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

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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

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SLIDE 1

Rate Equations for Graphs

Vincent Danos1 Tobias Heindel2 Ricardo Honorato-Zimmer3 Sandro Stucki4

1CNRS/ENS-PSL/INRIA, France 2TU Berlin, Germany 3CINV, Chile 4GU/Chalmers, Sweden

Virtual CMSB 2020 – Konstanz – 23 Sep 2020

sandro.stucki@gu.se @stuckintheory 1

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SLIDE 2

Mean field approximations (MFAs)

Question

What is the expected value E(F) of some observable F on a CTMC?

2

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SLIDE 3

Mean field approximations (MFAs)

Photo: J Ligero & I Barrios 2013 (Wikipedia).

2

slide-4
SLIDE 4

Mean field approximations (MFAs)

Question

What is the expected value E(F) of some observable F on a CTMC?

Example (reproduction) 2

kα

− − ⇀ 3

2

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SLIDE 5

Mean field approximations (MFAs)

Question

What is the expected value E(F) of some observable F on a CTMC?

Example (reproduction) 2B

kα

− − ⇀ 3B

2

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SLIDE 6

Mean field approximations (MFAs)

Question

What is the expected value E(F) of some observable F on a CTMC?

Example (reproduction) 2B

kα

− − ⇀ 3B

The function [B] counts the number of occurrences of B.

d dt E[B] = kα E[2B] = kα E([B]([B] − 1))

(meanfield)

2

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SLIDE 7

Mean field approximations (MFAs)

Question

What is the expected value E(F) of some observable F on a CTMC?

Example (reproduction) 2B

kα

− − ⇀ 3B

The function [B] counts the number of occurrences of B.

d dt E[B] = kα E[2B] = kα E([B]([B] − 1))

(meanfield) ≃ kα E([B][B]) ≃ kα E[B] E[B] (approximation)

2

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SLIDE 8

Mean field approximations (MFAs)

Question

What is the expected value E(F) of some observable F on a CTMC?

Example (reproduction) 2B

kα

− − ⇀ 3B

The function [B] counts the number of occurrences of B.

d dt E[B] = kα E[2B] = kα E([B]([B] − 1))

(meanfield) ≃ kα E([B][B]) ≃ kα E[B] E[B] (approximation)

d dt[B] ≃ kα[B]2

(thermodynamic limit)

2

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SLIDE 9

CRNs are Graph Transformation Systems (GTSs)

Reaction/rule

kα

− − ⇀ Observable B :=

3

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SLIDE 10

CRNs are Graph Transformation Systems (GTSs)

Reaction/rule

kα

− − ⇀ Observable B := MFA/Rate equation

d dt

=

d dt[B] = 3

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SLIDE 11

CRNs are Graph Transformation Systems (GTSs)

Reaction/rule

kα

− − ⇀ Observable B := MFA/Rate equation

d dt

= −kα + · · ·

d dt[B] = −kα[2B] + · · · 3

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SLIDE 12

CRNs are Graph Transformation Systems (GTSs)

Reaction/rule

kα

− − ⇀ Observable B := MFA/Rate equation

d dt

= −2kα + · · ·

d dt[B] = −2kα[2B] + · · · 3

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SLIDE 13

CRNs are Graph Transformation Systems (GTSs)

Reaction/rule

kα

− − ⇀ Observable B := MFA/Rate equation

d dt

= −2kα + kα + · · ·

d dt[B] = −2kα[2B] + kα[2B] + · · · 3

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SLIDE 14

CRNs are Graph Transformation Systems (GTSs)

Reaction/rule

kα

− − ⇀ Observable B := MFA/Rate equation

d dt

= −2kα + 2kα + · · ·

d dt[B] = −2kα[2B] + 2kα[2B] + · · · 3

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SLIDE 15

CRNs are Graph Transformation Systems (GTSs)

Reaction/rule

kα

− − ⇀ Observable B := MFA/Rate equation

d dt

= −2kα + 3kα

d dt[B] = −2kα[2B] + 3kα[2B] 3

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SLIDE 16

CRNs are Graph Transformation Systems (GTSs)

Reaction/rule

kα

− − ⇀ Observable B := MFA/Rate equation

d dt

= kα

d dt[B] = kα[2B] 3

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SLIDE 17

CRNs are Graph Transformation Systems (GTSs)

