Preferential attachment without vertex growth: emergence of the giant component

We study the following preferential attachment variant of the classical Erdos-Renyi random graph process. Starting with an empty graph on n vertices, new edges are added one-by-one, and each time an edge is chosen with probability roughly proportional to the product of the current degrees of its endpoints (note that the vertex set is fixed). We determine the asymptotic size of the giant component in the supercritical phase, confirming a conjecture of Pittel from 2010. Our proof uses a simple method: we condition on the vertex degrees (of a multigraph variant), and use known results for the configuration model.

💡 Research Summary

The paper investigates a dynamic random graph model in which the vertex set is fixed while edges are added one by one according to a preferential attachment rule. Formally, starting from an empty graph on n vertices, at each step an unordered pair {v,w} of distinct vertices is selected with probability proportional to (d_v+α)(d_w+α), where d_v denotes the current degree of vertex v and α>0 is a parameter. Loops are allowed in a multigraph variant, but the simple‑graph version discards them. When α→∞ the rule becomes uniform and the process reduces to the classical Erdős–Rényi G(n,m) model; for finite α the rule favours high‑degree vertices, creating a “rich‑get‑richer’’ effect without changing the number of vertices.

The main focus is the size of the largest connected component (the giant component) in the supercritical regime, i.e. when the number of edges m exceeds the critical value
m_c = nα²/(α+1).
Earlier work by Pittel (2010) proved that around m≈m_c the largest component jumps from Θ(log n) to Θ(n) and gave a finite‑size scaling result under the relatively strong condition ε³n→∞, where ε = (m/m_c)−1. However, the precise linear growth rate of the giant component for ε = O(1) and only ε³n→∞ remained conjectural.

The authors resolve this conjecture and extend it in several directions. Their strategy consists of three key steps:

1. Multigraph reduction. They introduce a convenient multigraph version G*{α,n,m} that permits loops and multiple edges. A transfer lemma (Theorem 2.2) shows that any high‑probability statement for G*{α,n,m} also holds for the simple‑graph process G_{α,n,m} provided m = O(n) and α = Ω(1). The proof compares the normalising constants of the two processes and shows they stay asymptotically equal during the first O(n) steps.

2. Conditioning on the degree sequence. For the multigraph, conditioning on its degree sequence d = (d_1,…,d_n) yields exactly the configuration model G(d): a random multigraph obtained by pairing “half‑edges’’ according to d. This equivalence (Theorem 2.4) is a standard fact for configuration models but had not been exploited for this preferential‑attachment process before.

3. Asymptotic degree distribution. Using a continuous‑time construction, the authors prove that the degree of each vertex in G*_{α,n,m} is asymptotically independent and follows a negative‑binomial distribution NB(α, p) with p ≈ 2m/(nα+2m). Consequently the empirical degree sequence concentrates around this law, and the required regularity conditions for the configuration model are satisfied.

With the degree distribution in hand, the authors invoke known results for the configuration model (Molloy–Reed criterion and its refinements). They obtain that for any ε = O(1) with ε³n→∞, \