Rate Equations for Graphs Vincent Danos 1 Tobias Heindel 2 Ricardo - PowerPoint PPT Presentation

Rate Equations for Graphs Vincent Danos 1 Tobias Heindel 2 Ricardo Honorato-Zimmer 3 Sandro Stucki 4 1 CNRS/ENS-PSL/INRIA, France 2 TU Berlin, Germany 3 CINV, Chile 4 GU/Chalmers, Sweden Virtual CMSB 2020 – Konstanz – 23 Sep 2020 sandro.stucki@gu.se @stuckintheory 1

Mean field approximations (MFAs) Question What is the expected value E ( F ) of some observable F on a CTMC? 2

Mean field approximations (MFAs) Photo: J Ligero & I Barrios 2013 (Wikipedia). 2

Mean field approximations (MFAs) Question What is the expected value E ( F ) of some observable F on a CTMC? Example (reproduction) k α 2 − − ⇀ 3 2

Mean field approximations (MFAs) Question What is the expected value E ( F ) of some observable F on a CTMC? Example (reproduction) k α 2 B − − ⇀ 3 B 2

Mean field approximations (MFAs) Question What is the expected value E ( F ) of some observable F on a CTMC? Example (reproduction) k α 2 B − − ⇀ 3 B The function [ B ] counts the number of occurrences of B . d dt E [ B ] = k α E [ 2 B ] = k α E ([ B ]([ B ] − 1 )) (meanfield) 2

Mean field approximations (MFAs) Question What is the expected value E ( F ) of some observable F on a CTMC? Example (reproduction) k α 2 B − − ⇀ 3 B The function [ B ] counts the number of occurrences of B . d dt E [ B ] = k α E [ 2 B ] = k α E ([ B ]([ B ] − 1 )) (meanfield) ≃ k α E ([ B ][ B ]) ≃ k α E [ B ] E [ B ] (approximation) 2

Mean field approximations (MFAs) Question What is the expected value E ( F ) of some observable F on a CTMC? Example (reproduction) k α 2 B − − ⇀ 3 B The function [ B ] counts the number of occurrences of B . d dt E [ B ] = k α E [ 2 B ] = k α E ([ B ]([ B ] − 1 )) (meanfield) ≃ k α E ([ B ][ B ]) ≃ k α E [ B ] E [ B ] (approximation) dt [ B ] ≃ k α [ B ] 2 d (thermodynamic limit) 2

CRNs are Graph Transformation Systems (GTSs) Reaction/rule Observable k α − − ⇀ B := 3

CRNs are Graph Transformation Systems (GTSs) Reaction/rule Observable k α − − ⇀ B := MFA/Rate equation d = dt d dt [ B ] = 3

CRNs are Graph Transformation Systems (GTSs) Reaction/rule Observable k α − − ⇀ B := MFA/Rate equation d = − k α + · · · dt d dt [ B ] = − k α [ 2 B ] + · · · 3

CRNs are Graph Transformation Systems (GTSs) Reaction/rule Observable k α − − ⇀ B := MFA/Rate equation d = − 2 k α + · · · dt d dt [ B ] = − 2 k α [ 2 B ] + · · · 3

CRNs are Graph Transformation Systems (GTSs) Reaction/rule Observable k α − − ⇀ B := MFA/Rate equation d = − 2 k α + k α + · · · dt d dt [ B ] = − 2 k α [ 2 B ] + k α [ 2 B ] + · · · 3

CRNs are Graph Transformation Systems (GTSs) Reaction/rule Observable k α − − ⇀ B := MFA/Rate equation d = − 2 k α + 2 k α + · · · dt d dt [ B ] = − 2 k α [ 2 B ] + 2 k α [ 2 B ] + · · · 3

CRNs are Graph Transformation Systems (GTSs) Reaction/rule Observable k α − − ⇀ B := MFA/Rate equation d = − 2 k α + 3 k α dt d dt [ B ] = − 2 k α [ 2 B ] + 3 k α [ 2 B ] 3

CRNs are Graph Transformation Systems (GTSs) Reaction/rule Observable k α − − ⇀ B := MFA/Rate equation d = k α dt d dt [ B ] = k α [ 2 B ] 3

CRNs are Graph Transformation Systems (GTSs) Reaction/rule Observable k α − − ⇀ B := MFA/Rate equation d = k α ≃ k α dt dt [ B ] = k α [ 2 B ] ≃ k α [ B ] 2 d 3

Bunnies with families Rules k β k γ − − ⇀ − − ⇀ Observables B := C := S := 4

Bunnies with families Rules k β k γ − − ⇀ − − ⇀ Observables B := C := S := MFA/Rate equation d = dt d dt [ B ] = 4

Bunnies with families Rules k β k γ − − ⇀ − − ⇀ Observables B := C := S := MFA/Rate equation d = k β + · · · dt d dt [ B ] = k β [ 2 B ] + · · · 4

Bunnies with families Rules k β k γ − − ⇀ − − ⇀ Observables B := C := S := MFA/Rate equation d = k β + k γ dt d dt [ B ] = k β [ 2 B ] + k γ [ C ] 4

Bunnies with families Rules k β k γ − − ⇀ − − ⇀ Observables B := C := S := MFA/Rate equation d ≃ k β + k γ dt dt [ B ] ≃ k β [ B ] 2 + k γ [ C ] d 4

