Mobile Geometric Graphs: Detection, Isolation and Percolation Perla - PowerPoint PPT Presentation

Mobile Geometric Graphs: Detection, Isolation and Percolation Perla Sousi 1 Based on joint works with Yuval Peres, Alistair Sinclair, Alexandre Stauffer 1 Statistical Laboratory, University of Cambridge Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Random Geometric Graph (Boolean Model) Nodes: Poisson point process in R d , intensity λ Edges: ∃ ( u , v ) ⇐ ⇒ d ( u , v ) ≤ r ( intersecting balls of radius r / 2) Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Random Geometric Graph (Boolean Model) Nodes: Poisson point process in R d , intensity λ Edges: ∃ ( u , v ) ⇐ ⇒ d ( u , v ) ≤ r ( intersecting balls of radius r / 2) Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Mobile Geometric Graph Nodes move as independent Brownian Motions Obtain stationary sequence of graphs ( G s ) s ≥ 0 Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Mobile Geometric Graph Nodes move as independent Brownian Motions Obtain stationary sequence of graphs ( G s ) s ≥ 0 time 0 Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Mobile Geometric Graph Nodes move as independent Brownian Motions Obtain stationary sequence of graphs ( G s ) s ≥ 0 time 0 time s Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Detection in a mobile geometric graph Target particle initially at origin T det = 1st time some node within distance r of target Want to study P ( T det > t ) P (target not detected at fixed s ) = P ( T det > 0) = e − λπ r 2 time 0 Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Detection in a mobile geometric graph Target particle initially at origin T det = 1st time some node within distance r of target Want to study P ( T det > t ) P (target not detected at fixed s ) = P ( T det > 0) = e − λπ r 2 time 0 time T det Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Detection of a non-mobile target Lemma (Classical result of stochastic geometry) Let ξ be a standard Brownian motion and W r ( t ) = ∪ s ≤ t B ( ξ ( s ) , r ) , the Wiener sausage up to time t. Then P ( T det > t ) = exp( − λ E [ vol ( W r ( t ))]) . Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Detection of a non-mobile target Lemma (Classical result of stochastic geometry) Let ξ be a standard Brownian motion and W r ( t ) = ∪ s ≤ t B ( ξ ( s ) , r ) , the Wiener sausage up to time t. Then P ( T det > t ) = exp( − λ E [ vol ( W r ( t ))]) . Proof. Π = { X i } : Poisson process( λ ) in R d ξ i : Brownian motion of X i Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Detection of a non-mobile target Lemma (Classical result of stochastic geometry) Let ξ be a standard Brownian motion and W r ( t ) = ∪ s ≤ t B ( ξ ( s ) , r ) , the Wiener sausage up to time t. Then P ( T det > t ) = exp( − λ E [ vol ( W r ( t ))]) . Proof. Π = { X i } : Poisson process( λ ) in R d ξ i : Brownian motion of X i Let Φ = { X i : ∃ s ≤ t s.t. X i + ξ i ( s ) ∈ B (0 , r ) } . Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Detection of a non-mobile target Lemma (Classical result of stochastic geometry) Let ξ be a standard Brownian motion and W r ( t ) = ∪ s ≤ t B ( ξ ( s ) , r ) , the Wiener sausage up to time t. Then P ( T det > t ) = exp( − λ E [ vol ( W r ( t ))]) . Proof. Π = { X i } : Poisson process( λ ) in R d ξ i : Brownian motion of X i Let Φ = { X i : ∃ s ≤ t s.t. X i + ξ i ( s ) ∈ B (0 , r ) } . Φ is a thinned Poisson process of intensity Λ( x ) = λ P ( x ∈ ∪ s ≤ t B ( ξ ( s ) , r )) . Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Detection of a non-mobile target Lemma (Classical result of stochastic geometry) Let ξ be a standard Brownian motion and W r ( t ) = ∪ s ≤ t B ( ξ ( s ) , r ) , the Wiener sausage up to time t. Then P ( T det > t ) = exp( − λ E [ vol ( W r ( t ))]) . Proof. Π = { X i } : Poisson process( λ ) in R d ξ i : Brownian motion of X i Let Φ = { X i : ∃ s ≤ t s.t. X i + ξ i ( s ) ∈ B (0 , r ) } . Φ is a thinned Poisson process of intensity Λ( x ) = λ P ( x ∈ ∪ s ≤ t B ( ξ ( s ) , r )) . P ( T det > t ) = Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Detection of a non-mobile target Lemma (Classical result of stochastic geometry) Let ξ be a standard Brownian motion and W r ( t ) = ∪ s ≤ t B ( ξ ( s ) , r ) , the Wiener sausage up to time t. Then P ( T det > t ) = exp( − λ E [ vol ( W r ( t ))]) . Proof. Π = { X i } : Poisson process( λ ) in R d ξ i : Brownian motion of X i Let Φ = { X i : ∃ s ≤ t s.t. X i + ξ i ( s ) ∈ B (0 , r ) } . Φ is a thinned Poisson process of intensity Λ( x ) = λ P ( x ∈ ∪ s ≤ t B ( ξ ( s ) , r )) . P ( T det > t ) = P (Φ( R d ) = 0) = Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Detection of a non-mobile target Lemma (Classical result of stochastic geometry) Let ξ be a standard Brownian motion and W r ( t ) = ∪ s ≤ t B ( ξ ( s ) , r ) , the Wiener sausage up to time t. Then P ( T det > t ) = exp( − λ E [ vol ( W r ( t ))]) . Proof. Π = { X i } : Poisson process( λ ) in R d ξ i : Brownian motion of X i Let Φ = { X i : ∃ s ≤ t s.t. X i + ξ i ( s ) ∈ B (0 , r ) } . Φ is a thinned Poisson process of intensity Λ( x ) = λ P ( x ∈ ∪ s ≤ t B ( ξ ( s ) , r )) . P ( T det > t ) = P (Φ( R d ) = 0) = exp( − Λ( R d )) . Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Detection in a mobile geometric graph t � 2 π log t (1 + o (1)) , d = 2 E [ vol ( W r ( t ))] = c d r d − 2 t (1 + o (1)) , d ≥ 3 Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Detection in a mobile geometric graph t � 2 π log t (1 + o (1)) , d = 2 E [ vol ( W r ( t ))] = c d r d − 2 t (1 + o (1)) , d ≥ 3 T f det : detection time when target motion is f deterministic. Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Detection in a mobile geometric graph t � 2 π log t (1 + o (1)) , d = 2 E [ vol ( W r ( t ))] = c d r d − 2 t (1 + o (1)) , d ≥ 3 T f det : detection time when target motion is f deterministic. Theorem (Peres, Sinclair, S., Stauffer (2010)) Let f be a continuous motion of the target. Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Detection in a mobile geometric graph t � 2 π log t (1 + o (1)) , d = 2 E [ vol ( W r ( t ))] = c d r d − 2 t (1 + o (1)) , d ≥ 3 T f det : detection time when target motion is f deterministic. Theorem (Peres, Sinclair, S., Stauffer (2010)) Let f be a continuous motion of the target. In dimension 1 P ( T f det > t ) ≤ P ( T 0 det > t ) , for all t . Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Detection in a mobile geometric graph t � 2 π log t (1 + o (1)) , d = 2 E [ vol ( W r ( t ))] = c d r d − 2 t (1 + o (1)) , d ≥ 3 T f det : detection time when target motion is f deterministic. Theorem (Peres, Sinclair, S., Stauffer (2010)) Let f be a continuous motion of the target. In dimension 1 P ( T f det > t ) ≤ P ( T 0 det > t ) , for all t . In dimension 2 as t → ∞ � t � P ( T f det > t ) ≤ exp − 2 πλ log t (1 + o (1)) . Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Detection in a mobile geometric graph t � 2 π log t (1 + o (1)) , d = 2 E [ vol ( W r ( t ))] = c d r d − 2 t (1 + o (1)) , d ≥ 3 T f det : detection time when target motion is f deterministic. Theorem (Peres, Sinclair, S., Stauffer (2010)) Let f be a continuous motion of the target. In dimension 1 P ( T f det > t ) ≤ P ( T 0 det > t ) , for all t . In dimension 2 as t → ∞ � t � P ( T f det > t ) ≤ exp − 2 πλ log t (1 + o (1)) . In dimension 3 and above as t → ∞ P ( T f − λα ( d ) c d r d − 2 t (1 + o (1)) � � det > t ) ≤ exp . Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Wiener sausage with drift It turned out that the right way to look at that was through the detection problem. Using this connection we showed Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Wiener sausage with drift It turned out that the right way to look at that was through the detection problem. Using this connection we showed Theorem (Peres, S. (2011)) Let ( ξ ( s )) s be a standard Brownian motion in d dimensions and f a deterministic function, f : R + → R d . Then for all r > 0 and all t we have that E [ vol ( ∪ s ≤ t B ( ξ ( s ) + f ( s ) , r ))] ≥ E [ vol ( ∪ s ≤ t B ( ξ ( s ) , r ))] . Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof Π = { X i } :Poisson process( λ ) in R d X i performs an independent random walk ζ i with transition kernel Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof Π = { X i } :Poisson process( λ ) in R d X i performs an independent random walk ζ i with transition kernel p ( x , y ) = 1 ( � x − y � < 1) . c ( d ) Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation