ARTICLE
https://doi.org/10.1038/s42005-021-00605-4 OPEN Higher-order simplicial synchronization of coupled topological signals ✉ ✉ ✉ Reza Ghorbanchian1, Juan G. Restrepo 2 , Joaquín J. Torres 3 & Ginestra Bianconi 1,4
Simplicial complexes capture the underlying network topology and geometry of complex systems ranging from the brain to social networks. Here we show that algebraic topology is a fundamental tool to capture the higher-order dynamics of simplicial complexes. In particular we consider topological signals, i.e., dynamical signals defined on simplices of different dimension, here taken to be nodes and links for simplicity. We show that coupling between signals defined on nodes and links leads to explosive topological synchronization in which fi fi
1234567890():,; phases de ned on nodes synchronize simultaneously to phases de ned on links at a dis- continuous phase transition. We study the model on real connectomes and on simplicial complexes and network models. Finally, we provide a comprehensive theoretical approach that captures this transition on fully connected networks and on random networks treated within the annealed approximation, establishing the conditions for observing a closed hysteresis loop in the large network limit.
1 School of Mathematical Sciences, Queen Mary University of London, London, UK. 2 Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO, USA. 3 Departamento de Electromagnetismo y Física de la Materia and Instituto Carlos I de Física Teórica y Computacional, Universidad de ✉ Granada, Granada, Spain. 4 The Alan Turing Institute, The British Library, London, UK. email: [email protected]; [email protected]; ginestra. [email protected]
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igher-order networks1–4 are attracting increasing atten- which is the network whose nodes corresponds to the links or the Htion as they are able to capture the many-body interac- original network46. tions of complex systems ranging from brain to social In this work, we show that by adopting a global adaptive networks. Simplicial complexes are higher-order networks that coupling of dynamics47–49 the coupled synchronization dynamics encode the network geometry and topology of real datasets. Using of topological signals defined on nodes and links is explosive50, simplicial complexes allows the network scientist to formulate i.e., it occurs at a discontinuous phase transition in which the two new mathematical frameworks for mining data5–10 and for topological signals of different dimension synchronize at the same understanding these generalized network structures revealing the time. We also illustrate numerical evidence of this discontinuity underlying deep physical mechanisms for emergent on real connectomes and on simplicial complex models, including geometry11–15 and for higher-order dynamics16–33. In particular, the configuration model of simplicial complexes51 and the non- this very vibrant research activity is relevant in neuroscience to equilibrium simplicial complex model called Network Geometry analyze real brain data and its profound relation to with Flavor (NGF)12,13. We provide a comprehensive theory of dynamics1,6,15,34–37 and in the study of biological transport this phenomenon on fully connected networks offering a com- networks10,38. plete analytical understanding of the observed transition. This In networks, dynamical processes are typically defined over signals approach can be extended to random networks treated within the associated to the nodes of the network. In particular, the Kuramoto annealed network approximation. The analytical results reveal model39–43 investigates the synchronization of phases associated to that the investigated transition is discontinuous. the nodes of the network. This scenario can change significantly in thecaseofsimplicialcomplexes16,17,19. In fact, simplicial complexes Results and discussion can sustain dynamical signals defined on simplices of different Higher-order topological Kuramoto model of topological sig- dimension, including nodes, links, triangles, and so on, called topo- nals of a given dimension. Let us consider a simplicial complex fi logical signals. For instance, topological signals de ned on links can K formed by N[n] simplices of dimension n, i.e., N[0] nodes, N[1] fl fi represent uxes of interest in neuroscience and in biological trans- links, N[2] triangles, and so on. In order to de ne the higher-order portation networks. The interest on topological signals is rapidly synchronization of topological signals we will make use of alge- growing with new results related to signal processing17,19 and higher- braic topology (see the Appendix for a brief introduction) and 16,28 order topological synchronization . (Note that here higher-order specifically we indicate with B[n] the nth incidence matrix refers to the higher-order interactions existing between topological representing the nth boundary operator. signals and not to higher-order harmonics.) In particular, higher- The higher-order Kuramoto model generalizes the classic order topological synchronization16 demonstrates that topological Kuramoto model to treat synchronization of topological signals of signals (phases) associated to higher dimensional simplices can higher-dimension. The classic Kuramoto model describes the undergo a synchronization phase transition. These results open a synchonization transition for phases new uncharted territory for the investigation of higher-order θ ¼ðθ ; θ ; ¼ θ Þ ð Þ synchronization. 1 2 N½0 1 Higher-order topological signals defined on simplices of dif- = ferent dimension can interact with one another in non-trivial associated to nodes, i.e., simplices of dimension n 0 (see Fig. 1). fi ways. For instance, in neuroscience the activity of the cell body of The Kuramoto model is typically de ned on a network but it can a neuron can interact with synaptic activity which can be directly treat also synchronization of the phases associated to the nodes of affected by gliomes in the presence of brain tumors44. In order to a simplicial complex. Each node i has associated an internal ω shed light on the possible phase transitions that can occur when frequency i drawn from a given distribution, for instance a topological signals defined on nodes and links interact, here we build on the mathematical framework of higher-order topological synchronization proposed by Millán et al.16 and consider a syn- chronization model in which topological signals of different dimension are coupled. We focus in particular on the coupled synchronization of topological signals defined on nodes and links, but we note that the model can be easily extended to topological signals of higher dimension. The reason why we focus on topo- logical signals defined on nodes and links is threefold. First of all we can have a better physical intuition of topological signals defined on nodes (traditionally studied by the Kuramoto model) and links (like fluxes) that is relevant in brain dynamics44,45 and biological transport networks10,38. Secondly, although the cou- pled synchronization dynamics of nodes and links can be con- sidered as a special case of coupled synchronization dynamics of higher-order topological signals on a generic simplicial complex, this dynamics can be observed also on networks including only pairwise interactions. Indeed nodes and links are the simplices Fig. 1 Schematic representation of the Kuramoto and the higher-order that remain unchanged if we reduce a simplicial complex to its topological synchronization model. Panel a shows a network formed by network skeleton. Since currently there is more availability of nodes and links in which nodes (blue circles numbered from 1 to 8) sustain θ ∈ … network data than simplicial complex data, this fact implies that a dynamical variable (a phase i with i {1, 2, , 8}) whose synchronization the coupled dynamics studied in this work has wide applicability is captured by the Kuramoto model. Panel b shows a simplicial complex as it can be tested on any network data and network model. formed by nodes, links, and triangles (here shaded in orange) in which not θ Thirdly, defining the coupled dynamics of topological signals only nodes but also links sustain dynamical variables (indicated i for the ∈ … ϕ ∈ … defined on nodes and links can open new perspectives in nodes i {1, 2, , 8} and ij for the links [i, j] with i, j {1, 2, , 8}) whose exploiting the properties of the line graph of a given network coupled synchronization dynamics is captured by the higher-order topological Kuramoto model.
