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D-dimensional oscillators in simplicial structures: Odd and even dimensions display different synchronization scenarios
Abstract
From biology to social science, the functioning of a wide range of systems is the result of elementary interactions which involve more than two constituents, so that their description has unavoidably to go beyond simple pairwise-relationships. Simplicial complexes are therefore the mathematical objects providing a faithful representation of such systems. We here present a complete theory of synchronization of D-dimensional oscillators obeying an extended Kuramoto model, and interacting by means of 1- and 2- simplices. Not only our theory fully describes and unveils the intimate reasons and mechanisms for what was observed so far with pairwise interactions, but it also offers predictions for a series of rich and novel behaviors in simplicial structures, which include: (a) a discontinuous de-synchronization transition at positive values of the coupling strength for all dimensions, (b) an extra discontinuous transition at zero coupling for all odd dimensions, and (c) the occurrence of partially synchronized states at D=2 (and all odd D) even for negative values of the coupling strength, a feature which is inherently prohibited with pairwise-interactions. Furthermore, our theory untangles several aspects of the emergent behavior: the system can never fully synchronize from disorder, and is characterized by an extreme multi-stability, in that the asymptotic stationary synchronized states depend always on the initial conditions. All our theoretical predictions are fully corroborated by extensive numerical simulations. Our results elucidate the dramatic and novel effects that higher-order interactions may induce in the collective dynamics of ensembles of coupled D-dimensional oscillators, and can therefore be of value and interest for the understanding of many phenomena observed in nature, like for instance the swarming and/or flocking processes unfolding in three or more dimensions.
... The extension of the Kuramoto model to the sphere S d−1 of any dimension d has been developed in detail (see the various descriptions). 12,[41][42][43] Such models are relevant to swarming and flocking processes in three or more dimensions [44][45][46] and to opinion formation and consensus. 47 By means of an LFT applied to each node, we can again write the system equations asġ i = M i g i , where g i , M i are real (d + 1) × (d + 1) matrices, then partial integration is performed as before, reducing the N vector equations to the single matrix equatioṅ g = Mg consisting of (d + 1) 2 individual equations. ...
... where we can vary κ 2 and κ 3 in both magnitude and sign in order to determine the relative effect of the two-and three-body interactions. The three-body system (53) differs from that previously considered in d dimensions 44 [see Eq. (4)], where x j , x k is replaced by x i , x k and/or x i , x j , leading to significant differences in behavior, such as extreme multi-stability, which we do not observe here for positive couplings. In any case, we set = 0, in order that the system be partially integrable. ...
... For d = 2, these equations reduce to (72), which has been proposed as a model of three-body (two-simplex) interactions. 53,54 System (76) has also been analyzed in any dimension d, see Eq. (8) in Ref. 44, where the variables σ i , ρ correspond to our x i , X, respectively. Whereas we choose the frequency matrix to be independent of i, in order that the system be partially integrable, in Ref. 44 the corresponding matrix W i is selected with random entries for each i, leading to much more complicated properties, including novel behaviors such as discontinuous transitions. ...
Exact reduction by partial integration has been extensively investigated for the Kuramoto model by means of the Watanabe–Strogatz transform. This is the simplest of higher-dimensional reductions that apply to a hierarchy of models in spaces of any dimension, including Riccati systems. Linear fractional transformations enable the system equations to be expressed in an equivalent matrix form, where the variables can be regarded as time-evolution operators. This allows us to perform an exact integration at each node, which reduces the system to a single matrix equation, where the associated time-evolution operator acts over all nodes. This operator has group-theoretical properties, as an element of SU ( 1 , 1 ) ∼ SO ( 2 , 1 ) for the Kuramoto model, and SO ( d , 1 ) for higher-dimensional models on the unit sphere S d − 1. Parameterization of the group elements using subgroup properties leads to a further reduction in the number of equations to be solved and also provides explicit formulas for mappings on the unit sphere, which generalize the Möbius map on S 1. Exact dimensional reduction also applies to another class of much less-studied models on the unit sphere, with cubic nonlinearities, for which the governing equations can again be transformed into an equivalent matrix form by means of the unit map. Exact integration at each node proceeds as before, where now the time-evolution operator lies in SL ( d , R ). The matrix formulation leads to exact solutions in terms of the matrix exponential for trajectories that asymptotically approach fixed points. As examples, we investigate partially integrable models with combined pairwise and higher-order interactions.
... Other examples can be found in neuroscience [4][5][6][7][8][9], ecology [10,11], biology [12] and social sciences [13][14][15]. Higher order interactions have particularly important consequences in the propagation of epidemics [16][17][18] and synchronization of coupled oscillators [19][20][21][22][23][24][25][26][27][28][29][30][31]. ...
... These forms of interaction have very different implications for the dynamics of the Kuramoto model. Symmetric terms have been studied in [20,22,27,31] and result in multistability. Dai et al [22] have also considered the effects of symmetric interactions of arbitrary order in the multidimensional Kuramoto model. ...
... Symmetric terms have been studied in [20,22,27,31] and result in multistability. Dai et al [22] have also considered the effects of symmetric interactions of arbitrary order in the multidimensional Kuramoto model. Asymmetric interactions, on the other hand, were considered in [21,30,34,35] and were shown to give rise to bi-stability. ...
Higher order interactions can lead to new equilibrium states and bifurcations in systems of coupled oscillators described by the Kuramoto model. However, even in the simplest case of 3-body interactions there are more than one possible functional forms, depending on how exactly the bodies are coupled. Which of these forms is better suited to describe the dynamics of the oscillators depends on the specific system under consideration. Here we show that, for a particular class of interactions, reduced equations for the Kuramoto order parameter can be derived for arbitrarily many bodies. Moreover, the contribution of a given term to the reduced equation does not depend on its order, but on a certain effective order, that we define. We give explicit examples where bi and tri-stability is found and discuss a few exotic cases where synchronization happens via a third order phase transition.
... Examples can be found in neuroscience [25][26][27][28][29], ecology [30,31], biology [32] and social sciences [33,34]. Studies have shown that the inclusion of higher order interactions have important consequences in the propagation of epidemics [35][36][37] and synchronization [38][39][40][41][42][43][44][45]. ...
... The particular form of higher order interactions described above is the same as in [40] and we refer to it as asymmetric because of the way θ i enters the equations. Symmetric higher order interactions have also been considered [39,41] but we shall not discuss them here. ...
... We note that higher order interactions can be defined in different ways [39][40][41] and most of them are not amenable to exact treatment under the Ott-Antonsen ansatz. Symmetric interactions, for instance, can lead to multi-stability [39,41] which could lead to a proliferation of branches in the bifurcation manifolds. ...
Synchronization is an important phenomenon in a wide variety of systems comprising interacting oscillatory units, whether natural (like neurons, biochemical reactions, cardiac cells) or artificial (like metronomes, power grids, Josephson junctions). The Kuramoto model provides a simple description of these systems and has been useful in their mathematical exploration. Here we investigate this model combining two common features that have been observed in many systems: external periodic forcing and higher-order interactions among the elements. We show that the combination of these ingredients leads to a very rich bifurcation scenario that produces 11 different asymptotic states of the system, with competition between forced and spontaneous synchronization. We found, in particular, that saddle-node, Hopf and homoclinic manifolds are duplicated in regions of parameter space where the unforced system displays bi-stability.
... For a general description of synchronization with respect to higher-order interactions we refer to [4] (section 6) also [5] (section 4) and [9], and note that extensions of the Kuramoto model to higher-order networks have been extensively investigated, but generally with trajectories restricted to the unit circle S 1 [7, 10-14]. An exception is [15] where an extended D-dimensional Kuramoto model on the sphere is regarded as a three-body system with symmetric connectivity coefficients, in contrast to the antisymmetric couplings in our determinantal systems. As a point of comparison, it has been observed [15,16] that the behaviour of Kuramoto systems on the sphere depends on whether the dimension is even or odd, which is consistent with our observations. ...
... An exception is [15] where an extended D-dimensional Kuramoto model on the sphere is regarded as a three-body system with symmetric connectivity coefficients, in contrast to the antisymmetric couplings in our determinantal systems. As a point of comparison, it has been observed [15,16] that the behaviour of Kuramoto systems on the sphere depends on whether the dimension is even or odd, which is consistent with our observations. We find, for example, that synchronized nodes form equally spaced sequences of points on S d−1 , which for odd d = D are closed, whereas for even d the nodes form only a half-ring of points. ...
... Another point of comparison with previous work concerns multistability, which refers to the coexistence of stable multiple steady states, see the discussion in [17], chapter 3. Multistability has been observed in various higher-order models [8,10,11], and it is generally thought 'that higher-order interactions favour multistability' [4], but it remains an open question as to whether such properties are model-specific, or are general consequences of higher-order interactions. In fact, multistability does not appear in the models that we consider here, since the stable steady states are unique up to rotation, although 'extreme multistability' is known to occur in other higher-order models on the sphere [15]. Although this conclusion is based on numerical simulations, we prove it to be true for the special case N = d = 3 in section 3.3. ...
We construct a system of N interacting particles on the unit sphere in d-dimensional space, which has d-body interactions only. The equations have a gradient formulation derived from a rotationally-invariant potential of a determinantal form summed over all nodes, with antisymmetric coefficients. For d=3, for example, all trajectories lie on the 2-sphere and the potential is constructed from the triple scalar product summed over all oriented 2-simplices. We investigate the cases d=3,4,5 in detail, and find that the system synchronizes from generic initial values, for both positive and negative coupling coefficients, to a static final configuration in which the particles lie equally spaced on . Completely synchronized configurations also exist, but are unstable under the d-body interactions. We compare the relative effect of 2-body and d-body forces by adding the well-studied 2-body interactions to the potential, and find that higher-order interactions enhance the synchronization of the system, specifically, synchronization to a final configuration consisting of equally spaced particles occurs for all d-body and 2-body coupling constants of any sign, unless the attractive 2-body forces are sufficiently strong relative to the d-body forces. In this case the system completely synchronizes as the 2-body coupling constant increases through a positive critical value, with either a continuous transition for d=3, or discontinuously for d=5. Synchronization also occurs if the nodes have distributed natural frequencies of oscillation, provided that the frequencies are not too large in amplitude, even in the presence of repulsive 2-body interactions which by themselves would result in asynchronous behaviour.
