Effect of diluted connectivities on cluster synchronization of adaptively coupled oscillator networks
EEffect of diluted connectivities on cluster synchronization ofadaptively coupled oscillator networks
Simon Vock , Rico Berner , Serhiy Yanchuk , and Eckehard Sch¨oll Institute of Theoretical Physics, Technische Universit¨at Berlin,Hardenbergstraße 36, 10623 Berlin, Germany Institute of Mathematics, Technische Universit¨at Berlin, Straße des 17. Juni 136,10623 Berlin, Germany Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universit¨at,Philippstraße 13, 10115 Berlin, Germany Potsdam Institute for Climate Impact Research, Telegrafenberg A 31, 14473Potsdam, Germany
Abstract
Synchronization in networks of oscillatory units is an emergent phenomenon present in var-ious systems, such as biological, technological, and social systems. Many real-world systemshave adaptive properties, meaning that their connectivities change with time, depending onthe dynamical state of the system. Networks of adaptively coupled oscillators show varioussynchronization phenomena, such as hierarchical multifrequency clusters, traveling waves, orchimera states. While these self-organized patterns have been previously studied on all-to-all coupled networks, this work extends the investigations towards more complex networks,analyzing the influence of random network topologies for various degrees of dilution of theconnectivities. Using numerical and analytical approaches, we investigate the robustness ofmulticluster states on networks of adaptively coupled Kuramoto-Sakaguchi oscillators againstthe random dilution of the underlying network topology. Further, we utilize the master sta-bility approach for adaptive networks in order to highlight the interplay between adaptivityand topology.
In general terms, a complex dynamical network is a set of dynamical units (nodes) with connectionsbetween them (links), representing a relation or interaction among the individual elements. Innature as well as in technology, complex dynamical networks provide a framework with a broadrange of applications in physics, chemistry, biology, neuroscience, economy, social science, andmany more [1, 2].Collective behavior in dynamical networks is the emergent phenomenon of spontaneous ordereddynamics. One example of particular importance is synchronization [3, 4, 5, 6, 7, 2]. First recog-nized by Huygens in the 17th century [8], synchronization phenomena of coupled oscillators are ofgreat interest in science, nature, engineering, and social life. Depending on the dynamical proper-ties of a system, diverse synchronization patterns of varying complexity have been observed, suchas complete synchronization [9], cluster synchronization [10, 11, 12, 13, 14], and various forms ofpartial synchronization [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. Synchronization patterns arebelieved to play an important role in neural networks, for instance in the context of cognition andlearning [27, 28, 29] as well as in pathological conditions such as Parkinsons’s disease [30, 31, 32, 33]or epilepsy [34, 35, 36, 37, 38, 39, 40, 41].A central question in the study of synchronization on complex networks is whether such behavioris stable. A powerful analytic framework to study the stability of synchronized states is the masterstability approach [42]. Since its introduction, this approach has been extended to various types1 a r X i v : . [ n li n . AO ] J a n f networks, such as multilayer networks [43, 44], networks with time-delays [45, 46, 47, 48, 12, 49,50, 51, 52, 53, 54, 55], hypernetworks [56, 57], and very recently to adaptive networks [58]. Themaster stability approach allows for the separation of the effects of the local node dynamics fromthe effects of the network topology. This can be used to draw general conclusions on the stabilityof dynamical systems by analyzing the eigenvalues of the network connectivity matrix.Most of the previous studies have analyzed the dynamical processes that occur on static networks,describing fixed interaction structures that do not change with time. Real-world networks howeveroften change in time, adapting their structure in response to the network state [59, 60, 61]. Thistype of network is called adaptive or co-evolutionary, and combines topological evolution of thenetwork with dynamics on the network nodes. This behavior appears in many real-world appli-cations. For instance, power grids and traffic networks continuously change to meet the evolvingneeds of society. Further, adaptive behavior is of great importance in neural networks. Theseare networks of individual neurons connected by synapses that pass electrical