Patterns of non-normality in networked systems

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Patterns of non-normality in networked systems

Muolo, Riccardo; Asllani, Malbor; Fanelli, Duccio; Maini, Philip; Carletti, Timoteo

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Journal of Theoretical Biology

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2019

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Muolo, R, Asllani, M, Fanelli, D, Maini, P & Carletti, T 2019, 'Patterns of non-normality in networked systems',

Journal of Theoretical Biology, vol. 480, pp. 81-91.

<https://www.sciencedirect.com/science/article/pii/S0022519319302814?via%3Dihub>

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ContentslistsavailableatScienceDirect

Journal

of

Theoretical

Biology

journalhomepage:www.elsevier.com/locate/jtb

Patterns

of

non-normality

in

networked

systems

Riccardo

Muolo

a

_,

_Malbor

_Asllani

b,c,∗

_,

_Duccio

_Fanelli

d

_,

_Philip

_K.

_Maini

b

_,

_Timoteo

_Carletti

e a Systems Bioinformatics, Vrije Universiteit Amsterdam, De Boelelaan 1108, 1081 HZ, Amsterdam, the Netherlands

b Mathematical Institute, University of Oxford, Woodstock Rd, OX2 6GG Oxford, UK

c MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick V94 T9PX, Ireland

d Dipartimento di Fisica e Astronomia, Universitá di Firenze, INFN and CSDC, Via Sansone 1, 50019 Sesto Fiorentino, Firenze, Italy

e Department of Mathematics and naXys, Namur Institute for Complex Systems, University of Namur, rempart de la Vierge 8, B 50 0 0 Namur, Belgium

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 6 December 2018 Revised 5 July 2019 Accepted 8 July 2019 Available online 8 July 2019

Keywords: Pattern formation Turing instability Non-normal networks Reaction-diffusion systems

a

b

s

t

r

a

c

t

Severalmechanismshave beenproposed toexplain thespontaneous generationofself-organised pat-terns,hypothesisedtoplayaroleintheformationofmanyofthemagnificentpatternsobservedin Na-ture.Inseveralcasesofinterest,thesystemunderscrutinydisplaysahomogeneousequilibrium,which isdestabilised viaasymmetrybreaking instabilitywhichreflects thespecificity oftheproblembeing inspected.The Turinginstabilityisamongthe mostcelebratedparadigmsforpattern formation.Inits originalform,the diffusionconstants ofthetwo mobile speciesneedto bequite differentfromeach otherfortheinstabilitytodevelop.Unfortunately,thiscondition limitsthe applicabilityofthetheory. Toovercomethisimpediment,andwiththeambitiouslongtermgoaltoeventuallyreconciletheoryand experiments,wehereproposeanalternativemechanismforpromotingtheonsetofpattern.Tothisend amulti-species reactivemodelisstudied, assumingageneralizedtransportonadiscreteanddirected network-likesupport:theinstabilityistriggeredbythenon-normality ofthe embeddingnetwork.The non-normalcharacterofthedynamicsinstigatesashorttimeamplificationoftheimposedperturbation, thusmakingthesystemunstablefor achoiceofparametersthatwouldyield stabilityunder the con-ventionalscenario.Inotherwords,non-normalitypromotestheemergenceofpatternsincaseswherea classicallinearanalysiswouldnotpredictthem.Theimportanceofourresultreliesalsoonthefactthat non-normalnetworksarepervasivelyfound,motivatingthegeneralinterestofthemechanismdiscussed here.

© 2019ElsevierLtd.Allrightsreserved.

1. Introduction

Wearesurroundedbyspatiallyheterogeneouspatterns(Nicolis andPrigogine,1977;Murray,2001). Inmanyapplications,the in-terplay between microscopic entities is modelled by means of reaction-diffusion equations-partial differential equations which govern the deterministic evolution of the concentrations in a multi-species model, across time andspace. Homogeneous equi-libria ofa genericreaction-diffusion systemmayundergo a sym-metry breakinginstability,when exposedtoa heterogeneous per-turbation,andthisunderliestheself-organisationtheoryofpattern formation.

The instability can develop according to different modalities, depending on the specificity of the system under consideration. Amongexistingapproaches,thecelebratedTuringinstability

occu-∗ _{Corresponding author at: MACSI, Department of Mathematics and Statistics,}

University of Limerick, Limerick V94 T9PX, Ireland.

E-mail address: malbor.asllani@ul.ie (M. Asllani).

piesa prominentrole(Turing,1952). Theconditionforthe emer-gence of Turing patterns has been elegantly grounded in an in-terplay between slow diffusing activators and fast diffusing in-hibitors(GiererandMeinhardt,1972).Indeed,theparameterspace forwhichtheinstabilitymaterialisesshrinkstozerowhenthe ra-tioofthediffusioncoefficientstendsto1,anobservationthat lim-its therangeofapplicability ofthe theory,atleastin itsoriginal conception.Tostateit differently,theconditionsforTuring insta-bilityare solelymet ina smallregion ofthe available parameter space,instarkcontrastwiththediversityofpatternsthatare rou-tinelyfoundacrossdifferentfieldsandscales (Ball, 1999). Consid-eringmorethantwofamiliesofinteractingentitiesandaccounting fortheroleofendogenousorexogenousnoiseenhancethe robust-ness ofthe schemeand partially overcomethe above limitations (Pattietal.,2018;Arbel-Gorenetal.,2018).

Intheir pioneeringpaper,OthmerandScriven(1971)extended the Turing theory of pattern formation to reaction-diffusion sys-temsdefinedon severaldiscrete lattices, inarbitrarydimensions. More recently Nakao and Mikhailov (2010) extended the analy-sis to systems defined on complex networks, a setting that is

https://doi.org/10.1016/j.jtbi.2019.07.004

relevantto, for example,multi-speciessystems confined in com-partmentalisedgeometries(Holland andHastings, 2008),optically controlledbioreactors(Horsthemkeetal.,2004),bistablechemical networks(Kouvarisetal.,2016),embryonicdevelopment(Bignone, 2001; Schnabel, 2006) andneuroscience (Buscarino etal., 2016). Thedispersion relation,whichultimatelysignalsthe onsetofthe instability,is afunction ofthediscrete spectrumofthe Laplacian matrix,i.e.the diffusionoperator, associated withthe underlying network.Laplacianeigenvaluessetthespatialcharacteristicsofthe emerging patterns when the system under scrutinyis rooted on a heterogeneous support. Patterns for systems evolved on com-plex graphs yield a macroscopic segregation into activator-rich and activator-poor nodes (Nakao and Mikhailov, 2010). They are termedTuringpatterns,becauseoftheanalogywiththeir contin-uumcounterparts.Forundirected(symmetric)networks,the topol-ogydefinestherelevantdirectionsforthespreadingofthe pertur-bation, but cannot impact on the onset of the instability, which emergesundertheverysameconditionsthatapplywhenthe sys-temisdefinedona continuoussupport.Networkasymmetrycan howevertrigger thesystemunstable,seeding theemergence ofa generalised class of topology driven patterns, which extends be-yond the conventional Turing scenario (Asllani et al., 2014b). As suchtheydonothaveanycounterpartinthecaseofsystems de-finedoncontinuousdomains.