... In recent years, research focuses on the system's hidden geometry features [3,4] and their impact on dynamics. Notably, new dynamical phenomena appear that can be related to the higher-order connectivity and interactions supported by the system's hidden geometry, which is mathematically described by simplicial complexes [5][6][7][8][9][10][11]. ...
... Hence, the structure-dynamics interplay can be expected both because of the pairwise and higher-order interactions due to the actual architecture of simplicial complexes. More precisely, it has been demonstrated by studies of spin kinetics [7,8], contagious dynamics [11], and synchronisation processes [9,10,32] on var-ious simplicial complexes. Notably, in the field-driven magnetisation reversal on simplicial complexes [7,8], the antiferromagnetic interactions via links of the triangle faces provide strong geometric frustration effects that determine the shape of the hysteresis loop. ...
... The nature of synchronisation transition can depend on the process' sensitivity to the sign of interactions, time delay, and the frustration effects causing new phenomena [36][37][38][39][40][41]. Furthermore, the presence of higherorder interactions are shown to induce an abrupt desynchronisation, depending on the dimension of the dynamical variable, and the range of couplings [9,10]. It remains unexplored how the coincidental interactions of a different order, encoded by the faces of a large simplicial complex, cooperate during the synchronisation processes. ...
Recent studies of dynamic properties in complex systems point out the profound impact of hidden geometry features known as simplicial complexes, which enable geometrically conditioned many-body interactions. Studies of collective behaviours on the controlled-structure complexes can reveal the subtle interplay of geometry and dynamics. Here, we investigate the phase synchronisation dynamics under the competing interactions embedded on 1-simplex (edges) and 2-simplex (triangles) faces of a homogeneous 4-dimensional simplicial complex. Its underlying network is a 1-hyperbolic graph with the assortative correlations among the node's degrees and the spectral dimension that exceeds . We determine the time-averaged system's order parameter to characterise the synchronisation level. In the absence of higher interactions, the complete synchrony is continuously reached with the increasing positive pairwise interactions (), and a partial synchronisation for the negative couplings () with no apparent hysteresis. Similar behaviour occurs in the degree-preserved randomised network. In contrast, the synchronisation is absent for the negative pairwise coupling in the entirely random graph and simple scale-free networks. Increasing the strength of the triangle-based interactions gradually hinders the synchronisation promoted by pairwise couplings, and the non-symmetric hysteresis loop opens with an abrupt desynchronisation transition towards the branch. However, for substantial triangle-based interactions, the frustration effects prevail, preventing the complete synchronisation, and the abrupt transition disappears. These findings shed new light on the mechanisms by which the high-dimensional simplicial complexes in natural systems, such as human connectomes, can modulate their native synchronisation processes.
... Apart from a collection of dyadic interactions, cliques can be also regarded as the pair-wise projection of richer substructures representing group interactions (also known as 'higher-order' interactions), involving more than two agents (nodes) at once. In fact, from few years on, a growing branch of literature dedicated to the study of various dynamical processes involving group interactions [12][13][14][15][16][17][18] , has been showing that such interactions can heavily affect the dynamics, and neglecting them can therefore lead to wrong predictions. ...
... Once all the coefficients have been found by insertion of Eq. (14) in the original linear system of 2 n − 1 equations, we substitute them in Eq. (13) , as the number of (g, r + 1)cliques incident on node i in DðKÞ, computed as k ...
... total number of neighbors of i within, respectively, (0, ⋅)-cliques and (1, ⋅)-cliques.Substituting Eq.(14) in Eq.(13), and doing some algebra and combinatorics, we getMP I ¼ DP Ið15Þwhere we have defined the matrices M and D, of elements M ij ¼ ∑ ...
Contagion processes have been proven to fundamentally depend on the structural properties of the interaction networks conveying them. Many real networked systems are characterized by clustered substructures representing either collections of all-to-all pair-wise interactions (cliques) and/or group interactions, involving many of their members at once. In this work, focusing on interaction structures represented as simplicial complexes, we present a discrete-time microscopic model of complex contagion for a susceptible-infected-susceptible dynamics. Introducing a particular edge clique cover and a heuristic to find it, the model accounts for the higher-order dynamical correlations among the members of the substructures (cliques/simplices). The analytical computation of the critical point reveals that higher-order correlations are responsible for its dependence on the higher-order couplings. While such dependence eludes any mean-field model, the possibility of a bi-stable region is extended to structured populations.
... 39,40 Studies have shown that the inclusion of higher order interactions have important consequences in the propagation of epidemics [41][42][43] and synchronization. [44][45][46][47][48][49][50][51] Pairwise, or two-body, interactions correspond to the notion of an "interaction graph." For higher order interactions, this concept is replaced by that of a simplicial complex or hypergraph, which, besides nodes and edges, also contains triangles, tetrahedra, etc. ...
... 29,45 and here. The Kuramoto model with symmetric higher order terms, on the other hand, exhibits multi-stability 26,[45][46][47] and anomalous transitions to synchrony. 30 In this case, the effects of external forcing might lead to a proliferation of branches in the bifurcation manifolds. ...
Synchronization is an important phenomenon in a wide variety of systems comprising interacting oscillatory units, whether natural (like neurons, biochemical reactions, and cardiac cells) or artificial (like metronomes, power grids, and Josephson junctions). The Kuramoto model provides a simple description of these systems and has been useful in their mathematical exploration. Here, we investigate this model by combining two common features that have been observed in many systems: External periodic forcing and higher-order interactions among the elements. We show that the combination of these ingredients leads to a very rich bifurcation scenario that produces 11 different asymptotic states of the system, with competition between forced and spontaneous synchronization. We found, in particular, that saddle-node, Hopf, and homoclinic manifolds are duplicated in regions of parameter space where the unforced system displays bi-stability.
... Quite recently, Chandra, Girvan, and Ott [17] systematically examined the dynamics of the D-dimensional generalized Kuramoto model with heterogeneous natural rotations; in particular, they unveiled that the nature of phase transition for the generalized Kuramoto model with the odd number of dimensions is remarkably different from that in even dimensions. Since then, there has been a burst of appealing works devoted to the study of the D-dimensional generalized Kuramoto model and its variants [18][19][20][21][22][23][24][25]. ...
... Under the weak coupling limit K → 0, our model reduces to the D-dimensional Kuramoto phase model, which is akin to a similar classic construction of the seminal Kuramoto phase model from weakly coupled two-dimensional limit-cycle oscillators [47,48]. In this sense, our work puts the recent studies regarding the D-dimensional Kuramoto model [17][18][19][20][21][22][23][24][25] on a stronger footing by providing a much more general framework to consider the previous results, owing to no longer being constrained by fixed amplitude dynamics. Thus, our model may find strong potential for actual applications in a wider range of physical, biological, and technological systems involving quenched random rotation axes and frequencies, such as leading to a deeper understanding of the collective motion in threedimensional swarming systems with helical trajectories [13], the spatiotemporal alignment of beating cilia [53], the ferromagnetic resonance in biomagnetism [54], etc. ...
We introduce a new model consisting of globally coupled high-dimensional generalized limit-cycle oscillators, which explicitly incorporates the role of amplitude dynamics of individual units in the collective dynamics. In the limit of weak coupling, our model reduces to the D-dimensional Kuramoto phase model, akin to a similar classic construction of the well-known Kuramoto phase model from weakly coupled two-dimensional limit-cycle oscillators. For the practically important case of D=3, the incoherence of the model is rigorously proved to be stable for negative coupling but unstable for positive coupling ; the locked states are shown to exist if ; in particular, the onset of amplitude death is theoretically predicted. For , the discrete and continuous spectra for both locked states and amplitude death are governed by two general formulas. Our proposed D-dimensional model is physically more reasonable, because it is no longer constrained by fixed amplitude dynamics, which puts the recent studies of the D-dimensional Kuramoto phase model on a stronger footing by providing a more general framework for D-dimensional limit-cycle oscillators.
... synchronous state from incoherence increases monotonically with λ 2 . (see panels b,d,f,h) [42,43]. Some other scenarios, instead, point to novel and relevant features of the system, which certainly deserve more detailed investigations: for instance the fact that, despite the simplicity of the model, the simple combination between pure 1 and 2-simplices make it possible for the system to exhibit mono-stability and bi-stability regions, as well as the fact that in the region of negative λ 1 and positive λ 2 the system features an abrupt transition from the fully synchronized state to the unsynchronized one. ...
... Some other scenarios, instead, point to novel and relevant features of the system, which certainly deserve more detailed investigations: for instance the fact that, despite the simplicity of the model, the simple combination between pure 1 and 2-simplices make it possible for the system to exhibit mono-stability and bi-stability regions, as well as the fact that in the region of negative λ 1 and positive λ 2 the system features an abrupt transition from the fully synchronized state to the unsynchronized one. Furthermore, it is evident that the theoretical predictions (contained in the SI) and the numerical results are in very good agreement, and therefore the developed theory allows to unveil the origin of bi-stability in the Kuramoto model, as reported also in various other settings and circumstances [42,43]. But the most remarkable evidence that Figure 2 is communicating is the presence of synchronization features in the backward transition for negative values of the coupling strengths. ...
We give evidence that a population of pure contrarians globally coupled D-dimensional Kuramoto oscillators reaches a collective synchronous state when the interplay between the units goes beyond the limit of pairwise interactions. An exact solution for the description of the microscopic dynamics for forward and backward transitions is provided, which entails imperfect symmetry breaking of the population into a frequency-locked state featuring two clusters of different instantaneous phases. Our results lift the veil towards unlocking the power full potential of group interactions entailing multi-dimensional choices and novel dynamical states in many circumstances, such as in social systems.
... Synchronization is an important dynamical behavior of complex systems, the study results of which can help people better understand and explain the synchronization phenomena in the real world. Synchronization is divided into controlled synchronization [18][19][20] and self-coupled synchronization [21][22][23] . Controlled synchronization means that the variables or systems are synchronized after added some control strategies. ...
