Generalised De-Preferential Random Graphs

We consider some further generalizations of the novel random graph models as introduced by Bandyopadhyay and Sen \cite{BaSe2025} and find asymptotic for the degree of a fixed vertex and along with the asymptotic degree distribution. We show that in the \emph{case of the inverse power law} the order of these statistics is much slower than the case of the simple inverse function, which was considered in \cite{BaSe2025}. However, the results for the linear case remain exactly the same even after introducing a “shift” parameter.

💡 Research Summary

This paper, “Generalised De-Preferential Random Graphs,” presents a significant generalization of the de-preferential random graph models originally introduced by Bandyopadhyay and Sen. De-preferential attachment models describe network growth where new vertices are more likely to connect to existing vertices with lower degrees, contrasting with the well-known preferential attachment models. The authors extend previous work by introducing additional parameters into the attachment probability functions and provide a rigorous asymptotic analysis of vertex degrees and the empirical degree distribution for these generalized models.

The core of the paper defines two broad classes of generalized models. First, the Linear De-Preferential Model is extended by incorporating a “shift” parameter θ ≥ 1 and a slope parameter α (0 < α ≤ 1). The probability that a new vertex connects to an existing vertex v_i is proportional to (θ - αd_i(n)). Surprisingly, the main results (Theorem 2.1) show that the introduction of these parameters does not alter the fundamental asymptotic behavior found in the simpler case (θ=α=1). Specifically, the degree of a fixed vertex grows like m log n, satisfies a Central Limit Theorem, and the limiting empirical degree distribution follows a geometric distribution p_k = 1/2^k for k≥1.

Second, the Inverse Power Law De-Preferential Model generalizes the attachment probability to the form 1/(δ + d_i(n))^α, where α > 0 is an exponent and δ > -1 is a shift parameter. The asymptotic behavior here is markedly different and more sensitive to the model parameters. For the case m=1 (one edge per new vertex), Theorem 2.2 establishes that the degree of a fixed vertex grows at the much slower rate of (log n)^{1/(1+α)}, converging almost surely to a constant involving the Malthusian parameter λ* of an associated branching process. The limiting degree distribution is derived explicitly. For the case m>1, Theorem 2.4 shows that the degree growth is bounded between constants times m√(log n), which is slower than both the linear model’s log n growth and the inverse law’s (log n)^{1/(1+α)} growth for m=1.

The proofs rely heavily on sophisticated probabilistic embedding techniques. For m=1, the entire random graph process is embedded into a Crump-Mode-Jagers (CMJ) branching process (Theorem 4.1). This embedding allows the application of limit theorems for branching processes, notably a result by Nerman (Theorem 4.2), to analyze properties like the proportion of vertices with a given degree. For m>1, an Athreya-Karlin embedding is employed (Theorem 4.3). This technique couples the degree evolution of each vertex with independent Yule processes (pure birth processes) with birth rates λ_i = 1/(i+δ)^α. The asymptotic properties of these Yule processes (Theorem 4.4) and the growth of the sequence of random times {τ_n} at which the graph is observed (Proposition 4.5) are then combined to establish the main theorems.

A crucial technical component involves the asymptotic analysis of the normalizing constant D_n = Σ_i 1/(δ+d_i(n))^α, which ensures the attachment probabilities sum to one. Lemmas 4.1 and 4.2 provide essential bounds and almost sure limits for D_n, which underpin the proofs for both model classes.

In conclusion, this work substantially generalizes the mathematical framework for de-preferential attachment networks. It demonstrates the robustness of the linear model’s properties to parameterization while revealing the dramatic sensitivity of growth rates and degree distributions to the exponent α in the inverse power law model. The paper deepens the theoretical understanding of how anti-preferential connection mechanisms shape network structure, employing advanced tools from branching process theory and stochastic embedding.