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Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References
Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References
What do Coxeter Groups and Boolean Networks have in Common? Matthew - - PowerPoint PPT Presentation
What do Coxeter Groups and Boolean Networks have in Common? Matthew - - PowerPoint PPT Presentation
Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References What do Coxeter Groups and Boolean Networks have in Common? Matthew Macauley Department of Mathematical Sciences CLEMSON UNIVERSITY
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Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References Definitions Examples Applications Basic Questions
Graph Dynamical Systems – GDSs:
◮ A Graph Dynamical System (see [15]) is a triple consisting of: A graph Y with vertex set v[Y ] = {1, 2, . . . , n}. For each vertex i a state xi ∈ K (e.g. K = {0, 1}) and a Y -local function Fi : K n − → K n Fi(x = (x1, x2, . . . , xn)) = (x1, . . . , xi−1, fi(x[i]) | {z }
vertex function
, xi+1, . . . , xn) . An update scheme that governs how the maps Fi are assembled to a map F: K n − → K n. ◮ Typical choices of update schemes: Parallel: Generalized Cellular Automata F(x1, . . . , xn)i = fi(x[i]) Sequential: Sequential Dynamical Systems [FY , w] = Fw(k) ◦ Fw(k−1) ◦ · · · ◦ Fw(1) (w = w(1) · · · w(k) – a word on v[Y ]) (Local dynamics)
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Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References Definitions Examples Applications Basic Questions
An Example:
Graph Y = Circ4 State set K = {0, 1} System state x = (x1, x2, x3, x4) Restricted vertex state x[1] = (x1, x2, x4) Vertex functions: fi = nor3 : K 3 − → K by nor3(x, y, z) = (1 + x)(1 + y)(1 + z) Y -local maps: Nor1(x) = (nor3(x[1]), x2, x3, x4), etc. Update sequence π = (1, 2, 3, 4) SDS map: [NorY , π] = Nor4 ◦ Nor3 ◦ Nor2 ◦ Nor1 Sequential: [NorY , π](0, 0, 0, 0) = (1, 0, 1, 0) Parallel: Nor(0, 0, 0, 0) = (1, 1, 1, 1)
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Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References Definitions Examples Applications Basic Questions
- Remark. The global dynamics can be sensitive to changes in the update order.
◮ [NorCirc4, π] for given update sequences:
1000 0010 0100 0001 1010 0000 0101 0011 1011 0111 1111 1101 (1234) 0110 1110 1001 1100 (1423) 0000 1100 0110 0010 1000 0101 1010 1010 1101 1110 1111 1011 0111 1001 0100 0001
0111 1111 1101 1010 0000 0101 0010 1000 1110 1100 0110 0011 1001 1011 0001 0100 (1324)
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Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References Definitions Examples Applications Basic Questions
Applications of SDSs
Large complex networks [15].
- Epidemiology. Disease propogation over social contact graphs.
Agent-based transportation simulations. Packet flow in wireless networks.
Gene annotation (Functional Linkage Networks) [8] Transport computation on irregular grids (e.g., heat, radiation). Image processing and pattern recognition [3]. Discrete event simulations (e.g., chemical reaction networks) [6].
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Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References Definitions Examples Applications Basic Questions
Fundamental Questions
Let S∗ be the free monoid generated by v[Y ] (i.e., words in the vertex set of Y ). There are infinitely many words w = w1w2w3 · · · wn in S∗ . . . . . . but only finitely many functions Fn
2 −
→ Fn
2.
Question 1. When are two SDS maps [FY , w] and [FY , w′] “the same”? Question 2. What do we really mean by “the same”?
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Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References Definitions Examples Applications Basic Questions
◮ Question 2: What does it mean for two SDSs to be “the same”? (see [12])
Definition
Two SDSs are functionally equivalent if their SDS maps are identical as functions K n − → K n.
Definition
Two finite dynamical systems φ, ψ: K n − → K n are dynamically equivalent if there is a bijection h: K n − → K n such that ψ ◦ h = h ◦ φ . (i.e., phase spaces are isomorphic).
Definition
Two finite dynamical systems φ, ψ: K n → K n are cycle equivalent if there exists a bijection h: Per(φ) − → Per(ψ) such that ψ|Per(ψ) ◦ h = h ◦ φ|Per(φ) . (i.e., phase spaces are isomorphic when restricted to the periodic points).
