Abstract
Many networked systems have evolved to optimize performance of function. Much literature has considered optimization of networks by central planning, but investigations of network formation amongst agents connecting to achieve nonaligned goals are comparatively rare. Here we consider the dynamics of synchronization in populations of coupled nonidentical oscillators and analyze adaptations in which individual nodes attempt to rewire network topology to optimize nodespecific aims. We demonstrate that, even though individual nodes’ goals differ very widely, rewiring rules in which each node attempts to connect to the rest of the network in such a way as to maximize its influence on the system can enhance synchronization of the collective. The observed speedup of consensus finding in this competitive dynamics might explain enhanced synchronization in real world systems and shed light on mechanisms for improved consensus finding in society.
Introduction
Understanding the interplay between topology and dynamics on complex networks is one of the fundamental problems in complex systems science. One area, in which much progress has been made in recent years, is studying the dynamics of synchronization of chaotic or limit cycle oscillators on complex networks, which has many applications in the biology of interacting neurons or heart cells as well as social or technical systems in which people, lasers or power systems have been observed to produce collective rhythms^{1,2,3,4}. Of some importance for the interpretation of the results below is also that models of coupled oscillators have also been proposed to explore consensus finding in society^{5,6}.
Insights gained have revolved around the impact of network topology on synchronization, i.e. finding that adding long range connections in spatial systems can enhance synchronization^{7,8}, or observations that the synchronization transition is influenced by network heterogeneity on scalefree networks^{9,10}. Other research has investigated the impact of modularity on time scales to synchronization^{11}, weighting schemes that can enhance synchronization properties of networks^{12}, or works that reveal different paths to synchronization on different types of networks^{13}. More recently there has been interest in adaptive coupling schemes whose coevolution with oscillator dynamics can boost synchronization. In this area the focus has often been on adaptation rules that allow for rewiring which only use local information, e.g. demonstrating that coupling which grows in proportion to phase or state differences^{14,15} or rewiring towards more outofphase oscillators^{16} can enhance synchronization, or that forming connections between oscillators with similar phases can lead to a concurrent enhancement of percolation and synchronization^{17}.
A prominent question in the above research is to find “optimal” networks for synchronization^{12,14,18,19,20,21,22,23,24}. This issue gains an additional dimension of complexity when oscillators are not identical, such that correlations between oscillator type and topological position can influence the dynamics of synchronization. In this context it has been found that anticorrelated placements of oscillators on networks can enhance synchronization^{19,21,22,24} or lead to first order transitions on heterogeneous networks^{25}. In the last few years these findings have been understood via a collective coordinate approach^{24,26}. Very recently also the game theory of synchronization was considered, exploring the influence of agents’ choices between making an effort to get synchronized or freeriding and waiting for others to synchronize to them^{27}. The latter study is motivation to our work, but we focus on the network game of agents attempting to maximize their influence on the collective instead.
So far, most optimization schemes for complex networks have considered how a central planner can design a system to optimally achieve a certain target^{18,20,23,28}. In some complex systems a scenario of such central control does not apply. Instead, system structure may arise as a result of individuals striving to better systemic outcomes from their individual perspectives. Fewer studies have analyzed this gametheoretic situation in which nodes compete to structure a network to achieve individual targets which need not necessarily be aligned with the best tradeoff at the system level. This is, for instance, the case for synchronization, where node specific aims of achieving best local synchronization may be in conflict with global synchronization^{29}. Here, we consider a variant of this problem, in which individual nodes of a directed network attempt to rewire connections in order to maximize their influence on the collective such that they drag the frequency of collective synchronization towards their respective preferences. One might anticipate that such a system can never achieve global consensus or dynamic equilibrium. However, as elaborated below, we find that under certain conditions such individual rewiring aimed at alignment of the system with individually very diverse nodespecific goals actually tends to accelerate global consensus finding, e.g. synchronization.
