Some Typical Properties of the Spatial Preferred Attachment Model

Cooper, Colin; Frieze, Alan; Prałat, Paweł

doi:10.1007/978-3-642-30541-2_3

Cited by 28 publications

(42 citation statements)

References 32 publications

(31 reference statements)

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“…The SPA model produces scale-free networks, which exhibit many of the characteristics of reallife networks (see [1,19]). In [31], it was shown that the SPA model gave the best fit, in terms of graph structure, for a series of social networks derived from Facebook.…”

Section: Random Graph Modelsmentioning

confidence: 99%

“…We will use the geometry of the model to obtain a suitable partition that yields high modularity of G n . The following properties (proved many times; see, for example, [1,19]) are the only properties of the model that will be used in the proof: a.a.s. for every pair i, t such that 1 ≤ i ≤ t ≤ n we have that…”

Section: The Spatial Preferential Attachment Modelmentioning

confidence: 99%

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Typical distances in a geometric model for complex networks

Abdullah¹,

Bode²,

Fountoulakis³

2017

Internet Mathematics

153

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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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Section: Random Graph Modelsmentioning

confidence: 99%

Section: The Spatial Preferential Attachment Modelmentioning

confidence: 99%

Typical distances in a geometric model for complex networks

Abdullah¹,

Bode²,

Fountoulakis³

2017

Internet Mathematics

153

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“…In the case a = 1/2, the profile function ϕ only takes the values zero and one, thus the decision is not random and we connect two vertices whenever they are close enough. The degree-based preferential attachment model in discrete time for this choice of ϕ was introduced in [1] and further studied in [5] and [16]. This particular choice for the profile function helps to get a better understanding of the problems and properties of this model, see for example Section 5.…”

Section: The Age-based Spatial Preferential Attachment Networkmentioning

confidence: 99%

The age-dependent random connection model

et al. 2019

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We investigate a class of growing graphs embedded into the d-dimensional torus where new vertices arrive according to a Poisson process in time, are randomly placed in space and connect to existing vertices with a probability depending on time, their spatial distance and their relative ages. This simple model for a scale-free network is called the age-based spatial preferential attachment network and is based on the idea of preferential attachment with spatially induced clustering. We show that the graphs converge weakly locally to a variant of the random connection model, which we call the age-dependent random connection model. This is a natural infinite graph on a Poisson point process where points are marked by a uniformly distributed age and connected with a probability depending on their spatial distance and both ages. We use the limiting structure to investigate asymptotic degree distribution, clustering coefficients and typical edge lengths in the age-based spatial preferential attachment network.MSc classification (2010): Primary 05C80 Secondary 60K35.

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“…The geometric nature of the network implies that there is a high amount of local clustering [10]. In [3], logarithmic bounds on the directed diameter were given. In [9] it was shown that the effective undirected diameter is also logarithmically bounded.…”

Section: The Spa Modelmentioning

confidence: 99%

High Degree Vertices and Spread of Infections in Spatially Modelled Social Networks

Feldman

Janssen

2017

Lecture Notes in Computer Science

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We examine how the behaviour of high degree vertices in a network affects whether an infection spreads through communities or jumps between them. We study two stochastic susceptible-infectedrecovered (SIR) processes and represent our network with a spatial preferential attachment (SPA) network. In one of the two epidemic scenarios we adjust the contagiousness of high degree vertices so that they are less contagious. We show that, for this scenario, the infection travels through communities rather than jumps between them. We conjecture that this is not the case in the other scenario, when contagion is independent of the degree of the originating vertex. Our theoretical results and conjecture are supported by simulations.

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Some Typical Properties of the Spatial Preferred Attachment Model

Cited by 28 publications

References 32 publications

Typical distances in a geometric model for complex networks

Typical distances in a geometric model for complex networks

The age-dependent random connection model

High Degree Vertices and Spread of Infections in Spatially Modelled Social Networks

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