# Preferential attachment with fitness dependent choice

Yu.A. Malyshkin

^{} Tver State University

**DOI:** 10.26456/pcascnn/2021.13.483

* Original article*

**Abstract: ** We study the asymptotic behavior of the maximum degree in the preferential attachment tree model with a choice based on both the degree and fitness of a vertex. The preferential attachment models are natural models for complex networks (like neural networks, etc.) and constructed in the following recursive way. To each vertex is assigned a parameter that is called a fitness of a vertex. We start from two vertices and an edge between them. On each step, we consider a sample with repetition of d vertices, chosen with probabilities proportional to their degrees plus some parameter β>-1. Then we add a new vertex and draw an edge from it to the vertex from the sample with the highest product of fitness and degree. We prove that the maximum degree, dependent on parameters of the model, could exhibit three types of asymptotic behavior: sublinear, linear, and of n / lnn order, where n is the number of edges in the graph.

*Keywords: complex networks, random graphs, preferential attachment, power of choice, fitness *

- Yury A. Malyshkin – Ph. D., Docent, Applied Mathematics and Cybernetics Department, Tver State University

**Reference: **

**Malyshkin, Yu.A. ** Preferential attachment with fitness dependent choice /
Yu.A. Malyshkin //
Physical and chemical aspects of the study of clusters, nanostructures and nanomaterials. – Tver: TSU, 2021. — I. 13. — P. 483-494. DOI: 10.26456/pcascnn/2021.13.483. (In Russian).

**Full article (in Russian): **
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