a large body of work has already been devoted to the cgl equation , which reads : where is a complex field .  in the context of amplitude equations  , which are large - scale descriptions of physical systems passed ( and near ) symmetry - breaking instability thresholds ,the cgl equation has been recognized as the relevant equation for the slow modulations of oscillations in a continuous medium near a hopf bifurcation  .more generally , it appears in the description of spatially - extended systems when oscillations or waves are present .    under the form ( [ cgl ] ) ,the cgl equation has been reduced ( without loss of generality ) to its simplest from , with only two real parameters , and .the first term of the rhs is related to the linear instability mechanism which led to oscillations .the second term accounts for diffusion and dispersion , while the cubic nonlinear term insures if , otherwise other terms may be necessary the saturation of the linear instability and is involved in the renormalisation of the oscillation frequency .two important limits are worth mentioning : when , one has the real ginzburg - landau equation , which possesses a lyapunov functional and thus exhibits only relaxational dynamics  .when dispersion plays the essential role , as one recovers the nonlinear schrdinger equation  . in the general case , sustained spatio - temporally disordered regimes are observed in large regions of the parameter plane ( fig . 1 ) .the genericity of the cgl equation , associated to its relative simplicity , has made it one of the favorite playgrounds for testing ideas about spatiotemporal chaos in a rather realistic context  .it is only recently , though , that a comprehensive study has been undertaken , as it was realized that away from the intricacy of the bifurcation diagrams at small sizes , there exists a crossover size beyond which chaos becomes extensive and can be characterized by intensive quantities independent of system size , boundary conditions , and , to a large extent , initial conditions  . indeed ,when chaos is extensive , statistical approaches are legitimate and should provide rather simple descriptions . in this context ,bifurcation diagrams typically used for small dynamical systems are replaced by `` phase diagrams '' delimiting the regions of different statistical signatures in parameter space .  whereas the one - dimensional case is now rather well known  , the situation in two dimensions is much less satisfactory , mainly because previous work was devoted to punctual problems rather than to acquiring a global picture of the properties of the equation .here , thanks to current computer power , we provide a comprehensive overview of the two - dimensional cgl equation in the large - size limit  .we now introduce a few important features of the cgl equation before proceeding to the description of the phase diagram .  early work on cgl has dealt with the problem of the linear stability of its family of plane - wave solutions with and .all these solutions are unstable for ( newell criterion ) , a condition which defines the so - called `` benjamin - feir '' ( bf ) line ( fig . 1 ) .for , plane - wave solutions with are linearly stable  .the instability of the travelling wave solutions above the bf line is readily verified as to be linked to the `` gauge '' invariance of the equation , i.e. its invariance by an arbitrary phase shift ( ) . near the bf line ,the amplitude modes are strongly damped , `` slaved '' to the marginal phase mode , so that one often speaks of a _ phase instability_. this instability has been conjectured to lead to a disordered regime called _ phase turbulence _ , in which the field never reaches zero , so that the phase is defined everywhere . near the bf line, the phase gradient is expected to remain small , and a systematic expansion can be performed , leading to a description of the large - scale dynamics in terms of the phase only .the behavior of the resulting series of _ phase equations _ obtained by truncation of this expansion is actually not very well known , except for the kuramoto - sivashinsky ( ks ) equation to which these equations reduce infinitely close to the bf line . if phase turbulence ( i.e. spatiotemporal chaos ) has been established for the ks equation , it is the subject of an ongoing controversy away from the bf line , both for the phase equations and for the cgl equation itself  .another important feature of the cgl equation is the structure , nature , and role of `` defects '' , i.e. points in space - time where . at such points , the phase is not defined , and it varies by a multiple of when going around them . for space dimensions , defects are topologically constrainedthis has been recognized for a long time as one of the salient features of the two - dimensional cgl  . for , defects are points and can only appear and disappear by pairs . for small enough values , they appear in two different types , `` spirals '' and `` shock - line vertices '' ( fig . 