Spectra of adjacency and Laplacian matrices of inhomogeneous Erdős–Rényi random graphs

2019 ◽  
Vol 10 (01) ◽  
pp. 2150009 ◽  
Author(s):  
Arijit Chakrabarty ◽  
Rajat Subhra Hazra ◽  
Frank den Hollander ◽  
Matteo Sfragara
Keyword(s):  
Social Networks ◽  
Continuous Function ◽  
Random Graphs ◽  
Empirical Distribution ◽  
Laplacian Matrix ◽  
Laplacian Matrices ◽  
Matrix Formula ◽  
Spectral Distributions ◽  
Special Case

This paper considers inhomogeneous Erdős–Rényi random graphs [Formula: see text] on [Formula: see text] vertices in the non-sparse non-dense regime. The edge between the pair of vertices [Formula: see text] is retained with probability [Formula: see text], [Formula: see text], independently of other edges, where [Formula: see text] is a continuous function such that [Formula: see text] for all [Formula: see text]. We study the empirical distribution of both the adjacency matrix [Formula: see text] and the Laplacian matrix [Formula: see text] associated with [Formula: see text], in the limit as [Formula: see text] when [Formula: see text] and [Formula: see text]. In particular, we show that the empirical spectral distributions of [Formula: see text] and [Formula: see text], after appropriate scaling and centering, converge to deterministic limits weakly in probability. For the special case where [Formula: see text] with [Formula: see text] a continuous function, we give an explicit characterization of the limiting distributions. Furthermore, we apply our results to constrained random graphs, Chung–Lu random graphs and social networks.

scholarly journals EMPIRICAL DISTRIBUTIONS OF LAPLACIAN MATRICES OF LARGE DILUTE RANDOM GRAPHS

2012 ◽  
Vol 01 (03) ◽  
pp. 1250004 ◽  
Author(s):  
TIEFENG JIANG
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Spectral Properties ◽  
Empirical Distribution ◽  
Laplacian Matrix ◽  
Free Convolution ◽  
Laplacian Matrices ◽  
Empirical Distributions ◽  
Large N ◽  
Eigenvalues Of The Laplacian

We study the spectral properties of the Laplacian matrices and the normalized Laplacian matrices of the Erdös–Rényi random graph G(n, pn) for large n. Although the graph is simple, we discover some interesting behaviors of the two Laplacian matrices. In fact, under the dilute case, that is, pn ∈ (0, 1) and npn → ∞, we prove that the empirical distribution of the eigenvalues of the Laplacian matrix converges to a deterministic distribution, which is the free convolution of the semi-circle law and N(0, 1). However, for its normalized version, we prove that the empirical distribution converges to the semi-circle law.

Conjugate Laplacian matrices of a graph

2018 ◽  
Vol 10 (06) ◽  
pp. 1850082
Author(s):  
Somnath Paul
Keyword(s):  
Cartesian Product ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Discrete Mathematics ◽  
Laplacian Matrices ◽  
Cartesian Product Of Graphs ◽  
Laplacian Spectra ◽  
The Matrix ◽  
Matrix Formula ◽  
Product Of Graphs

Let [Formula: see text] be a simple graph of order [Formula: see text] Let [Formula: see text] and [Formula: see text] where [Formula: see text] and [Formula: see text] are two nonzero integers and [Formula: see text] is a positive integer such that [Formula: see text] is not a perfect square. In [M. Lepovi[Formula: see text], On conjugate adjacency matrices of a graph, Discrete Mathematics 307 (2007) 730–738], the author defined the matrix [Formula: see text] to be the conjugate adjacency matrix of [Formula: see text] if [Formula: see text] for any two adjacent vertices [Formula: see text] and [Formula: see text] for any two nonadjacent vertices [Formula: see text] and [Formula: see text] and [Formula: see text] if [Formula: see text] In this paper, we define conjugate Laplacian matrix of graphs and describe various properties of its eigenvalues and eigenspaces. We also discuss the conjugate Laplacian spectra for union, join and Cartesian product of graphs.

