Abstract
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 higherorder 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 phases defined on nodes synchronize simultaneously to phases defined on links at a discontinuous 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.
Similar content being viewed by others
Introduction
Higherorder networks^{1,2,3,4} are attracting increasing attention as they are able to capture the manybody interactions of complex systems ranging from brain to social networks. Simplicial complexes are higherorder networks that encode the network geometry and topology of real datasets. Using simplicial complexes allows the network scientist to formulate new mathematical frameworks for mining data^{5,6,7,8,9,10} and for understanding these generalized network structures revealing the underlying deep physical mechanisms for emergent geometry^{11,12,13,14,15} and for higherorder dynamics^{16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33}. In particular, this very vibrant research activity is relevant in neuroscience to analyze real brain data and its profound relation to dynamics^{1,6,15,34,35,36,37} and in the study of biological transport networks^{10,38}.
In networks, dynamical processes are typically defined over signals associated to the nodes of the network. In particular, the Kuramoto model^{39,40,41,42,43} investigates the synchronization of phases associated to the nodes of the network. This scenario can change significantly in the case of simplicial complexes^{16,17,19}. In fact, simplicial complexes can sustain dynamical signals defined on simplices of different dimension, including nodes, links, triangles, and so on, called topological signals. For instance, topological signals defined on links can represent fluxes of interest in neuroscience and in biological transportation networks. The interest on topological signals is rapidly growing with new results related to signal processing^{17,19} and higherorder topological synchronization^{16,28}. (Note that here higherorder refers to the higherorder interactions existing between topological signals and not to higherorder harmonics.) In particular, higherorder topological synchronization^{16} demonstrates that topological signals (phases) associated to higher dimensional simplices can undergo a synchronization phase transition. These results open a new uncharted territory for the investigation of higherorder synchronization.
Higherorder topological signals defined on simplices of different dimension can interact with one another in nontrivial ways. For instance, in neuroscience the activity of the cell body of a neuron can interact with synaptic activity which can be directly affected by gliomes in the presence of brain tumors^{44}. In order to shed light on the possible phase transitions that can occur when topological signals defined on nodes and links interact, here we build on the mathematical framework of higherorder topological synchronization proposed by Millán et al.^{16} and consider a synchronization 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 topological 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 dynamics^{44,45} and biological transport networks^{10,38}. Secondly, although the coupled synchronization dynamics of nodes and links can be considered as a special case of coupled synchronization dynamics of higherorder 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 that remain unchanged if we reduce a simplicial complex to its network skeleton. Since currently there is more availability of network data than simplicial complex data, this fact implies that the coupled dynamics studied in this work has wide applicability as it can be tested on any network data and network model. Thirdly, defining the coupled dynamics of topological signals defined on nodes and links can open new perspectives in exploiting the properties of the line graph of a given network which is the network whose nodes corresponds to the links or the original network^{46}.
In this work, we show that by adopting a global adaptive coupling of dynamics^{47,48,49} the coupled synchronization dynamics of topological signals defined on nodes and links is explosive^{50}, i.e., it occurs at a discontinuous phase transition in which the two topological signals of different dimension synchronize at the same time. We also illustrate numerical evidence of this discontinuity on real connectomes and on simplicial complex models, including the configuration model of simplicial complexes^{51} and the nonequilibrium simplicial complex model called Network Geometry with Flavor (NGF)^{12,13}. We provide a comprehensive theory of this phenomenon on fully connected networks offering a complete analytical understanding of the observed transition. This approach can be extended to random networks treated within the annealed network approximation. The analytical results reveal that the investigated transition is discontinuous.
Results and discussion
Higherorder topological Kuramoto model of topological signals of a given dimension
Let us consider a simplicial complex \({\mathcal{K}}\) formed by N_{[n]} simplices of dimension n, i.e., N_{[0]} nodes, N_{[1]} links, N_{[2]} triangles, and so on. In order to define the higherorder synchronization of topological signals we will make use of algebraic topology (see the Appendix for a brief introduction) and specifically we indicate with B_{[n]} the nth incidence matrix representing the nth boundary operator.
The higherorder Kuramoto model generalizes the classic Kuramoto model to treat synchronization of topological signals of higherdimension. The classic Kuramoto model describes the synchonization transition for phases
associated to nodes, i.e., simplices of dimension n = 0 (see Fig. 1). The Kuramoto model is typically defined on a network but it can treat also synchronization of the phases associated to the nodes of a simplicial complex. Each node i has associated an internal frequency ω_{i} drawn from a given distribution, for instance a normal distribution \({\omega }_{i} \sim {\mathcal{N}}({{{\Omega }}}_{0},1/{\tau }_{0})\). In absence of any coupling, i.e., in absence of pairwise interactions, every node oscillates at its own frequency. However in a network or in a simplicial complex skeleton the phases associated to the nodes follow the dynamical evolution dictated by the equation:
where here and in the following we use the notation \(\sin ({\bf{x}})\) to indicate the column vector where the sine function is taken elementwise. Note that here we have chosen to write this system of equations in terms of the incidence matrix B_{[1]}. However if we indicate with a the adjacency matrix of the network and with a_{ij} its matrix elements, this system of equations is equivalent to
valid for every node i of the network. For coupling constant σ = σ_{c} the Kuramoto model^{39,40,41} displays a continuous phase transition above which the order parameter
is nonzero also in the limit N_{[0]} → ∞.