or chemical signalsbetween them. It has been shown that the coupling weights between the individual neurons maybe potentiated or depressed, depending on the order of the spike times of post- and presynapticneuron [62, 63, 64]. This mechanism called spike timing-dependent plasticity is believed to play animportant role in temporal coding of information in the brain [65].In this article, we investigate the robustness of multicluster states on networks of adaptively coupledKuramoto-Sakaguchi oscillators against the random dilution of the underlying network topology.By randomly and successively deleting links, we observe a linear dependence of the cluster fre-quencies on the relative number of deleted links. We explain this linear dependence by a suitableapproximation of the oscillators frequencies. We show numerically that the shape of a multiclusterstate is preserved on networks of different sparsity, ranging from fully coupled to almost uncou-pled topologies. Further, we show that the multicluster states observed are multistable, meaningthat different multiclusters may emerge for different initial conditions. By utilizing the masterstability approach for adaptive networks, we present the effects induced by changes of the networktopology on the desynchronization of the phase synchronized state. We show that the resultingdesynchronization has strong implications for the emergence of multicluster states.This article is organized as follows. In section 2, we introduce the model used throughout this work.In section 3, we discuss the emergence and structural formation of multi-frequency-cluster states.By applying a suitable randomization process on the network connectivity, we investigate the effectof random dilution of links in the connectivity structure on multicluster states in section 4. Further,we apply the master stability approach for adaptive networks to investigate the interplay betweennodal dynamics, adaptivity, and a complex connectivity structure in section 5. We use these resultsto describe a desynchronization transition by changes in the network topology in section 6. We consider a network of N Kuramoto-Sakaguchi type phase oscillators with adaptive couplingweights, described by ˙ φ i = ω − σ N (cid:88) j =1 a ij κ ij sin ( φ i − φ j + α ) , (1)˙ κ ij = − (cid:15) ( κ ij + a ij sin ( φ i − φ j + β )) , (2)where φ i ( t ) ∈ [0 , π ) describes the phase of oscillator i ∈ { , . . . , N } , and κ ij ( t ) ∈ [ − ,
1] denotesthe coupling strength from oscillator j to i . The connectivity structure is given by the elements a ij ∈ { , } of the adjacency matrix A which is independent of time. Note that self-coupling doesnot influence the relative dynamics, which is why the N diagonal elements a ii are set to zero in allour simulations. Equation (1) describes a network of Kuramoto-Sakaguchi type phase oscillatorswith a diffusive coupling kernel sin ( φ i − φ j + α ) scaled by an overall coupling strength σ , and ω isthe intrinsic frequency. The parameter α can be considered as a phase lag of the interaction betweenthe oscillators [66] or even related to a synaptic propagation delay [67]. Equation (2) describes thedynamics of the coupling weights κ ij , and (cid:15) (cid:28) − sin ( φ i − φ j + β ), where β is a control parameter, enabling us to tune2 φ/π ∆ φ/π ∆ φ/π − s i n ( ∆ φ + β ) Figure 1: The adaptation function − sin(∆ φ + β ) used in system (1)–(2), for the parameter values(a) β = − π/ β = 0 (causal, effectively similar to spike timing-dependent plasticity),(c) β = π/ β = − π/ | ∆ φ | < π/ φ = φ i − φ j .The synchrony of the oscillators at a given time t is typically quantified by the Kuramoto-Daidoorder parameter [9, 68]. The complex n th order parameter for the state φ ( t ) = ( φ ( t ) , . . . , φ N ( t )) T is defined as Z n ( φ ( t )) = R n ( t ) e i ψ n ( t ) = 1 N N (cid:88) j =1 e i nφ j ( t ) . (3)The first order parameter Z can be thought of as the centroid of the N phases of the oscillators rep-resented on the unit circle, i.e., in the complex plane [69]. Here, ψ n ( t ) is the collective mean phase ofthe population, and the modulus R n ( t ) is given by the absolute value R n ( t ) = | (1 /N ) (cid:80) Nj =1 e i nφ j ( t ) | .The quantity n ∈ N is also referred to as n th moment of the order parameter. Note that herei = √− φ = ( a, . . . , a ) T for some a ∈ [0 , π ], R n = 1. If R n = 0, we consider the oscillators as incoherent with respect tothe n th moment. In this section, we show the emergence of different synchronization patterns for an all-to-all coupledbase topology a ij = 1 ( i (cid:54) = j ) starting from uniformly distributed random initial conditions ( φ i ∈ [0 , π ), κ ij ∈ [ − ,
