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Fixed Points and Feedback Cycles in Boolean Networks Adrien Richard - PowerPoint PPT Presentation

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Fixed Points and Feedback Cycles in Boolean Networks Adrien Richard Laboratoire I3S, CNRS & Universit e de Nice-Sophia Antipolis Joint work with Julio Aracena & Lilian Salinas Universidad de Concepci on, Chile Groupe de travail

  1. Remark If H ⊆ G then τ ( H ) ≤ τ ( G ) thus φ ( H ) ≤ 2 τ ( H ) ≤ 2 τ ( G ) → connexion with Network Coding from Information Theory ֒ Binary network coding problem Given a digraph G , is there exists H ⊆ G such that φ ( H ) = 2 τ ( G ) ? Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 12/25

  2. Remark If H ⊆ G then τ ( H ) ≤ τ ( G ) thus φ ( H ) ≤ 2 τ ( H ) ≤ 2 τ ( G ) → connexion with Network Coding from Information Theory ֒ Binary network coding problem Given a digraph G , is there exists H ⊆ G such that φ ( H ) = 2 τ ( G ) ? Surprisingly, the following question has deserved very few attention Given a digraph G , do we have φ ( G ) = 2 τ ( G ) ? Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 12/25

  3. Upper bounds on φ ( G σ ) Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 13/25

  4. In G σ the sign of a cycle (or path) is the product of the sign of its arcs τ + ( G σ ) := positive transversal number := minimum size of a set of vertices meeting every non-negative cycle Remark 1 τ + ≤ τ Remark 2 τ + is invariant under subdivisions of arcs preserving signs Remark 2 e.g. → replaced by →→ , or → replaced by →→ Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 13/25

  5. Theorem (signed version of the classical bound) [Aracena, 2008] For every signed digraph G σ φ ( G σ ) ≤ 2 τ + Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 14/25

  6. Theorem (signed version of the classical bound) [Aracena, 2008] For every signed digraph G σ φ ( G σ ) ≤ 2 τ + G σ has only negative cycles ⇒ τ + = 0 ⇒ φ ( G σ ) ≤ 1 Remark 1 Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 14/25

  7. Theorem (signed version of the classical bound) [Aracena, 2008] For every signed digraph G σ φ ( G σ ) ≤ 2 τ + G σ has only negative cycles ⇒ τ + = 0 ⇒ φ ( G σ ) ≤ 1 Remark 1 Also true for differential equation systems [Soul´ e 03] ! Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 14/25

  8. Theorem (signed version of the classical bound) [Aracena, 2008] For every signed digraph G σ φ ( G σ ) ≤ 2 τ + G σ has only negative cycles ⇒ τ + = 0 ⇒ φ ( G σ ) ≤ 1 Remark 1 Also true for differential equation systems [Soul´ e 03] ! Remark 2 We recover the classical upper-bound: 2 τ + ( G σ ) = 2 τ ( G ) φ ( G ) = max φ ( G σ ) ≤ max σ σ Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 14/25

  9. Theorem (signed version of the classical bound) [Aracena, 2008] For every signed digraph G σ φ ( G σ ) ≤ 2 τ + G σ has only negative cycles ⇒ τ + = 0 ⇒ φ ( G σ ) ≤ 1 Remark 1 Also true for differential equation systems [Soul´ e 03] ! Remark 2 We recover the classical upper-bound: 2 τ + ( G σ ) = 2 τ ( G ) φ ( G ) = max φ ( G σ ) ≤ max σ σ This is the state of the art for upper bounds that depend on the cycle structure Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 14/25

  10. Theorem (signed version of the classical bound) [Aracena, 2008] For every signed digraph G σ φ ( G σ ) ≤ 2 τ + G σ has only negative cycles ⇒ τ + = 0 ⇒ φ ( G σ ) ≤ 1 Remark 1 Also true for differential equation systems [Soul´ e 03] ! Remark 2 We recover the classical upper-bound: 2 τ + ( G σ ) = 2 τ ( G ) φ ( G ) = max φ ( G σ ) ≤ max σ σ This is the state of the art for upper bounds that depend on the cycle structure No lower bounds on φ ( G ) neither φ ( G σ ) ! Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 14/25

  11. The bound φ ≤ 2 τ + is very perfectible Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 15/25

  12. The bound φ ≤ 2 τ + is very perfectible · · · • • • • • • φ = 1 2 τ + ∼ 2 n/ 4 • • • • Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 15/25