The higherorder topological Kuramoto model^{16} describes synchronization of phases associated to the n dimensional simplices of a simplicial complex. Although the definition of the model applies directly to any value of n, here we consider specifically the case in which the higherorder Kuramoto model is defined on topological signals (phases) associated to the links
where \({\phi }_{{\ell }_{r}}\) indicates the phase associated to the rth link ℓ_{r} of the simplicial complex (see Fig. 1). The higherorder Kuramoto dynamics defined on simplices of dimension n > 0 is the natural extension of the standard Kuramoto model defined by Eq. (2). Let us indicate with \(\tilde{{\boldsymbol{\omega }}}\) the internal frequencies associated to the links of the simplicial complex, sampled for example from a normal distribution, \({\tilde{\omega }}_{\ell } \sim {\mathcal{N}}({{{\Omega }}}_{1},1/{\tau }_{1})\). The higherorder topological Kuramoto model is defined as
Once the synchronization dynamics is defined on higherorder topological signals of dimension n (here taken to be n = 1) an important question is whether this dynamics can be projected on (n + 1) and (n − 1) simplices. Interestingly, algebraic topology provides a clear solution to 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^{[+]}, respectively, on nodes and on triangles defined as
Note that in this case B_{[1]} acts as a discrete divergence and \({{\bf{B}}}_{[2]}^{\top }\) acts as a discrete curl. Interestingly, since the incidence matrices satisfy B_{[1]}B_{[2]} = 0 and \({{\bf{B}}}_{[2]}^{\top }{{\bf{B}}}_{[1]}^{\top }={\bf{0}}\) (see “Methods”) these two projected phases follow the uncoupled dynamics
where \({{\bf{L}}}_{[0]}={{\bf{B}}}_{[1]}{{\bf{B}}}_{[1]}^{\top }\) and \({{\bf{L}}}_{[2]}^{\,\text{down}\,}={{\bf{B}}}_{[2]}^{\top }{{\bf{B}}}_{[2]}\). These two projected dynamics undergo a continuous synchronization transition at σ_{c} = 0^{16} with order parameters
In Millán et al.^{16} an adaptive coupling between these two dynamics is considered formulating the explosive higherorder topological Kuramoto model in which the topological signal follows the set of coupled equations
The projected dynamics on nodes and triangles are now coupled by the modulation of the coupling constant σ with the order parameters \({R}_{1}^{\,\text{down}\,}\) and \({R}_{1}^{\,\text{up}\,}\), i.e. the two projected phases follow the coupled dynamics
This explosive higherorder topological Kuramoto model has been shown in Millán et al.^{16} to lead to a discontinuous synchronization transition on different models of simplicial complexes and on clique complexes of real connectomes.
Higherorder topological Kuramoto model of coupled topological signals of different dimension
Until now, we have captured synchronization occurring only among topological signals of the same dimension. However, signals of different dimension can be coupled to each other in nontrivial ways. In this work we will show how topological signals of different dimensions can be coupled together leading to an explosive synchronization transition. Specifically we focus on the coupling of the traditional Kuramoto model [Eq. (2)] to a higherorder topological Kuramoto model defined for phases associated to the links [Eq. (6)]. The coupling between these two dynamics is here performed considering the modulation of the coupling constant σ with the global order parameters of the node dynamics [defined in Eq. (4)] and the link dynamics [defined in Eq. (9)]. Specifically, we consider two models denoted as Model NodesLinks (NL) and Model NodesLinksTriangles (NLT). Model NL couples the dynamics of the phases of the nodes θ and of the links ϕ according to the following dynamical equations
The projected dynamics for ϕ^{[−]} and ϕ^{[+]} then obeys
Therefore the projection on the nodes ϕ^{[−]} of the phases ϕ associated to the links [Eq. (14)] is coupled to the dynamics of the phases θ [Eq. (12)] associated directly to nodes. However the projection on the triangles ϕ^{[+]} of the phases ϕ associated to the links is independent of ϕ^{[−]} and of θ as well. Model NLT also describes the coupled dynamics of topological signals defined on nodes and links but the adaptive coupling captured by the model is different. In this case the dynamical equations are taken to be
For Model NLT the projected dynamics for ϕ^{[−]} and for ϕ^{[+]} obeys
Therefore, as in Model NL, the dynamics of the projection ϕ^{[−]} of the phases ϕ associated to the links [Eq. (18)] is coupled to the dynamics of the phases θ associated directly to nodes [Eq. (16)] and vice versa. Moreover, the dynamics of the projection of the phases ϕ on the triangles ϕ^{[+]} [Eq. (19)] is now also coupled with the dynamics of ϕ^{[−]} [Eq. (18)] and vice versa. Here and in the following we will use the convenient notation (using the parameter r) to indicate both models NL and NLT with the same set of dynamical equations given by
which reduce to Eq. (13) for r = 1 and to Eq. (17) for r = 2.
We make two relevant observations:

First, the proposed coupling between topological signals of different dimension can be easily extended to signals defined on higherorder simplices providing a very general scenario for coupled dynamical processes on simplicial complexes.

Second, the considered coupled dynamics of topological signals defined on nodes and links can be also studied on networks with exclusively pairwise interactions where we assume that the number of simplices of dimension n > 1 is zero. Therefore in this specific case this topological dynamics can have important effects also on simple networks.
We have simulated Model NL and Model NLT on two main examples of simplicial complex models: the configuration model of simplicial complexes^{51} and the NGF^{12,13} (see Fig. 2). In the configuration model we have considered powerlaw distribution of the generalized degree with exponent γ < 3, and for the NGF model with have considered simplicial complexes of dimensions d = 3 whose skeleton is a powerlaw network with exponent γ = 3. In both cases we observe an explosive synchronization of the topological signals associated to the nodes and to the links. On finite networks, the discontinuous transition emerge together with the hysteresis loop formed by the forward and backward synchronization transition. However the two models display a notable difference. In Model NL we observe a discontinuity for R_{0} and \({R}_{1}^{\,\text{down}\,}\) at a nonzero coupling constant σ = σ_{c}; however, \({R}_{1}^{\,\text{up}\,}\) follows an independent transition at zero coupling (see Fig. 2, panels in the second and fourth column). In Model NLT, on the contrary, all order parameters R_{0}, \({R}_{1}^{\,\text{down}\,}\), and \({R}_{1}^{\,\text{up}\,}\) display a discontinuous transition occurring for the same non zero value of the coupling constant σ = σ_{c} (see Fig. 2 panels in the first and third column). This is a direct consequence of the fact that in Model NL the adaptive coupling leading to discontinuous phase transition only couples the phases ϕ^{[−]} and θ, while for Model NLT the coupling involves also the phases ϕ^{[+]}.
Additionally we studied both Model NL and Model NTL on two real connectomes: the human connectome^{52} and the Caenorhabditis elegans (C. elegans) connectome^{53} (see Fig. 3). Interestingly also for these real datasets we observe that in Model NL the explosive synchronization involves only the phases θ and ϕ^{[−]} while in Model NLT we observe that also ϕ^{[+]} undergoes an explosive synchronization transition at the same value of the coupling constant σ = σ_{c}.
Theoretical solution of the NL model
As mentioned earlier the higherorder topological Kuramoto model coupling the topological signals of nodes and links can be defined on simplicial complexes and on networks as well. In the following sections we exploit this property of the dynamics to provide an analytical understanding of the synchronization transition on uncorrelated random networks.