Model
Consider a system of N coupled oscillators with phases ϕ_{ i } and native frequencies ω_{ i } whose dynamics are specified by the wellknown Kuramoto dynamics^{30}
where the adjacency matrix of the coupling network has the properties a_{ ij } = 1 if there is a link from j to i and a_{ ij } = 0 otherwise, while κ denotes the coupling strength. The native frequencies ω_{ i } are drawn from some distribution, e.g. the uniform distribution between [−1, 1]. Below, we will assume that a node’s native frequency represents the nodes frequency preference. Depending on the structure of the coupling network and the coupling strength, the Kuramoto model on complex networks can exhibit a phase transition at some critical coupling κ_{ c } such that a macroscopic fraction of the oscillators are phase and frequency locked for κ > κ_{ c } while this is not the case below the critical coupling. Conventionally one defines the order parameter r
such that r ≈ 0 indicates lack of synchronization while values of r close to one indicate a highly synchronized state.
Summing over Eq. (1) shows that for symmetric coupling/undirected coupling networks a_{ ij } = a_{ ji } synchronization will always occur to the mean of the ensemble of native frequencies. However, it is important to note that this is generally not the case for directed coupling in which the collective (dynamical) frequency can be influenced by the structure of the coupling network. Exactly this is the basis of the model below in which we consider a set of selfinterested Kuramoto agents which attempt to rewire connections aiming to align collective behavior (i.e. the collective frequency) with their individual goals (i.e. their respective native frequencies). In principle, this problem constitutes a network formation game^{31}. However, the involved nonlinearities make an analytical calculation of equilibria for systems of interesting size unfeasible. Instead, in the spirit of Brede^{28} we consider a dynamics in which agents subsequently attempt to approach their aims by local rewiring.
More precisely, consider a set of N nodes each of which disposes of k_{out} links with which to influence the collective dynamics. After initially assigning connections at random (thus generating a random graph which is regular in outdegree), nodes then iterate the following process: (i) a random focus node x is chosen and one of the outconnections of the focus node, say x → y, is selected at random and a rewiring of this connection to a random target node z is considered; (ii) node x will change from influencing y to influencing z if the respective rewiring will result in a collective frequency of the ensemble that is closer to x’s native frequency than the collective frequency of the ensemble with the link x → y in place. If this is not the case, the rewiring is not performed and the network remains unaltered. Steps (i) and (ii) are iterated until some quasistationary equilibrium is reached. For comparison, we also consider a conventional optimization scheme in which links x → y are rewired to x → z if the respective rewiring results in a more synchronized system state. See Fig. 1 for a schematic illustration of the two rewiring schemes. Note that in both schemes the distribution of outdegrees remains regular throughout the procedure – each node has the same potential to influence the system – while the indegrees will change.
In the following we consider a numerical implementation of the above scheme. To estimate collective frequencies of the oscillator ensemble, numerical integration of Eq. (1) is performed using a RungeKutta 4th order method with step size 0.1 over a time interval [0, T] where we usually set T = 1000 which is long enough to reach equilibration after T/2. After discarding an initial transient up to T/2 collective frequencies are then estimated by \({\rm{\Omega }}=\mathrm{1/}N{\langle {\sum }_{i}{\dot{{\varphi }}}_{i}\rangle }_{[T\mathrm{/2,}T]}\), where 〈 ⋅ 〉_{[T/2,T]} stands for a time average over the interval [T/2, T]. We have experimented with different system sizes of systems up to N = 500 oscillators and different integration steps, but for reasons of limited computation time we only report qualitatively robust results for networks of N = 100 oscillators below. Since rewiring in our model is based on node frequencies, we also measure frequency synchronization via
which measures the ratio of the standard deviations σ of the distribution of the average instananeous frequencies \({\dot{{\varphi }}}_{i}\) over the period [T/2, T] relative to the standard deviation of native frequencies. Thus Δf = 0 indicates perfect frequency synchronization and Δf ≈ 1 corresponds to a state without frequency synchronization.