Reaction/rule

kα

− − ⇀ Observable B := MFA/Rate equation

d dt

= kα ≃ kα

d dt[B] = kα[2B] ≃ kα[B]2 3

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SLIDE 18

Bunnies with families

Rules

kβ

− − ⇀

kγ

− − ⇀ Observables B := C := S :=

4

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SLIDE 19

Bunnies with families

Rules

kβ

− − ⇀

kγ

− − ⇀ Observables B := C := S := MFA/Rate equation

d dt

=

d dt[B] = 4

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SLIDE 20

Bunnies with families

Rules

kβ

− − ⇀

kγ

− − ⇀ Observables B := C := S := MFA/Rate equation

d dt

= kβ + · · ·

d dt[B] = kβ[2B] + · · · 4

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SLIDE 21

Bunnies with families

Rules

kβ

− − ⇀

kγ

− − ⇀ Observables B := C := S := MFA/Rate equation

d dt

= kβ + kγ

d dt[B] = kβ[2B] + kγ[C] 4

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SLIDE 22

Bunnies with families

Rules

kβ

− − ⇀

kγ

− − ⇀ Observables B := C := S := MFA/Rate equation

d dt

≃ kβ + kγ

d dt[B] ≃ kβ[B]2 + kγ[C] 4

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SLIDE 23

Bunnies with families (cont.)

Rule

kβ

− − ⇀ Observable C :=

5

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SLIDE 24

Bunnies with families (cont.)

Rule

kβ

− − ⇀ Observable C := MFA/Rate equation

d dt

=

d dt[C] = 5

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SLIDE 25

Bunnies with families (cont.)

Rule

kβ

− − ⇀ Observable C := Refinement MFA/Rate equation

d dt

=

d dt[C] = 5

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SLIDE 26

Bunnies with families (cont.)

Rule

kβ

− − ⇀ Observable C := Refinement

kβ

− − ⇀ MFA/Rate equation

d dt

= −kβ + · · ·

d dt[C] = −kβ[C] + · · · 5

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SLIDE 27

Bunnies with families (cont.)

Rule

kβ

− − ⇀ Observable C := Refinement MFA/Rate equation

d dt

= −kβ + · · ·

d dt[C] = −kβ[C] + · · · 5

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SLIDE 28

Bunnies with families (cont.)

Rule

kβ

− − ⇀ Observable C := Refinement

kβ

− − ⇀ MFA/Rate equation

d dt

= −kβ − kβ + · · ·

d dt[C] = −kβ[C] − kβ[F0] + · · · 5

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SLIDE 29

Interlude: minimal gluings (overlaps)

∗ =                  , , , , , , ,                 

6

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SLIDE 30

Interlude: minimal gluings (overlaps)

∗ =                  , , , , , , ,                  The set of MGs grows quickly, even for small graphs.

  • ∗
  • = 44
  • ∗
  • = 101
  • ∗
  • = 381

6

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SLIDE 31

Case 1: irrelevant MGs

Rule

kβ

− − ⇀ Observable C := MFA/Rate equation

d dt

= −kβ − kβ + · · ·

d dt[C] = −kβ[C] − kβ[F0] + · · · 7

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SLIDE 32

Case 1: irrelevant MGs

Rule

kβ

− − ⇀ Observable C := Refinement MFA/Rate equation

d dt

= −kβ − kβ + · · ·

d dt[C] = −kβ[C] − kβ[F0] + · · · 7

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SLIDE 33

Case 1: irrelevant MGs

Rule

kβ

− − ⇀ Observable C := Refinement

kβ

− − ⇀ MFA/Rate equation

d dt

= −kβ − kβ + kβ + · · ·

d dt[C] = −kβ[C] − kβ[F0] + kβ[C] + · · · 7

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SLIDE 34

Case 1: irrelevant MGs

Rule

kβ

− − ⇀ Observable C := Refinement

kβ

− − ⇀ MFA/Rate equation

d dt

= −kβ + · · ·

d dt[C] = −kβ[F0] + · · · 7

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SLIDE 35

Case 1: irrelevant MGs

Rule

kβ

− − ⇀ Observable C := Refinement MFA/Rate equation

d dt

= −kβ + · · ·

d dt[C] = −kβ[F0] + · · · 7

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SLIDE 36

Case 1: irrelevant MGs

Rule

kβ

− − ⇀ Observable C := Refinement

kβ

− − ⇀ MFA/Rate equation

d dt

= −kβ + kβ + · · ·

d dt[C] = −kβ[F0] + kβ[F0] + · · · 7

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SLIDE 37

Case 1: irrelevant MGs

Rule

kβ

− − ⇀ Observable C := Refinement

kβ

− − ⇀ MFA/Rate equation

d dt

= · · ·

d dt[C] = · · · 7

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SLIDE 38

Case 2: underivable MGs (RHS only)