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ω N ðΩ ; =τ Þ normal distribution i 0 1 0 . In absence of any dynamics undergo a continuous synchronization transition at coupling, i.e., in absence of pairwise interactions, every node σ = 016 with order parameters c oscillates at its own frequency. However in a network or in a N½ 1 0 iϕ½À simplicial complex skeleton the phases associated to the nodes down ¼ ∑ i ; R1 e follow the dynamical evolution dictated by the equation: N½ i¼1 0 ð Þ > 9 θ_ ¼ ω À σ θ ; ð Þ N½2 ½þ B½1 sin B½ 2 up 1 iϕ 1 ¼ ∑ i : R1 e N½ i¼1 where here and in the following we use the notation sinðxÞ to 2 indicate the column vector where the sine function is taken In Millán et al.16 an adaptive coupling between these two elementwise. Note that here we have chosen to write this system dynamics is considered formulating the explosive higher-order of equations in terms of the incidence matrix B[1]. However if topological Kuramoto model in which the topological signal we indicate with a the adjacency matrix of the network and follows the set of coupled equations with a its matrix elements, this system of equations is ij ϕ_ ¼ ω~ À σ up > ð ϕÞ equivalent to R1 B½1 sin B½1 ð10Þ down > N ÀσR B½ sinðB ϕÞ: θ_ ¼ ω þ σ ∑ ðθ À θ Þ; ð Þ 1 2 ½2 i i aij sin j i 3 j¼1 The projected dynamics on nodes and triangles are now valid for every node i of the network. For coupling constant σ coupled by the modulation of the coupling constant σ with the = σ 39–41 down up c the Kuramoto model displays a continuous phase order parameters R1 and R1 , i.e. the two projected phases transition above which the order parameter follow the coupled dynamics
½À N½ _ up ½À 1 0 iθ ϕ ¼ B½ ω~ À σR L½ sin ϕ ; R ¼ ∑ e i ð4Þ 1 1 0 ð Þ 0 ¼ ½þ 11 N½0 i 1 ϕ_ ¼ > ω~ À σ down down ϕ½þ: B½2 R1 L½2 sin → ∞ isnon-zeroalsointhelimitN[0] . 16 This explosive higher-order topological Kuramoto model The higher-order topological Kuramoto model describes hasbeenshowninMillánetal.16 to lead to a discontinuous synchronization of phases associated to the n dimensional fi synchronization transition on different models of simplicial simplices of a simplicial complex. Although the de nition of complexes and on clique complexes of real connectomes. the model applies directly to any value of n,hereweconsider specifically the case in which the higher-order Kuramoto Higher-order topological Kuramoto model of coupled topo- model is defined on topological signals (phases) associated to logical signals of different dimension the links . Until now, we have captured synchronization occurring only among topological ϕ ¼ðϕ‘ ; ϕ‘ ; ¼ ϕ‘ Þ; ð Þ signals of the same dimension. However, signals of different 1 2 N½ 5 1 dimension can be coupled to each other in non-trivial ways. In ϕ ℓ where ‘ indicates the phase associated to the rth link r of the r this work we will show how topological signals of different simplicial complex (see Fig. 1). The higher-order Kuramoto dimensions can be coupled together leading to an explosive dynamics defined on simplices of dimension n > 0 is the natural synchronization transition. Specifically we focus on the cou- extension of the standard Kuramoto model defined by Eq. (2). pling of the traditional Kuramoto model [Eq. (2)] to a higher- Let us indicate with ω~ the internal frequencies associated to the order topological Kuramoto model defined for phases asso- links of the simplicial complex, sampled for example from a ciated to the links [Eq. (6)]. The coupling between these two ω~ N ðΩ ; =τ Þ normal distribution, ‘ 1 1 1 . The higher-order dynamics is here performed considering the modulation of the topological Kuramoto model is defined as coupling constant σ with the global order parameters of the _ > > node dynamics [defined in Eq. (4)] and the link dynamics ϕ ¼ ω~ À σB sinðB½ ϕÞÀσB½ sinðB ϕÞ: ð6Þ ½1 1 2 ½2 [defined in Eq. (9)]. Specifically, we consider two models Once the synchronization dynamics is defined on higher-order denoted as Model Nodes-Links (NL) and Model Nodes-Links- topological signals of dimension n (here taken to be n = 1) an Triangles (NLT). Model NL couples the dynamics of the phases important question is whether this dynamics can be projected on of the nodes θ and of the links ϕ accordingtothefollowing (n + 1) and (n − 1) simplices. Interestingly, algebraic topology dynamical equations provides a clear solution to this question. Indeed for n = 1, when _ down > θ ¼ ω À σR B½ sinðB θÞ; ð12Þ the dynamics describes the evolution of phases associated to the 1 1 ½1 ϕ[−] ϕ[+] links, one can consider the projection and , respectively, ϕ_ ¼ ω~ À σ > ð ϕÞÀσ ð > ϕÞ: ð Þ on nodes and on triangles defined as R0B½1 sin B½1 B½2 sin B½2 13 ϕ½À ¼ ϕ; The projected dynamics for ϕ[−] and ϕ[+] then obeys B½1 ð7Þ ½À ϕ½þ ¼ > ϕ: ϕ_ ¼ ω~ À σ ϕ½À; ð Þ B½2 B½1 R0L½0 sin 14 > Note that in this case B acts as a discrete divergence and B ½þ [1] ½2 ϕ_ ¼ > ω~ À σ down ϕ½þ: ð Þ acts as a discrete curl. Interestingly, since the incidence matrices B½2 L½2 sin 15 = > > ¼ “ ” − satisfy B[1]B[2] 0 and B½2B½1 0 (see Methods ) these two Therefore the projection on the nodes ϕ[ ] of the phases ϕ projected phases follow the uncoupled dynamics associated to the links [Eq. (14)] is coupled to the dynamics of _ ½À ½À the phases θ [Eq. (12)] associated directly to nodes. However ϕ ¼ B½ ω~ À σL½ sin ϕ ; + 1 0 ð Þ the projection on the triangles ϕ[ ] of the phases ϕ associated ½þ 8 − _ > down ½þ ϕ[ ] θ ϕ ¼ B½ ω~ À σL½ sin ϕ ; to the links is independent of and of as well. Model NLT 2 2 also describes the coupled dynamics of topological signals ¼ > down ¼ > fi where L½0 B½1B½1 and L½2 B½2B½2. These two projected de nedonnodesandlinksbuttheadaptive coupling captured