  13. The bound φ ≤ 2 τ + is very perfectible · · · • • • • • • φ = 1 2 τ + ∼ 2 n/ 4 • • • • We think that improvements could be obtained by considering negative cycles too. This is a difficult problem... What happen when there is only positive cycles ? ֒ → This essentially corresponds to the case where f is monotone Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 15/25

  14. Monotone networks Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 16/25

  15. { 0 , 1 } n is equipped with the usual partial order x ≤ y ⇐ ⇒ x i ≤ y i for all i f is monotone if for all x, y ∈ { 0 , 1 } n x ≤ y ⇒ f ( x ) ≤ f ( y ) Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 16/25

  16. { 0 , 1 } n is equipped with the usual partial order x ≤ y ⇐ ⇒ x i ≤ y i for all i f is monotone if for all x, y ∈ { 0 , 1 } n x ≤ y ⇒ f ( x ) ≤ f ( y ) Remark f is monotone ⇐ ⇒ G σ has only positive arcs φ ( G + ) = maximum number of fixed points in a monotone boolean network with G as interaction graph Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 16/25

  17. { 0 , 1 } n is equipped with the usual partial order x ≤ y ⇐ ⇒ x i ≤ y i for all i f is monotone if for all x, y ∈ { 0 , 1 } n x ≤ y ⇒ f ( x ) ≤ f ( y ) Remark f is monotone ⇐ ⇒ G σ has only positive arcs φ ( G + ) = maximum number of fixed points in a monotone boolean network with G as interaction graph Proposition If G σ is strong and has only positive cycles then φ ( G σ ) = φ ( G + ) Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 16/25

  18. Fixed points in monotone networks Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 17/25

  19. Theorem [Knaster-Tarski, 1928] If f is monotone then Fixe ( f ) is a non-empty lattice Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 17/25

  20. Theorem [Knaster-Tarski, 1928] If f is monotone then Fixe ( f ) is a non-empty lattice To go further we need another graph parameter about cycles ν ( G ) := packing number := maximum number of vertex-disjoint cycles Remark ν ≤ τ Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 17/25

  21. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 18/25

  22. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 Proof of the isomorphism • • • • • • • • Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 18/25

  23. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 Proof of the isomorphism FVS of size τ I • • • • • • • • Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 18/25

  24. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 Proof of the isomorphism FVS of size τ I • • • • • • • • Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 18/25

  25. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 ∀ x, y ∈ Fixe ( f ) x I ≤ y I ⇐ ⇒ x ≤ y Proof of the isomorphism FVS of size τ I • • • • • • • • Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 18/25

  26. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 ∀ x, y ∈ Fixe ( f ) x I ≤ y I ⇐ ⇒ x ≤ y Proof of the isomorphism Fixe ( f ) is isomorphic to L = { x I : x ∈ Fixe ( f ) } I • • • • • • • • Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 18/25

  27. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 ∀ x, y ∈ Fixe ( f ) x I ≤ y I ⇒ x ≤ y Proof of the isomorphism = I • • • • • • • • Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 18/25

  28. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 ∀ x, y ∈ Fixe ( f ) x I ≤ y I ⇒ x ≤ y Proof of the isomorphism = x I ≤ y I 0 1 0 1 1 0 I • • • • • • • • • • Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 18/25

  29. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 ∀ x, y ∈ Fixe ( f ) x I ≤ y I ⇒ x ≤ y Proof of the isomorphism = x I ≤ y I 0 1 0 1 1 0 I J • • • • • • • • • • Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 18/25

  30. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 ∀ x, y ∈ Fixe ( f ) x I ≤ y I ⇒ x ≤ y Proof of the isomorphism = x I ≤ y I 0 1 0 1 1 0 I i i J • • • • • • • • Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 18/25

  31. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 ∀ x, y ∈ Fixe ( f ) x I ≤ y I ⇒ x ≤ y Proof of the isomorphism = x I ≤ y I 0 1 0 1 1 0 I f i ( x ) ≤ f i ( y ) i i J • • • • • • • • Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 18/25

  32. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 ∀ x, y ∈ Fixe ( f ) x I ≤ y I ⇒ x ≤ y Proof of the isomorphism = x I ≤ y I 0 1 0 1 1 0 I x i ≤ y i i i J • • • • • • • • Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 18/25

  33. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 ∀ x, y ∈ Fixe ( f ) x I ≤ y I ⇒ x ≤ y Proof of the isomorphism = x I ≤ y I 0 1 0 1 1 0 I x J ≤ y J 0 1 1 1 J • • • • • • Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 18/25

