Higher-order simplicial synchronization of coupled topological signals

Reza Ghorbanchian,1 Juan G. Restrepo∗,2 Joaqúın J. Torres†,3 and Ginestra Bianconi‡1, 4

1School of Mathematical Sciences, Queen Mary University of London, London, E1 4NS, United Kingdom2Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO 80309, USA

3Departamento de Electromagnetismo y F́ısica de la Materia and Instituto Carlos I de F́ısica Teórica y Computacional,Universidad de Granada, 18071, Granada, Spain

4 The Alan Turing Institute, The British Library, London, United Kingdom

Simplicial complexes capture the underlying network topology and geometry of complex systemsranging from the brain to social networks. Here we show that algebraic topology is a fundamentaltool to capture the higher-order dynamics of simplicial complexes. In particular we consider topo-logical signals, i.e., dynamical signals defined on simplices of different dimension, here taken to benodes and links for simplicity. We show that coupling between signals defined on nodes and linksleads to explosive topological synchronization in which phases defined on nodes synchronize simulta-neously to phases defined on links at a discontinuous phase transition. We study the model on realconnectomes and on simplicial complexes and network models. Finally, we provide a comprehensivetheoretical approach that captures this transition on fully connected networks and on random net-works treated within the annealed approximation, establishing the conditions for observing a closedhysteresis loop in the large network limit.

I. INTRODUCTION

Higher-order networks [1–4] are attracting increasingattention as they are able to capture the many-body in-teractions of complex systems ranging from brain to so-cial networks. Simplicial complexes are higher-order net-works that encode the network geometry and topologyof real datasets. Using simplicial complexes allows thenetwork scientist to formulate new mathematical frame-works for mining data [5–10] and for understanding thesegeneralized network structures revealing the underlyingdeep physical mechanisms for emergent geometry [11–15]and for higher-order dynamics [16–33]. In particular, thisvery vibrant research activity is relevant in neuroscienceto analyse real brain data and its profound relation todynamics [1, 6, 15, 34–37] and in the study of biologicaltransport networks [10, 38].

In networks, dynamical processes are typically definedover signals associated to the nodes of the network. Inparticular, the Kuramoto model [39–43] investigates thesynchronization of phases associated to the nodes of thenetwork. This scenario can change significantly in thecase of simplicial complexes [16, 17, 19]. In fact, simpli-cial complexes can sustain dynamical signals defined onsimplices of different dimension, including nodes, links,triangles and so on, called topological signals. For in-stance, topological signals defined on links can representfluxes of interest in neuroscience and in biological trans-portation networks. The interest on topological signalsis rapidly growing with new results related to signal pro-cessing [17, 19] and higher-order topological synchroniza-tion [16, 28, 44]. In particular, higher-order topological

∗Correspoding author. Email:juanga@colorado.edu†Correspoding author. Email:jtorres@onsager.ugr.es‡Correspoding author. Email:ginestra.bianconi@gmail.com

synchronization [16] demonstrates that topological sig-nals (phases) associated to higher dimensional simplicescan undergo a synchronization phase transition. Theseresults open a new uncharted territory for the investiga-tion of higher-order synchronization.

Higher-order topological signals defined on simplicesof different dimension can interact with one another innon-trivial ways. For instance in neuroscience the activ-ity of the cell body of a neuron can interact with synap-tic activity which can be directly affected by gliomes inthe presence of brain tumors [45]. In order to shed lighton the possible phase transitions that can occur whentopological signals defined on nodes and links interact,here we build on the mathematical framework of higher-order topological synchronization proposed in Ref. [16]and consider a synchronization model in which topolog-ical signals of different dimension are coupled. We focusin particular on the coupled synchronization of topolog-ical signals defined on nodes and links, but we note thatthe model can be easily extended to topological signalsof higher dimension. The reason why we focus on topo-logical signals defined on nodes and links is three-fold.First of all we can have a better physical intuition oftopological signals defined on nodes (traditionally stud-ied by the Kuramoto model) and links (like fluxes) that isrelevant in brain dynamics [45, 46] and biological trans-port networks [10, 38]. Secondly, although the coupledsynchronization dynamics of nodes and links can be con-sidered as a special case of coupled synchronization dy-namics of higher-order topological signals on a genericsimplicial complex, this dynamics can be observed alsoon networks including only pairwise interactions. Indeednodes and links are the simplices that remain unchangedif we reduce a simplicial complex to its network skele-ton. Since currently there is more availability of networkdata than simplicial complex data, this fact implies thatthe coupled dynamics studied in this work has wide ap-plicability as it can be tested on any network data and

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FIG. 1: Schematic representation of the Kuramoto and the topological Kuramoto model. Panel (a) shows a networkin which nodes sustain a dynamical variable (a phase) whose synchronization is captured by the Kuramoto model. Panel (b)shows a simplicial complex in which not only nodes but also links sustain dynamical variables whose coupled synchronizationdynamics is captured by the higher-order topological Kuramoto model.

network model. Thirdly, defining the coupled dynam-ics of topological signals defined on nodes and links canopen new perspectives in exploiting the properties of theline graph of a given network which is the network whosenodes corresponds to the links or the original network[47].

In this work, we show that by adopting a global adap-tive coupling of dynamics inspired by Refs. [48–50] thecoupled synchronization dynamics of topological signalsdefined on nodes and links is explosive [51], i.e., it oc-curs at a discontinuous phase transition in which the twotopological signals of different dimension synchronize atthe same time. We also illustrate numerical evidenceof this discontinuity on real connectomes and on simpli-cial complex models including the configuration model ofsimplicial complexes [52] and the non-equilibrium simpli-cial complex model called Network Geometry with Flavor[12, 13]. We provide a comprehensive theory of this phe-nomenon on fully connected networks offering a completeanalytical understanding of the observed transition. Thisapproach can be extended to random networks treatedwithin the annealed network approximation. The ana-lytical results reveal that the investigated transition isdiscontinuous.