Startingfromthesepremises,weheresetouttoexploreanew situationthat is faced when the network of connections is non-normal(Trefethenand Embree, 2005; Asllani and Carletti, 2018), hence inherently asymmetric, but the ensuing dynamics proves stableunderastandard linearstabilityanalysis.Otherwisestated, wewilldealwithhomogeneous equilibriawhicharedeemed sta-bleandrely on the shorttime amplificationof the perturbation, asinstigatedbythe non-normalcharacterofthenetwork,to find apath towards theheterogeneous attractor.Non-normal patterns display a characteristic amplitude, comparable to that associated with their conventional homologues. Small initial perturbations, self-consistently amplified by the non-normality, suffice to cross thebarrierwhichseparatesthebasinsofattractionof,respectively, thehomogeneous andheterogeneous solutions.The lattercan be ideallypictured astheminimaofa generalisedenergylandscape, asschematicallyhighlightedinFig.1.Equally,smallperturbations cannot triggerthe instability when thesystem isinstead defined onasymmetrisedversionofthenon-normalsupport.Thenet ef-fectisacontractionofthebasinofattractionofthehomogeneous fixedpoint,asproducedbytheimposeddegreeofnon-normality.

The role of non-normality has been already studied in

Neubertet al., 2002, whereit was shownthat reactivity, i.e. the amplificationofthedisturbanceasseededbythenon-normal na-tureofthereaction terms,is anecessary conditionforthe onset ofclassicalTuring patterns,i.e.patternsensuing froma reaction-diffusion system defined on a continuous domain. Endogenous noise, or perpetual exogenous perturbations, can effectively con-tributetothestabilisationofthepatterns(Biancalanietal.,2010), the effectiveness increasing with the degree of non-normality of thereactioncomponent(Biancalanietal., 2017).Atvariance with theformerstudies,we shallhereafterfocusonthenon-normality asstemming fromthe couplingswhichdefine thespatialsupport oftheinvestigatedmodel.Thereasonforsuchachoiceistwofold: onthe one hand,existing reactionmodels do notexhibit alarge non-normalbehaviour(Ridolfi etal.,2011),andontheotherhand, a strong non-normality manifests as a natural property of em-pirical networks, as has been recently reported in Asllani et al., 2018. In conclusion, based on the above, we can show that the non-normal nature of the imposed spatial couplings contributes tosignificantly enlargingtheparameter spacewherepatternsare predictedto occur, potentially increasing the applicability of the theory.

Fig. 1. Attraction landscape. A schematic layout to depict the landscape of a generic reaction-diffusion system, defined on a symmetric network (a): the basin of attrac- tion associated with the homogeneous equilibrium, x ∗_{(light blue) is displayed; the}

latter extends considerably to eventually encompass a large fraction of orbits (as e.g. A 1 ). Only trajectories which are sufficiently distant from the homogeneous fixed

point (as e.g. B 1 ) can evolve towards a different, possibly nonhomogeneous, equilib-

rium x  _{(turquoise). Once the dynamics is made to flow on a non-normal support}

(panel b), the effective basin of attraction of the homogeneous fixed point shrinks considerably. This is the direct signature of the short time amplification of the im- posed perturbation, as stimulated by the non-normal character of the underlying support. The amplification makes it possible for the system to overcome the barrier as displayed in (c) and eventually results in the dynamical landscape, that is picto- rially exemplified in (b). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Thepaper isorganisedasfollows:inthe next sectionwe will introducethemathematicalframeworkandthenturn,inSection3, to discussingthe emergence of patternfora reactive model that implements a generalized transport scheme on a non-normal, network-likesupport.InSection4we willdeveloptheconceptof thepseudo-dispersionrelationforpredictingtheonsetofthe non-normalinstability.Thisservesasapowerfuldiagnostictoolbeyond the linearorderof approximation,asroutinely employed. Finally, inSection5wesummariseanddrawourconclusion.

2. Reaction-diffusionmodelsonsymmetricnetworksandtheir generalizationtodirectedsupports

We consider the coupled evolution of two species which are boundto relocateona directednetwork.We introducetheindex

i₌1_,...,N toidentifythenodesofthecollectionanddenotebyu_i

and

v

ithepositiveandnon-dimensionalisedconcentrationsofthe

specieson theithnode. Local, reactive dynamicsare assumedto begovernedbythenon-linearfunctions f

(

ui,

v

i

)

andg

(

ui,

v

i

)

.The

underlyingnetworkis characterisedby abinaryadjacencymatrix

node i, and zero otherwise. The species relocate across the net-work, following available edges. More specifically, and with ref-erence to species u,we postulate that the rate of change ofthe concentrationu_i,asreflectingtheinteractionwiththeneighboring nodej,ismodulatedbyDu

(

uj− ui

)

.Takingintoaccountthe

contri-butionofallneighboringnodes,thenetchangeofconcentrationat nodei,relativetospeciesu,isgivenbyDuNj=1Ai j

(

uj− ui

)

where

thesumisrestrictedtothesubsetofnodesjforwhichAij=0and

Duisthepositivetransportcoefficientforspeciesu.Collectingthis

informationtogether,thesystemunderexaminationobeysthe fol-lowingsetof2Nordinarydifferentialequations:



˙ u i= f

(

u i,

v

i

)

+D uNj=1L i ju j ˙

v

i= g

(

u i,

v

i

)

+D vNj=1L i j

v

j, (1)

where the“dot” denotes d

dt, Li j=Ai j−

δ

i jk

in

i are theelements of

theLaplacianmatrixLwhere

δ

_{i j=}1ifi₌j and0otherwise,and

kin i =



jAi jstandsfortheincomingdegreeofnodei,i.e.the

num-ber of connections pointing towards i. It is worth emphasising that the above operator is different from standard diffusion and it converges to the latter only in the limiting setting of a sym-metric (undirected) spatialsupport. It is worth emphasising that theaboveoperatorisdifferentfromstandarddiffusionandit con-verges to the latter only in the limiting setting of a symmetric (undirected)spatialsupport.Withsomeabuseoflanguage,wewill however occasionally refer to the implemented transport rule as diffusion. We also recall that standard reaction-diffusion systems on continuousdomains are customarilydescribedbya closedset of partial differential equations (PDEs) for concentrations. When discretising thePDEovera finitespatialperiodic mesh,the prob-lemcanbeequivalentlyrecastintheform(1),foraspecificchoice of the operator L, e.g. a (centred) second order finite differences matrixforasquarelattice,andbyproperlyrescaling thecoupling constants, soastoaccountfortheselectedmeshsize,e.g.the2D

lattice case. The use of the Laplace matrix L can thus be seen as a mathematical extension of the classical framework of regu-lar supports (Othmer and Scriven, 1971) to the one of complex networks.