Hyper-networks tend to perform better in representing multivariate relationships among nodes. Yet, due to the complexity of the hyper-network structure, research in synchronization dynamics is rarely involved. In this paper, a Kuramoto model more suitable for k-uniform hyper-networks is proposed. And the generalized Laplacian matrix expression of the k-uniform hyper-network is present. We use the eigenvalue ratio of the generalized Laplacian matrix to quantify synchronization. And we studied the effects of some important structure parameters on the synchronization of three types of k-uniform hyper-networks. And obtained different relationships between synchronization and these parameters. The results show the synchronization of the k-uniform hyper-networks is related to both structure and parameters. And as the size of the nodes increases, the synchronization ability gradually increases for ER random hyper-network, while that gradually decreases for NW small-world hyper-network and BA scale-free hyper-network. As the uniformity increases, the synchronization ability of all three types of uniform hyper-networks increases. In addition, when the structure and node size are fixed, the synchronization ability increases with the increase of the hyper-clustering coefficient in BA scale-free hyper-network and ER random hyper-network, while it decreases with the increase of the hyper-clustering coefficient in NW small-world hyper-network.
... Furthermore, in [24,27] it has been shown that by considering both pairwise and higher order interactions in the system, the latter ones can stabilize synchronized states even for repulsive pair-wise coupling, a situation prohibited in the classical Kuramoto model. Furthermore, it was recently shown that, even with both pairwise and triadic coupling being repulsive, the two-cluster state can exist when the triadic strength is sufficiently large [28]. ...
We have examined the synchronization and de-synchronization transitions observable in the Kuramoto model with a standard pair-wise first harmonic interaction plus a higher order (triadic) symmetric interaction for unimodal and bimodal Gaussian distributions of the natural frequencies. These transitions have been accurately characterized thanks to a self-consistent mean-field approach joined to accurate numerical simulations. The higher-oder interactions favour the formation of two cluster states, which emerge from the incoherent regime via continuous (discontinuos) transitions for unimodal (bimodal) distributions. Fully synchronized initial states give rise to two symmetric equally populated clusters at a angular distance , which increases for decreasing pair-wise couplings until it reaches (corresponding to an anti-phase configuration) where the cluster state disappears via a saddle-node bifurcation and reforms immediately after with a smaller angle . For bimodal distributions we have obtained detailed phase diagrams involving all the possible dynamical states in terms of standard and novel order parameters. In particular, the clustering order parameter, here introduced, appears quite suitable to characterize the two cluster regime. As a general aspect, hysteretic (non hysteretic) synchronization transitions, mostly mediated by the emergence of standing waves, are observable for attractive (repulsive) higher-order interactions.
... Hence, advanced topological structures, particularly simplicial complexes, are widely adopted to describe the many-body interactions, where asimplex is used to characterize a ( + 1)-body interaction [28,29]. Analyses and applications regarding the dynamics of these higherorder complex networks become recent hotspot and achieve fruitful theoretical and practical results in areas such as network synchronization [30][31][32][33][34], social contagion [35,36], protein interaction [37], https://doi. ...
Increasing evidences have demonstrated the essentiality of many-body interactions in modeling various systems in physics, biology, and social sciences. Control strategies are useful tools to steer real-world complex systems to desired targets. Existing works focus on open-loop control, which relies on predefined control signals, and closed-loop control, which requires an infinite-time duration, on conventional networks expressed by graphs. This work designs closed-loop controllers for higher-order complex networks characterized by simplicial complexes. Protocols including the linear feedback controller and a switching controller, which is a combination of a linear controller and a finite-time controller, are first adapted to higher-order networks to realize successful control tasks, with the rigorous upper bound of the control time theoretically derived and compared. Extensive numerical simulations on benchmark and real-world higher-order complex networks demonstrate the effectiveness of the control protocols and further provide insights on pinning control strategy to higher-order networks. These results shed light on a comprehensive discovery of controlling higher-order complex networks and have also applied values.
... The matrices W i become D × D anti-symmetric matrices containing the D(D − 1)/2 natural frequencies of each oscillator. It has been shown, in particular, [8] that the system exhibits discontinuous phase transitions in odd dimensions, which attracted a lot of attention [11][12][13]. ...
The Kuramoto model describes how coupled oscillators synchronize their phases as the intensity of the coupling increases past a threshold. The model was recently extended by reinterpreting the oscillators as particles moving on the surface of unit spheres in a D-dimensional space. Each particle is then represented by a D-dimensional unit vector; for D=2 the particles move on the unit circle and the vectors can be described by a single phase, recovering the original Kuramoto model. This multidimensional description can be further extended by promoting the coupling constant between the oscillators to a matrix K that acts on the unit vectors. As the coupling matrix changes the direction of the vectors, it can be interpreted as a generalized frustration, that tends to hinder synchronization. In a recent paper we have studied in detail the role of the coupling matrix for D=2. Here we extend this analysis to arbitrary dimensions. We show that, for identical oscillators, when the natural frequencies of all particles are set to zero, the system converges either to a stationary synchronized state, given by one of the real eigenvectors of K, or to an effective two-dimensional rotation, defined by one of the complex eigenvectors of K. The stability of these states depend on the set eigenvalues and eigenvectors of the coupling matrix, which controls the asymptotic behavior of the system and, therefore, can be used to manipulate these states. When the natural frequencies are not zero, synchronization depends on whether D is even or odd. In even dimensions the transition to synchronization is continuous and rotating states are replaced by active states, where the module of the order parameter oscillates while it rotates. If D is odd the phase transition is discontinuous and active states are suppressed for some distributions of natural frequencies.
... 23,24 These non-pairwise interactions include higher-order simplicial complexes involving simultaneous multinode interactions. [25][26][27] These more complex but also more insightful interaction mechanisms have recently been studied in many scientific studies, such as mathematics, 28 physics, 29 and computer science. 30 For instance, in Ref. ...
In neuronal network analysis on, for example, synchronization, it has been observed that the influence of interactions between pairwise nodes is essential. This paper further reveals that there exist higher-order interactions among multi-node simplicial complexes. Using a neuronal network of Rulkov maps, the impact of such higher-order interactions on network synchronization is simulated and analyzed. The results show that multi-node interactions can considerably enhance the Rulkov network synchronization, better than pairwise interactions, for involving more and more neurons in the network.
... Hypergraphs and simplicial complexes are instead the ideal framework to encapsulate the higher-order interactions taking place in complex systems, and can be viewed as generalizations of the concept of graphs, as they consider also manybody interactions encompassing more than two nodes [18]. Recently, local conditions for the emergence of a synchronous behaviour have been derived in [19], consensus-like dynamics have been studied [20]- [22], and a series of rich and novel behaviors were observed in networks of Kuramoto oscillators coupled through simplicial structures [23]- [25], suitable to describe undirected interactions. ...
A standard assumption in control of network dynamical systems is that its nodes interact through pairwise interactions, which can be described by means of a directed graph. However, in several contexts, multibody, directed interactions may occur, thereby requiring the use of directed hypergraphs rather then digraphs. For the first time, we propose a strategy, inspired by the classic pinning control on graphs, that is tailored for controlling network systems coupled through a directed hypergraph. By drawing an analogy with signed graphs, we provide sufficient conditions for controlling the network onto the desired trajectory provided by the pinner, and a dedicated algorithm to design the control hyperedges.
... Analysis of D ( ≥2)-dimensional Kuramoto dynamics on top of simplicial structures (1-and 2-simplices predominantly) is presented in [63]. Theoretical analysis and extensive numerical simulations are put forward wherein reasoning behind different synchronization patterns resulting from odd and even dimensions is explained [64]. ...
Network science has evolved into an indispensable platform for studying complex systems. But recent research has identified limits of classical networks, where links connect pairs of nodes, to comprehensively describe group interactions. Higher-order networks, where a link can connect more than two nodes, have therefore emerged as a new frontier in network science. Since group interactions are common in social, biological and technological systems, higher-order networks have recently led to important new discoveries across many fields of research. Here, we review these works, focusing in particular on the novel aspects of the dynamics that emerges on higher-order networks. We cover a variety of dynamical processes that have thus far been studied, including different synchronization phenomena, contagion processes, the evolution of cooperation and consensus formation. We also outline open challenges and promising directions for future research.
... Using phase reduction, it has been shown that the higher order interactions arise in generic oscillatory systems [23,24], which have significant influences on shaping collective dynamics that one can find beyond the typical pairwise interactions [25,26]. Investigating the dynamical effects induced by the simplicial complex has been a topic of high interest across a number of disciplines including brain dynamics, social interactions, data analysis, and so on [27][28][29][30][31][32][33]. ...
Many-body interactions between dynamical agents have caught particular attention in recent works that found wide applications in physics, neuroscience, and sociology. In this paper we investigate such higher order (nonadditive) interactions on collective dynamics in a system of globally coupled heterogeneous phase oscillators. We show that the three-body interactions encoded microscopically in nonlinear couplings give rise to added dynamic phenomena occurring beyond the pairwise interactions. The system in general displays an abrupt desynchronization transition characterized by irreversible explosive synchronization via an infinite hysteresis loop. More importantly, we give a mathematical argument that such an abrupt dynamic pattern is a universally expected effect. Furthermore, the origin of this abrupt transition is uncovered by performing a rigorous stability analysis of the equilibrium states, as well as by providing a detailed description of the spectrum structure of linearization around the steady states. Our work reveals a self-organized phenomenon that is responsible for the rapid switching to synchronization in diverse complex systems exhibiting critical transitions with nonpairwise interactions.
... A number of results along these lines have now been found. [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45] In particular, several researchers have explored a generalization of the Kuramoto model in which the oscillators move on spheres instead of the unit circle; the spheres could be either the ordinary twodimensional sphere or higher-dimensional spheres. In particular, a system of particles moving on the two-sphere has been used to model the orientation dynamics of swarms of drones flying around in three dimensions. ...