Results
Rewiring dynamics
By plotting some key system properties as a function of the number of rewiring iterations, Fig. 2 illustrates typical dynamics of the competitive influencemaximization for some specific system configurations for a network of N = 100 nodes, outdegree k = 6 and coupling strength κ = 0.23. The choice of a fixed outdegree reflects the idea that there is a limit to how many separate nodes can be influenced by any one agent in the system. For that level of coupling in random configurations typically partial synchronization is found corresponding to an order parameter of r ≈ 0.58. Then, starting the competitive rewiring process from this low level of synchronization essentially two possible outcomes are observed. First, for some starting configurations of networks and native frequencies global synchronization initially declines and then fluctuates around some low level concomitant with ongoing fluctuations in the average instantaneous frequency Ω. At the same time, the rewiring dynamics remains active with about 45% of suggested rewirings being accepted (Fig. 2c), and no global consensus is found, even after very large numbers of iterations. In these cases influence maximization results in the formation of a coreperiphery network which we will discuss in more detail below. Second, for some other initial conditions, concurrent enhancement of synchronization and influence maximization is found. In these cases the system always ends up in an almost fully synchronized state in which all nodes have been recruited into a frequency locked cluster. The saturation towards this state goes hand in hand with a drop off in rewiring activity so that eventually a static network configuration is approached, i.e. an equilibrium is reached that is stable in the sense of not being unilaterally changeable by the requirements of individual nodes. The discrete nature of possible outcomes is illustrated in panel (f) in which we plot the distribution of final values of order parameters estimated from a sample of 100 independent simulation runs. One notes that this distribution is clearly bimodal with peaks around r ≈ 0.35 and r ≈ 0.98 marking nonsynchronized and synchronized networks. As we will explore in more detail later, the high synchrony peak quickly grows to domination when larger coupling strengths κ > 0.23 are chosen such that, e.g., for κ = 0.27 all 100 evolved configurations grow to full synchronization. Interestingly, such configurations do not only show enhanced synchronization for the coupling strength at which they were evolved, they also show markedly improved synchronization levels for a large range of coupling strengths and an onset of macroscopic synchronization for much lower coupling than random networks, c.f. Fig. S1 in the Supplementary material.
We also see that for synchronizing and nonsynchronizing trajectories the network topology is affected in different ways. To illustrate this and follow the dynamics of network change, we monitor two characteristics of network organization. First, to get an impression of changes in the degree distributions, panel Fig. 2d shows the maximum indegree. Large fluctuations around a mean value distinctly above the average for random configurations are found for non synchronized situations, whereas after an initial transient a reduction in degree heterogeneity is observed when convergence towards synchronization occurs. In panel Fig. 2e we plot the strengths of correlations between linked native frequencies measured by Newman’s homophily coefficient^{32}. Saturation towards strong anticorrelations occurs when approaching full synchronization, and fluctuations about a slightly negatively anticorrelated native frequency arrangement are found in the opposite case.
