Spectral and localization properties of random bipartite graphs

Martinez-Martinez, C. T.; Méndez-Bermúdez, J. A.; Moreno, Yamir; Pineda-Pineda, Jair J.; Sigarreta, José M.

doi:10.1016/j.csfx.2020.100021

Cited by 16 publications

(23 citation statements)

References 22 publications

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“…In the case of complex networks, the use of RMT techniques might reveal universal properties. Among several studies available in the literature we can mention, as examples, that (1) the nearest-neighbor spacing distribution P ( s ) of the eigenvalues of the adjacency matrices of various network models follow Gaussian Orthogonal Ensemble (GOE) statistics [ 52 ]; (2) the P ( s ) and the entropic eigenfunction localization lengths of the adjacency matrices of Erdös-Rényi networks are universal for fixed average degrees [ 53 ]; (3) spectral and eigenfunction properties of multilayer networks [ 54 ], random rectangular graphs [ 55 ], and bipartite graphs [ 56 ] are universal for properly-defined scaling variables; and (4) RMT-based scaling analysis allows to predict the performance of network discovery algorithms [ 57 ].…”

Section: Spectral Analysismentioning

confidence: 99%

“…Moreover, we consider an important modification to the standard adjacency matrix definition: We consider weights for vertices and edges in the layers of the ATNs studied here. Our main motivation to include weights, particularly random weights ( i.e ., statistically independent random variables drawn from a normal distribution with zero mean and variance one), is to retrieve well-known random matrices in the appropriate limits to use RMT results as a reference, see for instance [ 53 – 56 ]. With this prescription, the adjacency matrix of a completely disconnected network becomes a diagonal random matrix, known in RMT as the Poisson limit, whereas a member of the GOE is recovered for a fully connected network.…”

Section: Spectral Analysismentioning

confidence: 99%

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Structural and spectral properties of generative models for synthetic multilayer air transportation networks

et al. 2021

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To understand airline transportation networks (ATN) systems we can effectively represent them as multilayer networks, where layers capture different airline companies, the nodes correspond to the airports and the edges to the routes between the airports. We focus our study on the importance of leveraging synthetic generative multilayer models to support the analysis of meaningful patterns in these routes, capturing an ATN’s evolution with an emphasis on measuring its resilience to random or targeted attacks and considering deliberate locations of airports. By resorting to the European ATN and the United States ATN as exemplary references, in this work, we provide a systematic analysis of major existing synthetic generation models for ATNs, specifically ANGEL, STARGEN and BINBALL. Besides a thorough study of the topological aspects of the ATNs created by the three models, our major contribution lays on an unprecedented investigation of their spectral characteristics based on Random Matrix Theory and on their resilience analysis based on both site and bond percolation approaches. Results have shown that ANGEL outperforms STARGEN and BINBALL to better capture the complexity of real-world ATNs by featuring the unique properties of building a multiplex ATN layer by layer and of replicating layers with point-to-point structures alongside hub-spoke formations.

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Section: Spectral Analysismentioning

confidence: 99%

Section: Spectral Analysismentioning

confidence: 99%

Structural and spectral properties of generative models for synthetic multilayer air transportation networks

et al. 2021

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“…Another widely known structural property of a network is the connectivity, denoted by α ∈ [0, 1], and defined as the ratio of current adjacent pairs over the total number of possible adjacent pairs (see [35]). In particular, for our geometric model,…”

Section: Structural Analysis Of the Geometric Modelmentioning

confidence: 99%

Multimetric Index to Evaluate Water Quality in Lagoons: A Biological and Geomorphological Approach

et al. 2021

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In recent years, Multimetric Indices (MMIs) have received a lot of attention thanks to their ability to develop integrative evaluations of water quality, particularly in lagoons. In this article, we propose a new MMI for determining the water quality in lagoons. The proposed index is composed of biotic and abiotic indicators, in particular macroinvertebrates, macrophytes and morphological indicators. The proposed index is based on a geometric representation of a phenomenon associated with an ecological system, the ecosystem elements are mapped as vertices of a network and the relationship between them is represented by the corresponding edges. We classify the status of water bodies, from very low to very high using the ecological quality ratio. We compare our index with different different indices that measure water quality, such as General Biotic Index (JP(G)), Macrophyte Index for River (MIR) and Shannon diversity index (H’) and validate our index with Pearson’s correlation coefficient. A strong correlation with the JP(G) and MIR indices (R2 = 0.8605 and R2=0.7661, respectively) is obtained. Although the proposed index is composed of other indices, the independence of the proposed index with respect to its component indices is proven and the structure of the geometric model associated to the proposed network is studied. A close relationship between the measure called medium articulation and the geometric model associated with the proposed index is highlighted, which allows to determine the missing relationships in the network using structural analysis. The proposed index presents a more comprehensive measure than most indices currently used and has the advantage in the scalability, since other existing indicators can be integrated into our model.

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“…We would like to mention that in contrast to the Shannon entropy which is a well accepted quantity to measure the degree of disorder in complex networks, the use of the ratio of consecutive eigenvalue spacings is relative recent in graph studies; see for example [25][26][27][28].…”

Section: Preliminariesmentioning

confidence: 99%

“…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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Spectral and localization properties of random bipartite graphs

Cited by 16 publications

References 22 publications

Structural and spectral properties of generative models for synthetic multilayer air transportation networks

Structural and spectral properties of generative models for synthetic multilayer air transportation networks

Multimetric Index to Evaluate Water Quality in Lagoons: A Biological and Geomorphological Approach

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

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