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during the last years , many scientists have scrutinized the world around us looking for regularities .
one of the most recent findings is the fact that many real systems can be cast into a common topological structure , called complex networks @xcite .
examples include biological @xcite social @xcite and technological @xcite systems that exhibit similar patterns of interconnections @xcite .
they are characterized by the existence of key elements in the network which drastically reduce the average distance between all of them , the so - called small - world property @xcite .
additionally , it turns out that for a large number of real - world systems , the probability that any given element ( node ) of the system interacts with ( is linked to ) @xmath0 other components , follows a power - law @xmath1 , with and exponent @xmath2 usually estimated between @xmath3 and @xmath4 .
these networks have been termed scale - free ( sf ) networks .
the above property was soon shown to be at the root of distinct behaviours when several dynamical processes are placed on top of sf networks .
these are the cases of percolation and epidemic spreading processes , intensively studied during the last years due to their practical relevance in different applications and the availability of analytical treatments @xcite .
the peculiar topological properties of the underlying network for these two processes lead to the absence of any percolation or spreading threshold in the thermodynamic limit , a previously well - established result for regular and random graphs .
it is then natural to ask whether or not and to what extent the topology of complex networks influences the behaviour of other dynamical processes . in particular , for biological and other applications , it would be relevant to consider the nodes of a given network as nonlinear dynamical systems .
the behaviour of an isolated generic dynamical system in the long - term limit can be described by stable fixed points , limit cycles or chaotic attractors @xcite .
however , we have learned in recent years that when many of such dynamical systems are coupled together , the details matter . in this way , the study of networks with both dynamical and structural complexity might shed light on a number of relevant open problems where nonlinearity and spatial complexity coexist .
dynamical complexity may manifest itself through self - organization , synchronization , the emergence of order , etc .    in this paper , we study the emergence of collective phase synchronization @xcite in scale - free networks and in simple topological configurations such as triangles , squares and pentagons with a variable number of internal connections , called network motifs . to this end , we study the model proposed by kuramoto several years ago and show how a large number of the system s constituents forms a common dynamical pattern , despite the intrinsic differences in their individual dynamics . on a theoretical level , we point out that the fitness for synchronization of network motifs correlates well with their conservation in the evolution of protein motifs @xcite , which might hint at new connections between graph theory , dynamics on networks and biological systems .
in order to study how topology influences collective dynamics , we assume that each network s component is an oscillator and that each interacts with the others following the kuramoto model @xcite .
this choice is due to the remarkable prestige achieved by synchronization as a cooperative phenomenon within nonlinear science and to the seniority , elegance and universality of this model .
specifically , each node @xmath5 is considered to be a planar rotor characterized by an angular phase , @xmath6 , and a natural frequency @xmath7 .
linked nodes interact with a coupling strength @xmath8 according to @xmath9 where @xmath10 is the number of neighbors of the rotor @xmath5 as given by the actual architecture of the underlying graph . for the present work ,
the natural frequencies and the initial values of @xmath6 have been randomly drawn from a uniform distribution @xmath11 with mean @xmath12 in the interval @xmath13 and @xmath14 , respectively .
synchronization occurs when @xmath8 exceeds a critical value , at which clusters of frequency - locked oscillators appear .
this state represents the emergence of cooperation between network s constituents .
the original kuramoto model corresponds to the simplest case of globally coupled ( all - to - all ) , equally weighted oscillators where the coupling strength @xmath15 to ensure that the model is well behaved in the thermodynamic limit @xcite . for this model and without any interaction , @xmath16 ,
the oscillators follow their own dynamics as determined by their natural frequencies and thus the system is unable to synchronize .
however , as @xmath17 increases the population of rotors becomes more coherent and few oscillators form a small cluster of ordered ( synchronized ) states .
if @xmath17 is further increased , the synchronized pack tends to recruit more and more oscillators and eventually the system settles into a unique and totally synchronous state .
the onset of synchronization occurs at a critical value of the coupling strength , @xmath18 .
the second - order phase transition is characterized by the order parameter @xmath19 which behaves when both @xmath20 and @xmath21 as @xmath22 for @xmath23 being @xmath24 .    in order to study the dynamics of the model on top of complex heterogeneous networks , we first generated sf nets using the ba procedure @xcite . in this model , starting from a set of @xmath25 nodes , one preferentially attaches each time step a newly introduced node to @xmath26 older ones .
the procedure is repeated @xmath27 times and a network made up of @xmath28 nodes with a power law degree distribution @xmath1 with @xmath29 and average connectivity @xmath30 builds up .
this network is a clear example of a highly heterogenous network in that the degree distribution has unbounded fluctuations when @xmath20 .
we have performed extensive simulations of the model @xcite through numerical integration of the equations of motion eq .
