prod_{u \in V(G)} F_V(d_u)$ and $X_Π(G) = \prod_{uv \in E(G)} F_E(d_u,d_v)$, where $uv$ denotes the edge of $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $F_V(x)$ and $F_E(x,y)$ are functions of the vertex degrees. Specifically, we find analytical inequalities involving these multiplicative indices. Also, we apply $X_Π(G)$ on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. We show that $\left< \ln X_Π(G) \right>$, normalized to the order of the network, scale with the corresponding average degree; here $\left< \cdot \right>$ denotes the average over an ensemble of random networks.

math.CO↗

arXiv2023

Localization properties of harmonic chains with correlated mass and spring disorder: Analytical approach

We study the localization properties of normal modes in harmonic chains with mass and spring weak disorder. Using a perturbative approach, an expression for the localization length is obtained, which is valid for arbitrary correlations of the disorder, and for practically the whole frequency band. In addition, we show how to generate effective mobility edges by the use of disorder with long range self-correlations and cross-correlations. Finally, the transport of phonons is also analyzed showing effective transparent windows that can be manipulated through the disorder correlations even for relative short chain sizes. Our results may have applications in modulating thermal transport, particularly in the design of thermal filters or in manufacturing high-thermal-conductivity materials.

cond-mat.dis-nn↗

arXiv2022

Stability mapping of bipartite tight-binding graphs with losses and gain: ${\cal PT}-$symmetry and beyond

We consider bipartite tight-binding graphs composed by $N$ nodes split into two sets of equal size: one set containing nodes with on-site loss, the other set having nodes with on-site gain. The nodes are connected randomly with probability $p$. We give a rationale for the relevance of such "throttle/brake" coupled systems (physically open systems) to grasp the stability issues of complex networks in areas such as biochemistry, neurons or economy, for which their modelling in terms of non-hermitian Hamiltonians is still in infancy. Specifically, we measure the connectivity between the two sets with the parameter $α$, which is the ratio of current adjacent pairs over the total number of possible adjacent pairs between the sets. For general undirected-graph setups, the non-hermitian Hamiltonian $H(γ,α,N)$ of this model presents pseudo-Hermiticity, where $γ$ is the loss/gain strength. However, we show that for a given graph setup $H(γ,α,N)$ becomes ${\cal PT}-$symmetric. In both scenarios (pseudo-Hermiticity and ${\cal PT}-$symmetric), depending on the parameter combination, the spectra of $H(γ,α,N)$ can be real even when it is non-hermitian. Thus, we numerically characterize the average fractions of real and imaginary eigenvalues of $H(γ,α,N)$ as a function of the parameter set $\{γ,α,N\}$. We demonstrate, for both setups, that there is a well defined sector of the $γα-$plane (which grows with $N$) where the spectrum of $H(γ,α,N)$ is predominantly real.

cond-mat.dis-nn↗

arXiv2022

Revan-degree indices on random graphs

Given a simple connected non-directed graph $G=(V(G),E(G))$, we consider two families of graph invariants: $RX_Σ(G) = \sum_{uv \in E(G)} F(r_u,r_v)$ (which has gained interest recently) and $RX_Π(G) = \prod_{uv \in E(G)} F(r_u,r_v)$ (that we introduce in this work); where $uv$ denotes the edge of $G$ connecting the vertices $u$ and $v$, $r_u$ is the Revan degree of the vertex $u$, and $F$ is a function of the Revan vertex degrees. Here, $r_u = Δ+ δ- d_u$ with $Δ$ and $δ$ the maximum and minimum degrees among the vertices of $G$ and $d_u$ is the degree of the vertex $u$. Particularly, we apply both $RX_Σ(G)$ and R$X_Π(G)$ on two models of random graphs: Erdös-Rényi graphs and random geometric graphs. By a thorough computational study we show that $\left< RX_Σ(G) \right>$ and $\left< \ln RX_Π(G) \right>$, normalized to the order of the graph, scale with the average Revan degree $\left< r \right>$; here $\left< \cdot \right>$ denotes the average over an ensemble of random graphs. Moreover, we provide analytical expressions for several graph invariants of both families in the dense graph limit.

