dynamics of networks 2 synchrony & balanced colourings ian stewart mathematics institute...

Dynamics of Networks 2Synchrony &

Balanced Colourings

Dynamics of Networks 2Synchrony &

Balanced Colourings

Ian Stewart

Mathematics InstituteUniversity of Warwick

Ian Stewart


UK-Japan Winter SchoolDynamics and Complexity

NetworkNetwork

A A networknetwork or or directed graphdirected graph

consists of a set of:consists of a set of:

•• nodesnodes or or verticesvertices or or cellscells

connected byconnected by

•• directeddirected edgesedges or or arrowsarrows

dxdx11/dt = f(/dt = f(xx11,,xx22,,xx44,,xx55))


dxdx33/dt = g(/dt = g(xx33,,xx11,,xx44))


dxdx55/dt = h(/dt = h(xx55,,xx44))

12453

Admissible ODEsAdmissible ODEs

Pattern of SynchronyPattern of Synchrony

Cells Cells cc, , dd are are synchronoussynchronous on some on some trajectory trajectory xx((tt) if) if

xxcc(t) = (t) = xxdd((tt) for all ) for all tt

Defines an equivalence relation ~ for whichDefines an equivalence relation ~ for whichc ~ d if and only if c ~ d if and only if cc, , dd are synchronous are synchronous

Call ~ a Call ~ a pattern of synchronypattern of synchrony

Colouring InterpretationColouring Interpretation

More intuitively, colour cells withMore intuitively, colour cells withdifferent different colourscolours, so that cells, so that cellshave the have the same coloursame colour if and only if they if and only if theyare synchronousare synchronous

Synchrony Space Synchrony Space oror Polydiagonal Polydiagonal~~ = { = {xx : : xxcc = = xxdd whenever whenever cc ~ ~ dd}}

This is the set of all cell states with the This is the set of all cell states with the pattern of synchrony ~pattern of synchrony ~

It forms a vector subspace of phase space It forms a vector subspace of phase space PP





dxdx55/dt = h(/dt = h(xx55,,xx44))12453





dxdx55/dt = h(/dt = h(xx55,,xx33))12453

xx11=x=x22 x x33=x=x44



dxdx55/dt = h(/dt = h(xx55,,xx33))12453

restricted equationsrestricted equations



dxdx55/dt = h(/dt = h(xx55,,xx33))12453

restricted to restricted to synchrony synchrony spacespace {(x {(x11,x,x11,x,x33,x,x33,x,x55)})}



dxdx55/dt = h(/dt = h(xx55,,xx33))12453

restricted to restricted to synchrony synchrony spacespace {(x {(x11,x,x11,x,x33,x,x33,x,x55)})}

153

quotient quotient networknetwork

States on the synchrony space satisfy the States on the synchrony space satisfy the restricted equations and have the pattern restricted equations and have the pattern of synchrony determined by the colouringof synchrony determined by the colouring

12453 The dynamics of these The dynamics of these states is determined by states is determined by the quotient networkthe quotient network

A pattern of synchrony ~ is A pattern of synchrony ~ is robustrobust if if ~~ is is

invariant under invariant under allall admissible vector fields admissible vector fields

That is: any initial condition with that pattern That is: any initial condition with that pattern continues to have that pattern for all timecontinues to have that pattern for all time

Robust SynchronyRobust Synchrony

A pattern of synchrony ~ (or the associated A pattern of synchrony ~ (or the associated colouring) is colouring) is balancedbalanced if cells with the same if cells with the same colour have input sets with the same colours colour have input sets with the same colours (of tail cells) up to some type-preserving (of tail cells) up to some type-preserving permutation (input isomorphism)permutation (input isomorphism)

TheoremTheoremA pattern of synchrony ~ is robust if and A pattern of synchrony ~ is robust if and

only if it is balancedonly if it is balanced

Balanced Patterns and ColouringsBalanced Patterns and Colourings

Balanced colouringBalanced colouring: cells with same colour : cells with same colour have have input setsinput sets with same colours with same colours

This colouring is not balancedThis colouring is not balanced

Nearest-neighbour couplingNearest-neighbour couplingBalanced 2-colouringBalanced 2-colouring

Example—the Square LatticeExample—the Square Lattice

For this pattern, interchanging black and white For this pattern, interchanging black and white along any upward sloping diagonal leads to along any upward sloping diagonal leads to another balanced 2-colouringanother balanced 2-colouring

