Fixing Boolean networks asynchronously Juilio Aracena and Lilian - PowerPoint PPT Presentation

Fixing Boolean networks asynchronously Juilio Aracena and Lilian Salinas Universidad de Concepci´ on, Chile Maximilien Gadouleau Durham University, UK Adrien Richard CNRS, Universit´ e Cˆ ote d’Azur, France S´ eminaire “Dynamique, Arithm´ etique, Combinatoire” ´ Equipe I2M de l’IML Marseille, le 13 mars 2018 Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 1/37

A Boolean network (BN) with n components is a function f : { 0 , 1 } n → { 0 , 1 } n x = ( x 1 , . . . , x n ) �→ f ( x ) = ( f 1 ( x ) , . . . , f n ( x )) . The dynamics is usually described by the successive iterations of f x → f ( x ) → f 2 ( x ) → f 3 ( x ) → · · · Fixed points correspond to stable states. Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 2/37

Example with n = 3 x f ( x ) 000 000 001 110  f 1 ( x ) = x 2 ∨ x 3 010 101  f 2 ( x ) = x 1 ∧ x 3 011 110  f 3 ( x ) = x 3 ∧ ( x 1 ∨ x 2 ) 100 001 101 100 110 101 111 100 Dynamics 010 011 101 111 000 110 100 001 Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 3/37

The interaction graph of f is the digraph G ( f ) on [ n ] := { 1 , . . . , n } s.t. j → i is an arc ⇐ ⇒ f i depends on x j . Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 4/37

Example f ( x ) x 000 000  001 110 f 1 ( x ) = x 2 ∨ x 3  010 101 f 2 ( x ) = x 1 ∧ x 3 011 110 100 001  f 3 ( x ) = x 3 ∧ ( x 1 ∨ x 2 ) 101 100 110 101 111 100 Dynamics Interaction graph 010 1 2 011 101 111 3 000 110 100 001 Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 5/37

Many applications , in particular: - Neural networks [McCulloch & Pitts 1943] - Gene networks [Kauffman 1969, Thomas 1973] - Network Coding [Riis 2007] Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 6/37

Synchronous dynamics: all components are updated at each step: x → f ( x ) → f 2 ( x ) → f 3 ( x ) → · · · Asynchronous : one component is updated at each step. ֒ → Update component i at state x means reach the state i → f i ( x ) := ( x 1 , . . . , x i − 1 , f i ( x ) , x i +1 , . . . , x n ) . x − Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 7/37

Synchronous dynamics: all components are updated at each step: x → f ( x ) → f 2 ( x ) → f 3 ( x ) → · · · Asynchronous : one component is updated at each step. ֒ → Update component i at state x means reach the state i → f i ( x ) := ( x 1 , . . . , x i − 1 , f i ( x ) , x i +1 , . . . , x n ) . x − The asynchronous graph Γ( f ) describes all the possible trajectories: the vertex set is { 0 , 1 } n and x → f i ( x ) for all x ∈ { 0 , 1 } n and i ∈ [ n ] . Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 7/37

Synchronous dynamics: all components are updated at each step: x → f ( x ) → f 2 ( x ) → f 3 ( x ) → · · · Asynchronous : one component is updated at each step. ֒ → Update component i at state x means reach the state i → f i ( x ) := ( x 1 , . . . , x i − 1 , f i ( x ) , x i +1 , . . . , x n ) . x − The asynchronous graph Γ( f ) describes all the possible trajectories: the vertex set is { 0 , 1 } n and x → f i ( x ) for all x ∈ { 0 , 1 } n and i ∈ [ n ] . It can be regarded as a Finite Deterministic Automta where 1. the alphabet is Σ := [ n ] ; 2. the set of states is Q := { 0 , 1 } n ; 3. the transition function δ : Q × Σ → Q is δ ( x, i ) := f i ( x ) . Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 7/37

Example 1 , 3 1 011 111 2 , 3 1 x f ( x ) 3 3 000 000 001 000 010 110 1 010 001 2 2 011 001 2 100 010 001 101 101 000 1 110 010 2 3 3 111 100 1 , 2 2 000 100 1 1 , 2 , 3 3 Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 8/37

Notation: If w = i 1 i 2 . . . i k ∈ [ n ] ∗ then f w ( x ) is the state obtained from x by updating successively the components i 1 , i 2 , . . . , i k , that is, f w ( x ) := ( f i k ◦ f i k − 1 ◦ · · · ◦ f i 1 )( x ) . Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 9/37

