The weighted tunable clustering in local-world networks with incremental behaviors

Since some realistic networks are influenced not only by increment behavior but also by tunable clustering mechanism with new nodes to be added to networks, it is interesting to characterize the model for those actual networks. In this paper, a weighted local-world model, which incorporates increment behavior and tunable clustering mechanism, is proposed and its properties are investigated, such as degree distribution and clustering coefficient. Numerical simulations are fit to the model characters and also display good right skewed scale-free properties. Furthermore, the correlation of vertices in our model is studied which shows the assortative property. Epidemic spreading process by weighted transmission rate on the model shows that the tunable clustering behavior has a great impact on the epidemic dynamic. Keywords: Weighted network, increment behavior, tun- able cluster, epidemic spreading.

💡 Research Summary

**
The paper introduces a weighted local‑world network model that simultaneously incorporates three realistic mechanisms often observed in empirical complex systems: (i) local‑world preferential attachment (LPA), (ii) strength‑preferential attachment (SPA), and (iii) an “increment behavior” whereby the weight of an existing edge increases by a fixed amount δ whenever a newly arriving node connects to both ends of that edge. In addition, the model features a tunable clustering mechanism: after a new node attaches to an existing node v_i, it creates an extra triad‑formation (TF) link to one of v_i’s neighbors with probability p. This TF step also triggers the increment behavior on the pre‑existing edge between the two neighbors, thereby coupling clustering and weight dynamics.

The growth process starts from c₀ complete local worlds, each containing n₀ nodes, and inter‑local‑world links that make the initial graph connected. At each discrete time step the network evolves as follows: with probability q a brand‑new local world (again a complete graph of size n₀) is introduced; the newcomer then creates m PA links to nodes in existing worlds according to the combined LPA‑SPA rule. With probability 1‑q a new node is added inside an existing world, again forming m PA links. After each PA link is created, a TF link is added with probability p, which closes a triangle and simultaneously increments the weight of the third side of that triangle by δ. All newly created edges start with weight w₀ = 1. When p = q = 0 the model reduces to the weighted Barabási‑Albert (BA) model.

Using a mean‑field approach the authors derive analytical expressions for several key quantities. The size N_i of a local world follows a power‑law distribution P(N) ∝ N^{‑γ} with exponent γ = 1 + q(n₀ − 1). The strength s_i and degree k_i of a node evolve according to ds_i/dt = A s_i t and dk_i/dt = C s_i t, where A = m(1 + p + 2pδ)/