COMMUNICATIONS PHYSICS | (2021) 4:120 | https://doi.org/10.1038/s42005-021-00605-4 | www.nature.com/commsphys 3 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00605-4 by the model is different. In this case the dynamical equations transition only couples the phases ϕ[−] and θ, while for Model are taken to be NLT the coupling involves also the phases ϕ[+]. _ > Additionally we studied both Model NL and Model NTL on θ ¼ ω À σ down B ðB θÞ; ð16Þ R1 ½1 sin ½1 two real connectomes: the human connectome52 and the 53 ϕ_ ¼ ω~ À σ up > ð ϕÞ Caenorhabditis elegans (C. elegans) connectome (see Fig. 3). R0R1 B½1 sin B½1 Interestingly also for these real datasets we observe that in Model ð17Þ Àσ down ð > ϕÞ: NL the explosive synchronization involves only the phases θ and R B½2 sin B½ 1 2 ϕ[−] while in Model NLT we observe that also ϕ[+] undergoes an − + For Model NLT the projected dynamics for ϕ[ ] and for ϕ[ ] explosive synchronization transition at the same value of the σ = σ obeys coupling constant c. _ ½À up ½À ð Þ ϕ ¼ B½ ω~ À σR R L½ sin ϕ ; 18 1 0 1 0 Theoretical solution of the NL model. As mentioned earlier the ½þ higher-order topological Kuramoto model coupling the topolo- ϕ_ ¼ > ω~ À σ down down ϕ½þ: ð Þ B½2 R1 L½2 sin 19 gical signals of nodes and links can be defined on simplicial Therefore, as in Model NL, the dynamics of the projection ϕ[−] complexes and on networks as well. In the following sections we of the phases ϕ associated to the links [Eq. (18)] is coupled to the exploit this property of the dynamics to provide an analytical dynamics of the phases θ associated directly to nodes [Eq. (16)] understanding of the synchronization transition on uncorrelated and vice versa. Moreover, the dynamics of the projection of the random networks. phases ϕ on the triangles ϕ[+] [Eq. (19)] is now also coupled with It is well known that the Kuramoto model is challenging to the dynamics of ϕ[−] [Eq. (18)] and vice versa. Here and in study analytically. Indeed the full analytical understanding of the the following we will use the convenient notation (using the model is restricted to the fully connected case, while on a generic parameter r) to indicate both models NL and NLT with the same sparse network topology the analytical approximation needs to set of dynamical equations given by rely on some approximations. A powerful approximation is the annealed network approximation41 which consists in approx- θ_ ¼ ω À σ down ð > θÞ; ð Þ R1 B½1 sin B½1 20 imating the adjacency matrix of the network with its expectation ÀÁ in a random uncorrelated network ensemble. In order to unveil _ up rÀ1 > ϕ ¼ ω~ À σR R B sinðB½ ϕÞ the fundamental theory that determines the coupled dynamics of 0 1 ½1 1 ð Þ ÀÁÀ 21 topological signals described by the higher-order Kuramoto Àσ down r 1 ð > ϕÞ; R1 B½2 sin B½2 model here we combine the annealed approximation with the Ott–Antonsen method43. This approach is able to capture the which reduce to Eq. (13) for r = 1 and to Eq. (17) for r = 2. coupled dynamics of topological signals defined on nodes and We make two relevant observations: links. In particular, the solution found to describe the dynamics of ● First, the proposed coupling between topological signals of topological signals defined on the links is highly non-trivial and it different dimension can be easily extended to signals is not reducible to the equations valid for the standard Kuramoto defined on higher-order simplices providing a very general model. Conveniently, the calculations performed in the annealead scenario for coupled dynamical processes on simplicial approximation can be easily recasted in the exact calculation valid complexes. in the fully connected case previous a rescaling of some of the ● Second, the considered coupled dynamics of topological parameters. The analysis of the fully connected network reveals signals defined on nodes and links can be also studied on that the discontinuous sychronization transition of the considered networks with exclusively pairwise interactions where we model is characterized by a non-trivial backward transition with a assume that the number of simplices of dimension n >1is well defined large network limit. On the contrary, the forward zero. Therefore in this specific case this topological transition is highly dependent on the network size and vanishes dynamics can have important effects also on simple in the large network limit, indicating that the incoherent state networks. remains stable for every value of the coupling constant σ in the large network limit. This implies that on a fully connected We have simulated Model NL and Model NLT on two main network the NL model does not display a closed hysteresis loop as examples of simplicial complex models: the configuration model 21 51 12,13 it occurs also for the model proposed in Skardal and Arenas . of simplicial complexes and the NGF (see Fig. 2). In the This scenario is here shown to extend also to sparse networks configuration model we have considered power-law distribution fi γ with nite second moment of the degree distribution while scale- of the generalized degree with exponent < 3, and for the NGF free networks display a well defined hysteresis loop in the large model with have considered simplicial complexes of dimensions network limit. d = 3 whose skeleton is a power-law network with exponent γ = 3. In both cases we observe an explosive synchronization of the Annealed dynamics. For the dynamics of the phases θ associated topological signals associated to the nodes and to the links. On — — finite networks, the discontinuous transition emerge together to the nodes Eq. (20) it is possible to proceed as in the tra- ditional Kuramoto model42,54,55. However, the annealed with the hysteresis loop formed by the forward and backward ϕ fi synchronization transition. However the two models display a approximation for the dynamics of the phases de ned in Eq. (21) needs to be discussed in detail as it is not directly