Geometrical and spectral study of 
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-skeleton graphs

Alonso, Lazaro; Méndez-Bermúdez, J. A.; Estrada, Ernesto

doi:10.1103/physreve.100.062309

Cited by 15 publications

(11 citation statements)

References 24 publications

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“…To address these research questions and break the bottleneck in the field of network analysis, we construct a node-based multifractal analysis (NMFA) framework. The traditional multifractal analyses (MFA) 15 – 19 of complex networks are used to observe the self-similarity of some complex networks at different scales based on renormalization procedures. However, when it comes to some real networks and small-world networks, the traditional MFA method fails to capture the structural scaling dependence (see Supplementary Note 4 ), which limits its applications.…”

Section: Introductionmentioning

confidence: 99%

Deciphering the generating rules and functionalities of complex networks

Xiao

Chen

Bogdan

2021

Sci Rep

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Network theory helps us understand, analyze, model, and design various complex systems. Complex networks encode the complex topology and structural interactions of various systems in nature. To mine the multiscale coupling, heterogeneity, and complexity of natural and technological systems, we need expressive and rigorous mathematical tools that can help us understand the growth, topology, dynamics, multiscale structures, and functionalities of complex networks and their interrelationships. Towards this end, we construct the node-based fractal dimension (NFD) and the node-based multifractal analysis (NMFA) framework to reveal the generating rules and quantify the scale-dependent topology and multifractal features of a dynamic complex network. We propose novel indicators for measuring the degree of complexity, heterogeneity, and asymmetry of network structures, as well as the structure distance between networks. This formalism provides new insights on learning the energy and phase transitions in the networked systems and can help us understand the multiple generating mechanisms governing the network evolution.

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Section: Introductionmentioning

confidence: 99%

Deciphering the generating rules and functionalities of complex networks

Xiao

Chen

Bogdan

2021

Sci Rep

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“…Here, we will follow a recently introduced approach under which the adjacency matrices of random graphs are represented by RMT ensembles; see the application of this approach on Erdös-Rényi graphs [26,29], RGGs and random rectangular graphs [30], β-skeleton graphs [31], multiplex and multilayer networks [32], and bipartite graphs [27]. Consequently, we define the elements of the adjacency matrix A of our random graph model as…”

Section: Preliminariesmentioning

confidence: 99%

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

Martinez-Martinez,

Mendez-Bermudez,

Rodrigues

et al. 2021

Preprint

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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 ∈ (0, ∞) and ρ(θ) as an uniform distribution in the interval θ ∈ [0, 2π). Then, two vertices are connected by an edge if their Euclidian distance is less or equal than the connection radius ℓ. We characterize the topological properties of this random graph model, which depends on the parameter set (n, σ, ℓ), by the use of the average degree k and the number of non-isolated vertices V×; 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 k(n, σ, ℓ) . Then, we look for the scaling properties of the normalized average measure X (where X stands for V×, r and S) over graph ensembles. We demonstrate that the scaling parameter of V× = V× /n is indeed k ; with V× ≈ 1 − exp(− k ). Meanwhile, the scaling parameter of both r and S is proportional to n −γ k with γ ≈ 0.16.

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“…Also, GOE-GUE transition depending on a parameter α, for α = 0, GOE and for α = 1, the ensemble is Gaussian unitary [84] and cross-over transition between Poisson-GOE-GUE [85]. However, the Brody distribution has been widely studied, and there are well-known results for the case of single layer networks to measure the transition/mixture of GOE and Poisson statistics and for real-world networks also [46,47,[52][53][54][55][56][57][59][60][61][62][63][64]. Hence we use Brody distribution in the current article.…”

Section: Model and Techniquesmentioning

confidence: 99%

“…For 0 ≤ p ≤ 1, NNSD shows an intermediate statistics between the Poisson and the GOE [46,47]. However, all the investigations on various spectral properties of adjacency matrices are confined to single layer networks only [52][53][54][55][56][57][59][60][61][62][63][64].…”

Section: Introductionmentioning

confidence: 99%

Random Matrix Analysis of Multiplex Networks

Raghav,

Jalan

2021

Preprint

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We investigate the spectra of adjacency matrices of multiplex networks under random matrix theory (RMT) framework. Through extensive numerical experiments, we demonstrate that upon multiplexing two random networks, the spectra of the combined multiplex network exhibit superposition of two Gaussian orthogonal ensemble (GOE)s for very small multiplexing strength followed by a smooth transition to the GOE statistics with an increase in the multiplexing strength. Interestingly, randomness in the connection architecture, introduced by random rewiring to 1D lattice, of at least one layer may govern nearest neighbor spacing distribution (NNSD) of the entire multiplex network, and in fact, can drive to a transition from the Poisson to the GOE statistics or vice versa. Notably, this transition transpires for a very small number of the random rewiring corresponding to the small-world transition. Ergo, only one layer being represented by the small-world network is enough to yield GOE statistics for the entire multiplex network. Spectra of adjacency matrices of underlying interaction networks have been contemplated to be related with dynamical behaviour of the corresponding complex systems, the investigations presented here have implications in achieving better structural and dynamical control to the systems represented by multiplex networks against structural perturbation in only one of the layers.

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Geometrical and spectral study of β -skeleton graphs

Cited by 15 publications

References 24 publications

Deciphering the generating rules and functionalities of complex networks

Deciphering the generating rules and functionalities of complex networks

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

Random Matrix Analysis of Multiplex Networks

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