  34. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 ∀ x, y ∈ Fixe ( f ) x I ≤ y I ⇒ x ≤ y Proof of the isomorphism = x I ≤ y I 0 1 0 1 1 0 I x J ≤ y J 0 1 1 1 J K • • • • • • Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 18/25

  35. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 ∀ x, y ∈ Fixe ( f ) x I ≤ y I ⇒ x ≤ y Proof of the isomorphism = x I ≤ y I 0 1 0 1 1 0 I x J ≤ y J 0 1 1 1 J x K ≤ y K K 0 1 0 1 1 0 Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 18/25

  36. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 Proof of 2 If Fixe ( f ) has a chain of size k then ν ≥ k − 1 Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 19/25

  37. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 Proof of 2 If Fixe ( f ) has a chain of size k then ν ≥ k − 1 x 5 = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x 4 = 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 x 3 = 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 x 2 = 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 19/25

  38. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 Proof of 2 If Fixe ( f ) has a chain of size k then ν ≥ k − 1 x 5 = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x 4 = 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 x 3 = 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 x 2 = 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∆( x 1 , x 2 ) ∆( x 2 , x 3 ) ∆( x 3 , x 4 ) ∆( x 4 , x 5 ) Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 19/25

  39. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 Proof of 2 If Fixe ( f ) has a chain of size k then ν ≥ k − 1 x 5 = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x 4 = 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 x 3 = 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 x 2 = 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∆( x 1 , x 2 ) ∆( x 2 , x 3 ) ∆( x 3 , x 4 ) ∆( x 4 , x 5 ) C 1 C 2 C 3 C 4 Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 19/25

  40. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 Proof of 2 If Fixe ( f ) has a chain of size k then ν ≥ k − 1 Thus Fixe ( f ) has no chains of length ν + 2 and so L x 5 = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x 4 = 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 x 3 = 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 x 2 = 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∆( x 1 , x 2 ) ∆( x 2 , x 3 ) ∆( x 3 , x 4 ) ∆( x 4 , x 5 ) C 1 C 2 C 3 C 4 Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 19/25

  41. Theorem [Erd˝ os, 1945] If X ⊆ { 0 , 1 } n has no chains of size ℓ + 1 then � n | X | ≤ the sum of the ℓ largest binomial coefficients � k Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 20/25

  42. Theorem [Erd˝ os, 1945] If X ⊆ { 0 , 1 } n has no chains of size ℓ + 1 then � n | X | ≤ the sum of the ℓ largest binomial coefficients � k Remark The case ℓ = 1 is Sperner’s lemma on antichains Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 20/25

  43. Theorem [Erd˝ os, 1945] If X ⊆ { 0 , 1 } n has no chains of size ℓ + 1 then � n | X | ≤ the sum of the ℓ largest binomial coefficients � k Corollary If f is monotone then � τ | Fixe ( f ) | − 2 ≤ the sum of the ν − 1 largest � k Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 20/25

  44. Theorem [Erd˝ os, 1945] If X ⊆ { 0 , 1 } n has no chains of size ℓ + 1 then � n | X | ≤ the sum of the ℓ largest binomial coefficients � k Corollary If f is monotone then � τ | Fixe ( f ) | − 2 ≤ the sum of the ν − 1 largest � k Proof Let L ⊆ { 0 , 1 } τ be a non-empty lattice isomorphic to Fixe ( f ) max b • no chains L of size ν + 2 • min a Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 20/25

  45. Theorem [Erd˝ os, 1945] If X ⊆ { 0 , 1 } n has no chains of size ℓ + 1 then � n | X | ≤ the sum of the ℓ largest binomial coefficients � k Corollary If f is monotone then � τ | Fixe ( f ) | − 2 ≤ the sum of the ν − 1 largest � k Proof Let L ⊆ { 0 , 1 } τ be a non-empty lattice isomorphic to Fixe ( f ) max b • no chains L \ { a, b } of size ν • min a Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 20/25

  46. Theorem [Erd˝ os, 1945] If X ⊆ { 0 , 1 } n has no chains of size ℓ + 1 then � n | X | ≤ the sum of the ℓ largest binomial coefficients � k Corollary If f is monotone then � τ | Fixe ( f ) | − 2 ≤ the sum of the ν − 1 largest � k Proof Let L ⊆ { 0 , 1 } τ be a non-empty lattice isomorphic to Fixe ( f ) max b • no chains � τ ≤ � L \ { a, b } the sum of the ν − 1 largest k of size ν • min a Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 20/25

  47. � τ � Corollary φ ( G + ) ≤ the sum of the ν − 1 largest + 2 k Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 21/25

