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Ernesto Estrada

Publications and source records attributed to Ernesto Estrada.

At least 37 records · Page 2Linked to original sources
arXiv2019

Geometrical and spectral study of $β$-skeleton graphs

We perform an extensive numerical analysis of $β$-skeleton graphs, a particular type of proximity graphs. In a $β$-skeleton graph (BSG) two vertices are connected if a proximity rule, that depends of the parameter $β\in(0,\infty)$, is satisfied. Moreover, for $β>1$ there exist two different proximity rules, leading to lune-based and circle-based BSGs. First, by computing the average degree of large ensembles of BSGs we detect differences, which increase with the increase of $β$, between lune-based and circle-based BSGs. Then, within a random matrix theory (RMT) approach, we explore spectral and eigenvector properties of randomly weighted BSGs by the use of the nearest-neighbor energy-level spacing distribution and the entropic eigenvector localization length, respectively. The RMT analysis allows us to conclude that a localization transition occurs at $β=1$.

physics.soc-ph↗
arXiv2019

Mathematical modeling for sustainable aphid control in agriculture via intercropping

Agricultural losses to pest represent an important challenge in a global warming scenario. Intercropping is an alternative farming practice that promotes pest control without the use of chemical pesticides. Here we develop a mathematical model to study epidemic spreading and control in intercropped agricultural fields as a sustainable pest management tool for agriculture. The model combines the movement of aphids transmitting a virus in an agricultural field, the spatial distribution of plants in the intercropped field, and the presence of `trap crops' in an epidemiological Susceptible-Infected-Removed (SIR) model. Using this model we study several intercropping arrangements without and with trap crops and find a new intercropping arrangement that improves significantly pest management in agricultural fields respect to the commonly used intercrop systems.

q-bio.PE↗
arXiv2019

Maximum walk entropy implies walk regularity

The notion of walk entropy $S^V(G,β)$ for a graph $G$ at the inverse temperature $β$ was put forward recently by Estrada et al. (2014) \cite{6}. It was further proved by Benzi \cite{1} that a graph is walk-regular if and only if its walk entropy is maximum for all temperatures $β\in I$, where $I$ is a set of real numbers containing at least an accumulation point. Benzi \cite{1} conjectured that walk regularity can be characterized by the walk entropy if and only if there is a $β>0$, such that $S^V(G,β)$ is maximum. Here we prove that a graph is walk regular if and only if the $S^V(G,β=1)=\ln n$. We also prove that if the graph is regular but not walk-regular $S^V(G,β)<\ln n$ for every $β>0$ and $\lim_{β\to 0} S^V(G,β)=\ln n=\lim_{β\to \infty} S^V(G,β)$. If the graph is not regular then $S^V(G,β) \leq \ln n-ε$ for every $β>0$, for some $ε>0$.

math-ph↗
arXiv2018

Quantum Interference, Graphs, Walks, and Polynomials

In this paper, we explore quantum interference in molecular conductance from the point of view of graph theory and walks on lattices. By virtue of the Cayley-Hamilton theorem for characteristic polynomials and the Coulson-Rushbrooke pairing theorem for alternant hydrocarbons, it is possible to derive a finite series expansion of the Green's function for electron transmission in terms of the odd powers of the vertex adjacency matrix or H{ü}ckel matrix. This means that only odd-length walks on a molecular graph contribute to the conductivity through a molecule. Thus, if there are only even-length walks between two atoms, quantum interference is expected to occur in the electron transport between them. However, even if there are only odd-length walks between two atoms, a situation may come about where the contributions to the QI of some odd-length walks are canceled by others, leading to another class of quantum interference. For non-alternant hydrocarbons, the finite Green's function expansion may include both even and odd powers. Nevertheless, QI can in some circumstances come about for non-alternants, from the cancellation of odd and even-length walk terms. We report some progress, but not a complete resolution of the problem of understanding the coefficients in the expansion of the Green's function in a power series of the adjacency matrix, these coefficients being behind the cancellations that we have mentioned. And we introduce a perturbation theory for transmission as well as some potentially useful infinite power series expansions of the Green's function.