We study a system of N identical interacting particles moving on the unit sphere in d-dimensional space. The particles are self-propelled and coupled all to all, and their motion is heavily overdamped. For d=2, the system reduces to the classic Kuramoto model of coupled oscillators; for d=3, it has been proposed to describe the orientation dynamics of swarms of drones or other entities moving about in three-dimensional space. Here, we use group theory to explain the recent discovery that the model shows low-dimensional dynamics for all N≥3 and to clarify why it admits the analog of the Ott–Antonsen ansatz in the continuum limit N→∞. The underlying reason is that the system is intimately connected to the natural hyperbolic geometry on the unit ball Bd. In this geometry, the isometries form a Lie group consisting of higher-dimensional generalizations of the Möbius transformations used in complex analysis. Once these connections are realized, the reduced dynamics and the generalized Ott–Antonsen ansatz follow immediately. This framework also reveals the seamless connection between the finite and infinite-N cases. Finally, we show that special forms of coupling yield gradient dynamics with respect to the hyperbolic metric and use that fact to obtain global stability results about convergence to the synchronized state.
... Substituting Eq. (14) in Eq. (13), and doing some algebra and combinatorics, we get ...
Contagion processes have been proven to fundamentally depend on the structural properties of the interaction networks conveying them. Many real networked systems -- especially social ones -- are characterized by clustered substructures representing either collections of all-to-all pair-wise interactions (cliques) and/or group interactions, involving many of their members at once. In this work we focus on interaction structures represented as simplicial complexes, in which a group interaction is identified with a face. We present a microscopic discrete-time model of complex contagion for which a susceptible-infected-susceptible dynamics is considered. Introducing a particular edge clique cover and a heuristic to find it, the model accounts for the high-order state correlations among the members of the substructures (cliques/simplices). The mathematical tractability of the model allows for the analytical computation of the epidemic threshold, thus extending to structured populations some primary features of the critical properties of mean-field models. Overall, the model is found in remarkable agreement with numerical simulations.
Coupled map lattices are a popular and computationally simpler model of pattern formation in nonlinear systems. In this work, we investigate three-site interactions with linear multiplicative coupling in one-dimensional coupled logistic maps that cannot be decomposed into pairwise interactions. We observe the transition to synchronization and the transition to long-range order in space. We coarse-grain the phase space in regions and denote them by spin values. We use two quantifiers the flip rate [Formula: see text] that quantify departure from expected band-periodicity as an order parameter. We also study a non-Markovian quantity, known as persistence [Formula: see text] to study dynamic phase transitions. Following transitions are observed. (a) Transition to two band attractor state: At this transition [Formula: see text] as well as [Formula: see text] shows a power-law decay in the range of coupling parameters. Here all sites reach one of the bands. The [Formula: see text] as well as [Formula: see text] decays as power-law with the decay exponent [Formula: see text] and [Formula: see text], respectively. (b) The transition from a fluctuating chaotic state to a homogeneous synchronized fixed point: Here both the quantifiers [Formula: see text] and [Formula: see text] show power-law decay with decay exponent [Formula: see text] and [Formula: see text], respectively. We compare the transitions with the case, where pairwise interactions are also present. The spatiotemporal evolution is analyzed as the coupling parameter is varied.
We study the asymptotic dynamics of the high-dimensional Kuramoto oscillators on the unit sphere with two- and three-body interactions that trigger competition between synchrony and non-synchrony. In this work, we find a critical threshold between interaction strengths for complete synchronizability. Moreover, critical slowing down is observed at this phase transition. Our main theorems are supplemented by several numerical experiments, which provide a qualitative insight beyond theoretical results.
We use Clifford's geometric algebra to extend the Stuart-Landau system to dimensions D>2 and give an exact solution of the oscillator equations in the general case. At the supercritical Hopf bifurcation marked by a transition from stable fixed-point dynamics to oscillatory motion, the Jacobian matrix evaluated at the fixed point has N=⌊D/2⌋ pairs of complex conjugate eigenvalues which cross the imaginary axis simultaneously. For odd D there is an additional purely real eigenvalue that does the same. Oscillatory dynamics is asymptotically confined to a hypersphere SD−1 and is characterised by extreme multistability, namely the coexistence of an infinite number of limiting orbits, each of which has the geometry of a torus TN on which the motion is either periodic or quasiperiodic. We also comment on similar Clifford extensions of other limit cycle oscillator systems and their generalizations.
This paper investigates a higher-order dynamical network on simplicial complexes, and the evolution of topological signals residing on simplices (e.g. nodes, links, triangles, etc) is governed by coupled higher-dimensional linear oscillators. We uncover the stability conditions of cluster synchronization of both the higher-order network and its projected systems corresponding to solenoidal and irrotational component of the dynamics, and the synchronization phenomena of the higher-order network and its projected systems can co-exist (i.e., symbiotic synchronization). In the framework of simplicial complexes, this study indicates that there are essential differences between the collective behaviours of phase oscillators and higher-dimensional oscillators.
We introduce a new model consisting of globally coupled high-dimensional generalized limit-cycle oscillators, which explicitly incorporates the role of amplitude dynamics of individual units in the collective dynamics. In the limit of weak coupling, our model reduces to the D-dimensional Kuramoto phase model, akin to a similar classic construction of the well-known Kuramoto phase model from weakly coupled two-dimensional limit-cycle oscillators. For the practically important case of D=3, the incoherence of the model is rigorously proved to be stable for negative coupling (K<0) but unstable for positive coupling (K>0); the locked states are shown to exist if K>0; in particular, the onset of amplitude death is theoretically predicted. For D≥2, the discrete and continuous spectra for both locked states and amplitude death are governed by two general formulas. Our proposed D-dimensional model is physically more reasonable, because it is no longer constrained by fixed amplitude dynamics, which puts the recent studies of the D-dimensional Kuramoto phase model on a stronger footing by providing a more general framework for D-dimensional limit-cycle oscillators.
Synchronization is one of the interesting collective behaviors of oscillators. It refers to a phenomenon in which some dynamically connected systems behave the same. In dynamic systems, as the bifurcation points of the system approach, the system slows down and returns to its steady-state later with a slight disturbance. The system's slowness before bifurcation points is called critical slowing down, which can be measured using early warning (EW) indicators. This paper considers two dynamic systems: the Kuramoto model and the Rössler system. Two networks are considered to show continuous and explosive synchronization in each system. Then the synchronization bifurcation point (SBP) of each case is indicated using the EW signals calculated using the time series of synchronization measures. EW signals measure the slowness in the synchronous benchmark time series. Two EW signals, skewness and kurtosis, are applied. The results show that the SBPs in various cases and systems can be predicted using the EW signals.
Higher-order interaction exists widely in real-world networks and dramatically affects the system’s phase transition and critical phenomena. In the study of epidemic spreading dynamics on higher-order networks, the discontinuous phase transition and bistable physical phenomena have attracted the attention of many scholars. However, epidemics are rarely spread individually but are spread on high-order networks so that multiple epidemics interact mutually. The dynamics of promoting transmission on higher-order networks still lack research. This paper explored the cooperative spreading of two epidemics in the simplicial complex, which simultaneously includes structural and dynamical reinforcement effects. We use the Markov chain and mean-field methods to predict the outbreak size and threshold for theoretical analysis. Increase the structural or dynamical reinforcement effect, and the spreading dynamics are promoted, i.e., a more significant outbreak size and smaller threshold. When the two reinforcement effects are strong, the system will have a discontinuous phase transition and hysteresis loop, and a large initial seed size will lead to the easy outbreak of the epidemic. For the greater initial seed size, the dynamical reinforcement effect affects the outbreak size in most cases.
Brain hubs are best connected central nodes in the human connectome that play a critical role in integrated brain dynamics. How the hubs perform their function vs different dynamical processes and the role of their higher-order connections involving different brain regions remains elusive. Here we simulate the phase synchronisation processes on the human connectome core network consisting of the eight brain hubs and all attached simplexes of different sizes. The leading pairwise interactions among neighbouring nodes are assumed, taking into account the natural weights of edges. Our results reveal that increasing the positive pairwise couplings promotes a continuous synchronisation while a weak partial synchronisation occurs for a wide range of negative couplings. The weights of edges stabilise the synchronisation process supporting the absence of hysteresis. Furthermore, the time evolution of the order parameter shows cyclic fluctuations induced by the concurrent evolution of phases associated with different groups of nodes. We show that these oscillations exhibit long-range temporal correlations and multifractality. The asymmetrical singularity spectra are determined, which vary with the time scale and depend on the weights of edges. These findings suggest a possible way that the brain functional geometry maintains a desirable low-level synchrony through complex patterns of phase fluctuations.
The Laplacian eigenvalue spectrum of a complex network contains a great deal of information about the network topology and dynamics, particularly affecting the network synchronization process and performance. This article briefly reviews the recent progress in the studies of network synchronizability, regarding its spectral criteria and topological optimization, and explores the role of higher-order topologies in measuring the optimal synchronizability of large-scale complex networks.
We consider systems of N particles interacting on the unit sphere in d-dimensional space with dynamics defined as the gradient flow of rotationally invariant potentials. The Kuramoto model on the sphere is a well-studied example of such a system but allows only pairwise interactions. Using the Kuramoto model as a guide, we construct n-body potentials from products and sums of rotation invariants, namely, bilinear inner products and multilinear determinants, which lead to a wide variety of higher-order systems with differing synchronization characteristics. The connectivity coefficients, which determine the strength of interaction between any set of n distinct nodes, have mixed symmetries, which follow from those of the symmetric inner product and the antisymmetric determinant. We investigate n-body systems in detail for n⩽6, both as isolated systems and in combination with lower-order systems, and analyze their properties as functions of the coupling constants. We show by example that in many cases, multistable states appear only when we forbid self-interactions within the system.
We give evidence that a population of pure contrarian globally coupled D-dimensional Kuramoto oscillators reaches a collective synchronous state when the interplay between the units goes beyond the limit of pairwise interactions. Namely, we will show that the presence of higher-order interactions may induce the appearance of a coherent state even when the oscillators are coupled negatively to the mean field. An exact solution for the description of the microscopic dynamics for forward and backward transitions is provided, which entails imperfect symmetry breaking of the population into a frequency-locked state featuring two clusters of different instantaneous phases. Our results contribute to a better understanding of the powerful potential of group interactions entailing multidimensional choices and novel dynamical states in many circumstances, such as in social systems.