math.CO↗

arXiv2021

Mean Sombor index

We introduce a degree-based variable topological index inspired on the power (or generalized) mean. We name this new index as the mean Sombor index: $mSO_α(G) = \sum_{uv \in E(G)} \left[\left( d_u^α+d_v^α\right) /2 \right]^{1/α}$. Here, $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $α\in \mathbb{R} \backslash \{0\}$. We also consider the limit cases $mSO_{α\to 0}(G)$ and $mSO_{α\to\pm\infty}(G)$. Indeed, for given values of $α$, the mean Sombor index is related to well-known topological indices such as the inverse sum indeg index, the reciprocal Randic index, the first Zagreb index, the Stolarsky--Puebla index and several Sombor indices. Moreover, through a quantitative structure property relationship (QSPR) analysis we show that $mSO_α(G)$ correlates well with several physicochemical properties of octane isomers. Some mathematical properties of mean Sombor indices as well as bounds and new relationships with known topological indices are also discussed.

math.CO↗

arXiv2021

Stolarsky-Puebla index

We introduce a degree-based variable topological index inspired on the Stolarsky mean (known as the generalization of the logarithmic mean). We name this new index as the Stolarsky-Puebla index: $SP_α(G) = \sum_{uv \in E(G)} d_u$, if $d_u=d_v$, and $SP_α(G) = \sum_{uv \in E(G)} \left[\left( d_u^α-d_v^α\right)/\left( α(d_u-d_v\right)\right]^{1/(α-1)}$, otherwise. Here, $uv$ denotes the edge of the network $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $α\in \mathbb{R} \backslash \{0,1\}$. Indeed, for given values of $α$, the Stolarsky-Puebla index reproduces well-known topological indices such as the reciprocal Randic index, the first Zagreb index, and several mean Sombor indices. Moreover, we apply these indices to random networks and demonstrate that $\left< SP_α(G) \right>$, normalized to the order of the network, scale with the corresponding average degree $\left< d \right>$.

math.CO↗

arXiv2021

Non-uniform random graphs on the plane: A scaling study

We consider random geometric graphs on the plane characterized by a non-uniform density of vertices. In particular, we introduce a graph model where $n$ vertices are independently distributed in the unit disc with positions, in polar coordinates $(l,θ)$, obeying the probability density functions $ρ(l)$ and $ρ(θ)$. Here we choose $ρ(l)$ as a normal distribution with zero mean and variance $σ\in(0,\infty)$ and $ρ(θ)$ as an uniform distribution in the interval $θ\in [0,2π)$. Then, two vertices are connected by an edge if their Euclidian distance is less or equal than the connection radius $\ell$. We characterize the topological properties of this random graph model, which depends on the parameter set $(n,σ,\ell)$, by the use of the average degree $\left\langle k \right\rangle$ and the number of non-isolated vertices $V_\times$; while we approach their spectral properties with two measures on the graph adjacency matrix: the ratio of consecutive eigenvalue spacings $r$ and the Shannon entropy $S$ of eigenvectors. First we propose a heuristic expression for $\left\langle k(n,σ,\ell) \right\rangle$. Then, we look for the scaling properties of the normalized average measure $\left\langle \overline{X} \right\rangle$ (where $X$ stands for $V_\times$, $r$ and $S$) over graph ensembles. We demonstrate that the scaling parameter of $\left\langle \overline{V_\times} \right\rangle=\left\langle V_\times \right\rangle/n$ is indeed $\left\langle k \right\rangle$; with $\left\langle \overline{V_\times} \right\rangle \approx 1-\exp(-\left\langle k \right\rangle)$. Meanwhile, the scaling parameter of both $\left\langle \overline{r} \right\rangle$ and $\left\langle \overline{S} \right\rangle$ is proportional to $n^{-γ} \left\langle k \right\rangle$ with $γ\approx 0.16$.

cond-mat.dis-nn↗

arXiv2021

Normalized Sombor indices as complexity measures of random graphs

We perform a detailed computational study of the recently introduced Sombor indices on random graphs. Specifically, we apply Sombor indices on three models of random graphs: Erdös-Rényi graphs, random geometric graphs, and bipartite random graphs. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the graph, scale with the graph average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random graphs and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the graph adjacency matrix.