2 ) .the shock - line vertices have mostly been considered as `` passive '' objects which play no important role .however accurate this statement may be , it remains that the spiral defects have attracted the most attention  . in spite of all these efforts ,no exact expression is available ; on the other hand , much is known about the core structure and the `` wings '' , i.e. the emitted outward - going waves .away from the core , these waves are asymptotic to a planewave with a well - defined wavenumber depending only on and .the stability properties of the planewave solution give rise to two important lines in the parameter plane . on the  line , , the maximum wavenumber of linearly stable planewaves ( fig . 1 ) .this line delimits the region of linear stability of the wave . to its left ,the wave is linearly unstable ( perturbations grow exponentially in phase space ) ; this is in fact a _ convective _ instability : a _ localized _ perturbation indeed grows , but is advected away from its initial position at the group velocity of the planewave . at this initial position, the solution relaxes to the planewave . according to aranson et al . , the planewave becomes _ absolutely _ unstable to the left of the  line : any initial perturbation grows at its initial location ( in addition to spreading in the direction of the wave ) .   with periodic boundary conditions and parameters and .( a ) : image of in color scale from ( dark red ) to ( light yellow ) ; ( b ) : lines ( green ) and ( red ) . , title="fig : " ] with periodic boundary conditions and parameters and .( a ) : image of in color scale from ( dark red ) to ( light yellow ) ; ( b ) : lines ( green ) and ( red ) . , title="fig : " ]  it is not exactly known how these stability limits of the planewave with wavenumber are related to the actual stability properties of the spiral solution and to its observability in an experimental context . in most of the region of the parameter plane of considered here , the spiral solution exists and is core - stable  . in consequence , it can be argued that its stability properties are essentially related to those of the asymptotic planewave in a _ semi - infinite _ domain ( with the core sitting at one end ) .we now formulate , at a somewhat conjectural level , the stability properties of the spiral solution and their consequences observable in experiments ( be they numerical or not ) . to the right of line ,one expects the spiral to resist a ( small ) amount of noise , due to its `` complete '' ( core and wings ) linear stability . between  and, perturbations are amplified but convected away from the core at the group velocity of the solution .numerical experiments have shown that the spiral is most sensitive to perturbations in the crossover region between the core and the wings  . at a given level of ( experimental ) noise ,perturbations coming from this region are the most dangerous ones .this convective instability in fact takes the form of growing oscillations of the _ modulus _ as one goes away from the core ( see fig .experimentally , these oscillations do not saturate , and the wave breaks down , creating more defects .this mechanism defines a maximum radius which limits the size of observable spirals , and depends on the instability rate and ( weakly ) on the noise level . approaching the limit of absolute instability ( line ), this diameter goes to zero , and beyond  the spiral is `` completely '' unstable and can not be observed in an experimental ( noisy ) context .finally , we note that if defects do play an important role in the two - dimensional cgl equation , as we will show below , their topological character is not crucial in determining the dynamical regimes : for , localized `` quasi - defects '' where remains locally very close to zero have been shown to be the key - ingredient in some disordered regimes  .but the topological constraint on defects for does provide them with a large domain of existence in parameter space , insuring their relevance in most of the regimes of interest .fig . 1 shows the phase diagram of the two - dimensional cgl equation established from a numerical exploration of systems of linear size of the order of with periodic boundary conditions , using a pseudospectral code .details about the integration scheme and the numerical protocol will be given elsewhere  .the various transition lines are discussed in detail in section 4 .  