II. RESULTS

A. Higher-order topological Kuramoto model oftopological signals of a given dimension

Let us consider a simplicial complex K formed by N[m]simplices of dimension m, i.e., N[0] nodes, N[1] links, N[2]triangles, and so on. In order to define the higher-ordersynchronization of topological signals we will make useof algebraic topology (see the Appendix for a brief intro-duction) and specifically we indicate with B[m] the m-thincidence matrix representing the m-th boundary opera-tor.The higher-order Kuramoto model generalizes the classicKuramoto model to treat synchronization of topologicalsignals of higher-dimension. The classic Kuramoto modeldescribes the synchonization transition for phases

θ = (θ1, θ2, . . . θN[0]) (1)

associated to nodes, i.e., simplices of dimension n = 0(see Figure 1). The Kuramoto model is typically definedon a network but it can treat also synchronization ofthe phases associated to the nodes of a simplicial com-plex. Each node i has associated an internal frequencyωi drawn from a given distribution, for instance a normal

3

distribution ωi ∼ N (Ω0, 1/τ0). In absence of any cou-pling, i.e., in absence of pairwise interactions, every nodeoscillates at its own frequency. However in a network orin a simplicial complex skeleton the phases associated tothe nodes follow the dynamical evolution dictated by theequation

θ̇ = ω − σB[1] sin(B>[1]θ

), (2)

where here and in the following we use the notation sin(x)to indicate the column vector where the sine function istaken element wise. Note that here we have chosen towrite this system of equations in terms of the incidencematrix B[1]. However if we indicate with a the adjacencymatrix of the network and with aij its matrix elements,this system of equations is equivalent to

θ̇i = ωi + σ

N∑j=1

aij sin(θj − θi), (3)

valid for every node i of the network. For coupling con-stant σ = σc the Kuramoto model [39–41] displays acontinuous phase transition above which the order pa-rameter

R0 =1

N[0]

∣∣∣∣∣∣N[0]∑i=1

eiθi

∣∣∣∣∣∣ (4)is non-zero also in the limit N[0] →∞.

The higher-order topological Kuramoto model [16] de-scribes synchronization of phases associated to the n di-mensional simplices of a simplicial complex. Althoughthe definition of the model applies directly to any valueof n, here we consider specifically the case in which thehigher-order Kuramoto model is defined on topologicalsignals (phases) associated to the links

φ = (φ`1 , φ`2 , . . . φ`N[1] ), (5)

where φ`r indicates the phase associated to the r-th linkof the simplicial complex (see Figure 1). The higher or-der Kuramoto dynamics defined on simplices of dimen-sion n > 0 is the natural extension of the standard Ku-ramoto model defined by Eq. (2). Let us indicate withω̃ the internal frequencies associated to the links of thesimplicial complex, sampled for example from a normaldistribution, ω̃` ∼ N (Ω1, 1/τ1). The higher-order topo-logical Kuramoto model is defined as

φ̇ = ω̃ − σB>[1] sin(B[1]φ)− σB[2] sin(B>[2]φ). (6)

Once the synchronization dynamics is defined on higher-order topological signals of dimension n (here taken to ben = 1) an important question is whether this dynamicscan be projected on (n + 1) and (n − 1) simplices. In-terestingly, algebraic topology provides a clear solutionto this question. Indeed for n = 1, when the dynamics

describes the evolution of phases associated to the links,one can consider the projection φ[−] and φ[+] respectivelyon nodes and on triangles defined as

φ[−] = B[1]φ,

φ[+] = B>[2]φ. (7)

Note that in this case B[1] acts as a discrete diver-

gence and B>[2] acts as a discrete curl. Interestingly,

since the incidence matrices satisfy B[1]B[down2] = 0 and

B>[2]B>[1] = 0 (see Methods V) these two projected phases

follow the uncoupled dynamics

φ̇[−] = B[1]ω̃ − σL[0] sinφ[−],φ̇[+] = B>[2]ω̃ − σL

down[2] sinφ

[+],

(8)

where L[0] = B[1]B>[1] and L

down[2] = B

>[2]B[2]. These two

projected dynamics undergo a continuous synchroniza-tion transition at σc = 0 [16] with order parameters

Rdown1 =1

N[0]

∣∣∣∣∣∣N[0]∑i=1

eiφ[−]i

∣∣∣∣∣∣ ,Rup1 =

1

N[2]

∣∣∣∣∣∣N[2]∑i=1

eiφ[+]i

∣∣∣∣∣∣ . (9)In Ref. [16] an adaptive coupling between these two dy-namics is considered formulating the explosive higher-order topological Kuramoto model in which the topolog-ical signal follows the set of coupled equations

φ̇ = ω̃ − σRup1 B>[1] sin(B[1]φ)

−σRdown1 B[2] sin(B>[2]φ). (10)

The projected dynamics on nodes and triangles are nowcoupled by the modulation of the coupling constant σwith the order parameters Rdown1 and R

up1 , i.e. the two

projected phases follow the coupled dynamics

φ̇[−] = B[1]ω̃ − σRup1 L[0] sinφ[−],φ̇[+] = B>[2]ω̃ − σR

down1 L

down[2] sinφ

[+].

(11)

This explosive higher-order topological Kuramoto modelhas been shown in Ref. [16] to lead to a discontinuous syn-chronization transition on different models of simplicialcomplexes and on clique complexes of real connectomes.

B. Higher-order topological Kuramoto model ofcoupled topological signals of different dimension

Until now, we have captured synchronization occurringonly among topological signals of the same dimension.

4

However, signals of different dimension can be coupled toeach other in non-trivial ways. In this work we will showhow topological signals of different dimensions can becoupled together leading to an explosive synchronizationtransition. Specifically we focus on the coupling of thetraditional Kuramoto model [Eq.(2)] to a higher-ordertopological Kuramoto model defined for phases associ-ated to the links [Eq.(6)]. The coupling between thesetwo dynamics is here performed considering the modu-lation of the coupling constant σ with the global orderparameters of the node dynamics [defined in Eq. (4)]and the link dynamics [defined in Eq. (9)]. Specifically,we consider two models denoted as Model NL (nodesand links) and model NLT (nodes, links, and triangles).Model NL couples the dynamics of the phases of thenodes θ and of the links φ according to the followingdynamical equations

θ̇ = ω − σRdown1 B[1] sin(B>[1]θ), (12)

φ̇ = ω̃ − σR0B>[1] sin(B[1]φ)− σB[2] sin(B>[2]φ).(13)

The projected dynamics for φ[−] and φ[+] then obey

φ̇[−] = B[1]ω̃ − σR0L[0] sinφ[−], (14)φ̇[+] = B>[2]ω̃ − σL

down[2] sinφ

[+]. (15)

Therefore the projection on the nodes φ[−] of the phasesφ associated to the links [Eq. (14)] is coupled to the dy-namics of the phases θ [Eq. (12)] associated directly tonodes. However the projection on the triangles φ[+] ofthe phases φ associated to the links is independent ofφ[−] and of θ as well. Model NLT also describes the cou-pled dynamics of topological signals defined on nodes andlinks but the adaptive coupling captured by the model isdifferent. In this case the dynamical equations are takento be

θ̇ = ω − σRdown1 B[1] sin(B>[1]θ), (16)

φ̇ = ω̃ − σR0Rup1 B>[1] sin(B[1]φ)

− σRdown1 B[2] sin(B>[2]φ). (17)