  48. � τ � Corollary φ ( G + ) ≤ the sum of the ν − 1 largest + 2 k � τ � τ � τ � � � τ/ 2 0 τ τ − 1 coefficients Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 21/25

  49. � τ � Corollary φ ( G + ) ≤ the sum of the ν − 1 largest + 2 k 2 τ � τ � τ � τ � � � τ/ 2 0 τ τ − 1 coefficients Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 21/25

  50. � τ � Corollary φ ( G + ) ≤ the sum of the ν − 1 largest + 2 k � τ � τ � τ � � � τ/ 2 0 τ ν − 1 coefficients τ − 1 coefficients φ ( G + ) = 2 τ ⇒ ν = τ Corollary Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 21/25

  51. � τ � Corollary φ ( G + ) ≤ the sum of the ν − 1 largest + 2 k The upper bound is interesting when ν is much more smaller that τ The largest gap known is ν log ν ≤ 30 τ [Seymour 93] Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 21/25

  52. � τ � Corollary φ ( G + ) ≤ the sum of the ν − 1 largest + 2 k The upper bound is interesting when ν is much more smaller that τ The largest gap known is ν log ν ≤ 30 τ [Seymour 93] For a fixed ν , τ cannot be arbitrarily large... Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 21/25

  53. � τ � Corollary φ ( G + ) ≤ the sum of the ν − 1 largest + 2 k The upper bound is interesting when ν is much more smaller that τ The largest gap known is ν log ν ≤ 30 τ [Seymour 93] For a fixed ν , τ cannot be arbitrarily large... Theorem [Reed-Robertson-Seymour-Thomas, 1995] There exists h : N → N such that, for every digraph G , τ ≤ h ( ν ) The upper-bound on h ( ν ) is astronomique (iterated use of Ramsey theorem) Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 21/25

  54. � τ � Corollary φ ( G + ) ≤ the sum of the ν − 1 largest + 2 k The upper bound is interesting when ν is much more smaller that τ The largest gap known is ν log ν ≤ 30 τ [Seymour 93] For a fixed ν , τ cannot be arbitrarily large... Theorem [Reed-Robertson-Seymour-Thomas, 1995] There exists h : N → N such that, for every digraph G , τ ≤ h ( ν ) The upper-bound on h ( ν ) is astronomique (iterated use of Ramsey theorem) φ ( G ) ≤ 2 τ ≤ 2 h ( ν ) Corollary Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 21/25

  55. � τ � Corollary φ ( G + ) ≤ the sum of the ν − 1 largest + 2 k The upper bound is interesting when ν is much more smaller that τ The largest gap known is ν log ν ≤ 30 τ [Seymour 93] For a fixed ν , τ cannot be arbitrarily large... Theorem [Reed-Robertson-Seymour-Thomas, 1995] There exists h : N → N such that, for every digraph G , τ ≤ h ( ν ) The upper-bound on h ( ν ) is astronomique (iterated use of Ramsey theorem) ν + 1 ≤ φ ( G ) ≤ 2 τ ≤ 2 h ( ν ) Corollary Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 21/25

  56. More on fixed points in monotone networks Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 22/25

  57. Special packing Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 22/25

  58. Special packing v • • u Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 22/25

  59. Special packing • u • v Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 22/25

  60. Special packing v • • P u Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 22/25

  61. Special packing • P ′ v • • P u Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 22/25

  62. Special packing • P ′ v • • P u We denote by ν ∗ ( G ) the maximum size of a special packing Remark ν ∗ ≤ ν ≤ τ Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 22/25

  63. A k -pattern in X ⊆ { 0 , 1 } n is a sequence ( x 1 , . . . , x k ) ∈ X k such that x p ≤ x q ( x 1 , . . . , x k ) ∈ X k and ⇐ ⇒ p � = q Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 23/25

  64. A k -pattern in X ⊆ { 0 , 1 } n is a sequence ( x 1 , . . . , x k ) ∈ X k such that x p ≤ x q ( x 1 , . . . , x k ) ∈ X k and ⇐ ⇒ p � = q Example ( e 1 , e 2 , e 3 ) is a 3 -pattern of { 0 , 1 } 3 e 1 = 011 e 2 = 101 e 3 = 110 e 1 = 100 e 2 = 010 e 3 = 001 Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 23/25

