Hyperbolic graph generator

Aldecoa, Rodrigo; Orsini, Chiara; Krioukov, Dmitri

doi:10.1016/j.cpc.2015.05.028

Cited by 52 publications

(63 citation statements)

References 12 publications

(12 reference statements)

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“…The iterations in the BA model cease when the graph has attained the requisite number n of vertices. The HGG model [41,42] produces a random graph of n vertices by initially fixing n vertices to n points on a hyperbolic disk. In the HGG model, the probability of existence of an edge between two vertices is proportional to the hyperbolic distance between the two points on the hyperbolic disk that correspond to these two vertices.…”

Section: Network Datasetsmentioning

confidence: 99%

“…In the HGG model, the probability of existence of an edge between two vertices is proportional to the hyperbolic distance between the two points on the hyperbolic disk that correspond to these two vertices. By tuning the input parameter γ, the HGG model can produce either a hyperbolic or a spherical random graph [41,42]. Specifically, the HGG model produces hyperbolic random graphs for γ ∈ [2, ∞) whereas spherical random graphs for γ = ∞.…”

Section: Network Datasetsmentioning

confidence: 99%

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Persistent homology of unweighted complex networks via discrete Morse theory

Kannan

Saucan

Roy

et al. 2019

Sci Rep

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Topological data analysis can reveal higher-order structure beyond pairwise connections between vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and undirected networks using persistent homology. Leveraging on the features of discrete Morse theory, our method not only captures the topology of the clique complex of such graphs via the concept of critical simplices, but also achieves close to the theoretical minimum number of critical simplices in several analyzed model and real networks. This leads to a reduced filtration scheme based on the subsequence of the corresponding critical weights, thereby leading to a significant increase in computational efficiency. We have employed our filtration scheme to explore the persistent homology of several model and real-world networks. In particular, we show that our method can detect differences in the higher-order structure of networks, and the corresponding persistence diagrams can be used to distinguish between different model networks. In summary, our method based on discrete Morse theory further increases the applicability of persistent homology to investigate the global topology of complex networks.

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Section: Network Datasetsmentioning

confidence: 99%

Section: Network Datasetsmentioning

confidence: 99%

Persistent homology of unweighted complex networks via discrete Morse theory

Kannan

Saucan

Roy

et al. 2019

Sci Rep

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show abstract

“…In Section 3.1, we review existing approaches in Euclidean space (Section 3.1.1); elliptic space, such as the surface of a three-dimensional sphere (Section 3.1.2); and other spaces (Section 3.1.3). In Section 3.2, we highlight properties of hyperbolic space and detail its promising potential application in the class of continuous latent space models for network data, inspired by recent network analysis methods which take advantage of a hyperbolic latent space (Krioukov et al, 2010;Asta and Shalizi, 2014;Aldecoa, Orsini and Krioukov, 2015). We will discuss various interesting aspects of the different choices for the geometry of the latent space in each subsection, but we also have provided a summary of some of the most important differences between Euclidean, elliptic, and hyperbolic space in Table 1.…”

Section: Latent Space Geometriesmentioning

confidence: 99%

The Geometry of Continuous Latent Space Models for Network Data

Smith¹,

Asta²,

Calder³

2019

Statist. Sci.

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We review the class of continuous latent space (statistical) models for network data, paying particular attention to the role of the geometry of the latent space. In these models, the presence/absence of network dyadic ties are assumed to be conditionally independent given the dyads' unobserved positions in a latent space. In this way, these models provide a probabilistic framework for embedding network nodes in a continuous space equipped with a geometry that facilitates the description of dependence between random dyadic ties. Specifically, these models naturally capture homophilous tendencies and triadic clustering, among other common properties of observed networks. In addition to reviewing the literature on continuous latent space models from a geometric perspective, we highlight the important role the geometry of the latent space plays on properties of networks arising from these models via intuition and simulation. Finally, we discuss results from spectral graph theory that allow us to explore the role of the geometry of the latent space, independent of network size. We conclude with conjectures about how these results might be used to infer the appropriate latent space geometry from observed networks.

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“…Similarity has also been proposed as a second mechanism, that together with popularity or preferential attachment, may result into complex networks with an hyperbolic topology [38]. In [2] it is described an hyperbolic graph generator. When SAT instances are modeled as graphs, many graph properties can be analyzed, such as the small-world property [43], the scale-free structure [5,6], the eigenvector centrality [26], or the self-similarity [3].…”

Section: Introductionmentioning

confidence: 98%

Generating SAT instances with community structure

Giráldez-Cru

Levy

2016

Artificial Intelligence

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Nowadays, modern SAT solvers are able to efficiently solve many industrial, or realworld, SAT instances. However, the process of development and testing of new SAT solving techniques is conditioned to the finite and reduced number of known industrial benchmarks. Therefore, new models of random SAT instances generation that capture realistically the features of real-world problems can be beneficial to the SAT community. In many works, the structure of industrial instances has been analyzed representing them as graphs and studying some of their properties, like modularity. In this work, we use the notion of modularity to define a new model of generation of random SAT instances with community structure, called Community Attachment. For high values of modularity (i.e., clear community structure), we realistically model pseudoindustrial random SAT formulas. This model also generates SAT instances very similar to classical random formulas using a low value of modularity. We also prove that the phase transition point, if exists, is independent on the modularity. We evaluate the adequacy of this model to real industrial SAT problems in terms of SAT solvers performance, and show that modern solvers do actually exploit this community structure. Finally, we use this generator to observe the connections between the modularity of the instance and some components of the solver, such as the variable branching heuristics or the clause learning mechanism.

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Hyperbolic graph generator

Cited by 52 publications

References 12 publications

Persistent homology of unweighted complex networks via discrete Morse theory

Persistent homology of unweighted complex networks via discrete Morse theory

The Geometry of Continuous Latent Space Models for Network Data

Generating SAT instances with community structure

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