For Model NLT the projected dynamics for φ[−] and forφ[+] obey

φ̇[−] = B[1]ω̃ − σR0Rup1 L[0] sinφ[−], (18)φ̇[+] = B>[2]ω̃ − σR

down1 L

down[2] sinφ

[+]. (19)

Therefore, as in Model NL, the dynamics of the projec-tion φ[−] of the phases φ associated to the links [Eq. (18)]is coupled to the dynamics of the phases θ associateddirectly to nodes [Eq. (16)] and vice versa. Moreover,the dynamics of the projection of the phases φ on thetriangles φ[+] [Eq. (19)] is now also coupled with the dy-namics of φ[−] [Eq. (18)] and vice versa. Here and in thefollowing we will use the convenient notation (using theparameter m) to indicate both models NL and NLT with

the same set of dynamical equations given by

θ̇ = ω − σRdown1 B[1] sin(B>[1]θ), (20)

φ̇ = ω̃ − σR0 (Rup1 )m−1

B>[1] sin(B[1]φ)

−σ(Rdown1

)m−1B[2] sin(B

>[2]φ), (21)

which reduce to Eqs. (13) for m = 1 and to Eqs. (17) form = 2.

We make two relevant observations:

• First, the proposed coupling between topologicalsignals of different dimension can be easily ex-tended to signals defined on higher-order simplicesproviding a very general scenario for coupled dy-namical processes on simplicial complexes.

• Second, the considered coupled dynamics of topo-logical signals defined on nodes and links can bealso studied on networks with exclusively pairwiseinteractions where we assume that the number ofsimplices of dimension n > 1 is zero. Thereforein this specific case this topological dynamics canhave important effects also on simple networks.

We have simulated Model NL and Model NLT on twomain examples of simplicial complex models: the config-uration model of simplicial complexes [52] and the Net-work Geometry with Flavor (NGF) [12, 13] (see Figure2). In the configuration model we have considered power-law distribution of the generalized degree with exponentγ < 3, and for the NGF model with have considered sim-plicial complexes of dimensions d = 3 whose skeleton isa power-law network with exponent γ = 3. In both caseswe observe an explosive synchronization of the topologi-cal signals associated to the nodes and to the links. Onfinite networks, the discontinuous transition emerge to-gether with the hysteresis loop formed by the forwardand backward synchronization transition. However thetwo models display a notable difference. In Model NLwe observe a discontinuity for R0 and R

down1 at a non-

zero coupling constant σ = σc, however Rup1 follows an

independent transition at zero coupling (see Figure 2,panels in the second and fourth column). In Model NLT,on the contrary, all order parameters R0, R

down1 , and

Rup1 display a discontinuous transition occurring for thesame non zero value of the coupling constant σ = σc (seeFigure 2 panels in the first and third column). This isa direct consequence of the fact that in Model NL theadaptive coupling leading to discontinuous phase transi-tion only couples the phases φ[−] and θ, while for ModelNLT the coupling involves also the phases φ[+].

Additionally we studied both Model NL and ModelNTL on two real connectomes: the human connectomeof Ref. [53] and the c. elegans connectome from Ref. [54](see Figure 3). Interestingly also for these real datasetswe observe that in Model NL the explosive synchroniza-tion involves only the phases θ and φ[−] while in ModelNLT we observe that also φ[+] undergoes an explosive

5

synchronization transition at the same value of the cou-pling constant σ = σc.

III. DISCUSSION

A. Theoretical solution of the NL model

As mentioned earlier the higher-order topological Ku-ramoto model coupling the topological signals of nodesand links can be defined on simplicial complexes and onnetworks as well. In this section we exploit this propertyof the dynamics to provide an analytical understandingof the synchronization transition on uncorrelated randomnetworks.

It is well known that the Kuramoto model is challeng-ing to study analytically. Indeed the full analytical un-derstanding of the model is restricted to the fully con-nected case, while on a generic sparse network topologythe analytical approximation needs to rely on some ap-proximations. A powerful approximation is the annealednetwork approximation [41] which consists in approxi-mating the adjacency matrix of the network with its ex-pectation in a random uncorrelated network ensemble. Inorder to unveil the fundamental theory that determinesthe coupled dynamics of topological signals described bythe higher-order Kuramoto model here we combine theannealed approximation with the Ott-Antonsen method[43]. This approach is able to capture the coupled dy-namics of topological signals defined on nodes and links.In particular the solution found to describe the dynamicsof topological signals defined on the links is highly nontrivial and it is not reducible to the equations valid forthe standard Kuramoto model. Conveniently, the cal-culations performed in the annealead approximation canbe easily recasted in the exact calculation valid in thefully connected case previous a rescaling of some of theparameters. The analysis of the fully connected networkreveals that the discontinuous sychronization transitionof the considered model is characterized by a non-trivialbackward transition with a well defined large networklimit. On the contrary the forward transition is highlydependent on the network size and vanishes in the largenetwork limit, indicating that the incoherent state re-mains stable for every value of the coupling constant σ inthe large network limit. This implies that on a fully con-nected network the NL model does not display a closedhysteresis loop as it occurs also for the model proposedin Ref. [21]. This scenario is here shown to extend alsoto sparse networks with finite second moment of the de-gree distribution while scale-free networks display a welldefined hysteresis loop in the large network limit.

B. Annealed dynamics

For the dynamics of the phases θ associated to thenodes - Eq. (20) - it is possible to proceed as in the tra-

ditional Kuramoto model [42, 55, 56]. However the an-nealed approximation for the dynamics of the phases φdefined in Eq. (21) needs to be discussed in detail as itis not directly reducible to previous results. To addressthis problem our aim is to directly define the annealedapproximation for the dynamics of the projected vari-ables φ[−] which, here and in the following are indicatedas

ψ = φ[−], (22)

in order to simplify the notation. Moreover we will indi-cate with N = N[0] the number of nodes in the network orin the simplicial complex skeleton. Here we focus on theNL Model defined on networks, i.e., we assume that thereare no simplices of dimension two. We provide an analyt-ical understanding of the coupled dynamics of nodes andlinks in the NL Model by determining the equations thatcapture the dynamics in the annealed approximation andpredict the value of the complex order parameters

R0eiΘ =

1

N

N∑i=1

eiθi ,

Rdown1 eiΨ =

1

N

N∑i=1

eiψi , (23)

(with R0, Rdown1 ,Θ and Ψ real) as a function of the cou-

pling constant σ.We notice that Eq. (14), valid for Model NL, can be

written as

ψ̇ = B[1]ω̃ − σR0L[0] sin(ψ). (24)

This equation can be also written elementwise as

ψ̇i = ω̂i + σR0

N∑j=1

aij [sin(ψj)− sin(ψi)] , (25)

where the vector ω̂ is given by

ω̂ = B[1]ω̃. (26)

Let us now consider in detail these frequencies in thecase in which the generic internal frequency ω̃` of a linkfollows a Gaussian distribution, specifically in the casein which ω̃` ∼ N (Ω1, 1/τ1) for every link `. Using thedefinition of the boundary operator on a link it is easyto show that the expectation of ω̂i is given by

〈ω̂i〉 =

∑ji

aij

Ω1. (27)Given that each node has degree ki, the covariance

matrix C is given by the graph Laplacian L[0] of thenetwork, i.e.