1] for all i, j = 1 , . . . , N ).System (1)–(2) generalizes Kuramoto-Sakaguchi type systems with fixed κ ij , and has attracteda lot of attention recently [70, 71, 72, 60, 73, 74, 75, 76, 77, 78, 79, 80, 61, 81, 82, 83]. In thissection, we briefly summarize the findings that have been reported in [61, 84, 85] and show differenttypes of multicluster states that emerge starting from random initial conditions. These states arecharacterized by strongly coupled oscillators within each cluster but weak couplings between them.Further, all oscillators in one cluster share a common frequency, while the frequencies between theclusters differ.In a multicluster state, the coupling weight matrix with elements κ ij can be divided into M ∈ N blocks, called clusters, each containing a number N µ ( µ = 1 , . . . , M ) of frequency synchronizedoscillators. We denote the entries of this coupling weight matrix by κ ij,µν , referring to the couplingweight from the j th oscillator in the ν th cluster to the i th oscillator in the µ th cluster. For the3emporal behavior of an oscillator in an M -cluster state we assume the form φ i,µ ( t ) = Ω µ t + a i,µ + s i,µ ( t ) , (4)where φ i,µ denotes the phase of oscillator i inside cluster µ , Ω µ is the collective frequency of thecluster, a i,µ ∈ [0 , π ) are phase lags, and function s i,µ ( t ) is considered to be a bounded functiondue to the interaction of the clusters.Despite the constant frequencies within a cluster, we can differentiate between three types of multi-clusters, depending on the oscillator phases [84, 85]. The first type is called splay-type multicluster(Fig. 2(a,c,e)). In this case, the coupling weights κ ij,µν are either constant or change periodicallyin time, depending on whether the oscillators φ i,µ and φ j,ν belong to the same ( µ = ν ), or a differ-ent ( µ (cid:54) = ν ) cluster, respectively. The amplitude of the coupling weights depends on the frequencydifference between the clusters, where a higher frequency difference leads to a smaller amplitude.The separation into three strongly coupled clusters can be seen in Fig. 2(a,c,e), as well as thehierarchical structure in the cluster size. Regarding the phases of the oscillators, the oscillators aredistributed on the unit circle such that the phases of each splay-type cluster fulfill the condition R ( a µ ) = 0 for µ = 1 , , a i,µ = a µ or the antipodal phase a i,µ = a µ + π , such that 2 a i,µ = 2 a µ for all i = 1 , . . . , N µ . Therefore, we call these states antipodal-type multicluster. In contrast to splaystates, the phase distribution of this type of state fulfills R ( a µ ) = 1, where µ = 1 ,
2. Note thatin-phase synchronous states belong to this type of clusters.A third possible type of multiclusters combines the previous two types, where the cluster can beeither of splay- or antipodal-type. These states are called mixed-type multicluster. For more detailswe refer the reader to [84, 85].The appearance of multicluster states suggests that certain one-cluster states serve as buildingblocks for more complex multicluster states. Formally, a one-cluster state is a frequency synchro-nized group of phase oscillators, described by φ i = Ω t + a i , with collective frequency Ω, relative phase shifts a i ∈ [0 , π ), and i = 1 , . . . , N . Figure 3 shows thecoupling matrices κ ij of all three possible one-cluster states on an all-to-all network of adaptivelycoupled phase oscillators (1)–(2). The first two types, namely the splay clusters and antipodalclusters, are described above, and serve as building blocks for multicluster states. The third type,Fig. 3(c), is called double-antipodal state. This type consists of two groups of antipodal phaseoscillators with a fixed phase lag between them. In contrast to splay- and antipodal clusters,double-antipodal states are unstable for the whole range of parameters, and are therefore unlikelyto be found as building blocks for multicluster states. In Section 3, we have described the generic appearance of multicluster states in system (1)–(2) onan all-to-all coupled network. Based on the oscillators’ phase relations within the clusters, we dis-tinguish between antipodal and splay-type multiclusters. We have shown that the oscillators canform groups of strongly connected units, where the interaction between the groups is weak com-pared to the interaction within the groups. This section investigates the robustness of multiclusterstates against the random dilution of the underlying network topology.We consider a network of N = 100 adaptively coupled phase oscillators, described by Eqs. (1)–(2).We fix the parameters α = 0 . β = − .