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Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References Definitions The word problem & Matsumoto’s theorem
Coxeter groups
Definition
A Coxeter group is a group with presentation s1, . . . , sn | s2
i = 1,
sis
m(si ,sj ) j
= 1 where mij := m(si, sj) ≥ 2 if i = j. m(s, t) = 2 = ⇒ st = ts (short braid relation) m(s, t) = 3 = ⇒ sts = tst (braid relation) m(s, t) = 4 = ⇒ stst = tsts (braid relation) . . . ◮ A Coxeter group is a generalized reflection group.
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Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References Definitions The word problem & Matsumoto’s theorem
The word problem for Coxeter groups
There are infinitely many words w = w1w2w3 · · · wn in S∗ . . . . . . and sometimes (usually), infinitely many group elements in W .
- Question. Given two words w = s1s2 · · · sn and w′ = s′
1s′ 2 · · · in S∗, when do they give
rise to the same group element? This question has an simple answer (see [4]): Matsumoto’s Theorem: Any two reduced expressions for the same group element differ by braid relations.
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Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References Acyclic orientations and source-to-sink moves Dynamics groups of SDSs The Root automaton of a Coxeter group
Role of acyclic orientations in Coxeter groups and SDSs
Coxeter groups Sequential dynamical systems Base graph ← → Coxeter graph Γ Dependency graph Y Acyc(Γ) ← → Coxeter elements SDS maps c = sπ(1)sπ(2) · · · sπ(n) [FY , π] = Fπ(n) ◦ · · · ◦ Fπ(2) ◦ Fπ(1). Source-to- ← → Conjugacy classes Cycle-equivalence classes sink moves
- f Coxeter elements
- f SDS maps
Aut(Γ) ← → Spectral classes Cycle-equivalence classes
- rbits
- f Coxeter elements
- f SDS maps (finer)
This can be extended beyond Coxeter elements using labeled heaps instead of acyclic
- rientations.
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Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References Acyclic orientations and source-to-sink moves Dynamics groups of SDSs The Root automaton of a Coxeter group
SDS dynamics revealed as quotients of Coxeter groups
◮ A sequence FY is w-independent if Per[FY , w] = Per[FY , w′] for all w and w′ in S∗.
Proposition
If FY is w-independent, then each Fi is bijective on P := Per(FY ). ◮ Let G(FY ) be the group of permutations of P generated by {F1, . . . , Fn}. This is called the dynamics group of FY . ◮ If K = F2, the dynamics group is the homomorphic image of a Coxeter group, because |Fi| ≤ 2 and |FiFj| = mij. (see [10]) ◮ If K = F2, the dynamics group is the homomorphic image of an Artin group.
- Question. Can we determine this homomorphism, i.e., the “extra relations”?
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Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References Acyclic orientations and source-to-sink moves Dynamics groups of SDSs The Root automaton of a Coxeter group
The root automaton of a Coxeter group (an infinite SDS)
Let Φ = Φ+ ⊔ Φ− be the set of all roots of a Coxeter group W . We can represent roots as vectors in Rn and partially order them by ≤ componentwise, to get the root poset. Each generator si ∈ S acts on Φ by reflection: z
si
− → z +
n
X
j=1
2 cos „ π mij « zjei . ◮ In summary, multiplication by si flips the sign of the i th entry and adds each neighboring state zj weighted by 2 cos(π/mij) ≥ 1. We can represent this as a local function Fi : Rn − → Rn: Fi : (z1, . . . zi−1, zi, zi+1, zn) − → (z1, . . . , zi−1, zi +
n
X
j=1
2 cos „ π mij « zj, zi+1, . . . , zn) .
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Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References Acyclic orientations and source-to-sink moves Dynamics groups of SDSs The Root automaton of a Coxeter group
The root automaton detecting reduced words
Consider a word w = sx1sx2sx3 · · · sxk−1sxk ∈ S∗. Start at the vector ex1 ∈ Φ+ (a positive root), and follow the paths labeled sx2, sx3, sx4, . . . . If sxk is the first instance of crossing over to the negative roots Φ−, then w is not a reduced expression, and moreover, sx1sx2sx3 · · · sxk−1sxk = sx2sx3 · · · sxk−1 . ◮ Thus, the root automaton detects reduced words.
