Perfect matchings and Hamiltonian cycles in the preferential attachment model

Frieze, Alan; Pérez‐Giménez, Xavier; Prałat, Paweł; Reiniger, Benjamin

doi:10.1002/rsa.20778

Cited by 14 publications

(26 citation statements)

References 36 publications

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“…In the context of random geometric graphs in the Euclidean plane, analogous results have been obtained [4,9,23]. The emergence of Hamilton cycles was also considered in other models, including the preferential attachment model [13] and the random d-regular graph model [27].…”

Section: Introductionmentioning

confidence: 61%

Hamilton cycles and perfect matchings in the KPKVB model

Fountoulakis

Mitsche

Müller

et al. 2021

Stochastic Processes and their Applications

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In this paper we consider the existence of Hamilton cycles and perfect matchings in a random graph model proposed by Krioukov et al. in 2010. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has been previously shown that this model has various properties associated with complex networks, including a power-law degree distribution, "short distances" and a non-vanishing clustering coefficient. The model is specified using three parameters: the number of nodes n, which we think of as going to infinity, and α, ν > 0, which we think of as constant. Roughly speaking α controls the power law exponent of the degree sequence and ν the average degree. Here we show that for every α < 1/2 and ν = ν(α) sufficiently small, the model does not contain a perfect matching with high probability, whereas for every α < 1/2 and ν = ν(α) sufficiently large, the model contains a Hamilton cycle with high probability.

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Section: Introductionmentioning

confidence: 61%

Hamilton cycles and perfect matchings in the KPKVB model

Fountoulakis

Mitsche

Müller

et al. 2021

Stochastic Processes and their Applications

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“…Some stronger expansion properties were recently obtained in [27]. However, whereas they presumably could be used to obtain some small improvements for an upper bound of q * (G n m ) (for specific values of m), we do not know how to show that q * (G n m ) → 0 as m → ∞.…”

Section: Theorem 13mentioning

confidence: 83%

Modularity of complex networks models

Prokhorenkova

Prałat

Райгородский

2017

Internet Mathematics

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Modularity is designed to measure the strength of division of a network into clusters (known also as communities). Networks with high modularity have dense connections between the vertices within clusters but sparse connections between vertices of different clusters. As a result, modularity is often used in optimization methods for detecting community structure in networks, and so it is an important graph parameter from a practical point of view. Unfortunately, many existing non-spatial models of complex networks do not generate graphs with high modularity; on the other hand, spatial models naturally create clusters. We investigate this phenomenon by considering a few examples from both sub-classes. We prove precise theoretical results for the classical model of random d-regular graphs as well as the preferential attachment model, and contrast these results with the ones for the spatial preferential attachment (SPA) model that is a model for complex networks in which vertices are embedded in a metric space, and each vertex has a sphere of influence whose size increases if the vertex gains an in-link, and otherwise decreases with time. The results obtained in this paper can be used for developing statistical tests for models selection and to measure statistical significance of clusters observed in complex networks.

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“…They also conjectured that it is asymmetric for k ≥ 3. More recently, Frieze, Pérez‐Giménez, Prałat and Reiniger studied the hamiltonicity and perfect matchings in uniform attachment graphs; they proved that whp the graph has a perfect matching for k ≥ 159 and it is Hamiltonian for k ≥ 3214.…”

Section: Introductionmentioning

confidence: 99%

On connectivity, conductance and bootstrap percolation for a random k‐out, age‐biased graph

Acan

Pittel

2019

Random Struct Algorithms

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A uniform attachment graph (with parameter k), denoted Gn,k in the paper, is a random graph on the vertex set [n], where each vertex v makes k selections from [v − 1] uniformly and independently, and these selections determine the edge set. We study several aspects of this graph. Our motivation comes from two similarly constructed, well‐studied random graphs: k‐out graphs and preferential attachment graphs. In this paper, we find the asymptotic distribution of its minimum degree and connectivity, and study the expansion properties of Gn,k to show that the conductance of Gn,k is of order false(normallognfalse)−1. We also study the bootstrap percolation on Gn,k, where r infected neighbors infect a vertex, and show that if the probability of initial infection of a vertex is negligible compared to false(normallognfalse)−rfalse/false(r−1false) then with high probability (whp) the disease will not spread to the whole vertex set, and if this probability exceeds false(normallognfalse)−rfalse/false(r−1false) by a sub‐logarithmical factor then the disease whp will spread to the whole vertex set.

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Perfect matchings and Hamiltonian cycles in the preferential attachment model

Cited by 14 publications

References 36 publications

Hamilton cycles and perfect matchings in the KPKVB model

Hamilton cycles and perfect matchings in the KPKVB model

Modularity of complex networks models

On connectivity, conductance and bootstrap percolation for a random k‐out, age‐biased graph

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