Example—the Square LatticeExample—the Square Lattice

The Origin of SpeciesThe Origin of Species

AllopatricAllopatric speciation speciationGeographical or other Geographical or other

barrier prevents gene-flowbarrier prevents gene-flow

Easily Easily understood understood and and WYSIWYGWYSIWYG


SympatricSympatric speciation speciationNo such barrier; No such barrier;

species split in the same species split in the same location even with location even with panmixispanmixis

SympatricSympatric speciation speciation appears paradoxical, but appears paradoxical, but can be viewed as a form of can be viewed as a form of symmetry-breakingsymmetry-breaking


Darwin’s FinchesDarwin’s Finches

Galápagos IslandsGalápagos Islands

Crossman / Los Hermanos


1. Large cactus finch (1. Large cactus finch (Geospiza conirostrisGeospiza conirostris))2. Large ground finch (2. Large ground finch (Geospiza magnirostrisGeospiza magnirostris))3. Medium ground finch (3. Medium ground finch (Geospiza fortisGeospiza fortis))4. Cactus finch (4. Cactus finch (Geospiza scandensGeospiza scandens))5. Sharp-beaked ground finch (5. Sharp-beaked ground finch (Geospiza difficilisGeospiza difficilis))6. Small ground finch (6. Small ground finch (Geospiza fuliginosaGeospiza fuliginosa))7. Woodpecker finch (7. Woodpecker finch (Cactospiza pallidaCactospiza pallida))8. Vegetarian tree finch (8. Vegetarian tree finch (Platyspiza crassirostrisPlatyspiza crassirostris))9. Medium tree finch (9. Medium tree finch (Camarhynchus pauperCamarhynchus pauper))10. Large tree finch (10. Large tree finch (Camarhynchus psittaculaCamarhynchus psittacula))11. Small tree finch (11. Small tree finch (Camarhynchus parvulusCamarhynchus parvulus))12. Warbler finch (12. Warbler finch (Certhidia olivaceaCerthidia olivacea))13. Mangrove finch (13. Mangrove finch (Cactospiza heliobatesCactospiza heliobates))


Interaction networkInteraction network — all-to-all coupled — all-to-all coupled in the panmictic casein the panmictic case


With generic nonlinear With generic nonlinear dynamics, instability of the dynamics, instability of the single-species state leads to single-species state leads to symmetry-breaking symmetry-breaking bifurcation. Universal features bifurcation. Universal features are:are:

Split is a Split is a jump bifurcationjump bifurcationMean phenotypes stay Mean phenotypes stay constantconstantUsual split is to two speciesUsual split is to two species


Simulation with 50 nodesSimulation with 50 nodes


Geospiza fuliginosaGeospiza fuliginosaGeospiza fortisGeospiza fortis

Albemarle/Albemarle/IsabelaIsabela

DaphneDaphne

Crossman/Crossman/Los HermanosLos Hermanos

Beak size in millimetresBeak size in millimetres

Character displacementCharacter displacementNot speciation, but might be Not speciation, but might be OK as a surrogateOK as a surrogate


J.Cohen and I.Stewart. Polymorphism viewed as phenotypic J.Cohen and I.Stewart. Polymorphism viewed as phenotypic symmetry-breaking, in symmetry-breaking, in Nonlinear Phenomena in Biological and Nonlinear Phenomena in Biological and Physical SciencesPhysical Sciences (eds. S.K.Malik, M.K.Chandra-sekharan, and (eds. S.K.Malik, M.K.Chandra-sekharan, and N.Pradhan), Indian National Science Academy, New Delhi 2000, N.Pradhan), Indian National Science Academy, New Delhi 2000, 1-631-63 I.Stewart. Self-organization in evolution: a mathematical I.Stewart. Self-organization in evolution: a mathematical perspective, Nobel Symposium Proceedings, perspective, Nobel Symposium Proceedings, Phil. Trans. Roy. Phil. Trans. Roy. Soc. Lond. ASoc. Lond. A 361 361 (2003) 1101-1123.(2003) 1101-1123.

I.Stewart., T.Elmhirst and J.Cohen. Symmetry-breaking I.Stewart., T.Elmhirst and J.Cohen. Symmetry-breaking as an origin of species, in as an origin of species, in Bifurcations, Symmetry, and PatternsBifurcations, Symmetry, and Patterns (eds. J. Buescu, S. Castro, A.P.S. Dias, and I. Labouriau), (eds. J. Buescu, S. Castro, A.P.S. Dias, and I. Labouriau), Birkhäuser, Basel 2003, 3-54.Birkhäuser, Basel 2003, 3-54. I.Stewart. Speciation: a case study in symmetric bifurcation I.Stewart. Speciation: a case study in symmetric bifurcation theory, theory, Univ. Iagellonicae Acta Math.Univ. Iagellonicae Acta Math. 4141 (2003) 67-88. (2003) 67-88.