Network organization
In fact, synchronized and nonsynchronized networks evolve to very distinct topologies. In Fig. 3a more detailed analysis and comparison to optimized and random networks is presented. All the evolved networks show degree distributions clearly distinct from random and optimized configurations, cf. Fig. 3a. The degree distributions of nonsynchronized networks are skewed and while some nodes tend to become disconnected there is a clear tendency for hub formation. In contrast, for the synchrony enhanced networks the support of the distributions is narrow, where a clear peak for nodes of degree three stands out. These results agree with previous findings for synchrony enhancement for identical^{33,34} and nonidentical^{19,20} oscillators. Also correlations between native frequencies and indegree show clear patterns, see Fig. 3b. For the synchronized and optimized networks a positive correlation is observed such that oscillators with native frequencies far off the mean frequency have large indegrees – as is required to draw them into the collective rhythm. However, note a small difference between the optimized and evolved networks for small native frequencies when the relationship reaches a plateau towards ω = 0 in one case and exhibits a sharp minimum in the other. The opposite correlation pattern holds true for nonsynchronized networks, for which links are concentrated around oscillators with native frequencies close to the center. In spite of a small tendency towards disassortative degree mixing measured by a = −0.16 ± 0.04^{32} one concludes that the evolved nonsynchronized networks have a coreperiphery architecture (see also Supplementary Figs S6 and S7 for more detail). Further analysis of synchronized clusters shows that synchronization prevails within the core (see Supplementary Fig. S3), but the majority of the offcentre native frequency oscillators have too little input from the core and remain unsynchronized. Synchronized systems typically exhibit strong disassortative mixing by degree, which can be concluded from Fig. 3f in which we plot the dependence of average neighbor degrees on native frequencies combined with information in panel (b). Disassortative mixing by degree has been observed to be a hallmark of highly synchronizable networks before^{35,36}, supporting our claim that competitive influence maximization tends to enhance synchronization. We note that mixing patterns are particularly strong in the competitively evolved networks (a = −0.73 ± 0.06 vs. a = −0.19 ± 0.05 for the optimized networks), hinting at further correlation patterns.
Indeed, nontrivial patterns are also found in correlations of native frequencies of linked oscillators. We illustrate them in panels (c) and (d) as average and variance of the distribution of native frequencies of nodes with given native frequencies. A richer picture is obtained by visualizations of the ω − ω correlation matrices in which is recorded the probability that a node of native frequency ω_{1} is linked in the evolved network to a node of native frequency ω_{2}, c.f. Fig. 3g–i. Here the xaxis denotes the ‘control’ node label for the frequency, as it seeks to drag the collective frequency to it, whereas the y – axis denotes the ‘receive’ node label. As observed before^{20,22,24}, arrangements are generally anticorrelated for synchronyenhanced networks, see panels (g) and (i) in which most probability mass is concentrated in the 2nd and 4th quadrant, but have failed to evolve towards a clear pattern in the unsynchronized networks, c.f. panel (h). Correlation patterns are further distinct for competitively evolved and optimized networks. In the former case, nodes with native frequency ω almost exclusively receive input from other nodes with native frequency in a narrow band of frequencies around −sgn(ω)(1 − ω); within this, nodes with ω ≈ 0 in the center are controlled and controlling nodes with ω = ±1 at the boundary of the native frequency distribution (hence also the strong peak in variance in panel (d) and the plateau in the dependence of 〈k_{in}〉 on ω in panel (b)). This connection pattern, which is also related to enhanced frequencies of reciprocated links (see Fig. 3e), arises from the particular way in which clusters of synchronized nodes can grow under the competitive rewiring dynamics. Such clusters initially mostly consist of oscillators in the center of the native frequency distribution, and then broaden out to include oscillators further and further away from the mean (see Supplementary Fig. S3 for more details).
This leads to the following picture of influence maximization. Oscillators generally tend to enhance their influence by linking to nodes with a large negative frequency distance. Further, nodes with many incoming links are more difficult to influence than nodes with lower indegree. Thus, a node’s attractiveness to acquire new connections is a tradeoff between its native frequency gap to the target and its indegree. Consequently, starting from a synchronized seed cluster of nodes with close to zero native frequency, nodes within the cluster initially tend to link to nodes at the extreme end of the frequency distribution. Nodes not yet part of the synchronized cluster can enhance their influence in two ways. The first is by establishing connections to other nodes with maximum native frequency gap, even if these might not be strongly enough connected to the synchronized cluster to influence it. Alternatively, their influence can be enhanced by linking to nodes within the synchronized cluster, thus being able to change the average rhythm of this cluster. What emerges is a tradeoff between frequency gap maximization and linking to large synchronized clusters, in that links to nodes in the synchronized cluster with maximized frequency gap are reciprocated. When indegrees of the farthest offcenter nodes have saturated, the next generation of closetocenter native frequency nodes recruited into the synchronized cluster will establish connections to nodes slightly off the extremes of the native frequency distribution, and so on, explaining the correlations shown in Fig. 3f.