  65. A k -pattern in X ⊆ { 0 , 1 } n is a sequence ( x 1 , . . . , x k ) ∈ X k such that x p ≤ x q ( x 1 , . . . , x k ) ∈ X k and ⇐ ⇒ p � = q Example ( e 1 , e 2 , e 3 ) is a 3 -pattern of { 0 , 1 } 3 e 1 = 011 e 2 = 101 e 3 = 110 e 1 = 100 e 2 = 010 e 3 = 001 Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 23/25

  66. A k -pattern in X ⊆ { 0 , 1 } n is a sequence ( x 1 , . . . , x k ) ∈ X k such that x p ≤ x q ( x 1 , . . . , x k ) ∈ X k and ⇐ ⇒ p � = q Example ( e 1 , e 2 , e 3 ) is a 3 -pattern of { 0 , 1 } 3 e 1 = 011 e 2 = 101 e 3 = 110 e 1 = 100 e 2 = 010 e 3 = 001 Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 23/25

  67. A k -pattern in X ⊆ { 0 , 1 } n is a sequence ( x 1 , . . . , x k ) ∈ X k such that x p ≤ x q ( x 1 , . . . , x k ) ∈ X k and ⇐ ⇒ p � = q Example ( e 1 , e 2 , e 3 ) is a 3 -pattern of { 0 , 1 } 3 e 1 = 011 e 2 = 101 e 3 = 110 e 1 = 100 e 2 = 010 e 3 = 001 Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 23/25

  68. A k -pattern in X ⊆ { 0 , 1 } n is a sequence ( x 1 , . . . , x k ) ∈ X k such that x p ≤ x q ( x 1 , . . . , x k ) ∈ X k and ⇐ ⇒ p � = q Example ( e 1 , e 2 , e 3 ) is a 3 -pattern of { 0 , 1 } 3 e 1 = 011 e 2 = 101 e 3 = 110 e 1 = 100 e 2 = 010 e 3 = 001 More generally ( e 1 , e 2 , . . . , e n ) is an n -pattern of { 0 , 1 } n Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 23/25

  69. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 3. L has no ( ν ∗ + 1) -pattern Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 24/25

  70. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 3. L has no ( ν ∗ + 1) -pattern Remark If L = { 0 , 1 } τ then L has a has a τ -pattern, so τ < ν ∗ + 1 . Remark Thus τ ≤ ν ∗ and since ν ∗ ≤ τ we deduce that ν ∗ = τ . Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 24/25

  71. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 3. L has no ( ν ∗ + 1) -pattern Remark If L = { 0 , 1 } τ then L has a has a τ -pattern, so τ < ν ∗ + 1 . φ ( G + ) = 2 τ ⇒ ν ∗ = τ Corollary Remark Thus τ ≤ ν ∗ and since ν ∗ ≤ τ we deduce that ν ∗ = τ . Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 24/25

  72. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 3. L has no ( ν ∗ + 1) -pattern Remark If L = { 0 , 1 } τ then L has a has a τ -pattern, so τ < ν ∗ + 1 . φ ( G + ) = 2 τ ⇒ ν ∗ = τ ⇒ ν = τ Corollary Remark Thus τ ≤ ν ∗ and since ν ∗ ≤ τ we deduce that ν ∗ = τ . Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 24/25

  73. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 3. L has no ( ν ∗ + 1) -pattern Remark If L = { 0 , 1 } τ then L has a has a τ -pattern, so τ < ν ∗ + 1 . φ ( G + ) = 2 τ ⇒ ν ∗ = τ ⇒ ν = τ Corollary Remark Thus τ ≤ ν ∗ and since ν ∗ ≤ τ we deduce that ν ∗ = τ . Theorem [Aracena-Salinas-R, 2016 + ] 2 ν ∗ ≤ φ ( G + ) Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 24/25

  74. Theorem [Aracena-Salinas-R, 2016 + ] If f is monotone then Fixe ( f ) a isomorphic to a subset L ⊆ { 0 , 1 } τ s.t. 1. L is a non-empty lattice 2. L has no chains of size ν + 2 3. L has no ( ν ∗ + 1) -pattern Remark If L = { 0 , 1 } τ then L has a has a τ -pattern, so τ < ν ∗ + 1 . φ ( G + ) = 2 τ ⇒ ν ∗ = τ ⇒ ν = τ Corollary Remark Thus τ ≤ ν ∗ and since ν ∗ ≤ τ we deduce that ν ∗ = τ . Theorem [Aracena-Salinas-R, 2016 + ] 2 ν ∗ ≤ φ ( G + ) ν ∗ = τ φ ( G + ) = 2 τ Corollary ⇐ ⇒ Adrien RICHARD Fixed Points in Boolean Networks Paris 2016 24/25

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