Bootstrap Percolation on Periodic Trees Milan Bradonjić work with Iraj Saniee Bell Labs, Alcatel-Lucent, Murray Hill, NJ 31 May ’13 AofA 2013, Spain 1 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Bootstrap Percolation Structure : A dynamic process defined on an underlying (deterministic or random) graph. Every node is in one of two states 0 or 1 ( inactive or active ). 2 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Bootstrap Percolation Dynamics : Random initial configuration at t = 0: Every node becomes active with prob. p i.i.d. Deterministic process for t ≥ 1: An inactive node becomes and stays active at time t if the number of its active neighbors at time t − 1 is ≥ θ , for given threshold θ . 3 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Applications Physics and Chemistry Some crystals modeled as lattices whose sites represent atoms with a certain spin. Ferromagnetism when a sufficient number of neighboring sites of each atom has the same spin. Advertising or Rumor Spreading An individual becomes influenced when a sufficient number of its close friends have already been influenced. 4 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Question An interesting phenomenon to study is metastability. Do there exist 0 < p ′ ≤ p ′′ < 1 such that: � ∀ p < p ′ � lim t →∞ P p ( V becomes fully active ) = 0 , and � ∀ p > p ′′ � lim t →∞ P p ( V becomes fully active ) = 1 ? Notice: We do not know in general if p ′ = p ′′ . 5 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Question An interesting phenomenon to study is metastability. Do there exist 0 < p ′ ≤ p ′′ < 1 such that: � ∀ p < p ′ � lim t →∞ P p ( V becomes fully active ) = 0 , and � ∀ p > p ′′ � lim t →∞ P p ( V becomes fully active ) = 1 ? Notice: We do not know in general if p ′ = p ′′ . 5 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Literature on Bootstrap Percolation RRG A similar model for adoption of new communication services on a random regular graph. Implicit critical thresholds for widespread adoption, Gersho and Mitra ’75. Reg.Tree Chalupa et all ’79, first formally to introduce bootstrap percolation, and derive the critical threshold on regular trees (Bethe lattices). Tree Critical threshold for non-regular trees, Balogh et all ’06. Z d Aizenman and Lebowitz ’88 studied metastability of bootstrap percolation on Z d . 6 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Literature on Bootstrap Percolation Z 2 The existence of a sharp threshold for bootstrap percolation in Z 2 , Holroyd ’03. Z d Recently generalized to d -dimensional lattices Z d , Balogh et all ’11. G n , ( d i ) Bootstrap percolation on random graphs with a given degree sequence, Amini ’10. G n , p Bootstrap percolation on random graphs G ( n , p ) by Luczak et all ’12+. Soc.Net. Formation of opinion in social networks. The percolation threshold is a fraction of the size of each neighborhood, Watts ’02. 7 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Motivation 1 The threshold for BP on a given graph is upper bounded by the threshold for BP on its spanning tree. 2 Bootstrap percolation has two thresholds on regular (homogeneous) trees Fontes and Schonmann ’08. Figure: Approximation for BP. 8 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Bootstrap Percolation on Periodic Trees 9 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Periodic Trees Definition Let ℓ, m 0 , m 1 , . . . , m ℓ − 1 ∈ N . A periodic tree T m 0 , m 1 ,..., m ℓ − 1 of periodicity ℓ is recursively defined as follows. Consider a node ∅ , called root. The nodes at the distance k mod ℓ from the root ∅ have degree m k + 1 for k = 0 , 1 , 2 , . . . . 