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Synchronizability determined by coupling strengths and topology on Complex Networks

We investigate in depth the synchronization of coupled oscillators on top of complex networks with different degrees of heterogeneity within the context of the Kuramoto model. In a previous paper [Phys. Rev. Lett. 98, 034101 (2007)], we unveiled how for fixed coupling strengths local patterns of synchronization emerge differently in homogeneous and heterogeneous complex networks. Here, we provide more evidence on this phenomenon extending the previous work to networks that interpolate between homogeneous and heterogeneous topologies. We also present new details on the path towards synchronization for the evolution of clustering in the synchronized patterns. Finally, we investigate the synchronization of networks with modular structure and conclude that, in these cases, local synchronization is first attained at the most internal level of organization of modules, progressively evolving to the outer levels as the coupling constant is increased. The present work introduces new parameters that are proved to be useful for the characterization of synchronization phenomena in complex networks.


I. INTRODUCTION
Studies on the emergence of collective and synchronized dynamics in large ensembles of coupled units have been carried out since the beginning of the nineties in different contexts and in a variety of fields, ranging from biology, ecology, and semiconductor lasers, to electronic circuits [1,2,3]. Collective synchronized dynamics has multiple applications in technology, and is a common framework to investigate the crucial features in the emergence of critical phenomena in natural systems. For instance, it is a relevant issue to fully understand some diseases that appear as the result of a sudden and undesirable synchronization of a large number of neuronal units [4]. Recently, synchronization phenomena have also been proved to be helpful outside the traditional fields where it applies, for instance, in sociology where it can be used to study the mechanisms leading to the formation of social collective behaviors [5,6].
Among the many models that have been proposed to address synchronization phenomena, one of the most successful attempts to understand them is due to Kuramoto [7,8], who capitalized on previous works by Winfree [9], and proposed a model system of nearly identical weakly coupled limit-cycle oscillators. The Kuramoto (KM) mean field case corresponding to a uniform, all-to-all and sinusoidal coupling is described by the equations of motion, sin (θ j − θ i ) (i = 1, ..., N ) . (1) where the factor 1/N is incorporated in order to ensure a good behavior of the model in the thermodynamic limit, N → ∞, ω i stands for the natural frequencies of the oscillators, and K is the coupling constant. Moreover, the coherence of the population of N oscillators is measured by the complex order parameter, where the modulus 0 ≤ r(t) ≤ 1 measures the phase coherence of the population and φ(t) is the average phase. In what follows, we will focus on the synchronization of coupled oscillators described by the dynamics Eq. (1), because of its validity as an approximation for a large number of nonlinear equations and its ubiquity in the nonlinear literature [10]. The KM approach to synchronization was a great breakthrough for the understanding of the emergence of synchronization in large populations of oscillators, in particular it presents a second-order phase transition from incoherence to synchronization, in the order parameter Eq.(2) for a critical value of the coupling constant. However, a large amount of real systems do not show a homogeneous pattern of interconnections among their parts [11,12] where the original KM assumptions apply.
Many real natural [13,14], social [15] and technological [16,17,18] systems conform as networks of nodes with connectivity patterns that diverge considerably from homogeneity, and are usually characterized by a scale-free degree distribution, P (k) ∼ k −γ (the degree k is the number of connections of a node). The study of processes taking place on top of scale-free networks has led to reconsider classical results obtained for regular lattices or random graphs due to the radical changes of the system's dynamics when the heterogeneity of the connectivity patterns can not be neglected [19,20,21,22,23,24]. In this case one has to deal with two sources of complexity, the nonlinear character of the dynamics and the complex structures of the substrate, which are usually entangled. A contemporary effort to attack this entangled problem was due to Watts and Strogatz, that in 1998, trying to understand the synchronization of cricket chirps, which show a high degree of coordination over long distances as though the insects where "invisibly" connected, end up with a seminal paper [25] about the small-world connectivity property. This work was the seed of the modern theory of complex networks [11,12]. Nevertheless, the understanding of the synchronization dynamics in complex networks still remains a challenge.