Cij = 〈ω̂iω̂j〉c =∑`,`′

〈[B[1]ω̃]i[B[1]ω̃]j

〉c

=[L[0]]ij

τ21=kiδij − aij

τ21, (28)

6

0 2 4 60

0.5

1

0 2 4 60

0.5

1

0 2 4 60

0.5

1

0 2 4 60

0.5

1

0 2 4 60

0.5

1

0 2 4 60

0.5

1

0 2 4 60

0.5

1

0 2 4 60

0.5

1

0 2 4 60

0.5

1

0 2 4 60

0.5

1

0 2 4 60

0.5

1

0 2 4 60

0.5

1

(d)(c)(b)(a)

(f) (g) (h)(e)

(i) (j) (k) (l)

FIG. 2: The Higher-order topological synchronization models (Models NL and NLT) coupling nodes and linkson simplicial complexes. The order parameters R0, R

down1 and R

up1 are plotted versus σ for the higher-order topological

synchronization Model NLT (panels (a)-(e)-(i) and (c)-(g)-(k)) and Model NL (panels (b)-(f)-(j) and (d)-(h)-(l)) defined overthe Network Geometry with Flavor [13] (panels (a)-(e)-(i) and (b)-(f)-(j)) and the configuration model of simplicial complexes[52] (panels (c)-(g)-(k) and (d)-(h)-(l)). The Network Geometry with Flavor on which we run the numerical results shown in(a) and (b) includes N[0] = 500 nodes and has flavor s = −1 and d = 3. The configuration model of simplicial complexes onwhich we run the numerical results shown in (c) and (d) includes N[0] = 500 nodes and has generalized degree distributionwhich is power-law with exponent γ = 2.8. In both Model NL and in Model NLT we have set Ω0 = Ω1 = 2 and τ0 = τ1 = 1.

where we have indicated with 〈. . .〉c the connected cor-relation. Therefore the variance of ω̂ in the annealedapproximation is〈

ω̂2i〉c

= 〈ω̂2i 〉 − 〈ω̂i〉2 =kiτ21. (29)

Moreover, the projected frequencies are actually corre-lated and for i 6= j we have

〈ω̂iω̂j〉c = 〈ω̂iω̂j〉 − 〈ω̂i〉 〈ω̂j〉 = −aijτ21. (30)

It follows that the frequencies ω̂ are correlated Gaussianvariables with average given by Eq. (27) and correlationmatrix given by the graph Laplacian. The fact that thefrequencies ω̂i are correlated is an important feature ofthe dynamics of ψ and, with few exceptions (e.g., [57]),this feature has remained relatively unexplored in thecase of the standard Kuramoto model. Additionally letus note that the average of ω̂ over all the nodes of thenetwork is zero. In fact

N∑i=1

ω̂i = 1T ω̂ = 1TB[1]ω = 0, (31)

where with 1 we indicate the N -dimensional column vec-tor of elements 1i = 1. By using the symmetry of theadjacency matrix, i.e. the fact that aij = aji, Eq. (31)

implies that the sum of ψ̇i over all the nodes of the net-work is zero, i.e.

N∑i=1

ψ̇i =∑Ni=1 ω̂i + σR0

∑i,j aij [sin(ψj)− sin(ψi)] = 0.

We now consider the annealed approximation consist-ing in substituting the adjacency matrix element aij withits expectation in an uncorrelated network ensemble

aij →kikj〈k〉N

, (32)

where ki indicates the degree of node i and 〈k〉 is theaverage degree of the network. Note that the consideredrandom networks can be both sparse [58] or dense [59]as long as they display the structural cutoff, i.e. ki �√〈k〉N for every node i of the network. In the annealed

7

0 50

0.5

1

0 50

0.5

1

0 50

0.5

1

0 50

0.5

1

0 50

0.5

1

0 50

0.5

1

0 2 40

0.5

1

0 50

0.5

1

0 50

0.5

1

0 2 40

0.5

1

0 50

0.5

1

0 50

0.5

1

(b) (c) (d)(a)

(f) (g) (h)(e)

(i) (j) (k) (l)

FIG. 3: The Higher-order topological synchronization models (Models NLT and NL) coupling nodes and linkson real connectomes. The order parameters R0, R

down1 and R

up1 are plotted versus σ on real connectomes. Panels (a)-

(e)-(i) and (b)-(f)-(j) show the numerical results on the human connectome [53] for Model NLT and Model NL respectively.Panels (c)-(g)-(k) and (d)-(h)-(i) show the numerical results on the c. elegans connectome [54] for Model NLT and Model NLrespectively. In both Model NLT and in Model NL we have set Ω0 = Ω1 = 2 and τ0 = τ1 = 1.

approximation we can put

〈ω̂i〉 ' kiΩ1

1− 2∑j>i

kj〈k〉N

. (33)Also, in the annealed approximation the dynamicalEq. (20) and Eq. (24) reduce to

θ̇ = ω − σRdown1 R̂0k · sin(θ − Θ̂), (34)ψ̇ = ω̂ + σR0R̂

down1 k sin Ψ̂− σR0k� sinψ, (35)

where � indicates the Hadamard product (element byelement multiplication) and where two auxiliary complexorder parameters are defined as

R̂0eiΘ̂ =

N∑i=1

ki〈k〉N

eiθi ,

R̂down1 eiΨ̂ =

N∑i=1

ki〈k〉N

eiψi , (36)

with R̂0, Θ̂, R̂down1 and Ψ̂ real.

C. The dynamics on a fully connected network

On a fully connected network in which each node hasdegree ki = N − 1 the dynamics of the NL Model is welldefined provided its parameter are properly rescaled. Inparticular we require a standard rescaling of the couplingconstant with the network size, given by

σ → σ/(N − 1) (37)

which guarantees that the interaction term in the dynam-ical equations has a finite contribution to the velocity ofthe phases.