As a prerequisite for the forthcoming analysis, we shall as-sumetheexistenceofahomogeneousequilibrium.Thisislabelled

(

u∗,

v

∗

)

, and satisfies the condition f

(

u∗,

v

∗

)

=g

(

u∗,

v

∗

)

=0. The solution

(

u_i,

v

i

)

=

(

u∗,

v

∗

)

foralli isfurtherassumedtobe stable

underhomogeneousperturbations.Asfirstintuitedby Turing,the aforementionedequilibriumcanbecomeunstableuponinjectionof a small heterogeneousperturbation, whichactivates the diffusive couplings.Thelatterareotherwisesilenced,aslongasthesolution stays homogeneous. Under specific conditions, non-homogeneous disturbances amplify: the systemis consequently driven towards different asymptoticattractors,termed in theliterature asTuring patterns.Toisolatetheconditionsthat maketheinstability possi-ble,oneproceedswithaconventionallinearstabilityanalysis.The keyidea istointroducea smallperturbationofthehomogeneous equilibrium,x=

(

u1,...,uN,

v

1,...,

v

N

)

T−

(

u∗,

v

∗

)

T andinsertitin

thegoverningEq.(1).Byexpandingtofirstorderinthe perturba-tion,weobtainthefollowinglinearsystem:

d

dt x=

(

J 0+L

)

x, (2)

wheretheJacobianmatrixJ0is

J 0=



f u1N f v1N g u1N g v1N



=J01N, (3) withJ0=



fu fv gu gv



the2× 2Jacobianofthereactionpart evalu-atedatthefixedpoint

(

u∗,

v

∗

₎

_{andwhere1NistheN× N identity}

matrix.The generalised Laplacian operator L is the matrix given by L =



D uL 0N 0N D vL



=



D u 0 0 D _v



L, (4)

thattakesintoaccountboththetransportcoefficientsandthe ge-ometryofthesupport.

Theeigenvalues

λ

ofthe2N× 2NmatrixJ0+Ldefinethefate ofthe perturbation. Ifthere exists (at least)one eigenvalue with apositiverealpart,thentheperturbationinitiallygrows exponen-tially,andthehomogeneous fixed pointispredictedunstable. An elegantproceduretocomputethespectrumofJ0+Linvolvesthe eigenvectorsoftheLaplacianmatrixL.Assumethelattertobe di-agonalisable,introduce itseigenvalues



(α) _{andassociated} eigen-vectors

φ

(α)_,as

jLi j

φ

(α)j =



(α)

φ

i(α),with

α

=1,...,N.Thenwe

can expand the perturbation on the basis of the Laplacian ma-trixand projectthe 2N× 2N Eq. (2)onto a collection of N inde-pendent 2× 2 problems, each associated to a given subspace, as spannedby the corresponding eigenvector. When the underlying networkissymmetric,theLaplacianeigenvalues



(α) _arerealand non-positive.Moreover,theeigenvectorsforman orthonormal ba-sis of the embedding manifold. The short time evolution of the imposed perturbation is exponential and the associated growth (ordamping) factors

λ

can be readily computedasa function of theeigenvalueentries



(α)_{,bysolvingthefollowingcharacteristic} problem: det



f u+D u



(α)−

λ

f v g u g v+D v



(α)−

λ



=0. (5)

Theeigenvaluewiththelargestrealpart,

λ

=max_α

λ

(

(α)

₎

_,

de-finesthesocalleddispersionrelation,whichcharacterisesthe re-sponse of the deterministic system(1) to external perturbations. The quantity



(α) _{is the analogue of the wavelength for a} spa-tial pattern in a system defined on a continuous regular lattice, where



(α)≡ −k2_{,andkstandsfortheusualspatialFouriermode.} ForthisreasontheconditionsfortheonsetofTuringpatternsare thesameasthoseobtainedforacontinuoussupportandforany (sufficiently large) symmetric network. When the dispersion re-lation

λ

(



(α)_{) is positive over a finiterange of}

_

(α)_{, the} pertur-bation istriggered unstable. The combinedaction of thereactive anddiffusive componentsproduces aself-consistent amplification of the smallinhomogeneities, driving the systemtowards a pat-ternedstationary stableattractor.Thisconstitutesthe natural ex-tensionofthe well-knownTuringpatterns,tosystemsdefinedon adiscrete,network-likesupport.Thediscretedispersionrelationis indeedidenticaltotheone forcontinuoussupport,exceptforthe factthat its domainofdefinitiontakesvaluesonthe finitesetof realeigenvalues



(α)_{.Theconditionsfortheonsetofthe} instabil-ityinthecontinuum anddiscrete settingshencecoincidefor suf-ficientlylarge networksand modulo smalldeviations that might eventually arise, ifthe discrete spectrum doesnot extend to the regionwherethedispersionrelationforthecontinuoussupportis positive.

A completely different scenario is, instead, faced when net-worksareassumedtobedirected,e.g.whentheedgesconnecting twoadjacent nodescanbe accessedinone directiononly. Break-ingthesymmetryimpliesdealingwithcomplexeigenvaluesofthe associated Laplacian,a modification that can significantly impact theonsetofdiffusion-driven, Turing-like,instabilities. Indeed,the imaginarycomponentof



(α) _{contributestotherealpartof}

λ

_,as followsfromtheself-consistentcondition(5).Itcanbeshownthat thisadditionalcontributionfavourstheoutbreakoftheinstability, thus enhancing the intrinsic abilityof the reaction-diffusion sys-temtodeveloporderedpatterns,ascomparedtoitscontinuum(or discreteandsymmetric)counterpart(Asllanietal.,2014b).

In this paper we are interestedin the emergence of patterns insystemsdefinedon anasymmetric support,whicharedeemed stableundera conventional linear stabilityanalysis, for anykind ofperturbations.Thelattercaninfactgrow,atshorttimes,ifthe underlyingnetworkisnon-normal,yieldingpatternsalsowhen lin-earstabilityreturnsanegativedispersionrelation,

λ

<0,asetting whereTuringinstabilitycannotdevelop.Theremainingpartofthis sectionis devotedto revisitingtheconcept ofnon-normality.We willinparticulardiscusshownon-normalityimpactstheevolution ofagenericlinearsystem,inarbitrarydimensions.Tothisend con-sideralinearsystem d

dtx=Mx,andassumeMto benon-normal.

Amatrixissaidtobenon-normalifitdoesnotcommutewithits adjoint(TrefethenandEmbree,2005).IfMisreal,thentakingthe adjointamountstocomputingthetransposeofthematrix.Hence,

Misnon-normalprovided[M,MT_{]≡ MM}T_{− M}T_{M=0,wherethe}

superscriptTstandsforthetransposeoperation.Observethat non-normalityisequivalenttothenon-existenceofasuitable orthogo-nalbasisofeigenvectors.