53, such that a multicluster state emerges from randominitial conditions for an all-to-all coupled network structure. In order to implement the dilution oflinks, we randomly and successively delete Q links ( Q ≤ ( N − N ) by choosing a set of Q indices( ij ) randomly that correspond to existing links in the adjacency matrix, i.e. a ij = 1. These linksare then removed, i.e., we set a ij = 0. The degree of dilution, i.e., the ratio of deleted links, isdefined as q = Q/ ( N ( N − nd e x i index j index j (cid:104) ˙ φ j (cid:105) φ j κ i j π π Figure 2: Multicluster states in a network of N = 200 adaptively coupled phase oscilla-tors Eqs. (1),(2). (a,b): snapshots of the phases φ j at t = 20000; (c,d): mean frequencies (cid:104) ˙ φ j (cid:105) = ( φ ( t + T ) − φ ( t )) /T with t = 10 000, T = 10 000; (e,f): snapshots of the couplingmatrices κ ij at t = 20000. In (a,c,e) a splay-type multicluster for α = 0 . π, β = 0 . π and in(b,d,f) an antipodal-type multicluster with α = 0 . π, β = − . π are displayed. Further parame-ters: (cid:15) = 0 . , ω = 1 , σ = 1 /N . i nd e x i (a) (b) (c) κ i j i ndex j i ndex j i ndex j Figure 3: Illustration of the coupling weights κ ij for all three existing types of one-cluster statesfor system (1)–(2). (a): splay state with α = 0 . π, β = 0 . π , (b): antipodal state with α =0 . π, β = − . π , (c): double-antipodal state with α = 0 . π, β = − . π . Further parameters: N = 50 , (cid:15) = 0 . , σ = 1 /N . After [85]. 5e apply two different numerical approaches to study the effects of dilution: (I) The system (1)–(2) is numerically solved for 11 000 time units and successively increasing q in each simulationrun, where the final multicluster state for q = 0 is set as the initial condition for all followingsimulations. We use this approach to investigate the robustness of a known multicluster stateagainst a random dilution of links. (II) We fix a set of 100 different random initial conditions. Foreach q we simulate the system dynamics for all 100 initial conditions and 11 000 time units.Figure 4 depicts three resulting states of system (1)–(2), obtained by the numerical simulationsdescribed above. The coupling weights κ ij are shown together with the corresponding mean fre-quencies (cid:104) ˙ φ j (cid:105) = ( φ i ( t + T ) − φ i ( t )) /T , for different values of q . Here, we choose t = 10 000 and T = 1000. Additionally, the phases φ j of the oscillators within the biggest cluster are representedon the unit circle. In the case of a fully coupled network ( q = 0, Fig. 4(a)), an antipodal-typemulticluster state emerges, consisting of three groups. The average frequencies of the oscillatorswithin each cluster are constant, as described in Sec. 3. Moreover, the oscillator phases withineach cluster possess the phase difference of either 0 or π ; hence the clusters are of the antipodaltype. The smaller clusters possess a mean frequency closer to the natural frequency ω = 1 due totheir smaller size.Figures 4(b) and 4(c) depict two possible states on networks with dilution degree (ratio of deletedlinks) q = 0 .
13, where the initial conditions were chosen as the multicluster state of Fig. 4(a)in panel (b) and as random initial conditions in panel (c). Note that the white dots in thecoupling matrix represent the deleted links. Our simulations show that despite the missing links,the oscillators can still organize themselves in multiclusters. The antipodal-type multicluster inFig. 4(b) has a similar shape as the multiclusters previously observed. In the case of missing linkshowever, the phases φ i within the clusters are slightly spread out in order to compensate for theheterogeneity in the network topology. The splay-type cluster in Fig. 4(c) is a different type ofstable state that can emerge in system (1)–(2) for q = 0 .
13. Note that in (b) and (c), the sameparameter values are used. This shows that different types of multicluster patterns may emergefrom random initial conditions. Therefore, we observe that multiclusters may exhibit multistability.In order to further investigate the robustness of multiclusters against dilution of the connectivities,we present the distribution ρ (color coded) of the mean frequencies (cid:104) ˙ φ j (cid:105) of the oscillators versusthe fraction of deleted links q in Fig. 5. We have obtained these results by applying the twonumerical approaches described above. In Fig. 5(I) we show the mean frequencies correspondingto the numerical procedure (I) where the multicluster state depicted in Fig. 4(a) is set as initialcondition. Note that here the data for each step of q is an average of all N oscillators. Fora wide range of q , three distinct clusters are visible. For large values of q , where the networkbecomes sparse, the frequency clusters are not clearly separated. The qualitative shape of theinitial multicluster state is preserved on networks of different degrees of dilution, ranging fromfully coupled to almost uncoupled topologies. The number of clusters stays the same on networkswith varying numbers of links, however, the collective mean frequencies of the oscillators adapt tothe changes in the coupling topology, in a linear relation with q .The linear dependence of the cluster frequencies on the relative number of deleted links q can beexplained as follows. For increasing dilution q , each link has an equal probability to be cut off.Therefore, on average, in each cluster there exist (1 − q ) · N µ ( N µ −
1) links. Furthermore, byassuming (cid:15) (cid:28)
1, the collective frequency of each cluster can be roughly approximated up to zerothorder in (cid:15) by Ω µ ≈ ω + σ (cid:80) N µ j =1 a ij,µµ sin( a i,µ − a j,µ + β ) sin( a i,µ − a j,µ + α ) [84]. Let us consider anapproximately antipodal cluster, i.e., a i,µ − a j,µ ≈ π . Then Ω µ ≈ ω + σ sin( β ) sin( α ) r i,µ where r i,µ = (cid:80) N µ j =1 a ij,µµ is the i th row sum restricted to the µ th cluster. By averaging over i = 1 , . . . , N µ ,we end up with the approximationΩ µ ≈ ω + σ (1 − q )( N µ −
1) sin( β ) sin( α ) . (5)The latter expression explains the linear dependence of the cluster frequency on the ratio of deletedlinks. Furthermore, Eq. (5) shows that the slope of the linear relation depends on the cluster size.This is in agreement with the findings in Fig. 5(I). Note that splay clusters can be treated similarly.In Fig. 5(II), we show the distribution of the collective mean frequencies versus q for random initialconditions according to the numerical procedure (II). Note that here the data for each step of q is an6 nd e x i index j index j index j κ i j (a) (b) (c) (cid:104) ˙ φ j (cid:105) Figure 4: Three multicluster states of system (1)–(2) for different ratios of deleted links q . In thebottom and middle panels, the mean frequencies (cid:104) ˙ φ j (cid:105) = ( φ ( t + T ) − φ ( t )) /T with t = 10 000, T = 1000, and snapshots of the coupling weights κ ij at t = 11 000 are shown, respectively. Thephases of the oscillators within the biggest cluster at t = 11 000 represented on the unit circleare displayed in the top panel. (a) q = 0, (b) and (c) q = 0 .