( [ eq1 ] ) . in the case of random sf networks
the global dynamics of the system is qualitatively the same as for the original kuramoto model .
the phase diagram of the system is shown in fig .  [ figure1 ] for a network of @xmath31 nodes and several values of the average connectivity @xmath32 , where @xmath33 is the order parameter as given by eq.([eq2 ] ) .
starting from small values of the coupling strength , the interactions do not overcome the tendency of each rotor to oscillate according to its individual dynamics . in this state ,
the behaviour of the system is completely incoherent and no synchronization is achieved .
this picture persists until a certain critical value @xmath34 is crossed . at this point
some elements lock their phase and a cluster of synchronized nodes comes up .
this constitutes the onset of synchronization . beyond this value
, there are several groups or clusters within which the nodes are either synchronized and locked in phase or still governed by their intrinsic dynamics and thus in an asynchronous state .
the first groups add to @xmath33 and it departs from zero as @xmath8 is increased beyond @xmath34 .
finally , after further increasing the value of @xmath8 , more and more nodes cluster around the mean phase and the system eventually settles into a completely synchronized state where @xmath35 .    on the other hand , fig .
[ figure1 ] also provides evidence that the critical point at which synchronization is attained in these sf networks is not zero @xcite .
this is more clearly appreciated if one eliminates the dependency observed in the figure of the parameter @xmath33 on the average connectivity of the network @xmath36 .
if in eq.([eq1 ] ) the term @xmath8 is substituted by @xmath37 , the @xmath36-dependency is avoided and all curves in fig .
[ figure1 ] collapse into a single one , showing a critical coupling strength @xmath38 signaling that even for large values of the average connectivity , @xmath34 is not equal to zero .
additionally , finite - size scaling analysis @xcite shows that the transition remains of the second - order type as for the case of fully connected networks .
let s now turn our attention to small geometrical structures known as motifs @xcite instead of looking at the whole network .
these structures can be defined as graph components which are observed in a given network more frequently than in a completely random graph with identical @xmath39 . triangles and rectangular loops are among these graph components , also known as cycles .
they are important because they express the degree of redundancy and multiplicity of paths among nodes in the topology of the network and reveal the existence of hierarchical levels .
hence , the extension of the study of synchronization phenomena to motifs is also relevant .
we have computed for all motifs up to @xmath40 nodes the probability , @xmath41 , that synchronization occurs .
this probability is an increasing function of @xmath8 , and is calculated for each @xmath8 by randomly drawing the values of the natural frequencies @xmath7 from a uniform distribution in the interval @xmath13 .
starting from @xmath42 , one averages , for every @xmath8 , over many realizations of @xmath7 and computes the relative number of simulations where synchronization is accomplished , @xmath41 .
we define @xmath43 , which varies from motif to motif , as the value of @xmath8 beyond which @xmath44 .
the lower @xmath43 is for a motif , the better it synchronizes . in fig.[figure2 ] , we have represented @xmath41 as a function of @xmath8 for several motifs .
the simplest case of the dimer ( @xmath45 in the table ) can be deduced analytically . using the same notation as in eq .
( [ eq1 ] ) , here we have :    @xmath46    to solve these equations , we introduce the new variables ,    @xmath47    thus , @xmath48 is the phase difference between the rotors , @xmath49 is the difference in the natural frequencies and @xmath50 is a dimensionless constant equal to twice the ratio @xmath51 . from eq.([eq3 ] ) we obtain ,    @xmath52    the integration of eq .
( [ eq7 ] ) leads to three solutions depending on the value of @xmath50 .
the three distinct results of the integral in eq .
( [ eq7 ] ) indicate a different functional behaviour of the system when @xmath53 , @xmath54 , and when @xmath55 .
in fact , @xmath54 is the critical point . in consequence , from the definition of @xmath50 in eq .
