A Multi-type Preferential Attachment Tree

Rosengren, Sebastian

doi:10.24166/im.05.2018

Cited by 8 publications

(9 citation statements)

References 11 publications

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“…Early works in this direction are [8,9]. Recently, in [10], multi-type preferential attachment trees were studied. In [10], the results of [11] on multi-type continuous time branching processes were applied to describe the evolution of the network.…”

Section: Introductionmentioning

confidence: 99%

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A Continuous-Time Network Evolution Model Describing 2- and 3-Interactions

Fazekas

Barta

2021

Mathematics

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A continuous-time network evolution model is considered. The evolution of the network is based on 2- and 3-interactions. 2-interactions are described by edges, and 3-interactions are described by triangles. The evolution of the edges and triangles is governed by a multi-type continuous-time branching process. The limiting behaviour of the network is studied by mathematical methods. We prove that the number of triangles and edges have the same magnitude on the event of non-extinction, and it is eαt, where α is the Malthusian parameter. The probability of the extinction and the degree process of a fixed vertex are also studied. The results are illustrated by simulations.

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Section: Introductionmentioning

confidence: 99%

“…We emphasize that in our model all the nodes have the same type. Thus, despite [7,10], the theory of multi-type branching processes will be applied for groups of nodes and not just for the nodes.…”

Section: Introductionmentioning

confidence: 99%

A Continuous-Time Network Evolution Model Describing 2- and 3-Interactions

Fazekas

Barta

2021

Mathematics

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“…In that paper the theory of general branching processes, so called Crump-Mode-Jagers processes (see [8]) is applied to obtain asymptotic theorems. In paper [9], the idea of preferential attachment is combined with the evolution mechanism of a multi-type continuous time branching process.…”

Section: Introductionmentioning

confidence: 99%

Simulation results on a triangle-based network evolution model

Fazekas¹,

Barta²,

Noszály³

2020

AMI

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We study a continuous time network evolution model. We consider the collaboration of three individuals. In our model, it is described by three connected vertices, that is by a triangle. During the evolution new collaborations, that is new triangles are created. The reproduction of the triangles is governed by a continuous time branching process. The long time behaviour of the number of triangles, edges and vertices is described. In this paper, we highlight the asymptotic behaviour of the network by simulation results.

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“…There are various random graph models with multi-type vertices, see e.g. [1,2,13], which have been investigated. In all of these models, the types of vertices are chosen with a dynamics strongly related to the evolution of the graph.…”

Section: Introductionmentioning

confidence: 99%

Barabási–Albert random graph with multiple type edges and perturbation

Backhausz

Rozner

2019

Acta Math. Hungar.

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In this paper we introduce the perturbed version of the Barabási-Albert random graph with multiple type edges and prove the existence of the (generalized) asymptotic degree distribution. Similarly to the non-perturbed case, the asymptotic degree distribution depends on the almost sure limit of the proportion of edges of different types. However, if there is perturbation, then the resulting degree distribution will be deterministic, which is a major difference compared to the non-perturbed case. one has to understand quantities like the number of vertices of a given type with a given number of edges. This kind of analysis is performed in the papers mentioned above. Now we consider models where it is not the vertices, but the edges that have different types. This can be used to model different kind of relationships between the individuals -in a social, biological or financial network, connections are usually not of the same nature, and this can be important from the point of view of contagion or epidemic spread. In our model, the type of an edge is an element of a fixed finite set, and it does not change with time. It is chosen randomly when the edge is born, with a distribution that depends on the current state of the graph and on the types of the edges going out from the neighbours of the new vertex. A general family of preferential attachment random graphs with multi-type edges has been examined in our previous work [3]. Growing networks with two different types of edges can be considered as directed graphs. In this case the type of an edge is its orientation. That is, when a new vertex is born, then it is attached to the graph with an edge directed from the new vertex to the already existing ones or directed from the existing vertices to the new one, and this corresponds to the two different types. Directed preferential attachment models were introduced and examined in [5,14], but with different dynamics than in [3]. In this paper we are interested in a version of robustness in preferential attachment graph models with multi-type edges. The goal is to compare (i) a model in which the probability of choosing a type is exactly the proportion of the current type among the edges going out from the endpoint of the new edge; and (ii) its modified version, when, after this step, types can change with certain probability. In particular, we introduce perturbation in the multi-type Barabási-Albert random graph, and prove that this shows different phenomena than the original version. That is, errors in the dynamics of multi-type random graphs can lead to essential changes in the asymptotic behaviour of the model. The multi-type Barabási-Albert random graph model has been described in [3], which is a generalization of the Barabási-Albert random graph model introduced in [4], specified in [6]. We prove the existence of the asymptotic degree distribution in the perturbed Barabási-Albert random graph, and we also provide recurrence equations for the asymptotic degree distribution. The main difference between the perturbed and...

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A Multi-type Preferential Attachment Tree

Cited by 8 publications

References 11 publications

A Continuous-Time Network Evolution Model Describing 2- and 3-Interactions

A Continuous-Time Network Evolution Model Describing 2- and 3-Interactions

Simulation results on a triangle-based network evolution model

Barabási–Albert random graph with multiple type edges and perturbation

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