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The largest subcritical component in inhomogeneous random graphs of preferential attachment type

Peter Mörters, Nick Schleicher

TL;DR

The paper analyzes the size of the largest subcritical component in an inhomogeneous random graph with a preferential-attachment-type kernel in the subcritical regime γ<1/2, 0<β<β_c. It identifies a polynomial scaling for the largest component, n^{ρ_-}, where ρ_- = 1/2 − sqrt((1/2−γ)^2+β(2γ−1)), and develops a local approximation by killed branching random walks to capture both typical and untypical vertex behavior. The authors prove a lower bound via a Galton–Watson embedding and a key main lemma, and an matching upper bound by coupling the graph to killed BRWs with large-deviation controls, highlighting a self-similar, self-contained mechanism that yields a larger subcritical component than the maximal degree. This work extends understanding beyond rank-one kernels and suggests universality of the subcritical component scaling in preferential-attachment-type graphs.

Abstract

We identify the size of the largest connected component in a subcritical inhomogeneous random graph with a kernel of preferential attachment type. The component is polynomial in the graph size with an explicitly given exponent, which is strictly larger than the exponent for the largest degree in the graph. This is in stark contrast to the behaviour of inhomogeneous random graphs with a kernel of rank one. Our proof uses local approximation by branching random walks going well beyond the weak local limit and novel results on subcritical killed branching random walks.

The largest subcritical component in inhomogeneous random graphs of preferential attachment type

TL;DR

The paper analyzes the size of the largest subcritical component in an inhomogeneous random graph with a preferential-attachment-type kernel in the subcritical regime γ<1/2, 0<β<β_c. It identifies a polynomial scaling for the largest component, n^{ρ_-}, where ρ_- = 1/2 − sqrt((1/2−γ)^2+β(2γ−1)), and develops a local approximation by killed branching random walks to capture both typical and untypical vertex behavior. The authors prove a lower bound via a Galton–Watson embedding and a key main lemma, and an matching upper bound by coupling the graph to killed BRWs with large-deviation controls, highlighting a self-similar, self-contained mechanism that yields a larger subcritical component than the maximal degree. This work extends understanding beyond rank-one kernels and suggests universality of the subcritical component scaling in preferential-attachment-type graphs.

Abstract

We identify the size of the largest connected component in a subcritical inhomogeneous random graph with a kernel of preferential attachment type. The component is polynomial in the graph size with an explicitly given exponent, which is strictly larger than the exponent for the largest degree in the graph. This is in stark contrast to the behaviour of inhomogeneous random graphs with a kernel of rank one. Our proof uses local approximation by branching random walks going well beyond the weak local limit and novel results on subcritical killed branching random walks.

Paper Structure

This paper contains 7 sections, 16 theorems, 92 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem 2
  3. Proof of Proposition \ref{['twokilled']}
  4. Proof of Proposition \ref{['mainlemma']}
  5. Proof of Theorem 3.
  6. Proof of Proposition \ref{['coupling2']}
  7. Proof of Proposition \ref{['ldp']}

Key Result

Theorem 1

Let $S_n(i)$ be the size of the connected component of vertex $i\in V_n$ in the inhomogeneous random graph of preferential attachment type in the subcritical regime. If $o_n\in V_n$ is such that $\frac{o_n}{n}\to u\in(0,1]$, then for all $x>0$, where and $Y$ is a positive random variable satisfying

Figures (2)

  • Figure 1: Illustration of Proposition \ref{['mainlemma']}: The vertices $u_1, \ldots, u_4$ are successively explored, the exploration of $u_1$ is depicted. The exploration yields particles in the entire interval $[bum,m]$ but only the red particles located in $[bm,m]$ are included in $\mathcal{X}_1$. A logarithmic scale is used on the abscissa.
  • Figure 2: Branching particles are marked in blue. The positions on $[0,\infty)$ of the frozen particles, which are marked in red, yield the point process $\xi$.

Theorems & Definitions (30)

  • Theorem 1: Early typical vertices
  • Theorem 2: Untypically early vertices
  • Theorem 3: Largest subcritical component
  • Remark 1.1
  • Proposition 1
  • Proposition 2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 20 more