A straightforward manipulation (Trefethen andEmbree, 2005) yieldsthefollowingequation fortheevolutionofthenormofthe perturbation



δ

x



: d



δ

x



dt =

δ

xT_H

(

_M

)

δ

_x



δ

x



(6) whereH

(

M

)

=M+MT

2 identifies theHermitianpartofM.The evo-lutionofthe perturbation atshort timesis ultimatelyset by the so-called numerical abscissa,

ω

=sup

σ

(

H

(

M

)

)

, where

σ

(

H

(

M

))

standsfor the spectrum of H(M). If

ω

>0, then the non-normal matrixis termed reactive, and perturbations may display an ini-tial,transientgrowth(Fig.2).Thedegreeofnon-normal amplifica-tion can be quantified through diverseindicators (Trefethen and Embree, 2005). In our setting, the reaction-diffusion system is amenable to its linear homologue (2), when operating in the vicinity of a stationary stable homogeneous attractor. Asymme-try, a key ingredient for non-normality, can be hence accommo-datedinthematrix thatencodesforpaired couplings,so trigger-ing a short time amplification of the imposed perturbation that willproveinstrumental forageneralised classofdiffusion-driven instabilities.

Fig. 2. Non-normal dynamics. Time evolution for the norm of x , solution of the sta- ble linear system ˙ x = Ax for a normal (blue) and non-normal (red) system. In both cases the system is stable, meaning that the spectral abscissa is negative α( A ) ≡ sup  σ( A ) < 0 . The norm  x ( t )  can however undergo a short time amplification if the numerical abscissa is positive, ω( A ) ≡ sup σ( H ) > 0 where H = A + A ∗_{/ 2 is}

the Hermitian part of A . The norm of the solution is bounded below by the Kreiss constant K(A) ≤ sup t≥0 x (t) /  x (0) ( Trefethen and Embree, 2005 ). (For interpre-

tation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

3. Non-normalpatternsforstablesystems

Without loss of generality, we shall assume in the following theBrusselatormodel(PrigogineandNicolis,1967;Prigogineand Lefever,1968; Boland etal., 2008), asareferencescheme involv-ing a cubic reaction. The Brusselatoris a paradigmof non-linear dynamics,anditisoftenemployedintheliteratureasareference modelforself-organisation,synchronisationandpatternformation. This choice amounts to setting f

(

x,y

)

=1−

(

b+1

)

x+cx2_{y and} g

(

x_,y

)

₌bx_{− cx}2_{ywherebandcdenotepositivenon-dimensional} parameters.The systemadmitsa trivialhomogeneous fixed point for

(

ui,

v

i

)

=

(

1,b/c

)

,

∀

i,thatisstableprovided c>b− 1.Wecan

easily determinethe conditionsforaTuring instability.Assuming asastartingpointa symmetricspatialsupport,we concludethat thediffusioncoefficientsneedtocomplywiththenecessary con-dition Du<Dv for patterns toemerge. A straightforward

calcula-tionenablesustoconcludethatc_<D_v/Du

(

b+1− 2√b

)

,asan

ad-ditional constraint on the parameters, for the Turing instability. The above conditions allow us to isolate the domain in the pa-rameterspace(b,c) whereTuringpatternsarepredictedtooccur. The region of interest is displayed in Fig. 3, with a green shad-ing, anditis delimitedby the curvesc=b− 1 (dashedline) and

c₌D_v/Du

(

b+1− 2

√

b

)

(solid line) (region(iii)). It is worth not-ingthatthelatterregionreflectstheinstabilityconditionsonboth continuousanddiscretesupports.

Consider now the Brusselator model defined on a directed, non-normalnetwork. Inthefollowing wechoose tooperatewith the class of non-normalnetworks introduced in Asllani and Car-letti,2018.Therecipeforthenetworkgenerationgoesasfollows: Start witha weighted directed 1D ring, andassume the weights toberandomlydrawnfromauniformdistributionU[0,

γ

],where

γ

is a (large) scalarparameter. Further, labelthe nodes with an ascendingindex,whencirculatingclockwiseacrossthering.Then,

Fig. 3. Different domains in the parameter plane ( b , c ) that yield pattern formation for the Brusselator model. Distinct domains of interest are isolated in the refer- ence plane ( b , c ) and depicted with different colour codes. In region ( iii ) (shaded in green), a Turing instability develops for the Brusselator model defined on a contin- uous or symmetric discrete support. In region ( ii ) (yellow), topology driven patterns emerge: the homogeneous fixed point is triggered unstable by the directed nature of the spatial support; here patterns can be stationary or wave-like. Finally, red dots define region ( i ), where non-normality drives the onset of pattern formation, notwithstanding the prediction of the linear stability analysis that deems the homo- geneous fixed point stable, and thus resilient to external perturbation. In white is the region where patterns cannot be established. The Turing region is bounded from below by the curve c = b − 1 (dashed line) and from above by c = Dv

Du(b + 1 − 2

√

b)

(solid curve). The underlying non-normal network (generated according to the pro- cedure discussed in the main body of the paper) is made up of N = 100 nodes. The model parameters are D u = 0 . 5 and D v = 1 . 925 . the initial conditions for u (resp.

v ) are uniformly drawn from an N -dimensional sphere of radius δ= 0 . 2 centred u ∗

(resp. v∗_{). (For interpretation of the references to colour in this figure legend, the}

foralli=1,...,N,wecreatewithprobabilityp1,0<p1<1,aweak linkoforder1,namelymuchsmallerthantheweights onthe di-rected ring, that bridges the

(

i+1

)

th node to the ith one. This procedure returns a non-normalnetwork which displaysa richer topology ascomparedtothat oftheinitial directskeleton. More-over,its degreeofnon-normalitycorrelatespositivelywith

γ

:the larger

γ

themorepronounced theinherentnon-normality,as re-vealed through standard quantitative indicators (Asllani and Car-letti, 2018). In the following we shall operate with

γ

=50 and

p1=0.4.

Asalreadystated,theimaginarycomponentoftheeigenvalues of the Laplacian matrix seeds an enlargement of the region of parameter space where the instability is predicted to occur. The portionoftheparameterplanewheretheinstabilityextendsis de-pictedinyellowinFig.3andlabelledasregion(ii).Here,the dis-persion relationispositiveandtheensuing patternsbear a topo-logicalimprint,asthey are ultimatelyreflecting thedirectionality oftheembeddingspatialsupport.Itisworthemphasisingthatthe Brusselator model evolved on a symmetric network or a contin-uous supportcannot develop Turingpatternsforthesamechoice of parameters. The boundaries of the region where topological patternsdevelopcanbetracedanalytically(Asllanietal.,2014b).