physics.chem-ph↗
arXiv2018

Machine Learning Analysis of Complex Networks in Hyperspherical Space

A complex network is a condensed representation of the relational topological framework of a complex system. A main reason for the existence of such networks is the transmission of items through the entities of these complex systems. Here, we consider a communicability function that accounts for the routes through which items flow on networks. Such a function induces a natural embedding of a network in a Euclidean high-dimensional sphere. We use one of the geometric parameters of this embedding, namely the angle between the position vectors of the nodes in the hyperspheres, to extract structural information from networks. Such information is extracted by using machine learning techniques, such as nonmetric multidimensional scaling and K-means clustering algorithms. The first allows us to reduce the dimensionality of the communicability hyperspheres to 3-dimensional ones that allow network visualization. The second permits to cluster the nodes of the networks based on their similarities in terms of their capacity to successfully deliver information through the network. After testing these approaches in benchmark networks and compare them with the most used clustering methods in networks we analyze two real-world examples. In the first, consisting of a citation network, we discover citation groups that reflect the level of mathematics used in their publications. In the second, we discover groups of genes that coparticipate in human diseases, reporting a few genes that coparticipate in cancer and other diseases. Both examples emphasize the potential of the current methodology for the discovery of new patterns in relational data.

physics.soc-ph↗
arXiv2018

Communication Melting in Graphs and Complex Networks

Complex networks are the representative graphs of interactions in many complex systems. Usually, these interactions are abstractions of the communication/diffusion channels between the units of the system. Real complex networks, e.g. traffic networks, reveal different operation phases governed by the dynamical stress of the system. Here we show how, communicability, a topological descriptor that reveals the efficiency of the network functionality in terms of these diffusive paths, could be used to reveal the transitions mentioned. By considering a vibrational model of nodes and edges in a graph/network at a given temperature (stress), we show that the communicability function plays the role of the thermal Green's function of a network of harmonic oscillators. After, we prove analytically the existence of a universal phase transition in the communicability structure of every simple graph. This transition resembles the melting process occurring in solids. For instance, regular-like graphs resembling crystals, melts at lower temperatures and display a sharper transition between connected to disconnected structures than the random spatial graphs, which resemble amorphous solids. Finally, we study computationally this graph melting process in some real-world networks and observe that the rate of melting of graphs changes either as an exponential or as a power-law with the inverse temperature. At the local level we discover that the main driver for node melting is the eigenvector centrality of the corresponding node, particularly when the critical value of the inverse temperature approaches zero. These universal results sheds light on many dynamical diffusive-like processes on networks that present transitions as traffic jams, communication lost or failure cascades.

physics.soc-ph↗
arXiv2017

Centralities in Simplicial Complexes

Complex networks can be used to represent complex systems which originate in the real world. Here we study a transformation of these complex networks into simplicial complexes, where cliques represent the simplices of the complex. We extend the concept of node centrality to that of simplicial centrality and study several mathematical properties of degree, closeness, betweenness, eigenvector, Katz, and subgraph centrality for simplicial complexes. We study the degree distributions of these centralities at the different levels. We also compare and describe the differences between the centralities at the different levels. Using these centralities we study a method for detecting essential proteins in PPI networks of cells and explain the varying abilities of the centrality measures at the different levels in identifying these essential proteins. The paper is written in a self-contained way, such that it can be used by practitioners of network theory as a basis for further developments.

cs.SI↗
arXiv2017

Indirect Long-range Interactions and Network Synchronization

The dynamical behavior of networked complex systems is shaped not only by the direct links among the units, but also by the long-range interactions occurring through the many existing paths connecting the network nodes. In this work, we study how synchronization dynamics is influenced by these long-range interactions, formulating a model of coupled oscillators that incorporates this type of interactions through the use of $d-$path Laplacian matrices. We study synchronizability of these networks by the analysis of the Laplacian spectra, both theoretically and numerically, for real-world networks and artificial models. Our analysis reveal that in all networks long-range interactions improve network synchronizability with an impact that depends on the original structure, for instance it is greater for graphs having a larger diameter. We also investigate the effects of edge removal in graphs with long-range interactions and, as a major result, find that the removal process becomes more critical, since also the long-range influence of the removed link disappears.