The Model NL on fully connected networks requiresalso some specific model dependent rescalings associatedto the dynamics on networks. Indeed in order to have afinite expectation 〈ω̂i〉 of the projected frequencies ω̂i anda finite of the covariance matrix C, [given by Eqs. (27)and (28), respectively] we require that on a fully con-nected network both Ω1 and τ1 are rescaled accordingto

Ω1 → Ω1/N,τ1 → τ1

√N − 1. (38)

8

Considering these opportune rescalings and noticingthat the order parameters obey R̂0 = R0, R̂

down1 =

Rdown1 , Θ = Θ̂, and Ψ = Ψ̂, we obtain that Model NLdictated by Eqs. (34)-(35) can be rewritten here as

θ̇ = ω − σRdown1 R0 sin(θ −Θ), (39)ψ̇ = ω̂ + σR0R

down1 sin Ψ− σR0 sinψ, (40)

with R0, Rdown1 ,Θ and Ψ given by Eq. (23) and

Cij = 〈ω̂iω̂j〉c = δij −1

N − 1. (41)

D. Solution of the dynamical equations in theannealed approximation

1. General framework for obtaining the solution of theannealed dynamical equations

In this section we will provide the analytic solutionsfor the order parameter of the higher-order topologicalsynchronization studied within the annealed approxima-tion, i.e., captured by Eqs. (34) and (35). In particularfirst we will find an expression of the order parametersR0 of the dynamics associated to the nodes (Eq. (34))and subsequently in the next paragraph we will derivethe expression for the order parameter Rdown1 associatedto the projection on the nodes of the topological signaldefined on the links (Eq. 35)). By combining the tworesults it is finally possible to uncover the discontinuousnature of the transition.

2. Dynamics of the phases of the nodes

When we investigate Eq. (34) we notice that thisequation can be easily reduced to the equation for thestandard Kuramoto model treated within the annealedapproximation [42] if one performs a rescaling of thecoupling constant σ R0 → σ. Therefore we can treatthis model similarly to the known treatment of the stan-dard Kuramoto model [40–42]. Specifically, starting fromEq. (34) and using a rescaling of the phases θ accordingto

θi → θi − Ω0t, (42)

it is possible to show that we can set Θ = 0 and thereforeEq. (34) reduces to the well-known annealed expressionfor the standard order Kuramoto model given by

θ̇ = ω − Ω01− σRdown1 R̂0k · sin(θ). (43)

Assuming that the system of equations reaches asteady state in which both Rdown1 and R̂0 become timeindependent, the order parameters of this system of equa-tions in the coherent state R̂0 > 0 and R

down1 > 0 can be

found to obey [40, 42, 51, 55]

R̂0 =

N∑i=1

ki〈k〉N

∫|ĉi|

9

Making the ansatz

f̂ (i)m (ω̂i, t) = [bi(ω̂i, t)]m (51)

we can derive the equation for the evolution of bi =bi(ω̂i, t) given by

∂tbi + ibiκi + σkiR01

2(b2i − 1) = 0. (52)

Since we showed before that the average value of ψ̇i overnodes is zero, we look for non-rotating stationary solu-tions of Eq. (52), ∂tbi = 0. As long as R0 > 0 thesestationary solutions are given by

bi = −idi ±√

1− d2i , (53)

where di is given by

di =ω̂i

σkiR0+ R̂down1 sin Ψ̂. (54)

By inserting this expression into Eq. (49) we get the ex-pression of the order parameters given the projected fre-

quencies ω̂, in the coherent phase in which R0 > 0

R̂down1 cos Ψ̂ =

N∑i=1

ki〈k〉N

√1− d2i θ(1− d

2i ),

R̂down1 sin Ψ̂ =

N∑i=1

ki〈k〉N

{√d2i − 1χ(di) + di

},

Rdown1 cos Ψ =

N∑i=1

1

N

√1− d2i θ(1− d

2i ),

Rdown1 sin Ψ =

N∑i=1

1

N

{√d2i − 1χ(di) + di

}, (55)

where, indicating by θ(x) the Heaviside function, we havedefined

χ(di) = [−θ(di − 1) + θ(−1− di)]. (56)

Finally, if the projected frequencies ω̂ are not known wecan average the result over the marginal frequency dis-tribution of the projected frequency ω̂i given by Gi(ω̂)getting

R̂down1 cos Ψ̂ =

N∑i=1

ki〈k〉N

∫|di|≤1

dω̂iGi(ω̂i)

√1−

(ω̂i

σR0ki+ R̂down1 sin Ψ̂

)2,

R̂down1 sin Ψ̂ = −N∑j=0

ki〈k〉N

∫di>1

dω̂iGi(ω̂i)

√(ω̂i

σR0ki+ R̂down1 sin Ψ̂

)2− 1

+

N∑i=1

ki〈k〉N

∫di

10

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

R0

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

R0

0.0 0.5 1.0 1.5 2.00

0.2

0.4

0.6

0.8

1

R1d

ow

n

0.0 0.5 1.0 1.5 2.00.0

0.1

0.2

0.3

0.4

0.5

R1d

ow

n

(a) (b)

(c) (d)

FIG. 4: Comparison between the simulation results of the NL Model and its solution in the annealed approxi-mation The order parameters R0 and R

down1 of the NL Model are shown as a function of σ for a Poisson network with average

degree c = 12 and for an uncorrelated scale-free network with minimum degree m = 6 and power-law exponent γ = 2.5. Bothnetworks have N = 1600 nodes. The symbols indicate the simulation results for the forward (cyan diamonds) and the backward(green circles) synchronization transition. The solid lines indicate the analytical solution for the backward transition obtainedby integrating Eq. (55).

from direct numerical integration of Eqs. (20) and (25)and the steady state solutions obtained from the numer-ical solution of Eqs. (55). The backward transition isfully captured by our theory, while the next paragraphswill clarify the theoretical expectations for the forwardtransition.

E. Solution on the fully connected network

The integration of Eq. (57) requires the knowledge ofthe marginal distributions Gi(ω̂) which does not have ingeneral a simple analytical expression. However, in thefully connected networks with Gaussian distribution ofthe internal frequency of nodes and links this calcula-tion simplifies significantly. Indeed, when the link fre-quencies are sampled from a Gaussian distribution withmean Ω1/N and standard deviation 1/(τ1

√N − 1), the

marginal frequency distribution Gi(ω̂) of the internal fre-quency ω̂i of a node i in a fully connected network is given

by (see Methods for details)

Gi(ω̂) =τ1√2π/c̄

exp

[−τ21 c̄

(ω̂i − 〈ω̂i〉)2

2

], (58)

where c̄ = NN−1 . By considering Ω0 = Ω1 = 〈ω̂i〉 = 0, andperforming a direct integration of Eqs. (57) we obtain(see Methods section for details) the closed system ofequations for R0 and R

down1

1 = σRdown1 h(σ2R20(R

down1 )