In recent years, scientists have addressed the problem of synchronization on complex networks capitalizing on the Master Stability Function (MSF) formalism [26] which allows to study the stability of the fully synchronized state [27,28,29,30,31,32,33,34]. The MSF is the result of a linear stability analysis for a completely synchronized system. While the MSF approach is useful to get a first insight into what is going on in the system as far as the stability of the synchronized state is concerned, it tells nothing about how synchronization is attained and whether or not the system under study exhibits a transition similar to the original KM. To this end, one must rely on numerical calculations and explore the entire phase diagram. Surprisingly, there are only a few works that have dealt with the study of the whole synchronization dynamics in specific scenarios [35,36,37,38,39,40] as compared with those where the MSF is used, given that the onset of synchronization is reacher in its behavioral repertoire than the state of complete synchronization.
In a previous work [41], we have shown how, for fixed coupling strengths, local patterns of synchronization emerge differently in homogeneous and heterogeneous complex networks, driving the process towards a certain degree of global synchronization following different paths. In this paper, we extend the previous work to different topologies, even those with modular structure, and report more results supporting the previous claim. First, we extend the analysis carried out in [41] to networks in which the degree of heterogeneity can be tuned between the two limits of random scale-free networks and random graphs with a Poisson degree distribution. Second, in order to get further insights about the role of the structural properties on the route towards complete synchronization, we study the same dynamics on top of networks with a non-random structure at the mesoscopic level, i.e., networks with communities. The results support the usefulness of the tools developed and highlight the relevance of synchronization phenomena to study in detail the relationship between structure and function in complex networks.

II. KM MODEL ON COMPLEX NETWORKS
Let us now focus on the paradigmatic Kuramoto model. In order to manage with the KM on top of complex topologies we reformulate eq. (1) to the form where Λ ij is the coupling strength between pairs of connected oscillators and A ij is the connectivity matrix (A ij = 1 if i is linked to j and 0 otherwise). The original Kuramoto model introduced above assumed mean-field interactions so that A ij = 1, ∀i = j (all-to-all) and Λ ij = K/N, ∀i, j.
The first problem when dealing with the KM in complex networks is the definition of the dynamics. In the seminal paper by Kuramoto [7], eq. (1), the coupling term in the right hand side of eq. (3) is an intensive magnitude. The dependence on the number of oscillators N is avoided by choosing Λ ij = K/N . This prescription turns out to be essential for the analysis of the system in the thermodynamic limit N → ∞. However, choosing Λ ij = K/N the dynamics of the KM in a complex network becomes dependent on N . Therefore, in the thermodynamic limit, the coupling term tends to zero except for those nodes with a degree that scales with N [42].
A second prescription consists of taking Λ ij = K/k i (where k i is the degree of node i) so that Λ ij is a weighted interaction factor that also makes intensive the right hand side of Eq. (3). This form has been used to solve the so-called paradox of heterogeneity that states that the heterogeneity in the degree distribution, which often reduces the average distance between nodes, may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength [34]. One should consider this result carefully because it refers to the stability of the fully synchronized state (see below) not to the whole evolution of synchronization in the network. More important, the inclusion of weights in the interaction strongly affects the original KM dynamics in complex networks because it imposes a dynamic homogeneity that could mask the real topological heterogeneity of the network.
Finally, the prescription Λ ij = K [39,43,44], which may seem more appropriate, also presents some conceptual problems because the sum in the right hand side of eq. (3) could eventually diverge in the thermodynamic limit if synchronization is achieved. To our understanding, the most accurate interpretation of the KM dynamics in complex networks should preserve the essential fact of treating the heterogeneity of the network independently of the interaction dynamics, and at the same time, should remain calculable in the thermodynamic limit. Taking into account these factors, the interaction Λ ij in complex networks should be inversely proportional to the largest degree of the system Λ ij = K/k max = λ keeping in this way the original formulation of the KM valid in the thermodynamic limit (in SF networks k max ∼ N 1/(γ−1) ). In addition, the same order parameter, eq. (2), can be used to describe the coherence of the synchronized state. Since k max is constant for a given network, the physical meaning of this prescription is a re-scaling of the time units involved in the dynamics. Note, however, that for a proper comparison of the synchronizability of different complex networks, the global and local measures of coherence should be represented according to their respective time scales. Therefore, given two complex networks A and B with k max = k A and k max = k B respectively, the comparison between observable must be done for the same effective coupling K A /k A = K B /k B = λ. With this formulation in mind eq. (3) reduces to independently of the specific topology of the network. This allow us to study the dynamics of eq. (4) over different topologies in order to compare the results and properly inspect the interplay between topology and dynamics in what concerns to synchronization.