10 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Periodic Trees The schematic presentation of T 3 , 2 . ❜ ❜ ❜ ❜ ❅� � ❅ � ❅ ❅ � r r � ❅ � ❅ � ❅ ❅ � ❜ ❜ ❅ � ❜ ❜ ❜ ❜ ❅� � ❅ � ❅ ❅ � ❜ r r � ❅ � ❅ � ❅ ❅ � ❜ ❜ ❅ � ❜ r ❜ � ❅ � ❅ ❜ ❜ ❅ � ❅ � ❅ � r r � ❅ �❅ � � ❅ ❅ ❜ � ❅ ❜ ❜ ❜ ❜ � ❅ ❜ ❜ ❅ � ❅ � ❅ � r r � ❅ � ❅ � ❅ � ❅ ❜ ❜ ❜ ❜ 11 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Periodic Oriented Trees Definition Let ℓ, m 0 , m 1 , . . . , m ℓ − 1 ∈ N . A periodic oriented tree � T m 0 , m 1 ,..., m ℓ − 1 of periodicity ℓ is recursively defined as follows. Consider a node ∅ , called root, with out-degree m 0 . The nodes at the distance k mod ℓ from the root ∅ have out-degree m k and in-degree 1 for k = 1 , 2 , 3 , . . . . 12 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Periodic Oriented Trees The schematic presentation of � T 3 , 2 . ❜ ❜ ❜ ❜ ❅ ■� ✒ � ❅ ■ ✒ � ❅ ❅ � r r � ❅ ■ � ✒ ❅ ✠ � ❘ ❅ ❅ � ❜ ❜ ❅ � ❜ ❜ ❜ ❜ ■� ❅ ✒ � ❅ ■ � ✒ ❅ ❜ ❅ � ✻ r r � ❅ ■ ✒ � ❅ ✠ � ❘ ❅ ❅ � ❜ ❜ ❅ ✛ ✲ � ❜ r ❜ � ❅ � ❅ ❜ ❜ ■ ❅ ❅ � ✠ ❅ ❘ � ✒ � ❄ r r � ❅ ✠❅ � ✠ � ❘ ❅ � ❘ ❅ ❜ � ❅ ❜ ❜ ❜ ❜ � ❅ ❜ ❜ ❅ ■ ✒ � ❅ � ✠ ❘ ❅ � r r � ❅ � ❅ � ✠ ❘ ❅ ✠ � ❘ ❅ ❜ ❜ ❜ ❜ 13 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Main Results The existence of and a closed form system of equations for the critical threshold p f ( θ ) ∈ ( 0 , 1 ) such that a.a.s. 1 if p > p f ( θ ) then the periodic tree becomes (eventually) fully active, 2 if p < p f ( θ ) then a periodic tree does not become (eventually) fully active. 14 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Lemma Given n , θ ∈ N such that 2 ≤ θ ≤ n − 1 and x ∈ [ 0 , 1 ] let n � � n x k ( 1 − x ) n − k − x . � Φ n , p ,θ ( x ) := p + ( 1 − p ) k k = θ There exists p c ∈ ( 0 , 1 ) such that the equation Φ n , p ,θ ( x ) = 0 (i) does not have real roots in ( 0 , 1 ) for any p > p c , (ii) has either one or two real roots in ( 0 , 1 ) for any p ≤ p c . x = 1 is always a solution of Φ n , p ,θ ( x ) = 0 for any p ∈ ( 0 , 1 ) . 15 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Lemma Given n , θ ∈ N such that 2 ≤ θ ≤ n − 1 and x ∈ [ 0 , 1 ] let n � � n x k ( 1 − x ) n − k − x . � Φ n , p ,θ ( x ) := p + ( 1 − p ) k k = θ There exists p c ∈ ( 0 , 1 ) such that the equation Φ n , p ,θ ( x ) = 0 (i) does not have real roots in ( 0 , 1 ) for any p > p c , (ii) has either one or two real roots in ( 0 , 1 ) for any p ≤ p c . x = 1 is always a solution of Φ n , p ,θ ( x ) = 0 for any p ∈ ( 0 , 1 ) . 15 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Figure: The curve Φ n , p ,θ ( x ) for n = 7 , θ = 5 , p = 0 . 4 (no roots). 16 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Figure: The curve Φ n , p ,θ ( x ) for n = 7 , θ = 5 , p = 0 . 3 (two roots). 17 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Main result First concentrate on a tree of periodicity ℓ = 2, then generalize. Theorem Given a , b ∈ N and 2 ≤ θ < a , b consider BP on periodic tree T a , b with the initial probability p. There exists p f ∈ ( 0 , 1 ) such that a.a.s. (i) T a , b becomes completely active for p ≥ p f , (ii) T a , b does not become completely active for p < p f . 18 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Main result First concentrate on a tree of periodicity ℓ = 2, then generalize. Theorem Given a , b ∈ N and 2 ≤ θ < a , b consider BP on periodic tree T a , b with the initial probability p. There exists p f ∈ ( 0 , 1 ) such that a.a.s. (i) T a , b becomes completely active for p ≥ p f , (ii) T a , b does not become completely active for p < p f . 18 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
Proof Outline p f on � 1 Derive the percolation threshold � T a , b . 2 Relate this to the percolation threshold p f on T a , b . 19 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
BP on (oriented) � T a , b 1 Call V a (resp. V b ) the set of nodes of degree a + 1 (resp. b + 1). 2 Dynamics of BP on � T a , b captured by � ζ t ( v ) , � η t ( u ) ∈ { 0 , 1 } for every v ∈ V a and u ∈ V b at time t = 0 , 1 , . . . . 3 Given symmetry and dynamical rules the marginal distribution that a node v ∈ V a at time t is active is the same for all nodes in V a . 20 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees
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