From social interactions to the human brain, higher-order networks are key to describe the underlying network geometry and topology of many complex systems. While it is well known that network structure strongly affects its function, the role that network topology and geometry has on the emerging dynamical properties of higher-order networks is yet to be clarified. In this perspective, the spectral dimension plays a key role since it determines the effective dimension for diffusion processes on a network. Despite its relevance, a theoretical understanding of which mechanisms lead to a finite spectral dimension, and how this can be controlled, still represents a challenge and is the object of intense research. Here, we introduce two nonequilibrium models of hyperbolic higher-order networks and we characterize their network topology and geometry by investigating the intertwined appearance of small-world behavior, δ-hyperbolicity, and community structure. We show that different topological moves, determining the nonequilibrium growth of the higher-order hyperbolic network models, induce tuneable values of the spectral dimension, showing a rich phenomenology which is not displayed in random graph ensembles. In particular, we observe that, if the topological moves used to construct the higher-order network increase the area/volume ratio, then the spectral dimension continuously decreases, while the opposite effect is observed if the topological moves decrease the area/volume ratio. Our work reveals a new link between the geometry of a network and its diffusion properties, contributing to a better understanding of the complex interplay between network structure and dynamics.
Recent studies of dynamic properties in complex systems point out the profound impact of hidden geometry features known as simplicial complexes, which enable geometrically conditioned many-body interactions. Studies of collective behaviors on the controlled-structure complexes can reveal the subtle interplay of geometry and dynamics. Here we investigate the phase synchronization (Kuramoto) dynamics under the competing interactions embedded on 1-simplex (edges) and 2-simplex (triangles) faces of a homogeneous four-dimensional simplicial complex. Its underlying network is a 1-hyperbolic graph with the assortative correlations among the node's degrees and the spectral dimension that exceeds ds=4. By numerically solving the set of coupled equations for the phase oscillators associated with the network nodes, we determine the time-averaged system's order parameter to characterize the synchronization level. Our results reveal a variety of synchronization and desynchronization scenarios, including partially synchronized states and nonsymmetrical hysteresis loops, depending on the sign and strength of the pairwise interactions and the geometric frustrations promoted by couplings on triangle faces. For substantial triangle-based interactions, the frustration effects prevail, preventing the complete synchronization and the abrupt desynchronization transition disappears. These findings shed new light on the mechanisms by which the high-dimensional simplicial complexes in natural systems, such as human connectomes, can modulate their native synchronization processes.
Synchronization processes play critical roles in the functionality of a wide range of both natural and man-made systems. Recent work in physics and neuroscience highlights the importance of higher-order interactions between dynamical units, i.e., three- and four-way interactions in addition to pairwise interactions, and their role in shaping collective behavior. Here we show that higher-order interactions between coupled phase oscillators, encoded microscopically in a simplicial complex, give rise to added nonlinearity in the macroscopic system dynamics that induces abrupt synchronization transitions via hysteresis and bistability of synchronized and incoherent states. Moreover, these higher-order interactions can stabilize strongly synchronized states even when the pairwise coupling is repulsive. These findings reveal a self-organized phenomenon that may be responsible for the rapid switching to synchronization in many biological and other systems that exhibit synchronization without the need of particular correlation mechanisms between the oscillators and the topological structure.
The dynamics of network social contagion processes such as opinion formation and epidemic spreading are often mediated by interactions between multiple nodes. Previous results have shown that these higher-order interactions can profoundly modify the dynamics of contagion processes, resulting in bistability, hysteresis, and explosive transitions. In this paper, we present and analyze a hyperdegree-based mean-field description of the dynamics of the susceptible-infected-susceptible model on hypergraphs, i.e., networks with higher-order interactions, and illustrate its applicability with the example of a hypergraph where contagion is mediated by both links (pairwise interactions) and triangles (three-way interactions). We consider various models for the organization of link and triangle structures and different mechanisms of higher-order contagion and healing. We find that explosive transitions can be suppressed by heterogeneity in the link degree distribution when links and triangles are chosen independently or when link and triangle connections are positively correlated when compared to the uncorrelated case. We verify these results with microscopic simulations of the contagion process and with analytic predictions derived from the mean-field model. Our results show that the structure of higher-order interactions can have important effects on contagion processes on hypergraphs.
The complexity of many biological, social and technological systems stems from the richness of the interactions among their units. Over the past decades, a great variety of complex systems has been successfully described as networks whose interacting pairs of nodes are connected by links. Yet, in face-to-face human communication, chemical reactions and ecological systems, interactions can occur in groups of three or more nodes and cannot be simply described just in terms of simple dyads. Until recently, little attention has been devoted to the higher-order architecture of real complex systems. However, a mounting body of evidence is showing that taking the higher-order structure of these systems into account can greatly enhance our modeling capacities and help us to understand and predict their emerging dynamical behaviors. Here, we present a complete overview of the emerging field of networks beyond pairwise interactions. We first discuss the methods to represent higher-order interactions and give a unified presentation of the different frameworks used to describe higher-order systems, highlighting the links between the existing concepts and representations. We review both the measures designed to characterize the structure of these systems, and the models proposed in the literature to generate synthetic structures, such as random and growing simplicial complexes, bipartite graphs and hypergraphs. We then introduce and discuss the rapidly growing research on higher-order dynamical systems and on dynamical topology. We focus on novel emergent phenomena characterizing landmark dynamical processes, such as diffusion, spreading, synchronization and games, when extended beyond pairwise interactions. We elucidate the relations between higher-order topology and dynamical properties, and conclude with a summary of empirical applications, providing an outlook on current modeling and conceptual frontiers.
We present an analytical description for the collective dynamics of oscillator ensembles with higher-order coupling encoded by simplicial structure, which serves as an illustrative and insightful paradigm for brain function and information storage. The novel dynamics of the system, including abrupt desynchronization and multistability, are rigorously characterized and the critical points that correspond to a continuum of first-order phase transitions are found to satisfy universal scaling properties. More importantly, the underlying bifurcation mechanism giving rise to multiple clusters with arbitrary ensemble size is characterized using a rigorous spectral analysis of the stable cluster states. As a consequence of SO2 group symmetry, we show that the continuum of abrupt desynchronization transitions result from the instability of a collective mode under the nontrivial antisymmetric manifold in the high-dimensional phase space.
Simplicial complexes constitute the underlying topology of interacting complex systems including among the others brain and social interaction networks. They are generalized network structures that allow to go beyond the framework of pairwise interactions and to capture the many-body interactions between two or more nodes strongly affecting dynamical processes. In fact, the simplicial complexes topology allows to assign a dynamical variable not only to the nodes of the interacting complex systems but also to links, triangles, and so on. Here we show evidence that the dynamics defined on simplices of different dimensions can be significantly different even if we compare dynamics of simplices belonging to the same simplicial complex. By investigating the spectral properties of the simplicial complex model called ‘network geometry with flavor’ (NGF) we provide evidence that the up and down higher-order Laplacians can have a finite spectral dimension whose value depends on the order of the Laplacian. Finally we discuss the implications of this result for higher-order diffusion defined on simplicial complexes showing that the n -order diffusion dynamics have a return type distribution that can depends on n as it is observed in NGFs.
Chimera states arising in the classic Kuramoto system of two-dimensional phase-coupled oscillators are transient but they are “long” transients in the sense that the average transient lifetime grows exponentially with the system size. For reasonably large systems, e.g., those consisting of a few hundred oscillators, it is infeasible to numerically calculate or experimentally measure the average lifetime, so the chimera states are practically permanent. We find that small perturbations in the third dimension, which make system “slightly” three dimensional, will reduce dramatically the transient lifetime. In particular, under such a perturbation, the practically infinite average transient lifetime will become extremely short because it scales with the magnitude of the perturbation only logarithmically. Physically, this means that a reduction in the perturbation strength over many orders of magnitude, insofar as it is not zero, would result in only an incremental increase in the lifetime. The uncovered type of fragility of chimera states raises concerns about their observability in physical systems.
In the past 20 years network science has proven its strength in modeling many real-world interacting systems as generic agents, the nodes, connected by pairwise edges. Nevertheless, in many relevant cases, interactions are not pairwise but involve larger sets of nodes at a time. These systems are thus better described in the framework of hypergraphs, whose hyperedges effectively account for multibody interactions. Here we propose and study a class of random walks defined on such higher-order structures and grounded on a microscopic physical model where multibody proximity is associated with highly probable exchanges among agents belonging to the same hyperedge. We provide an analytical characterization of the process, deriving a general solution for the stationary distribution of the walkers. The dynamics is ultimately driven by a generalized random-walk Laplace operator that reduces to the standard random-walk Laplacian when all the hyperedges have size 2 and are thus meant to describe pairwise couplings. We illustrate our results on synthetic models for which we have full control of the high-order structures and on real-world networks where higher-order interactions are at play. As the first application of the method, we compare the behavior of random walkers on hypergraphs to that of traditional random walkers on the corresponding projected networks, drawing interesting conclusions on node rankings in collaboration networks. As the second application, we show how information derived from the random walk on hypergraphs can be successfully used for classification tasks involving objects with several features, each one represented by a hyperedge. Taken together, our work contributes to unraveling the effect of higher-order interactions on diffusive processes in higher-order networks, shedding light on mechanisms at the heart of biased information spreading in complex networked systems.
Complex networks have been successfully used to describe the spread of diseases in populations of interacting individuals. Conversely, pairwise interactions are often not enough to characterize social contagion processes such as opinion formation or the adoption of novelties, where complex mechanisms of influence and reinforcement are at work. Here we introduce a higher-order model of social contagion in which a social system is represented by a simplicial complex and contagion can occur through interactions in groups of different sizes. Numerical simulations of the model on both empirical and synthetic simplicial complexes highlight the emergence of novel phenomena such as a discontinuous transition induced by higher-order interactions. We show analytically that the transition is discontinuous and that a bistable region appears where healthy and endemic states co-exist. Our results help explain why critical masses are required to initiate social changes and contribute to the understanding of higher-order interactions in complex systems.