math.CO↗

arXiv2021

Analytical and computational study of the variable inverse sum deg index

A large number of graph invariants of the form $\sum_{uv \in E(G)} F(d_u,d_v)$ are studied in mathematical chemistry, where $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, and $d_u$ is the degree of the vertex $u$. Among them the variable inverse sum deg index $ISD_a$, with $F(d_u,d_v)=1/(d_u^a+d_v^a)$, was found to have applicative properties. The aim of this paper is to obtain new inequalities for the variable inverse sum deg index, and to characterize graphs extremal with respect to them. Some of these inequalities generalize and improve previous results for the inverse sum deg index. In addition, we computationally validate some of the obtained inequalities on ensembles of random graphs and show that the ratio $\left\langle ISD_a(G) \right\rangle/n$ ($n$ being the order of the graph) depends only on the average degree $\left\langle d \right\rangle$.

math.CO↗

arXiv2021

Analytical and computational properties of the variable symmetric division deg index

The aim of this work is to obtain new inequalities for the variable symmetric division deg index $SDD_α(G) = \sum_{uv \in E(G)} (d_u^α/d_v^α+d_v^α/d_u^α)$, and to characterize graphs extremal with respect to them. Here, $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $α\in \mathbb{R}$. Some of these inequalities generalize and improve previous results for the symmetric division deg index. In addition, we computationally apply the $SDD_α(G)$ index on random graphs and show that the ratio $\left\langle SDD_α(G) \right\rangle/n$ ($n$ being the order of the graph) depends only on the average degree $\left\langle d \right\rangle$.

math.CO↗

arXiv2020

Topological versus spectral properties of random geometric graphs

In this work we perform a detailed statistical analysis of topological and spectral properties of random geometric graphs (RGGs); a graph model used to study the structure and dynamics of complex systems embedded in a two dimensional space. RGGs, $G(n,\ell)$, consist of $n$ vertices uniformly and independently distributed on the unit square, where two vertices are connected by an edge if their Euclidian distance is less or equal than the connection radius $\ell \in [0,\sqrt{2}]$. To evaluate the topological properties of RGGs we chose two well-known topological indices, the Randić index $R(G)$ and the harmonic index $H(G)$. While we characterize the spectral and eigenvector properties of the corresponding randomly-weighted adjacency matrices by the use of random matrix theory measures: the ratio between consecutive eigenvalue spacings, the inverse participation ratios and the information or Shannon entropies $S(G)$. First, we review the scaling properties of the averaged measures, topological and spectral, on RGGs. Then we show that: (i) the averaged--scaled indices, $\left\langle R(G) \right\rangle$ and $\left\langle H(G) \right\rangle$, are highly correlated with the average number of non-isolated vertices $\left\langle V_\times(G) \right\rangle$; and (ii) surprisingly, the averaged--scaled Shannon entropy $\left\langle S(G) \right\rangle$ is also highly correlated with $\left\langle V_\times(G) \right\rangle$. Therefore, we suggest that very reliable predictions of eigenvector properties of RGGs could be made by computing topological indices.

cond-mat.dis-nn↗

arXiv2020

Computational and analytical studies of the Randić index in Erdös-Rényi models

In this work we perform computational and analytical studies of the Randić index $R(G)$ in Erdös-Rényi models $G(n,p)$ characterized by $n$ vertices connected independently with probability $p \in (0,1)$. First, from a detailed scaling analysis, we show that $\left\langle \overline{R}(G) \right\rangle = \left\langle R(G)\right\rangle/(n/2)$ scales with the product $ξ\approx np$, so we can define three regimes: a regime of mostly isolated vertices when $ξ< 0.01$ ($R(G)\approx 0$), a transition regime for $0.01 < ξ< 10$ (where $0 10$ ($R(G)\approx n/2$). Then, motivated by the scaling of $\left\langle \overline{R}(G) \right\rangle$, we analytically (i) obtain new relations connecting $R(G)$ with other topological indices and characterize graphs which are extremal with respect to the relations obtained and (ii) apply these results in order to obtain inequalities on $R(G)$ for graphs in Erdös-Rényi models.

cond-mat.dis-nn↗