Connectivity of random irrigation networks Nicolas Broutin, Luc Devroye, abor Lugosi September 4, 2012 Nicolas Fraiman, and G ´
X n be i.i.d. uniform on 0 1 d and graph such that the vertices are X 1 X j iff X i r . bluetooth networks” (2004). Petrioli ”A new approach to device discovery and scatternet formation in The model was introduced by Ferraguto, Mambrini, Panconesi, and They are locally sparsified random geometric graphs. Such graphs are also called bluetooth graphs. Irrigation graph X j Start with a connected graph on n vertices. X i We consider the case when the underlying graph is a random geometric neighbors at random (without replacement). An irrigation subgraph is obtained when that each vertex selects c Main question: How large does c need to be for G n r c to be connected?
X n be i.i.d. uniform on 0 1 d and graph such that the vertices are X 1 X j iff X i r . bluetooth networks” (2004). Petrioli ”A new approach to device discovery and scatternet formation in The model was introduced by Ferraguto, Mambrini, Panconesi, and They are locally sparsified random geometric graphs. Such graphs are also called bluetooth graphs. Irrigation graph X j Start with a connected graph on n vertices. X i We consider the case when the underlying graph is a random geometric neighbors at random (without replacement). Main question: How large does c need to be for G n r c to be connected? An irrigation subgraph is obtained when that each vertex selects c
Irrigation graph Start with a connected graph on n vertices. neighbors at random (without replacement). Such graphs are also called bluetooth graphs. They are locally sparsified random geometric graphs. The model was introduced by Ferraguto, Mambrini, Panconesi, and Petrioli ”A new approach to device discovery and scatternet formation in bluetooth networks” (2004). Main question: How large does c need to be for G n r c to be connected? An irrigation subgraph is obtained when that each vertex selects c We consider the case when the underlying graph is a random geometric graph such that the vertices are X 1 ,..., X n be i.i.d. uniform on [ 0 , 1 ] d and X i ∼ X j iff � X i − X j � < r .
Irrigation graph Start with a connected graph on n vertices. neighbors at random (without replacement). They are locally sparsified random geometric graphs. The model was introduced by Ferraguto, Mambrini, Panconesi, and Petrioli ”A new approach to device discovery and scatternet formation in bluetooth networks” (2004). Main question: How large does c need to be for G n r c to be connected? An irrigation subgraph is obtained when that each vertex selects c We consider the case when the underlying graph is a random geometric graph such that the vertices are X 1 ,..., X n be i.i.d. uniform on [ 0 , 1 ] d and X i ∼ X j iff � X i − X j � < r . Such graphs are also called bluetooth graphs.
Irrigation graph Start with a connected graph on n vertices. neighbors at random (without replacement). They are locally sparsified random geometric graphs. The model was introduced by Ferraguto, Mambrini, Panconesi, and Petrioli ”A new approach to device discovery and scatternet formation in bluetooth networks” (2004). Main question: How large does c need to be for G n r c to be connected? An irrigation subgraph is obtained when that each vertex selects c We consider the case when the underlying graph is a random geometric graph such that the vertices are X 1 ,..., X n be i.i.d. uniform on [ 0 , 1 ] d and X i ∼ X j iff � X i − X j � < r . Such graphs are also called bluetooth graphs.
Irrigation graph Start with a connected graph on n vertices. neighbors at random (without replacement). They are locally sparsified random geometric graphs. The model was introduced by Ferraguto, Mambrini, Panconesi, and Petrioli ”A new approach to device discovery and scatternet formation in bluetooth networks” (2004). An irrigation subgraph is obtained when that each vertex selects c We consider the case when the underlying graph is a random geometric graph such that the vertices are X 1 ,..., X n be i.i.d. uniform on [ 0 , 1 ] d and X i ∼ X j iff � X i − X j � < r . Such graphs are also called bluetooth graphs. Main question: How large does c need to be for G ( n , r , c ) to be connected?
r t where Irrigation graph n 1 d 2 d Vol B 0 1 2 d and 1 d d log n r t 1 r 0, G n r is connected whp if Penrose (1997) showed that We only consider values of r above this level. • G ( n , r , c ) is a subgraph of a random geometric graph G ( n , r ) , so we need G ( n , r ) to be connected.
Irrigation graph n and 2 We only consider values of r above this level. • G ( n , r , c ) is a subgraph of a random geometric graph G ( n , r ) , so we need G ( n , r ) to be connected. • Penrose (1997) showed that ∀ ε > 0, G ( n , r ) is connected whp if r ≥ ( 1 + ε ) r t where ) 1 / d ( log n r t = θ d θ d = ( 2 d Vol B ( 0 , 1 )) 1 / d .
Irrigation graph
Irrigation graph
Irrigation graph
Previous results Theorem ( Fenner and Frieze, 1982) Theorem ( Dubhashi, Johansson, Haggstrom, Panconesi, Sozio, 2007) For constant r the graph G n r 2 is connected whp. Theorem ( Crescenzi, Nocentini, Pietracaprina, Pucci, 2009) In dimension d 2 , such that if r log n n and c log 1 r then G n r c is connected whp. For r = ∞ , the graph G ( n , r , 2 ) (the random 2 -out graph) is connected whp.
