On the growth of components with non fixed excesses

Denote by an $l$-component a connected graph with $l$ edges more than vertices. We prove that the expected number of creations of $(l+1)$-component, by means of adding a new edge to an $l$-component in a randomly growing graph with $n$ vertices, tends to 1 as $l,n$ tends to $\infty$ but with $l = o(n^{1/4})$. We also show, under the same conditions on $l$ and $n$, that the expected number of vertices that ever belong to an $l$-component is $\sim (12l)^{1/3} n^{2/3}$.

💡 Research Summary

The paper investigates the evolution of connected components whose excess (the number of edges minus the number of vertices) is not fixed but grows with the size of the graph. The authors work within the continuous‑time random graph model G(n,t), where each edge of the complete graph Kₙ receives an independent random “arrival time” Tₑ. At time t the graph consists of all edges with Tₑ ≤ t. A (k,k + ℓ) graph is a connected graph on k vertices with k + ℓ edges; its excess is ℓ.

The central object of study is the expected number α(ℓ;k) of times that a new edge is added to an ℓ‑component of order k, thereby creating an (ℓ + 1)‑component. Summing over all possible orders yields α_ℓ = ∑_{k=1}^{n}α(ℓ;k), the total expected number of ℓ → ℓ + 1 transitions during the whole growth process.

To compute α(ℓ;k) the authors first need the number c(k,k + ℓ) of labelled connected (k,k + ℓ) graphs. This quantity is given by the Bender‑Canfield‑McKay extension of Wright’s formula. In particular, c(k,k + ℓ) = w_ℓ·s₃·π^{‑½}·e^{12ℓ}·ℓ^{‑5/2}·k^{k+(3ℓ‑1)/2}·exp(∑_{i≥1} r_i ℓ^{i+1}k^{‑i})·(1 + O(ℓ³k^{‑2}+ℓ^{½}k^{‑½}+…)), where w_ℓ involves Gamma functions and the constants r_i are known.

Lemma 1 combines the combinatorial choices (choosing the ℓ‑component and the new edge) with the probability that the component exists at a given time. After integrating over t, the exact expression (1) for α(ℓ;k) is obtained. For the regime ℓ = O(k^{2/3}) the authors simplify this to the asymptotic form (2): α(ℓ;k) ≈ ½·ρ_ℓ·k^{(3ℓ+1)/2}·n^{ℓ+1}·exp(−k³/(24n²)+ℓk²/(8n²)+ℓk²/n)·(1 + small error terms), with ρ_ℓ = ½·s₃·π^{‑½}·e^{12ℓ}·ℓ^{‑5/2}(1+O(1/ℓ)).

The next step is to evaluate α_ℓ. Lemma 3 treats the sum S = ∑_{k=1}^{n}k^{a}exp(−k³/(24n²)+ℓk²/(8n²)+ℓk²/n), where a = (3ℓ+1)/2, by Laplace’s method. After the change of variables t = 2n^{2/3}e^{z}, the dominant contribution comes from the saddle point z₀ ≈ (1/3)ln(a+1). Expanding the exponent around z₀ and evaluating the Gaussian integral yields S ∼ 2^{a+1}·3^{(a‑2)/3}·Γ(a+1/3)·n^{2(a+1)/3}. Substituting this estimate back into the expression for α_ℓ and simplifying the many constants leads to α_ℓ ∼ ρ_ℓ·2²·3^{ℓ+3/2}·3^{‑ℓ+1/2}·Γ(ℓ+½)^{‑1}. All the factors cancel except a term that tends to 1, provided ℓ → ∞, n → ∞ and ℓ = o(n^{1/4}). Consequently, Theorem 4: In a randomly growing graph on n vertices, the expected number of ℓ → ℓ + 1 transitions tends to 1 under the stated scaling. This recovers the result of Janson (2000) but extends it to the case where ℓ is not fixed.

Having established the transition count, the authors turn to the total number V_ℓ of vertices that ever belong to an ℓ‑component. Using the relationship between α_ℓ and the size distribution of ℓ‑components (essentially a branching‑process argument), they obtain E