reducible to notable difference. In Model NL we observe a discontinuity for R0 down σ = σ up previous results. To address this problem our aim is to directly and R1 at a non-zero coupling constant c; however, R1 define the annealed approximation for the dynamics of the pro- follows an independent transition at zero coupling (see Fig. 2, jected variables ϕ[−] which, here and in the following, are indi- panels in the second and fourth column). In Model NLT, on the down up cated as contrary, all order parameters R0, R1 , and R1 display a discontinuous transition occurring for the same non zero value of ψ ¼ ϕ½À; ð22Þ σ = σ fi the coupling constant c (see Fig. 2 panels in the rst and third column). This is a direct consequence of the fact that in in order to simplify the notation. Moreover we will indicate with = Model NL the adaptive coupling leading to discontinuous phase N N[0] the number of nodes in the network or in the simplicial
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Fig. 2 The higher-order topological synchronization models coupling nodes and links on simplicial complexes. The hysteresis loop for the down up σ synchronization order parameters R0, R1 , and R1 are plotted versus for the higher-order topological synchronization Model Nodes-Links-Triangles (NLT) (panels a, e, i and c, g, k) and Model Nodes-Links (NL) (panels b, f, j and d, h, l)defined over the Network Geometry with Flavor13 (panels a, e, i and b, f, j) and the configuration model of simplicial complexes51 (panels c, g, k and d, h, l). The green lines indicate the backward transitions and the cyan lines indicate the forward transitions. The Network Geometry with Flavor on which we run the numerical results shown in a, b, e, f, i, j includes N[0] = 500 nodes and has flavor s = −1 and d = 3. The configuration model of simplicial complexes on which we run the numerical results shown in c, d, g, h, k, l includes N[0] = 500 nodes and has generalized degree distribution which 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. complex skeleton. Here we focus on the NL Model defined on given by networks, i.e., we assume that there are no simplices of dimension ω^ ¼ ∑ À ∑ Ω : ð Þ two. We provide an analytical understanding of the coupled i aij aij 1 27 dynamics of nodes and links in the NL Model by determining the ji equations that capture the dynamics in the annealed approx- Given that each node has degree ki, the covariance matrix C is imation and predict the value of the complex order parameters given by the graph Laplacian L of the network, i.e. DE [0]DE iΘ 1 N iθ ¼ ∑ i ; ¼ ω^ ω^ ¼ ∑ ½ ω~ ½ ω~ R0e e Cij i j B½1 B½1 N i¼1 ð Þ c ‘;‘0 i j c 23 ð Þ iΨ 1 N iψ ½ δ À 28 down ¼ ∑ i ; L½0 k a R1 e e ¼ ij ¼ i ij ij ; N i¼1 τ2 τ2 1 1 (with R ; R down ; Θ, and Ψ real) as a function of the coupling 0 1 where we have indicated with hi¼ the connected correlation. constant σ. c Therefore the variance of ω^ in the annealed approximation is We notice that Eq. (14), valid for Model NL, can be written as
2 k ψ_ ¼ B ω~ À σ L ðψÞ: ð Þ ω^2 ¼ ω^2 À ω^ ¼ i : ð Þ ½1 R0 ½0 sin 24 i c i i τ2 29 1 This equation can be also written elementwise as hi Moreover, the projected frequencies are actually correlated and N for i ≠ j we have ψ_ ¼ ω^ þ σR ∑ a sinðψ ÞÀsinðψ Þ ; ð25Þ DEDE DE i i 0 ¼ ij j i a j 1 ω^ ω^ ¼ ω^ ω^ À ω^ ω^ ¼À ij : ð Þ i j i j i j τ2 30 where the vector ω^ is given by c 1 ω^ ω^ ¼ B½ ω~: ð26Þ It follows that the frequencies are correlated Gaussian 1 variables with average given by Eq. (27) and correlation matrix ω^ Let us now consider in detail these frequencies in the case in given by the graph Laplacian. The fact that the frequencies i are which the generic internal frequency ω~‘ of a link follows a correlated is an important feature of the dynamics of ψ and, with Gaussian distribution, specifically in the case in which ω~‘ few exceptions56, this feature has remained relatively unexplored N ðΩ ; =τ Þ ℓ fi 1 1 1 for every link . Using the de nition of the boundary in the case of the standard Kuramoto model. Additionally let us ω^ ω^ operator on a link it is easy to show that the expectation of i is note that the average of over all the nodes of the network is
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Fig. 3 The higher-order topological synchronization models coupling nodes and links on real connectomes. The hysteresis loop for the synchronization down up σ order parameters R0, R1 , and R1 are plotted versus on real connectomes. The green lines indicate the backward transitions and the cyan lines indicate the forward transitions. Panels a, e, i and b, f, j show the numerical results on the human connectome52 for Model Nodes-Links-Triangles (NLT) and Model Nodes-Links (NL) respectively. Panels c, g, k and d, h, l show the numerical results on the C. elegans connectome53 for Model NLT and Model
NL, respectively. In both Model NLT and in Model NL we have set Ω0 = Ω1 = 2 and τ0 = τ1 = 1. zero. In fact multiplication) and where two auxiliary complex order para- fi N meters are de ned as ∑ ω^ ¼ Tω^ ¼ T ω ¼ ; ð Þ i 1 1 B½1 0 31 N i¼1 iΘ^ k iθ ^ ¼ ∑ i i ; R0e e i¼1 hik N where with 1 we indicate the N-dimensional column vector of ð36Þ = N elements 1i 1. By using the symmetry of the adjacency matrix, down iΨ^ k iψ ^ ¼ ∑ i i ; = ψ_ R1 e e i.e. the fact that aij aji, Eq. (31) implies that the sum of i over i¼1 hik N all the nodes of the network is zero, i.e. ^ ; Θ^; ^ down Ψ^ N N with R0 R1 and real. ∑ ψ_ ¼ ∑ ω^ þ σR ∑ a ½sinðψ ÞÀsinðψ Þ ¼ 0: ¼ i ¼ i 0 ; ij j i i 1 i 1 i j The dynamics on a fully connected network. On a fully con- = − We now consider the annealed approximation consisting in nected network in which each node has degree ki N 1 the substituting the adjacency matrix element aij with its expectation dynamics of the NL Model is well defined provided its parameter in an uncorrelated network ensemble are properly rescaled. In particular, we require a standard k k rescaling of the coupling constant with the network size, given by a ! i j ; ð32Þ ij hik N σ ! σ=ðN À 1Þð37Þ hi where ki indicates the degree of node i and k is the average which guarantees that the interaction term in the dynamical degree of the network. Note that the considered random networks equations has a finite contribution to the velocity of the phases. 