III. HOMOGENEOUS VERSUS HETEROGENEOUS TOPOLOGIES
Recent results have shed light on the influence of the local interactions' topology on the route to synchronization [36,40]. However, in these studies at least two parameters (clustering and average path length) vary along the studied family of networks. This paired evolution, although yielding an interesting interplay between the two topological parameters, makes it difficult to distinguish what effects were due to one or other factors. Here, we would like to address first what is the influence of heterogeneity, keeping the number of degrees of freedom to a minimum for the comparison to be meaningful. The family of networks used in the present section are comparable in their clustering, average distance and correlations so that the only difference relies on the degree distribution, ranging from a Poissonian type to a scale-free distribution. Later on in this paper, we will relax these constraints and study networks in which the main topological feature is given at the mesoscopic scale, i.e., networks with community structure.
Therefore, let us first scrutinize and compare the synchronization patterns in Erdös-Rényi (ER) and Scale-Free (SF) networks. For this purpose we make use of the model proposed in [45] that allows a smooth interpolation between these two extremal topologies. Besides, we introduce a new parameter to characterize the synchronization paths to unravel their differences. The results reveal that the synchronizability of these networks does depend on the coupling between units, and hence, that general statements about their synchronizability are eventually misleading. Moreover, we show that even in the incoherent solution, r = 0, the system is self-organizing towards synchronization. We will analyze in detail how this self-organization is attained.
The first numerical study about the onset of synchronization of Kuramoto oscillators in SF networks [39] revealed the great propense of SF networks to synchronization, which is revealed by a non-zero but very small critical value λ c [46]. Besides, it was observed that at the synchronized state, r = 1, hubs are extremely robust to perturbations since the recovery time of a node as a function of its degree follows a power law with exponent −1. However, how do SF networks compare with homogeneous networks and what are the roots of the different behaviors observed?
We first concentrate on global synchronization for the Kuramoto model Eq. (4). For this we follow the evolution of the order parameter r, Eq. (2), as λ increases, to capture the global coherence of the synchronization in networks. We will perform this analysis on the family of networks generated with the model introduced in [45]. This model generates a one-parameter family of networks labeled by α ∈ [0, 1]. The parameter α measures the degree of heterogeneity of the final networks so that α = 0 corresponds to the heterogeneous BA network and α = 1 to homogeneous ER graphs. For in- and k = 6 (N l = 3 · 10 4 ) and were generated using the model introduced in [45] termediate values of α one obtains networks that have been grown combining both preferential attachment and homogeneous random linking so that each mechanism is chosen with probabilities (1 − α) and α, respectively. It is worth stressing that the growth mechanism preserves the total number of links, N l , and nodes, N , for a proper comparison between different values of α. Specifically, assuming the final size of the network to be N , the network is build up starting from a fully connected core of m 0 nodes and a set S(0) of N − m 0 unconnected nodes. Then, at each time step, a new node (not selected before) is chosen from S(0) and linked to m other nodes. Each of the m links is attached with probability α to a randomly chosen node (avoiding self-connections) from the whole set of N − 1 remaining nodes and with probability (1 − α) following a linear preferential attachment strategy [47]. After repeating this process N − m 0 times, networks interpolating between the limiting cases of ER (α = 1) and SF (α = 0) topologies are generated [45]. Furthermore, with this procedure, the degree of heterogeneity of the grown networks varies smoothly between the two limiting cases. The curves r(λ) for several network topologies ranging from ER to SF are shown in Fig.1. We have performed extensive numerical simulations of eq. (4) for each network substrate starting from λ = 0 and increasing it up to λ = 0.4 with δλ = 0.02. A large number (at least 500) of different network realizations and initial conditions were considered for every value of λ in order to obtain an accurate phase diagram. The natural frequencies ω i and the initial values of θ i were randomly drawn from a uniform distribution in the interval (−1/2, 1/2) and (−π, π), respectively. Fig.1 reveals the differences in the critical behavior as a function of the substrate heterogeneity. The global coherence of the synchronized state, represented by r, shows that the onset of synchronization first occurs for SF networks. As the net- work substrate becomes more homogeneous the critical point λ c shifts to larger values and the system seems to be less synchronizable. On the other hand, it is also clear that the route to complete synchronization, r = 1, is faster for homogeneous networks. That is, when λ > λ c (α) the growth rate of r increases with α. To inspect in depth the critical parameters of the system dynamics we perform a finite size scaling analysis. This allows to determine with precision the curve λ c (α) and study the critical behavior near the synchronization transition.