Moreinteresting forourcurrentstudy,iswhathappensabove region(ii).Here,thedispersionrelation,theratewhichgovernsthe exponentialevolutionoftheperturbation,isnegative, thus imply-inglinearstability.Patterns,resemblingthosegeneratedinsidethe domain of (topological orTuring-like) instability,are instead ob-served, when integratingnumerically the governing equations. A red dotisplottedinFig.3ifthepatternsare obtained,upon nu-merical integration, for the corresponding values of the reaction parameters(b,c).TheresultsdisplayedinFig.3refer toone real-isation ofthe dynamics.Repeatingthe analysisto account foran extended setofindependentnumericalexperiments yields equiv-alentconclusions.Theamplitudeofthenoiseisassumedsosmall that patterns cannot set in when the non-normal networkis re-placed withits symmetrised analogue (see alsoFig. 9 wherethe emergence of this novel class of patterns is studied in terms of the pseudo-dispersion relation). Denote by A the adjacency ma-trix of the non-normal network, as obtained via the procedure discussed above. Then,

(

A+AT

₎

_{/2is the adjacencymatrixwhich}

characterisestheassociatedsymmetricnetwork.Weanticipatethat propensity to self-organisationexhibitedby the systemstems ul-timately from the characteristic dynamics as displayed by linear non-normal systems atshort times.We shallreturn to elaborate onthispointinthefollowing.

In Fig.4the dispersion relationisshownfordifferent choices of the parameters (b, c). Selected points in the parameter space are displayedwitha symbolicmarker(respectively circle, square, triangle)inFig.3.Inthetop-leftpanelofFig.4thedispersion re-lation

λ

isplottedagainsttherealpartoftheLaplacianeigenvalues



,whenoperatinginsidetheregionofconventionalTuringorder. Thecontinuum dispersionrelation(solidblueline)predicts insta-bility,i.e.ittakespositivevalues.Greensymbols,obtainedforthe Brusselatormodelevolvedonasymmetrisednetworksupport(as defined above), follow, asexpected, the profile of the dispersion relationforthecontinuoussupport.Redcirclesreferinsteadtothe modeloperatedonanon-normal,hencedirected,network,ofthe type discussedabove. Againthesystemispredictedunstable,the estimated growth rate of the perturbation being larger as com-paredto the symmetriccase. Inthe insets annexedtothe panel, thetimeevolutionofthenormoftheperturbationisrepresented, forbothsymmetric(top, greencurve)andasymmetric(down,red curve)settings.Inbothcasesthecurvesgrowandeventuallyreach an equilibrium plateau, the signature of the existence of a non-homogeneoussolution.Thecorrespondingpatterns,asobtainedin theasymptoticregimeoftheevolution,aredisplayedintheboxes

annexedimmediatelybelowthecorrespondingdispersionrelation. The nodes of the collection, arranged in a 2−dimensional circu-lararray for the sake ofvisual representation,are coloured with anappropriate codewhich reflectsthesteadystate concentration ofspecies u. The systemsegregates intonodes rich/poor in acti-vators/inhibitors,thepolarisationbeingmorenotableinthe asym-metricsetting(bottomfigure). Inthe second plotofthetop row, the parameters are set so asto havethe systeminitiated in re-gion (ii) of Fig. 3 (square symbol). The system defined on sym-metricgraphsisnowstable:thenormoftheperturbationdamps steadilyandthe homogeneous fixed point proves resilient to ex-ogenousperturbationoftheinitialstate(thedispersionrelationis negative).No patternscanhencedevelop,asdisplayedinthe sec-ond panel of the second row. Conversely, thedispersion relation computedfortheBrusselatoronthedirectednetwork(redcircles) signalstheinstability,whicheventuallymaterialisesinatopology driven pattern, as displayed inthe bottom middle panel. Finally, forregion(i)weobtaintheresultenclosedintheredboxofFig.4. Thedispersion relationsare nownegative,thus implyingthat the perturbation should asymptotically vanish.While no patternsare observedforthesystemintegratedonasymmetricsupport,thisis manifestlynotthecasewhentheunderlyinggraphisassumedto benon-normal.The shorttime growthoftheperturbation,as re-portedinthesmall(lower)insetofthetop rightpanel,stabilises ina macroscopicpattern(lowerpanel ofthe columnenclosed in red)whichsharesstrikingsimilarities(intermsofdensity distribu-tionandcorrespondingamplitude) withthe homologouspatterns obtainedinsidetheregionoflinearinstability.Fromthese observa-tionsitappearsalreadyevidentthatthemechanismthatdrivesthe emergence of non-normal patternfrom the asymptotically stable homogeneoussolutionresidesintheabilitytoamplifythe pertur-bationimposedatshorttime.Theidea,thatwewillnowexplore, isthattheshorttimeamplification,astriggeredbynon-normality, makes the system jump across the barrierthat separates homo-geneousandheterogeneousattractors,thusyieldingaspatially ex-tendedpattern,whichcouldnotbeanticipatedviaalinearstability analysis.

Tofurtherinvestigatethisissue,weintroduceaquantitative in-dicator, A, for measuring the amplitude of the patterns that are eventually established by means of the alternative routes high-lightedabove.Thisisdefinedas:

A =



i

(

u ∞ i − u∗

)

2+[

(

v

∞i −

v

∗

)

2



, (7) where u∞_i (resp.

v

∞

i ) is the asymptotic stationary value of ui(t)

(resp.

v

i

(

t

)

) at node i. In Fig. 5, the scalar quantity A averaged

overalarge sampleofindependentrealisations,

A



,isplottedas afunction of b,at fixedvalue of c. Circles (red)refer tothe sys-temevolvedon thenon-normal network, thesame network em-ployedintheanalysisthat ledtoFig.3.Squares(green)standfor thevaluesof

A



recordedwhenthesystemisstudiedonthe sym-metrisedversion ofthenetwork. Ascan beappreciated byvisual inspectionofFig.5,non-normalitydrivesasignificantenlargement oftheregionoftheparameterspacethatyieldsheterogeneous pat-terns(seethelightredregioninFig.5),beyondthedomainwhere topologicalpatterns(orpatternofdirectionality)are foundto oc-cur(see thelight blueregion inFig.5). Notice alsothat the pat-terns sustainedby non-normalityhave a characteristicamplitude comparableto that associated withconventional Turing patterns, asquantifiedby

A



.Thesameconclusionisreachedupon inspec-tion of the annexed histograms. Performing a vertical cut, for a value of b atthe threshold (vertical linelocated at the point A) oftheleftmostbifurcation,onefindsthatnon-normalpatternsare characterised by a bimodal distribution of the associated ampli-tude.Thecoexistenceofhomogeneous,

A



∼ 0, andpatterned so-lutions,

A



=O

(

1

)

, suggests that the transitionis of first order.

