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Maximum Shallow Clique Minors in Preferential Attachment Graphs Have Polylogarithmic Size

Authors Jan Dreier , Philipp Kuinke , Peter Rossmanith

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Author Details

Jan Dreier
  • Department of Computer Science, RWTH Aachen University, Germany
Philipp Kuinke
  • Department of Computer Science, RWTH Aachen University, Germany
Peter Rossmanith
  • Department of Computer Science, RWTH Aachen University, Germany

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Jan Dreier, Philipp Kuinke, and Peter Rossmanith. Maximum Shallow Clique Minors in Preferential Attachment Graphs Have Polylogarithmic Size. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.14

Abstract

Preferential attachment graphs are random graphs designed to mimic properties of real word networks. They are constructed by a random process that iteratively adds vertices and attaches them preferentially to vertices that already have high degree. We prove various structural asymptotic properties of this graph model. In particular, we show that the size of the largest r-shallow clique minor in Gⁿ_m is at most log(n)^{O(r²)}m^{O(r)}. Furthermore, there exists a one-subdivided clique of size log(n)^{1/4}. Therefore, preferential attachment graphs are asymptotically almost surely somewhere dense and algorithmic techniques developed for structurally sparse graph classes are not directly applicable. However, they are just barely somewhere dense. The removal of just slightly more than a polylogarithmic number of vertices asymptotically almost surely yields a graph with locally bounded treewidth.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Random graphs
Keywords
  • Random Graphs
  • Preferential Attachment
  • Sparsity
  • Somewhere Dense

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