( [ eq5 ] ) , the critical coupling of this system corresponds to    @xmath56    recalling that the @xmath57s are drawn from a uniform probability distribution in the interval @xmath13 with a mean equal to zero , i.e. , @xmath58 , we have that the synchronization condition reduces to @xmath59 .
the probability @xmath41 is then equal to @xmath60 , for @xmath61 , and @xmath62 for @xmath63 .
thus , for the dimer we have @xmath64 . for the other motifs , @xmath41
has been calculated numerically by means of a @xmath65-order runge - kutta method .
the results are summarized in fig.[figure2 ] for several motifs ( see also table  [ table1 ] below ) .    from fig .
[ figure2 ] , one realizes that for a fixed motif s size , the way in which connections are established determines the synchronization threshold of that motif configuration .
for instance , configurations @xmath66 , @xmath67 , and @xmath68 correspond to motifs made up of four oscillators with a variable number of connections among them . however , their @xmath43 are quite different .
this indicates that the details of the local and internal connections matter . in particular , we observe that the higher the interconnection between motif s constituents , the lower @xmath43 .
we now show how the present study may provide useful hints to a better understanding of several experimental observations in biological systems .
recent advances in network analysis and modeling have provided a promising approach to understanding basic biological organization @xcite . in this
context , quantitative evidence has been recently reported @xcite that in _ saccharomyces cerevisiae _ , the proteins organized in definite cohesive patterns of interaction , and these patterns themselves , are conserved in the evolution across species in a substantially higher degree than those that do not participate in such specific motifs .
a second finding @xcite is that the conservation of proteins in distinct topological motifs correlates with their interconnectedness .
these observations take place in the context of activities occurring within a living cell and which no doubt , are of extreme complexity .
non - linear dynamics prevails at this level of organization , where units that interact according to simple rules can generate unexpected complex patterns @xcite . on a theoretical level
, one may hypothesize that the observed persistence across evolution is due to some mechanism aimed at optimizing the cooperation between neighboring nodes
. a first approach would then be to see if the same structures favor the synchronization as a cooperative phenomenon .
table 1 summarizes the results found in @xcite and the values of @xmath43 for the kuramoto s model in the same motifs .
it is surprising the existence of a possible correlation between conservation and fitness for synchronization of network motifs .
namely , the lower @xmath43 , the higher the natural conservation rate . from table  [ table1 ] , one observes that : @xmath69 @xmath70 , for the chain - like motifs ; @xmath71 @xmath72 , for motifs ordered in increasing degree of complexity , i.e. , when one can go from the lower configuration @xmath73 to next one by adding a new link arbitrarily ; and @xmath74 @xmath75 , when interconnectedness between motif constituents increases .
this may indicate that motifs displaying an improved fitness to develop cooperative activities are preserved across evolution with a higher probability .
the results here obtained for topological motifs may be seen on a more general basis concerning the architecture of real biological and social networks .
it is known that for most of these networks , the probability of finding cycles ( motifs ) in their structure is higher than that expected from a completely random graph with the same connectivity distribution . for social and biological networks , friendship or business relationships and natural selection
seem to be at the origin of such cycles . from this perspective
, one may assume that these networks have been shaped during their evolution by some kind of optimization mechanism that improves on a local level their ability to develop cooperation .
if a new link is created and afterwards it reveals as not beneficial , then it is removed at later times .    in summary
, we have studied the synchronization of kuramoto s phase coupled oscillators for the cases of scale - free networks and motifs . the results obtained indicate that the architecture of random sf networks allows synchronization at a small value of the coupling strength @xmath8 .
additionally , motifs with high interconnectedness show the lower synchronization thresholds . moreover , the results presented here for the kuramoto s model hold for other non - linear dynamical systems , including chaotic oscillators @xcite .
our results then lead to a twofold conclusion . on one hand
, they suggest that non - linear mechanisms may be key ingredients for the understanding of the evolution of networks at a local scale . on the other hand ,
as recently unraveled for protein and other biological networks , the real topology of the systems under analysis is worth taking into account @xcite .
finally , we point out that the kind of models studied here together with the spatial complexity of the underlying net is relevant for cell biology .
a recent example can be found in the discovery that an ultradian clock ( oscillator ) shapes genome expression in the yeast _ saccharomyces cerevisiae _ @xcite .
we study the phase synchronization of kuramoto s oscillators in small parts of networks known as motifs . 
 we first report on the system dynamics for the case of a scale - free network and show the existence of a non - trivial critical point . 
 we compute the probability that network motifs synchronize , and find that the fitness for synchronization correlates well with motif s interconnectedness and structural complexity . 
 possible implications for present debates about network evolution in biological and other systems are discussed .