Applications of random graphs

2017 ◽  
Author(s):  
A.C.C. Coolen ◽  
A. Annibale ◽  
E.S. Roberts
Keyword(s):  
Social Networks ◽  
Random Graphs ◽  
Interaction Network ◽  
Power Grids ◽  
Protein Protein Interaction ◽  
Topological Features ◽  
First Case ◽  
The Stability ◽  
Protein Protein Interaction Network

This chapter reviews graph generation techniques in the context of applications. The first case study is power grids, where proposed strategies to prevent blackouts have been tested on tailored random graphs. The second case study is in social networks. Applications of random graphs to social networks are extremely wide ranging – the particular aspect looked at here is modelling the spread of disease on a social network – and how a particular construction based on projecting from a bipartite graph successfully captures some of the clustering observed in real social networks. The third case study is on null models of food webs, discussing the specific constraints relevant to this application, and the topological features which may contribute to the stability of an ecosystem. The final case study is taken from molecular biology, discussing the importance of unbiased graph sampling when considering if motifs are over-represented in a protein–protein interaction network.

scholarly journals Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach

2021 ◽  
Author(s):  
Yves Achdou ◽  
Jiequn Han ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions ◽  
Benjamin Moll
Keyword(s):  
Continuous Time ◽  
Wealth Distribution ◽  
Closed Form Solution ◽  
Form Solution ◽  
Heterogeneous Agent ◽  
Saving Behavior ◽  
Income And Wealth Distribution ◽  
Heterogeneous Agent Models ◽  
Special Case

Abstract We recast the Aiyagari-Bewley-Huggett model of income and wealth distribution in continuous time. This workhorse model – as well as heterogeneous agent models more generally – then boils down to a system of partial differential equations, a fact we take advantage of to make two types of contributions. First, a number of new theoretical results: (i) an analytic characterization of the consumption and saving behavior of the poor, particularly their marginal propensities to consume; (ii) a closed-form solution for the wealth distribution in a special case with two income types; (iii) a proof that there is a unique stationary equilibrium if the intertemporal elasticity of substitution is weakly greater than one. Second, we develop a simple, efficient and portable algorithm for numerically solving for equilibria in a wide class of heterogeneous agent models, including – but not limited to – the Aiyagari-Bewley-Huggett model.

scholarly journals A characterization of a map whose inverse limit is an arc

2021 ◽  
pp. 1-17
Author(s):  
SINA GREENWOOD ◽  
SONJA ŠTIMAC
Keyword(s):  
Continuous Function ◽  
Inverse Limit ◽  
Splitting Sequence

Abstract For a continuous function $f:[0,1] \to [0,1]$ we define a splitting sequence admitted by f and show that the inverse limit of f is an arc if and only if f does not admit a splitting sequence.

scholarly journals Congruence pairs of principal MS-algebras and perfect extensions

2020 ◽  
Vol 70 (6) ◽  
pp. 1275-1288
Author(s):  
Abd El-Mohsen Badawy ◽  
Miroslav Haviar ◽  
Miroslav Ploščica
Keyword(s):  
Boolean Algebra ◽  
Congruence Lattices ◽  
Descriptive Solution ◽  
The One ◽  
Special Case ◽  
Congruence Pair

AbstractThe notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for K2-algebras [6], is introduced. It is proved that the congruences of the principal MS-algebras L correspond to the MS-congruence pairs on simpler substructures L°° and D(L) of L that were associated to L in [4].An analogy of a well-known Grätzer’s problem [11: Problem 57] formulated for distributive p-algebras, which asks for a characterization of the congruence lattices in terms of the congruence pairs, is presented here for the principal MS-algebras (Problem 1). Unlike a recent solution to such a problem for the principal p-algebras in [2], it is demonstrated here on the class of principal MS-algebras, that a possible solution to the problem, though not very descriptive, can be simple and elegant.As a step to a more descriptive solution of Problem 1, a special case is then considered when a principal MS-algebra L is a perfect extension of its greatest Stone subalgebra LS. It is shown that this is exactly when de Morgan subalgebra L°° of L is a perfect extension of the Boolean algebra B(L). Two examples illustrating when this special case happens and when it does not are presented.

Intersections of quotient rings and Prüfer v-multiplication domains

2016 ◽  
Vol 15 (08) ◽  
pp. 1650149 ◽  
Author(s):  
Said El Baghdadi ◽  
Marco Fontana ◽  
Muhammad Zafrullah
Keyword(s):  
Integral Domain ◽  
Algebraic Approach ◽  
Quotient Field ◽  
Topological Characterization ◽  
Locally Finite ◽  
Star Operation ◽  
Star Operations ◽  
Quotient Rings ◽  
Special Case

