Golnaz Ghasemiesfeh
Jie Gao
ABSTRACT
Diseases, information and rumors could spread fast in social networks exhibiting the small world property. In the diffusion of these 'simple contagions', which can spread through a single contact, a small network diameter and the existence of weak ties in the network play important roles. Recent studies by sociologists [Centola and Macy 2007] have also explored 'complex contagions' in which multiple contacts are required for the spread of contagion. [Centola and Macy 2007] and [Romero et al. 2011] have shown that complex contagions exhibit different diffusion patterns than simple ones. In this paper, we study three small world models and provide rigorous analysis on the diffusion speed of a k-complex contagion, in which a node becomes active only when at least k of its neighbors are active. Diffusion of a complex contagion starts from a constant number of initial active nodes. We provide upper and lower bounds on the number of rounds it takes for the entire network to be activated. Our results show that compared to simple contagions, weak ties are not as effective in spreading complex contagions due to the lack of simultaneous active contacts; and the diffusion speed depends heavily on the the way weak ties are distributed in a network.
We show that in Newman-Watts model with Ө(n) random edges added on top of a ring structure, the diffusion speed of a 2-complex contagion is Ω(3√n and O(5√n4logn) with high probability. In Kleinberg's small world model (in which Ө(n) random edges are added with a spatial distribution inversely proportional to the grid distance to the power of 2, on top of a 2-dimensional grid structure), the diffusion speed of a 2-complex contagion is O(log3.5n) and Ω (log n/log logn) with high probability. We also show a similar result for Kleinberg's hierarchical network model, in which random edges are added with a spatial distribution on their distance in a tree hierarchy. In this model the diffusion is fast with high probability bounded by O(logn) and Ω(logn/log logn), when the number of random edges issued by each node is Ө(log2n). We also generalize these results to k-complex contagions.
- Adler, J. 1991. Bootstrap percolation. Physica A: Statistical and Theoretical Physics 171, 3, 453--470.Google Scholar
Cross Ref
- Balogh, J., Peres, Y., and Pete, G. 2006. Bootstrap percolation on infinite trees and non-amenable groups. Combinatorics, Probability and Computing 15, 5, 715--730. Google ScholarDigital Library
- Balogh, J. and Pittel, B. 2007. Bootstrap percolation on the random regular graph. Random Struct. Algorithms 30, 1-2, 257--286.Google ScholarCross Ref
- Baxter, G. J., Dorogovtsev, S. N., Goltsev, A. V., and Mendes, J. F. F. 2010. Bootstrap percolation on complex networks. Phys. Rev. E 82, 011103.Google ScholarCross Ref
- Blume, L., Easley, D., Kleinberg, J., Kleinberg, R., and Tardos, E. 2011. Which networks are least susceptible to cascading failures? In Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science. FOCS '11. IEEE Computer Society, Washington, DC, USA, 393--402. Google ScholarDigital Library
- Bollobás, B. and Chung, F. R. K. 1988. The diameter of a cycle plus a random matching. SIAM J. Discret. Math. 1, 3, 328--333. Google ScholarDigital Library
- Bradonjic, M. and Saniee, I. 2012. Bootstrap percolation on random geometric graphs. arXiv:1201.2953v3.Google Scholar
- Centola, D. and Macy, M. 2007. Complex Contagions and the Weakness of Long Ties. American Journal of Sociology 113, 3, 702--734.Google ScholarCross Ref
- Chalupa, J., Leath, P. L., and Reich, G. R. 1979. Bootstrap percolation on a bethe lattice. Journal of Physics C: Solid State Physics 12, 1, L31.Google ScholarCross Ref
- Chen, N. 2008. On the approximability of influence in social networks. In Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms. SODA '08. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1029--1037. Google ScholarDigital Library
- Coleman, J. S., Katz, E., and Menzel, H. 1966. Medical Innovation: A Diffusion Study. Bobbs-Merrill Co.Google Scholar
- D.Watts and S.Strogatz. 1998. Collective dynamics of 'small-world' networks. Nature 393, 6684, 409--410.Google Scholar
- Easley, D. and Kleinberg, J. 2010. Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, New York, NY, USA. Google ScholarDigital Library
- Ellison, G. 1993. Learning, local interaction, and coordination. Econometrica 61, 5, 1047--1071.Google ScholarCross Ref
- Flaxman, A. D. and Frieze, A. M. 2007. The diameter of randomly perturbed digraphs and some applications. Random Struct. Algorithms 30, 4, 484--504. Google ScholarDigital Library