Rule

kβ

− − ⇀ Observable C := MFA/Rate equation

d dt

= · · ·

d dt[C] = · · · 8

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SLIDE 39

Case 2: underivable MGs (RHS only)

Rule

kβ

− − ⇀ Observable C := Refinement MFA/Rate equation

d dt

= · · ·

d dt[C] = · · · 8

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SLIDE 40

Case 2: underivable MGs (RHS only)

Rule

kβ

− − ⇀ Observable C := Refinement

  • kβ

− − ⇀ MFA/Rate equation

d dt

= · · ·

d dt[C] = · · · 8

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SLIDE 41

Case 2: underivable MGs (RHS only)

Rule

kβ

− − ⇀ Observable C := Refinement

  • kβ

− − ⇀ MFA/Rate equation

d dt

= · · ·

d dt[C] = · · · 8

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SLIDE 42

Case 3: relevant derivable MGs

Rule

kβ

− − ⇀ Observable C := MFA/Rate equation

d dt

= · · ·

d dt[C] = · · · 9

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SLIDE 43

Case 3: relevant derivable MGs

Rule

kβ

− − ⇀ Observable C := Refinement MFA/Rate equation

d dt

= · · ·

d dt[C] = · · · 9

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SLIDE 44

Case 3: relevant derivable MGs

Rule

kβ

− − ⇀ Observable C := Refinement

kβ

− − ⇀ MFA/Rate equation

d dt

= kβ + · · ·

d dt[C] = kβ[2B] + · · · 9

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SLIDE 45

Case 3: relevant derivable MGs

Rule

kβ

− − ⇀ Observable C := Refinement

kβ

− − ⇀ MFA/Rate equation

d dt

= 2kβ + · · ·

d dt[C] = 2kβ[2B] + · · · 9

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SLIDE 46

Case 3: relevant derivable MGs

Rule

kγ

− − ⇀ Observable C := Refinement MFA/Rate equation

d dt

= 2kβ + · · ·

d dt[C] = 2kβ[2B] + · · · 9

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SLIDE 47

Case 3: relevant derivable MGs

Rule

kγ

− − ⇀ Observable C := Refinement

kγ

− − ⇀ MFA/Rate equation

d dt

= 2kβ + kγ + · · ·

d dt[C] = 2kβ[2B] + kγ[C] + · · · 9

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SLIDE 48

Case 3: relevant derivable MGs

Rule

kγ

− − ⇀ Observable C := Refinement

kγ

− − ⇀ MFA/Rate equation

d dt

= 2kβ + 2kγ

d dt[C] = 2kβ[2B] + 2kγ[C] 9

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SLIDE 49

Case 3: relevant derivable MGs

Rule

kγ

− − ⇀ Observable C := Refinement

kγ

− − ⇀ MFA/Rate equation

d dt

≃ 2kβ + 2kγ

d dt[C] ≃ 2kβ[B]2 + 2kγ[C] 9

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SLIDE 50

Bunnies with families (cont.)

Rules

kβ

− − ⇀

kγ

− − ⇀ Observables B := C := MFA/Rate equations

d dt

≃ kβ + kγ

d dt[B] ≃ kβ[B]2 + kγ[C] d dt

≃ 2kβ + 2kγ

d dt[C] ≃ 2kβ[B]2 + 2kγ[C] 10

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SLIDE 51

Bunnies with families (cont.)

Rules

kβ

− − ⇀

kγ

− − ⇀ Observables B := C := S := MFA/Rate equations

d dt

≃ kβ + kγ

d dt

= 0

d dt

≃ 2kβ + 2kγ

d dt

= 0

d dt

= 4(kβ + kγ) + 4kγ + 4kγ + 8kγ

10

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SLIDE 52

Bunnies with families (cont.)

Rules

kβ

− − ⇀

kγ

− − ⇀ Observables B := C := S := MFA/Rate equations

d dt[B] ≃ kβ[B]2 + kγ[C] d dt[F1] = 0 d dt[C] ≃ 2kβ[B]2 + 2kγ[C] d dt[F2] = 0 d dt[S] = 4(kβ + kγ)[C] + 4kγ[S] + 4kγ[F1] + 8kγ[F2] 10

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SLIDE 53

Two-legged DNA walker

kF,E kB,C kF,C kB,E G1 := G2 := G3 := V = 1

2

  • kF,E E[G1] + kF,C E[G2] − kB,E E[G3] − kB,C E[G2]
  • 11
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SLIDE 54

Minimal gluings

12

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SLIDE 55

Two-legged DNA walker (cont.)

