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The partial duplication random graph with edge deletion

Felix Hermann, Peter Pfaffelhuber

Abstract

We study a random graph model in continuous time. Each vertex is partially copied with the same rate, i.e.\ an existing vertex is copied and every edge leading to the copied vertex is copied with independent probability $p$. In addition, every edge is deleted at constant rate, a mechanism which extends previous partial duplication models. In this model, we obtain results on the degree distribution, which shows a phase transition such that either -- if $p$ is small enough -- the frequency of isolated vertices converges to 1, or there is a positive fraction of vertices with unbounded degree. We derive results on the degrees of the initial vertices as well as on the sub-graph of non-isolated vertices. In particular, we obtain expressions for the number of star-like subgraphs and cliques.

The partial duplication random graph with edge deletion

Abstract

We study a random graph model in continuous time. Each vertex is partially copied with the same rate, i.e.\ an existing vertex is copied and every edge leading to the copied vertex is copied with independent probability . In addition, every edge is deleted at constant rate, a mechanism which extends previous partial duplication models. In this model, we obtain results on the degree distribution, which shows a phase transition such that either -- if is small enough -- the frequency of isolated vertices converges to 1, or there is a positive fraction of vertices with unbounded degree. We derive results on the degrees of the initial vertices as well as on the sub-graph of non-isolated vertices. In particular, we obtain expressions for the number of star-like subgraphs and cliques.

Paper Structure

This paper contains 12 sections, 11 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model and main results
  3. Proof of Theorem 1
  4. Birth-death processes with disasters and $p$-jump processes
  5. Properties of the piecewise deterministic jump process $X$
  6. Proof of Theorem \ref{['thm:deg-dis-whole-graph']}
  7. Proof of Proposition \ref{['prop:limitOfI_x']}
  8. Proof of Theorem 2
  9. Two auxiliary functions
  10. The size of the graph
  11. Some martingales
  12. Proof of Theorem \ref{['thm:functionals']}

Key Result

Lemma 2.3

Let $X = (X_t)_{t\geq 0}$ be a Markov process on $[0,1]$ jumping at rate 1 from $X_t = x$ to $px$, in between jumps evolving according to $\dot X_t=pX_t(1-X_t)-\delta X_t$. Furthermore, let i.e. the probability generating function of the degree distribution at time $t$. Then, for all $t\geq0$ and $x\in[0,1]$, writing $\mathbb E_x[.] := \mathbb E[.|X_0=x]$,

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (28)

  • Definition 2.1: Partial duplication graph process with edge deletion
  • Remark 2.2
  • Lemma 2.3: Connection of $PD(p,\delta)$ and a piecewise-deterministic Markov process
  • Theorem 1: Limit of the degree distribution
  • Remark 2.4: Interpretations
  • Proposition 2.5: Limit of degree distribution on the set of non-isolated vertices
  • Remark 2.6: Interpretation
  • Remark 2.7: Convergence of moments
  • Theorem 2: Binomial moments, cliques and degrees
  • Remark 2.8: Interpretations
  • ...and 18 more