Phase transitions in dynamical random graphs

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Abstract

We study a large-time limit of a Markov process whose states are finite graphs. The number of the vertices is described by a supercritical branching process, and the dynamics of edges is determined by the rates of appending and deleting. We find a phase transition in our model similar to the one in the random graph model G (n,p). We derive a formula for the line of critical parameters which separates two different phases: one is where the size of the largest component is proportional to the size of the entire graph, and another one, where the size of the largest component is at most logarithmic with respect to the size of the entire graph. In the supercritical phase we find the asymptotics for the size of the largest component.

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Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Probability Theory and Statistics

Keywords

  • inhomogeneous random graphs, phase transitions
Original languageEnglish
Pages (from-to)1007-1032
JournalJournal of Statistical Physics
Volume123
Issue number5
Publication statusPublished - 2006
Publication categoryResearch
Peer-reviewedYes