13 (different initial conditions: (b)multicluster state of panel (a), (c): random initial conditions). Other parameters: N = 100, (cid:15) = 0 . ω = 1, α = 0 . π , β = − . π , σ = 1 /N . (cid:104) ˙ φ j (cid:105) ρ ( (cid:104) ˙ φ j (cid:105) ) q (a) (b) q (I) multi-cluster (c) (II) random Figure 5: Collective mean frequencies (cid:104) ˙ φ j (cid:105) vs the ratio of deleted links q for system (1)–(2). (I):distribution of mean frequencies (cid:104) ˙ φ j (cid:105) , where the multicluster state obtained at q = 0 serves asinitial condition for all simulations with q >
0. (II): distribution of mean frequencies averaged over100 realizations of random initial conditions for each q . The red lines (a)–(c) mark the states shownin Fig. 4. Note (cid:104) ˙ φ j (cid:105) = ( φ j ( t + T ) − φ j ( t )) /T , where t = 10 000 and T = 1000. Parameters: N = 100, (cid:15) = 0 . ω = 1, α = 0 . π , β = − . π , σ = 1 /N .7verage of 100 numerical runs. Hence the color code represents the fraction of oscillators that lie ina corresponding frequency band. For a wide range of q , the frequencies roughly show three maximaof ρ ( (cid:104) ˙ φ j (cid:105) ), indicating 3-cluster states, see also Fig. 8 in the Appendix as an example. Further, weobserve 2-cluster states of splay type, corresponding to a fourth, smaller maximum of ρ ( (cid:104) ˙ φ j (cid:105) ), whichvanishes for q > . q ,the overall frequencies increase linearly, and eventually converge to (cid:104) ˙ φ j (cid:105) (cid:12)(cid:12)(cid:12) q =1 = 1. In this case, allthe nodes are uncoupled, and oscillate with their natural frequency of Ω = 1. These results can becompared with reference [86], where the dynamical states on networks of adaptively coupled phaseoscillators are studied in the ( α, β ) parameter space for different degree of network dilution. Thereit has been shown that splay- and antipodal-type clusters emerge in the corresponding region ofthe parameter space for all-to-all coupled networks. For decreasing number of links, i.e., increasing q , the network first loses its ability to reach splay-type states, and subsequently loses its ability tosynchronize at all. These findings suggest that the fourth maximum in Fig. 5(II) exists due to theexistence of splay-type multiclusters since it vanishes for q > . κ ij in Fig. 6. We note that the multicluster state at q = 0 was chosenas the initial condition for all simulations with q >
0. In agreement with our prior observations,three clusters are visible for densely connected networks. For increasing q , meaning sparser net-works, the frequency clusters dissolve in a hierarchical manner, where clusters consisting of feweroscillators vanish prior to those containing a large number of oscillators. Further, we observe thatthe dissolution of a cluster starts with the uncoupling of single oscillators from the cluster and acontinuous decrease in cluster size. This process continues until the cluster vanishes. A similarplot corresponding to Fig. 5(II) with random initial conditions can be found in the Appendix, seeFig. 8.In this section, we have shown that randomly diluting the network topology leads to a desynchro-nization of multiclusters. It has been observed that system (1)–(2) preserves the qualitative shapeof multicluster states on random networks of different sparsity. Hence, multiclusters are robustagainst topological perturbations. Further, system (1)–(2) possesses a high degree of multistabilityalso for diluted network topologies. In the following section, we investigate the stability of synchro-nized states by means of the master stability function for networks with adaptive couplings [58].With this we show that the presence of adaptive couplings can be used to influence the stabilityof system (1)–(2). Further, we show that the destabilization of the phase-synchronized state hasimplications for the emergence of multicluster states. In the preceding section, we have described the desynchronization of multicluster states for adecreasing number of links. It has been shown that system (1)–(2) keeps its ability to form stronglycoupled groups of oscillators from random initial conditions on networks with increasing degree ofdilution, while the fraction of desynchronized oscillators increases for sparse networks. Since theindividual clusters in a multicluster state are effectively uncoupled from each other, the stabilityof these strongly coupled subnetworks may play an important role in the stability of multiclusterstates.In this and the next section, we aim to gain analytic insights into the stability of in-phase syn-chronized states, i.e., φ i = Ω t , i = 1 , . . . , N . Note that the in-phase synchronized states belongto the class of antipodal states, see Sec. 3, and share the same dynamical properties [84, 85]. Weassume that the network topology expressed by the adjacency matrix possesses a constant row sum r , i.e., r = (cid:80) Nj =1 a ij for all i . Following the master stability approach for networks with adaptivecoupling weights as developed in [58], we derive the master stability function for networks of adap-tively coupled phase oscillators. This allows us to separate the effects of the local node dynamicsfrom the effects of adaptivity and from the network topology. Hence, we are able to draw gen-eral conclusions on the stability of the in-phase synchronized state, for almost arbitrary complexnetwork topologies. With this, the master stability approach allows us to study the interplay of8 nd e x i i nd e x i i nd e x i i nd e x i i nd e x i index j index j index j index jq = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . κ ij Figure 6: Multicluster synchronization in system (1)–(2) with decreasing ratio of randomly deletedlinks q = Q/ ( N ( N − q = 0 is used as initial condition for allsimulations with q >
0. Snapshots of the coupling matrices κ ij at t = 11 000 for different valuesof q are presented. In each panel the oscillators are ordered according to their mean frequency,and subsequently their phases. Parameters: α = 0 . π , β = − . π , ω = 1, (cid:15) = 0 . σ = 1 /N , N = 100. 9odal dynamics, adaptivity, and complex network structures. The approach, further, enables usto control the stability of synchronized states, depending on the changes in the coupling structure.In the following, we provide a brief discussion of the master stability function for the adaptiveKuramoto-Sakaguchi network (1)–(2). Using the results from [58], the stability of the synchronousstate of system (1)–(2) is governed by the two linearized differential equations for perturbations ofthe synchronous state in the new coordinates ζ ∈ R and κ ∈ R dd t (cid:18) ζκ (cid:19) = (cid:18) µσ cos( α ) sin( β ) − σ sin( α ) − (cid:15)µ cos( β ) − (cid:15) (cid:19) (cid:18) ζκ (cid:19) , (6)where µ ∈ C denotes all eigenvalues of the Laplacian matrix L = r I N − A corresponding to thenetwork described by the adjacency matrix A . Here, I N is the N -dimensional identity matrix. Thecharacteristic polynomial in λ of (6) is of degree two and reads λ + ( (cid:15) − σµ cos( α ) sin( β )) λ − (cid:15)σµ sin( α + β ) = 0 . (7)The master stability function is given by Λ( σµ ) = max(Re( λ ) , Re( λ )) where λ and λ are the twosolutions of the quadratic polynomial (7). Using the master stability function, we can deduce thelocal stability of the in-phase state (also for the antipodal states) directly from the connectivitystructure given by the adjacency matrix. In particular, if there exists at least one Laplacianeigenvalue such that Λ( σµ ) >
0, the state is locally unstable. If for all Laplacian eigenvaluesΛ( σµ ) < L there exists always one eigenvalue µ = 0 which leads to λ = 0 and λ = − (cid:15) .The zero eigenvalue of the matrix in (6) corresponds to the shift symmetry of system (1)–(2) andis not considered in the above stability condition.In order to get insight into the form of the master stability function, we consider the boundaryof the region in σµ parameter space that corresponds to stable local dynamics. The boundary isgiven by λ = i γ with γ ∈ R . Plugging this into equation (7), we obtain the function σµ = a ( γ ) + i b ( γ )with a ( γ ) = (cid:15) γ (cos α sin β − sin( α + β )) γ cos α sin β + (cid:15) sin ( α + β ) ,b ( γ ) = γ cos α sin β + (cid:15) γ sin( α + β ) γ cos α sin β + (cid:15) sin ( α + β ) . Hence, the boundary in the complex σµ -plane is parametrically described as a cubic function ofthe real value γ . Due to the symmetry of the master stability function, a condition to observe anontrivial shape of the boundary is that the function σµ ( γ ) possesses three crossings with the realaxis, i.e., two positive real solutions for b ( γ ) = 0. The three crossings are given by γ = 0 andas real solutions γ and γ of γ cos α sin β = − (cid:15) sin( α + β ). From this, we deduce the existencecondition for three crossings as sin( α + β ) / (cos α sin β ) < (cid:15) > a ( γ ) = a ( γ ). Ifthere are three crossings, the master stability function possesses a stability island in the complexplane. A phenomenon induced by the existence of a stability island is in the focus of the nextsection. The emergence of stability islands in the master stability function enables us to destabilize syn-chronous states in various ways. In the following, we demonstrate a desynchronization by modifyingthe network topology. A similar approach has been studied on non-adaptive, delay-coupled systemsin [48, 50]. There it is shown that randomly added inhibitory links lead to a desynchronizationtransition on regular ring topologies. We extend these studies towards adaptively coupled systems,and use random network topologies. 