Fig. 4. The dispersion relation and the ensuing patterns in different regions of the parameter plane. Turing patterns (b, c) = (26 , 61 . 1) (left column), topology-driven patterns

(b, c) = (24 , 61 . 1) (middle column), patterns of non-normality (b, c) = (22 . 5 , 61 . 1) (right column, enclosed in the red box). The position of the selected working points is marked in Fig. 3 , with, respectively, a triangle, a square and a circle. In all cases, the dispersion relation is drawn for the non-normal (directed) network (red symbols) and the symmetrised network (green symbols). Recall that the instability is set once the dispersion relation is positive. The insets in the panels that form the first row, show the time evolution of the norm of the system for the non-normal network (red curves) and the symmetric settings (green curves). Here, the system is initialised δ-close to the equilibrium (see caption of Fig. 3 ), with δ= 0 . 2 . The remaining parameters, as well as the network employed, are those used in Fig. 3 . The ensuing patterns are displayed for, respectively, the symmetric setting (second row), and the non-normal network. In particular, in the third row, the patterns are depicted when representing the network with a stylised lattice layout (as for the symmetric setting). In the fourth row, the edges of the networks are instead shown. Nodes displaying a low concentration are assigned to the central portion of the cluster.. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Thehistogramobtainedforasymmetric settingreturnsinsteada uniquepeak,atzero

A



.Thisisalsotruewhentheverticalcutis performedwell insidetheregionofnon-normalpatterns(vertical linelocatedatB).Non-normalpatternsareinsteadassociatedwith macroscopicamplitudes.

TheresultsofFig.5revealanotherimportantfact:patternsare alsofoundforsymmetricnetworks,inasmallregionoutside(but in proximity of) the Turingdomain. Such patterns are againdue tonon-normality,asstemmingfromthereactionterm.Indeedthe Jacobian ofthe Brusselatorisnon-normalforanychoice ofband

Fig. 5. On the amplitude of the ensuing patterns. Main panel (centre): the average pattern amplitude A  is plotted as a function of the parameter b , for c = 61 . 1 . Red circles refer to the system evolved on a non-normal network, while green squares stand for its symmetrised analogue. The grey region corresponds to parameters giving rise to Turing instability and the blue one to instability driven by the network directionality. Finally, the domain coloured in pink highlights the region where the homogeneous fixed point is stable according to a linear stability analysis, but where patterns can emerge due to non-normality. The insets show histograms of pattern amplitudes, for two specific values of b , namely b ∼ 20.5 (A), b ∼ 22.5 (B). Red symbols refer to data collected when the Brusselator model evolves on a non-normal support. The green histograms are computed from simulations that assume symmetric support. The remaining parameters have been set to D u = 0 . 5 , D v = 1 . 925 , δ= 0 . 2 . Simulations have been performed

using the same networks as in Fig. 3 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

c,andthisisa sufficientconditiontotriggerthesystemunstable, for a perturbation of modest size, also beyond the strict bound-aries, obtainedunder thelinear stabilityframework. The positive interferenceoftwodistinct sourcesofnon-normality,respectively associated withthe reactionand diffusionparts, enlarges the re-giondeputedtotheinstability,asalreadyremarked.Interestingly, patterns ofnon-normalitycan alsoexist forsystems displayinga symmetricJacobian,whentheembeddingspatialsupportismade non-normal(AsllaniandCarletti,2018).

Toshed further light on the mechanismthat seeds the insta-bilityatnegativedispersionrelation,andsoelaborateontherole played by non-normality, we study the response of the system, on both symmetric and asymmetric supports, at different values of the size of the initial perturbations applied on the homoge-neous equilibrium. In the left panel of Fig. 6, we plot the aver-age patternamplitudes as a function of

δ

, fora given choice of

b: in the case of symmetric networks (green symbols) patterns can only develop formacroscopic perturbations (

δ

>

δ

crit=0.83).

Fig. 6. On the stability domain. Left panel: average pattern amplitude as a function of the size of the initial perturbation δ, for b = 22 . 56 . The average is computed over 500 independent realisations of the dynamics. Right panel: the size of the stability domain δcrit is computed for, respectively, the Brusselator model defined on the non-normal network (red symbols) and on its symmetric analogue (green symbols), and plotted as a function of b . The remaining parameters are set to c = 61 . 1 , D u = 0 . 5 , D v = 1 . 925 .

For the analysis we employed the networks used in Fig. 3 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 7. Transient amplification and linearised dynamics. Left panel: we show the time evolution of the norm of the perturbation for the linearised dynamics (dotted line) and for the full non-linear model (solid line). For the linear system, we can see the transient growth and the subsequent decrease towards the asymptotic equilibrium. The non-linear system stabilises eventually to a value of the norm which is different from zero. Here, δ= 0 . 2 . Right panel: (δ) := max t||  x(t)|| −||  x(0)|| as a function of the perturbation size δ, for non-normal (red symbols) and symmetric networks (green symbols). The parameters are set to b = 22 , c = 61 . 1 , D u = 0 . 5 , D v = 1 . 925 . The networks

employed are those used in Fig. 3 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Ontheother hand,fornon-normalnetworks, aperturbationwith

δ

>

δ

crit =0.005sufficesfortheinitiationofthepatterns.Forany

bwedefine

δ

_crittobethesmallestvalueoftheinitialperturbation forwhichpatternsemerge.In therightpanel ofFig. 6,thevalue of

δ

crit isplottedagainstb:thebasinofattractionofthe

homoge-neousstate shrinksconsiderablywhenthesystemisdefinedona non-normalsupport,an observationwhichhelpsexplain the aug-mented propensity ofnon-normal systems forpattern formation. Theshrinkingofthebasinofattractionofthehomogeneousfixed point followsthe transientgrowth ofthe perturbation, asdriven by non-normality.In the left panel ofFig 7 the evolution of the normof theperturbation is tracked underthe linear approxima-tion,which follows Eq. (2).The norm grows at shorttimes, and thenconvergestozero,asitshouldsince thehomogeneousfixed point is linearlystable. The evolution of the perturbation as ob-tainedforthefullnon-linear modelis,instead,quitedifferent,as canbevisuallyappreciated.Non-linearitieseventuallystabilisethe normtoa non-zerovalue whichwasmadeaccessibletothe sys-tem under the transient evolution. In the right panel of Fig. 7,

(

δ

)

:=maxt

||

x

(

t

)

||

−

||

x

(

0

)

||

isrepresentedasafunctionofthe

perturbationsize

δ

for,respectively, thenon-normalnetwork(red symbols)andthesymmetricone (greensymbols).The twocurves runparalleltoeachotheronthelog-logplotandtherelativeshift isrationalisedasfollows:Consider,forexample,therightpanelof

Fig.6, where

δ

_crit is plottedagainst b andfocus in particular on

b=22.For thecase ofa non-normal network, we readilyobtain (

δ

crit)nnࣃ0.05, while(

δ

crit)symࣃ0.6. On theother hand, thevalue

of

(0.05) (i.e.a directmeasure ofthe transientgrowthinduced bynon-normalityfor

δ

ࣃ0.05,under thelinearapproximation) is about0.4(see rightpanel ofFig.7,plottedforb=22),i.e.ofthe correctorderofmagnitudetoexplaintheobserveddiscrepancy be-tween(

δ

_crit)symand(

δ

crit)nn.