It is well known that the Kuramoto model is challenging to study analytically. Indeed the full analytical understanding of the model is restricted to the fully connected case, while on a generic sparse network topology the analytical approximation needs to rely on some approximations. A powerful approximation is the annealed network approximation^{41} which consists in approximating the adjacency matrix of the network with its expectation in a random uncorrelated network ensemble. In order to unveil the fundamental theory that determines the coupled dynamics of topological signals described by the higherorder Kuramoto model here we combine the annealed approximation with the Ott–Antonsen method^{43}. This approach is able to capture the coupled dynamics of topological signals defined on nodes and links. In particular, the solution found to describe the dynamics of topological signals defined on the links is highly nontrivial and it is not reducible to the equations valid for the standard Kuramoto model. Conveniently, the calculations performed in the annealead approximation can be easily recasted in the exact calculation valid in the fully connected case previous a rescaling of some of the parameters. The analysis of the fully connected network reveals that the discontinuous sychronization transition of the considered model is characterized by a nontrivial backward transition with a well defined large network limit. On the contrary, the forward transition is highly dependent on the network size and vanishes in the large network limit, indicating that the incoherent state remains stable for every value of the coupling constant σ in the large network limit. This implies that on a fully connected network the NL model does not display a closed hysteresis loop as it occurs also for the model proposed in Skardal and Arenas^{21}. This scenario is here shown to extend also to sparse networks with finite second moment of the degree distribution while scalefree networks display a well defined hysteresis loop in the large network limit.
Annealed dynamics
For the dynamics of the phases θ associated to the nodes—Eq. (20)—it is possible to proceed as in the traditional Kuramoto model^{42,54,55}. However, the annealed approximation for the dynamics of the phases ϕ defined in Eq. (21) needs to be discussed in detail as it is not directly reducible to previous results. To address this problem our aim is to directly define the annealed approximation for the dynamics of the projected variables ϕ^{[−]} which, here and in the following, are indicated as
in order to simplify the notation. Moreover we will indicate with N = N_{[0]} the number of nodes in the network or in the simplicial complex skeleton. Here we focus on the NL Model defined on networks, i.e., we assume that there are no simplices of dimension two. We provide an analytical understanding of the coupled dynamics of nodes and links in the NL Model by determining the equations that capture the dynamics in the annealed approximation and predict the value of the complex order parameters
(with \({R}_{0},{R}_{1}^{\,\text{down}\,},{{\Theta }}\), and Ψ real) as a function of the coupling constant σ.
We notice that Eq. (14), valid for Model NL, can be written as
This equation can be also written elementwise as
where the vector \(\hat{{\boldsymbol{\omega }}}\) is given by
Let us now consider in detail these frequencies in the case in which the generic internal frequency \({\tilde{\omega }}_{\ell }\) of a link follows a Gaussian distribution, specifically in the case in which \({\tilde{\omega }}_{\ell } \sim {\mathcal{N}}({{{\Omega }}}_{1},1/{\tau }_{1})\) for every link ℓ. Using the definition of the boundary operator on a link it is easy to show that the expectation of \({\hat{\omega }}_{i}\) is given by
Given that each node has degree k_{i}, the covariance matrix C is given by the graph Laplacian L_{[0]} of the network, i.e.
where we have indicated with \({\left\langle \ldots \right\rangle }_{c}\) the connected correlation. Therefore the variance of \(\hat{\omega }\) in the annealed approximation is
Moreover, the projected frequencies are actually correlated and for i ≠ j we have
It follows that the frequencies \(\hat{{\boldsymbol{\omega }}}\) are correlated Gaussian variables with average given by Eq. (27) and correlation matrix given by the graph Laplacian. The fact that the frequencies \({\hat{\omega }}_{i}\) are correlated is an important feature of the dynamics of ψ and, with few exceptions^{56}, this feature has remained relatively unexplored in the case of the standard Kuramoto model. Additionally let us note that the average of \(\hat{\omega }\) over all the nodes of the network is zero. In fact
where with 1 we indicate the Ndimensional column vector of elements 1_{i} = 1. By using the symmetry of the adjacency matrix, i.e. the fact that a_{ij} = a_{ji}, Eq. (31) implies that the sum of \({\dot{\psi }}_{i}\) over all the nodes of the network is zero, i.e.
We now consider the annealed approximation consisting in substituting the adjacency matrix element a_{ij} with its expectation in an uncorrelated network ensemble
where k_{i} indicates the degree of node i and \(\left\langle k\right\rangle\) is the average degree of the network. Note that the considered random networks can be both sparse^{57} or dense^{58} as long as they display the structural cutoff, i.e. \({k}_{i}\ll \sqrt{\left\langle k\right\rangle N}\) for every node i of the network. In the annealed approximation we can put
Also, in the annealed approximation the dynamical Eq. (20) and Eq. (24) reduce to
where ⊙ indicates the Hadamard product (element by element multiplication) and where two auxiliary complex order parameters are defined as
with \({\hat{R}}_{0},\hat{{{\Theta }}},{\hat{R}}_{1}^{\,\text{down}\,}\) and \(\hat{{{\Psi }}}\) real.
The dynamics on a fully connected network
On a fully connected network in which each node has degree k_{i} = N − 1 the dynamics of the NL Model is well defined provided its parameter are properly rescaled. In particular, we require a standard rescaling of the coupling constant with the network size, given by
which guarantees that the interaction term in the dynamical equations has a finite contribution to the velocity of the phases.
The Model NL on fully connected networks requires also some specific model dependent rescalings associated to the dynamics on networks. Indeed, in order to have a finite expectation \(\left\langle {\hat{\omega }}_{i}\right\rangle\) of the projected frequencies \({\hat{\omega }}_{i}\) and a finite of the covariance matrix C [given by Eqs. (27) and (28), respectively], we require that on a fully connected network both Ω_{1} and τ_{1} are rescaled according to
Considering these opportune rescalings and noticing that the order parameters obey \({\hat{R}}_{0}={R}_{0}\), \({\hat{R}}_{1}^{\,\text{down}}={R}_{1}^{\text{down}\,}\), \({{\Theta }}=\hat{{{\Theta }}}\), and \({{\Psi }}=\hat{{{\Psi }}}\), we obtain that Model NL dictated by Eqs. (34)–(35) can be rewritten here as
with \({R}_{0},\,{R}_{1}^{\,\text{down}\,},{{\Theta }}\) and Ψ given by Eq. (23) and
Solution of the dynamical equations in the annealed approximation
General framework for obtaining the solution of the annealed dynamical equations
In this section we will provide the analytic solutions for the order parameter of the higherorder topological synchronization studied within the annealed approximation, i.e., captured by Eqs. (34) and (35). In particular, first we will find an expression of the order parameters R_{0} of the dynamics associated to the nodes (Eq. (34)) and subsequently in the next paragraph we will derive the expression for the order parameter \({R}_{1}^{\,\text{down}\,}\) associated to the projection on the nodes of the topological signal defined on the links (Eq. (35)). By combining the two results it is finally possible to uncover the discontinuous nature of the transition.