10e consider a network of adaptively coupled phase oscillators, described by the set of differentialequations (1)–(2). The oscillators are assumed to have a natural frequency ω = 1. In the follow-ing, we investigate the stability of phase-synchronized states on networks with random adjacencymatrices, first by using the master stability function, and second by numerical integration. Wechoose the parameters α and β , such that there exists a stability island of the master stabilityfunction in the complex plane. We then prepare random initial conditions, i.e., φ i (0) ∈ [0 , π )and κ ij (0) ∈ [ − , N = 100 and t = 30000for three different random adjacency matrices, corresponding to directed, connected networks withdifferent node in-degree. In order to guarantee the existence of synchronized states, we assumethat the adjacency matrices have a constant row sum r = (cid:80) Nj =1 a ij . Note that this conditionis not preserved by the dilution procedure used in Sect. 3. We apply the following procedure toconstruct the directed random networks. For each node i of the N nodes, r links are randomlypicked from the set that consists of all possible links from nodes j (cid:54) = i to node i . For the selectedlinks a ij are set to 1. This procedure results in a directed random network with N nodes and aconstant row sum (in-degree) r . We note that the row sum r defines the ratio of deleted links q = Q/ ( N ( N − − r/ ( N − σµ ) for the parameters α = 0 . π and β = 0 . π , where the black dots indicate the scaled Laplacian eigenvalues σµ i of three randomadjacency matrices with different row sums r . We use this to determine the stability of the in-phasesynchronized state for the given coupling topology, and show the corresponding numerical resultsbelow.We observe that for sparse random networks with r = 3 ( q = 0 . σµ i for all Laplacian eigenvalues lie within the stable region Λ( σµ ) < (cid:104) ˙ φ j (cid:105) , see Fig. 7(g). Moreover, the oscillators φ j either share the same phase a i ≈ a , or the antipodal phase a i ≈ a + π , indicating an antipodal-typecluster. We note that the in-phase synchronous states considered by the master stability functionΛ belong to the class of antipodal states and share the same local stability properties [85]. If thenumber of links is increased, e.g. to r = 50 ( q = 0 . r = 99 ( q = 0), the values σµ i are either located at 0 or1, see Fig. 7(c). Since the eigenvalues located at σµ = 1 lie in the unstable region, the in-phasesynchronized state is unstable. In this case, we again observe the emergence of an antipodal-typemulticluster, see Fig. 7(f,i,l).In this section, we have presented the effects induced by changes of the network topology on thedesynchronization of the phase-synchronized state. While the phase-synchronized state is stablefor sparse networks, it is destabilized for an increasing number of links. We have described thisbehavior by means of the master stability function for networks with adaptive coupling weights.Modifying the adjacency matrix affects the Laplacian eigenvalues µ i , and consequently changes thelargest Lyapunov exponents of the network. We use the presence of bounded stable regions (stabil-ity islands in the complex plane) of the master stability function in order to successively shift theLaplacian eigenvalues from the stable into the unstable regime by changing the network connec-tivities. We show that the resulting desynchronization has strong implications for the emergenceof multicluster states. We have investigated the emergence of cluster synchronization on networks of adaptively coupledKuramoto-Sakaguchi oscillators with complex topologies. Specifically, we have focused on therobustness of the multifrequency cluster states against diluted connectivities. We have shown the11 j i nd e x i I m ( σ µ ) Re( σµ )ΛRe( σµ ) Re( σµ )index j index j (cid:104) ˙ φ j (cid:105) κ ij index j Figure 7: Dynamics of a network of N = 100 adaptively coupled Kuramoto-Sakaguchi oscillatorswith random adjacency matrices with different constant row sums r = (cid:80) Nj =1 a ij , and randominitial conditions φ i (0) ∈ [0 , π } and κ ij (0) ∈ [ − ,
1] for i, j = 1 . . . N . The simulations resultsare shown for the three values (a,d,g,j) r = 3, (b,e,h,k) r = 50, (c,f,i,l) r = 99. The panels show:(a,b,c) the master stability function color coded, together with σµ i , where µ i are the N Laplacianeigenvalues corresponding to each adjacency matrix; the inset in (a) depicts a blow-up of the markedarea, where eigenvalues lie close to the border of the stability island; in (d,e,f) snapshots of theoscillators phases φ j at t = 30000; in (g,h,i) the mean frequencies (cid:104) ˙ φ j (cid:105) = ( φ j ( t + T ) − φ j ( t )) /T ,where t = 25 000 and T = 5000; in (j,k,l) snapshots of the coupling weights κ ij , where the couplingweights are color coded. The oscillators are ordered according to their mean frequencies (cid:104) ˙ φ j (cid:105) , andsubsequently their phases φ j . Other parameters: α = 0 . π , β = 0 . π , ω = 1, σ = 1 /N , (cid:15) = 0 . q =0 .
