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We study a class models of correlated random networks in which vertices are characterized by \textit{hidden variables} controlling the establishment of edges between pairs of vertices. We find analytical expressions for the main topological…

Disordered Systems and Neural Networks · Physics 2009-11-10 Marian Boguna , Romualdo Pastor-Satorras

In spatial networks vertices are arranged in some space and edges may cross. When arranging vertices in a 1-dimensional lattice edges may cross when drawn above the vertex sequence as it happens in linguistic and biological networks. Here…

Discrete Mathematics · Computer Science 2020-02-24 Lluís Alemany-Puig , Ramon Ferrer-i-Cancho

Specify a randomized algorithm that, given a very large graph or network, extracts a random subgraph. What can we learn about the input graph from a single subsample? We derive laws of large numbers for the sampler output, by relating…

Statistics Theory · Mathematics 2017-10-13 Peter Orbanz

Random intersection graphs (RIGs) are an important random structure with applications in social networks, epidemic networks, blog readership, and wireless sensor networks. RIGs can be interpreted as a model for large randomly formed…

Discrete Mathematics · Computer Science 2015-05-19 Milan Bradonjic , Aric Hagberg , Nicolas W. Hengartner , Allon G. Percus

When each vertex is assigned a set, the intersection graph generated by the sets is the graph in which two distinct vertices are joined by an edge if and only if their assigned sets have a nonempty intersection. An interval graph is an…

Combinatorics · Mathematics 2017-02-13 Jeong Han Kim , Sang June Lee , Joohan Na

Many machine learning algorithms used for dimensional reduction and manifold learning leverage on the computation of the nearest neighbours to each point of a dataset to perform their tasks. These proximity relations define a so-called…

Statistical Mechanics · Physics 2020-07-22 Vittorio Erba , Sebastiano Ariosto , Marco Gherardi , Pietro Rotondo

Across the sciences, the statistical analysis of networks is central to the production of knowledge on relational phenomena. Because of their ability to model the structural generation of networks, exponential random graph models are a…

Data Analysis, Statistics and Probability · Physics 2015-05-27 Bruce A. Desmarais , Skyler J. Cranmer

We propose and analyze a graph model to study the connectivity of interdependent networks. Two interdependent networks of arbitrary topologies are modeled as two graphs, where every node in one graph is supported by supply nodes in the…

Discrete Mathematics · Computer Science 2018-09-10 Jianan Zhang , Eytan Modiano

We study the family of network models derived by requiring the expected properties of a graph ensemble to match a given set of measurements of a real-world network, while maximizing the entropy of the ensemble. Models of this type play the…

Statistical Mechanics · Physics 2009-11-10 Juyong Park , M. E. J. Newman

There has been an increased interest in applying machine learning techniques on relational structured-data based on an observed graph. Often, this graph is not fully representative of the true relationship amongst nodes. In these settings,…

Machine Learning · Statistics 2022-08-05 Florence Regol , Soumyasundar Pal , Jianing Sun , Yingxue Zhang , Yanhui Geng , Mark Coates

Recently, random graphs in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices have attracted much attention. Here, we present a specific realization of a class of random…

Mathematical Physics · Physics 2009-11-13 Xinping Xu , Feng Liu

In this paper we explore mathematical tools that can be used to relate directed and undirected random graph models to each other. We identify probability spaces on which a directed and an undirected graph model are equivalent, and…

Probability · Mathematics 2025-03-03 Mike van Santvoort , Pim van der Hoorn

The problem of predicting links in large networks is an important task in a variety of practical applications, including social sciences, biology and computer security. In this paper, statistical techniques for link prediction based on the…

Applications · Statistics 2021-09-01 Francesco Sanna Passino , Anna S. Bertiger , Joshua C. Neil , Nicholas A. Heard

Let $\mathcal{V}$ and $\mathcal{U}$ be the point sets of two independent homogeneous Poisson processes on $\mathbb{R}^d$. A graph $\mathcal{G}_\mathcal{V}$ with vertex set $\mathcal{V}$ is constructed by first connecting pairs of points…

Probability · Mathematics 2024-11-20 Maria Deijfen , Riccardo Michielan

Random graph models are frequently used as a controllable and versatile data source for experimental campaigns in various research fields. Generating such data-sets at scale is a non-trivial task as it requires design decisions typically…

Data Structures and Algorithms · Computer Science 2020-03-03 Manuel Penschuck , Ulrik Brandes , Michael Hamann , Sebastian Lamm , Ulrich Meyer , Ilya Safro , Peter Sanders , Christian Schulz

The random graph of Erdos and Renyi is one of the oldest and best studied models of a network, and possesses the considerable advantage of being exactly solvable for many of its average properties. However, as a model of real-world networks…

Statistical Mechanics · Physics 2007-05-23 M. E. J. Newman

We introduce a natural generalization of the Erd\H{o}s-R\'enyi random graph model in which random instances of a fixed motif are added independently. The binomial random motif graph $G(H,n,p)$ is the random (multi)graph obtained by adding…

Combinatorics · Mathematics 2019-07-30 Michael Anastos , Peleg Michaeli , Samantha Petti

We consider the problem of determining the proportion of edges that are discovered in an Erdos-Renyi graph when one constructs all shortest paths from a given source node to all other nodes. This problem is equivalent to the one of…

Statistical Mechanics · Physics 2009-11-13 Vincent D. Blondel , Jean-Loup Guillaume , Julien M. Hendrickx , Raphael M. Jungers

We investigate the connectivity of a wireless sensor network secured by the heterogeneous key predistribution scheme under an independent on/off channel model. The heterogeneous scheme induces an inhomogeneous random key graph, denoted by…

Optimization and Control · Mathematics 2017-10-09 Rashad Eletreby , Osman Yağan

We consider the number of crossings in a graph which is embedded randomly on a convex set of points. We give an estimate to the normal distribution in Kolmogorov distance which implies a convergence rate of order $n^{-1/2}$ for various…

Combinatorics · Mathematics 2022-08-26 Santiago Arenas-Velilla , Octavio Arizmendi