57 58 can be both sparse orp denseffiffiffiffiffiffiffiffiffiffi as long as they display the The Model NL on fully connected networks requires also some hi structural cutoff, i.e. ki k N for every node i of the network. specific model dependent rescalings associated to the dynamics In the annealed approximation we can put fi ω^ on networks. Indeed, in order to have a nite expectation i of k the projected frequencies ω^ and a finite of the covariance matrix ω^ ’ Ω À ∑ j : ð Þ i i ki 1 1 2 33 C [given by Eqs. (27) and (28), respectively], we require that on a j>i hik N fully connected network both Ω1 and τ1 are rescaled according to Also, in the annealed approximation the dynamical Eq. (20) Ω ! Ω = ; and Eq. (24) reduce to 1 1 N pffiffiffiffiffiffiffiffiffiffiffiffi ð38Þ τ ! τ À : θ_ ¼ ω À σ down ^ Á ðθ À Θ^Þ; ð Þ 1 1 N 1 R1 R0k sin 34
down Considering these opportune rescalings and noticing that the ψ_ ¼ ω^ þ σ ^ Ψ^ À σ ψ; ð Þ down R0R1 k sin R0k sin 35 ^ ¼ ^ ¼ down Θ ¼ Θ^ order parameters obey R0 R0, R1 R1 , , and where ⊙ indicates the Hadamard product (element by element Ψ ¼ Ψ^, we obtain that Model NL dictated by Eqs. (34)–(35) can
6 COMMUNICATIONS PHYSICS | (2021) 4:120 | https://doi.org/10.1038/s42005-021-00605-4 | www.nature.com/commsphys COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00605-4 ARTICLE be rewritten here as density function ρðiÞðψ; tjω^Þ that node i is associated to a projected θ_ ¼ ω À σ down ðθ À ΘÞ; ð Þ phase of the link equal to ψ. Since in the annealed approximation R1 R0 sin 39 ψ i has a dynamical evolution dictated by Eq. (35) the probability ψ_ ¼ ω^ þ σ down Ψ À σ ψ; ð40Þ density function obeys the continuity equation R0R1 sin R0 sin ÂÃÁ ∂ ρðiÞðψ; jω^Þþ∂ ρðiÞðψ; jω^Þ ¼ ð Þ with R ; R down ; Θ and Ψ given by Eq. (23) and t t ψ t vi 0 46 0 1 DE ¼ ω^ ω^ ¼ δ À 1 : ð Þ with associated velocity v given by Cij i j ij 41 i c N À 1 ¼ κ À σ ψ ; ð Þ vi i R0ki sin i 47 Solution of the dynamical equations in the annealed where we have defined κ as approximation i down General framework for obtaining the solution of the annealed κ ¼ ω^ þ σk R R^ sin Ψ^: ð48Þ dynamical equations. In this section we will provide the analytic i i i 0 1 solutions for the order parameter of the higher-order topological In this case the complex order parameters are given by synchronization studied within the annealed approximation, i.e., Z N captured by Eqs. (34) and (35). In particular, first we will find an ^ down iΨ ¼ ∑ ki ψρðiÞðψ; jω^Þ iψ; R1 e d t e expression of the order parameters R of the dynamics associated i¼1 hik N 0 Z ð49Þ to the nodes (Eq. (34)) and subsequently in the next paragraph we iΨ~ N 1 ð Þ iψ down down ¼ ∑ ψρ i ðψ; jω^Þ : will derive the expression for the order parameter R asso- R1 e d t e 1 i¼1 N ciated to the projection on the nodes of the topological signal defined on the links (Eq. (35)). By combining the two results it is In order to solve the continuity equation we follow finally possible to uncover the discontinuous nature of the Ott–Antonsen43 and we express ρðiÞðψ; tjω^Þ in the Fourier basis as transition. 1 ð Þ ρðiÞðψ; jω^Þ¼ 1 þ ∑ ^ i ðω^ ; Þ imψ þ : : : ð Þ t 1 f m i t e c c 50 Dynamics of the phases of the nodes. When we investigate Eq. (34) 2π m¼1 we notice that this equation can be easily reduced to the equation for the standard Kuramoto model treated within the annealed Making the ansatz 42 approximation if one performs a rescaling of the coupling ^ðiÞ f ðω^ ; tÞ¼½b ðω^ ; tÞm ð51Þ constant σR0 → σ. Therefore we can treat this model similarly to m i i i the known treatment of the standard Kuramoto model40–42. we can derive the equation for the evolution of b ¼ b ðω^ ; tÞ given Specifically, starting from Eq. (34) and using a rescaling of the i i i by phases θ according to θ ! θ À Ω ; ð Þ 1 2 i i 0t 42 ∂ b þ ib κ þ σk R ðb À 1Þ¼0: ð52Þ t i i i i 0 2 i it is possible to show that we can set Θ = 0 and therefore Eq. (34) ψ_ reduces to the well-known annealed expression for the standard Since we showed before that the average value of i over nodes order Kuramoto model given by is zero, we look for non-rotating stationary solutions of Eq. (52), ∂ b = 0. As long as R > 0 these stationary solutions are given by θ_ ¼ ω À Ω À σ down ^ Á ðθÞ: ð Þ t i 0 01 R1 R0k sin 43 qffiffiffiffiffiffiffiffiffiffiffiffiffi ¼Ài À 2; ð53Þ Assuming that the system of equations reaches a steady state in bi di ± 1 di down ^ which both R1 and R0 become time independent, the order ^ where di is given by parameters of this system of equations in the coherent state R0 > 0 down and R > 0 can be found to obey40,42,50,54 ω^ down 1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ i þ ^ Ψ^: ð Þ u ! di σ R1 sin 54 Z u 2 kiR0 N k t ω À Ω R^ ¼ ∑ i dωgðωÞ 1 À 0 ; 0 ¼ hi ^ down By inserting this expression into Eq. (49) we get the expression i 1 k N j^c j<1 σk R R i vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 0 1 ð Þ of the order parameters given the projected frequencies ω^, in the u !44 Z u 2 coherent phase in which R0 >0 1 N t ω À Ω R ¼ ∑ dωgðωÞ 1 À 0 ; qffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ^ down down N k N i¼1 j^c j<1 σk R R ^ Ψ^ ¼ ∑ i À 2θð À 2Þ; i i 0 1 R1 cos 1 di 1 di i¼1 hik N ^ qffiffiffiffiffiffiffiffiffiffiffiffiffi where ci indicates N ^ down Ψ^ ¼ ∑ ki 2 À χð Þþ ; ω À Ω R1 sin hi di 1 di di ^c ¼ 0 : ð Þ i¼1 k N i σ ^ down 45 qffiffiffiffiffiffiffiffiffiffiffiffiffi ð55Þ kjR0R1 N down Ψ ¼ ∑ 1 À 2θð À 2Þ; R1 cos 1 di 1 di and g(ω) is the Gaussian distribution with expectation Ω and i¼1 N 0 qffiffiffiffiffiffiffiffiffiffiffiffiffi standard deviation 1. N down Ψ ¼ ∑ 1 2 À χð Þþ ; R1 sin di 1 di di i¼1 N Dynamics of the phases of the links projected on the nodes. In this paragraph we will derive the expression of the order parameters where, indicating by θ(x) the Heaviside function, we have defined down R down and R^ which, together with Eq. (44), will provide the 1 1 χðd Þ¼½Àθðd À 1ÞþθðÀ1 À d Þ: ð56Þ annealed solution of our model. To start with we assume that the i i i ω^ frequencies are known. In this case we can express the order Finally, if the projected frequencies ω^ are not known we can down ^ down parameters R1 and R1 as a function of the probability average the result over the marginal frequency distribution of the