as in the one - dimensional case , two types of disordered regimes can be distinguished , depending on whether they exhibit defects or not . to the right of the line l in the parameter plane , _ phase turbulence _( no defects ) is observed , whereas _ defect turbulence _occurs to the left of line t.   between the bf and the l lines , spatiotemporally chaotic regimes of phase turbulence where no defect occurs are observed . with periodic boundary conditions , the total phase gradient across the system ( the `` winding number '' )is conserved .this introduces a new invariant to the problem .most results reported here ( in particular the location of line l ) are for the case of zero winding number .   with periodic boundary conditions and parameters and .( a ) : field in a sub - system of linear size ; color scale from ( dark red ) to ( light yellow ) ; ( b ) : phase field in the whole system ( color scale from dark red to light yellow over the total range of variation of the phase ).,title="fig : " ] with periodic boundary conditions and parameters and . (a ) : field in a sub - system of linear size ; color scale from ( dark red ) to ( light yellow ) ; ( b ) : phase field in the whole system ( color scale from dark red to light yellow over the total range of variation of the phase ).,title="fig : " ]  in phase turbulence , the solution consists of a disordered cellular structure , ( best seen in plots of or ) slowly evolving in time ( fig .the typical size of the cells diverges like when approaching the bf line ; this size is in fact of the order of the wavelength of the most unstable mode in the corresponding ks equation . if the correlations of the modulus still decay rapidly , those of the phase decay slowly , with power - law - like behavior .this is apparent in the large - scale modulations of the phase field ( fig .3b ) . in section 5, we discuss the effective large - scale model for phase turbulence and the asymptotic behavior of the correlations in phase turbulence .even though they are certainly important to better understanding the dynamics of phase turbulence , the `` elementary processes '' at play in this regime are not known . for , it has been shown that propagative structures are the objects triggering the breakdown of phase turbulence  . here, no equivalent has so far been found ( see below ) .one should also investigate whether the evolution of the cellular structure involves some of the elementary events observed in the coarsening of soap froths  ( even though these can not account for all the dynamics here , since the cellular structure is statistically stationary ) .knowledge of the local dynamics is necessary to build a large - scale effective description of two - dimensional phase turbulence in the spirit of the work of chow and hwa  for the one - dimensional ks equation .defect turbulence is the most chaotic regime of the two - dimensional cgl equation : correlations decay exponentially , with short correlation lengths and times .depending mostly on , the space - time signature of the solutions varies . for large ,the density of defects is large , they come and go rapidly , and they rarely form spirals ( fig . 4 ) .indeed , it can be argued that defects _per se _ are not crucial features in this case .`` amplitude turbulence '' is a better name for such spatiotemporal chaos regimes .furthermore , increasing toward the nonlinear schrdinger equation limit , _ pulses _ become the relevant objects : the solutions consist of localized regions where . approaching the bf line , the defect density decreases , the characteristic scales increase , and spiralscan be observed .in fact well - developed spirals can only be observed , in the defect turbulence region , to the right of the  line ( see the discussion of the transition lines in the next section ) .   with parameters ( a - b ) and ( c - d ) .there are 268 defects in the first case , but no well - formed spirals are observed ; in the second case , on the other hand , spirals are clearly visible .( a ) : field ; color scale from ( dark red ) to ( light yellow ) ; ( b ) : ( green ) and ( red ) lines ; ( c ) : field ; color scale from ( dark red ) to ( light yellow ) ; ( d ) : ( green ) and ( red ) lines.,title="fig : " ] with parameters ( a - b ) and ( c - d ) .there are 268 defects in the first case , but no well - formed spirals are observed ; in the second case , on the other hand , spirals are clearly visible .( a ) : field ; color scale from ( dark red ) to ( light yellow ) ; ( b ) : ( green ) and ( red ) lines ; ( c ) : field ; color scale from ( dark red ) to ( light yellow ) ; ( d ) : ( green ) and ( red ) lines.,title="fig : " ]   with parameters ( a - b ) and ( c - d ) .there are 268 defects in the first case , but no well - formed spirals are observed ; in the second case , on the other hand , spirals are clearly visible .( a ) : field ; color scale from ( dark red ) to ( light yellow ) ; ( b ) : ( green ) and ( red ) lines ; ( c ) : field ; color scale from ( dark red ) to ( light yellow ) ; ( d ) : ( green ) and ( red ) lines.,title="fig : " ] with parameters ( a - b ) and ( c - d ) .there are 268 defects in the first case , but no well - formed spirals are observed ; in the second case , on the other hand , spirals are clearly visible .