Spectra of random networks in the weak clustering regime

The asymptotic behaviour of dynamical processes in networks can be expressed as a function of spectral properties of the corresponding adjacency and Laplacian matrices. Although many theoretical results are known for the spectra of traditional configuration models, networks generated through these models fail to describe many topological features of real-world networks, in particular non-null values of the clustering coefficient. Here we study effects of cycles of order three (triangles) in network spectra. By using recent advances in random matrix theory, we determine the spectral distribution of the network adjacency matrix as a function of the average number of triangles attached to each node for networks without modular structure and degree-degree correlations. Implications to network dynamics are discussed. Our findings can shed light in the study of how particular kinds of subgraphs influence network dynamics.

💡 Research Summary

The paper addresses a fundamental gap in network theory: the influence of clustering, specifically triangles, on the spectral properties of random graphs and, consequently, on dynamical processes that depend on these spectra. Traditional configuration models generate locally tree‑like graphs with zero clustering, which limits their applicability to many real‑world systems where the clustering coefficient is non‑zero. To overcome this, the authors adopt the Newman‑Miller “expected degree” model, which allows each vertex i to have an expected number of single edges s_i and an expected number of triangles t_i. Single edges are formed with Poisson mean s_i s_j / Σ_q s_q, while triangles are formed with Poisson mean 2 t_i t_j t_k / (Σ_q t_q)^2. Consequently the expected degree of a vertex is k_i = s_i + 2 t_i, and the global transitivity T can be expressed in terms of the moments of the joint distribution p_{st}.

The analytical core proceeds in two stages. First, the authors define a modularity matrix B = A – ⟨A⟩, where ⟨A⟩ is the ensemble average adjacency matrix. Because the off‑diagonal entries of B have zero mean, its spectral density ρ(λ) can be obtained from the Stieltjes transform of the average resolvent ⟨(λI – B)^{-1}⟩. By recursively removing the last row and column of the matrix (a technique originally used for the standard configuration model), they derive a self‑consistent equation for the diagonal element γ_λ(s_n, t_n) = ⟨