Regimes for different coupling strengths
Next we proceed with a more systematic exploration of the dependence of the evolved networks on the coupling strengths. A summary of results is reported in Fig. 4 which also includes comparisons to random networks in the absence of optimization. Panels (a) and (b) give results of the phase and frequency synchronization order parameters on the coupling strength; the remaining panels illustrate the change of various network properties with coupling. Noticing the two plateaus and the distinct very low coupling phase in panels (c–f) of 4 it becomes clear that three regimes of coupling strengths exist: (i) a regime of very low coupling κ ≤ 0.02 in which networks with very strong positive assortment by native frequency are formed, (ii) a regime of low to intermediate coupling 0.03 ≤ κ ≤ 0.17 in which coreperiphery type networks as described above emerge, and (iii) a regime of fairly high coupling κ ≥ 0.2 in which competitive influence maximization enhances synchronization. In regime (i) due to large maximum indegrees (cf. 4c) networks become very heterogeneous, short loops and triangles are avoided (cf. 4d,e), and the strongest hallmark of network organization becomes a preponderant tendency of oscillators of like native frequency to link to each other (cf. 4f). Networks in this regime have no clearly defined unique core. Network structures pertaining to regime (ii) and (iii) have been discussed above for the specific case of synchronizing and nonsynchronizing networks evolved for κ = 0.23, but in Fig. 4 we notice that their properties are indeed typical for the corresponding regimes. Hence, in regime (ii) networks are distinguished by cores comprised of closetothecenter native frequency oscillators, whereas the large peripheral parts of the network retain their closeto random organization. In regime (iii) hallmarks of network organization known to be associated with enhanced synchronization emerge. This rough subdivision of phases and the tendency for enhancement of synchronization with influence maximization in phase (iii) are not unique to uniform distributions of native frequencies. To ensure the robustness of our result, we also investigated Gaussian and bimodal distributions of native frequencies and find similar behaviour in these cases (see Supplementary Fig. S2 for details).
Partial rewiring
It is also of interest to investigate situations in which not all Kuramoto agents participate in the rewiring process. Accordingly, we designed experiments in which only a fraction ρ of all oscillators is selected for active rewiring, i.e. only selected oscillators competitively rewire their outconnections whereas outconnections of all other oscillators remain fixed during the simulation. We also explore scenarios with a bias for oscillator selection: either by only selecting a fraction ρ of oscillators with native frequencies close to the center of the distribution or by selecting oscillators starting at the extremes of the distribution. Technically, this selection is achieved by ordering nodes by ascending absolute value of their native frequencies \({\{{\omega }_{i}\}}_{i=0}^{N1}\) such that ω_{ i } ≥ ω_{ j } all i > j, and either setting the first i = 0, ..., ρN oscillators (center first) or the last oscillators i = (1 − ρ)N, ..., N (extremes first) active for rewiring. For maximum sensitivity to ρ experiments were carried out in the transition region with κ = 0.23. Results are illustrated in Fig. 5a in which the dependence of the phase synchronization order parameter r on the density of active oscillators ρ is shown. Since the distribution of outcomes of the competitive influence maximization is bimodal (see Fig. 2f), we also define a fraction of synchronizing configurations by counting the fraction n_{ s } of all runs in which networks have achieved r > 0.7 (note that any threshold in between the high and low synchrony peaks can be chosen here) and plot n_{ s } vs ρ in Fig. 5b. Inspecting Fig. 5a,b several observations stand out. First, substantial enhancement of synchrony can already be achieved when only a small fraction of oscillators participates in the rewiring. Second, the relationship between ρ and measures of synchronization is generally not monotonic: a certain share of oscillators participating is required to achieve high synchronization, but effects of competition when too many oscillators participate has a slightly detrimental effect on the level of synchronization achieved. Third, we notice that biased selection of rewiring oscillators influences outcomes. If oscillators with native frequencies close to the center are preferentially chosen, high levels of synchrony can be achieved with fewer actively participating oscillators, but the level of achieved phase synchronization is lower, since some high native frequency oscillators could not be recruited into the synchronized cluster. In contrast, biasing towards off the center oscillators for rewiring can boost synchronization beyond random selection, but a larger fraction of oscillators (compared to centrefirst bias) needs to be selected to achieve this.