Bunnies with families (cont.) Rule Observable k β − − ⇀ C := 5

Bunnies with families (cont.) Rule Observable k β − − ⇀ C := MFA/Rate equation d = dt d dt [ C ] = 5

Bunnies with families (cont.) Rule Observable k β − − ⇀ C := Refinement MFA/Rate equation d = dt d dt [ C ] = 5

Bunnies with families (cont.) Rule Observable k β − − ⇀ C := Refinement k β − − ⇀ MFA/Rate equation d = − k β + · · · dt d dt [ C ] = − k β [ C ] + · · · 5

Bunnies with families (cont.) Rule Observable k β − − ⇀ C := Refinement MFA/Rate equation d = − k β + · · · dt d dt [ C ] = − k β [ C ] + · · · 5

Bunnies with families (cont.) Rule Observable k β − − ⇀ C := Refinement k β − − ⇀ MFA/Rate equation d = − k β − k β + · · · dt d dt [ C ] = − k β [ C ] − k β [ F 0 ] + · · · 5

Interlude: minimal gluings (overlaps)   ,            , ,      ∗ = , ,              , ,    6

Interlude: minimal gluings (overlaps)   ,            , ,      ∗ = , ,              , ,    The set of MGs grows quickly, even for small graphs. � � � � � � � � ∗ � = 44 ∗ � = 101 � � � � � � � � � � � � � = 381 ∗ � � � � � 6

Case 1: irrelevant MGs Rule Observable k β C := − − ⇀ MFA/Rate equation d = − k β − k β + · · · dt d dt [ C ] = − k β [ C ] − k β [ F 0 ] + · · · 7

Case 1: irrelevant MGs Rule Observable k β C := − − ⇀ Refinement MFA/Rate equation d = − k β − k β + · · · dt d dt [ C ] = − k β [ C ] − k β [ F 0 ] + · · · 7

Case 1: irrelevant MGs Rule Observable k β C := − − ⇀ Refinement k β − − ⇀ MFA/Rate equation d = − k β − k β + k β + · · · dt d dt [ C ] = − k β [ C ] − k β [ F 0 ] + k β [ C ] + · · · 7

Case 1: irrelevant MGs Rule Observable k β C := − − ⇀ Refinement k β − − ⇀ MFA/Rate equation d = − k β + · · · dt d dt [ C ] = − k β [ F 0 ] + · · · 7

Case 1: irrelevant MGs Rule Observable k β C := − − ⇀ Refinement MFA/Rate equation d = − k β + · · · dt d dt [ C ] = − k β [ F 0 ] + · · · 7

Case 1: irrelevant MGs Rule Observable k β C := − − ⇀ Refinement k β − − ⇀ MFA/Rate equation d = − k β + k β + · · · dt d dt [ C ] = − k β [ F 0 ] + k β [ F 0 ] + · · · 7

Case 1: irrelevant MGs Rule Observable k β C := − − ⇀ Refinement k β − − ⇀ MFA/Rate equation d = · · · dt d dt [ C ] = · · · 7

Case 2: underivable MGs (RHS only) Rule Observable k β C := − − ⇀ MFA/Rate equation d = · · · dt d dt [ C ] = · · · 8

Case 2: underivable MGs (RHS only) Rule Observable k β C := − − ⇀ Refinement MFA/Rate equation d = · · · dt d dt [ C ] = · · · 8

� Case 2: underivable MGs (RHS only) Rule Observable k β C := − − ⇀ Refinement k β − − ⇀ MFA/Rate equation d = · · · dt d dt [ C ] = · · · 8

� Case 2: underivable MGs (RHS only) Rule Observable k β C := − − ⇀ Refinement k β − − ⇀ MFA/Rate equation d = · · · dt d dt [ C ] = · · · 8

Case 3: relevant derivable MGs Rule Observable k β C := − − ⇀ MFA/Rate equation d = · · · dt d dt [ C ] = · · · 9

Case 3: relevant derivable MGs Rule Observable k β C := − − ⇀ Refinement MFA/Rate equation d = · · · dt d dt [ C ] = · · · 9

Case 3: relevant derivable MGs Rule Observable k β C := − − ⇀ Refinement k β − − ⇀ MFA/Rate equation d = k β + · · · dt d dt [ C ] = k β [ 2 B ] + · · · 9

Case 3: relevant derivable MGs Rule Observable k β C := − − ⇀ Refinement k β − − ⇀ MFA/Rate equation d = 2 k β + · · · dt d dt [ C ] = 2 k β [ 2 B ] + · · · 9

Case 3: relevant derivable MGs Rule Observable k γ C := − − ⇀ Refinement MFA/Rate equation d = 2 k β + · · · dt d dt [ C ] = 2 k β [ 2 B ] + · · · 9

Case 3: relevant derivable MGs Rule Observable k γ C := − − ⇀ Refinement k γ − − ⇀ MFA/Rate equation d = 2 k β + k γ + · · · dt d dt [ C ] = 2 k β [ 2 B ] + k γ [ C ] + · · · 9

Case 3: relevant derivable MGs Rule Observable k γ C := − − ⇀ Refinement k γ − − ⇀ MFA/Rate equation d = 2 k β + 2 k γ dt d dt [ C ] = 2 k β [ 2 B ] + 2 k γ [ C ] 9