Generate equations using relevant gluings, first try. . .

d dt

= kF,E −kB,C − kF,C + kB,E

13

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SLIDE 56

Two-legged DNA walker (cont.)

Generate equations using relevant gluings, first try. . .

d dt

= kF,E −kB,C − kF,C + kB,E

d dt

= −kF,E + kB,C + kF,C − . . .

13

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SLIDE 57

Two-legged DNA walker (cont.)

Generate equations using relevant gluings, first try. . .

d dt

= kF,E −kB,C − kF,C + kB,E

d dt

= −kF,E + kB,C + kF,C − . . .

d dt

= kF,E − kB,C − kF,C + . . .

13

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SLIDE 58

Two-legged DNA walker (cont.)

Generate equations using relevant gluings, first try. . .

d dt

= kF,E −kB,C − kF,C + kB,E

d dt

= −kF,E + kB,C + kF,C − . . .

d dt

= kF,E − kB,C − kF,C + . . .

d dt

= −kF,E + kB,C + kF,C − . . .

13

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SLIDE 59

Two-legged DNA walker (cont.)

Generate equations using relevant gluings, first try. . .

d dt

= kF,E −kB,C − kF,C + kB,E

d dt

= −kF,E + kB,C + kF,C − . . .

d dt

= kF,E − kB,C − kF,C + . . .

d dt

= −kF,E + kB,C + kF,C − . . .

d dt

= . . .

13

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SLIDE 60

Invariants and Solution

= =

d dt

= kF,E −kB,C −kF,C +kB,E

d dt

= −kF,E +kB,C +kF,C −kB,E

14

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SLIDE 61

Invariants and Solution

= =

d dt

= kF,E −kB,C −kF,C +kB,E

d dt

= −kF,E +kB,C +kF,C −kB,E Steady state: (kF,E + kB,E) E[G0] = (kF,C + kB,C) E[G2]

14

slide-62
SLIDE 62

Invariants and Solution

= =

d dt

= kF,E −kB,C −kF,C +kB,E

d dt

= −kF,E +kB,C +kF,C −kB,E Steady state: (kF,E + kB,E) E[G0] = (kF,C + kB,C) E[G2] E[G0] + E[G2] = 1

14

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SLIDE 63

Invariants and Solution

= =

d dt

= kF,E −kB,C −kF,C +kB,E

d dt

= −kF,E +kB,C +kF,C −kB,E Steady state: (kF,E + kB,E) E[G0] = (kF,C + kB,C) E[G2] E[G0] + E[G2] = 1 V = 1

2

  • (kF,E − kB,E) E[G0] + (kF,C − kB,C) E[G2]
  • = (kF,C + kB,C)(kF,E − kB,E) + (kF,E + kB,E)(kF,C − kB,C)

2(kF,E + kB,E + kF,C + kB,C)

14

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SLIDE 64

Full details. . .

. . . are in the paper.

L R T G H

f α g1 g2 β

L R S T

f1 g1 α† γ†

L R S T G H

f α f1 g1 f2 γ g2 β

d dt Ep[F] = −

  • α∈R

k(α)

  • µ∈α∗LF

Ep[ˆ µ] +

  • α∈R

k(α)

  • µ∈α∗RF

Ep[ˆ α†(µ1)].

15

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SLIDE 65

Fragger

Web app https://rhz.github.io/fragger/ Source code https://github.com/rhz/graph-rewriting/

16

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SLIDE 66

Related and future work

Site graph rewriting Differential semantics of the Kappa language.

  • Derived via abstract interpretation of ground CRN (“fragmentation”).
  • [Feret et al., 2009, Danos et al., 2010, Harmer et al., 2010].

Moment semantics Generalization to other graph-like structures.

  • Direct derivation of MFAs (no ground CRN) incl. higher moments.
  • Preliminary: support for negative application conditions (NACs).
  • Open problems: truncation; approximate model reduction.
  • [Danos et al., 2014, Danos et al., 2015a, Danos et al., 2015b].

Rule algebra Alternative approach leveraging algebraic structure of rules.

  • Developed independently by Behr and others.
  • Powerful, very general approach based on representation theory.
  • Supports irreversible systems and NACs.
  • Future work: better understand the relation between the two approaches.
  • [Behr et al., 2016, Behr and Krivine, 2020, Behr et al., 2020a, Behr et al., 2020b].

17

slide-67
SLIDE 67

Thank you!