Dynamics of the phases of the nodes
When we investigate Eq. (34) we notice that this equation can be easily reduced to the equation for the standard Kuramoto model treated within the annealed approximation^{42} if one performs a rescaling of the coupling constant σR_{0} → σ. Therefore we can treat this model similarly to the known treatment of the standard Kuramoto model^{40,41,42}. Specifically, starting from Eq. (34) and using a rescaling of the phases θ according to
it is possible to show that we can set Θ = 0 and therefore Eq. (34) reduces to the wellknown annealed expression for the standard order Kuramoto model given by
Assuming that the system of equations reaches a steady state in which both \({R}_{1}^{\,\text{down}\,}\) and \({\hat{R}}_{0}\) become time independent, the order parameters of this system of equations in the coherent state \({\hat{R}}_{0}\,> \, 0\) and \({R}_{1}^{\,\text{down}\,}\,> \, 0\) can be found to obey^{40,42,50,54}
where \({\hat{c}}_{i}\) indicates
and g(ω) is the Gaussian distribution with expectation Ω_{0} and standard deviation 1.
Dynamics of the phases of the links projected on the nodes
In this paragraph we will derive the expression of the order parameters \({R}_{1}^{\,\text{down}\,}\) and \({\hat{R}}_{1}^{\,\text{down}\,}\) which, together with Eq. (44), will provide the annealed solution of our model. To start with we assume that the frequencies \(\hat{{\boldsymbol{\omega }}}\) are known. In this case we can express the order parameters \({R}_{1}^{\,\text{down}\,}\) and \({\hat{R}}_{1}^{\,\text{down}\,}\) as a function of the probability density function \({\rho }^{(i)}(\psi ,t \hat{{\boldsymbol{\omega }}})\) that node i is associated to a projected phase of the link equal to ψ. Since in the annealed approximation ψ_{i} has a dynamical evolution dictated by Eq. (35) the probability density function obeys the continuity equation
with associated velocity v_{i} given by
where we have defined κ_{i} as
In this case the complex order parameters are given by
In order to solve the continuity equation we follow Ott–Antonsen^{43} and we express \({\rho }^{(i)}(\psi ,t \hat{{\boldsymbol{\omega }}})\) in the Fourier basis as
Making the ansatz
we can derive the equation for the evolution of \({b}_{i}={b}_{i}({\hat{\omega }}_{i},t)\) given by
Since we showed before that the average value of \({\dot{\psi }}_{i}\) over nodes is zero, we look for nonrotating stationary solutions of Eq. (52), ∂_{t}b_{i} = 0. As long as R_{0} > 0 these stationary solutions are given by
where d_{i} is given by
By inserting this expression into Eq. (49) we get the expression of the order parameters given the projected frequencies \(\hat{{\boldsymbol{\omega }}}\), in the coherent phase in which R_{0} > 0
where, indicating by θ(x) the Heaviside function, we have defined
Finally, if the projected frequencies \(\hat{{\boldsymbol{\omega }}}\) are not known we can average the result over the marginal frequency distribution of the projected frequency \({\hat{\omega }}_{i}\) given by \({G}_{i}(\hat{{\boldsymbol{\omega }}})\)
and an analogous equations for \({R}_{1}^{\,\text{down}\,}\sin ({{\Psi }})\) (not shown). We note that in the case of distributions g(ω) and \({G}_{i}(\hat{\omega })\) that are symmetric around their means the above equations always admit the solution \({{\Psi }}=\hat{{{\Psi }}}=0\). Such values of the phases are also confirmed by direct numerical integration of the NL model.
These equations together with Eq. (44) capture the steadystate behavior of the higherorder Kuramoto model coupling topological signals defined on nodes and links within the annealed approximation in the coherent synchronized phase. Note that by derivation, these equations cannot capture the asynchronous phase which is instead always a trivial solution of the dynamical equations corresponding to \({R}_{0}={R}_{1}^{\,\text{down}\,}=0\). Finally we observe that for the NL Model as well as for the standard Kuramoto model on random networks, it is expected that the annealed approximation is more accurate for networks that are connected and are sufficiently dense.
To illustrate the applicability of the theoretical analysis, we consider two examples of connected networks with N = 1600 nodes: a Poisson network with average degree c = 12 and an uncorrelated scalefree network with minimum degree m = 6 and powerlaw exponent γ = 2.5 In Fig. 4 we compare the values of R_{0}, \({R}_{1}^{\,\text{down}\,}\) obtained from direct numerical integration of Eqs. (20) and (25) and the steadystate solutions obtained from the numerical solution of Eq. (55). The backward transition is fully captured by our theory, while the next paragraphs will clarify the theoretical expectations for the forward transition.