COMMUNICATIONS PHYSICS | (2021) 4:120 | https://doi.org/10.1038/s42005-021-00605-4 | www.nature.com/commsphys 7 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00605-4
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Fig. 4 Comparison between the simulation results of the Nodes-Links (NL) Model and its solution in the annealed approximation. The hysteresis loop down σ = for the synchronization order parameters R0 and R1 of the NL Model are shown as a function of for a Poisson network with average degree c 12 (a, c) and for an uncorrelated scale-free network with minimum degree m = 6 and power-law exponent γ = 2.5 (b, d). Both networks have N = 1600 nodes. The symbols indicate the simulation results for the forward (cyan diamonds) and the backward (green circles) synchronization transition. The solid black lines indicate the analytical solution for the backward transition obtained by integrating Eq. (55).
ω^ ðω^Þ = projected frequency i given by Gi uncorrelated scale-free network with minimum degree m 6 and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi power-law exponent γ = 2.5 In Fig. 4 we compare the values of Z N ω^ 2 down ^ down Ψ^ ¼ ∑ ki ω^ ðω^ Þ À i þ ^down Ψ^ ; R0, R1 obtained from direct numerical integration of Eqs. (20) R1 cos d iGi i 1 R1 sin i¼1 hik N j j ≤ σR k and (25) and the steady-state solutions obtained from the di 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 i Z numerical solution of Eq. (55). The backward transition is fully N ω^ 2 ^ down Ψ^ ¼À∑ ki ω^ ðω^ Þ i þ ^down Ψ^ À captured by our theory, while the next paragraphs will clarify the R1 sin d iGi i R1 sin 1 j¼0 hik N σR k di > 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 i theoretical expectations for the forward transition. Z N ω^ 2 þ ∑ ki ω^ ðω^ Þ i þ ^down Ψ À d iGi i R1 sin 1 i¼1 hik N À σR k Solution on the fully connected network. The integration of Eq. di < 1 0 i Z 1 (57) requires the knowledge of the marginal distributions G ðω^Þ N k ω^ down i þ ∑ i dω^ G ðω^ Þ i þ R^ sin Ψ ; which does not have in general a simple analytical expression. ¼ hi i i i σ 1 i 1 k N À1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR0ki Z However, in the fully connected networks with Gaussian dis- 2 N 1 ω^ down tribution of the internal frequency of nodes and links this cal- R down cos Ψ ¼ ∑ dω^ G ðω^ Þ 1 À i þ R^ sin Ψ^ ; 1 ¼ i i i σ 1 fi fi i 1 N jd j ≤ 1 R0ki culation simpli es signi cantly. Indeed, when the link frequencies i Ω ð57Þ are sampled from a Gaussianpffiffiffiffiffiffiffiffiffiffiffiffi distribution with mean 1/N and =ðτ À Þ standard deviation 1 1 N 1 , the marginal frequency dis- ðω^Þ ω^ and an analogous equations for R down sinðΨÞ (not shown). We tribution Gi of the internal frequency i of a node i in a fully 1 “ ” note that in the case of distributions g(ω) and G ðω^Þ that are connected network is given by (see Methods for details) i "# symmetric around their means the above equations always admit τ ðω^ À ω^ Þ2 Ψ ¼ Ψ^ ¼ G ðω^Þ¼pffiffiffiffiffiffiffiffiffiffi1 exp Àτ2c i i ; ð58Þ the solution 0. Such values of the phases are also i π= 1 confirmed by direct numerical integration of the NL model. 2 c 2
These equations together with Eq. (44) capture the steady-state where c ¼ N . By considering Ω ¼ Ω ¼ ω^ ¼ 0; and per- behavior of the higher-order Kuramoto model coupling topolo- NÀ1 0 1 i forming a direct integration of Eq. (57) we obtain (see “Methods” gical signals defined on nodes and links within the annealed section for details) the closed system of equations for R and approximation in the coherent synchronized phase. Note that by 0 R down derivation, these equations cannot capture the asynchronous 1 2 phase which is instead always a trivial solution of the dynamical 1 ¼ σR down h σ2R2ðRdownÞ ; ¼ down ¼ 1 0 1 equations corresponding to R0 R1 0. Finally we observe pffiffi ÀÁ ð59Þ down ¼ σ τ σ2τ2 2 ; that for the NL Model as well as for the standard Kuramoto R1 R0 1 ch 1R0 model on random networks, it is expected that the annealed approximation is more accurate for networks that are connected where the scaling function h(x) is given by fi rffiffiffi hi and are suf ciently dense. π À = x x To illustrate the applicability of the theoretical analysis, we hðxÞ¼ e x 4 I þ I ; ð60Þ 2 0 4 1 4 consider two examples of connected networks with N = 1600 = fi nodes: a Poisson network with average degree c 12 and an with I0 and I1 indicating the modi ed Bessel functions. The
8 COMMUNICATIONS PHYSICS | (2021) 4:120 | https://doi.org/10.1038/s42005-021-00605-4 | www.nature.com/commsphys COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00605-4 ARTICLE
1.0
