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Graph Models for Sensor‘s Networks A personal and partial view

Graph Models for Sensor‘s Networks A personal and partial view. J. Diaz. Partially supported by the EU within the 6th FP: DELIS. Sensor network. Large networks of simple sensors Usually deployed randomly Very prone to failures Use broadcast paradigms to communicate with other sensors

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Graph Models for Sensor‘s Networks A personal and partial view

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  1. Graph Models for Sensor‘s NetworksA personal and partial view J. Diaz Partially supported by the EU within the 6th FP: DELIS

  2. Sensor network • Large networks of simple sensors • Usually deployed randomly • Very prone to failures • Use broadcast paradigms to communicate with other sensors • Collect information and send it to base station • Must focus on power conservation, instead of QoS. Energy=distance2

  3. Sensor network • Static: once deployed do not change. • Dynamic: could move by themselves or by an external agent.

  4. Sensor node processing storage sensing unit transceiver power unit

  5. Sensor node localization mobility processing storage sensing unit transceiver power unit energy scaravenging

  6. Sensor characterictics: • consume low power • autonomous • operate in high volumetric densities • adaptive to environment • cheap

  7. Transceiver unit • Radio Frequency (RF) • Opticallaser beam(smart dust)

  8. Transceiver unit • Radio Frequency (RF) more expensive and larger. Interference omnidirectional antenna directional antenna • Opticallaser beam(smart dust)

  9. Transceiver unit • Radio Frequency (RF) more expensive and larger. Interference omnidirectional antenna directional antenna • Opticallaser beam(smart dust) need line of sight for communication no interference

  10. Ex. Smart Dust:

  11. Graph Models for static networks • Omnidirectional RF • Directional RF, smart dust

  12. Random Geometric Graphs (RGG) E.N. Gilbert: Random Plane Networks J. Soc. Ind. Appl. Math. 9 (4) 533-543, 1961.

  13. Random Geometric Graphs (RGG) E.N. Gilbert: Random Plane Networks J. Soc. Ind. Appl. Math. 9 (4) 533-543, 1961. … To construct a random plane network, pick points from the plane by a Poisson process with density D points per unit area. Next joint each each pair of points by a line if they are at distance less that r. …

  14. Random Geometric Graphs (RGG) • Scale down to I=[0,1]2

  15. Random Geometric Graphs (RGG) • Scale down to I=[0,1]2 • Springle n points u.a.r. on I (n large).

  16. Random Geometric Graphs (RGG) • Scale down to I=[0,1]2 • Springle n points u.a.r. on I (n large). • Given a communication radius r, two points are connected if they are at distance ≤r.

  17. RGG

  18. RGG r

  19. RGG

  20. RGG

  21. G(n,r) Asymptotic Results: • Threshold: Given G(n,r), r(n) and property Q, wish to find smallest rQ(n) s.t. Q holds w.h.p.

  22. G(n,r) Asymptotic Results: • Threshold: Given G(n,r), r(n) and property Q, wish to find smallest rQ(n) s.t. Q holds w.h.p. • Thm (Goel, Rai, Krishnamachari-04). Any monotone Q of G(n,r), has a threshold.

  23. G(n,r) Asymptotic Results: • Connectivity(Penrose-97, Gupta-Kumar-98): Let then

  24. G(n,r) Asymptotic Results: • Connectivity(Penrose-97, Gupta-Kumar-98): Let then • is a sharp a threshold for connectivity.

  25. G(n,r) Asymptotic Results: • Chromatic number:W.h.p c(G(n,rc))= Q(log n)

  26. G(n,r) Asymptotic Results: • Chromatic number:W.h.p c(G(n,rc))= Q(log n) • Clique number:W.h.p w(G(n,rc))= Q(log n)

  27. G(n,r) Asymptotic Results: • Chromatic number:W.h.p c(G(n,rc))= Q(log n) • Clique number:W.h.p w(G(n,rc))= Q(log n) If r<rc (sparse case) c / w —> 1 in prob. If r≥rc (dense case) c / w —> 1.103 a.s. C. McDiarmird RSA-2003

  28. G(n,r) Asymptotic Results: • Average degree (Penrose-97):At rc the average degree of a node is Q(log n)

  29. G(n,r) Asymptotic Results: • Average degree (Penrose-97):At rc the average degree of a node is Q(log n) I.e. in G(n, rc) each ball contains Q(log n) nodes.

