Very large cliques in a scale-free random graph

Abstract: In this short article we consider a preferential attachment random graph model with edge steps, studied by Alves, Ribeiro and Sanchis. Starting with an initial graph $\mathbb{G}<em>1$ formed by a vertex with a self-loop attached to it, the model evolves as follows. At every subsequent (discrete) time step, either with probability $p$ we add a vertex to the graph and connect it to exactly one of the older vertices selected with probability proportional to its degree, or with probability $1-p$ we add one edge between two existing vertices, both selected (independently) with probability proportional to their degrees. Let $ω(\mathbb{G})$ be the clique number of a graph $\mathbb{G}$, i.e.\ the number of vertices in a largest complete subgraph of $\mathbb{G}</em>{}$. Alves, Ribeiro and Sanchis showed that, for any given $\varepsilon&gt;0$, we have $$ω(\mathbb{G}<em>{2t})\geq t<sup>{\frac{1-p}{2-p}(1-\varepsilon)}$$ with high probability (i.e.\ with probability tending to $1$ as $t\rightarrow \infty$). Here we strengthen this bound by showing that, for any function $f:\mathbb{N}\mapsto \mathbb{N}$ that satisfies $f(t)\rightarrow \infty$ as $t\rightarrow \infty$, with high probability [ω(\mathbb{G}{2t}) = Ω\left(t^{\frac{1-p}{2-p}}\Big(\log^{\frac{1}{2-p}}(t)f(t)\Big)^{-1}\right).]

• The paper strengthens lower bounds on the clique number by proving that w(G₂ₜ) exceeds t^(2-p) divided by polylog factors with high probability.
• It employs refined probabilistic concentration inequalities and martingale techniques to analyze degree growth among strategically selected older vertices.
• The results enhance our understanding of clustering in preferential attachment models and raise open questions on achieving optimal constant-factor bounds.

Clique Growth in Preferential Attachment Random Graphs with Edge Steps

Model Definition and Background

The study examines a scale-free random graph constructed via a generalized preferential attachment mechanism introduced by Alves, Ribeiro, and Sanchis. The process iteratively evolves a graph beginning from a single vertex with a self-loop. At each time step, with probability $p$, a new vertex is added and attached to an existing vertex selected with probability proportional to its degree; with probability $1-p$, an edge is introduced between a pair of existing vertices, both chosen independently proportional to their degrees. This model produces networks exhibiting heavy-tailed degree distributions and clustering properties akin to real-world networks.

The central object of study is the clique number $w(G)$, representing the size of the largest complete subgraph. Previous work established bounds on $w(G_{2t})$ for this model, showing that it lies between $t^{2-p}(1-\epsilon)$ and $t^{2-p}\log t$ with high probability as $t \to \infty$.

Main Contributions

The paper delivers a nontrivial strengthening of lower bounds for the clique number in this preferential attachment model with edge steps. For any function $f$ with $f(t) \to \infty$, the authors prove that

$w(G_{2t}) \geq t^{2-p} / (\log^{2-p}(t) f(t))$

with high probability, up to polylogarithmic corrections. This result refines prior bounds by eliminating dependence on a fixed $1-p$0 and demonstrating that a clique of nearly the order $1-p$1 persists, modulo subpolynomial corrections.

The approach leverages probabilistic concentration inequalities, notably Chernoff and Azuma-Hoeffding bounds, applied to the binomial sampling and martingale constructions inherent to the graph process. By restricting attention to vertices born within carefully chosen intervals and requiring these vertices to possess sufficiently high degrees, the authors show they are likely to be mutually connected due to the edge-step mechanism. The analysis accounts for the degree growth of individual vertices and conditions on their birth time, quantifying the probability that a vertex fails to achieve the requisite degree threshold.

Technical Results

• Theorem 1.1 asserts that for a slow-growing diverging function $1-p$2, and sufficiently large $1-p$3, $1-p$4 exceeds $1-p$5 with probability $1-p$6.
• The proof exploits a refined selection of a set of "old" vertices, whose degree accumulation is analyzed via martingales. The likelihood that any given vertex from this interval has too low a degree is shown to be $1-p$7.
• The probability that such sets of vertices fail to induce a clique is exponentially small due to repeated edge-step opportunities, and application of union bounds confirms the existence of large cliques.

The quantitative statements tighten the previously known lower bounds and nearly match the upper bounds, up to logarithmic and subpolynomial factors. The authors highlight the still-open problem of determining optimal constant-factor bounds for the clique number and pose the question of whether there exists a constant $1-p$8 such that $1-p$9 is bracketed between $w(G)$0 and $w(G)$1 with high probability.

Practical and Theoretical Implications

The result has implications for understanding clustering and community structure in scale-free networks generated by preferential attachment, and specifically delineates how large cliques can arise in such models as the network grows. This bears relevance for network resilience, information flow, and complexity in real-world systems modeled by preferential attachment, such as social or communication networks.

On the theoretical side, the work advances the probabilistic combinatorics of scale-free graphs, providing sharper asymptotic estimates for maximal clique sizes. The techniques signal further possibilities for analyzing extremal subgraph properties (such as large independent sets or chromatic numbers) in similarly defined random graph models.

Future Directions

• The determination of tight constant-factor bounds for the clique number remains open, and determining explicit dependence of these constants on model parameter $w(G)$2 is a target for future research.
• Extending the results to more general attachment kernels or to models incorporating deletion mechanisms could broaden applicability.
• The connections between maximal clique growth and other structural properties (such as diameter or community detection) warrant further investigation in both mathematical and algorithmic contexts.

Conclusion

The paper rigorously improves lower bounds for the maximum clique size in scale-free random graphs generated by preferential attachment with edge steps, showing that very large cliques exist almost surely and quantifying their growth rates with respect to $w(G)$3 and $w(G)$4. The methods deployed sharpen probabilistic estimates for extremal graph substructures and suggest further avenues for analysis in random network theory, particularly regarding community and clustering phenomena.