Coauthors

  • Vincent Danos, CNRS & ENS-PSL
  • Tobias Heindel, TU Berlin
  • Ricardo Honorato-Zimmer, CINV

Checkout the Fragger web-app!

https://rhz.github.io/fragger/ https://github.com/rhz/graph-rewriting/

18

slide-68
SLIDE 68

Backup slides

18

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SLIDE 69

Rate equations

For Petri nets:

d dt E([A]) = −

  • α∈R

k(α) ρα(A)

  • (u,n)∈ρα

E(u)n +

  • α∈R

k(α) γα(A)

  • (u,n)∈γα

E(u)n

19

slide-70
SLIDE 70

Rate equations

For Petri nets:

d dt E([A]) = −

  • α∈R

k(α) ρα(A)

  • (u,n)∈ρα

E(u)n +

  • α∈R

k(α) γα(A)

  • (u,n)∈γα

E(u)n More generally,

  • S a countable set (state),
  • RS probabilities and observables, topology),
  • Q : RS → RS′ a continuous linear map (transition matrix).

d dtpT = pTQ d dt Ep(f) = d dtpTf = pTQf = Ep(Qf)

(Q f)(x) :=

y qxy(f(y) − f(x)) 19

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SLIDE 71

Rate equations

Suppose

  • A a linear subspace of RS with basis B, and
  • B is jump-closed: QB ⊆ A.

Q g =

h∈B αg,hh d dt Ep(g) = h∈B αg,h Ep(h)

  • B0 ⊆ B such that poly(B0) = A

h =

φ∈B0 βh,φ φ d dt Ep(g) ≃

  • h∈B

αg,h

  • φ∈B0

βh,φ

  • u∈φ

Ep(u)

20

slide-72
SLIDE 72

Rate equations

So, in general:

d dt Ep(g) =

  • h∈B

αg,h

  • φ∈B0

βh,φ

  • u∈φ

Ep(u) For Petri nets:

d dt E([A]) = −

  • α∈R

k(α) ρα(A)

  • (u,n)∈ρα

E(u)n +

  • α∈R

k(α) γα(A)

  • (u,n)∈γα

E(u)n

  • B0 is the set of species.

21

slide-73
SLIDE 73

Rate equations for graphs

  • B0 is the set of connected graphs

d dt E([F]) = −

  • α∈R

k(α)

  • µ∈α∗LF
  • φ∈Φ(ˆ

µ)

  • u∈φ

E(u) +

  • α∈R

k(α)

  • µ∈α∗RF
  • φ∈Φ(ˆ

µ†)

  • u∈φ

E(u) ∗ =                  , , , , , , ,                 

22

slide-74
SLIDE 74

Behr, N., Danos, V., and Garnier, I. (2016). Stochastic mechanics of graph rewriting. In Proc. 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2016), New York, NY, USA, page 46–55. ACM. Behr, N., Danos, V., and Garnier, I. (2020a). Combinatorial conversion and moment bisimulation for stochastic rewriting systems. Logical Methods in Computer Science, 16. Behr, N. and Krivine, J. (2020). Rewriting theory for the life sciences: A unifying framework for CTMC semantics. In Gadducci, F . and Kehrer, T., editors, Proc. Graph Transformation, 13th International Conference, (ICGT 2020), Bergen, Norway, volume 12150 of

  • LNCS. Springer.

Behr, N., Saadat, M. G., and Heckel, R. (2020b). Commutators for stochastic rewriting systems: Theory and implementation in Z3.

18

slide-75
SLIDE 75

CoRR arXiv:2003.11010. Danos, V., Feret, J., Fontana, W., Harmer, R., and Krivine, J. (2010). Abstracting the differential semantics of rule-based models: exact and automated model reduction. In Jouannaud, J.-P ., editor, Proc. 25th Annual IEEE Symposium on Logic in Computer Science (LICS 2010), Edinburgh, Scotland, UK, volume 0, pages 362–381. IEEE Computer Society. Danos, V., Heindel, T., Honorato-Zimmer, R., and Stucki, S. (2014). Approximations for stochastic graph rewriting. In Merz, S. and Pang, J., editors, Proc. Formal Methods and Software Engineering – 16th International Conference on Formal Engineering Methods (ICFEM 2014), Luxembourg, Luxembourg, volume 8829 of LNCS, pages 1–10. Springer. Danos, V., Heindel, T., Honorato-Zimmer, R., and Stucki, S. (2015a). Computing approximations for graph transformation systems.

18

slide-76
SLIDE 76

In Hildebrandt, T. and Miculan, M., editors, 2nd International Workshop on Meta Models for Process Languages (MeMo 2015), Grenoble, France. Pre-proceedings, pages 33–43. Electronic version available at http:

//users.dimi.uniud.it/~marino.miculan/Papers/MeMo15-preproc.pdf.

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