Critical Density Thresholds and Complexity in Wireless Networks Bhaskar Krishnamachari School of Electrical and Computer Engineering Cornell University Department of Electrical Engineering University of Southern California (Fall 2002)

Large Scale Wireless Networks: The Vision  The “many - tiny” principle: wireless networks of thousands of inexpensive miniature devices capable of computation, communication and sensing  For smart spaces, environmental monitoring... Berkeley Mote From Pister et al., Berkeley Smart Dust Project

Tomorrow’s Embedded Wireless Systems 2 From Manges et al., Oak Ridge National Laboratory, Instrumentation and Controls Division ORNL Telesensor Chip 2 Berkeley Dust Mote 1 1 From Pister et al., Berkeley Smart Dust Project

Challenges  Unattended Operation: adaptive, self-configuration mechanisms  Severe energy constraints : no battery replacement  Large Scale : possibly thousands of nodes  Are there qualitatively new phenomena at this scale? Berkeley Mote From Pister et al., Berkeley Smart Dust Project

Complex Systems  Ordered global behavior and structure “emerging” from multiple local interactions.

From the Oxford Cryodetector Group Phase Transitions  Emergent structure - abrupt change in a global system property at a critical level of local interactions. Example: Superconductivity

Connectivity in a Multi-hop Network © 2002 Bhaskar Krishnamachari All nodes increasing transmission range R simultaneously

© 2002 Bhaskar Krishnamachari Phase Transition for Connectivity Communication radius RProbability (Network is Connected) Communication radius R Probability (Network is Connected) undesirable regime desirable regime Energy-efficient operating point n = 20

© 2002 Bhaskar Krishnamachari Phase Transition for Connectivity Communication radius R Probability (Network is Connected)

© 2002 Bhaskar Krishnamachari Phase Transition for Connectivity x1x1x1x1 x2x2x2x2 x3x3x3x3 x4x4x4x4R  2D model of Gupta & Kumar (1998) shows log n density threshold; involves continuum percolation theory.  Simpler 1D Model. Consider Poisson arrivals with rate. Connectivity between nodes within range R.  Probability that first n nodes form a connected network:  This model also shows an analogous phase transition with an O(log n) density threshold function.

© 2002 Bhaskar Krishnamachari Phase Transition for Bi-Connectivity  2 node-disjoint paths between all pairs of nodes. n = 100 Communication radius R Probability (Network is Connected)

© 2002 Bhaskar Krishnamachari Bernoulli Random Graphs G(n,p)  Studied by mathematicians since 50’s (Erdos, Renyi, Bollobas, Spencer, others)

© 2002 Bhaskar Krishnamachari Phase Transitions in Random Graphs  A number of zero-one laws developed over the years. E.g. (Fagin 1976): for all first order graph properties A,  (Friedgut 1996): ALL monotone graph properties undergo sharp phase transitions.  Examples:  k-Connectivity  k-Colorability  Hamiltonian cycle

© 2002 Bhaskar Krishnamachari Fixed Radius Random Graphs G(n,R)  A model for multi-hop wireless networks  Density parameter: communication radius R

© 2002 Bhaskar Krishnamachari Phase Transition for Hamiltonian Cycle Formation  Does there exist a cycle in the network graph that visits each node exactly once? Communication radius R Probability (Hamiltonian cycle exists) n = 100

© 2002 Bhaskar Krishnamachari A Wireless Sensor Tracking Problem  Given: Multiple sensors and targets  Sensors can only communicate locally  Sensors can only “see” targets locally  Need 3 communicating sensors tracking each target sensor target

© 2002 Bhaskar Krishnamachari Communication between sensors within range R sensor target R A Wireless Sensor Tracking Problem

© 2002 Bhaskar Krishnamachari Communication graph sensor target A Wireless Sensor Tracking Problem

© 2002 Bhaskar Krishnamachari sensor target S A Wireless Sensor Tracking Problem Targets visible within range S

© 2002 Bhaskar Krishnamachari Visibility graph sensor target A Wireless Sensor Tracking Problem

© 2002 Bhaskar Krishnamachari Decision problem: can all targets can be tracked by three communicating sensors ? sensor target A Wireless Sensor Tracking Problem

© 2002 Bhaskar Krishnamachari Phase Transition for Sensor Tracking Problem Probability of Tracking all Targets Sensing rangeCommunication range s = 17 t = 5 %

© 2002 Bhaskar Krishnamachari Broadcast Scheduling/Channel Allocation  No nodes within 2 hops of each other can be allocated the same channel.  Are k channels enough ? Less likely with more edges in network graph, i.e. with higher transmission range.  NP-Hard in general, polynomial-time for 1D model

© 2002 Bhaskar Krishnamachari Broadcast Scheduling/Channel Allocation Communication/Interference range R Probability (k channels suffice) n = 100 1D model

© 2002 Bhaskar Krishnamachari Broadcast Scheduling/Channel Allocation  If k channels are used, and W is the aggregate available bandwidth/data-rate, the maximum network- wide throughput is T = n(W/k)  Let  1 = max(# of 1-hop neighbors),  2 = max(# of 2- hop neighbors), then:

© 2002 Bhaskar Krishnamachari Broadcast Scheduling/Channel Allocation n = 100 1D model 99% connectivity threshold Operating point w/ maximum throughput

