Spatial Models for Virtual Networks

Janssen, Jeannette

doi:10.1007/978-3-642-13962-8_23

Cited by 12 publications

(11 citation statements)

References 35 publications

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“…Finally, consider the case where pA 1 ≤ 1/2. For α ∈ (1 − pA 1 , 1) we have the exact same expression for E(E 1 ) as for the case where pA 1 > 1/2, as given in Equation (14). Thus where the last step follows since…”

Section: 3mentioning

confidence: 95%

Geometric graph properties of the spatial preferred attachment model

Janssen

Prałat

Wilson

2013

Advances in Applied Mathematics

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The spatial preferred attachment (SPA) model is a model for networked information spaces such as domains of the World Wide Web, citation graphs, and on-line social networks. It uses a metric space to model the hidden attributes of the vertices. Thus, vertices are elements of a metric space, and link formation depends on the metric distance between vertices. We show, through theoretical analysis and simulation, that for graphs formed according to the SPA model it is possible to infer the metric distance between vertices from the link structure of the graph. Precisely, the estimate is based on the number of common neighbours of a pair of vertices, a measure known as co-citation. To be able to calculate this estimate, we derive a precise relation between the number of common neighbours and metric distance. We also analyze the distribution of edge lengths, where the length of an edge is the metric distance between its end points. We show that this distribution has three different regimes, and that the tail of this distribution follows a power law.

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Section: 3mentioning

confidence: 95%

Geometric graph properties of the spatial preferred attachment model

Janssen

Prałat

Wilson

2013

Advances in Applied Mathematics

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“…The use of β in past works on the Waxman graphs has allowed for β > 1, at which point the function (2) must be truncated, creating a corner in the deterrence function shape. This is undesirable from the point of view of parametric estimation, so we restrict †Some examples of use of the Waxman graph include (Waxman, 1988;Thomas and Zegura, 1994;Zegura et al, 1996Zegura et al, , 1997Doar and Leslie, 1993;Wei and Estrin, 1994;Salama et al, 1997;Verma et al, 1998;Matta and Guo, 1999;Shaikh et al, 1999;Fortz and Thorup, 2004;Neve and Mieghem, 2000;Rastogi et al, 2001;Wu et al, 2000;Guo and Matta, 2003;Kuipers, 2004;Kaiser and Hilgetag, 2004b,a;Gunduz et al, 2004;Carzaniga et al, 2004;Lua et al, 2005;Holzer et al, 2005;Wang et al, 2005;Ahuja et al, 2009;Huang et al, 2007;Malladi et al, 2007;Tran and Pham, 2009;Fang et al, 2011a,b;Costa et al, 2010;Janssen, 2010;Davis et al, 2014).…”

Section: Background and Related Workmentioning

confidence: 99%

Estimating the Parameters of the Waxman Random Graph

Roughan

Tuke

Parsonage

2019

Lecture Notes in Computer Science

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The Waxman random graph is a generalisation of the simple Erdős-Rényi or Gilbert random graph. It is useful for modelling physical networks where the increased cost of longer links means they are less likely to be built, and thus less numerous than shorter links. The model has been in continuous use for over two decades with many attempts to select parameters which match real networks. In most the parameters have been arbitrarily selected, but there are a few cases where they have been calculated using a formal estimator. However, the performance of the estimator was not evaluated in any of these cases. This paper presents both the first evaluation of formal estimators for the parameters of these graphs, and a new Maximum Likelihood Estimator with O(n) computational time complexity that requires only link lengths as input.arXiv:1506.07974v2 [math.ST] 29 Jun 2015 * Note that our parameterisation differs from that in much of the literature, for reasons to be explained in the following sections.

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“…to become friends on Facebook, and scientific papers mostly refer to papers on similar topics. See [8] for more details about applying spatial graphs to model complex networks.…”

Section: Geometric Chung-lu Modelmentioning

confidence: 99%

An unsupervised framework for comparing graph embeddings

Kamiński

Prałat

Théberge³

2019

Journal of Complex Networks

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Graph embedding is a transformation of vertices of a graph into set of vectors. Good embeddings should capture the graph topology, vertex-to-vertex relationship, and other relevant information about graphs, subgraphs, and vertices. If these objectives are achieved, they are meaningful, understandable, and compressed representations of networks. They also provide more options and tools for data scientists as machine learning on graphs is still quite limited. Finally, vector operations are simpler and faster than comparable operations on graphs.The main challenge is that one needs to make sure that embeddings well describe the properties of the graphs. In particular, the decision has to be made on the embedding dimensionality which highly impacts the quality of an embedding. As a result, selecting the best embedding is a challenging task and very often requires domain experts.In this paper, we propose a "divergence score" that can be assign to various embeddings to distinguish good ones from bad ones. This general framework provides a tool for an unsupervised graph embedding comparison. In order to achieve it, we needed to generalize the well-known Chung-Lu model to incorporate geometry which is interesting on its own rights. In order to test our framework, we did a number of experiments with synthetic networks as well as real-world networks, and various embedding algorithms. each vertex such that nearby vertices are more likely to share an edge than those far from each other. In a good embedding most of the network's edges can be predicted from the coordinates of the vertices. For example, in [9] protein interaction networks are embedded in low-dimension Euclidean space. Unfortunately, in the absence of a general-purpose representation for graphs, very often graph embedding requires domain experts to craft features or to use specialized feature selection algorithms. Having said that, there are some graph embedding algorithms that work without any prior or additional information other than graph structure. However, these are randomized algorithms that are usually not so stable; that is, the outcome is often drastically different despite the fact that all the algorithm parameters remain the same.Consider a graph G = (V, E) on n vertices, and several embeddings of its vertices to some multidimensional spaces (possibly in different dimensions). The main question we try to answer in this paper is: how do we evaluate these embeddings? Which one is the best and should be used? In order to answer these questions, we propose a general framework that assigns the divergence score to each embedding which, in an unsupervised learning fashion, distinguishes good from bad embeddings. In order to benchmark embeddings, we generalize well-known Chung-Lu random graph model to incorporate geometry. The model is interesting on its own and should be useful for many other problems and tools. In order to test our algorithm, we experiment with synthetic networks as well as real-world networks, and various embedding algorithms.The paper is...

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Spatial Models for Virtual Networks

Cited by 12 publications

References 35 publications

Geometric graph properties of the spatial preferred attachment model

Geometric graph properties of the spatial preferred attachment model

Estimating the Parameters of the Waxman Random Graph

An unsupervised framework for comparing graph embeddings

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