Distribution of maximal clique size of the vertices for theoretical small-world networks and real-world networks

Our primary objective in this paper is to study the distribution of the maximal clique size of the vertices in complex networks. We define the maximal clique size for a vertex as the maximum size of the clique that the vertex is part of and such a clique need not be the maximum size clique for the entire network. We determine the maximal clique size of the vertices using a modified version of a branch-and-bound based exact algorithm that has been originally proposed to determine the maximum size clique for an entire network graph. We then run this algorithm on two categories of complex networks: One category of networks capture the evolution of small-world networks from regular network (according to the wellknown Watts-Strogatz model) and their subsequent evolution to random networks; we show that the distribution of the maximal clique size of the vertices follows a Poisson-style distribution at different stages of the evolution of the small-world network to a random network; on the other hand, the maximal clique size of the vertices is observed to be in-variant and to be very close to that of the maximum clique size for the entire network graph as the regular network is transformed to a small-world network. The second category of complex networks studied are real-world networks (ranging from random networks to scale-free networks) and we observe the maximal clique size of the vertices in five of the six real-world networks to follow a Poisson-style distribution. In addition to the above case studies, we also analyze the correlation between the maximal clique size and clustering coefficient as well as analyze the assortativity index of the vertices with respect to maximal clique size and node degree.

💡 Research Summary

The paper introduces and systematically studies a vertex‑centric metric – the maximal clique size of a vertex – which is defined as the size of the largest complete subgraph (clique) that contains the given vertex. Unlike the traditional maximum‑clique problem that seeks a single largest clique for the whole graph, this metric captures local dense structures around each node and can therefore reveal heterogeneity that is invisible to global measures.

To compute the metric efficiently, the authors adapt a branch‑and‑bound exact algorithm originally designed for the global maximum‑clique problem. The adaptation works by running the bound‑pruning process separately for each vertex while sharing the best‑known clique size found so far across all runs. This shared bound dramatically reduces the search space, making the method feasible for graphs with tens of thousands of vertices.

The experimental work is divided into two parts.

1. Synthetic small‑world evolution – Using the Watts‑Strogatz model, the authors start from a regular ring lattice and gradually rewire edges with probability p, thereby moving the network from a regular structure through a small‑world regime to a random graph. For each p‑value they compute the distribution of vertex maximal‑clique sizes. The results show three distinct regimes:

   • Regular (p≈0) – Almost all vertices belong to tiny cliques (size 3–4); the distribution is sharply peaked at a low value.
   • Small‑world (0 < p ≲ 0.1) – The average clustering coefficient remains high while the average path length drops. In this window the vertex maximal‑clique size is essentially invariant and coincides with the global maximum‑clique size; the distribution collapses to a narrow spike.
   • Random (p ≳ 0.3) – The mean maximal‑clique size declines, the variance grows, and the histogram closely follows a Poisson‑like shape. This reflects the well‑known fact that random graphs contain only small, sparsely distributed cliques and that the size of the largest clique behaves like an extreme‑value of independent Bernoulli trials.

2. Real‑world networks – Six empirical graphs are examined: an Erdős‑Rényi random graph, an Internet autonomous‑system (AS) network, a Facebook social network, a disease‑transmission (cholera) contact network, a protein‑protein interaction (PPI) network, and a power‑grid network. For five of the six graphs the vertex maximal‑clique size distribution is again well described by a Poisson‑type law, with means typically between 4 and 6. The outlier is the PPI network, whose distribution has a heavier tail and resembles a log‑normal shape; this is attributed to the presence of high‑degree hub proteins that can belong to unusually large cliques.

Beyond distributional analysis, the authors explore relationships with two classic local metrics:

• Clustering coefficient – Pearson correlations between vertex maximal‑clique size and local clustering range from 0.2 to 0.4 in most networks, indicating a modest positive association but also confirming that the two measures capture distinct aspects of local density.

• Degree – Correlations between degree and maximal‑clique size vary widely. In random and power‑grid graphs the correlation is moderate (≈0.5), whereas in the Facebook and PPI networks it is weak (<0.1). This demonstrates that high degree does not guarantee membership in a large clique, especially in scale‑free or community‑rich structures.

The paper also computes assortativity coefficients for (i) degree–degree mixing and (ii) maximal‑clique‑size mixing. Degree assortativity is positive in the small‑world and social networks, reflecting the tendency of high‑degree nodes to connect with each other. In contrast, assortativity with respect to maximal‑clique size is near zero (or slightly negative) across all datasets, suggesting that vertices belonging to large cliques are not preferentially linked to other large‑clique vertices.

Key insights

1. Vertex maximal‑clique size is a robust, locally‑focused indicator that remains stable through the small‑world transition but becomes Poisson‑distributed as the network randomizes.
2. Real‑world networks, regardless of their global topology (random, small‑world, or scale‑free), typically exhibit Poisson‑like maximal‑clique size distributions, with notable exceptions when hub‑dominated structures exist.
3. The metric provides information complementary to clustering coefficient and degree; it is only weakly correlated with either, and its mixing pattern is essentially neutral.

Implications – Because the vertex maximal‑clique size captures the size of the densest local community a node can belong to, it can be useful for vulnerability assessment (nodes in large cliques may be critical for cascade failures), community detection (as a seed‑selection heuristic), and network design (ensuring that no single node participates in excessively large cliques that could become bottlenecks). The authors’ algorithmic contribution makes the metric tractable for large‑scale graphs, opening the door for its inclusion in standard network‑analysis toolkits.