Solution on the fully connected network
The integration of Eq. (57) requires the knowledge of the marginal distributions \({G}_{i}(\hat{\omega })\) which does not have in general a simple analytical expression. However, in the fully connected networks with Gaussian distribution of the internal frequency of nodes and links this calculation simplifies significantly. Indeed, when the link frequencies are sampled from a Gaussian distribution with mean Ω_{1}/N and standard deviation \(1/({\tau }_{1}\sqrt{N1})\), the marginal frequency distribution \({G}_{i}(\hat{\omega })\) of the internal frequency \({\hat{\omega }}_{i}\) of a node i in a fully connected network is given by (see “Methods” for details)
where \(\bar{c}=\frac{N}{N1}\). By considering \({{{\Omega }}}_{0}={{{\Omega }}}_{1}=\left\langle {\hat{\omega }}_{i}\right\rangle =0,\) and performing a direct integration of Eq. (57) we obtain (see “Methods” section for details) the closed system of equations for R_{0} and \({R}_{1}^{\,\text{down}\,}\)
where the scaling function h(x) is given by
with I_{0} and I_{1} indicating the modified Bessel functions. The numerical solution of Eq. (59) reveals the following picture: for low values of σ, only the incoherent solution \({R}_{0}={R}_{1}^{\,\text{down}\,}=0\) exists. At a positive value of σ, two solutions of Eq. (59) appear at a bifurcation point, with the upper solution corresponding to a stable synchronized state and the lower solution to an unstable synchronized solution. For larger values of σ, the values of R_{0} and \({R}_{1}^{\,\text{down}\,}\) corresponding to the upper solution approach one (full phase synchronization), while those for the lower solution approach zero asymptotically, thus indicating that the incoherent state never loses stability. Indeed, it can be easily checked (see “Methods” for details) that for large σ the unstable solution of Eq. (59) has asymptotic behavior
with J_{0} and J_{1} constants given by
Therefore, the unstable branch approaches the trivial solution \({R}_{0}={R}_{1}^{\,\text{down}\,}=0\) only asymptotically for σ → ∞. This implies that the trivial solution remains stable for every possible value of σ although as σ increases it describes the stationary state of an increasingly smaller set of initial conditions.
This scenario is confirmed by numerical simulations (see Fig. 5) showing that the backward transition is captured very well by our theory and does not display notable finitesize effects. The forward transition, instead, displays remarkable finitesize effects. Indeed, as σ increases, the system remains in the incoherent state until it explosively synchronizes at a positive value of σ and reaches the stable synchronized branch. However the incoherent state is stable in the limit N → ∞, and this forward transition is the result of finitesize fluctuations that push the system above the unstable branch, causing the observed explosive transition. This is consistent with the fact that for larger values of N, which have smaller finitesize fluctuations, the system remains in the incoherent state for larger values of σ.
Therefore, while a closed hysteresis loop is not present in the NL model defined on fully connected networks, we observe fluctuationdriven hysteresis, in which finitesize fluctuations of the zero solution drive the system towards the synchronized solution, creating an effective hysteresis loop.
Hysteresis on homogeneous and scalefree networks
In this section we discuss how the scenario found for the fully connected network can be extended to random networks with given degree distribution. We will start from the selfconsistent Eq. (57) obtained within the annealed approximation model. These equations display a saddle point bifurcation with the emergence of two nontrivial solutions describing a stable and an unstable branch of these selfconsistent equations. These solutions always exist in combination with the trivial solution \({R}_{0}={R}_{1}^{\,\text{down}\,}=0\) describing the asynchronous state. Two scenarios are possible: either the unstable branch converges to the trivial solution only in the limit σ → ∞ or it converges to the trivial solution at a finite value of σ. In the first case, the scenario is the same as the one observed for the fully connected network, and the trivial solution remains stable for any finite value of σ. In this case the forward transition is not obtained in the limit N → ∞ and the transition observed on finite networks is only caused by finitesize effects. In the second case the trivial solution loses its stability at a finite value of σ. Therefore the forward transition is not subjected to strong finitesize effects and we expect to see a forward transition also in the N → ∞ limit. in order to determine which network topologies can sustain a nontrivial hysteresis loop we expand Eq. (57) for 0 < R_{0} ≪ 1, \(0\,<\,{\hat{R}}_{0}\ll 1\), and \(0\,<\,{R}_{1}^{\,\text{down}\,}\ll 1\) under the hypothesis that the distributions g(ω) and \({G}_{i}(\hat{\omega })\) are symmetric and unimodal. Under these hypothesis it is easy to show that Eq. (57) predict an unstable solution in which R_{0} and \({R}_{1}^{\,\text{down}\,}\) scale with σ according to
with J_{0} and J_{1} constants given by
As long as the network does not have vanishing J_{0} and J_{1} the unstable branch converges to the trivial solution \({R}_{0}={R}_{1}^{\,\text{down}\,}\) only in the limit σ → ∞. This happens for instance for Gaussian distribution of the internal frequency of the links and converging second moment \(\left\langle {k}^{2}\right\rangle\) of the degree distribution. However, when the second moment diverges, i.e., the network is scalefree with \(\left\langle {k}^{2}\right\rangle \to \infty\) as N → ∞, then R_{0} and R_{1} can converge to the trivial solution \({R}_{0}={R}_{1}^{\,\text{down}\,}=0\) also for finite σ. This analysis suggests that the scenario described for the fully connected network remains valid for sparse (connected) networks as long as the degree distribution does not have a diverging second moment, while a stable hysteresis loop can be observed for scalefree networks.
Conclusions
Until recently the synchronization phenomenon has been explored only in the context of topological signals associated to the nodes of a network. However, the growing interest in simplicial complexes opens the perspective of investigating synchronization of higherorder topological signals, for instance associated to the links of the discrete networked structure. Here we uncover how topological signals associated to nodes and links can be coupled to one another giving rise to an explosive synchronization phenomenon involving both signals at the same time. The model has been tested on real connectomes and on major examples of simplicial complexes (the configuration model^{51} of simplicial complex and the NGF^{13}). Moreover, we provide an analytical solution of this model that provides a theoretical understanding of the mechanism driving the emergence of this discontinuous phase transition and the mechanism responsible for the emergence of a closed hysteresis loop. This work can be extended in different directions including the study of the desynchronization dynamics of this coupled higherorder synchronization and the duality of this model with the same model defined on the line graph of the same network.
Methods
Definition of simplicial complexes
Simplicial complexes represent higherorder networks whose interactions include two or more nodes. These manybody interactions are captured by simplices. An ndimensional simplex α is a set of n + 1 nodes
For instance a node is a 0dimensional simplex, a link is a onedimensional simplex, a triangle is a twodimensional simplex, a tetrahedron is a threedimensional 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 tetrahedron are 4 nodes, 6 links, and 4 triangles. A simplicial complex is a set of simplices closed under the inclusion of the faces of each simplex. Any simplicial complex can be reduced to its simplicial complex skeleton, which is the network formed by the simplicial complex nodes and links. Simplices have a relevant topological and geometrical interpretation and constitute the topological structures studied by discrete algebraic topology. Therefore representing the manybody interactions of a complex system with a simplicial complex opens the very fertile opportunity to use the tools of algebraic topology^{5,59} to study the topology of the system under investigation. In this work we show that algebraic topology can also shed significant light on the role that topology has on higherorder synchronization.