The Kuramoto model, originally proposed to model the dynamics of many interacting oscillators, has been used and generalized for a wide range of applications involving the collective behavior of large heterogeneous groups of dynamical units whose states are characterized by a scalar angle variable. One such application in which we are interested is the alignment of orientation vectors among members of a swarm. Despite being commonly used for this purpose, the Kuramoto model can only describe swarms in two dimensions, and hence the results obtained do not apply to the often relevant situation of swarms in three dimensions. Partly based on this motivation, as well as on relevance to the classical, mean-field, zero-temperature Heisenberg model with quenched site disorder, in this paper we study the Kuramoto model generalized to D dimensions. We show that in the important case of three dimensions, as well as for any odd number of dimensions, the D-dimensional generalized Kuramoto model for heterogeneous units has dynamics that are remarkably different from the dynamics in two dimensions. In particular, for odd D the transition to coherence occurs discontinuously as the interunit coupling constant K is increased through zero, as opposed to the D=2 case (and, as we show, also the case of even D) for which the transition to coherence occurs continuously as K increases through a positive critical value K_{c}. We also demonstrate the qualitative applicability of our results to related models constructed specifically to capture swarming and flocking dynamics in three dimensions.
We provide a short introduction to the field of topological data analysis (TDA) and discuss its possible relevance for the study of complex systems. TDA provides a set of tools to characterise the shape of data, in terms of the presence of holes or cavities between the points. The methods, based on the notion of simplicial complexes, generalise standard network tools by naturally allowing for many-body interactions and providing results robust under continuous deformations of the data. We present strengths and weaknesses of current methods, as well as a range of empirical studies relevant to the field of complex systems, before identifying future methodological challenges to help understand the emergence of collective phenomena.
Networks provide a powerful formalism for modeling complex systems, by representing the underlying set of pairwise interactions. But much of the structure within these systems involves interactions that take place among more than two nodes at once; for example, communication within a group rather than person-to-person, collaboration among a team rather than a pair of co-authors, or biological interaction between a set of molecules rather than just two. We refer to these type of simultaneous interactions on sets of more than two nodes as higher-order interactions; they are ubiquitous, but the empirical study of them has lacked a general framework for evaluating higher-order models. Here we introduce such a framework, based on link prediction, a fundamental problem in network analysis. The traditional link prediction problem seeks to predict the appearance of new links in a network, and here we adapt it to predict which (larger) sets of elements will have future interactions. We study the temporal evolution of 19 datasets from a variety of domains, and use our higher-order formulation of link prediction to assess the types of structural features that are most predictive of new multi-way interactions. Among our results, we find that different domains vary considerably in their distribution of higher-order structural parameters, and that the higher-order link prediction problem exhibits some fundamental differences from traditional pairwise link prediction, with a greater role for local rather than long-range information in predicting the appearance of new interactions.
The dynamics of networks of neuronal cultures has been recently shown to be strongly dependent on the network geometry and in particular on their dimensionality. However, this phenomenon has been so far mostly unexplored from the theoretical point of view. Here we reveal the rich interplay between network geometry and synchronization of coupled oscillators in the context of a simplicial complex model of manifolds called Complex Network Manifold. The networks generated by this model combine small world properties (infinite Hausdorff dimension) and a high modular structure with finite and tunable spectral dimension. We show that the networks display frustrated synchronization for a wide range of the coupling strength of the oscillators, and that the synchronization properties are directly affected by the spectral dimension of the network.
Synchronization occurs in many natural and technological systems, from cardiac pacemaker cells to coupled lasers. In the synchronized state, the individual cells or lasers coordinate the timing of their oscillations, but they do not move through space. A complementary form of self-organization occurs among swarming insects, flocking birds, or schooling fish; now the individuals move through space, but without conspicuously altering their internal states. Here we explore systems in which both synchronization and swarming occur together. Specifically, we consider oscillators whose phase dynamics and spatial dynamics are coupled. We call them swarmalators, to highlight their dual character. A case study of a generalized Kuramoto model predicts five collective states as possible long-term modes of organization. These states may be observable in groups of sperm, Japanese tree frogs, colloidal suspensions of magnetic particles, and other biological and physical systems in which self-assembly and synchronization interact.
The structure of scientific collaborations has been the object of intense study both for its importance for innovation and scientific advancement, and as a model system for social group coordination and formation thanks to the availability of authorship data. Over the last years, complex networks approach to this problem have yielded important insights and shaped our understanding of scientific communities. In this paper we propose to complement the picture provided by network tools with that coming from using simplicial descriptions of publications and the corresponding topological methods. We show that it is natural to extend the concept of triadic closure to simplicial complexes and show the presence of strong simplicial closure. Focusing on the differences between scientific fields, we find that, while categories are characterized by different collaboration size distributions, the distributions of how many collaborations to which an author is able to participate is conserved across fields pointing to underlying attentional and temporal constraints. We then show that homological cycles, that can intuitively be thought as hole in the network fabric, are an important part of the underlying community linking structure.
Ecologists have long sought a way to explain how the remarkable biodiversity observed in nature is maintained. On the one hand, simple models of interacting competitors cannot produce the stable persistence of very large ecological communities. On the other hand, neutral models, in which species do not interact and diversity is maintained by immigration and speciation, yield unrealistically small fluctuations in population abundance, and a strong positive correlation between a species' abundance and its age, contrary to empirical evidence. Models allowing for the robust persistence of large communities of interacting competitors are lacking. Here we show that very diverse communities could persist thanks to the stabilizing role of higher-order interactions, in which the presence of a species influences the interaction between other species. Although higher-order interactions have been studied for decades, their role in shaping ecological communities is still unclear. The inclusion of higher-order interactions in competitive network models stabilizes dynamics, making species coexistence robust to the perturbation of both population abundance and parameter values. We show that higher-order interactions have strong effects in models of closed ecological communities, as well as of open communities in which new species are constantly introduced. In our framework, higher-order interactions are completely defined by pairwise interactions, facilitating empirical parameterization and validation of our models.
The lack of a formal link between neural network structure and its emergent function has hampered our understanding of how the brain processes information. We have now come closer to describing such a link by taking the direction of synaptic transmission into account, constructing graphs of a network that reflect the direction of information flow, and analyzing these directed graphs using algebraic topology. Applying this approach to a local network of neurons in the neocortex revealed a remarkably intricate and previously unseen topology of synaptic connectivity. The synaptic network contains an abundance of cliques of neurons bound into cavities that guide the emergence of correlated activity. In response to stimuli, correlated activity binds synaptically connected neurons into functional cliques and cavities that evolve in a stereotypical sequence toward peak complexity. We propose that the brain processes stimuli by forming increasingly complex functional cliques and cavities.
Synchronization of an ensemble of oscillators is an emergent phenomenon
present in several complex systems, ranging from social and physical to
biological and technological systems. The most successful approach to describe
how coherent behavior emerges in these complex systems is given by the
paradigmatic Kuramoto model. This model has been traditionally studied in
complete graphs. However, besides being intrinsically dynamical, complex
systems present very heterogeneous structure, which can be represented as
complex networks. This report is dedicated to review main contributions in the
field of synchronization in networks of Kuramoto oscillators. In particular, we
provide an overview of the impact of network patterns on the local and global
dynamics of coupled phase oscillators. We cover many relevant topics, which
encompass a description of the most used analytical approaches and the analysis
of several numerical results. Furthermore, we discuss recent developments on
variations of the Kuramoto model in networks, including the presence of noise
and inertia. The rich potential for applications is discussed for special
fields in engineering, neuroscience, physics and Earth science. Finally, we
conclude by discussing problems that remain open after the last decade of
intensive research on the Kuramoto model and point out some promising
directions for future research.
Detecting meaningful structure in neural activity and connectivity data is
challenging in the presence of hidden nonlinearities, where traditional
eigenvalue-based methods may be misleading. We introduce a novel approach to
matrix analysis, called clique topology, that extracts features of the data
invariant under nonlinear monotone transformations. These features can be used
to detect both random and geometric structure, and depend only on the relative
ordering of matrix entries. We then analyzed the activity of pyramidal neurons
in rat hippocampus, recorded while the animal was exploring a two-dimensional
environment, and confirmed that our method is able to detect geometric
organization using only the intrinsic pattern of neural correlations.
Remarkably, we found similar results during non-spatial behaviors such as wheel
running and REM sleep. This suggests that the geometric structure of
correlations is shaped by the underlying hippocampal circuits, and is not
merely a consequence of position coding. We propose that clique topology is a
powerful new tool for matrix analysis in biological settings, where the
relationship of observed quantities to more meaningful variables is often
nonlinear and unknown.
From the formation of animal flocks to the emergence of coordinated motion in bacterial swarms, populations of motile organisms at all scales display coherent collective motion. This consistent behaviour strongly contrasts with the difference in communication abilities between the individuals. On the basis of this universal feature, it has been proposed that alignment rules at the individual level could solely account for the emergence of unidirectional motion at the group level. This hypothesis has been supported by agent-based simulations. However, more complex collective behaviours have been systematically found in experiments, including the formation of vortices, fluctuating swarms, clustering and swirling. All these (living and man-made) model systems (bacteria, biofilaments and molecular motors, shaken grains and reactive colloids) predominantly rely on actual collisions to generate collective motion. As a result, the potential local alignment rules are entangled with more complex, and often unknown, interactions. The large-scale behaviour of the populations therefore strongly depends on these uncontrolled microscopic couplings, which are extremely challenging to measure and describe theoretically. Here we report that dilute populations of millions of colloidal rolling particles self-organize to achieve coherent motion in a unique direction, with very few density and velocity fluctuations. Quantitatively identifying the microscopic interactions between the rollers allows a theoretical description of this polar-liquid state. Comparison of the theory with experiment suggests that hydrodynamic interactions promote the emergence of collective motion either in the form of a single macroscopic 'flock', at low densities, or in that of a homogenous polar phase, at higher densities. Furthermore, hydrodynamics protects the polar-liquid state from the giant density fluctuations that were hitherto considered the hallmark of populations of self-propelled particles. Our experiments demonstrate that genuine physical interactions at the individual level are sufficient to set homogeneous active populations into stable directed motion.