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down Fig. 5 The backward and the forward discontinuous phase transition on fully connected networks. The order parameters R0 (circles) and R1 (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 Eq. (59). Simulations (shown as data point) are here obtained by integrating numerically Eqs. (34) and (35) for a fully connected network of N = 500 (cyan circles), N = 1000 (green squares), and N = 2000 (purple diamonds) with Ω0 = Ω1 = 0 and (rescaled) τ0 = τ1 = 1. The backward transition is perfectly captured by the theoretical prediction (solid black line) and is affected by finite-size effects very marginally. The forward transition is instead driven by stochastic fluctuations and moves to higher values of σ as the network size increases. This is in agreement with the fact that the unstable branch of the self-consistent solution (black dashed line) does not cross the x-axis for any finite value of the coupling constant σ. numerical solution of Eq. (59) reveals the following picture: for state is stable in the limit N → ∞, and this forward transition is σ ¼ down ¼ fi fl low values of , only the incoherent solution R0 R1 0 the result of nite-size uctuations that push the system above the exists. At a positive value of σ, two solutions of Eq. (59) appear at unstable branch, causing the observed explosive transition. This is a bifurcation point, with the upper solution corresponding to a consistent with the fact that for larger values of N, which have stable synchronized state and the lower solution to an unstable smaller finite-size fluctuations, the system remains in the σ σ synchronized solution. For larger values of , the values of R0 and incoherent state for larger values of . R down corresponding to the upper solution approach one (full Therefore, while a closed hysteresis loop is not present in the 1 fi phase synchronization), while those for the lower solution NL model de ned on fully connected networks, we observe fl fi fl approach zero asymptotically, thus indicating that the incoherent uctuation-driven hysteresis, in which nite-size uctuations of state never loses stability. Indeed, it can be easily checked (see the zero solution drive the system towards the synchronized “Methods” for details) that for large σ the unstable solution of Eq. solution, creating an effective hysteresis loop. (59) has asymptotic behavior ¼ σÀ2 ; R0 J0 ð61Þ Hysteresis on homogeneous and scale-free networks. In this down ¼ σÀ1 ; section we discuss how the scenario found for the fully connected R1 J1 network can be extended to random networks with given degree with J0 and J1 constants given by distribution. We will start from the self-consistent Eq. (57) hiÀ Âà π 2 À1 obtained within the annealed approximation model. These J ¼ Gð0Þgð0Þ ; ð62Þ 0 2 equations display a saddle point bifurcation with the emergence hi of two non-trivial solutions describing a stable and an unstable π À1 ¼ ð Þ : ð Þ branch of these self-consistent equations. These solutions always J1 g 0 63 2 ¼ down ¼ exist in combination with the trivial solution R0 R1 0 Therefore, the unstable branch approaches the trivial solution describing the asynchronous state. Two scenarios are possible: ¼ down ¼ σ → ∞ R0 R1 0 only asymptotically for . This implies that either the unstable branch converges to the trivial solution only in the trivial solution remains stable for every possible value of σ the limit σ → ∞ or it converges to the trivial solution at a finite although as σ increases it describes the stationary state of an value of σ. In the first case, the scenario is the same as the one increasingly smaller set of initial conditions. observed for the fully connected network, and the trivial solution This scenario is confirmed by numerical simulations (see remains stable for any finite value of σ. In this case the forward Fig. 5) showing that the backward transition is captured very well transition is not obtained in the limit N → ∞ and the transition by our theory and does not display notable finite-size effects. The observed on finite networks is only caused by finite-size effects. In forward transition, instead, displays remarkable finite-size effects. the second case the trivial solution loses its stability at a finite Indeed, as σ increases, the system remains in the incoherent state value of σ. Therefore the forward transition is not subjected to until it explosively synchronizes at a positive value of σ and strong finite-size effects and we expect to see a forward transition reaches the stable synchronized branch. However the incoherent also in the N → ∞ limit. in order to determine which network
COMMUNICATIONS PHYSICS | (2021) 4:120 | https://doi.org/10.1038/s42005-021-00605-4 | www.nature.com/commsphys 9 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00605-4 topologies can sustain a non-trivial hysteresis loop we expand Eq. simplices closed under the inclusion of the faces of each simplex. Any simplicial ≪ ^ down complex can be reduced to its simplicial complex skeleton, which is the network (57) for 0 < R0 1, 0 < R0 1, and 0
Methods Definition of simplicial complexes. Simplicial complexes represent higher-order networks whose interactions include two or more nodes. These many-body interactions are captured by simplices. An n-dimensional simplex α is a set of n + 1 nodes Fig. 6 The boundary operators and their representation in terms of the α ¼½ ; ; ¼ ; : ð Þ i0 i1 in 66 incidence matrices. Panels (a) and (b) describe the action of the boundary For instance a node is a 0-dimensional simplex, a link is a one-dimensional operator on an oriented link and on an oriented triangle respectively. Panel simplex, a triangle is a two-dimensional simplex, a tetrahedron is a three- (c) shows a toy example of a simplicial complex and panel (d) indicates its dimensional simplex, and so on. A face of a simplex is the simplex formed by a ∂ proper subset of the nodes of the original simplex. For instance, the faces of a incidence matrices B[1] and B[2] representing the boundary operators 1 and tetrahedron are 4 nodes, 6 links, and 4 triangles. A simplicial complex is a set of ∂2, respectively.
10 COMMUNICATIONS PHYSICS | (2021) 4:120 | https://doi.org/10.1038/s42005-021-00605-4 | www.nature.com/commsphys COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00605-4 ARTICLE follows that the incidence matrices obey with "# > > À2À B½ B½ þ ¼ 0; B½ þ B½ ¼ 0; ð73Þ π 2 1 n n 1 n 1 n ¼ hi k ðΩ Þ 1 ∑ ð ω^ Þ ; J0 k hi g 0 kiGi i for any n >0. 2 k N i "# ð Þ À1 81 π k2 Higher-order Laplacians. Using the incidence matrices it is natural to generalize J ¼ gðΩ Þ : 1 0 2 hik the definition of the graph Laplacian This confirms the theoretical framework revealing that in this dynamics there is ¼ > ð Þ L½0 B½1B½1 74 ¼ ^ ¼ down ¼ always a trivial solution R0 R0 R1 0 while Eqs. (44) and (57) are 17,19,60 σ σ to the higher-order Laplacian L[n] (also called combinatorial Laplacians) that characterized by a saddle-point instability so that for > c two additional can be represented as a N[n] × N[n] matrix given by solutions emerge, a stable solution and an unstable solution. The stable solution describes the synchronized phase and captures the backward transition. As ¼ down þ up ð Þ L½n L½n L½n 75 long as the second moment of the degree distribution does not diverge, the ¼ ^ ¼ down ¼ with unstable solution converges to the trivial solution R0 R0 R1 0 only for σ → ∞. down > down L½ ¼ B½ B½ ; The asymptotic scaling for R and R given by Eq. (80) can be adapted to n n n ð Þ 0 1 up > 76 capture the asymptotic scaling of the fully connected case with a suitable rescaling L ¼ B½ þ B½ þ ; ½n n 1 n 1 of the model parameters of the model, obtaining Eqs. (61) and (63). for n > 0. The higher-order Laplacian can be proven to be independent of the orientation of the simplices as long as the simplicial complex has an orientation ðω^Þ ω^ Marginal distributions in the fully connected case. The distribution G1 of induced by a labeling of the nodes. is a Gaussian distribution with averages given by Eq. (27) and covariance matrix C The most celebrated property of higher-order Laplacian is that the degeneracy given by Eq. (28). The covariance matrix has N − 1 eigenvalues given by λ ¼ 1=τ2 L β 1 of the zero eigenvalue of the n Laplacian [n] is equal to the Betti number n and and one zero eigenvalue λ = 0 corresponding to the eigenvector that their corresponding eigenvectors localize around the corresponding n- pffiffiffiffi pffiffiffiffi > dimensional cavities of the simplicial complex. The higher-order Laplacians can be 1= N ¼ð1; 1; ¼ ; 1Þ = N: ð82Þ fi 17 used to de ne higher-order diffusion and can display a higher-order spectral This means that we should always have dimension on network geometries. Here we are particularly interested in the use of incidence matrices and higher-order Laplacians to define higher-order topological N ½ω^ À ω^ ∑ n pffiffiffiffi n ¼ 0; ð83Þ synchronization. n¼1 N a constraint that we can introduce as a delta function in the expression for the joint Steady-state solution of the annealed equations for the NL Model. Here we distribution G^ðω^Þ of the vector ω^. Here we note that under these hypotheses and study Eqs. (44) and (57) assuming that the distributions g(ω) and G ðω^Þ are ^ i assuming that the distribution of the frequenciespffiffiffiffiffiffiffiffiffiffiffiffi of the links is a Gaussian with unimodal functions symmetric about their means. Setting Ψ ¼ Ψ ¼ 0 and con- Ω =ðτ À Þ ðω^Þ average 1/N and standard deviation 1 1 N 1 the marginal probability Gi sidering the change of variables z ¼ ω=ðσR R down Þ, y ¼ ω^=ðσR Þ, Eq. (44) can be ω^ 0 1 0 of i can be expressed as Eq. (58). written as Given that the covariance matrix has a zero eigenvalue we can express the joint Z ^ðω^Þ N k2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi Gaussian distribution G as ¼ σ down ∑ i ðΩ þ σ ^ down Þ À 2 ; 1 R1 g 0 z kiR0R1 1 z dz i¼1 hik N À N ½ω^ À ω^ Z 1 ^ðω^Þ¼C ÀFðω^Þδ ∑ n pffiffiffiffi n ; ð Þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi G e 84 N k ¼ ¼ σ^ down ∑ i ðΩ þ σ ^ down Þ À 2 ; n 1 N R0 R0R1 g 0 z kiR0R1 1 z dz i¼1 N À1 where δ(x) indicates the delta function and where Fðω^Þ and C are given by while Eq. (57) reduce to τ2 N ÀÁ Z Fðω^Þ¼ 1 ∑ ω^ À ω^ 2; 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi n n N k 2 n¼1 down ¼ σ ∑ i ð ω^ þ σ Þ À 2 ; ð Þ R1 R0 Gi i y R0ki 1 y dy NÀ1 85 i¼1 N À τ Z 1 C ¼ pffiffiffiffiffi1 : 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi down N k π ^ ¼ σ ∑ i ðhiω þ σ Þ À 2 : 2 R1 R0 Gi i y R0ki 1 y dy i¼1 hik N À The marginal probability G ðω^Þ is given by 1 i Z We notice that the equations for R ; R^ and R down do not depend on the Y 0 0 1 ðω^Þ¼ ω^ ^ðω^Þ: ð Þ ^ down Gi d nG 86 order parameter R1 so we can obtain a fully analytical solution of the model n≠i without solving the last equation. The above equations depend on the By expressing the delta function in Eq. (84) in its integral form ω ðω^ Þ distribution g( ) and the set of marginal distributions Gi i . However we can Z 1 2 =hi fi 1 i ð À Þ show that, provided k k is nite, the solution of these equations does not δðx; yÞ¼ dze z x y ð87Þ ¼ ^ ¼ down ¼ fi σ 2π À1 converge to the trivial solution R0 R0 R1 0forany nite value of . Indeed we are now going to show that the unstable branch of the solution these we get the final expression for the marginal distribution Eq. (58), in fact, by putting equations converges to the trivial solution only in the limit σ → ∞. Assuming 0 < c\;=\;N/(N–1), we have ≪ ^ down Z Z R0 1, 0 < R0 1, and 0 COMMUNICATIONS PHYSICS | (2021) 4:120 | https://doi.org/10.1038/s42005-021-00605-4 | www.nature.com/commsphys 11 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00605-4 References 34. Severino, F. P. U. et al. The role of dimensionality in neuronal network 1. Giusti, C., Ghrist, R. & Bassett, D. S. Two’s company, three (or more) is a dynamics. Sci. Rep. 6, 29640 (2016). simplex. J. Comput. Neurosci. 41,1–14 (2016). 35. Saggar, M. et al. 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Nature Communications. 12, 1255 (2021). R.G., J.G.R., J.J.T., and G.B. have all contributed in the design of the project, and in the 33. Millán, A. P., Torres, J. J. & Bianconi, G. Synchronization in network numerical implementations of the algorithm. J.G.R., J.J.T., and G.B. have contributed to geometries with finite spectral dimension. Phys. Rev. E 99, 022307 (2019). the theoretical derivations and the writing of the manuscript. 12 COMMUNICATIONS PHYSICS | (2021) 4:120 | https://doi.org/10.1038/s42005-021-00605-4 | www.nature.com/commsphys COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00605-4 ARTICLE Competing interests Open Access This article is licensed under a Creative Commons Attri- The authors declare no competing interests. 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