1. 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

2. 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

3. 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 ?

4. 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?

5. 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

6. 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

7. 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

8. 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

9. 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.

10. 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

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

12. 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

13. 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

14. 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

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

16. 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