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The spread of fire on a random multigraph

Part of: Combinatorial probability Graph theory Limit theorems

Published online by Cambridge University Press:  22 July 2019

Christina Goldschmidt  and
Eleonora KreačIć
Christina Goldschmidt*
Affiliation:
University of Oxford
Eleonora KreačIć*
Affiliation:
University of Oxford
*
*Postal address: Department of Statistics, University of Oxford, 24-29 St Giles’, Oxford OX1 3LB, UK.
*Postal address: Department of Statistics, University of Oxford, 24-29 St Giles’, Oxford OX1 3LB, UK.
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Abstract

We study a model for the destruction of a random network by fire. Suppose that we are given a multigraph of minimum degree at least 2 having real-valued edge lengths. We pick a uniform point from along the length and set it alight; the edges of the multigraph burn at speed 1. If the fire reaches a vertex of degree 2, the fire gets directly passed on to the neighbouring edge; a vertex of degree at least 3, however, passes the fire either to all of its neighbours or none, each with probability ${\textstyle{1 \over 2}}$. If the fire goes out before the whole network is burnt, we again set fire to a uniform point. We are interested in the number of fires which must be set in order to burn the whole network, and the number of points which are burnt from two different directions. We analyse these quantities for a random multigraph having n vertices of degree 3 and α(n) vertices of degree 4, where α(n)/n → 0 as n → ∞, with independent and identically distributed standard exponential edge lengths. Depending on whether $\alpha(n) \gg \sqrt{n}$ or $\alpha(n)=O(\sqrt{n})$, we prove that, as n → ∞, these quantities converge jointly in distribution when suitably rescaled to either a pair of constants or to (complicated) functionals of Brownian motion. We use our analysis of this model to make progress towards a conjecture of Aronson, Frieze and Pittel (1998) concerning the number of vertices which remain unmatched when we use the Karp–Sipser algorithm to find a matching on the Erdős–Rényi random graph.

Keywords

Random multigraphKarp-Sipser algorithmdifferential equations methodreflected stochastic differential equation

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60F17: Functional limit theorems; invariance principles 05C80: Random graphs 05C70: Factorization, matching, partitioning, covering and packing 05C85: Graph algorithms
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 1 , March 2019 , pp. 1 - 40
Copyright
© Applied Probability Trust 2019 

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References

Aronson, J., Frieze, A. and Pittel, B. G. (1998). Maximum matchings in sparse random graphs: Karp-Sipser revisited. Random Structures Algorithms 12, 111–177.Google Scholar
Arratia, R., Barbour, A. D. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society, Zürich.CrossRefGoogle Scholar
Bender, E. A. and Canfield, E. R. (1978). The asymptotic number of labeled graphs with given degree sequences. J. Combinatorial Theory A 24, 296–307.CrossRefGoogle Scholar
Bertoin, J. and Goldschmidt, C. (2004). Dual random fragmentation and coagulation and an application to the genealogy of Yule processes. In Mathematics and Computer Science III, Birkhäuser, Basel, pp. 295–308.CrossRefGoogle Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.Google Scholar
Bohman, T. and Frieze, A. (2011). Karp-Sipser on random graphs with a fixed degree sequence. Combinatorics Prob. Comput. 20, 721–741.Google Scholar
Bollobás, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Europ. J. Combinatorics 1, 311–316.CrossRefGoogle Scholar
Darling, R. W. R. and Norris, J. R. (2008). Differential equation approximations for Markov chains. Prob. Surveys 5, 37–79.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. John Wiley, New York.CrossRefGoogle Scholar
Joos, F., Perarnau, G., Rautenbach, D. and Reed, B. (2018). How to determine if a random graph with a fixed degree sequence has a giant component. Prob. Theory Relat. Fields 170, 263–310.CrossRefGoogle Scholar
Kang, W. and Williams, R. J. (2007). An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries. Ann. Appl. Prob. 17, 741–779.CrossRefGoogle Scholar
Karp, R. M. and Sipser, M. (1981). Maximum matchings in sparse random graphs. In Proc. 22nd Annual Symposium on Foundations of Computer Science, IEEE, pp. 364–375.Google Scholar
Pilipenko, A. (2014). An Introduction to Stochastic Differential Equations with Reflection (Lectures Pure Appl. Math. 1). Potsdam University Press.Google Scholar
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion (Fundamental Principles Math. Sci. 293), 3rd edn. Springer, BerlinGoogle Scholar
Stroock, D. W. and Varadhan, S. R. S. (1971). Diffusion processes with boundary conditions. Commun. Pure Appl. Math. 24, 147–225.CrossRefGoogle Scholar
van der Hofstad, R. (2017). Random Graphs and Complex Networks (Camb. Ser. Statist. Probabilistic Math. 43), Vol. 1. Cambridge University Press.Google Scholar
van der Hofstad, R. (2019). Random Graphs and Complex Networks, Vol. 2. In preparation. Available at http://www.win.tue.nl/~rhofstad/NotesRGCN.html.Google Scholar
Wormald, N. C. (1979). Some problems in the enumeration of labelled graphs. Doctoral Thesis, University of Newcastle, Australia.Google Scholar
Wormald, N. C. (1981). The asymptotic connectivity of labelled regular graphs. J. Combinatorial Theory B 31, 156–167.CrossRefGoogle Scholar
Wormald, N. C. (1981). The asymptotic distribution of short cycles in random regular graphs. J. Combinatorial Theory B 31, 168–182.CrossRefGoogle Scholar
Wormald, N. C. (1995). Differential equations for random processes and random graphs. Ann. Appl. Prob. 5, 1217–1235.Google Scholar