Let [Formula: see text] be an integral domain with quotient field [Formula: see text]. Call an overring [Formula: see text] of [Formula: see text] a subring of [Formula: see text] containing [Formula: see text] as a subring. A family [Formula: see text] of overrings of [Formula: see text] is called a defining family of [Formula: see text], if [Formula: see text]. Call an overring [Formula: see text] a sublocalization of [Formula: see text], if [Formula: see text] has a defining family consisting of rings of fractions of [Formula: see text]. Sublocalizations and their intersections exhibit interesting examples of semistar or star operations [D. D. Anderson, Star operations induced by overrings, Comm. Algebra 16 (1988) 2535–2553]. We show as a consequence of our work that domains that are locally finite intersections of Prüfer [Formula: see text]-multiplication (respectively, Mori) sublocalizations turn out to be Prüfer [Formula: see text]-multiplication domains (PvMDs) (respectively, Mori); in particular, for the Mori domain case, we reobtain a special case of Théorème 1 of [J. Querré, Intersections d’anneaux intègers, J. Algebra 43 (1976) 55–60] and Proposition 3.2 of [N. Dessagnes, Intersections d’anneaux de Mori — exemples, Port. Math. 44 (1987) 379–392]. We also show that, more than the finite character of the defining family, it is the finite character of the star operation induced by the defining family that causes the interesting results. As a particular case of this theory, we provide a purely algebraic approach for characterizing P vMDs as a subclass of the class of essential domains (see also Theorem 2.4 of [C. A. Finocchiaro and F. Tartarone, On a topological characterization of Prüfer [Formula: see text]-multiplication domains among essential domains, preprint (2014), arXiv:1410.4037]).

Laplacian Energy of Digraphs and a Minimum Laplacian Energy Algorithm

2015 ◽  
Vol 26 (03) ◽  
pp. 367-380 ◽  
Author(s):  
Xingqin Qi ◽  
Edgar Fuller ◽  
Rong Luo ◽  
Guodong Guo ◽  
Cunquan Zhang
Keyword(s):  
Oriented Graph ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Spectral Graph Theory ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Spectral Graph ◽  
Laplacian Energy ◽  
Energy Formula ◽  
Matrix Formula

In spectral graph theory, the Laplacian energy of undirected graphs has been studied extensively. However, there has been little work yet for digraphs. Recently, Perera and Mizoguchi (2010) introduced the directed Laplacian matrix [Formula: see text] and directed Laplacian energy [Formula: see text] using the second spectral moment of [Formula: see text] for a digraph [Formula: see text] with [Formula: see text] vertices, where [Formula: see text] is the diagonal out-degree matrix, and [Formula: see text] with [Formula: see text] whenever there is an arc [Formula: see text] from the vertex [Formula: see text] to the vertex [Formula: see text] and 0 otherwise. They studied the directed Laplacian energies of two special families of digraphs (simple digraphs and symmetric digraphs). In this paper, we extend the study of Laplacian energy for digraphs which allow both simple and symmetric arcs. We present lower and upper bounds for the Laplacian energy for such digraphs and also characterize the extremal graphs that attain the lower and upper bounds. We also present a polynomial algorithm to find an optimal orientation of a simple undirected graph such that the resulting oriented graph has the minimum Laplacian energy among all orientations. This solves an open problem proposed by Perera and Mizoguchi at 2010.

Characterization of Fake News Based on Subjectivity Lexicons

2020 ◽  
Vol 1 (4) ◽  
pp. 419-441
Author(s):  
Caio L.M. Jeronimo ◽  
Leandro B. Marinho ◽  
Cclaudio E.C. Carmpelo ◽  
Adriano Veloso ◽  
Allan S. Da Costa Melo
Keyword(s):  
Social Networks ◽  
Brazilian Portuguese ◽  
News Article ◽  
Bag Of Words ◽  
Fake News ◽  
Feature Vectors ◽  
Textual Content

While many works investigate spread patterns of fake news in social networks, we focus on the textual content. Instead of relying on syntactic representations of documents (aka Bag of Words) as many works do, we seek more robust representations that may better differentiate fake from legitimate news. We propose to consider the subjectivity of news under the assumption that the subjectivity levels of legitimate and fake news are significantly different. For computing the subjectivity level of news, we rely on a set subjectivity lexicons for both Brazilian Portuguese and English languages. We then build subjectivity feature vectors for each news article by calculating the Word Mover's Distance (WMD) between the news and these lexicons considering the embedding the news words lie in, in order to analyze and classify the documents. The results demonstrate that our method is robust, especially in scenarios where training and test domains are different.

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