Dependence on connectivity
As a last point we address the dependence of results on the connectivity of networks, where up till now k_{out} = 6 has been used. For this purpose, experiments are repeated for outdegrees ranging between k_{out} = 3 and k_{out} = 10 where saturation is observed. To enable sensible comparisons between different connectivities, coupling strengths for each connectivity were tuned to give the same degree of phase synchronization for outdegree regular random networks as for the reference case of k_{out} = 6. In this way, for each connectivity experiments started with the same initial degree of phase synchronization. Then, competitive influence maximizing rewiring was carried out and the stationary values of r and the fraction of synchronizing configurations n_{ s } were recorded. Results are displayed in Fig. 5c,d, where we notice that influence maximization increasingly leads to concurrent enhancement of synchronization the larger the connectivity of the network. In fact, we find transitionlike behaviour: for low connectivities no enhancement of synchronization through influence maximization can be achieved, whereas for larger connectivities and large enough coupling strengths almost all configurations allowed for influence maximization to substantially boost synchronization.
Discussion
In this paper we have considered a process of network formation between nonidentical Kuramoto oscillators based on individual oscillators competitively rewiring their outconnections with an aim to maximize their influence on the collective. For this we identified oscillators’ native frequencies with their goals and subsequently considered a scenario in which each oscillator tries to arrange its connections with the aim to make the system synchronize to its native frequency. We have shown that outcomes of such competitive influencemaximizing rewiring depend on the coupling strength and connectivity. Regimes for low and intermediate coupling exist, in which networks are rewired towards coreperiphery architectures in which at least part of the closeto center oscillators are recruited to a frequency locked cluster. More interestingly, however, for all the scenarios we investigated, a regime exists in which competitive influence between agents of very widely divergent goals drives the system toward a synchronyenhanced state. We find that typical network configurations in this state bear many of the hallmarks of network organization previously associated with synchronyenhancement^{20,22,24,29}. Additionally, however, networks also carry a signature resulting from the growth process of the largest synchronized cluster, namely particular correlations of native frequencies of adjacent oscillators.
Traditionally, as pioneered by^{37} influence maximization has been typically considered in the very different context of one agent attempting to convert opinions of sets of agents with binary opinions in probabilistic settings. In this paper we are the first to consider the other extreme of the spectrum: a set of many agents with widely divergent goals competitively attempting to maximize their individual impact on the system. As we show for the particular case of nonlinearly diffusively coupled Kuramoto agents, such a process can actually enhance consensus formation. Interestingly, this is already the case if only a small fraction of agents take part in the rewiring and can be optimized if only a certain fraction of agents is active.
We believe our results are of importance in at least two respects. First, to our knowledge, synchronyenhancement has mostly been considered in a context where a central planner has the power to arrange system structure^{18,20,23}. Here we show how enhanced synchrony might arise as a byproduct of influence maximization which might be a more realistic pathway for systems comprised of many selfinterested heterogeneous agents. Second, results of this work would be of interest in the wider context of consensus formation in social dynamics^{38}. One wonders if our particular results in the context of the Kuramoto model can be generalized to typical models of consensus formation, which we believe is an interesting direction for future work.
References
 1.
Pikovsky, A., Rosenblum, M. & Kurths, J. Synchronization: A Universal Concept in Nonlinear Sciences. (Cambridge University Press, Cambridge, UK, 2003).
 2.
Arenas, A., DiazGuilera, A., Kurths, J., Moreno, Y. & Zhou, C. Synchronization in complex networks. Phys. Rep. 469, 93–153 (2008).