We assume a scaling relation of the form where f (x) is as usual a universal scaling function bounded as x → ±∞ and ν and β are critical exponents to be determined. The detailed analysis performed for both SF and ER topologies shows that the critical value of the effective coupling, λ c , corresponds in scale-free networks to λ SF c = 0.051, and in random networks to λ ER c = 0.122, accordingly with Fig.1. In both cases, the transition strongly recalls the classical transition of the original KM [7] with a critical exponent near 1/2 for the SF network [39]. For intermediate values of α, the results show that the critical point shifts to larger values as the degree of heterogeneity increases. They are shown in Table I together with some topological properties of the networks.
The differences between ER and SF topologies observed when looking at global patterns of synchronization motivate a more detailed study of the synchronization onset for both topologies. The original work by Kuramoto pointed out that at the onset of synchronization small clusters of locked oscillators emerge and that the recruitment of more oscillators into these clusters as the coupling is increased makes it larger the global coherence r of the system. Obviously the emergence of these clusters would depend on the underlying topology which drives the possible configurations that locked oscillators would eventually form. To see how this initial coherence is achieved we propose a new order parameter, r link . This parameter measures the local construction of the synchronization patterns [48] and allows for the exploration of how global synchronization is attained. We define being Γ i the set of neighbors of node i. The parameter r link measures the fraction of all possible links that are synchronized in the network. The averaging time ∆t should be taken large enough in order to obtain good measures of the degree of coherence between each pair of physically connected nodes. Besides, r link is computed after the system relaxes at some large time t r . Note that in the limit of all-to-all coupling the information provided by r link is exactly the same that the one provided by r because in this case r link ∝ r 2 . Therefore, no additional information would be provided by this new parameter in the all-to-all case. Here, however, it turns out to be the key parameter to characterize how synchronization emerges at a local scale.
In Fig.2 we represent the evolution of both order parameters, r and r link , as a function of the coupling strength λ for several values of α. The behavior of r link shows a change in synchronizability between ER and SF and provides additional information to that reported by r. Interestingly, the nonzero values of r link for λ ≤ λ c indicate the existence of some local synchronization patterns even in the regime of global incoherence (r ≈ 0). Right at the onset of synchronization for the SF network limit, its r link value deviates from that of the ER recovering the known result about the synchronization of SF networks for lower values of the coupling. In this region, while the synchronization patterns continue to grow for the ER network at the same rate, the formation of locally synchronized structures occurs at a faster rate in the SF network. Finally, when the incoherent solution in the ER network destabilizes, the growing in its synchronization pattern increases drastically up to values of r link comparable to those obtained in SF networks and even higher. For intermediate values of α, the results show that the effect of varying the heterogeneity of the underlying network is twofold. On one hand, the more heterogeneous the network is, the smaller the values of λ needed for the onset of synchronization. Conversely, the increase in the degree of heterogeneity results in larger values of λ in order to achieve complete synchronization. In short, as the heterogeneity is increased, the onset of synchronization is anticipated, but at the same time, the appearance of the fully synchronized state is delayed.
