Complex Contagions Models in Opportunistic Mobile Social Networks

1. Yunsheng Wang Dept. of Computer Science, Kettering University Jie Wu Dept. of Computer and Info. Sciences, Temple University Complex Contagions Models in Opportunistic Mobile Social Networks

2. Outline • Opportunistic Mobile Social Networks • Existing Complex Contagions Schemes • Proposed Hierarchical Complex Contagions Schemes • Simulation • Conclusion

3. Opportunistic Mobile Social Networks • New types of Delay Tolerant Networks (DTNs) • Occasionally connected networks • Frequent network partition

4. Information Dissemination • Most of the recent work considers simple contagions in OMSNs • The disease infection or information propagation only requires one “activated” source to infect the opinion-free neighbors • Spreading of beliefs or behaviors is slower, the willingness to participate may require independent affirmation or reinforcement from multiple sources.

5. Existing Complex Contagions Schemes • Ring Lattice-based Complex Contagions (RLCC) • Grid-based Complex Contagions (GCC)

6. Ring Lattice-based Complex Contagions (RLCC) • Watts and Strogatz’s Small World Model • n nodes are placed on a ring lattice, and nodes within ring distance 2 are connected by a local link (or strong tie). The long-range links (or weak ties) are the links randomly rewired in this ring lattice.

7. The Strength of Weak Ties • Efficient in simple contagion • Not efficient in complex contagions, simply due to the lack of multiple, collective contacts. • Fast diffusion of complex contagion requires not only long bridges, but also “wide” ones.

8. Weakness of RLCC • Complex contagions require more rewiring in order to benefit from randomization. • The number of links that need to be randomly rewired increases exponentially with the number required to form a bridge.

9. Grid-based Complex Contagions (GCC) • Kleinberg’s Small World Model • A constant number of random additional long ties will be given to each node with the harmonic distribution.

10. Weakness of GCC • When the contagion is merely minimally complex, it would require a substantially large number of random long-range outgoing edges to even create one single ‘bridge’ to diffuse the contagion. • In the GCC, the additional long-range links may not be significant in helping diffusion of complex contagions.

11. Proposed Hierarchical Complex Contagions Schemes • Tree-based Complex Contagions (TCC) • Clique-based Complex Contagions (CCC) • Hypercube-based Complex Contagions (HCC)

12. Tree-based Complex Contagions (TCC) • Complex contagions is a complex broadcasting process. • In network science, tree-based structure is one of the most intuitive representations to study the broadcast process.

13. TCC • The children nodes under the same parent connect with each other, and the children nodes also connect with the cousin nodes of their parent nodes. • The initial active nodes can be any two linked nodes in this binary tree.

14. Clique-based Complex Contagions (CCC) • In social networks, there is a small amount of individuals, who are more popular than other people. • K-clique: the set of the popular people. • Star structure: other individuals connect to all nodes in this k-clique.

15. Hypercube-based Complex Contagions (HCC) • In the social network, many pairs of nodes have multiple common friends. • The matching pair property of the balanced hypercube. • Balanced hypercube is a load-balanced graph.

16. Properties of HCC • Property 1: In an m-dimensional balanced hypercube, nodes can be partitioned into a set of matching pairs v = (a0, a1, ..., am−1) and v′ = (a0 + 2, a1, ..., am−1). • In order to active all nodes, it only needs one matching pair of nodes to be the initial activated nodes. • Property 2: An m-dimensional balanced hypercube has 22m nodes, each of which has 2m adjacent nodes. • Since, each node has 2m adjacent nodes, HCC is a fault-tolerant approach. HCC also does not have the bottleneck problem.

17. Analysis • Number of links: connectivity of the network • Diameter: communication delay • Node Degree: the number of connections the node has to other nodes • Bisection width: the minimum number of communication links that can be removed to break it into two equally-sized disconnected networks

18. Discussion • TCC reduces the diameter compared with RLCC (increase the spread speed) • CCC has a higher bisection width and lower diameter compared with RLCC and GCC. But, it increases the number of links and node degree dramatically (not suitable for sparse networks) • HCC is a more balanced model, which has a higher bisection width and smaller diameter, while the number of links and node degree are not big (a congestion-free model: fault tolerant and load balancing)

19. Simulation • One message 2-complex contagions • Multiple messages 2-complex contagions • 20%, 40%, 60%, 80%, and 100% nodes activated • 1,024-node network with 1,048,576 contacts

20. Delivery Time

21. Delivery Ratio

22. Conclusion and Future Work • Complex Contagions • RLCC • GCC • TCC • CCC • HCC • Extend to a-complex contagions in directed networks

23. Thank you! Questions ?