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Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References Acyclic orientations and source-to-sink moves Dynamics groups of SDSs The Root automaton of a Coxeter group
The SDS root automaton of type H5.
a b c d e 5 H5 = a, b, c, d, e | a2 = b2 = c2 = d2 = e2, (ab)5, (bc)3 = (cd)3 = (de)3, (ac)2 = (ad)2 = (ae)2, (bd)2 = (be)2 = (ce)2
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Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References
Summary
◮ People have studied various forms of composed dynamical systems (asychronous cel- lular automata, sequential dynamical systems, Boolean networks, etc), using “traditional techniques.” Borrowing ideas from Coxeter theory is a relatively new and novel approach to view these systems (see [10, 12] Some of the combinatorial aspects of Coxeter groups themselves are quite new (within the past 5 years). Some of the question arising in these discrete dynamical systems have led to new research and new results on Coxeter groups (see [9]).
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Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References
Acknowledgments
Collaborators: Chris Barrett, Tom Boothbyg , Jeffrey Burkettug , Morgan Eichwaldug , Dana Ernst, Richard Green, Anders Hansson, V.S. Anil Kumar, Reinhard Laubenbacher, Jon McCammond, Henning S. Mortveit, Christian Reidys. Institutions: Clemson, Colorado, Harvey Mudd, Institute for Combinatorics at Nankai University, Institute for Systems Biology, Los Alamos National Lab, Plymouth State, Montana, University of California Santa Barbara, Virginia Tech, Washington. Special thanks: Clemson University http://www.clemson.edu Virginia Tech http://www.vt.edu
Viginia Bioinformatics Institute http://www.vbi.vt.edu Network Dynamics and Simulation Science Laboratory http://ndssl.vbi.vt.edu
Ilya Shmulevich’s research group @ The Institute for Systems Biology http://[shmulevich.]systemsbiology.net/ University of Washington REU in Mathematics (Sara Billey & James Morrow) www.math.washington.edu/~reu/
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Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References
References
[1]
- C. L. Barrett, H. S. Mortveit, and C. M. Reidys. Elements of a theory of simulation III: Equivalence of SDS. Appl. Math. Comput.
122, 2001:325–340. [2]
- C. L. Barrett, H. S. Mortveit, and C. M. Reidys. Elements of a theory of simulation IV: Fixed points, invertibility and equivalence.
- Appl. Math. Comput. 134, 2003:153–172.
[3]
- J. Besag. On the statistical analysis of dirty pictures. J. Royal Statist. Soc. Ser. B 48, 1986:259–302.
[4]
- A. Bj¨
- rner and F. Brenti. Combinatorics of Coxeter Groups. Springer Verlag GTM, 2005.
[5]
- T. Boothby, J. Burkett, M. Eichwald, D.C. Ernst, R.M. Green, and M. Macauley. On the cyclically fully commuative elements of
Coxeter groups, Submitted, 2009. [6]
- D. T. Gillespie. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 1977:2340–2361.
[7]
- A. ˚
- A. Hansson, H. S. Mortveit, and C. M. Reidys. On asynchronous cellular automata. Adv. Comp. Sys. 8 (4), 2005:521–538.
[8]
- U. Karaoz, T. Murali, S. Letovsky, Y. Zheng, C. Ding, C. R. Cantor, S. Kasif. Whole-genome annotation by using evidence integration
in functional-linkage networks. Proc. Nat. Acad. Sci. 101, 2004:2888–2893. [9]
- M. Macauley, J. McCammond, and H. S. Mortveit. Order independence in asynchronous cellular automata. J. Cell. Autom. 3 (1),
2008:37–56. [10]
- M. Macauley, J. McCammond, and H. S. Mortveit. Dynamics groups of finite asynchronous cellular automata. Submitted, 2009.
[11]
- M. Macauley, H. S. Mortveit. Admissible Coxeter sequences and their applications.. Submitted, 2009.
[12]
- M. Macauley, H. S. Mortveit. Cycle equivalence of graph dynamical systems. Nonlinearity 22, 2009:421–436.
[13]
- M. Macauley, H. S. Mortveit. On enumeration of conjugacy classes of Coxeter elements. Proc. Amer. Math. Soc. 136, 2008:4157–
4165. [14]
- M. Macauley, and H. S. Mortveit. Update sequence stability in graph dynamical systems. Submitted, 2009.
[15]
- H. S. Mortveit, C. M. Reidys. An introduction to sequential dynamical systems. Springer Verlag, 2007.