physics.soc-ph↗
arXiv2017

Two-walks degree assortativity in graphs and networks

Degree ssortativity is the tendency for nodes of high degree (resp.low degree) in a graph to be connected to high degree nodes (resp. to low degree ones). It is sually quantified by the Pearson correlation coefficient of the degree-degree correlation. Here we extend this concept to account for the effect of second neighbours to a given node in a graph. That is, we consider the two-walks degree of a node as the sum of all the degrees of its adjacent nodes. The two-walks degree assortativity of a graph is then the Pearson correlation coefficient of the two-walks degree-degree correlation. We found here analytical expression for this two-walks degree assortativity index as a function of contributing subgraphs. We then study all the 261,000 connected graphs with 9 nodes and observe the existence of assortative-assortative and disassortative-disassortative graphs according to degree and two-walks degree, respectively. More surprinsingly, we observe a class of graphs which are degree disassortative and two-walks degree assortative. We explain the existence of some of these graphs due to the presence of certain topological features, such as a node of low-degree connected to high-degree ones. More importantly, we study a series of 49 real-world networks, where we observe the existence of the disassortative-assortative class in several of them. In particular, all biological networks studied here were in this class. We also conclude that no graphs/networks are possible with assortative-disassortative structure.

physics.soc-ph↗
arXiv2017

What is the meaning of the graph energy after all?

For a simple graph $G=(V,E)$ with eigenvalues of the adjacency matrix $λ_{1}\geqλ_{2}\geq\cdots\geqλ_{n}$, the energy of the graph is defined by $E(G)=\sum_{j=1}^{n}|λ_{j}|$. Myriads of papers have been published in the mathematical and chemistry literature about properties of this graph invariant due to its connection with the energy of (bipartite) conjugated molecules. However, a structural interpretation of this concept in terms of the contributions of even and odd walks, and consequently on the contribution of subgraphs, is not yet known. Here, we find such interpretation and prove that the (adjacency) energy of any graph (bipartite or not) is a weighted sum of the traces of even powers of the adjacency matrix. We then use such result to find bounds for the energy in terms of subgraphs contributing to it. The new bounds are studied for some specific simple graphs, such as cycles and fullerenes. We observe that including contributions from subgraphs of sizes not bigger than 6 improves some of the best known bounds for the energy, and more importantly gives insights about the contributions of specific subgraphs to the energy of these graphs.

math.CO↗
arXiv2017

Path Laplacian operators and superdiffusive processes on graphs. I. One-dimensional case

We consider a generalization of the diffusion equation on graphs. This generalized diffusion equation gives rise to both normal and superdiffusive processes on infinite one-dimensional graphs. The generalization is based on the $k$-path Laplacian operators $L_{k}$, which account for the hop of a diffusive particle to non-nearest neighbours in a graph. We first prove that the $k$-path Laplacian operators are self-adjoint. Then, we study the transformed $k$-path Laplacian operators using Laplace, factorial and Mellin transforms. We prove that the generalized diffusion equation using the Laplace- and factorial-transformed operators always produce normal diffusive processes independently of the parameters of the transforms. More importantly, the generalized diffusion equation using the Mellin-transformed $k$-path Laplacians $\sum_{k=1}^{\infty}k^{-s}L_{k}$ produces superdiffusive processes when $1<s<3$.

math.FA↗
arXiv2017

Random Neighborhood Graphs as Models of Fracture Networks on Rocks: Structural and Dynamical Analysis

We propose a new model to account for the main structural characteristics of rock fracture networks (RFNs). The model is based on a generalization of the random neighborhood graphs to consider fractures embedded into rectangular spaces. We study a series of 29 real-world RFNs and find the best fit with the random rectangular neighborhood graphs (RRNGs) proposed here. We show that this model captures most of the structural characteristics of the RFNs and allows a distinction between small and more spherical rocks and large and more elongated ones. We use a diffusion equation on the graphs in order to model diffusive processes taking place through the channels of the RFNs. We find a small set of structural parameters that highly correlates with the average diffusion time in the RFNs. We found analytically some bounds for the diameter and the algebraic connectivity of these graphs that allow to bound the diffusion time in these networks. We also show that the RRNGs can be used as a suitable model to replace the RFNs in the study of diffusion-like processes. Indeed, the diffusion time in RFNs can be predicted by using structural and dynamical parameters of the RRNGs. Finally, we also explore some potential extensions of our model to include variable fracture apertures, the possibility of long-range hops of the diffusive particles as a way to account for heterogeneities in the medium and possible superdiffusive processes, and the extension of the model to 3-dimensional space.

