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Large-scale behavior of the partial duplication random graph

Felix Hermann, Peter Pfaffelhuber

Abstract

The following random graph model was introduced for the evolution of protein-protein interaction networks: Let $\mathcal G = (G_n)_{n=n_0, n_0+1,...}$ be a sequence of random graphs, where $G_n = (V_n, E_n)$ is a graph with $|V_n|=n$ vertices, $n=n_0,n_0+1,...$ In state $G_n = (V_n, E_n)$, a vertex $v\in V_n$ is chosen from $V_n$ uniformly at random and is partially duplicated. Upon such an event, a new vertex $v'\notin V_n$ is created and every edge $\{v,w\} \in E_n$ is copied with probability~$p$, i.e.\ $E_{n+1}$ has an edge $\{v',w\}$ with probability~$p$, independently of all other edges. Within this graph, we study several aspects for large~$n$. (i) The frequency of isolated vertices converges to~1 if $p\leq p^* \approx 0.567143$, the unique solution of $pe^p=1$. (ii) The number $C_k$ of $k$-cliques behaves like $n^{kp^{k-1}}$ in the sense that $n^{-kp^{k-1}}C_k$ converges against a non-trivial limit, if the starting graph has at least one $k$-clique. In particular, the average degree of a vertex (which equals the number of edges -- or 2-cliques -- divided by the size of the graph) converges to $0$ iff $p<0.5$ and we obtain that the transitivity ratio of the random graph is of the order $n^{-2p(1-p)}$. (iii) The evolution of the degrees of the vertices in the initial graph can be described explicitly. Here, we obtain the full distribution as well as convergence results.

Large-scale behavior of the partial duplication random graph

Abstract

The following random graph model was introduced for the evolution of protein-protein interaction networks: Let be a sequence of random graphs, where is a graph with vertices, In state , a vertex is chosen from uniformly at random and is partially duplicated. Upon such an event, a new vertex is created and every edge is copied with probability~, i.e.\ has an edge with probability~, independently of all other edges. Within this graph, we study several aspects for large~. (i) The frequency of isolated vertices converges to~1 if , the unique solution of . (ii) The number of -cliques behaves like in the sense that converges against a non-trivial limit, if the starting graph has at least one -clique. In particular, the average degree of a vertex (which equals the number of edges -- or 2-cliques -- divided by the size of the graph) converges to iff and we obtain that the transitivity ratio of the random graph is of the order . (iii) The evolution of the degrees of the vertices in the initial graph can be described explicitly. Here, we obtain the full distribution as well as convergence results.

Paper Structure

This paper contains 13 sections, 11 theorems, 88 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model and results
  3. Model
  4. Results
  5. Preparation
  6. Some recursions
  7. An auxiliary process
  8. Martingales, the Gamma function and a recursion
  9. Proof of Theorem \ref{['T1']}
  10. A time-continuous partial duplication graph
  11. A duality relationship between $\mathcal{X}$ and $\mathcal{G}$
  12. Proof of Theorem \ref{['T1']}
  13. Proof of Theorems \ref{['T2']} and \ref{['T3']}

Key Result

Theorem 1

Let $p^\ast$ be the (unique) solution of $pe^p=1$ (or $p + \log p =0$). Then, the following dichotomy holds:

Figures (1)

  • Figure 1: Illustration of one step in the PDn random graph; see also Definition \ref{['def:PDn']}. At time $n=6$ (since there are 6 vertices in the graph on the left), the vertex $v$ is picked uniformly at random. It is copied, giving rise to the new vertex $v'$, together with all potential edges to neighbors of $v$ (see the dashed lines in the middle). Then, every dashed line is kept independently of the others with probability $p$. The result is the random graph with $n=7$ vertices on the right.

Theorems & Definitions (38)

  • Definition 2.1: Graph, degree, clique
  • Remark 2.2: Relationships
  • Definition 2.3: Partial duplication random graph
  • Remark 2.4: Basic observations
  • Remark 2.5: Related random graph models
  • Remark 2.6: Notation
  • Theorem 1: Frequency of isolated vertices
  • Remark 2.7: Connections to work by Bebek et al (2006)
  • Theorem 2: Cliques, stars
  • Remark 2.8: Dependence on the initial graph
  • ...and 28 more