  30. G(n,r) Asymptotic Results: • Cover time (Arvin, Ercal:ICALP-2005): If rct2≥8log n /n then whp G(n,rct) has a cover timeof Q(n log n). If rct2 ≤ log n /pn then whp G(n,rct) has an infinite cover time.

  31. G(n,r) Asymptotic Results: • Cover time (Arvin, Ercal-2005): If rct2≥8log n /n then whp G(n,rct) has a cover timeof Q(n log n). If rct2 ≤ log n /pn then whp G(n,rct) has an infinite cover time. Therefore rct is a threshold for having n log n cover time

  32. Proximity graph G(n,f(n)) • Scale down to I=[0,1]2 • Springle n vertices u.a.r on I • Connect each vertex v with the f(n) nearest neighbors (euclidian distance) A measure of the number of nodes needed to connect a network

  33. ExampleG(n,3)

  34. ExampleG(n,3)

  35. G(n,f(n)) Asymptotic Results: • (Fan-Xue, Kumar-03) Let n= min number of neighbors of any node.If n ≤0.0074 log n, then whp the graph is disconnected. If n≥5.117log n, then whp the graph is strongly connected.

  36. G(n,f(n)) Asymptotic Results: • (Fan-Xue, Kumar-03) Let n= min number of neighbors of any node. If n ≤0.0074 log n, then whp the graph is disconnected. If n≥5.117log n, then whp the graph is stronglyconnected. • Open problem: Any monotone property has a sharp threshold property?

  37. G(n,f(n)) Asymptotic Results: • (Fan-Xue, Kumar-03) Let n= min number of neighbors of any node. If n ≤0.0074 log n, then whp the graph is disconnected. If n≥5.117log n, then whp the graph is stronglyconnected. • Open problem: Any monotone property has a sharp threshold property? • Open problem: Compute the cover time

  38. Random Sector Graphs (RSG) • For unicasting RF or optical (smart dust)

  39. Random Sector Graphs (RSG) • For unicasting RF or optical (smart dust) • Fix angle a. Let Xn={x1,..,xn} i.u.d. points in I, let Bn={b1,..bn} a sequence of i.u.d. angles, let {ri} a sequence in [0,1]. Ga(Xn,Bn,rn) is a random sector graph, where (x,y) is an arc iff y in Sx.

  40. Random Sector Graphs (RSG) • For unicasting RF or optical (smart dust) • Fix angle a. Let Xn={x1,..,xn} i.u.d. points in I, let Bn={b1,..bn} a sequence of i.u.d. angles, let {ri} a sequence in [0,1]. Ga(Xn,Bn,rn) is a random sector graph, where (x,y) is an arc iff y in Sx. D.,Petit,Serna IEEE Trans. MobiComp 2004

  41. Model for RSG Each sensor x covers a sector Sx, defined by r and a (parameters of the system) Sx r a b x

  42. Random Sector Graphs (RSG) • Ga(Xn,Bn,rn)is a digraph

  43. Random Sector Graphs (RSG) • Ga(Xn,Bn,rn)is a digraph • If x5 is not in Sx1, to communicate from x1 to x5:

  44. Random Sector Graphs (RSG) • Connectivity: Sharp threshold at

  45. Random Sector Graphs (RSG) • Connectivity: Sharp threshold at • Undirected chromatic number: Fix rc If a< p then c(G)= Q(ln n/lnln n) whp If a> p then c(G)= Q(ln n) whp DSSS-05

  46. General communication issues Each sensor must transmet information to one or more base stations (BTS). BTS may have to send information to sensors.

  47. RG(S)G as models of sensor‘s network

  48. RG(S)G as models of sensor‘s network Add the following probabilities: • p0 that sensor i is operative • pb that sensor i can communicate with BTS • pc that sensor i can communicate with sensor j, where (i,j) is an edge

  49. RG(S)G as models of sensor‘s network The BTS can be (more frequently): • One or more differenciated static points within I • Mobile agents crossing I • A flying agent over I

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