The diameter of a cycle plus a random matching
SIAM Journal on Discrete Mathematics
The small-world phenomenon: an algorithmic perspective
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Maximizing the spread of influence through a social network
Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining
Analyzing Kleinberg's (and other) small-world Models
Proceedings of the twenty-third annual ACM symposium on Principles of distributed computing
Bootstrap Percolation on Infinite Trees and Non-Amenable Groups
Combinatorics, Probability and Computing
The diameter of randomly perturbed digraphs and some applications
Random Structures & Algorithms
On the approximability of influence in social networks
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Convergence to equilibrium in local interaction games
ACM SIGecom Exchanges
Networks, Crowds, and Markets: Reasoning About a Highly Connected World
Networks, Crowds, and Markets: Reasoning About a Highly Connected World
Proceedings of the 20th international conference on World wide web
Which Networks are Least Susceptible to Cascading Failures?
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Influential nodes in a diffusion model for social networks
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Competitive contagion in networks
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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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.