physics.geo-ph↗
arXiv2016

Accounting for the Role of Long Walks on Networks via a New Matrix Function

We introduce a new matrix function for studying graphs and real-world networks based on a double-factorial penalization of walks between nodes in a graph. This new matrix function is based on the matrix error function. We find a very good approximation of this function using a matrix hyperbolic tangent function. We derive a communicability function, a subgraph centrality and a double-factorial Estrada index based on this new matrix function. We obtain upper and lower bounds for the double-factorial Estrada index of graphs, showing that they are similar to those of the single-factorial Estrada index. We then compare these indices with the single-factorial one for simple graphs and real-world networks. We conclude that for networks containing chordless cycles---holes---the two penalization schemes produce significantly different results. In particular, we study two series of real-world networks representing urban street networks, and protein residue networks. We observe that the subgraph centrality based on both indices produce significantly different ranking of the nodes. The use of the double factorial penalization of walks opens new possibilities for studying important structural properties of real-world networks where long-walks play a fundamental role, such as the cases of networks containing chordless cycles.

math.CO↗
arXiv2016

Epidemic Spreading in Random Rectangular Networks

The use of network theory to model disease propagation on populations introduces important elements of reality to the classical epidemiological models. The use of random geometric graphs (RGG) is one of such network models that allows for the consideration of spatial properties on disease propagation. In certain real-world scenarios -like in the analysis of a disease propagating through plants- the shape of the plots and fields where the host of the disease is located may play a fundamental role on the propagation dynamics. Here we consider a generalization of the RGG to account for the variation of the shape of the plots/fields where the hosts of a disease are allocated. We consider a disease propagation taking place on the nodes of a random rectangular graph (RRG) and we consider a lower bound for the epidemic threshold of a Susceptible-Infected-Susceptible (SIS) or Susceptible-Infected-Recovered (SIR) model on these networks. Using extensive numerical simulations and based on our analytical results we conclude that (ceteris paribus) the elongation of the plot/field in which the nodes are distributed makes the network more resilient to the propagation of a disease due to the fact that the epidemic threshold increases with the elongation of the rectangle. These results agree with accumulated empirical evidence and simulation results about the propagation of diseases on plants in plots/fields of the same area and different shapes.

physics.soc-ph↗
arXiv2016

Exploring the "Middle Earth" of Network Spectra via a Gaussian Matrix Function

We study a Gaussian matrix function of the adjacency matrix of artificial and real-world networks. In particular, we study the Gaussian Estrada index---an index characterizing the importance of eigenvalues close to zero. This index accounts for the information contained in the eigenvalues close to zero in the spectra of networks. Here we obtain bounds for this index in simple graphs, proving that it reaches its maximum for star graphs followed by complete bipartite graphs. We also obtain formulas for the Estrada Gaussian index of Erdős-Rényi random graphs as well as for the Barabási-Albert graphs. We also show that in real-world networks this index is related to the existence of important structural patterns, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of the evolutionary processes giving rise to them. In general, the Gaussian matrix function of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs.

physics.soc-ph↗
arXiv2015

Core-satellite Graphs. Clustering, Assortativity and Spectral Properties

Core-satellite graphs (sometimes referred to as generalized friendship graphs) are an interesting class of graphs that generalize many well known types of graphs. In this paper we show that two popular clustering measures, the average Watts-Strogatz clustering coefficient and the transitivity index, diverge when the graph size increases. We also show that these graphs are disassortative. In addition, we completely describe the spectrum of the adjacency and Laplacian matrices associated with core-satellite graphs. Finally, we introduce the class of generalized core-satellite graphs, and we analyze the spectral properties of such graphs.

cs.SI↗