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Structure of percolating clusters in random clustered networks.

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The study reveals that percolating clusters in random clustered networks remain clustered at the percolation threshold. Their assortativity varies with network details, showing both disassortative and assortative behaviors.

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Area of Science:

  • Network science
  • Statistical physics
  • Complex systems analysis

Background:

  • Percolating clusters (PC) are fundamental in understanding phase transitions in disordered systems.
  • Random clustered networks (RCN) offer a model to study network structure with inherent clustering.
  • Characterizing the structural properties of PCs on clustered networks is crucial for network theory.

Purpose of the Study:

  • To investigate the structural properties, specifically clustering and assortativity, of percolating clusters (PC) on random clustered networks (RCN).
  • To determine how network clustering influences the PC's properties at the percolation threshold.
  • To analyze the dependence of PC assortativity on the specific configuration of the RCN.

Main Methods:

  • Utilizing generating functions to derive analytical expressions for the clustering and assortative coefficients of the PC.
  • Employing both analytical derivations and numerical simulations to validate findings.
  • Applying renormalization group techniques to reconcile observations with fractal network properties.

Main Results:

  • The percolating cluster (PC) in highly clustered random clustered networks (RCN) exhibits clustering even at the percolation threshold.
  • The assortativity of the PC is contingent upon the specific RCN construction.
  • The PC demonstrates disassortative behavior under Poisson degree and triangle distributions but becomes assortative when nodes have a uniform small degree concentrated in triangles.

Conclusions:

  • The structural properties of percolating clusters are significantly influenced by the underlying network's clustering.
  • The seemingly contradictory assortativity results highlight the nuanced relationship between network topology and emergent cluster properties.
  • Despite local variations, fractal renormalization schemes confirm the inherently disassortative nature of fractal percolating clusters.