Cooperative Connectivity in Wireless Networks: Full Connect. & Percolation

1. S A R r Annual Conference of ITA ACITA 2009 d1 Full Connectivity and Percolation in Large Cooperative Wireless Networks r d2 Çağatay Çapar*, Dennis Goeckel*, Benyuan Liu†, Don Towsley*, and Liaoruo Wang‡ *Univ. of Massachusetts Amherst, †Univ. of Massachusetts Lowell, ‡ Cornell University What is Cooperation? Noncoherent Cooperation Model Sender Cooperation Each node itself can reach a radius r determined by: Receiver Cooperation (Pt : transmit power, a:path-loss exponent, t :power threshold for communication). R A The disconnected three-node network is now fully connected thanks to cooperation. Full Connectivity: Any node can talk to anyone. Percolation: Any given node belongs to an infinite cluster with positive probability. • Basic Idea: • S wants to talk to R, but R is too far away. • A and S cooperate and shout together and reach R. • R wants to talk to S, but S is too far away. • A and S cooperate and listen together and hear R. Power Summing: Two nodes can cooperatively connect to the third if: S We analyze to what extent cooperation can help full connectivity and percolation of large wireless networks. A group of nodes A can connect to B if : A Typical Proof Sketch Extended Networks Negative (α > 2 in 2-D): Positive (α≤ 2 in 2-D): We study cooperative extended networks which means: Network is infinitely large with a fixed finite node density λ. Nodes are distributed according to a Poisson distribution. Nodes can cooperate according to the noncoherent transmission model explained. • Find a very dense • group over a large • area (happens • with probability 1) • Connect that group • Now, use that initial • group to connect all • nodes in a larger • square. • Can show, with • appropriately sized • squares, that this • will continue without • fail. • There exists a • distance d such that, for a node with no neighbors within d, • the node is almost • surely disconnected. • There exists a node • with no neighbors • within d almost • surely somewhere • in the network. Process: Two nodes with radius r cooperate to reach farther than r, pull another node and form a bigger cluster, and so on… d (In both cases, the large size of the network is crucial: either for finding a large dense group to start the collaboration, or finding an isolated node.) Results