Oriented simplices and boundary map
In algebraic topology simplices are oriented. For instance a link α = [i, j] has the opposite sign of the link [j, i], i.e.,
Similarly to higherorder simplices we associate an orientation such that
where σ(π) indicates the parity of the permutation π. It is good practice to use as orientation of the simplices the orientation induced by the labeling of the nodes, i.e., giving, for example, a positive orientation to any simplex
where
This will ensure that the spectral properties of the higherorder Laplacians that will be defined later are independent of the labeling of the nodes. Given a simplicial complex, a nchain consists of the elements of a free abelian group \({{\mathcal{C}}}_{n}\) with basis formed by the set of all oriented nsimplices. Therefore every element of \({{\mathcal{C}}}_{n}\) can be uniquely expressed as a linear combination of the basis elements (nsimplices) with coefficients in \({{\mathbb{Z}}}\). The boundary operator ∂_{n} is a linear operator \({\partial }_{n}:{{\mathcal{C}}}_{n}\to {{\mathcal{C}}}_{n1}\) whose action is determined by the action on each nsimplex of the simplicial complex given by
As a concrete example, in Fig. 6 we demonstrate the action of the boundary operator on links and triangles. A celebrated property of the boundary operator is that the boundary of a boundary is null, i.e.
for any n > 0. This relation can be directly proven by using Eq. (71). Let us consider a simplicial complex \({\mathcal{K}}\). Let us indicate with N_{[n]} the number of simplices of the simplicial complex with generic dimension n. Given a basis for the linear space of nchains \({{\mathcal{C}}}_{n}\) and for the linear space of (n − 1)chains \({{\mathcal{C}}}_{n1}\) formed by an ordered list of the n simplices and (n − 1) simplices of the simplicial complex, the boundary operator ∂_{n} can be represented as N_{[n−1]} × N_{[n]} incidence matrix B_{[n]}. In Fig. 6 we show a twodimensional simplicial complex and its corresponding incidence matrices B_{[1]} and B_{[2]}. Given that the boundary matrices obey Eq. (72) it follows that the incidence matrices obey
for any n > 0.
Higherorder Laplacians
Using the incidence matrices it is natural to generalize the definition of the graph Laplacian
to the higherorder Laplacian L_{[n]} (also called combinatorial Laplacians)^{17,19,60} that can be represented as a N_{[n]} × N_{[n]} matrix given by
with
for n > 0. The higherorder Laplacian can be proven to be independent of the orientation of the simplices as long as the simplicial complex has an orientation induced by a labeling of the nodes.
The most celebrated property of higherorder Laplacian is that the degeneracy of the zero eigenvalue of the n Laplacian L_{[n]} is equal to the Betti number β_{n} and that their corresponding eigenvectors localize around the corresponding ndimensional cavities of the simplicial complex. The higherorder Laplacians can be used to define higherorder diffusion^{17} and can display a higherorder spectral dimension on network geometries. Here we are particularly interested in the use of incidence matrices and higherorder Laplacians to define higherorder topological synchronization.
Steadystate solution of the annealed equations for the NL Model
Here we study Eqs. (44) and (57) assuming that the distributions g(ω) and \({G}_{i}(\hat{\omega })\) are unimodal functions symmetric about their means. Setting \({{\Psi }}=\hat{{{\Psi }}}=0\) and considering the change of variables \(z=\omega /(\sigma {R}_{0}{R}_{1}^{\,\text{down}\,})\), \(y=\hat{\omega }/(\sigma {R}_{0})\), Eq. (44) can be written as
while Eq. (57) reduce to
We notice that the equations for \({R}_{0},{\hat{R}}_{0}\) and \({R}_{1}^{\,\text{down}\,}\) do not depend on the order parameter \({\hat{R}}_{1}^{\,\text{down}\,}\) so we can obtain a fully analytical solution of the model without solving the last equation. The above equations depend on the distribution g(ω) and the set of marginal distributions \({G}_{i}({\hat{\omega }}_{i})\). However we can show that, provided \(\left\langle {k}^{2}\right\rangle /\left\langle k\right\rangle\) is finite, the solution of these equations does not converge to the trivial solution \({R}_{0}={\hat{R}}_{0}={R}_{1}^{\,\text{down}\,}=0\) for any finite value of σ. Indeed we are now going to show that the unstable branch of the solution these equations converges to the trivial solution only in the limit σ → ∞. Assuming 0 < R_{0} ≪ 1, \(0\,<\,{\hat{R}}_{0}\ll 1\), and \(0\,<\,{R}_{1}^{\,\text{down}\,}\ll 1\) we can expand the functions \(g(z\sigma {k}_{i}{\hat{R}}_{0}{R}_{1}^{\,\text{down}\,})\) and G_{i}(yσR_{0}k_{i}) as
Stopping at the first order of this expansion we get
This equations lead to the following scaling of R_{0} and \({R}_{1}^{\,\text{down}\,}\) with σ
with
This confirms the theoretical framework revealing that in this dynamics there is always a trivial solution \({R}_{0}={\hat{R}}_{0}={R}_{1}^{\,\text{down}\,}=0\) while Eqs. (44) and (57) are characterized by a saddlepoint instability so that for σ > σ_{c} two additional solutions emerge, a stable solution and an unstable solution. The stable solution describes the synchronized phase and captures the backward transition. As long as the second moment of the degree distribution does not diverge, the unstable solution converges to the trivial solution \({R}_{0}={\hat{R}}_{0}={R}_{1}^{\,\text{down}\,}=0\) only for σ → ∞.
The asymptotic scaling for R_{0} and \({R}_{1}^{\,\text{down}\,}\) given by Eq. (80) can be adapted to capture the asymptotic scaling of the fully connected case with a suitable rescaling of the model parameters of the model, obtaining Eqs. (61) and (63).
Marginal distributions in the fully connected case
The distribution \({G}_{1}(\hat{{\boldsymbol{\omega }}})\) of \(\hat{{\boldsymbol{\omega }}}\) is a Gaussian distribution with averages given by Eq. (27) and covariance matrix C given by Eq. (28). The covariance matrix has N − 1 eigenvalues given by \(\lambda =1/{\tau }_{1}^{2}\) and one zero eigenvalue λ = 0 corresponding to the eigenvector
This means that we should always have
a constraint that we can introduce as a delta function in the expression for the joint distribution \(\hat{G}(\hat{{\boldsymbol{\omega }}})\) of the vector \(\hat{{\boldsymbol{\omega }}}\). Here we note that under these hypotheses and assuming that the distribution of the frequencies of the links is a Gaussian with average Ω_{1}/N and standard deviation \(1/({\tau }_{1}\sqrt{N1})\) the marginal probability \({G}_{i}(\hat{\omega })\) of \({\hat{\omega }}_{i}\) can be expressed as Eq. (58).