2),

Rdown1 = σR0τ1√c̄h(σ2τ21R

20

), (59)

where the scaling function h(x) is given by

h(x) =

√π

2e−x/4

[I0

(x4

)+ I1

(x4

)], (60)

with I0 and I1 indicating the modified Bessel functions.The numerical solution of Eqs. (59) reveals the followingpicture: for low values of σ, only the incoherent solution

11

R0 = Rdown1 = 0 exists. At a positive value of σ, two

solutions of Eqs. (59) appear at a bifurcation point, withthe upper solution corresponding to a stable synchronizedstate and the lower solution to an unstable synchronizedsolution. For larger values of σ, the values of R0 andRdown1 corresponding to the upper solution approach one(full phase synchronization), while those for the lower so-lution approach zero asymptotically, thus indicating thatthe incoherent state never loses stability. Indeed, it canbe easily checked (see Methods for details) that for largeσ the unstable solution of Eqs. (59) has asymptotic be-havior

R0 = σ−2J0,

Rdown1 = σ−1J1, (61)

with J0 and J1 constants given by

J0 =[π

2

]−2[G(0)g(0)]

−1, (62)

J1 =[g(0)

π

2

]−1. (63)

Therefore the unstable branch approaches the trivial so-lution R0 = R

down1 = 0 only asymptotically for σ → ∞.

This implies that the trivial solution remains stable forevery possible value of σ although as σ increases it de-scribes the stationary state of an increasingly smaller setof initial conditions.

This scenario is confirmed by numerical simulations(see Figure 5) showing that the backward transition iscaptured very well by our theory and does not displaynotable finite size effects. The forward transition, in-stead, displays remarkable finite size effects. Indeed, asσ increases, the system remains in the incoherent stateuntil it explosively synchronizes at a positive value of σand reaches the stable synchronized branch. However theincoherent state is stable in the limit N → ∞, and thisforward transition is the result of finite size fluctuationsthat push the system above the unstable branch, causingthe observed explosive transition. This is consistent withthe fact that for larger values of N , which have smallerfinite size fluctuations, the system remains in the inco-herent state for larger values of σ.

Therefore, while a closed hysteresis loop is not presentin the NL model defined on fully connected networks, weobserve fluctuation-driven hysteresis, in which finite-sizefluctuations of the zero solution drive the system towardsthe synchronized solution, creating an effective hysteresisloop.

F. Hysteresis on homogeneous and scale-freenetworks

In this section we discuss how the scenario found forthe fully connected network can be extended to randomnetworks with given degree distribution. We will startfrom the self-consistent Eqs. (57) obtained within the an-nealed approximation model. These equations display

a saddle point bifurcation with the emergence of twonon-trivial solutions describing a stable and an unsta-ble branch of these self-consistent equations. These solu-tions always exist in combination with the trivial solutionR0 = R

down1 = 0 describing the asynchronous state. Two

scenarios are possible: either the unstable branch con-verges to the trivial solution only in the limit σ → ∞or it converges to the trivial solution at a finite value ofσ. In the first case, the scenario is the same as the oneobserved for the fully connected network, and the trivialsolution remains stable for any finite value of σ. In thiscase the forward transition is not obtained in the limitN → ∞ and the transition observed on finite networksis only caused by finite size effects. In the second casethe trivial solution loses its stability at a finite value ofσ. Therefore the forward transition is not subjected tostrong finite size effects and we expect to see a forwardtransition also in the N → ∞ limit. in order to deter-mine which network topologies can sustain a non-trivialhysteresis loop we expand Eqs. (57) for 0 < R0 � 1,0 < R̂0 � 1, and 0 < Rdown1 � 1 under the hypothesisthat the distributions g(ω) and Gi(ω̂) are symmetric andunimodal. Under these hypothesis it is easy to show thatEqs. (57) predict an unstable solution in which R0 andRdown1 scale with σ according to

R0 = σ−2J0,

Rdown1 = σ−1J1, (64)

with J0 and J1 constants given by

J0 = 〈k〉

[π

2

〈k2〉

〈k〉

]−2 [g(Ω0)

1

N

∑i

kiGi(〈ω̂i〉)

]−1,

J1 =

[g(Ω0)

π

2

〈k2〉

〈k〉

]−1. (65)

As long as the network does not have vanishing J0 andJ1 the unstable branch converges to the trivial solutionR0 = R

down1 only in the limit σ → ∞. This happens

for instance for Gaussian distribution of the internal fre-quency of the links and converging second moment 〈k2〉of the degree distribution. However, when the secondmoment diverges, i.e., the network is scale free with〈k2〉 → ∞ as N → ∞, then R0 and R1 can convergeto the trivial solution R0 = R

down1 = 0 also for finite σ.

This analysis suggests that the scenario described for thefully connected network remains valid for sparse (con-nected) networks as long as the degree distribution doesnot have a diverging second moment, while a stable hys-teresis loop can be observed for scale-free networks.

IV. CONCLUSIONS

Until recently the synchronization phenomenon hasbeen explored only in the context of topological signals

12

0 1 2 3 4 5 6 7 8 90.0

0.2

0.4

0.6

0.8

1.0

R0

0 1 2 3 4 5 6 7 8 90.0

0.2

0.4

0.6

0.8

1.0

R1d

ow

n

FIG. 5: The backward and the forward discontinuous phase transition on fully connected networks The orderparameters R0 (circles) and R

down1 (squares) are plotted as a function of the coupling constant σ on a fully connected network.

The solid and the dashed lines indicate the stable branch and the unstable branch predicted by Eqs.(59). Simulations (shownas data point) are here obtained by integrating numerically Eqs. (34) and (35) for a fully connected network of N = 500 (cyancircles), N = 1000 (green squares), and N = 2000 (purple diamonds) with Ω0 = Ω1 = 0 and (rescaled) τ0 = τ1 = 1. Thebackward transition is perfectly captured by the theoretical prediction and is affected by finite size effects very marginally. Theforward transition is instead driven by stochastic fluctuations and moves to higher values of σ as the network size increases.

associated to the nodes of a network. However, the grow-ing interest in simplicial complexes opens the perspectiveof investigating synchronization of higher order topologi-cal signals, for instance associated to the links of the dis-crete networked structure. Here we uncover how topolog-ical signals associated to nodes and links can be coupledto one another giving rise to an explosive synchroniza-tion phenomenon involving both signals at the same time.The model has been tested on real connectomes and onmajor examples of simplicial complexes (the configura-tion model [52] of simplicial complex and the NetworkGeometry with Flavor [13]). Moreover, we provide an an-alytical solution of this model that provides a theoreticalunderstanding of the mechanism driving the emergenceof this discontinuous phase transition and the mechanismresponsible for the emergence of a closed hysteresis loop.This work can be extended in different directions includ-ing the study of the de-synchronization dynamics of thiscoupled higher-order synchronization and the duality ofthis model with the same model defined on the line graph

of the same network.

