On the Critical Total Power for k-Connectivity in Wireless Networks Honghai Zhang, Jennifer C. Hou Department of Computer Science University of Illinois at Urbana-Champaign

Wireless Networks Characteristics of wireless networks Nodes may be mobile Energy is often supplied by batteries Benefit of reducing transmission power Extending network lifetime Increasing network capacity Mitigating MAC-level interference

How to Reduce Transmission Power Two ways of reducing transmission power All nodes use a minimum common power All nodes may choose different power Which one is better? Intuitively, the latter is better How do we define “better”? Total power consumption Motivate the audience How much power can be saved by using power control compared with using common power ?

Problem Statement We investigate the critical total power required to maintain asymptotic k-connectivity on a unit square S=[0,1]2 . We consider the heterogeneous case in which each node can choose its own transmission power. Formulation: Wt,i : critical transmission power node i uses Rt,i: corresponding transmission range of node i Wt,i = Rt,ia. Total power: Problem: Given the assumption that wireless nodes are distributed on S according to a Poisson point process with density l, how does the minimal power Wc scale with l as l  infinity?

Network Model Nodes are randomly distributed as a Poisson point process with density n in a unit-area square A good approximation for uniformly random distribution with n nodes 1 1

Network Model (con’t) Xi : location of node i Ri : transmission range of node i Link (j,i) exists if Rj  | Xi – Xj | : transmission power of node i Total power: k-connectivity: requires removing at least k nodes to disconnect the network Critical total power Wc: minimum total power W for maintaining k-connectivity 1 Xj Rj Ri Xi 1

Previous Studies All nodes choose the common power [Gupta & Kumar 98] studied the critical transmission range rn for 1-connectivity [Wan & Yi 04] for k-connectivity All nodes choose different power [Blough 02] Critical total power for 1-connectivity Our study: critical total power for k-connectivity

Major Results Main theory: Comparison with common power The critical total power for maintaining k-connectivity is with probability approaching 1 Comparison with common power The critical total power for k-connectivity with common power is Allowing power control reduces the total power by a factor of

Sketched Proof of the Main Theory Main theory: the critical total power for maintaining k-connectivity is with high probability We derive a lower bound based on the necessary condition that every node has to be able to reach at least k nearest nodes in order to maintain strong k-connectivity. We derive an upper bound based on an assertion that the network is strongly k-connected if every node can reach k other nodes in each of its four quadrants.

Proof of the Lower Bound Every node has to be able to reach at least k nearest nodes Ri : the distance from node i to its kth nearest neighbor is a lower bound for the critical power Ri r

Proof of the Lower Bound (con’t) Goal We have got Suffices to show Chebyshev Inequality

Proof of the Upper Bound Each node has a coordinate system centered at itself All x-axes and y-axes are in parallel, respectively Lemma 1: The network has k-connectivity if every node can reach at least k nearest neighbors in each of the four quadrants of its own coordinate system If a quadrant contains less than k nodes, it is sufficient to reach all of them. Ri

Proof of Lemma 1 (for k=1) (xi, yi): Node i’s location Proof using contradiction Choose the pair of nodes A, B such that No path from A to B, and A, B have the minimum Goal: find another pair of nodes X, Y such that No path from X to Y, and

Proof of Lemma 1 for k=1 Assume B is in the first quadrant of A’s coordinate system A must be able to reach the nearest neighbor C in the same quadrant If C is not on x-axis or y-axis, then (C,B) is the pair needed for contradiction No path from C to B The case for C on x-axis or y-axis in in the paper! B C A x

Proving Assume B C C’’ 45o A B’ C’ D x Contradiction!

Conclusion The critical total power for k-connectivity is Using power control can save the power by a factor of compared with using common power This can make a difference between bounded value and infinite value