Stable Sets of Threshold-Based Cascades on the Erdős-Rényi Random Graphs

  • Ching-Lueh Chang
  • Yuh-Dauh Lyuu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7056)


Consider the following reversible cascade on the Erdős-Rényi random graph G(n,p). In round zero, a set of vertices, called the seeds, are active. For a given ρ ∈ ( 0,1 ], a non-isolated vertex is activated (resp., deactivated) in round t ∈ ℤ +  if the fraction f of its neighboring vertices that were active in round t − 1 satisfies f ≥ ρ (resp., f < ρ). An irreversible cascade is defined similarly except that active vertices cannot be deactivated. A set of vertices, S, is said to be stable if no vertex will ever change its state, from active to inactive or vice versa, once the set of active vertices equals S. For both the reversible and the irreversible cascades, we show that for any constant ε > 0, all p ∈ [ (1 + ε) (ln (e/ρ))/n,1 ] and with probability 1 − n − Ω(1), every stable set of G(n,p) has size O(⌈ρn⌉) or n − O(⌈ρn⌉).


Random Graph Neighboring Vertex Discrete Apply Mathematic Active Vertex Simple Undirected Graph 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Ching-Lueh Chang
    • 1
  • Yuh-Dauh Lyuu
    • 2
    • 3
  1. 1.Department of Computer Science and EngineeringYuan Ze UniversityTaoyuanTaiwan
  2. 2.Department of Computer Science and Information EngineeringNational Taiwan UniversityTaipeiTaiwan
  3. 3.Department of FinanceNational Taiwan UniversityTaipeiTaiwan

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