As mentioned in theIntroduction, Turing patternsrequirethe inhibitortodiffusefasterthantheactivator.Moreover,theratioof thediffusivities D_v/Du shouldbe sufficientlylarge forthepattern

tomaterialise in an extended region ofparameter space.Forthe Brusselator,sendingDv/Du→1contextuallyimpliessettingb and

c very large forthe patterns to set in via a conventional Turing instability.Networkasymmetry,andtheassociatednon-normality, enableustoextendtheregionwherepatternsarereportedto oc-curtosmallerratiosofDv/Du→1,withoutforcingthereactive

pa-rameterstoextremelylargevalues.AsanexampleseeFig.8where

Fig. 8. Patterns of non-normality for D v /D u ∼ 1 . We plot the average pattern am- plitude A  as a function of the parameter b , for c = 23 . 72 and D v /D u = 1 . 5 , for the Brusselator model evolving on the non-normal network used in Fig. 3 . Values of b lying in the blue region yield instability driven patterns due to the network directionality. The region coloured in pink highlights the interval in b where the homogeneous fixed point is linearly stable and patterns can emerge due to non- normality. Each point is the average over 100 independent replicas. The remaining model parameters are the same as those used in Fig. 3 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

weshowthepatternamplitudeforDv=0.75andDu=0.5,

result-ing in D_v/Du=1.5, for values of the parameters b and c of the

same orderofthe onesused inFig. 5.Patterns of non-normality do exist for a large range of valuesof the b parameter (observe alsothatinthissettingTuringpatternscannotdevelopona sym-metricsupport).Thepatternsareofanon-neglibilemagnitudeup tob∼ 21. Belowthisvalue themagnitudereduces,asdepictedin

Fig.8.Thisisaspuriouseffectduetothefactthat,forb_∈[20,21], thereisacoexistenceoftwodistinct solutions,one corresponding tothemacroscopicpatternandtheother tothepersisting homo-geneousstate(seealsoFig.5).

4. Pseudo-dispersionrelation

In the above discussion we have shown that patterns can emergeduetothenon-normalityoftheunderlyingnetworkwhen thedispersion relationwoulddeemthehomogeneousfixed point

Fig. 9. Dispersion relation and pseudo-dispersion relation. Left panel: Fixing the parameters (b, c) = (22 . 5 , 61 . 1) (see also Fig. 4 ) we plot the dispersion relation for the non-normal (directed) network (red circles) and symmetrised network (green squares). Empty symbols stand for the pseudo-dispersion relation, with an identical code for the assigned colours. The system cannot exhibit Turing patterns because the dispersion relations are consistently negative. However the pseudo-dispersion relation is positive when dealing with a non-normal network. This implies that the system undergoes an initial amplification which can eventually form a macroscopic pattern. Right panel: using the same symbol/colour scheme as in the left panel, we show the maximum of the dispersion relation as a function of b . We can see that the pseudo-dispersion relation for the non-normal network (empty red circles) is positive for a much larger parameter range. Hence, non-normality of the embedding support favours the emergence of patterns. Here δ= 0 . 2 and each point is the average over 50 independent replicas. The remaining parameters, as well as the networks selection, are those used in Fig. 3 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

stable. Our conclusionsrely on a direct numerical integration of the governingequations andwe lacka predictive tool that could eventually beemployedtoanticipatethephenomenon.Inthe fol-lowing we shall propose an extended definition of the disper-sion relation that builds on the concept of the pseudospectrum (TrefethenandEmbree,2005).Webeginbyrecallingthedefinition ofthepseudospectrum,

σ



(

M

)

,ofamatrixM,namely:

σ



(

M

)

=

σ

(

M+E

)

,

∀

||

E

||

<



, (8)

where

σ

(

M

)

denotes the spectrum of the matrix M and



is a scalar positive defined quantity. The pseudospectrum is an im-portant and relatively recent mathematical tool which has been successfullyapplied to differentdisciplinarycontexts where non-normality occurs (Trefethen and Embree, 2005). The Kreiss con-stant, alower boundforthe lineargrowthofanimposed pertur-bation,canbecomputedfromthepseudospectrum.

While the behaviour of a linear (non-normal) systemis rela-tively well understood in theframework of pseudospectrum the-ory,extendingthistonon-linearmodelsischallenging.Considera genericnon-linearsystem,x˙=f

(

x

)

,wheref:Rn_→Rn_isa

vector-valuednon-linearfunction.

TheproblemoflocalstabilityistackledbyperformingaTaylor expansionaboutagivenfixedpointx∗,f

(

x∗

)

=0.Theevolutionof theperturbation



neartheequilibriumx∗isthengivenby:

˙



=



Df

(

x∗

)

+ 1

2!



T_D2_f

₍

_x∗

₎

_

_{+. . . , (9)} where Df

(

x∗

)

and D2f

(

x∗

)

are, respectively, the Jacobian matrix and the Hessian tensor1 _{of the non-linear function f, both}

eval-uated atthefixed point.Keeping onlythe firsttermonthe right handsidecorrespondstoperformingalinearanalysis,i.e. comput-ing thespectrumofDf

(

x∗

)

,andstraightforwardlyyieldsthe defi-nitionofthedispersionrelation,asintroducedabove.

Accounting for the second term inthe expansion results in a quadraticproblemthatcannotbesolvedexactly.Wecan,however, makesomeprogressbyreplacing



T_D2

f

(

x∗

)

/2withaconstant fac-tor that depends on the equilibrium point and also on the

ini-1 With a slight abuse of notation we denote with such symbol the term



i j∂i j f l( x ∗)ij for each component f l of the vector f = (f1 , .. . f n) .

tialperturbation,



0=x

(

0

)

− x∗.InthiswayEq.(9)canbewritten againasalinearsystem

˙



=



Df

(

x∗

)

+ 1 2!



T 0D 2 f

(

x∗

)





≡ [J

(

x∗

)

+P

(

x∗,



0

)

]



, (10) where J is the Jacobian and P stands for a perturbation that is small,butnot negligible.The spectrum ofJ₊P can thus provide informationaboutthepossiblegrowth oftheinitial perturbation. Inanalogy withthedispersion relation,andborrowingthe name fromthepseudospectrum,wedefinethepseudo-dispersionrelation

λ

tobethelargestrealpartoftheeigenvaluesofJ+P.Apositive valuefor

λ

 impliesthat



growsintime.Theperturbationhence isamplifiedandthesystemcanpossiblyheadtowardsadifferent attractor.