These results undoubtedly point out that statements about synchronizability are dependent on the coupling strength value. To shed new light on this phenomenon, we have studied the characteristics of the synchronization patterns along the evolution of r link . Following the usual picture, synchronization patterns are formed by pairs of oscillators, physically connected, whose phase difference in the stationary state tends to zero. In order to determine which pairs of nodes are truly synchronized we should determine the coherence of their dynamics. Note that eq.(6) is the average dynamical coherence between every pair of linked nodes and then the synchronization degree of every pair of connected oscillators can be written in terms of a symmetric matrix Then one has to analyze each matrix term D ij in order to label a link (i, j) as synchronized or not. As introduced above, from the computation of r link one determines the fraction of physical links that are synchronized so that one would expect that 2r link · N l elements of the matrix D are D ij = 1, while the remaining elements are D ij = 0. However, this is not the real situation since the network dynamics is not well defined in terms of a fully synchronized cluster and a set of completely incoherent oscillators. On the other hand, the worst scenario would be found if there were 2N l elements of matrix D so that D ij = r link , implying that all the physically connected pairs are equally synchronized and hence the parameter r link could not be interpreted as the fraction of links that are dynamically coherent and no information about the topological patterns of synchronization could be extracted from matrix D. The situation found is not as simple as the former possibility and not so dramatic as the latter. The contributions D ij of the N l elements of matrix D that correspond to physical links can be ordered from the highest to the lowest one. We have checked that for two situations, corresponding to the onset of synchronization (λ = 0.05) and when high global coherence (λ = 0.13) is observed for a SF network, synchronized links can be clearly identified. For the onset of synchronization, a subset of nearly 20% of links displaying coherent dynamics with high degree of synchronization, D ij > 0.8, is well separated from the behavior of the remaining links as a dramatic decrease of D ij takes place. In this sense, it is clear that the dynamics of a 20% of the possible pairs can be regarded as synchronized which is in agreement with the obtained value r link = 0.25 for λ = 0.05 and support that although macroscopic coherence is not observed (r ≃ 0 at this point) the system is seen to walk towards it. For λ = 0.13 (r link ≃ 0.82) a plateau of nearly 75% of links is observed, thus revealing the high degree of global coherence, r ≃ 0.7, at this point. Therefore, the shape of the ranked D ij curves confirm that r link gives the fraction of synchronized links and thus the latter allows to obtain information about synchronized patterns from D.
To determine exactly which pairs of nodes are regarded as synchronized, the matrix D is filtered using a threshold T such that the fraction of synchronized pairs equals r link . In this way T is a moving threshold so that if D ij > T oscillators i and j are considered synchronized. The value of T depends on the particular realization and is determined by means of an iterative scheme starting from T = 1. Decreasing it with δT = 0.01 one computes the amount of links that fulfills the condition. In this way, the value of T progressively decreases and more pairs of oscillators are chosen. The process lasts until T is such that the fraction of chosen links is equal to the desired value r link previously computed from D. Finally, when the synchronized links are identified the clusters of synchronized nodes are reconstructed. shows that for values of λ for which the incoherent solution r = 0 is stable, the networks have developed a largest cluster of synchronized pairs of oscillators involving 50% of the nodes, and an equal number of smaller synchronization clusters. From this point on, the behavior of both GC and N c depends on the specific value of α. When heterogeneity dominates, the GC grows and the number of smaller clusters goes down, whereas for less heterogeneous networks the growth of GC is more abrupt and nodes are incorporated to it more faster. Moreover, the results highlight the fact that although heterogeneous networks exhibit more coherence in terms of r and r link , the microscopic evolution of the synchronization patterns is faster in homogeneous networks, being these networks far more locally synchronizable than the heterogeneous ones once λ > λ c .
The observed differences in the behavior at a local scale are rooted in the growth of the GC. For homogeneous topologies, many small clusters of synchronized pairs of oscillators (note in Fig.3 the large number of clusters formed when a 15% of the links are synchronized) merge together to form a GC when the effective coupling is increased. This coalescence of many small clusters results in a giant component made up of almost the size of the system once the incoherent state destabilizes. On the other hand, for heterogeneous graphs, the growth of the giant component is more smooth and the oscillators form new pairs starting from a core made up of half the nodes of the network. That is, in one case (ER-like networks), almost all the nodes of the network takes part of the giant component from the beginning and latter on, when λ is increased, what is added to the GC are the links among these nodes that were missing in the original cluster of synchronized nodes. For SF-like networks, the mechanism is the opposite. Nodes are added to the GC together with most of their links, resulting in a growth of r link much slower than for the homogeneous topologies.
The above picture is confirmed in Fig.4, where we have represented the evolution of the local synchronization patterns of the giant components in ER and SF networks for several values of λ [49]. It is clear that when r ≃ 0 the two networks follow different paths toward synchronization. In particular, the giant component for the SF network seems to retain the topological features of the substrate network, while this is not the case for the ER network (for instance, the small-world prop-erty is clearly lacking).