The emergence of explosive synchronization has been reported as an abrupt
transition in complex networks of first-order Kuramoto oscillators. In this
Letter, we demonstrate that the nodes in a second-order Kuramoto model, perform
a cascade of transitions toward a synchronous macroscopic state, which is a
novel phenomenon that we call \textit{cluster explosive synchronization}. We
provide a rigorous analytical treatment using a mean-field analysis in
uncorrelated networks. Our findings are in good agreement with numerical
simulations and fundamentally deepen the understanding of microscopic
mechanisms toward synchronization.
An imperative condition for the functioning of a power-grid network is that its power generators remain synchronized. Disturbances can prompt desynchronization, which is a process that has been involved in large power outages. Here we derive a condition under which the desired synchronous state of a power grid is stable, and use this condition to identify tunable parameters of the generators that are determinants of spontaneous synchronization. Our analysis gives rise to an approach to specify parameter assignments that can enhance synchronization of any given network, which we demonstrate for a selection of both test systems and real power grids. Because our results concern spontaneous synchronization, they are relevant both for reducing dependence on conventional control devices, thus offering an additional layer of protection given that most power outages involve equipment or operational errors, and for contributing to the development of "smart grids" that can recover from failures in real time.
We study a variety of mixed synchronous/incoherent ("chimera") states in sev-eral heterogeneous networks of coupled phase oscillators. For each network, the recently-discovered Ott-Antonsen ansatz is used to reduce the number of variables in the PDE governing the evolution of the probability density function by one, re-sulting in a time-evolution PDE for a variable with as many spatial dimensions as the network. Bifurcation analysis is performed on the steady states of these PDEs. The results emphasise the commonality of the dynamics of the different networks, and provide stability information that was previously inferred.
We describe a new model for synchronization of neuronal oscillators that is based on the observation that certain species of fireflies are able to alter their free-running period. We show that by adding adaptation to standard oscillator models it is possible to observe the frequency alteration. One consequence of this is the perfect synchrony between coupled oscillators. Stability and some analytic results are included along with numerical simulations.
We study collaboration networks in terms of evolving, self-organizing
bipartite graph models. We propose a model of a growing network,
which combines preferential edge attachment with the bipartite structure,
generic for collaboration networks. The model depends exclusively
on basic properties of the network, such as the total number of collaborators
and acts of collaboration, the mean size of collaborations, etc.
The simplest model defined within this framework already allows us
to describe many of the main topological characteristics (degree
distribution, clustering coefficient, etc.) of one-mode projections
of several real collaboration networks, without parameter fitting.
We explain the observed dependence of the local clustering on degree
and the degree\degree correlations in terms of the "aging" of
collaborators and their physical impossibility to participate in
an unlimited number of collaborations.
Collective motion, where large numbers of individuals move synchronously together, is achieved when individuals adopt interaction rules that determine how they respond to their neighbors' movements and positions. These rules determine how group-living animals move, make decisions, and transmit information between individuals. Nonetheless, few studies have explicitly determined these interaction rules in moving groups, and very little is known about the interaction rules of fish. Here, we identify three key rules for the social interactions of mosquitofish (Gambusia holbrooki): (i) Attraction forces are important in maintaining group cohesion, while we find only weak evidence that fish align with their neighbor's orientation; (ii) repulsion is mediated principally by changes in speed; (iii) although the positions and directions of all shoal members are highly correlated, individuals only respond to their single nearest neighbor. The last two of these rules are different from the classical models of collective animal motion, raising new questions about how fish and other animals self-organize on the move.
We investigate chimera states in a ring of identical phase oscillators coupled in a time-delayed and spatially nonlocal fashion. We find novel clustered chimera states that have spatially distributed phase coherence separated by incoherence with adjacent coherent regions in antiphase. The existence of such time-delay induced phase clustering is further supported through solutions of a generalized functional self-consistency equation of the mean field. Our results highlight an additional mechanism for cluster formation that may find wider practical applications.
We consider a generalization of the Kuramoto model in which the oscillators are coupled to the mean field with random signs. Oscillators with positive coupling are "conformists"; they are attracted to the mean field and tend to synchronize with it. Oscillators with negative coupling are "contrarians"; they are repelled by the mean field and prefer a phase diametrically opposed to it. The model is simple and exactly solvable, yet some of its behavior is surprising. Along with the stationary states one might have expected (a desynchronized state, and a partially-synchronized state, with conformists and contrarians locked in antiphase), it also displays a traveling wave, in which the mean field oscillates at a frequency different from the population's mean natural frequency.
Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included.
We analyze a large system of globally coupled phase oscillators whose natural frequencies are bimodally distributed. The dynamics of this system has been the subject of long-standing interest. In 1984 Kuramoto proposed several conjectures about its behavior; ten years later, Crawford obtained the first analytical results by means of a local center manifold calculation. Nevertheless, many questions have remained open, especially about the possibility of global bifurcations. Here we derive the system's stability diagram for the special case where the bimodal distribution consists of two equally weighted Lorentzians. Using an ansatz recently discovered by Ott and Antonsen, we show that in this case the infinite-dimensional problem reduces exactly to a flow in four dimensions. Depending on the parameters and initial conditions, the long-term dynamics evolves to one of three states: incoherence, where all the oscillators are desynchronized; partial synchrony, where a macroscopic group of phase-locked oscillators coexists with a sea of desynchronized ones; and a standing wave state, where two counter-rotating groups of phase-locked oscillators emerge. Analytical results are presented for the bifurcation boundaries between these states. Similar results are also obtained for the case in which the bimodal distribution is given by the sum of two Gaussians.
Simple models of infectious diseases tend to assume random mixing of individuals, but real interactions are not random pairwise encounters: they occur within various types of gatherings such as workplaces, households, schools, and concerts, best described by a higher-order network structure. We model contagions on higher-order networks using group-based approximate master equations, in which we track all states and interactions within a group of nodes and assume a mean-field coupling between them. Using the susceptible-infected-susceptible dynamics, our approach reveals the existence of a mesoscopic localization regime, where a disease can concentrate and self-sustain only around large groups in the network overall organization. In this regime, the phase transition is smeared, characterized by an inhomogeneous activation of the groups. At the mesoscopic level, we observe that the distribution of infected nodes within groups of the same size can be very dispersed, even bimodal. When considering heterogeneous networks, both at the level of nodes and at the level of groups, we characterize analytically the region associated with mesoscopic localization in the structural parameter space. We put in perspective this phenomenon with eigenvector localization and discuss how a focus on higher-order structures is needed to discern the more subtle localization at the mesoscopic level. Finally, we discuss how mesoscopic localization affects the response to structural interventions and how this framework could provide important insights for a broad range of dynamics.
From fireflies to cardiac cells, synchronization governs important aspects of nature, and the Kuramoto model is the staple for research in this area. We show that generalizing the model to oscillators of dimensions higher than 2 and introducing a positive feedback mechanism between the coupling and the global order parameter leads to a rich and novel scenario: the synchronization transition is explosive at all even dimensions, whilst it is mediated by a time-dependent, rhythmic, state at all odd dimensions. Such a latter circumstance, in particular, differs from all other time-dependent states observed so far in the model. We provide the analytic description of this novel state, which is fully corroborated by numerical calculations. Our results can, therefore, help untangle secrets of observed time-dependent swarming and flocking dynamics that unfold in three dimensions, and where this novel state could thus provide a fresh perspective for as yet not understood formations.
The higher-order interactions of complex systems, such as the brain, are captured by their simplicial complex structure and have a significant effect on dynamics. However, the existing dynamical models defined on simplicial complexes make the strong assumption that the dynamics resides exclusively on the nodes. Here we formulate the higher-order Kuramoto model which describes the interactions between oscillators placed not only on nodes but also on links, triangles, and so on. We show that higher-order Kuramoto dynamics can lead to an explosive synchronization transition by using an adaptive coupling dependent on the solenoidal and the irrotational component of the dynamics.
The Kuramoto model of coupled phase oscillators is often used to describe synchronization phenomena in nature. Some applications, e.g., quantum synchronization and rigid-body attitude synchronization, involve high-dimensional Kuramoto models where each oscillator lives on the n-sphere or SO(n). These manifolds are special cases of the compact, real Stiefel manifold St(p,n). Using tools from optimization and control theory, we prove that the generalized Kuramoto model on St(p,n) converges to a synchronized state for any connected graph and from almost all initial conditions provided (p,n) satisfies p≤23n−1 and all oscillator frequencies are equal. This result could not have been predicted based on knowledge of the Kuramoto model in complex networks over the circle. In that case, almost global synchronization is graph dependent; it applies if the network is acyclic or sufficiently dense. This paper hence identifies a property that distinguishes many high-dimensional generalizations of the Kuramoto models from the original model.
Collective behavior in large ensembles of dynamical units with nonpairwise interactions may play an important role in several systems ranging from brain function to social networks. Despite recent work pointing to simplicial structure, i.e., higher-order interactions between three or more units at a time, their dynamical characteristics remain poorly understood. Here we present an analysis of the collective dynamics of such a simplicial system, namely coupled phase oscillators with three-way interactions. The simplicial structure gives rise to a number of novel phenomena, most notably a continuum of abrupt desynchronization transitions with no abrupt synchronization transition counterpart, as well as extensive multistability whereby infinitely many stable partially synchronized states exist. Our analysis sheds light on the complexity that can arise in physical systems with simplicial interactions like the human brain and the role that simplicial interactions play in storing information.
Previous results have shown that a large class of complex systems consisting of many interacting heterogeneous phase oscillators exhibit an attracting invariant manifold. This result has enabled reduced analytic system descriptions from which all the long term dynamics of these systems can be calculated. Although very useful, these previous results are limited by the restriction that the individual interacting system components have one-dimensional dynamics, with states described by a single, scalar, angle-like variable (e.g., the Kuramoto model). In this paper, we consider a generalization to an appropriate class of coupled agents with higher-dimensional dynamics. For this generalized class of model systems, we demonstrate that the dynamics again contain an invariant manifold, hence enabling previously inaccessible analysis and improved numerical study, allowing a similar simplified description of these systems. We also discuss examples illustrating the potential utility of our results for a wide range of interesting situations.
