scholarly journals Computational Properties of General Indices on Random Networks

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1341 ◽  
Author(s):  
R. Aguilar-Sánchez ◽  
I. F. Herrera-González ◽  
J. A. Méndez-Bermúdez ◽  
José M. Sigarreta
Keyword(s):  
Matrix Theory ◽  
Random Network ◽  
Scaling Law ◽  
Network Models ◽  
Network Size ◽  
Topological Indices ◽  
Random Networks ◽  
Theory Approach ◽  
Complexity Measures ◽  
Computational Properties

We perform a detailed (computational) scaling study of well-known general indices (the first and second variable Zagreb indices, M1α(G) and M2α(G), and the general sum-connectivity index, χα(G)) as well as of general versions of indices of interest: the general inverse sum indeg index ISIα(G) and the general first geometric-arithmetic index GAα(G) (with α∈R). We apply these indices on two models of random networks: Erdös–Rényi (ER) random networks GER(nER,p) and random geometric (RG) graphs GRG(nRG,r). The ER random networks are formed by nER vertices connected independently with probability p∈[0,1]; while the RG graphs consist of nRG vertices uniformly and independently distributed on the unit square, where two vertices are connected by an edge if their Euclidean distance is less or equal than the connection radius r∈[0,2]. Within a statistical random matrix theory approach, we show that the average values of the indices normalized to the network size scale with the average degree k of the corresponding random network models, where kER=(nER−1)p and kRG=(nRG−1)(πr2−8r3/3+r4/2). That is, X(GER)/nER≈X(GRG)/nRG if kER=kRG, with X representing any of the general indices listed above. With this work, we give a step forward in the scaling of topological indices since we have found a scaling law that covers different network models. Moreover, taking into account the symmetries of the topological indices we study here, we propose to establish their statistical analysis as a generic tool for studying average properties of random networks. In addition, we discuss the application of specific topological indices as complexity measures for random networks.

scholarly journals Normalized Sombor Indices as Complexity Measures of Random Networks

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 976
Author(s):  
R. Aguilar-Sánchez ◽  
J. Méndez-Bermúdez ◽  
José Rodríguez ◽  
José Sigarreta
Keyword(s):  
Random Matrix ◽  
Adjacency Matrix ◽  
Matrix Theory ◽  
Computational Study ◽  
Random Networks ◽  
Geometric Graphs ◽  
Theory Approach ◽  
Complexity Measures ◽  
Random Geometric Graphs ◽  
Highly Correlated

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

scholarly journals Stochastic Models of Emerging Infectious Disease Transmission on Adaptive Random Networks

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Navavat Pipatsart ◽  
Wannapong Triampo ◽  
Charin Modchang
Keyword(s):  
Infectious Disease ◽  
Disease Transmission ◽  
Random Network ◽  
Network Models ◽  
Transmission Probability ◽  
Stochastic Simulations ◽  
Random Networks ◽  
Cumulative Number ◽  
Network Adaptation ◽  
Infectious Disease Dynamics

We presented adaptive random network models to describe human behavioral change during epidemics and performed stochastic simulations of SIR (susceptible-infectious-recovered) epidemic models on adaptive random networks. The interplay between infectious disease dynamics and network adaptation dynamics was investigated in regard to the disease transmission and the cumulative number of infection cases. We found that the cumulative case was reduced and associated with an increasing network adaptation probability but was increased with an increasing disease transmission probability. It was found that the topological changes of the adaptive random networks were able to reduce the cumulative number of infections and also to delay the epidemic peak. Our results also suggest the existence of a critical value for the ratio of disease transmission and adaptation probabilities below which the epidemic cannot occur.

scholarly journals Modeling Network Dynamics

2020 ◽  
pp. 145-171
Keyword(s):  
Communication Networks ◽  
Random Network ◽  
Network Dynamics ◽  
Network Models ◽  
Random Networks ◽  
Social Phenomena ◽  
Disease Propagation ◽  
The Social ◽  
Wide Range ◽  
Model Dynamics

Social relationships and the social networks over these relationships do not occur arbitrarily. However, the random networks dealt with in this chapter are important tools for modeling the networks of these systems. The authors use random networks to understand and to model dynamics regarding the whole social structure. Random network models became the topic of several studies independently from social network analysis in the 1950s. These models were used in the analysis of a wide range of social and non-social phenomena, from electrical and communication networks to the speed and manner of disease propagation. This chapter explores the modeling network dynamics of random networks.

scholarly journals Information Entropy of Tight-Binding Random Networks with Losses and Gain: Scaling and Universality

Entropy ◽  
2019 ◽  
Vol 21 (1) ◽  
pp. 86 ◽  
Author(s):  
C. Martínez-Martínez ◽  
J. Méndez-Bermúdez
Keyword(s):  
Numerical Simulations ◽  
Network Model ◽  
Information Entropy ◽  
Spectral Properties ◽  
Random Network ◽  
Tight Binding ◽  
Network Connectivity ◽  
Network Size ◽  
Random Networks ◽  
Scaling And Universality