- Goyal, S. and Kearns, M. 2012. Competitive contagion in networks. In Proceedings of the 44th symposium on Theory of Computing. STOC '12. ACM, New York, NY, USA, 759--774. Google ScholarDigital Library
- Granovetter, M. 1973. The Strength of Weak Ties. The American Journal of Sociology 78, 6, 1360--1380.Google ScholarCross Ref
- Granovetter, M. 1978. Threshold models of collective behavior. The American Journal of Sociology 83, 6, 1420--1443.Google ScholarCross Ref
- Granovetter, M. S. 1974. Getting a Job: A Study of Contacts and Careers. Harvard University Press.Google Scholar
- Holroyd, A. E. 2003. Sharp metastability threshold for two-dimensional bootstrap percolation. Probability Theory and Related Fields 125, 195--224. 10.1007/s00440-002-0239-x.Google ScholarCross Ref
- Kempe, D., Kleinberg, J., and Tardos, É. 2003. Maximizing the spread of influence through a social network. In Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining. KDD '03. ACM, New York, NY, USA, 137--146. Google ScholarDigital Library
- Kempe, D., Kleinberg, J., and Tardos, É. 2005. Influential nodes in a diffusion model for social networks. In Proceedings of the 32nd international conference on Automata, Languages and Programming. ICALP'05. Springer-Verlag, Berlin, Heidelberg, 1127--1138. Google ScholarDigital Library
- Kleinberg, J. 2000. The small-world phenomenon: an algorithm perspective. In STOC '00: Proceedings of the thirty-second annual ACM symposium on Theory of computing. 163--170. Google ScholarDigital Library
- Kleinberg, J. M. 2001. Small-world phenomena and the dynamics of information. In NIPS. 431--438.Google Scholar
- Macdonald, J. S. and Macdonald, L. D. 1964. Chain migration, ethnic neighborhood formation and social networks. The Milbank Memorial Fund Quarterly 42, 1, 82--97.Google ScholarCross Ref
- Martel, C. and Nguyen, V. 2004. Analyzing kleinberg's (and other) small-world models. In Proceedings of the twenty-third annual ACM symposium on Principles of distributed computing. PODC '04. ACM, New York, NY, USA, 179--188. Google ScholarDigital Library
- Milgram, S. 1967. The small world problem. Psychology Today 1.Google Scholar
- Montanari, A. and Saberi, A. 2009. Convergence to equilibrium in local interaction games. SIGecom Exch. 8, 1, 11:1--11:4. Google ScholarDigital Library
- Newman, M. E. J. and Watts, D. J. 1999. Scaling and percolation in the small-world network model. Physical Review E 60, 6, 7332--7342.Google ScholarCross Ref
- Rapoport, A. 1953. Spread of information through a population with socio-structural bias: I. Assumption of transitivity. Bulletin of Mathematical Biology 15, 4, 523--533.Google ScholarCross Ref
- Romero, D. M., Meeder, B., and Kleinberg, J. 2011. Differences in the mechanics of information diffusion across topics: idioms, political hashtags, and complex contagion on twitter. In Proceedings of the 20th international conference on World wide web. WWW'11. ACM, New York, NY, USA, 695--704. Google ScholarDigital Library
- van Enter, A. C. D. 1987. Proof of straley's argument for bootstrap percolation. Journal of Statistical Physics 48, 943--945. 10.1007/BF01019705.Google ScholarCross Ref
- Young, H. P. 2000. The diffusion of innovations in social networks. Economics Working Paper Archive 437, The Johns Hopkins University,Department of Economics. May.Google Scholar
Index Terms
-
Complex contagion and the weakness of long ties in social networks: revisited
-
Recommendations
-
Complex contagion and the weakness of long ties in social networks: revisited
EC '13: Proceedings of the fourteenth ACM conference on Electronic commerceDiseases, information and rumors could spread fast in social networks exhibiting the small world property. In the diffusion of these 'simple contagions', which can spread through a single contact, a small network diameter and the existence of weak ties ...
-
Complex Contagions in Kleinberg's Small World Model
ITCS '15: Proceedings of the 2015 Conference on Innovations in Theoretical Computer ScienceComplex contagions describe diffusion of behaviors in a social network in settings where spreading requires influence by two or more neighbors. In a k-complex contagion, a cluster of nodes are initially infected, and additional nodes become infected in ...
-
Modeling Twitter as weighted complex networks using retweets
ASONAM '16: Proceedings of the 2016 IEEE/ACM International Conference on Advances in Social Networks Analysis and MiningSocial Networks are evolving rapidly to have a significant number of users using or joining their platforms daily, and a huge volume of information exchange and flow continuously. As human is the center of all information traffic, the information ...
Comments