( a ) : field ; color scale from ( dark red ) to ( light yellow ) ; ( b ) : ( green ) and ( red ) lines ; ( c ) : field ; color scale from ( dark red ) to ( light yellow ) ; ( d ) : ( green ) and ( red ) lines.,title="fig : " ]   cellular structures also appear in the two - dimensional cgl equation in the form of quasi - frozen arrangements of spiral defects surrounded by shock lines . in these states ,the field is generally completely stationary in time .the network of these lines form the cells of these spatially - disordered states ( fig .non - spiral defects lie at the shock - line vertices , sometimes also along the shock lines themselves , in metastable arrangements . because the timescales involved are very long , it is actually difficult to decide when these structures stop evolving .residual , intermittent , local rearrangements less and less frequent along time are observed , and this relaxation process is reminiscent of that taking place in glasses .in fact , much remains to be done in order to decide to what extent these dynamical states are glassy states .a first study along these lines can be found in huber  .a particular point of interest is to investigate whether some kind of aging phenomena are taking place in these frozen structures .the frozen states are easily observed in the region of the parameter space to the right of line t , where they are the only asymptotic solutions possessing defects .their total domain of existence in the plane can be estimated on the basis of the stability properties of spirals , as discussed in section 2 .the frozen states do not exist to the left of line , since there the spirals are absolutely unstable . on the other hand, nothing precludes their existence to the right of .the size of the cells is not limited , except in the presence of noise , since in this case , between  and , spirals have a maximum radius . in practice ,the dynamical `` history '' which led to a given frozen structure greatly influences the distribution of sizes of cells in the structure ( see the discussion in section 4.1 below ) .   with parameters and .( a ) : field ; color scale from ( dark red ) to ( light yellow ) ; ( b ) : lines ( green ) and ( red).,title="fig : " ] with parameters and . (a ) : field ; color scale from ( dark red ) to ( light yellow ) ; ( b ) : lines ( green ) and ( red).,title="fig : " ]  the actual observation of frozen states in the region between lines  and t , where defect turbulence exists , is not easy , though , because these states are metastable with respect to defect turbulence .coming , for example , from a frozen asymptotic state to the right of line t , the parameters have to be changed `` adiabatically '' to prevent the nucleation of defect turbulence .even then , the necessary rearrangements of the cellular structure , which involve the rapid motion of some defects , most often trigger the `` melting '' of the frozen structure .frozen structures are most easily observed far to the right of line t , and especially to the right of .their domain of existence probably extends to large values of ( except maybe for large ) . at any rate , along the axis, it extends to the real ginzburg - landau ( ) limit , where the spirals become the vortex excitations of the xy model  .the respective domains of existence of the three disordered phases described above are delimited by the lines bf , t , l , and .we discuss now the nature of the various ( phase ) transitions observed when crossing these lines and comment on the relative stability of the disordered phases .as already mentioned , line t delimits the ( numerically estimated ) domain of existence of _ sustained _ regimes of defect turbulence . starting from a defect turbulence regime , _ increasing , this highly chaotic regime is maintained until line t is crossed ; defect turbulence is only transient then , and is followed by the nucleation of a frozen state ( fig .6 ) . as observed by huber et al . , this transition is indeed reminiscent of a first - order phase transition . depending both on the amplitude ofthe `` quench '' beyond line t ( i.e. the distance of the current parameters to line t ) and the value of the crossing point , the duration of the transient varies widely .the smaller the quench and the larger , the longer the transient . as a matter of fact ,line t can be seen as the line where this transient is infinite .the nucleation process involves the appearance of a sufficiently large spiral core . to the right of line t , the spiral may then grow , but its size is limited to a maximum radius ( fig .this radius results from the interaction between the outward - going spiral waves and the strong , finite - amplitude fluctuations characteristic of the defect turbulence `` bath '' surrounding it .these fluctuations trigger