# What I tell you three times is true: Bootstrap percolation in small worlds

Leo Hamed Amini, Nikolaos Fountoulakis

Research output: Chapter in Book/Report/Conference proceedingConference contribution

12 Citations (Scopus)

### Abstract

A bootstrap percolation process on a graph G is an "infection" process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round each uninfected node which has at least r infected neighbours becomes infected and remains so forever. The parameter r ≥ 2 is fixed. We analyse this process in the case where the underlying graph is an inhomogeneous random graph, which exhibits a power-law degree distribution, and initially there are a(n) randomly infected nodes. The main focus of this paper is the number of vertices that will have been infected by the end of the process. The main result of this work is that if the degree sequence of the random graph follows a power law with exponent β, where 2 <β <3, then a sublinear number of initially infected vertices is enough to spread the infection over a linear fraction of the nodes of the random graph, with high probability. More specifically, we determine explicitly a critical function ac (n) such that ac (n) = o(n) with the following property. Assuming that n is the number of vertices of the underlying random graph, if a(n) ≫ ac (n), then the process does not evolve at all, with high probability as n grows, whereas if a(n) ≫ ac (n), then there is a constant ε > 0 such that, with high probability, the final set of infected vertices has size at least εn. This behaviour is in sharp contrast with the case where the underlying graph is a G(n,p) random graph with p = d/n. Recent results of Janson, Łuczak, Turova and Vallier have shown that if the number of initially infected vertices is sublinear, then with high probability the size of the final set of infected vertices is approximately equal to a(n). That is, essentially there is lack of evolution of the process. It turns out that when the maximum degree is o(n1/(β - 1)), then ac (n) depends also on r. But when the maximum degree is Θ(n1/(β - 1)), then ac(n) = n β-2/β-1.

Original language English (US) Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 462-474 13 7695 LNCS https://doi.org/10.1007/978-3-642-35311-6_34 Published - 2012 Yes 8th International Workshop on Internet and Network Economics, WINE 2012 - Liverpool, United KingdomDuration: Dec 10 2012 → Dec 12 2012

### Publication series

Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 7695 LNCS 03029743 16113349

### Other

Other 8th International Workshop on Internet and Network Economics, WINE 2012 United Kingdom Liverpool 12/10/12 → 12/12/12

### Fingerprint

Bootstrap Percolation
Small World
Random Graphs
Maximum Degree
Graph in graph theory
Vertex of a graph
Approximately equal
Degree Sequence
Power-law Distribution
Degree Distribution
Infection
Power Law
Exponent
Subset

### Keywords

• bootstrap percolation
• contagion
• power-law random graphs

### ASJC Scopus subject areas

• Computer Science(all)
• Theoretical Computer Science

### Cite this

Amini, L. H., & Fountoulakis, N. (2012). What I tell you three times is true: Bootstrap percolation in small worlds. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7695 LNCS, pp. 462-474). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7695 LNCS). https://doi.org/10.1007/978-3-642-35311-6_34

What I tell you three times is true : Bootstrap percolation in small worlds. / Amini, Leo Hamed; Fountoulakis, Nikolaos.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 7695 LNCS 2012. p. 462-474 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7695 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Amini, LH & Fountoulakis, N 2012, What I tell you three times is true: Bootstrap percolation in small worlds. in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). vol. 7695 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7695 LNCS, pp. 462-474, 8th International Workshop on Internet and Network Economics, WINE 2012, Liverpool, United Kingdom, 12/10/12. https://doi.org/10.1007/978-3-642-35311-6_34
Amini LH, Fountoulakis N. What I tell you three times is true: Bootstrap percolation in small worlds. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 7695 LNCS. 2012. p. 462-474. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-35311-6_34
Amini, Leo Hamed ; Fountoulakis, Nikolaos. / What I tell you three times is true : Bootstrap percolation in small worlds. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 7695 LNCS 2012. pp. 462-474 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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