Triggering cascades on undirected connected graphs

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Abstract

Consider the following cascading process on an undirected connected graph G(V,E). A set S of vertices, called the seeds, are active initially. Thereafter, an inactive vertex is activated when at least a ρ fraction of its neighbors is active, where ρ(0,1]. The cascading process proceeds asynchronously until no more vertices can be activated. This paper proves the existence of at most (22+3)ρ|V| seeds that can activate all vertices at the end.

Highlights

► We consider threshold-based cascades on connected graphs. ► A degree-d vertex is activated when it has at least ρd active neighbors. ► We find bounds on the minimum number of seeds activating all vertices eventually. ► We use techniques of Ackerman et al. [Theoret. Comput. Sci. 411 (44–46) (2010) 4017–4022].

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      Citation Excerpt :

      We contribute to this question by showing the following results. For many more references to and discussion of related work see [12,13]. We proceed to the second proof.

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    Supported in part by the National Science Council of Taiwan under grant 99-2218-E-155-014-MY2.

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