J.Cohen and I.Stewart. Polymorphism viewed as phenotypic J.Cohen and I.Stewart. Polymorphism viewed as phenotypic symmetry-breaking, in symmetry-breaking, in Nonlinear Phenomena in Biological and Nonlinear Phenomena in Biological and Physical SciencesPhysical Sciences (eds. S.K.Malik, M.K.Chandra-sekharan, and (eds. S.K.Malik, M.K.Chandra-sekharan, and N.Pradhan), Indian National Science Academy, New Delhi 2000, N.Pradhan), Indian National Science Academy, New Delhi 2000, 1-631-63 I.Stewart. Self-organization in evolution: a mathematical I.Stewart. Self-organization in evolution: a mathematical perspective, Nobel Symposium Proceedings, perspective, Nobel Symposium Proceedings, Phil. Trans. Roy. Phil. Trans. Roy. Soc. Lond. ASoc. Lond. A 361 361 (2003) 1101-1123.(2003) 1101-1123.

I.Stewart., T.Elmhirst and J.Cohen. Symmetry-breaking I.Stewart., T.Elmhirst and J.Cohen. Symmetry-breaking as an origin of species, in as an origin of species, in Bifurcations, Symmetry, and PatternsBifurcations, Symmetry, and Patterns (eds. J. Buescu, S. Castro, A.P.S. Dias, and I. Labouriau), (eds. J. Buescu, S. Castro, A.P.S. Dias, and I. Labouriau), Birkhäuser, Basel 2003, 3-54.Birkhäuser, Basel 2003, 3-54. I.Stewart. Speciation: a case study in symmetric bifurcation I.Stewart. Speciation: a case study in symmetric bifurcation theory, theory, Univ. Iagellonicae Acta Math.Univ. Iagellonicae Acta Math. 4141 (2003) 67-88. (2003) 67-88.

Equilibrium StatesEquilibrium Statesx*x*is an is an equilibrium equilibrium state for an admissible state for an admissible ff

if and only ifif and only ifff((xx*) = 0*) = 0

Associated Pattern of SynchronyAssociated Pattern of SynchronyUse the cell coordinates of Use the cell coordinates of xx* to define a * to define a

colouring: then colouring: then cc and and dd have the same colour if have the same colour if and only ifand only if

xx**cc = = xx**dd

Determines an equivalence relation ~Determines an equivalence relation ~xx**

Rigid EquilibriaRigid Equilibria

An equilibrium An equilibrium xx* is * is hyperbolichyperbolic if the Jacobian D if the Jacobian Dff||

xx** has no eigenvalues on the imaginary axis (and in has no eigenvalues on the imaginary axis (and in

particular is nonsingular).particular is nonsingular).

TheoremTheorem If x* is hyperbolic and If x* is hyperbolic and gg = = ff++pp is a small is a small

admissible perturbation of admissible perturbation of ff, then , then gg has a has a uniqueunique zero zero yy* near * near xx*.*.

HyperbolicityHyperbolicity

Let Let x*x*be a hyperbolic equilibrium state for an be a hyperbolic equilibrium state for an admissible admissible ff. Then the pattern of synchrony ~. Then the pattern of synchrony ~xx* * is is rigidrigid

if it is the same as ~if it is the same as ~yy* * for the nearby equilibrium y* of for the nearby equilibrium y* of

any sufficiently small perturbation any sufficiently small perturbation gg..

RigidityRigidity

More carefully: if More carefully: if x*x*is a hyperbolic equilibrium state for an is a hyperbolic equilibrium state for an admissible admissible ff, we can define its , we can define its rigid pattern of synchronyrigid pattern of synchrony to to be ~be ~xx**

rigrig, where , where cc ~ ~xx**rigrig d d if and only if if and only if

yycc = = yydd

for every nearby equilibrium for every nearby equilibrium yy* of any sufficiently small * of any sufficiently small perturbation perturbation gg..

Then ~Then ~xx**rigrig is rigid. is rigid.

RigidityRigidity

TheoremTheorem [Rigid implies balanced] [Rigid implies balanced]Let Let x* x* be a hyperbolic equilibrium state for be a hyperbolic equilibrium state for

an admissible an admissible ff. Then ~. Then ~xx**rigrig is balanced. is balanced.

The proof is nontrivial: see Golubitsky, Stewart, and The proof is nontrivial: see Golubitsky, Stewart, and Török. Török.