13 in Fig. 4(c)). This result is in agreement with the findings presented in [86], where thedynamical states on networks of adaptively coupled phase oscillators have been studied in the( α, β ) parameter space for different network sparsities. We have shown in numerical investigationsthat the multicluster states observed are multistable with regard to initial conditions, meaningthat different multiclusters may emerge for different initial conditions. By depicting asymptoticstates for different values of the ratio of deleted links q , we have shown that the qualitative shapeof a given multicluster state is preserved on networks of different degree of dilution. The numberof clusters stays the same for a wide range of q , however, the frequencies of the oscillators adaptto the changes in the coupling topology, in a linear relation with q . The latter effect has beenanalytically described.Since in-phase synchronous and antipodal states have the same local stability properties, we haveextended our investigations by an analytical approach describing the stability of in-phase synchro-nized states in Sec. 5. We have analyzed the master stability function for networks of Kuramoto-Sakaguchi oscillators with adaptive coupling weights using the novel methods presented in [58]. Wehave observed the emergence of bounded regions that lead to stable synchronous dynamics in themaster stability function, representing stability islands in the complex plane. We have analyticallydescribed the stability border, and provided a condition for the emergence of stability islands. Dueto the shape of these stability islands, it is possible to destabilize in-phase synchronized states byincreasing the number of links within the network. We have shown that such a destabilization hasimplications for the emergence of multicluster states. By tailoring the system configurations suchthat a stability island emerges in the master stability function, we have observed stable multiclusterstates emerging from random initial conditions for those topological configuration with unstablein-phase synchronized states. In previous work [58], it has been shown that such a counterintuitivedesynchronization effect is also possible by increasing the overall coupling strength.In this work, we have shown the emergence of multicluster states on networks of adaptively cou-pled Kuramoto-Sakaguchi oscillators with random coupling topologies. Due to the adaptivity ofthe coupling weights, the individual oscillators are able to adapt their frequencies and form stronglycoupled groups of frequency-synchronized units. We have shown by numerical and analytical inves-tigations how random coupling topologies affect the emergence of these synchronization patterns.While the structural shape of multiclusters stays the same, the overall synchrony in a network de-clines with decreasing number of links, since more and more oscillators decouple from the systemand exhibit incoherent dynamics. Nonetheless, we have shown by means of the master stabilityfunction that some network configurations allow for stable in-phase synchronized dynamics for verysparse random networks. While our investigations are restricted to networks with uniform degreedistributions, they may be generalized to more realistic network models, such as small-world orscale-free networks. The master stability approach serves as a universal tool, describing the stabil-13ty of in-phase synchronized states. We have shown that the destabilization of fully synchronizedstates has implications for the emergence of multicluster states. In order to obtain a more completepicture, this approach may be extended towards splay-type synchrony. Our findings on the effectsof diluted connectivities upon multicluster states illustrate the complex interplay between topol-ogy and adaptivity. By applying the master stability approach, we have shown that the adaptiveproperties of a network can have a huge impact on the stability of fully and partially synchronizedstates since the presence of adaptivity provides a feedback mechanism that can change the stability;intuitively this is similar to an additional effective phase lag. Acknowledgments
This work was supported by the German Research Foundation DFG, Project Nos. 411803875 and440145547.
A Appendix
Figure 4(b,c) shows that system (1)–(2) is multistable with regard to initial conditions. This meansthat depending upon the initial state different realizations of multicluster states can emerge. Weillustrate this feature in Fig. 8, where we show the coupling matrices κ ij for different values of q .Note that the presented coupling matrices correspond to the dynamics shown in Fig. 5(II). References [1] M. E. J. Newman:
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