© 2002 Bhaskar Krishnamachari Broadcast Scheduling/Channel Allocation Communication/Interference range R Probability (k channels suffice) n = 30 2D model 99% connectivity threshold 29

Communication/Interference radius R © 2002 Bhaskar Krishnamachari Broadcast Scheduling/Channel Allocation n = 30 2D model 99% connectivity threshold Operating point w/ maximum throughput Expected normalized network throughput T/W

Communication/Interference radius R © 2002 Bhaskar Krishnamachari Broadcast Scheduling/Channel Allocation n = 100 2D model 99% connectivity threshold Operating point w/ maximum throughput Expected normalized network throughput T/W

 Min-k-neighbor property: all nodes have at least k neighbors.  Let A i = event that node i has at least k neighbors, then where (for R  0.5, ignoring edge effects) © 2002 Bhaskar Krishnamachari Min-k-Neighbor

 The min-k-neighbor property undergoes a zero-one phase transition asymptotically at transmission radius  For finite n, the probability that all neighbors have at least k neighbors is > 1 - , if © 2002 Bhaskar Krishnamachari Critical Threshold for Min-k-Neighbor

© 2002 Bhaskar Krishnamachari Critical Threshold for Min-k-neighbor n = 100  Probability that all nodes have at least 2 neighbors Communication radius R

© 2002 Bhaskar Krishnamachari Probabilistic Flooding Probability( all nodes receive packet) Query forwarding probability q  Each node forwards packet with probability q n = 100 Resource-efficient operating point

© 2002 Bhaskar Krishnamachari Phase Transitions and Computational Complexity: Constraint Satisfaction  In the early 90’s, AI researchers found phase transitions in NP-complete constraint satisfaction problems like SAT (Cheeseman et al. 1991, Selman et al. 1992, Hogg et al. 1994)  SAT: given a binary logic formula in CNF (conjunction of OR clauses), is there a truth assignment which makes the formula true? ( a  b  c )  ( d   e  f )  (  a  c   e )

© 2002 Bhaskar Krishnamachari Phase Transition in 3-SAT  Selman et al. (1994):

© 2002 Bhaskar Krishnamachari Phase Transition in 3-SAT  (Friedgut 1999):  constant c k s.t. , formulas with at most (1-  )c k n clauses are satisfiable w.h.p. and formulas with at least (1+  )c k n are unsatisfiable w.h.p.  Critical ratio of clauses to variables for 3-SAT: empirically ~ 4.24, theoretically between and 4.601

© 2002 Bhaskar Krishnamachari Worst-case vs. Average Complexity  3-SAT remains NP-complete even for random instances with ratios between 1/3 and 7(n 2 -3n+2).  (Frieze et al. 1996): Polynomial heuristic finds satisfying solutions w.h.p. if ratio less than  (Bearne et al. 1998): Can prove unsatisfiability in polynomial time w.h.p. if ratio more than n/log(n).  (Williams and Hogg, 1994): First-order, algorithm- independent analysis of CSPs showing that average computational effort peaks at phase transition point.

© 2002 Bhaskar Krishnamachari Distributed Constraint Satisfaction in Wireless Networks  Many NP-complete problems in wireless networks can be formulated as distributed constraint satisfaction problems (DCSP).  A DCSP consists of agents, each with a set of variables they need to set values to. There are intra- agent constraints and inter-agent constraints on these values.

© 2002 Bhaskar Krishnamachari Hamiltonian Cycle Formation  Does there exist a cycle in the network graph that visits each node exactly once?

© 2002 Bhaskar Krishnamachari Hamiltonian Cycle Formation Communication radius R n = 100

© 2002 Bhaskar Krishnamachari Hamiltonian Cycle Formation Communication radius R n = 100

© 2002 Bhaskar Krishnamachari Conflict Free Resource Allocation  Allocating channels with 1-hop constraint. 3 channels

© 2002 Bhaskar Krishnamachari Conflict Free Resource Allocation 3 channels

© 2002 Bhaskar Krishnamachari Conflict Free Resource Allocation Interference Range More bandwidth

© 2002 Bhaskar Krishnamachari Decision problem: can all targets can be tracked by three distinct communicating sensors ? sensor target Sensor Tracking Problem

© 2002 Bhaskar Krishnamachari Sensor Tracking Problem  Decision Problem: Can each target be tracked by a set of three distinct communicating sensors?  NP-complete for arbitrary communication and visibility graphs, polynomial solvable when the communication graph is complete.  Can be formulated as a DCSP:

© 2002 Bhaskar Krishnamachari Sensor Tracking Problem Mean computational costProbability of Tracking Sensing range Communication range Sensing range Communication range

© 2002 Bhaskar Krishnamachari Sensor Tracking Problem Mean communication costProbability of Tracking Sensing range Communication range Sensing range

© 2002 Bhaskar Krishnamachari Conclusions  Phase transition analysis appears to be a promising approach for determining resource-efficient operating points for various global network properties.  Also helpful in reducing the average computational cost of distributed algorithms for constraint satisfaction in wireless networks.  It may be possible to incorporate this approach into online, self-configuring mechanisms.

Acknowledgements  Professor Stephen Wicker (Cornell ECE)  Professor Bart Selman (Cornell CS)  Dr. Ramon Bejar (Cornell CS)  Dr. Carla Gomes (Cornell CS)  Marc Pearlman (Cornell ECE) © 2002 Bhaskar Krishnamachari

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