Normalization Phenomena in Asynchronous Networks
In this work we study a diffusion process in a network that consists of two types of vertices: inhibitory vertices (those obstructing the diffusion) and excitatory vertices (those facilitating the diffusion). We consider a continuous time model in which every edge of the network draws its transmission time randomly. For such an asynchronous diffusion process it has been recently proven that in Erdős-Rényi random graphs a normalization phenomenon arises: whenever the diffusion starts from a large enough (but still tiny) set of active vertices, it only percolates to a certain level that depends only on the activation threshold and the ratio of inhibitory to excitatory vertices. In this paper we extend this result to all networks in which the percolation process exhibits an explosive behaviour. This includes in particular inhomogeneous random networks, as given by Chung-Lu graphs with degree parameter \(\beta \in (2,3)\).
KeywordsRandom Graph Percolation Process Active Neighbor Normalization Phenomenon Explosive Process
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- 3.Balogh, J., Bollobás, B., Duminil-Copin, H., Morris, R.: The sharp threshold for bootstrap percolation in all dimensions. Transactions of the American Mathematical Society (2012)Google Scholar
- 4.Balogh, J., Bollobás, B., Morris, R.: Bootstrap percolation in three dimensions. The Annals of Probability, 1329–1380 (2009)Google Scholar
- 8.Breskin, I., Soriano, J., Moses, E., Tlusty, T.: Percolation in living neural networks. Physical Review Letters (2006)Google Scholar
- 12.Einarsson, H., Lengler, J., Panagiotou, K., Mousset, F., Steger, A.: Bootstrap percolation with inhibition (2014). arXiv preprint arXiv:1410.3291
- 13.Faloutsos, M., Faloutsos, P., and Faloutsos, C. On power-law relationships of the internet topology. In: ACM SIGCOMM Computer Communication Review, vol. 29, pp. 251–262. ACM (1999)Google Scholar
- 14.Grimmett, G., Stirzaker, D.: One Thousand Exercises in Probability. OUP Oxford (2001)Google Scholar
- 16.Kempe, D., Kleinberg, J., Tardos, É.: Maximizing the spread of influence through a social network. In: ACM SIGKDD Knowledge Discovery and Data Mining, pp. 137–146. ACM (2003)Google Scholar
- 21.Ugander, J., Karrer, B., Backstrom, L., Marlow, C.: The anatomy of the facebook social graph (2011). arXiv preprint arXiv:1111.4503