Cumulative Merging Percolation: A Novel Long-Range Percolation Process in Networks

Alapfogalmak

Cumulative Merging Percolation (CMP) is a novel long-range percolation process in networks where nodes can merge into clusters even if they are topologically distant, leading to the formation of a giant CMP cluster that does not coincide with the topological connected components.

Kivonat

The content introduces a general formulation of Cumulative Merging Percolation (CMP), a new class of percolation processes in networks where nodes can merge into clusters even if they are topologically distant. This is in contrast to standard percolation where clusters coincide with topologically connected components.
The key highlights are:

CMP is defined by an iterative merging procedure where nodes can join a cluster if their topological distance is less than the interaction range of the cluster, which is a function of the cluster mass.
CMP can be seen as a long-range percolation process, as clusters do not need to be topologically connected.
The authors analyze a specific class of degree-ordered CMP models on power-law distributed networks, where the node masses are set to their degrees and the interaction range is a function of the ratio between the cluster mass and a control parameter.
The authors develop a general scaling theory to understand the behavior of the size of the giant CMP cluster (CMPGC) as a function of the control parameter.
Two specific forms of the interaction range are studied in detail - algebraically and logarithmically growing. These lead to rich phase transition scenarios with competition between different mechanisms for the formation of the CMPGC.
Numerical simulations confirm the analytical predictions, including the observation of crossover phenomena between different regimes.

Statisztikák

The average distance between a node of degree k and its closest node with degree at least k is given by d(k) ≃ 1 + a(γ) ln(k/kmin), where a(γ) = (γ - 3) / ln(κ) and κ = ⟨k^2⟩ / ⟨k⟩ - 1 is the network branching factor.
The fraction of active (non-removed) nodes is given by ϕ = (ka/kmin)^(1-γ).

Idézetek

"CMP is a truly long-range percolation process. This specification (long-range percolation) is often used for models where, in a lattice, additional links connecting sites separated by any euclidean distance are added, with a probability depending on the distance."
"CMP has been recently studied in a specific degree-ordered case [18] to elucidate the behavior of the SIS model on random uncorrelated networks with power-law degree distribution P(k) ∼ k^-γ. By means of a scaling approach and numerical simulations, it was shown that the long-range nature of the model guarantees, for any γ > 2, the existence of a percolating cluster for any value of the control parameter (i.e., the degree threshold that determines node removal), at variance with what happens for the short-range counterpart [19, 20]."

Főbb Kivonatok

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

by Lorenzo Ciri... : arxiv.org 05-01-2024

https://arxiv.org/pdf/2203.01014.pdf

Mélyebb kérdések

How would the CMP process behave if nodes were activated at random rather than in a degree-ordered way

If nodes were activated at random in the CMP process instead of in a degree-ordered way, it would lead to a different behavior in the formation of clusters. Random activation would introduce a stochastic element into the process, potentially resulting in clusters that are not based on the nodes' degrees. This randomness could lead to a more unpredictable and varied cluster formation compared to the degree-ordered activation. The structure and size of the clusters may vary significantly, and the overall network dynamics could be more complex to analyze due to the lack of a systematic approach in node activation.

What changes would occur if the node masses were assigned differently, e.g., mi ≠ ki

If the node masses were assigned differently, for example, if mi ≠ ki, it would impact the way clusters are formed in the CMP process. The assignment of node masses plays a crucial role in determining how nodes interact and merge to form clusters. If the masses were not directly related to the nodes' degrees, it could introduce a new level of complexity into the merging process. Nodes with different masses would interact based on their assigned values, potentially leading to clusters that are not solely dependent on the nodes' degrees. This change could alter the overall structure of the network and the characteristics of the clusters formed.

Can the CMP framework be extended to model other types of network dynamics beyond epidemic spreading, such as cascading failures or information diffusion

Yes, the CMP framework can be extended to model various types of network dynamics beyond epidemic spreading. For example, it can be applied to study cascading failures in networks, where the failure of one node triggers a chain reaction of failures in interconnected nodes. By incorporating specific rules for node activation and merging based on the concept of cumulative merging percolation, the CMP framework can simulate how cascading failures propagate through a network. Additionally, the CMP model can be adapted to analyze information diffusion processes in networks, where nodes share and spread information to their neighbors. By defining interactions and merging mechanisms that mimic information diffusion dynamics, the CMP framework can provide insights into how information spreads and influences different parts of a network.