This study about the patterns of self-organization towards synchronization reveals that the quantitative difference about the macroscopic behavior, shown by the computation of the evolution of the global coherence r for ER and SF networks, has its roots on a qualitatively different route at the microscopic level of description. The use of the new parameter r link which involves the computation of the degree of coherence between each pair of linked nodes is a useful tool for describing such differences. Moreover, the results suggest that the degree of heterogeneity of the network is the key ingredient to explain the two different routes observed. The technique developed to extract the synchronization patterns allows the analysis of the topological features of such clusters of nodes. We can compute the average measures of relevant quantities such as the clustering coefficient or the degree distribution, and see how these magnitudes evolve from the uncoupled limit, where no synchronization occurs, to the coherent regime where the synchronized network coincides with the underlying substrate. It is then relevant to explore the regions where the onset of syn- chronization takes place and characterize topologically these emergent synchronized clusters. In Fig.5 the evolution of the average clustering coefficient c sync of the giant synchronized cluster referred to c network in the underlying network, is plotted as a function of λ for both the BA and ER networks. It is worth mentioning that the results depicted in the figure have been computed taking into account that nodes with degree 1 does not contribute to the clustering coefficient of the GC, as c is not properly defined for these nodes. The results are illustrative of the local organization of synchronized nodes. The figure shows that for both topologies the clustering decreases as the coupling is increased beyond their respective λ c or, in other words, as the giant component grows by the addition of new synchronized pairs of nodes. However, the effects of the two different routes to complete synchronization observed for ER and SF networks are well appreciated from the results. For the heterogeneous network there is a smooth decrease of the clustering coefficient for λ > λ SF c and the effects of the emergence of global coherence are not dramatic in what refers to the behavior of c sync . This is because in this case the giant component mainly grows by recruiting new synchronized nodes and their links. On the other hand, for the ER graph the behavior observed for λ < λ ER c , i.e. when no macroscopic coherence is observed, is interrupted by a sudden jump near its critical value. In fact, for λ > λ ER c the clustering of the synchronized cluster quickly approaches the value of c of the substrate network. This effect becomes clear if one has in mind the coalescence of small clusters, which happens around the critical point for ER graphs. In fact, taking into account the giant synchronized component on ER for λ < λ ER c , implies to consider one of the several disjoint synchronized clusters of similar sizes that are in this region. Moreover, the coalescence process leads to the formation of a giant cluster that contains almost all the nodes of the network (see Fig. 3), but a number of links significantly smaller. Hence, when the clusters collapse into a much larger one, the topological features change dramatically as observed from the evolution of the clustering coefficient.
All the results reported above point out that the ultimate reason behind the two different routes to complete synchronization is the heterogeneous character of the SF network and the role played by the hubs. The natural cohesion that hubs provide to SF network prevents the existence of independent macroscopic clusters of synchrony as occurs for ER networks. It is then interesting to study how these hubs participate in the formation of the final synchronized state. For this, we first study the evolution with λ of the composition of the synchronized cluster in terms of the degree of its components. In [41], we reported the probability that a node with degree k belongs to the giant synchronized cluster as a function of its degree k and the coupling λ for the SF network. This probability turns out to be an increasing function of k for every value of λ and therefore the more connected a node is, the more likely it takes part in the cluster of synchronized links. In particular, the results confirm the hypothesis made above that the hubs participate from the very beginning on the formation of the synchronized cluster. A similar result was obtained in [51], where Zhou and Kurths studied the hierarchical organization in complex networks, using the MSF and a mean-field approach in the weak coupling limit. The above characterization of the synchronized cluster in terms of the degree of its component can be completed studying their effective degree, k int . The effective degree of a synchronized node is the number of links it shares with other nodes belonging to the same synchronized cluster. Obviously, at the complete synchronized regime a node with degree k will have k int = k. We have plotted in Fig.6 the quantity k int /k (the fraction of links that a node has with synchronized neighbors) as a function of λ and the degree k of the nodes (α = 0). The results reveal that although hubs are the first to take part of the synchronized cluster, their neighbors are progressively incorporated to the cluster as λ grows. Besides, if a node with small k is synchronized the probability that its neighbors are also synchronized grows very fast with λ which is an effect of the network topology. These results further support the statement about the essential role played by the hubs in the recruitment of oscillators into the synchronized group and in the emergence of complete synchronization in SF networks.

IV. SYNCHRONIZATION IN STRUCTURED NETWORKS
In light of the results of the above section we have extended the study beyond unstructured networks to structured or modular networks. This is a limiting situation in which the local structure may greatly affect the dynamics, irrespective of whether or not we deal with homogeneous or heterogeneous networks and then they constitute a perfect framework for testing the new order parameter r link introduced in the last section.