 3.
Dörfler, F. & Bullo, F. Synchronization in complex networks of phase oscillators: A survey. Automatica 50, 1539–1564 (2014).
 4.
Rodrigues, A., Peron, T. K. D., Peng, J. & Kurths, J. The kuramoto model in complex networks. Physics Reports 610, 1–98 (2016).
 5.
Pluchino, A., Latora, V. & Rapisarda, A. Changing opinions in a changing world: A new perspective in sociophysics. Int. J. Mod. Phys. C 16, 515–531 (2005).
 6.
Pluchino, A., Boccaletti, S., Latora, V. & Rapisarda, A. Opinion dynamics and synchronization in a network of scientific collaborations. Physica A 372, 316–325 (2006).
 7.
Watts, D. J. & Strogatz, S. H. Collective dynamics of ‘smallworld’ networks. Nature (London) 393, 440 (1999).
 8.
Baharona, M. & Pecora, L. T. Synchronization in smallworld systems. Phys. Rev. Lett. 89, 054101 (2002).
 9.
Moreno, Y. & Pacheco, A. F. Synchronization of kuramoto oscillators in scalefree networks. Europhysics Letters 68, 603–609 (2004).
 10.
Lee, D.S. Synchronization transition in scalefree networks: Clusters of synchrony. Phys. Rev. E 72, 026208 (2005).
 11.
Arenas, A., DiazGuilera, A., Latora, V. & PerezVicente, C. J. Synchronization reveals topological scales in complex networks. Phys. Rev. Lett. 96, 114102 (2006).
 12.
Chavez, M., Hwang, D. U., Amann, A., Hentschel, H. G. & Boccaletti, S. Synchronization is enhanced in weighted complex networks. Phys. Rev. Lett. 94, 218701 (2005).
 13.
GómezGardeñes, J., Moreno, Y. & Arenas, A. Paths to synchronization on complex networks. Phys. Rev. Lett. 98, 034101 (2007).
 14.
Zhou, C. & Kurths, J. Dynamical weights and enhanced synchronization in adaptive complex networks. Phys. Rev. Lett. 96, 164102 (2006).
 15.
Ren, Q. & Zhao, J. Adaptive coupling and enhanced synchronization in coupled phase oscillators. Phys. Rev. E 76, 016207 (2007).
 16.
Papadopoulos, L., Kim, J., Kurths, J. & Bassett, D. S. Development of structural correlations and synchronization from adaptive rewiring in networks of kuramoto oscillators. Chaos 27, 073115 (2017).
 17.
Eom, Y.H., Boccaletti, S. & Caldarelli, G. Concurrent enhancement of percolation and synchronization in adaptive networks. Sci. Rep. 6, 27111–330 (2016).
 18.
Donetti, L., Hurtado, P. I. & Munoz, M. A. Entangled networks, synchronization, and optimal network topology. Phys. Rev. Lett. 95, 188701 (2005).
 19.
Brede, M. Synchronyoptimized networks of nonidentical kuramoto oscillators. Phys. Lett. A 372, 2618–2622 (2008).
 20.
Brede, M. Construction principles for highly synchronizable sparse directed networks. Phys. Lett. A 372, 5305–5308 (2008).
 21.
Brede, M. Optimal synchronization on strongly connected directed networks. Eur. Phys. J. B 74, 217–225 (2010).
 22.
Kelly, D. & Gottwald, G. A. On the topology of synchrony optimized networks of a kuramotomodel with nonidentical oscillators. Chaos 21, 025110 (2011).
 23.
Skardal, P. S., Taylor, D. & Sun, J. Optimal synchronization of complex networks. Phys. Rev. Lett. 113, 144101 (2014).
 24.
Pinto, R. S. & Saa, A. Optimal synchronization of kuramoto oscillators: A dimensional reduction approach. Phys. Rev. E 92, 062801 (2015).
