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Liudmila Ostroumova Prokhorenkova - One of the best experts on this subject based on the ideXlab platform.
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Clustering Properties of Spatial Preferential Attachment Model
arXiv: Social and Information Networks, 2018Co-Authors: Lenar Iskhakov, Liudmila Ostroumova Prokhorenkova, Maksim Mironov, Bogumił Kamiński, Paweł PrałatAbstract:In this paper, we study the clustering properties of the Spatial Preferential Attachment (SPA) model introduced by Aiello et al. in 2009. This model naturally combines geometry and Preferential Attachment using the notion of spheres of influence. It was previously shown in several research papers that graphs generated by the SPA model are similar to real-world networks in many aspects. For example, the vertex degree distribution was shown to follow a power law. In the current paper, we study the behaviour of C(d), which is the average local clustering coefficient for the vertices of degree d. This characteristic was not previously analyzed in the SPA model. However, it was empirically shown that in real-world networks C(d) usually decreases as d^{-a} for some a>0 and it was often observed that a=1. We prove that in the SPA model C(d) decreases as 1/d. Furthermore, we are also able to prove that not only the average but the individual local clustering coefficient of a vertex v of degree d behaves as 1/d if d is large enough. The obtained results are illustrated by numerous experiments with simulated graphs.
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Local Clustering Coefficient of Spatial Preferential Attachment Model
arXiv: Probability, 2017Co-Authors: Lenar Iskhakov, Liudmila Ostroumova Prokhorenkova, Maksim Mironov, Paweł PrałatAbstract:In this paper, we study the clustering properties of the Spatial Preferential Attachment (SPA) model introduced by Aiello et al. in 2009. This model naturally combines geometry and Preferential Attachment using the notion of spheres of influence. It was previously shown in several research papers that graphs generated by the SPA model are similar to real-world networks in many aspects. For example, the vertex degree distribution was shown to follow a power law. In the current paper, we study the behavior of C(d), which is the average local clustering coefficient for the vertices of degree d. This characteristic was not previously analyzed in the SPA model. However, it was empirically shown that in real-world networks C(d) usually decreases as d^{-a} for some a>0 and it was often observed that a=1. We prove that in the SPA model C(d) degreases as 1/d.
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Local Clustering Coefficient in Generalized Preferential Attachment Models
Internet Mathematics, 2017Co-Authors: Alexander M. Krot, Liudmila Ostroumova ProkhorenkovaAbstract:In this paper, we analyze the local clustering coefficient of Preferential Attachment models. A general approach to Preferential Attachment was introduced in earlier, where a wide class of models (PA-class) was defined in terms of constraints that are sufficient for the study of the degree distribution and the clustering coefficient. It was previously shown that the degree distribution in all models of the PA-class follows a power law. Also, the global clustering coefficient was analyzed and a lower bound for the average local clustering coefficient was obtained. We expand the results by analyzing the local clustering coefficient for the PA-class of models. Namely, we analyze the behavior of C(d) which is the average local clustering for the vertices of degree d.
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WAW - Assortativity in Generalized Preferential Attachment Models
Lecture Notes in Computer Science, 2016Co-Authors: Alexander M. Krot, Liudmila Ostroumova ProkhorenkovaAbstract:In this paper, we analyze assortativity of Preferential Attachment models. We deal with a wide class of Preferential Attachment models (PA-class). It was previously shown that the degree distribution in all models of the PA-class follows a power law. Also, the global and the average local clustering coefficients were analyzed. We expand these results by analyzing the assortativity property of the PA-class of models. Namely, we analyze the behavior of \(d_{nn}(d)\) which is the average degree of a neighbor of a vertex of degree d.
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Assortativity in Generalized Preferential Attachment Models
arXiv: Probability, 2016Co-Authors: Alexander M. Krot, Liudmila Ostroumova ProkhorenkovaAbstract:In this paper, we analyze assortativity of Preferential Attachment models. We deal with a wide class of Preferential Attachment models (PA-class). It was previously shown that the degree distribution in all models of the PA-class follows a power law. Also, the global and the average local clustering coefficients were analyzed. We expand these results by analyzing the assortativity property of the PA-class of models. Namely, we analyze the behavior of $d_{nn}(d)$ which is the average degree of a neighbor of a vertex of degree $d$.
Remco Van Der Hofstad - One of the best experts on this subject based on the ideXlab platform.
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Consistent Estimation in General Sublinear Preferential Attachment Trees
arXiv: Statistics Theory, 2017Co-Authors: Fengnan Gao, Aad Van Der Vaart, Rui M. Castro, Remco Van Der HofstadAbstract:We propose an empirical estimator of the Preferential Attachment function $f$ in the setting of general Preferential Attachment trees. Using a supercritical continuous-time branching process framework, we prove the almost sure consistency of the proposed estimator. We perform simulations to study the empirical properties of our estimators.
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Consistent estimation in general sublinear Preferential Attachment trees
Electronic Journal of Statistics, 2017Co-Authors: Fengnan Gao, Aad Van Der Vaart, Rui M. Castro, Remco Van Der HofstadAbstract:We propose an empirical estimator of the Preferential Attachment function f in the setting of general sublinear Preferential Attachment trees. Using a supercritical continuous-time branching process framework, we prove the almost sure consistency of the proposed estimator.We perform simulations to study the empirical properties of our estimators.