Let us observe that the spectrum of J+P is not, in general, a small perturbation of the spectrum of J. We cannot therefore usethe toolsdeveloped inAsllani et al., 2014a;Hataand Nakao, 2017butrelyinsteadonacontinuationmethod(seeAppendixA). Let us denote by

η

the norm of P and observe that if



0→0 then

η

_→0.Letusdenoteby

μ

_ηtheeigenvalueofJ₊P_,suchthat

||

P

||

=

η

. Notice that, indoing this, we are abusing notation be-causeP - andthus its norm- dependson



0,so different initial perturbationscanproducedifferentP_,butallwiththesamenorm. Wecan,however,account forthisfactby averaging

μ

_η over sev-eralinitialconditionsthatsharethecondition

||

P

||

=

η

.For

η

=0,

we have that

μ

0 is an eigenvalue of J andthus we can find an eigenvalue



(α)_{oftheLaplacianmatrixsuchthat}

_μ

0=

μ

0

(

(α)

)

. Then varyingcontinuously s∈(0,

η

) we canfollow, with continu-ity,

μ

s(



(α)). In Fig. 9 (left panel) we show 

μ

s(



(α)), for s=0

ands₌

η

_,asafunctionof



(α) _{computedwiththisprocedureand} fortheparametervaluescorrespondingto thesettingoftheright panelofFig. 4. Werecallthat forthisparametersetting,the sys-temdefined on a non-normal network exhibits patterns of non-normality,whileitdoesnotwhenconfinedonitssymmetrised ho-mologue.Remarkably,thepseudo-dispersionrelationcapturesthis difference, the latter being positive for the non-normal network (emptyredcircles)whileitisnegativeforthesymmetricnetwork (emptygreen squares). In the right panel of Fig.9 we show the dispersion relationand the pseudo-dispersion relationas a func-tionofthemodelparameterbwhilefixingalltheremainingones. The procedure outlined here can hence be regarded as a viable

approachforpredictingthestabilityofthesystembeyondthe con-ventionallinearorderofapproximation.

5. Conclusions

Theaimofthispaperwastodiscussageneralisedrouteto pat-ternformation. To thisend we studied a reactive systemthat is designedtoexplore anetworkedspace andconsideredthe inter-esting setting where the network is non-normal. The concept of non-normalityis quite intriguing: a multidimensional linear sys-tem,ruled byanon-normal matrix,can displaya shorttime am-plification for the norm of an injected perturbation, also in the case when the latter is predicted to fade away asymptotically, because of the stability assumption. For the case at hand, the non-normalityoftheembeddingsupportcombineswiththe non-normalityinheritedbythereactioncomponent,totriggerthe sys-temunstable,evenifthecorrespondingdispersionrelationis neg-ative.WorkingwiththeBrusselatormodel,forillustrativepurposes foritsrecognisedpedagogicalvalue,weprovedthatnon-normality yieldsacontractionofthebasinofattractionofthehomogeneous fixedpoint, andconsequentlyactsasthemain driverforthe on-setoftheinstability.Thisphenomenoncanbequantitatively anal-ysed based on the pseudo-dispersion relation, an effective tool for predicting the fate of the perturbation beyond conventional linear analysis and accounting for non-linear corrections. Taken together, making the network of couplings non-normal favours thespontaneousdrivetomacroscopicorganisation,anobservation that could potentially help in overcoming the limitations intrin-sicto other paradigmatic approaches to pattern formation. Non-normalitycould resultin anovel routeto theemergence of self-organisedstructures, widespread patternsof complexityof broad applied and fundamental relevance (Pastor-Satorras and Vespig-nani, 2010). These conclusionsare even more relevant giventhe recentaccount on the ubiquity of non-normality across different fieldsof investigations (Asllani et al., 2018). To conclude, we re-markthat we haveheresolely focused ona purely deterministic setting.Itistemptingtospeculatethatnon-normalityina stochas-ticframework (thenoise sourcebeingexogenousorendogenous), willeventually yield a more robust andflexible route to pattern formation in dynamicsmodelling (Biancalani etal., 2017; Fanelli etal.,2017;Nicolettietal.,2018).

Acknowledgements

R.M. wouldlike tothank theErasmus+ program,Università di FirenzeandUniversité deNamurforfundinghismasterinternship inthegroupofProfessorT.C.

AppendixA. Continuationmethod

The computation ofthe eigenvalues/eigenvectors ofJ₊P can-notbe tackledwithaperturbative schemebecause,aspreviously mentioned,thecorrectiononJisnot sufficientlysmall.Apossible waytoovercomethisproblemistoconsiderthemodifiedmatrix

J

(

s

)

:=J+s P, (A1)

wheres∈[0,1].HenceJ

(

s

)

interpolatesbetweentheextremecases

J,theknownsystem, andJ+P thesystemunderscrutiny.Forall

swe areinterestedin findingthe eigenvalues,hereby castintoa diagonalmatrix D

(

s

)

, andleft/right eigenvectors,again organised intoamatrixform,U

(

s

)

andV

(

s

)

_,wherewe explicitlyemphasise theirdependenceons.Insummary,wewanttosolve

J

(

s

)

V

(

s

)

=V

(

s

)

D

(

s

)

andU

(

s

)

J

(

s

)

=D

(

s

)

U

(

s

)

. (A2)

Letusnowassumestobeavariableanddifferentiatethe pre-viousequationswithrespecttos.Undertherequirementof regu-larityweobtainforV

(

s

)

(andsimilarlyforU

(

s

)

):

PV

(

s

)

+J

(

s

)

d ds V

(

s

)

=



d ds V

(

s

)



D

(

s

)

+V

(

s

)

d ds D

(

s

)

. (A3)

LeftmultiplybyU

(

s

)

toobtain

UPV+DUd ds V=U



d ds V



D+UVd ds D, (A4)

where,to simplifynotation, weremovedthe explicitdependence ons.

Thisrelation,andtheotherobtainedforU,arematrix differen-tialequationsinvolvingtheunknownmatrixfunctionsU_,VandD. ConsideringthediagonaltermsofEq.(A4),andrecalling thatDis adiagonalmatrix,weobtain:2

(

UPV

)

ii+DiiUil d ds Vli=Uil



d ds Vli



Dii+UilVlm d ds Dmi, (A5)

thatis,theterminvolvingthederivativeofVcancels outandwe endupwiththederivativeofD:

UilVli d ds Dii=

(

UPV

)

ii, (A6) thatis d ds

μ

i

(

s

)

=

(

UPV

)

ii

(

UV

)

ii , (A7)

wherewereintroducedtheeigenvalues

μ

i

(

s

)

=Dii.

Consideringnow thecomponentijof Eq.(A4), wecan obtain, aftersomestraightforwardcomputation,

Uil d ds Vl j=−

(

UPV

)

i j

μ

i

(

s

)

−

μ

j

(

s

)

+

(

UV

)

i j

μ

i

(

s

)

−

μ

j

(

s

)

(

UPV

)

j j

(

UV

)

j j , (A8) and



d ds Uil



Vl j=

(

UPV

)

i j

μ

i

(

s

)

−

μ

j

(

s

)

−

(

UV

)

i j

μ

i

(

s

)

−

μ

j

(

s

)

(

UPV

)

ii

(

UV

)

ii . (A9)

Undertheassumptionofnon-degenerateeigenvaluestheabove matrixODEcanbenumericallysolvedstartingfromtheinitial con-ditions

μ

i(0) (eigenvalues of J), U0 and V0 (resp. left and right eigenvectors ofJ), up tos=1 providingin thiswaytherequired continuationoftheeigenvalues.Proceedingalongthislineand us-ing a first order method to solve the ODE returns the pseudo-dispersionrelationsdisplayedinFig.9.

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