 25.
GomezGardenes, J., Gomez, S. & Arenas, A. Explosive synchronization transitions in scalefree networks. Phys. Rev. Lett. 106, 128701 (2011).
 26.
Gottwald, G. A. Model reduction for networks of coupled oscillators. Chaos 25, 053111 (2015).
 27.
Antonioni, A. & Cardillo, A. Coevolution of synchronization and cooperation in costly networked interactions. Phys. Rev. Lett. 118, 238301 (2017).
 28.
Brede, M. Optimal synchronization in space. Phys. Rev. E 81, 025202 (2010).
 29.
Brede, M. Locals vs global synchronization in networks of nonidentical kuramoto oscillators. Eur. Phys. J. B 62, 87–94 (2008).
 30.
Kuramoto, Y. Chemical Oscillations, Waves and Turbulence. (Springer, Berlin, 1984).
 31.
Jackson, M. O. A survey of models of network formation: Stability and efficiency. Working paper 1161, California Institute of Technology, Division of the Humanities and Social Sciences (2003).
 32.
Newman, M. E. J. Networks: An Introduction. (Oxford University Press, Oxford, UK, 2010).
 33.
Nishikawa, T., Motter, A. E., Lai, Y.C. & Hoppensteadt, F. C. Heterogeneity in oscillator networks: Are smaller worlds easier to synchronize? Phys. Rev. Lett. 91, 014101 (2003).
 34.
Motter, A. E., Zhou, C. & Kurths, J. Network synchronization, diffusion, and the paradox of heterogeneity. Phys. Rev. E 71, 016116 (2005).
 35.
di Bernardo, M., Garofalo, F. & Sorrentino, F. Synchronizability and synchronization dynamics of weighed and unweighed scale free networks with degree mixing. International Journal of Bifurcation and Chaos 17, 2419–2434 (2007).
 36.
Chavez, M., Hwang, D.U., Martinerie, J. & Boccaletti, S. Degree mixing and the enhancement of synchronization in complex weighted network. Phys. Rev. E 74, 066107 (2006).
 37.
Kempe, D., Kleinberg, J. & Tardos, E. Maximizing the spread of influence through a social network. Proceedings of the Nineth International Conference on Knowledge discovery and Data Mining (KDD), Washington, DC, USA 137–146 (2003).
 38.
Castellano, C., Fortunato, S. & Loreto, V. Statistical physics of social dynamics. Rev. Mod. Phys. 81, 591–646 (2009).
Acknowledgements
The authors acknowledge the use of the IRIDIS High Performance Computing Facility, and associated support services at the University of Southampton, in the completion of this work. ACK was supported by a Chief Defence Scientist Fellowship and thanks the University of Southampton for its hospitality when this work was initiated. M.S. was supported by an EPSRC DTC grant (EP/G03690X/1).
Author information
Affiliations
Contributions
M.B., M.S., and A.K. conceived the experiment(s), M.B. conducted the experiment(s), M.B., M.S., and A.K. analyzed the results. All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Competing Interests
The authors declare that they have no competing interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Brede, M., Stella, M. & Kalloniatis, A.C. Competitive influence maximization and enhancement of synchronization in populations of nonidentical Kuramoto oscillators. Sci Rep 8, 702 (2018). https://doi.org/10.1038/s4159801718961z
Received:
Accepted:
Published:
Further reading

Analysis of Influence Maximization in Temporal Social Networks
IEEE Access (2019)

How Does Active Participation Affect Consensus: Adaptive Network Model of Opinion Dynamics and Influence Maximizing Rewiring
Complexity (2019)

Investigating Peer and Sorting Effects within an Adaptive Multiplex Network Model
Games (2019)

Maximizing threehop influence spread in social networks using discrete comprehensive learning artificial bee colony optimizer
Applied Soft Computing (2019)

Distance Entropy Cartography Characterises Centrality in Complex Networks
Entropy (2018)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.