the most unstable mode of the spiral solutions , i.e. the oscillations of which are the signature of the nonlinear stage of the convective instability , and influence the spiral wave _ inward_. we stress that this is different from the problem usually considered when studying convective instabilities . here a semi - infinite convectively - unstable medium is put in contact `` downstream '' with a turbulent medium .the balance between the ( destabilizing ) turbulent fluctuations and the ( regularizing ) advection of perturbations by the spiral waves takes place at the nonlinear level , so that the radius can not be determined from the stability properties of the spiral alone  .approaching line t from the right , decreases .numerical experiments  show that the line t where ( not shown in fig .1 ) is located to the _ left _ of line t. note that , in contrast with line t , line t is defined _ via _ a local dynamical phenomenon . in the region between lines t and t , `` fully developed '' defect turbulence decays to mixed states , i.e. mostly - frozen structures in which some localized patches of turbulence subsist ( fig .6b ) . numerically speaking, this residual turbulence does not seem to vanish at long times ; extensive statistical data has to be compiled in order to decide whether this remains true in the infinite - time limit , in which case lines t and t are distinct . in the other case , one must conclude that line t , in the thermodynamic limit , moves to the left to coalesce with line t. however , in a similar fashion to what happens with line l in phase turbulence ( see next section ) , line t is numerically well - defined for all practical purposes , and it is only a theoretical point to know whether it is distinct from t in the infinite - time limit .we note finally that the frozen states nucleated this way ( i.e. from the spontaneous decay of defect turbulence ) possess a maximal cell size given by .   with parameters and .( a ) : two spirals have been nucleated and have reached their maximal size ; ( b ) : asymptotic state consisting of a frozen structure of spirals with maximum radius with some residual turbulence for , , and .this asymptotic state is typical of the region between lines t and t ( compare with fig . 5 ) .( snapshots of field , color scale from ( dark red ) to ( light yellow ) ) . note the oscillations of away from the spiral cores , near their maximum radius , which are the signature of the convective instability of the planewave.,title="fig : " ] with parameters and .( a ) : two spirals have been nucleated and have reached their maximal size ; ( b ) : asymptotic state consisting of a frozen structure of spirals with maximum radius with some residual turbulence for , , and .this asymptotic state is typical of the region between lines t and t ( compare with fig . 5 ) .( snapshots of field , color scale from ( dark red ) to ( light yellow ) ) . note the oscillations of away from the spiral cores , near their maximum radius , which are the signature of the convective instability of the planewave.,title="fig : " ]  the transition from defect turbulence to frozen states is hysteretic : coming from a frozen state , and _ decreasing _ , it is , in principle , possible to keep _ completely _ frozen structures past line t , ( even if `` partially '' frozen structures , such as the one shown in fig .6b , coexist in the region between lines t and t ) .as mentioned in section 3 , line  is only an absolute and approximate limit of the hysteresis region : in theory , the effects of the curvature of the waves and the spiral core should be taken into account , and in practice , in this region of parameter space , the frozen states are easily destroyed by perturbations , so that it is extremely difficult to observe the frozen states far to the left of line t. in fact , the real limit of existence of frozen states might be actually determined by the properties of the _ nonlinear _ stage of the convective instability of the waves and possibly also by the stability properties of the shock - line vertices , the other key - component of frozen structures .this very intricate situation will be examined in detail in .the distance between lines t and  is thus only an approximate measure of the _ maximal _ width of the hysteresis loop .line l is the ( numerically - determined ) line beyond which ( to its left ) phase turbulence is only transient . to its right , phase turbulence can be observed for as big a system and as long a time as current computers allow. a brief discussion of the existence of phase turbulence in the infinite - size , infinite - time limit is given in the next section , but , numerically speaking , line l is rather well defined , with the probability of breakdown of phase turbulence being finite to its left and essentially zero to its right .   with parameters and coming from a phase turbulent system at and .