Many complex networks in nature are modular, i.e. composed of certain subgraphs with differentiated internal and ex-ternal connectivity that form communities [12,50]. The use of modular networks where a proper comparison in synchronizability can be performed (same number of nodes and links) restricts us to the consideration of synthetic structured networks. To this end, we make use of a common benchmark of random network with community structure, first proposed by Newman [52] considering one hierarchical level and later extended to several hierarchical levels [37,38].
The modular network structure we build is as follows: in a set of N nodes, we prescribe n compartments that will represent our first community organizational level, and m compartments, each one embedding four different compartments of the first level, that define the second organizational level of the network. The internal degree of nodes at the first level z in1 and the internal degree of nodes at the second level z in2 keep an average degree z in1 + z in2 + z out = k so that these networks are strictly homogeneous in the sense of the degree distribution , P (k) = δ(k− k ). Networks with two hierarchical levels are labeled as z in1 -z in2 , e.g. a network with i-j means i links with the nodes of its first hierarchical community level (more internal), j links with the rest of communities that form the second hierarchical level (more external) and ( k − i − j) links with any community of the rest of the network.
Synchronization processes on top of modular networks of this type have been recently studied as a mechanism for community detection [37,53]. In [37], the authors studied the situation in which starting from a set of homogeneous (in terms of the natural frequencies) Kuramoto oscillators with different initial conditions the system evolves after a transient of time to the synchronized state. It was shown that the community structure is progressively unveiled at the same time the system's dynamics evolves toward the coherent state and the synchronization is attained. In particular, the nodes belonging to the first community level are the first to get synchronized, subsequently the second level nodes achieve the frequency entrainment and finally the whole system shows global synchronization.
Here we adopt a different perspective since we will consider as previously a set of non-identical Kuramoto oscillators with random assignment of natural frequencies and hence the final degree of system's synchronization will depend on the strength of the coupling. It is then interesting to study how the degree of synchronization evolves as a function of λ and whether the coherence between nodes is progressively distributed following the hierarchy imposed by the underlying topology. For this, we make use of the order parameters r, eq. (2), and r link , eq (6), to characterize the synchronization transition on two slightly different modular networks with two well defined hierarchical levels, 13 − 4 and 15 − 2, being this difference the cohesion of the internal community core, 13 links out of 15 possible neighbors or 15 links (i.e., all-toall) at the most internal level. Both networks have N = 256 and k = 18. Fig.7 shows the results for both kinds of networks revealing that the path towards synchronization as a function of the interaction is again affected by the structure. They also show that the information provided by r link is essential to unveil the synchronization process. While the global synchronization parameter r is reflecting that the 13 − 4 struc- The networks are synthetically built with an a priori community structure. The network size is 256 nodes and the number of links is 4608. We prescribe 16 compartments that will represent our first community organizational level, and four compartments each one embedding four different compartments of the above first level, that define the second organizational level of the network. Each node has 18 links distributed between its first community level, the second, and the whole network at random. The network 13-4 has 13 internal connections in its first hierarchical level, 4 external connections in its second hierarchical level, and 1 connection with any other community in the network. The generation of the 15-2 structure is equivalent. The curves show that although 13-4 has always a better global synchronization, 15-2 has better local synchronization as shown by r link . ture globally synchronizes always better, r link tells us again about the local synchronization. It shows that local synchronization is indeed favored in the 15 − 2 structure since r link is larger for this topology for small values of λ where the system is locally forming synchronized clusters. This result, not captured by the macroscopic indicator r, is expected since the internal cohesion of communities at the first hierarchical level is larger for the 15-2 than for the 13-4. The evolution of r link shows that when the coupling λ is increased the number of links synchronized in the 13 − 4 network becomes larger than in the 15 − 2 structure revealing that complete synchronization is then favored by the presence of more external links connecting the first level communities.   change at the microscopic level is due to the readjustment of links that connect synchronized nodes. In fact, as Fig. 9 illustrates for both networks, in this region of λ values, the number of links connecting synchronized nodes of the third level decreases while the number of those ascribed to the second level raises. That is, the synchronization process takes place in such a way that the first to synchronize are the nodes of the inner community level, then the second and so on until the whole network gets synchronized. The relevant fact is that in order for r link and r to grow, the nodes and links of the second level adjust their phases at the expense of those of the outer layer, the third level. This is also reflected in the number of clusters of synchronized links (N c ), i.e., the network appears like if the nodes of the third level were "temporarily" disconnected. Moreover, as the 13−4 network has more links connecting the first and the second hierarchical levels, N links2 raises faster in this network than in the 15 − 2.