Notation: If w = i 1 i 2 . . . i k ∈ [ n ] ∗ then f w ( x ) is the state obtained from x by updating successively the components i 1 , i 2 , . . . , i k , that is, f w ( x ) := ( f i k ◦ f i k − 1 ◦ · · · ◦ f i 1 )( x ) . Definition 1. A word w ∈ [ n ] ∗ fixes f if ∀ x ∈ { 0 , 1 } n , f w ( x ) is a fixed point of f. The fixing length λ ( f ) is the min length of a word fixing f . Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 9/37

Notation: If w = i 1 i 2 . . . i k ∈ [ n ] ∗ then f w ( x ) is the state obtained from x by updating successively the components i 1 , i 2 , . . . , i k , that is, f w ( x ) := ( f i k ◦ f i k − 1 ◦ · · · ◦ f i 1 )( x ) . Definition 1. A word w ∈ [ n ] ∗ fixes f if ∀ x ∈ { 0 , 1 } n , f w ( x ) is a fixed point of f. The fixing length λ ( f ) is the min length of a word fixing f . Definition 2. A word w fixes a family F of BNs if it fixes each f ∈ F . The fixing length λ ( F ) is the min length of a word fixing F . Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 9/37

Example: 1231 is fixing (and no shorter word is fixing, thus λ ( f ) = 4 ). 1 , 3 1 011 111 2 , 3 1 x f ( x ) 3 3 000 000 001 000 010 110 1 010 001 2 2 011 001 2 100 010 001 101 101 000 1 110 010 2 3 3 111 100 1 , 2 2 000 100 1 1 , 2 , 3 3 Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 10/37

Remarks 1. f is fixable only if f has a fixed point. 2. If f has a unique fixed point then: w fixes f ⇐ ⇒ w is synchronizing . 3. A family F is fixable if and only if each f ∈ F is fixable. Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 11/37

Remarks 1. f is fixable only if f has a fixed point. 2. If f has a unique fixed point then: w fixes f ⇐ ⇒ w is synchronizing . 3. A family F is fixable if and only if each f ∈ F is fixable. Theorem 1 [Bollob´ as, Gotsman and Shamir 1993] There is a positive fraction φ ( n ) of fixable BNs with n components: n →∞ φ ( n ) = 1 − 1 lim e ≥ 0 . 64 . Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 11/37

Example of fixable families 1. F M ( n ) : Monotone BNs ( 2 Θ( √ n 2 n ) ): ∀ x, y ∈ { 0 , 1 } n , x ≤ y ⇒ f ( x ) ≤ f ( y ) . Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 12/37

Example of fixable families 1. F M ( n ) : Monotone BNs ( 2 Θ( √ n 2 n ) ): ∀ x, y ∈ { 0 , 1 } n , x ≤ y ⇒ f ( x ) ≤ f ( y ) . 2. F A ( n ) : BNs with an Acyclic interaction graph ( 2 Θ(2 n ) ). Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 12/37

Example of fixable families 1. F M ( n ) : Monotone BNs ( 2 Θ( √ n 2 n ) ): ∀ x, y ∈ { 0 , 1 } n , x ≤ y ⇒ f ( x ) ≤ f ( y ) . 2. F A ( n ) : BNs with an Acyclic interaction graph ( 2 Θ(2 n ) ). 3. F I ( n ) : Increasing BNs ( 2 n 2 n − 1 ): ∀ x ∈ { 0 , 1 } n , x ≤ f ( x ) . Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 12/37

Example of fixable families 1. F M ( n ) : Monotone BNs ( 2 Θ( √ n 2 n ) ): ∀ x, y ∈ { 0 , 1 } n , x ≤ y ⇒ f ( x ) ≤ f ( y ) . 2. F A ( n ) : BNs with an Acyclic interaction graph ( 2 Θ(2 n ) ). 3. F I ( n ) : Increasing BNs ( 2 n 2 n − 1 ): ∀ x ∈ { 0 , 1 } n , x ≤ f ( x ) . 4. F P ( n ) : Monotone BNs whose interaction graph is a Path ( 2 n ! ). Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 12/37

Theorem [Aracena, Gadouleau, R., Salinas 2018+] Networks F max f ∈F λ ( f ) λ ( F ) Θ( n 2 ) Acyclic F A ( n ) n Θ( n 2 ) Path F P ( n ) n Θ( n 2 ) Θ( n 2 ) Increasing F I ( n ) Ω( n 2 ) O ( n 3 ) Monotone F M ( n ) Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 13/37

Acyclic networks Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 14/37

1 2 3 4 5 G ( f ) stabilization 6 7 8 9 10 w := 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 is a fixing word Aracena, Gadouleau, Richard , Salinas Fixing monotone Boolean networks asynchronously Marseille 2018-02-22 15/37