Acknowledgements

This work is partially founded by SUPERSTRIPESOnlus. This research utilized Queen Mary’s Ap-ocrita HPC facility, supported by QMUL Research-IT. http://doi.org/10.5281/zenodo.438045. G.B. ac-knowledge support from the Royal Society IEC\NSFC\191147. J.J.T. acknowledges financial support fromthe Spanish Ministry of Science and Technology, andthe Agencia Española de Investigación (AEI) undergrant FIS2017-84256-P (FEDER funds) and from theConsejeŕıa de Conocimiento, Investigación y Univer-sidad, Junta de Andalućıa and European RegionalDevelopment Fund, Refs. A-FQM-175-UGR18 andSOMM17/6105/UGR.

http://doi.org/10.5281/zenodo.438045

13

Author contributions

All authors have contributed in the design of theproject, in the numerical implementations of the algo-rithm, the theoretical derivations and the writing of themanuscript.

Code Availability

All codes are available upon request to the Authors.

Data Availability

The connectome network dataset used in this study arefreely available: the Homo sapiens dataset comes fromRef. [53] the C.elegans dataset comes from Ref. [54].

Competing interests

The authors declare no competing interests.

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METHODS

V. SIMPLICIAL COMPLEXES AND HIGHERORDER LAPLACIANS

A. Definition of simplicial complexes

Simplicial complexes represent higher-order networkswhose interactions include two or more nodes. Thesemany-body interactions are captured by simplices. A n-dimensional simplex α is a set of n+ 1 nodes

α = [i0, i1, . . . , in]. (66)

For instance a node is a 0-dimensional simplex, a linkis a 1-dimensional simplex, a triangle is a 2-dimensionalsimplex, a tetrahedron is a 3-dimensional simplex, andso on. A face of a simplex is the simplex formed by aproper subset of the nodes of the original simplex. For

instance the faces of a tetrahedron are 4 nodes, 6 linksand 4 triangles. A simplicial complex is a set of simplicesclosed under the inclusion of the faces of each simplex.Any simplicial complex can be reduced to its simplicialcomplex skeleton, which is the network formed by thesimplicial complex nodes and links. Simplices have a rel-evant topological and geometrical interpretation and con-stitute the topological structures studied by discrete al-gebraic topology. Therefore representing the many-bodyinteractions of a complex system with a simplicial com-plex opens the very fertile opportunity to use the toolsof algebraic topology [5, 60] to study the topology of thesystem under investigation. In this work we show thatalgebraic topology can also shed significant light on therole that topology has on higher-order synchronization.

B. Oriented simplices and boundary map

In algebraic topology simplices are oriented. For in-stance a link α = [i, j] has the opposite sign of the link[j, i], i.e.,

[i, j] = −[j, i]. (67)

Similarly to higher order simplices we associate an orien-tation such that

[i0, i1, . . . , in] = (−1)σ(π)[iπ(0), iπ(1), . . . , iπ(n)], (68)

where σ(π) indicates the parity of the permutation π.It is good practice to use as orientation of the simplicesthe orientation induced by the labelling of the nodes, i.e.,giving, for example, a positive orientation to any simplex

[i0, i1, . . . , in], (69)

where

i0 < i1 < i2 . . . < in. (70)

This will ensure that the spectral properties of the higher-order Laplacians that will be defined later are indepen-dent of the labelling of the nodes. Given a simplicial com-plex, a n-chain consists of the elements of a free abeliangroup Cn with basis formed by the set of all oriented n-simplices. Therefore every element of Cn can be uniquelyexpressed as a linear combination of the basis elements(n-simplices) with coefficients in Z2. The boundary op-erator ∂n is a linear operator ∂n : Cn → Cn−1 whoseaction is determined by the action on each n-simplex ofthe simplicial complex given by

∂n[i0, i1 . . . , in] =

n∑p=0

(−1)p[i0, i1, . . . , ip−1, ip+1, . . . , in].(71)

As a concrete example, in Figure 6 we demonstrate theaction of the boundary operator on links and triangles.

15

FIG. 6: The boundary operators and their representation in terms of the incidence matrices. Panel (a) and (b)describe the action of the boundary operator on an oriented link and on an oriented triangle respectively. Panel (c) shows atoy example of a simplicial complex and panel (d) indicates its incidence matrices B[1] and B[2] representing the boundaryoperators ∂1 and ∂2 respectively.

A celebrated property of the boundary operator is thatthe boundary of a boundary is null, i.e.

∂n∂n+1 = 0 (72)

for any n > 0. This relation can be directly proven byusing Eq. (71). Let us consider a simplicial complex K.Let us indicate with N[n] the number of simplices of thesimplicial complex with generic dimension n. Given a ba-sis for the linear space of n-chains Cn and for the linearspace of (n − 1)-chains Cn−1 formed by an ordered listof the n simplices and (n− 1) simplices of the simplicialcomplex, the boundary operator ∂n can be representedas N[n−1] × N[n] incidence matrix B[n]. In Figure 6 weshow a 2-dimensional simplicial complex and its corre-sponding incidence matrices B[1] and B[2]. Given thatthe boundary matrices obey Eq. (72) it follows that theincidence matrices obey

B[n]B[n+1] = 0, B>[n+1]B

>[n] = 0, (73)

for any n > 0.

C. Higher order Laplacians

Using the incidence matrices it is natural to generalizethe definition of the graph Laplacian

L[0] = B[1]B>[1] (74)

to the higher-order Laplacian L[n](also called combina-torial Laplacians) [17, 19, 61] that can be represented asa N[n] ×N[n] matrix given by

L[n] = Ldown[n] + L

up[n] (75)

with

Ldown[n] = B>[n]B[n],

Lup[n] = B[n+1]B>[n+1], (76)

for n > 0. The higher-order Laplacian can be proven tobe independent of the orientation of the simplices as longas the simplicial complex has an orientation induced bya labelling of the nodes.

The most celebrated property of higher-order Lapla-cian is that the degeneracy of the zero eigenvalue of the

16

n Laplacian L[n] is equal to the Betti number βn and thattheir corresponding eigenvectors localize around the cor-responding n-dimensional cavities of the simplicial com-plex. The higher-order Laplacians can be used to definehigher-order diffusion [17] and can display a higher-orderspectral dimension on network geometries. Here we areparticularly interested in the use of incidence matricesand higher-order Laplacians to define higher-order topo-logical synchronization.