This paper considers a recently introduced D-dimensional generalized Kuramoto model for many (N≫1) interacting agents, in which the agent states are D-dimensional unit vectors. It was previously shown that, for even (but not odd) D, similar to the original Kuramoto model (D=2), there exists a continuous dynamical phase transition from incoherence to coherence of the time asymptotic attracting state (time t→∞) as the coupling parameter K increases through a critical value which we denote Kc(+)>0. We consider this transition from the point of view of the stability of an incoherent state, where an incoherent state is defined as one for which the N→∞ distribution function is time-independent and the macroscopic order parameter is zero. In contrast with D=2, for even D>2, there is an infinity of possible incoherent equilibria, each of which becomes unstable with increasing K at a different point K=Kc. Although there are incoherent equilibria for which Kc=Kc(+), there are also incoherent equilibria with a range of possible Kc values below Kc(+), (Kc(+)/2)≤Kc<Kc(+). How can the possible instability of incoherent states arising at K=Kc<Kc(+) be reconciled with the previous finding that, at large time (t→∞), the state is always incoherent unless K>Kc(+)? We find, for a given incoherent equilibrium, that, if K is rapidly increased from K<Kc to Kc<K<Kc(+), due to the instability, a short, macroscopic burst of coherence is observed, in which the coherence initially grows exponentially, but then reaches a maximum, past which it decays back into incoherence. Furthermore, after this decay, we observe that the equilibrium has been reset to a new equilibrium whose Kc value exceeds that of the increased K. Thus, this process, which we call “Instability-Mediated Resetting,” leads to an increase in the effective Kc with continuously increasing K, until the equilibrium has been effectively set to one for which Kc≈Kc(+). Thus, instability-mediated resetting leads to a unique critical point of the t→∞ time asymptotic state (K=Kc(+)) in spite of the existence of an infinity of possible pretransition incoherent states.
Recently there is a surge of interest in network geometry and topology. Here we show that the spectral dimension plays a fundamental role in establishing a clear relation between the topological and geometrical properties of a network and its dynamics. Specifically we explore the role of the spectral dimension in determining the synchronization properties of the Kuramoto model. We show that the synchronized phase can only be thermodynamically stable for spectral dimensions above four and that phase entrainment of the oscillators can only be found for spectral dimensions greater than two. We numerically test our analytical predictions on the recently introduced model of network geometry called complex network manifolds, which displays a tunable spectral dimension.
Many complex systems find a convenient representation in terms of networks: structures made by pairwise interactions (links) of elements (nodes). For many biological and social systems, elementary interactions involve, however, more than two elements, and simplicial complexes are more adequate to describe such phenomena. Moreover, these interactions often change over time. Here, we propose a framework to model such an evolution: the simplicial activity driven model, in which the building block is a simplex of nodes representing a multiagent interaction. We show analytically and numerically that the use of simplicial structures leads to crucial structural differences with respect to the activity driven model, a paradigmatic temporal network model involving only binary interactions. It also impacts the outcome of paradigmatic processes modeling disease propagation or social contagion. In particular, fluctuations in the number of nodes involved in the interactions can affect the outcome of models of simple contagion processes, contrarily to what happens in the activity driven model.
Simplicial complexes are increasingly used to understand the topology of complex systems as different as brain networks and social interactions. It is therefore of special interest to extend the study of percolation to simplicial complexes. Here we propose a topological theory of percolation for discrete hyperbolic simplicial complexes. Specifically, we consider hyperbolic manifolds in dimension d=2 and d=3 formed by simplicial complexes, and we investigate their percolation properties in the presence of topological damage, i.e., when nodes, links, triangles or tetrahedra are randomly removed. We show that in d=2 simplicial complexes there are four topological percolation problems and in d=3 there are six. We demonstrate the presence of two percolation phase transitions characteristic of hyperbolic spaces for the different variants of topological percolation. While most of the known results on percolation in hyperbolic manifolds are in d=2, here we uncover the rich critical behavior of d=3 hyperbolic manifolds, and show that triangle percolation displays a Berezinskii-Kosterlitz-Thouless (BKT) transition. Finally, we provide evidence that topological percolation can display a critical behavior that is unexpected if only node and link percolation are considered.
We report on a novel collective state, occurring in globally coupled nonidentical oscillators in the proximity of the point where the transition from the system’s incoherent to coherent phase converts from explosive to continuous. In such a state, the oscillators form quantized clusters, where neither their phases nor their instantaneous frequencies are locked. The oscillators’ instantaneous speeds are different within the clusters, but they form a characteristic cusped pattern and, more importantly, they behave periodically in time so that their average values are the same. Given its intrinsic specular nature with respect to the recently introduced Chimera states, the phase is termed the Bellerophon state. We provide an analytical and numerical description of Bellerophon states, and furnish practical hints on how to seek them in a variety of experimental and natural systems.
Percolation and synchronization are two phase transitions that have been extensively studied since already long ago. A classic result is that, in the vast majority of cases, these transitions are of the second-order type, i.e. continuous and reversible. Recently, however, explosive phenomena have been reported in com- plex networks' structure and dynamics, which rather remind first-order (discontinuous and irreversible) transitions. Explosive percolation, which was discovered in 2009, corresponds to an abrupt change in the network's structure, and explosive synchronization (which is concerned, instead, with the abrupt emergence of a collective state in the networks' dynamics) was studied as early as the first models of globally coupled phase oscillators were taken into consideration. The two phenomena have stimulated investigations and de- bates, attracting attention in many relevant fields. So far, various substantial contributions and progresses (including experimental verifications) have been made, which have provided insights on what structural and dynamical properties are needed for inducing such abrupt transformations, as well as have greatly enhanced our understanding of phase transitions in networked systems. Our intention is to offer here a monographic review on the main-stream literature, with the twofold aim of summarizing the existing results and pointing out possible directions for future research.
In the past years, network theory has successfully characterized the
interaction among the constituents of a variety of complex systems, ranging
from biological to technological, and social systems. However, up until
recently, attention was almost exclusively given to networks in which all
components were treated on equivalent footing, while neglecting all the extra
information about the temporal- or context-related properties of the
interactions under study. Only in the last years, taking advantage of the
enhanced resolution in real data sets, network scientists have directed their
interest to the multiplex character of real-world systems, and explicitly
considered the time-varying and multilayer nature of networks. We offer here a
comprehensive review on both structural and dynamical organization of graphs
made of diverse relationships (layers) between its constituents, and cover
several relevant issues, from a full redefinition of the basic structural
measures, to understanding how the multilayer nature of the network affects
processes and dynamics.
In conjunction with the study of the perturbations of the statistical properties of nuclear spectra produced by interactions which are odd under time reversal, it was found that the case in which the odd part is very large can be treated analytically. While of no immediate physical interest, the precise results obtained serve as a check on the adequacy of the Monte-Carlo calculations reported in the preceding paper. With the help of the theory of random matrices, analytical results are obtained for the distribution of widths, the level density, two-level cluster function and the distribution of the spacing between adjacent energy levels.
Random walks on a graph reflect many of its topological and spectral
properties, such as connectedness, bipartiteness and spectral gap magnitude. In
the first part of this paper we define a stochastic process on simplicial
complexes of arbitrary dimension, which reflects in an analogue way the
existence of higher dimensional homology, and the magnitude of the
high-dimensional spectral gap originating in the works of Eckmann and Garland.
The second part of the paper is devoted to infinite complexes. We present a
generalization of Kesten's result on the spectrum of regular trees, and of the
connection between return probabilities and spectral radius. We study the
analogue of the Alon-Boppana theorem on spectral gaps, and exhibit a
counterexample for its high-dimensional counterpart. We show, however, that
under some assumptions the theorem does hold - for example, if the
codimension-one skeletons of the complexes in question form a family of
expanders.
Our study suggests natural generalizations of many concepts from graph
theory, such as amenability, recurrence/transience, and bipartiteness. We
present some observations regarding these ideas, and several open questions.
Coupled biological and chemical systems, neural networks, social interacting species, the Internet and the World Wide Web, are only a few examples of systems composed by a large number of highly interconnected dynamical units. The first approach to capture the global properties of such systems is to model them as graphs whose nodes represent the dynamical units, and whose links stand for the interactions between them. On the one hand, scientists have to cope with structural issues, such as characterizing the topology of a complex wiring architecture, revealing the unifying principles that are at the basis of real networks, and developing models to mimic the growth of a network and reproduce its structural properties. On the other hand, many relevant questions arise when studying complex networks’ dynamics, such as learning how a large ensemble of dynamical systems that interact through a complex wiring topology can behave collectively. We review the major concepts and results recently achieved in the study of the structure and dynamics of complex networks, and summarize the relevant applications of these ideas in many different disciplines, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering.
The Kuramoto model describes a large population of coupled limit-cycle oscillators whose natural frequencies are drawn from some prescribed distribution. If the coupling strength exceeds a certain threshold, the system exhibits a phase transition: some of the oscillators spontaneously synchronize, while others remain incoherent. The mathematical analysis of this bifurcation has proved both problematic and fascinating. We review 25 years of research on the Kuramoto model, highlighting the false turns as well as the successes, but mainly following the trail leading from Kuramoto’s work to Crawford’s recent contributions. It is a lovely winding road, with excursions through mathematical biology, statistical physics, kinetic theory, bifurcation theory, and plasma physics.
We review the observations and the basic laws describing the essential
aspects of collective motion -- being one of the most common and spectacular
manifestation of coordinated behavior. Our aim is to provide a balanced
discussion of the various facets of this highly multidisciplinary field,
including experiments, mathematical methods and models for simulations, so that
readers with a variety of background could get both the basics and a broader,
more detailed picture of the field. The observations we report on include
systems consisting of units ranging from macromolecules through metallic rods
and robots to groups of animals and people. Some emphasis is put on models that
are simple and realistic enough to reproduce the numerous related observations
and are useful for developing concepts for a better understanding of the
complexity of systems consisting of many simultaneously moving entities. As
such, these models allow the establishing of a few fundamental principles of
flocking. In particular, it is demonstrated, that in spite of considerable
differences, a number of deep analogies exist between equilibrium statistical
physics systems and those made of self-propelled (in most cases living) units.
In both cases only a few well defined macroscopic/collective states occur and
the transitions between these states follow a similar scenario, involving
discontinuity and algebraic divergences.