Arbitrary degree distribution and high clustering in networks of locally interacting agents

Many real world networks, such as social networks, are primarily formed through local interactions between agents. Additionally, in contrast with common network models, social and biological networks exhibit a high degree of clustering. Here we construct a class of network growth models based on local interactions on a metric space, capable of producing arbitrary degree distributions as well as a naturally high degree of clustering akin to biological networks. As a specific example, we study the case of random- walking agents, though most results hold for any linear stochastic dynamics. Agents form bonds when they meet at designated locations we refer to as “rendezvous points.” The spatial distribution of the rendezvous points determines key characteristics of the network such as the degree distribution. For any arbitrary (monotonic) degree distribution, we are able to analytically solve for the required rendezvous point distribution.

💡 Research Summary

The paper introduces a novel class of network growth models that generate arbitrary degree distributions while naturally producing high clustering, by grounding link formation in local interactions of agents moving in a metric space. The authors focus on a concrete example: agents performing isotropic random walks in a two‑dimensional continuous domain. Links are created only when two agents meet at designated “rendezvous points” (RPs); each encounter results in a bond with a small probability λ. The spatial‑temporal density of RPs, denoted Γ(x,t), is the key control variable.

Mathematically, the agents’ probability density φ_i(x,t) satisfies a linear Fokker‑Planck (diffusion) equation with a point source at the agent’s start location. The Green’s function G(x,t;x_i,t_0) solves this equation. The probability that agents i and j become connected by a final time T is, to first order in λ, A_{ij}=λ∬ G_i(x,t) Γ(x,t) G_j(x,t) dt d²x. This expression can be interpreted either as entries of a weighted adjacency matrix or as bond probabilities for an ensemble of unweighted graphs.

The expected degree of a node i, k_i=∑j A{ij}, becomes in the continuum limit k(x_i,t_0,T)=λ∬ G_i(x,t) Γ(x,t) dt d²x. Applying the adjoint diffusion operator L† to both sides yields the pivotal relation L†k(x,t,T)=λ θ(T−t) Γ(x,t). Thus the degree field k(x,t,T) and the RP density Γ are directly linked via a linear differential operator. This result enables two complementary tasks: (1) given a prescribed RP distribution, compute the resulting degree distribution; (2) given any monotonic target degree distribution P(k), solve analytically for the RP distribution that would generate it.

Assuming rotational symmetry (Γ(r,t) with r=|x|) simplifies the analysis because k depends only on r. The degree distribution is related to the radial node density dN/dr through P(k)= (dN/dr)/(dk/dr). Combining this with the operator relation above, the authors derive explicit formulas for Γ(r,t) that yield power‑law degree distributions P(k)∝k^{−γ}. For γ>1 the RP density takes the form Γ(r,t)=L†