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On Reversible Cascades in Scale-Free and Erdős-Rényi Random Graphs

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Abstract

Consider the following cascading process on a simple undirected graph G(V,E) with diameter Δ. In round zero, a set S⊆V of vertices, called the seeds, are active. In round i+1, i∈ℕ, a non-isolated vertex is activated if at least a ρ∈(0,1] fraction of its neighbors are active in round i; it is deactivated otherwise. For k∈ℕ, let min-seed(k)(G,ρ) be the minimum number of seeds needed to activate all vertices in or before round k. This paper derives upper bounds on min-seed(k)(G,ρ). In particular, if G is connected and there exist constants C>0 and γ>2 such that the fraction of degree-k vertices in G is at most C/k γ for all k∈ℤ+, then min-seed(Δ)(G,ρ)=O(⌈ρ γ−1|V|⌉). Furthermore, for n∈ℤ+, p=Ω((ln(e/ρ))/(ρn)) and with probability 1−exp(−n Ω(1)) over the Erdős-Rényi random graphs G(n,p), min-seed(1)(G(n,p),ρ)=O(ρn).

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  1. Department of Computer Science and Engineering, Yuan Ze University, Taoyuan, Taiwan

    Ching-Lueh Chang & Chao-Hong Wang

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  1. Ching-Lueh Chang
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Correspondence to Ching-Lueh Chang.

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The authors are supported in part by the National Science Council of Taiwan under grant 99-2218-E-155-014-MY2.

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Chang, CL., Wang, CH. On Reversible Cascades in Scale-Free and Erdős-Rényi Random Graphs. Theory Comput Syst 52, 303–318 (2013). https://doi.org/10.1007/s00224-012-9387-2

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