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A preferential attachment model with random initial degrees

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Arkiv för Matematik

Abstract

In this paper, a random graph process {G(t)} t≥1 is studied and its degree sequence is analyzed. Let {W t } t≥1 be an i.i.d. sequence. The graph process is defined so that, at each integer time t, a new vertex with W t edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on G(t-1), the probability that a given edge of vertex t is connected to vertex i is proportional to d i (t-1)+δ, where d i (t-1) is the degree of vertex i at time t-1, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent τ=min{τW,τP}, where τW is the power-law exponent of the initial degrees {W t } t≥1 and τP the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze.

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Authors and Affiliations

  1. Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden

    Maria Deijfen

  2. Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, P.O. Box 5031, NL-2600 GA, Delft, The Netherlands

    Henri van den Esker & Gerard Hooghiemstra

  3. Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, NL-5600 MB, Eindhoven, The Netherlands

    Remco van der Hofstad

Authors
  1. Maria Deijfen
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  2. Henri van den Esker
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  3. Remco van der Hofstad
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  4. Gerard Hooghiemstra
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Correspondence to Remco van der Hofstad.

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Deijfen, M., van den Esker, H., van der Hofstad, R. et al. A preferential attachment model with random initial degrees. Ark Mat 47, 41–72 (2009). https://doi.org/10.1007/s11512-007-0067-4

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  • DOI: https://doi.org/10.1007/s11512-007-0067-4

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