Given that the covariance matrix has a zero eigenvalue we can express the joint Gaussian distribution \(\hat{G}(\hat{{\boldsymbol{\omega }}})\) as
where δ(x) indicates the delta function and where \({\mathcal{F}}(\hat{{\boldsymbol{\omega }}})\) and \({\mathcal{C}}\) are given by
The marginal probability \({G}_{i}(\hat{\omega })\) is given by
By expressing the delta function in Eq. (84) in its integral form
we get the final expression for the marginal distribution Eq. (58), in fact, by putting c\;=\;N/(N–1), we have
Data availability
The connectome network dataset used in this study are freely available: the Homo sapiens dataset comes from ref. ^{52} and the C. elegans dataset comes from ref. ^{53}.
Code availability
All codes are available upon request to the corresponding authors.
References
Giusti, C., Ghrist, R. & Bassett, D. S. Two’s company, three (or more) is a simplex. J. Comput. Neurosci. 41, 1–14 (2016).
Battiston, F. et al. Networks beyond pairwise interactions: structure and dynamics. Phys. Rep. 874, 1–92 (2020).
Torres, L., Blevins, A. S., Bassett, D. S. & EliassiRad, T. The why, how, and when of representations for complex systems. Preprint at https://arxiv.org/abs/2006.02870 (2020).
Salnikov, V., Cassese, D. & Lambiotte, R. Simplicial complexes and complex systems. Eur. J. Phys. 40, 014001 (2018).
Otter, N., Porter, M. A., Tillmann, U., Grindrod, P. & Harrington, H. A. A roadmap for the computation of persistent homology. EPJ Data Sci. 6, 17 (2017).
Petri, G., Scolamiero, M., Donato, I. & Vaccarino, F. Topological strata of weighted complex networks. PLoS ONE 8, e66506 (2013).
Massara, G. P., Di Matteo, T. & Aste, T. Network filtering for big data: triangulated maximally filtered graph. J. Complex Netw. 5, 161–178 (2016).
Sreejith, R., Mohanraj, K., Jost, J., Saucan, E. & Samal, A. Forman curvature for complex networks. J. Stat. Mech. Theory Exp. 2016, 063206 (2016).
KartunGiles, A. P. & Bianconi, G. Beyond the clustering coefficient: a topological analysis of node neighbourhoods in complex networks. Chaos Solitons Fractals X 1, 100004 (2019).
Rocks, J. W., Liu, A. J. & Katifori, E. Revealing structurefunction relationships in functional flow networks via persistent homology. Phys. Rev. Res. 2, 033234 (2020).
Wu, Z., Menichetti, G., Rahmede, C. & Bianconi, G. Emergent complex network geometry. Sci. Rep. 5, 1–12 (2015).
Bianconi, G. & Rahmede, C. Emergent hyperbolic network geometry. Sci. Rep. 7, 41974 (2017).
Bianconi, G. & Rahmede, C. Network geometry with flavor: from complexity to quantum geometry. Phys. Rev. E 93, 032315 (2016).
Dankulov, M. M., Tadić, B. & Melnik, R. Spectral properties of hyperbolic nanonetworks with tunable aggregation of simplexes. Phys. Rev. E 100, 012309 (2019).
Tadić, B., Andjelković, M. & Melnik, R. functional geometry of human connectomes. Sci. Rep. 9, 1–12 (2019).
Millán, A. P., Torres, J. J. & Bianconi, G. Explosive higherorder kuramoto dynamics on simplicial complexes. Phys. Rev. Lett. 124, 218301 (2020).
Torres, J. J. & Bianconi, G. Simplicial complexes: higherorder spectral dimension and dynamics. J. Phys. Complex. 1, 015002 (2020).
Reitz, M. & Bianconi, G. The higherorder spectrum of simplicial complexes: a renormalization group approach. J. Phys. A Math. Theor. 53, 295001 (2020).
Barbarossa, S. & Sardellitti, S. Topological signal processing over simplicial complexes. IEEE Trans. Signal Process. https://doi.org/10.1109/TSP.2020.2981920 (2020).
Landry, N. & Restrepo, J. G. The effect of heterogeneity on hypergraph contagion models. Chaos 30, 103117 (2020).
Skardal, P. S. & Arenas, A. Abrupt desynchronization and extensive multistability in globally coupled oscillator simplexes. Phys. Rev. Lett. 122, 248301 (2019).
Skardal, P. S. & Arenas, A. Higherorder interactions in complex networks of phase oscillators promote abrupt synchronization switching. Communications Physics. 3, 218 (2020).
Iacopini, I., Petri, G., Barrat, A. & Latora, V. Simplicial models of social contagion. Nat. Commun. 10, 1–9 (2019).
Taylor, D. et al. Topological data analysis of contagion maps for examining spreading processes on networks. Nat. Commun. 6, 1–11 (2015).
Lucas, M., Cencetti, G. & Battiston, F. Multiorder laplacian for synchronization in higherorder networks. Phys. Rev. Res. 2, 033410 (2020).
Zhang, Y., Latora, V. & Motter, A. E. Unified treatment of dynamical processes on generalized networks: higherorder, multilayer, and temporal interactions. Preprint at https://arxiv.org/abs/2010.00613 (2020).
Skardal, P. S. & Arenas, A. Memory selection and information switching in oscillator networks with higherorder interactions. J. Phys. Complexity 2, 015003 (2020).
DeVille, L. Consensus on simplicial complexes, or: the nonlinear simplicial Laplacian. Preprint at https://arxiv.org/abs/2010.07421 (2020).
Carletti, T., Fanelli, D. & Nicoletti, S. Dynamical systems on hypergraphs. JPhys Complexity 1, 035006 (2020).
Millán, A. P., Torres, J. J. & Bianconi, G. Complex network geometry and frustrated synchronization. Sci. Rep. 8, 1–10 (2018).
Mulas, R., Kuehn, C. & Jost, J. Coupled dynamics on hypergraphs: master stability of steady states and synchronization. Phys. Rev. E 101, 062313 (2020).