D. Steady-state solution of the annealed equationsfor the NL Model

Here we study Eqs. (44), (57) assuming that the dis-tributions g(ω) and Gi(ω̂) are unimodal functions sym-

metric about their means. Setting Ψ = Ψ̂ = 0 andconsidering the change of variables z = ω/(σR0R

down1 ),

y = ω̂/(σR0), Eqs. (44) can be written as

1 = σRdown1

N∑i=1

k2i〈k〉N

∫ 1−1g(Ω0 + zσkiR̂0R

down1 )

√1− z2dz,

R0 = σR̂0Rdown1

N∑i=1

kiN

∫ 1−1g(Ω0 + zσkiR̂0R

down1 )

√1− z2dz,

while Eqs. (57) reduce to

Rdown1 = σR0

N∑i=1

kiN

∫ 1−1Gi(〈ω̂i〉+ yσR0ki)

√1− y2dy,

R̂down1 = σR0

N∑i=1

k2i〈k〉N

∫ 1−1Gi(〈ω〉i + yσR0ki)

√1− y2dy.

We notice that the equations for R0, R̂0 and Rdown1 do not

depend on the order parameter R̂down1 so we can obtaina fully analytical solution of the model without solvingthe last equation. The above equations depend on thedistribution g(ω) and the set of marginal distributionsGi(ω̂i). However we can show that, provided 〈k2〉/〈k〉 isfinite, the solution of these equations does not converge tothe trivial solution R0 = R̂0 = R

down1 = 0 for any finite

value of σ. Indeed we are now going to show that theunstable branch of the solution these equations convergesto the trivial solution only in the limit σ →∞. Assuming0 < R0 � 1, 0 < R̂0 � 1 and 0 < Rdown1 � 1 we canexpand the functions g(zσkiR̂0R

down1 ) and Gi(yσR0ki)

as

g(Ω0 + zσkiR̂0Rdown1 ) ' g(Ω0) +

g′′(Ω0)2

(zσkiR̂0Rdown1 )

2

Gi(〈ω̂i〉+ yσR0ki) ' Gi(〈ω̂i〉) +G′′i (〈ω̂i〉)

2(yσR0ki)

2

Stopping at the first order of this expansion we get

1 = σRdown1 g(Ω0)π

2

〈k2〉

〈k〉, (77)

R0 = σR̂0Rdown1 g(Ω0)

π

2〈k〉 , (78)

Rdown1 = σR̂0π

2

1

N

∑i

kiGi(〈ω̂i〉). (79)

This equations lead to the following scaling of R0 andRdown1 with σ

R0 = σ−2J0,

Rdown1 = σ−1J1, (80)

with

J0 = 〈k〉

[π

2

〈k2〉

〈k〉

]−2 [g(Ω0)

1

N

∑i

kiGi(〈ω̂i〉)

]−1,

J1 =

[g(Ω0)

π

2

〈k2〉

〈k〉

]−1. (81)

This confirms the theoretical framework revealing thatin this dynamics there is always a trivial solution R0 =R̂0 = R

down1 = 0 while Eqs. (44), (57) are characterized

by a saddle-point instability so that for σ > σc two addi-tional solutions emerge, a stable solution and an unstablesolution. The stable solution describes the synchronizedphase and captures the backward transition. As long asthe second moment of the degree distribution does notdiverge, the unstable solution converges to the trivial so-lution R0 = R̂0 = R

down1 = 0 only for σ →∞.

The asymptotic scaling for R0 and Rdown1 given by

Eq. (80) can be adapted to capture the asymptotic scal-ing of the fully connected case with a suitable rescaling ofthe model parameters of the model, obtaining Eqs. (61),(63).

17

E. Marginal distributions in the fully connectedcase

The distribution G1(ω̂) of ω̂ is a Gaussian distributionwith averages given by Eq. (27) and covariance matrixC given by Eq.(28). The covariance matrix has N − 1eigenvalues given by λ = 1/τ21 and one zero eigenvalueλ = 0 corresponding to the eigenvector

1/√N = (1, 1, . . . , 1)>/

√N. (82)

This means that we should always have

N∑n=1

[ω̂n − 〈ω̂n〉]√N

= 0, (83)

a constraint that we can introduce as a delta functionin the expression for the joint distribution Ĝ(ω̂) of thevector ω̂. Here we note that under these hypotheses andassuming that the distribution of the frequencies of thelinks is a Gaussian with average Ω1/N and standard de-viation 1/(τ1

√N − 1) the marginal probability Gi(ω̂) of

ω̂i can be expressed as Eq. (58).Given that the covariance matrix has a zero eigenvalue

we can express the joint Gaussian distribution Ĝ(ω̂) as

Ĝ(ω̂) = Ce−F(ω̂)δ

(N∑n=1

[ω̂n − 〈ω̂n〉]√N

), (84)

where δ(x) indicates the delta function and where F(ω̂)and C are given by

F(ω̂) = τ21

2

N∑n=1

(ω̂n − 〈ω̂n〉)2 ,

C =(

τ1√2π

)N−1. (85)

The marginal probability Gi(ω̂) is given by

Gi(ω̂) =

∫ ∏n 6=i

dω̂nĜ(ω̂). (86)

By expressing the delta function in Eq. (84) in its integralform

δ(x, y) =1

2π

∫ ∞−∞

dzeiz(x−y) (87)

we get the final expression for the marginal distributionEq. (58), in fact we have

G(i)1 (ω̂) =

C2π

∫dz

∫ ∏n 6=i

dω̂ne−F(ω̂) exp

[iz

(N∑n=1

[ω̂n − 〈ω̂n〉]√N

)]

=e−τ

21

[ω̂i−〈ω̂i〉]2

2π

∫dz exp

[− z

2

2τ21 c̄+ iz

[ω̂i − 〈ω̂i〉]√N

]=

τ1√2π/c̄

exp

[−τ21 c̄

(ω̂i − 〈ω̂i〉)2

2

]. (88)

I IntroductionII ResultsA Higher-order topological Kuramoto model of topological signals of a given dimension B Higher-order topological Kuramoto model of coupled topological signals of different dimension

III DiscussionA Theoretical solution of the NL modelB Annealed dynamicsC The dynamics on a fully connected networkD Solution of the dynamical equations in the annealed approximation1 General framework for obtaining the solution of the annealed dynamical equations2 Dynamics of the phases of the nodes3 Dynamics of the phases of the links projected on the nodes

E Solution on the fully connected networkF Hysteresis on homogeneous and scale-free networks

IV Conclusions Acknowledgements Author contributions Code Availability Data Availability Competing interests ReferencesV Simplicial complexes and higher order LaplaciansA Definition of simplicial complexesB Oriented simplices and boundary mapC Higher order LaplaciansD Steady-state solution of the annealed equations for the NL ModelE Marginal distributions in the fully connected case