( a - c ) : snapshots of field at time in color scale from ( dark red ) to ( light yellow ) . note the growing `` bubble '' of defect turbulence , whose diameter increases linearly with time ( ( d ) : square root of the surface of the bubble along time).,title="fig : " ] with parameters and coming from a phase turbulent system at and .( a - c ) : snapshots of field at time in color scale from ( dark red ) to ( light yellow ) .note the growing `` bubble '' of defect turbulence , whose diameter increases linearly with time ( ( d ) : square root of the surface of the bubble along time).,title="fig : " ]   with parameters and coming from a phase turbulent system at and .( a - c ) : snapshots of field at time in color scale from ( dark red ) to ( light yellow ) . note the growing `` bubble '' of defect turbulence , whose diameter increases linearly with time ( ( d ) : square root of the surface of the bubble along time).,title="fig : " ] with parameters and coming from a phase turbulent system at and .( a - c ) : snapshots of field at time in color scale from ( dark red ) to ( light yellow ) . note the growing `` bubble '' of defect turbulence , whose diameter increases linearly with time ( ( d ) : square root of the surface of the bubble along time).,title="fig : " ]  the breakdown of phase turbulence is also a nucleation process ( fig .a pair of defects is nucleated by some fluctuation on one side of a cell , triggering a `` chain - reaction '' leading to the quasi - deterministic invasion of the system by a growing bubble of the defect turbulence phase .the diameter of this bubble increases linearly with time ( fig .a priori , three sections of line l have to be considered , delimited by the crossing points with lines t and  ( or , rather , the line actually delimiting the existence of frozen states ) . below line t , the only possible regime is defect turbulence , so that the breakdown of phase turbulence can only lead to this regime . between lines  and t ,defect turbulence and frozen states coexist , and the breakdown could lead to either state . in fact , the nucleation events at the origin of the breakdown are always highly chaotic , and only defect turbulence arises ( this is not surprising , considering the metastability of frozen states in this region ) .the third portion of line l , above its crossing with line t , offers an interesting possibility : there , only frozen states are expected asymptotically .but the breakdown of phase turbulence first triggers a defect turbulence transient which then itself nucleates a frozen state .no direct transition from phase turbulence to frozen states seems possible , although it is conceivable that one might observe , in a large system , a nucleating frozen state within the growing bubble of defect turbulence invading phase turbulence .finally , we note , not surprisingly , that the breakdown of phase turbulence is also an hysteretic transition . crossing line l from left to right, one remains in either defect turbulence or a frozen state .after this brief description of the `` phase diagram '' , we comment on important related points as well as some theoretical problems currently under investigation .the phase diagram of fig . 1 summarizes numerical results . as such , even though customary precautions have been taken ( e.g. to insure the extensivity of chaos in the disordered regimes ) , it does not represent the `` true '' phase diagram , i.e. that of the infinite - size , infinite - time , `` thermodynamic '' limit .this question , which also arises in the case , has been recently investigated  , in particular with respect to the existence of phase turbulence in the thermodynamic limit ( represented by line l here ) .indeed , the very existence of phase turbulence is questioned . as for the case ,it is currently impossible to make a definitive statement , on the basis of numerical simulations alone , as to whether line l coalesces with the bf line in the thermodynamic limit .a careful analysis of statistical data about phase turbulence in one and two dimensions for various system sizes and integration times is under way and will be reported elsewhere  .extrapolation of size effects , though , seems to lead to the conclusion that phase turbulence might not exist in the thermodynamic limit .it remains nevertheless that for all practical purposes ( numerical or experimental ) , there exists a domain of parameter space where phase turbulence is statistically stationary and subsists even in very large systems for times as long as desired . at any rate , fig .1 is representative of system sizes and integration times accessible to current computers ( say up to linear size l of the order of and integration times up to ) , and line l is probably slightly shifted for much larger systems .the status of line t , which delimits the domain of existence