Cumulative Merging Percolation: A long-range percolation process in networks

TL;DR

Cumulative Merging Percolation (CMP) extends standard percolation to long-range interactions on networks, allowing clusters to form from topologically disconnected components via a mass-dependent interaction range. The authors develop a general scaling framework for degree-ordered CMP and analyze two representative interaction-range forms: algebraic and logarithmic. For power-law networks with exponent $\gamma$, CMP exhibits distinct regimes: $\gamma \le 3$ yields a CMP giant component for all finite control parameters, while $\gamma > 3$ reveals two competing mechanisms—extended DOP and merging of distant isolated nodes—producing crossovers and a phase diagram in parameter space. The results interpolate between known DOP behavior and new long-range effects, with analytic predictions (including Lambert $W$-function appearances in the algebraic case) confirmed by numerical simulations, and provide a framework potentially applicable to SIS-like epidemic dynamics on weighted networks.

Abstract

Percolation on networks is a common framework to model a wide range of processes, from cascading failures to epidemic spreading. Standard percolation assumes short-range interactions, implying that nodes can merge into clusters only if they are nearest-neighbors. Cumulative Merging Percolation (CMP) is an new percolation process that assumes long-range interactions, such that nodes can merge into clusters even if they are topologically distant. Hence in CMP percolation clusters do not coincide with the topological connected components of the network. Previous work has shown that a specific formulation of CMP features peculiar mechanisms for the formation of the giant cluster, and allows to model different network dynamics such as recurrent epidemic processes. Here we develop a more general formulation of CMP in terms of the functional form of the cluster interaction range, showing an even richer phase transition scenario with competition of different mechanisms resulting in crossover phenomena. Our analytic predictions are confirmed by numerical simulations.

Figures (7)

• Figure 1: Visual representation of a CMP process on a graph with $k_a=3$ and $r(m)=m/k_a$. (a) All nodes in the graph are shown. Colors depend on the degree $k$. (b) Initial configuration of the merging process, with $k_a=3$, each node forming a cluster. Empty circles are inactive nodes. Colored dashed regions represent the interaction range of each active node. (c) Intermediate configuration of the merging process. Dark red regions represent clusters. (d) Final configuration of the merging process. From (c) to (d) the long-range nature of the process plays a crucial role. Note that the process ends since the interaction range of the cluster with $r=2$ does not reach any node of the cluster with $r=22/3$.
• Figure 2: Analytical results for algebraically growing interaction range. Plot of $k_2^*$ (a) and $N_2^*$ (b) as a function of $\gamma$ for algebraically growing interaction range and several values of $\alpha$.
• Figure 3: Comparison of analytical and simulation results for the size of the CMP largest cluster as a function of $k_a$, for algebraically growing interaction range and different combinations of $\gamma$ and $\alpha$ values: (a) $\gamma=3.7$ and $\alpha=0.5$; (b) $\gamma=4$ and $\alpha=5$. The red dashed line is the scaling with exponent $1-\gamma$, the green dot-dashed line is the scaling with exponent $2(2-\gamma)$ and the blue dashed line is the prediction of $S_{\text{CMP}}^{(2)}$ given by Eq. \ref{['eq:size_2']} where $k_x=\omega k_a$ and $\omega$ is given by Eq. \ref{['eq:omega_exact_solution_W']}. In panel (a) we also report the results of a simulation of the DOP process on the network with size $N=10^7$.
• Figure 4: Phase diagram of CMP with logarithmically growing interaction range in the $(\delta,\gamma)$ plane. In the shadowed region below the blue solid line, the second mechanism does not activate at all. In the region between the solid and dashed lines the second mechanism activates, but it is subleading compared to the first mechanism (as long as the first mechanism is at work). Above the dashed line the second mechanism activates and is leading with respect to the extended DOP mechanism.
• Figure 5: Analytical results for logarithmically growing interaction range. (a) Three-dimensional representation of $\mu(\gamma, \delta)$ of Eq. \ref{['exponent_mu']}. (b) Plot of $k_2^*$ and $N_2^*$ as a function of $\gamma$ for $\delta=2$. (c) Plot of $k_2^*$ and $N_2^*$ as a function of $\gamma$ for $\delta=20$. In both panels b) and c) the condition $\delta>b(\gamma)$ is verified. The red vertical lines represent the value of $\gamma$ for which $\delta=b(\gamma)$ and thus $k_2^*$ and $N_2^*$ diverge.