We have further inspected the synchronization path in modular networks. This can be easily done and visualized by the representation of the filtered matrix D. It implies to reassign the values of matrix D so that D ij = 1 if D ij > T , and D ij = 0 otherwise. Plotting this filtered matrix for different values of the coupling λ one can easily determine which links are the first to synchronize since the form of the adjacency matrix (that includes all the physical links between nodes) is also easy to interpret because of its nested structure. Fig.10 shows how the community structure determines the internal organization of the system in the route towards full synchronization for the 13 − 4 network. For this study we have computed the value of the filtered matrix D for a number of initial conditions and then took its average value so that D ij ∈ [0, 1] accounts for the synchronization strength of the network link (i, j). The results point out that link synchronization depends on the organizational level they belong to. Those connecting nodes belonging to the same first level community are the fastest (in terms of the coupling strength λ) to reach full synchronization. For larger values of λ full synchronization is attained progressively for the subsequent organizational levels. Then, one can conclude that the inner the link is the faster it gets synchronized in agreement with previous studies reported above [37].

V. CONCLUSIONS
In this paper we have explored several issues about synchronization in complex networks of Kuramoto phase oscillators. Our main concern has been the study of the synchronization patterns that emerge as the coupling between nonidentical oscillators increases. We have described the degree of synchronization between each pair of connected oscillators. The use of a new parameter, r link , allows to reconstruct the synchronization clusters from the dynamical data. We have studied how the underlying topology (ranging from homogeneous to heterogeneous structures) affects the evolution of synchronization patterns. The results reveal that the route towards full synchronization depends strongly on whether one deals with homogeneous or heterogenous topologies. In particular, it has been shown that a giant cluster of synchronization in heterogeneous networks comes out from a unique core formed by highly connected nodes (hubs) whereas for homogeneous networks several synchronization clusters of similar size can coexist. In the latter case, a coalescence of these clusters is observed in the synchronization path which is macroscopically manifested by the sudden growth of global coherence. Another important effect of the underlying topology is manifested in an anticipated onset of global coherence for heterogeneous networks with respect to more homogeneous topologies. However, the latter reaches the state of full synchronization at lower values of the coupling strength, therefore showing that statements about synchronizability of complex networks are relative to the region of the phase diagram where they operate. Additionally, we have shown that these systems are seen to organize towards synchronization even when no macroscopic signs of global coherence is observed.
Finally, the framework of structured networks has provided a useful benchmark for testing the validity of the new parameter r link and the information obtained from the computation of matrix D. The results obtained by means of these quantities allow to conclude that for modular networks synchronization is first locally attained at the most internal level of organization and, as the coupling is increased, it progressively evolves toward outer shells of the network. The latter process is however achieved at the expense of partially readjusting some pairs of synchronized nodes between the inner and outer community levels. Besides, we have obtained evidences that a high cohesion at the first level communities produce a high degree of local synchronization although it delays the appearance of the global coherent state.
This study has extended the previous findings about the paths towards synchronization in complex networks [41], and provides a deeper understanding of phase synchronization phenomena on top of complex topologies. In general, the work supports the idea that in the absence of analytical tools to confront the resolution of non-linear dynamical models in complex networks, the introduction of new parameters to describe the statistical properties of the emergence of local patterns is needed as they give novel and useful information that might guide our comprehension of these phenomena. On more general grounds, this work adds to other recent findings [54,55] about the topology emerging from dynamical processes. The evidences that are being accumulated point to a dynamical organization, both at the local and global scales, that is driven by the underlying topology. Whether or not this intriguing regularity has something to do with the ubiquity of complex heterogeneous networks in Nature is not clear yet. More works in this direction are needed, but we think that they may ultimately lead to uncover important universal relations between the structure and function of complex natural systems that form networks. Another issue to explore in future works concerns the behavior of non-linear dynamical systems on top of directed networks [56], which will allow deeper insights into the behavior of natural systems.