Stronger Result—TransversalityStronger Result—TransversalityAldis (unpublished) has proved a similar theorem with Aldis (unpublished) has proved a similar theorem with ‘hyperbolic’ replaced by ‘transverse’, which means that ‘hyperbolic’ replaced by ‘transverse’, which means that the Jacobian Dthe Jacobian Dff||xx** has no zero eigenvalues (is has no zero eigenvalues (is

nonsingular). This is stronger because nonsingular). This is stronger because Hopf bifurcationHopf bifurcation points (nonzero imaginary eigenvalues) are permitted.points (nonzero imaginary eigenvalues) are permitted.

Rigid Periodic StatesRigid Periodic States

Is there an analogous theorem for periodic Is there an analogous theorem for periodic states?states?

A periodic state x(t) is A periodic state x(t) is hyperbolichyperbolic if its linearized if its linearized Poincaré map (Floquet map) has no eigenvalues on the Poincaré map (Floquet map) has no eigenvalues on the unit circle.unit circle.

A pattern of synchrony, or of phase shifts, is A pattern of synchrony, or of phase shifts, is rigidrigid if it if it persists after any sufficiently small admissible persists after any sufficiently small admissible perturbation.perturbation.

Rigid Synchrony ConjectureRigid Synchrony Conjecture

ConjectureConjectureIf x(t) is a hyperbolic periodic state, then If x(t) is a hyperbolic periodic state, then

rigidly synchronous cells have synchronous input rigidly synchronous cells have synchronous input cells (up to some input isomorphism).cells (up to some input isomorphism).

TheoremTheoremIf the RSC is true, and x(t) is a hyperbolic If the RSC is true, and x(t) is a hyperbolic

periodic state, then the relation of periodic state, then the relation of rigid rigid synchronysynchrony is balanced. is balanced.

Rigid Phase ConjectureRigid Phase Conjecture

ConjectureConjectureIf x(t) is a hyperbolic periodic state, then If x(t) is a hyperbolic periodic state, then

rigidly phase-related cells have phase-related rigidly phase-related cells have phase-related input cells (up to some input isomorphism) with input cells (up to some input isomorphism) with the same phase relations.the same phase relations.

TheoremTheoremIf the RPC is true, then the quotient If the RPC is true, then the quotient

network (for the relation of rigid synchrony) has a network (for the relation of rigid synchrony) has a cyclic group of global symmetries, and the phase cyclic group of global symmetries, and the phase relations are among those for a dynamical relations are among those for a dynamical system with this symmetry .system with this symmetry .

Dynamics of NetworksDynamics of Networksto be continued...to be continued...

Dynamics of NetworksDynamics of Networksto be continued...to be continued...

Ian Stewart


Ian Stewart



NetworkNetworkEach cell has a Each cell has a cell-typecell-type and each arrow has an and each arrow has an arrow-typearrow-type, allowing us , allowing us to require the cells or to require the cells or arrows concerned to arrows concerned to have ‘the same’ have ‘the same’ structure. In effect these structure. In effect these are are labelslabels on the cells on the cells and arrows. Abstractly and arrows. Abstractly they are specified by they are specified by equivalence relationsequivalence relations on on the set of cells and the the set of cells and the set of arrows.set of arrows.

NetworkNetworkArrows may form Arrows may form loopsloops (same head and tail), (same head and tail), and there may be and there may be multiple arrowsmultiple arrows (connecting (connecting the same pair of cells).the same pair of cells).

Special case: Special case: regular homogeneous networksregular homogeneous networks. .

These have one type of cell, one type of arrow, These have one type of cell, one type of arrow, and the number of arrows entering each cell is and the number of arrows entering each cell is the same.the same.

This number is the This number is the valencyvalency of the network. of the network.

Regular Homogeneous Regular Homogeneous NetworkNetworkThis is a regular homogeneous network of valency 3. This is a regular homogeneous network of valency 3.

12345

Network EnumerationNetwork EnumerationNN vv=1=1 vv=2=2 vv=3=3 vv=4=4 vv=5=5 vv=6=6

11 11 11 11 11 11 11

22 33 66 1010 1515 2121 2828

33 77 4444 180180 590590 15821582 37243724

44 1919 475475 69156915 6342063420 412230412230 20808272080827

55 4747 68746874 444722444722 104072268104072268 265076184265076184 34056654123405665412

66 130130 126750126750 4324260443242604 55696772105569677210 355906501686355906501686 1350853483470413508534834704

Number of topologically distinct regular homogeneous Number of topologically distinct regular homogeneous networks on networks on NN cells with valency cells with valency vv

Network Network DynamicsDynamics

To any network we associate a class of To any network we associate a class of admissible vector fieldsadmissible vector fields, defining , defining admissible admissible ODEsODEs, which consists of those vector fields, which consists of those vector fields

FF((xx))

That respect the network structure, and the That respect the network structure, and the corresponding ODEscorresponding ODEs

ddxx/d/dtt = = FF((xx))

What does ‘respect the network structure’ mean?