Previous results such that if log 1 r c and n log n r 2 , Theorem ( Fenner and Frieze, 1982) In dimension d Theorem ( Crescenzi, Nocentini, Pietracaprina, Pucci, 2009) om, Panconesi, Sozio, 2007) then G n r c is connected whp. For r = ∞ , the graph G ( n , r , 2 ) (the random 2 -out graph) is connected whp. Theorem ( Dubhashi, Johansson, H ¨ aggstr ¨ For constant r the graph G ( n , r , 2 ) is connected whp.
Previous results Theorem ( Fenner and Frieze, 1982) om, Panconesi, Sozio, 2007) Theorem ( Crescenzi, Nocentini, Pietracaprina, Pucci, 2009) log n n and For r = ∞ , the graph G ( n , r , 2 ) (the random 2 -out graph) is connected whp. Theorem ( Dubhashi, Johansson, H ¨ aggstr ¨ For constant r the graph G ( n , r , 2 ) is connected whp. In dimension d = 2 , ∃ α , β such that if √ r ≥ α c ≥ β log ( 1 / r ) , then G ( n , r , c ) is connected whp.
c t then G n r c is connected whp. c t then G n r c is disconnected whp. c t does not depend on Main result then 1 if c 1 if c 2log n Theorem and n or d . There exists a constant γ ∗ > 0 such that for all γ ≥ γ ∗ and ε ∈ ( 0 , 1 ) , if √ ) 1 / d ( log n r ∼ γ c t = loglog n ,
c t does not depend on Main result and then 2log n Theorem or d . n There exists a constant γ ∗ > 0 such that for all γ ≥ γ ∗ and ε ∈ ( 0 , 1 ) , if √ ) 1 / d ( log n r ∼ γ c t = loglog n , • if c ≥ ( 1 + ε ) c t then G ( n , r , c ) is connected whp. • if c ≤ ( 1 − ε ) c t then G ( n , r , c ) is disconnected whp.
Main result Theorem n and 2log n then There exists a constant γ ∗ > 0 such that for all γ ≥ γ ∗ and ε ∈ ( 0 , 1 ) , if √ ) 1 / d ( log n r ∼ γ c t = loglog n , • if c ≥ ( 1 + ε ) c t then G ( n , r , c ) is connected whp. • if c ≤ ( 1 − ε ) c t then G ( n , r , c ) is disconnected whp. c t does not depend on γ or d .
Below the threshold Theorem n The smallest possible components are cliques of size c 1. ) 1 / d and ( log n Let γ ≥ γ ∗ and ε ∈ ( 0 , 1 ) . If r = γ c ≤ ( 1 − ε ) c t then G ( n , r , c ) is disconnected whp.
Below the threshold Theorem n components are cliques of ) 1 / d and ( log n Let γ ≥ γ ∗ and ε ∈ ( 0 , 1 ) . If r = γ c ≤ ( 1 − ε ) c t then G ( n , r , c ) is disconnected whp. • The smallest possible size c + 1.
Below the threshold Theorem n components are cliques of ) 1 / d and ( log n Let γ ≥ γ ∗ and ε ∈ ( 0 , 1 ) . If r = γ c ≤ ( 1 − ε ) c t then G ( n , r , c ) is disconnected whp. • The smallest possible size c + 1.
Isolated (c+1)-cliques X i I Q is the number of isolated c Q Then N Let I Q be the indicator of the event that Q is an isolated clique. Q i j r X j 1 c Q n 1 Q n given by be the random family of subsets of 1 Let 1 -cliques. • We show that there exists an isolated ( c + 1 ) -clique whp.
Let I Q be the indicator of the event that Q is an isolated clique. Isolated (c+1)-cliques Then N Q I Q is the number of isolated c 1 -cliques. • We show that there exists an isolated ( c + 1 ) -clique whp. • Let F be the random family of subsets of { 1 ,..., n } given by } : | Q | = c + 1 , � X i − X j � < r ∀ i , j ∈ Q { } { 1 ,..., n F = Q ⊂ .
Isolated (c+1)-cliques • We show that there exists an isolated ( c + 1 ) -clique whp. • Let F be the random family of subsets of { 1 ,..., n } given by } : | Q | = c + 1 , � X i − X j � < r ∀ i , j ∈ Q { } { 1 ,..., n F = Q ⊂ . • Let I ( Q ) be the indicator of the event that Q is an isolated clique. Then N = ∑ Q ∈ F I ( Q ) is the number of isolated ( c + 1 ) -cliques.
Isolated (c+1)-cliques Let D be the event described above. We use the second-moment method and prove that P N 1 D 0 E N 1 D 2 E N 2 1 D 1 • We need some regularity on the uniformly distributed points. For every 1 ≤ j ≤ n α nr 2 < # { i : X i ∈ B ( X j , r ) } < β nr 2 .
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