Gambuzza, L. et al. The master stability function for synchronization in simplicial complexes. Nature Communications. 12, 1255 (2021).
Millán, A. P., Torres, J. J. & Bianconi, G. Synchronization in network geometries with finite spectral dimension. Phys. Rev. E 99, 022307 (2019).
Severino, F. P. U. et al. The role of dimensionality in neuronal network dynamics. Sci. Rep. 6, 29640 (2016).
Saggar, M. et al. Towards a new approach to reveal dynamical organization of the brain using topological data analysis. Nat. Commun. 9, 1–14 (2018).
Giusti, C., Pastalkova, E., Curto, C. & Itskov, V. Clique topology reveals intrinsic geometric structure in neural correlations. Proc. Natl Acad. Sci. USA 112, 13455–13460 (2015).
Reimann, M. W. et al. Cliques of neurons bound into cavities provide a missing link between structure and function. Front. Comput. Neurosci. 11, 48 (2017).
RuizGarcia, M. & Katifori, E. Topologically controlled emergent dynamics in flow networks. Preprint at https://arxiv.org/abs/2001.01811 (2020).
Strogatz, S. H. From kuramoto to crawford: exploring the onset of synchronization in populations of coupled oscillators. Phys. D Nonlinear Phenom. 143, 1–20 (2000).
Boccaletti, S., Pisarchik, A. N., Del Genio, C. I. & Amann, A. Synchronization: from Coupled Systems to Complex Networks (Cambridge University Press, 2018).
Rodrigues, F. A., Peron, T. K. D., Ji, P. & Kurths, J. The kuramoto model in complex networks. Phys. Rep. 610, 1–98 (2016).
Restrepo, J. G., Ott, E. & Hunt, B. R. Onset of synchronization in large networks of coupled oscillators. Phys. Rev. E 71, 036151 (2005).
Ott, E. & Antonsen, T. M. Low dimensional behavior of large systems of globally coupled oscillators. Chaos 18, 037113 (2008).
Araque, A. et al. Gliotransmitters travel in time and space. Neuron 81, 728–739 (2014).
Huang, W. et al. A graph signal processing perspective on functional brain imaging. Proc. IEEE 106, 868–885 (2018).
Evans, T. S. & Lambiotte, R. Line graphs of weighted networks for overlapping communities. Eur. Phys. J. B 77, 265–272 (2010).
D’Souza, R. M., GómezGardeñes, J., Nagler, J. & Arenas, A. Explosive phenomena in complex networks. Adv. Phys. 68, 123–223 (2019).
Zhang, X., Boccaletti, S., Guan, S. & Liu, Z. Explosive synchronization in adaptive and multilayer networks. Phys. Rev. Lett. 114, 038701 (2015).
Dai, X. et al. Discontinuous transitions and rhythmic states in the ddimensional kuramoto model induced by a positive feedback with the global order parameter. Phys. Rev. Lett. 125, 194101 (2020).
Boccaletti, S. et al. Explosive transitions in complex networks’ structure and dynamics: percolation and synchronization. Phys. Rep. 660, 1–94 (2016).
Courtney, O. T. & Bianconi, G. Generalized network structures: the configuration model and the canonical ensemble of simplicial complexes. Phys. Rev. E 93, 062311 (2016).
Hagmann, P. et al. Mapping the structural core of human cerebral cortex. PLoS Biol. 6, e159 (2008).
Varshney, L. R., Chen, B. L., Paniagua, E., Hall, D. H. & Chklovskii, D. B. Structural properties of the Caenorhabditis elegans neuronal network. PLoS Comput. Biol. 7, e1001066 (2011).
Ichinomiya, T. Frequency synchronization in a random oscillator network. Phys. Rev. E 70, 026116 (2004).
Lee, D.S. Synchronization transition in scalefree networks: clusters of synchrony. Phys. Rev. E 72, 026208 (2005).
Skardal, P. S., Restrepo, J. G. & Ott, E. Frequency assortativity can induce chaos in oscillator networks. Phys. Rev. E 91, 060902 (2015).
Anand, K. & Bianconi, G. Entropy measures for networks: toward an information theory of complex topologies. Phys. Rev. E 80, 045102 (2009).
SeyedAllaei, H., Bianconi, G. & Marsili, M. Scalefree networks with an exponent less than two. Phys. Rev. E 73, 046113 (2006).
Ghrist, R. W. Elementary Applied Topology, Vol. 1 (Createspace Seattle, 2014).
Horak, D. & Jost, J. Spectra of combinatorial Laplace operators on simplicial complexes. Adv. Math. 244, 303–336 (2013).
Acknowledgements
This research utilized Queen Mary’s Apocrita HPC facility, supported by QMUL ResearchIT (https://doi.org/10.5281/zenodo.438045). G.B. acknowledge support from the Royal Society IEC\NSFC \191147. J.J.T. acknowledges financial support from the Spanish Ministry of Science and Technology, and the Agencia Española de Investigación (AEI) under grant FIS201784256P (FEDER funds) and from the Consejería de Conocimiento, Investigación y Universidad, Junta de Andalucía and European Regional Development Fund, Refs. AFQM175UGR18 and SOMM17/6105/UGR.
Author information
Authors and Affiliations
Contributions
R.G., J.G.R., J.J.T., and G.B. have all contributed in the design of the project, and in the numerical implementations of the algorithm. J.G.R., J.J.T., and G.B. have contributed to the theoretical derivations and the writing of the manuscript.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests. G.B. is a Guest Editor for the Focus Collection on Higher Order Interaction Networks in Communications Physics, but was not involved in the editorial review of, or the decision to publish this article.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Ghorbanchian, R., Restrepo, J.G., Torres, J.J. et al. Higherorder simplicial synchronization of coupled topological signals. Commun Phys 4, 120 (2021). https://doi.org/10.1038/s42005021006054
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s42005021006054
This article is cited by

Synchronization analyze of kuniform hypernetworks
Scientific Reports (2024)

Higherorder nonMarkovian social contagions in simplicial complexes
Communications Physics (2024)

Persistent Dirac for molecular representation
Scientific Reports (2023)

Higherorder temporal interactions promote the cooperation in the multiplayer snowdrift game
Science China Information Sciences (2023)

Connecting Hodge and SakaguchiKuramoto through a mathematical framework for coupled oscillators on simplicial complexes
Communications Physics (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.