Admissible Admissible ODEsODEs

The The input setinput set II((cc)) of a cell of a cell cc is the set of all arrows is the set of all arrows whose head is whose head is cc..

This This includesincludes multiple multiple arrows and loops. arrows and loops.


Choose coordinates Choose coordinates xxcc R Rkk for each cell for each cell cc. .

(We use (We use RRkk for simplicity, and because we for simplicity, and because we consider only consider only locallocal bifurcation). Then bifurcation). Then

ddxxcc/d/dtt = = ffcc ((xxcc,,xxTT((II ( (cc))))))

where where TT((II((cc)))) is the tuple of tail cells of is the tuple of tail cells of II((cc))..


ddxx11/d/dtt = = ffcc ((xx11,,xx11, , xx22, , xx33, , xx33, , xx44, , xx55, , xx55, , xx55))

12345

ddxxcc/d/dtt = = ffcc ((xxcc,,xxTT((II ( (cc))))))

Admissible Admissible ODEsODEs for the example network:for the example network:

ddxx11/d/dtt = = ff((xx11, , xx22,, xx22,, xx33) )




ddxx55/d/dtt = = ff((xx55, , xx22,, xx44,, xx44))

Where Where ff satisfies the symmetry condition satisfies the symmetry condition

ff((xx,,uu,,vv,,ww)) is symmetric in is symmetric in uu, , vv, , ww

12345

dxdx11/dt = f/dt = f11((xx11,x,x22,x,x44,x,x55))

dxdx22/dt = f/dt = f22((xx22,x,x11,x,x33,x,x55))

dxdx33/dt = f/dt = f33((xx33,x,x11,x,x44))

dxdx44/dt = f/dt = f44((xx44,x,x22,x,x44))

dxdx55/dt = f/dt = f55((xx55,x,x44))12453

domain conditiondomain condition

dxdx11/dt = f(/dt = f(xx11,x,x22,x,x44,x,x55))

dxdx22/dt = f(/dt = f(xx22,x,x11,x,x33,x,x55))

dxdx33/dt = g(/dt = g(xx33,x,x11,x,x44))

dxdx44/dt = g(/dt = g(xx44,x,x22,x,x44))

dxdx55/dt = h(/dt = h(xx55,x,x44))12453

pullback conditionpullback condition





dxdx55/dt = h(/dt = h(xx55,,xx44))12453

Vertex groupVertex group symmetry symmetry

[with M.Golubitsky and M.Nicol] Some curious phenomena [with M.Golubitsky and M.Nicol] Some curious phenomena in coupled cell networks, in coupled cell networks, J. Nonlin. SciJ. Nonlin. Sci. . 1414 (2004) 207- (2004) 207-236236..

[with M.Golubitsky and A.Török] Patterns of synchrony in [with M.Golubitsky and A.Török] Patterns of synchrony in coupled cell networks with multiple arrows, coupled cell networks with multiple arrows, SIAM J. Appl. SIAM J. Appl. Dyn. Sys.Dyn. Sys. 44 (2005) 78-100. [DOI: 10.1137/040612634] (2005) 78-100. [DOI: 10.1137/040612634]

[with M.Golubitsky] Nonlinear dynamics of networks: the [with M.Golubitsky] Nonlinear dynamics of networks: the groupoid formalism, groupoid formalism, Bull. Amer. Math. SocBull. Amer. Math. Soc. . 4343 (2006) (2006) 305-364.305-364.

[[with M.Parker] Periodic dynamics of coupled cell networks I: with M.Parker] Periodic dynamics of coupled cell networks I: rigid patterns of synchrony and phase relations, rigid patterns of synchrony and phase relations, Dynamical SystemsDynamical Systems 2222 (2007) 389-450. (2007) 389-450.

[with M.Parker] Periodic dynamics of coupled cell networks [with M.Parker] Periodic dynamics of coupled cell networks II: cyclic symmetry, II: cyclic symmetry, Dynamical SystemsDynamical Systems 2323 (2008) 17-41. (2008) 17-41.

ReferencesReferences

Dynamics of Networksto be continued...

Dynamics of Networksto be continued...

Ian Stewart


Ian Stewart



dynamics of networks 2 synchrony & balanced colourings ian stewart mathematics institute...

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