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diameters in Preferential Attachment models
Journal of Statistical Physics, 2010Co-Authors: Sander Dommers, Remco Van Der Hofstad, Gerard HooghiemstraAbstract:In this paper, we investigate the diameter in Preferential Attachment (PA-) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PA-models. There is a substantial amount of literature proving that, quite generally, PA-graphs possess power-law degree sequences with a power-law exponent τ>2.
Egor Samosvat - One of the best experts on this subject based on the ideXlab platform.
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Recency-based Preferential Attachment models
Journal of Complex Networks, 2016Co-Authors: Liudmila Ostroumova Prokhorenkova, Egor SamosvatAbstract:Preferential Attachment models were shown to be very effective in predicting such important properties of real-world networks as the power-law degree distribution, small diameter, etc. Many different models are based on the idea of Preferential Attachment: LCD, Buckley-Osthus, Holme-Kim, fitness, random Apollonian network, and many others. Although Preferential Attachment models reflect some important properties of real-world networks, they do not allow to model the so-called recency property. Recency property reflects the fact that in many real networks nodes tend to connect to other nodes of similar age. This fact motivated us to introduce a new class of models – recency-based models. This class is a generalization of fitness models, which were suggested by Bianconi and Barabasi. Bianconi and Barabasi extended Preferential Attachment models with pages’ inherent quality or fitness of nodes. When a new node is added to the graph, it is joined to some already existing nodes that are chosen with probabilities proportional to the product of their fitness and incoming degree. We generalize fitness models by adding a recency factor to the attractiveness function. This means that pages are gaining incoming links according to their attractiveness, which is determined by the incoming degree of the page, its inherent popularity (some page-specific constant) and age (new pages are gaining new links more rapidly). We analyze different properties of recency-based models. For example, we show that some distributions of inherent popularity lead to the power-law degree distribution.
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Recency-based Preferential Attachment models
arXiv: Probability, 2014Co-Authors: Liudmila Ostroumova Prokhorenkova, Egor SamosvatAbstract:Preferential Attachment models were shown to be very effective in predicting such important properties of real-world networks as the power-law degree distribution, small diameter, etc. Many different models are based on the idea of Preferential Attachment: LCD, Buckley-Osthus, Holme-Kim, fitness, random Apollonian network, and many others. Although Preferential Attachment models reflect some important properties of real-world networks, they do not allow to model the so-called recency property. Recency property reflects the fact that in many real networks vertices tend to connect to other vertices of similar age. This fact motivated us to introduce a new class of models - recency-based models. This class is a generalization of fitness models, which were suggested by Bianconi and Barabasi. Bianconi and Barabasi extended Preferential Attachment models with pages' inherent quality or fitness of vertices. When a new vertex is added to the graph, it is joined to some already existing vertices that are chosen with probabilities proportional to the product of their fitness and incoming degree. We generalize fitness models by adding a recency factor to the attractiveness function. This means that pages are gaining incoming links according to their attractiveness, which is determined by the incoming degree of the page (current popularity), its inherent quality (some page-specific constant) and age (new pages are gaining new links more rapidly). We analyze different properties of recency-based models. In particular, we show that some distributions of inherent quality lead to the power-law degree distribution.
Steffen Staab - One of the best experts on this subject based on the ideXlab platform.
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Time-invariant degree growth in Preferential Attachment network models.
Physical review. E, 2020Co-Authors: Jun Sun, Matúš Medo, Steffen StaabAbstract:Preferential Attachment drives the evolution of many complex networks. Its analytical studies mostly consider the simplest case of a network that grows uniformly in time despite the accelerating growth of many real networks. Motivated by the observation that the average degree growth of nodes is time invariant in empirical network data, we study the degree dynamics in the relevant class of network models where Preferential Attachment is combined with heterogeneous node fitness and aging. We propose an analytical framework based on the time invariance of the studied systems and show that it is self-consistent only for two special network growth forms: the uniform and the exponential network growth. Conversely, the breaking of such time invariance explains the winner-takes-all effect in some model settings, revealing the connection between the Bose-Einstein condensation in the Bianconi-Barabási model and similar gelation in superlinear Preferential Attachment. Aging is necessary to reproduce realistic node degree growth curves and can prevent the winner-takes-all effect under weak conditions. Our results are verified by extensive numerical simulations.
Fengnan Gao - One of the best experts on this subject based on the ideXlab platform.
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Consistent Estimation in General Sublinear Preferential Attachment Trees
arXiv: Statistics Theory, 2017Co-Authors: Fengnan Gao, Aad Van Der Vaart, Rui M. Castro, Remco Van Der HofstadAbstract:We propose an empirical estimator of the Preferential Attachment function $f$ in the setting of general Preferential Attachment trees. Using a supercritical continuous-time branching process framework, we prove the almost sure consistency of the proposed estimator. We perform simulations to study the empirical properties of our estimators.
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Consistent estimation in general sublinear Preferential Attachment trees
Electronic Journal of Statistics, 2017Co-Authors: Fengnan Gao, Aad Van Der Vaart, Rui M. Castro, Remco Van Der HofstadAbstract:We propose an empirical estimator of the Preferential Attachment function f in the setting of general sublinear Preferential Attachment trees. Using a supercritical continuous-time branching process framework, we prove the almost sure consistency of the proposed estimator.We perform simulations to study the empirical properties of our estimators.