We study the localization properties of the eigenvectors, characterized by their information entropy, of tight-binding random networks with balanced losses and gain. The random network model, which is based on Erdős–Rényi (ER) graphs, is defined by three parameters: the network size N, the network connectivity α , and the losses-and-gain strength γ . Here, N and α are the standard parameters of ER graphs, while we introduce losses and gain by including complex self-loops on all vertices with the imaginary amplitude i γ with random balanced signs, thus breaking the Hermiticity of the corresponding adjacency matrices and inducing complex spectra. By the use of extensive numerical simulations, we define a scaling parameter ξ ≡ ξ ( N , α , γ ) that fixes the localization properties of the eigenvectors of our random network model; such that, when ξ < 0.1 ( 10 < ξ ), the eigenvectors are localized (extended), while the localization-to-delocalization transition occurs for 0.1 < ξ < 10 . Moreover, to extend the applicability of our findings, we demonstrate that for fixed ξ , the spectral properties (characterized by the position of the eigenvalues on the complex plane) of our network model are also universal; i.e., they do not depend on the specific values of the network parameters.

scholarly journals SURVIVING OPINIONS IN SZNAJD MODELS ON COMPLEX NETWORKS

2005 ◽  
Vol 16 (11) ◽  
pp. 1785-1792 ◽  
Author(s):  
F. A. RODRIGUES ◽  
L. DA F. COSTA
Keyword(s):  
Complex Networks ◽  
Scaling Law ◽  
Network Size ◽  
Random Networks ◽  
Scale Free ◽  
Scale Free Networks ◽  
Network Topologies

The Sznajd model has been largely applied to simulate many sociophysical phenomena. In this paper, we applied the Sznajd model with more than two opinions on three different network topologies and observed the evolution of surviving opinions after many interactions among the nodes. As result, we obtained a scaling law which depends of the network size and the number of possible opinions. We also observed that this scaling law is not the same for all network topologies, being quite similar between scale-free networks and Sznajd networks but different for random networks.

scholarly journals Analysis of Basic Features in Dynamic Network Models

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 681 ◽  
Author(s):  
Pedro Zufiria ◽  
Iker Barriales-Valbuena
Keyword(s):  
Random Network ◽  
Density Ratio ◽  
Dynamic Network ◽  
Network Models ◽  
Clustering Coefficient ◽  
Edge Density ◽  
Mathematical Framework ◽  
Complexity Measures ◽  
Dynamic Network Models ◽  
Link Density

Time evolving Random Network Models are presented as a mathematical framework for modelling and analyzing the evolution of complex networks. This framework allows the analysis over time of several network characterizing features such as link density, clustering coefficient, degree distribution, as well as entropy-based complexity measures, providing new insight on the evolution of random networks. First, some simple dynamic network models, based only on edge density, are analyzed to serve as a baseline reference for assessing more complex models. Then, a model that depends on network structure with the aim of reflecting some characteristics of real networks is also analyzed. Such model shows a more sophisticated behavior with two different regimes, one of them leading to the generation of high clustering coefficient/link density ratio values when compared with the baseline values, as it happens in many real networks. Simulation examples are discussed to illustrate the behavior of the proposed models.

Application of Lattice Imaging Techniques to Amorphous Films

1975 ◽  
Vol 33 ◽  
pp. 8-9
Author(s):  
S. R. Herd ◽  
P. Chaudhari
Keyword(s):  
Nearest Neighbor ◽  
Random Network ◽  
Imaging Techniques ◽  
Network Models ◽  
Accurate Method ◽  
Direct Transmission ◽  
Lattice Imaging ◽  
Transmission Electron ◽  
Lattice Planes ◽  
Nearest Neighbor Distances

Electron diffraction and direct transmission have been used extensively to study the local atomic arrangement in amorphous solids and in particular Ge. Nearest neighbor distances had been calculated from E.D. profiles and the results have been interpreted in terms of the microcrystalline or the random network models. Direct transmission electron microscopy appears the most direct and accurate method to resolve this issue since the spacial resolution of the better instruments are of the order of 3Å. In particular the tilted beam interference method is used regularly to show fringes corresponding to 1.5 to 3Å lattice planes in crystals as resolution tests.

Random-network models of the conductance of disordered condensed matter

1976 ◽  
Vol 13 (4) ◽  
pp. 1720-1727 ◽  
Author(s):  
Paul Erdös ◽  
Stephen B. Haley
Keyword(s):  
Random Network ◽  
Condensed Matter ◽  
Network Models

Visualization of tetrahedral disordering in amorphous germanium through local atomic motifs

2018 ◽  
Vol 51 (6) ◽  
pp. 1544-1550
Author(s):  
Aly Rahemtulla ◽  
Bruno Tomberli ◽  
Stefan Kycia
Keyword(s):  
Random Network ◽  
Network Models ◽  
Refined Model ◽  
Amorphous Germanium ◽  
Crystalline Materials ◽  
X Ray ◽  
Alignment Procedure ◽  
X Ray Scattering ◽  
Structural Details ◽  
Continuous Random Network

The atomic arrangements in amorphous solids, unlike those in crystalline materials, remain elusive. The details of atom ordering are under debate even in simplistic random network models. This work presents further advancements in the local atomic motif (LAM) method, first through the introduction of an optimized alignment procedure providing a clearer image of the angular ordering of atoms in a model. Secondly, by applying stereographic projections with LAMs, the angular ordering within coordination shells can be quantified and investigated. To showcase the new capabilities, the LAM method is applied to amorphous germanium, the archetype of covalent amorphous systems. The method is shown to dissect structural details of amorphous germanium (a-Ge) from the continuous random network (CRN) model and a reverse Monte Carlo (RMC) refined model fitted to high-resolution X-ray scattering measurements. The LAMs reveal well defined dihedral ordering in the second shell. The degree of dihedral ordering is observed to be coupled to bond length distances in the CRN model. This coupling is clearly not present within the RMC refined model. The LAMs reveal inclusions of third-shell atoms occupying interstitial positions in the second shell in both models.

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