Overview This paper is about information theoretic (rate-distortion based) clustering of sensor networks. How should information emanate from a sensor network? How should bandwidth/power be allocated to sensor nodes? If the final application is detection/classification based on the received data, how should the above change? Based on looking just at the topology of an optimized network, can we tell something about what the network measures?
( compressed version of ) Information Setup : Central node - Node 0 : Sensor node – Node i (i=1,...,N) Task: The final collector of all information, interested in: 1. Measure random variable 2. Compress (quantize + entropy code) and communicate: : Set of nodes that have sent information to node i Case 1 : Case 2 : Case 1 : Each piece of data Case 2 : The average or sum statistic of data (Re-compression is allowed) (what is the total number of aliens? – we can also handle other linear combinations, several statistics, etc.) (does sensor i think there is an alien around?)
We assume we know the capacity matrix between nodes. (bits) constrains the point to point bandwidth between i,j. Communication happens during well defined time intervals. We assume the capacity matrix remains unchanged over a reasonable duration (>> one time interval). Bandwidth is constrained: Node i can send at most bits to node j inside a time interval. Routing is constrained: Every node i (i=1,...,N) can transmit to at most one other node inside a time interval (fan out = 1). Wireless Network Setup (We can also operate under more general frameworks, under some capacity scaling constraints – please see the routing over a depth- two tree example.)
Routing Setup Routing is over a tree 1 3 4 2 5 0 depth of routing tree (Nodes 1,3 send their r.v.’s to node 4, which combines the received information with its r.v. (case 1 or case 2), and sends everything to node 0....) 6
Problem Statement: Find the Optimal Information Flow Find the jointly optimal compression, detection (case 1, case 2), and routing strategy for the given: i.e., minimize the total distortion at node 0, subject to constraints. : total distortion observed for case 1. : total distortion observed for case 2. We will find optimal solutions for each case and compare them.
Example Deployed nodes Optimal Case 1 routing: Optimal Case 2 routing: ? (>,<,=)
Mini FAQ A: No, we use a good upper bound. Practical (achievable) distortion D for encoding any with variance under rate constraint R<=C: <= D <= Q: Don’t you need to know the distribution of the r.v. before you compress, do rate allocation, etc.? Using this bound, optimal rate allocation can be done using the “reverse water- filling theorem”. Q: If case 2, shouldn’t the sensor network always send the linear combination since A: No. There is a penalty for collecting information within the sensor network due to capacity constraints. The routing problem is combinatorial in the general case. i.e., isn’t the routing problem trivial?
Toy Scenario (given routing) Intra-network bandwidth is sufficient to achieve exponential improvements. (Skipping many details, reverse water filling, dropping of coefficients, etc.) 2 i 1 N 0 …… (i=2,..N), Intra-network bandwidth is the bottleneck. Setup: (a) (b)
Optimal Clustering: Harder Scenario Arbitrary routing tree of depth two, with a fan-in constraint.... cluster 1 cluster L 0 N(1) nodes N(L) nodes Intra-cluster bandwidth for cluster i, (or any function of N(i)) ’s given,. Dynamic Programming ~ How many clusters? N(i)? Which nodes are the cluster heads?
Harder Scenario contd. (W=2.5) ( for W=2.5 and W=5.0, N=40) Range of exponential gains for case 2. (Beyond this range little penalty for case1 optimal routing even if the actual scenario is case 2.)
Optimal Clustering: Hardest Scenario Arbitrary Heuristic, steepest descent algorithm ( Central node is at the center)
Hardest Scenario (contd.) ( Central node is at the center)
Conclusion Optimal clustering of capacity constrained wireless sensor networks. Intra-network bandwidth is very important. Without sufficient intra-network bandwidth, no gains for sending statistics instead of the individual data in case 2. We can solve a dual problem of network lifetime maximization under the constant fidelity. We can comply with “scaling laws” and find optimal clusters. Based on looking just at the topology of an optimized network, can we tell something about what the network does? (image from http://www.sruweb.com/~walsh/neuron.jpg)