of sustained regimes of defect turbulence , is subjected to similar remarks .statistical data about the probability of breakdown ( to a frozen state ) should be cumulated in order to estimate the position of line t in the thermodynamic limit .this should be completed by a detailed study of the variation of with parameters and to provide a precise determination of line t. the question , raised in section 4.1 , of whether lines t and t coalesce in the thermodynamic limit , could thus be addressed .the existence of defect turbulence is not in question , though , as line t is bound to be situated to the right of line  ( which limits absolutely the domain of existence of frozen states ) .cellular structures appear both in phase turbulence ( fig . 3 ) and, of course , as frozen states ( fig . 5 ) . in phase turbulence ,the dynamics produces statistically stationary configurations which can be studied along the lines of what is usually done for , say , the coarsening of soap froths  . for example, a first step would be to estimate the statistical properties of cells their sizes , their number of sides and a second stage could consist in determining the local events of which their dynamical evolution is composed . as mentioned in section 3.2 , this should provide a better understanding of the elementary processes involved in phase turbulence , as well as a better statistical description of this regime .this could also pave the way to a simple `` particle model '' sharing the same statistical properties , similar to the work of rost and krug  on the ks equation . in the case of the frozen states ,as discussed earlier , the characteristics of the structures depend strongly on the `` dynamical history '' that led to them , so that an investigation of their geometrical properties must be correlated to their origin and formation .admittedly , the phase diagram of fig .1 is not quite complete .work remains to be done , in particular on the three following points :  * the , `` nonlinear schrdinger '' , limit needs to be clarified : where are lines  and t located ?how does the _ core _ instability of spirals intervene in the dynamical regimes ? * the region , and especially the prolongation of line t , should be investigated , as well as the limit .* it is known that , at least for , the cgl equation exhibits well - behaved disordered regimes in some parts of the half - plane  , in spite of the absence of the usual nonlinear saturation mechanism .such a possibility should also be explored in the two - dimensional case . *the processes of nucleation of the spiral cores should be studied in detail , at least from a statistical point of view . in particular , the minimal core size that can possibly lead to the growth of a spiral should be estimated , as well as the probability of such an event .this would help define a line to the right of line t beyond which nucleation is immediate ( corresponding to the line `` nuc '' in ) , as well as clarify the nature of the frozen states with residual turbulence observed experimentally between lines t and t.   the possibility of large - scale descriptions of deterministic spatiotemporal chaos in terms of langevin - like , ( nonlinear ) stochastic equations , is a question currently being debated , essentially because one can then hope to apply methods of statistical mechanics  .one central point is to investigate to what extent and under what conditions the local chaotic fluctuations are equivalent to a `` noise '' at large scales .the cgl equation offers , here also , a good testground for such questions . in this spirit, phase turbulence has been proposed  to be described , at large scales , by the noisy burgers or kardar - parisi - zhang ( kpz )  equation , which is ( among other things ) a model for the kinetic roughening of stochastic interfaces .indeed , since the phase is always defined in this regime , one can consider the evolution of the phase field only ( at least on an experimental level ) , which , in turn , can be seen as the progression of a -dimensional interface in a -dimensional medium ( unwinding the phase advance on the real axis ) .numerical results for and seem to confirm the validity of the kpz ansatz  .the kpz picture also reveals the asymptotic behavior of correlations in phase turbulence . in particular , spatial correlations of the phase should decrease either algebraically ( `` linear regime '' of kpz ) or like a stretched exponential ( `` nonlinear regime '' ) .  to go beyond this type of numerical observation, the effective large - scale stochastic equation has to be built from the original model .an important step toward this aim has been achieved recently for the one - dimensional kuramoto - sivashinsky equation ( which is also believed to be described by kpz at large scales  ) . carefully studying the elementary mechanisms at the origin of spatiotemporal chaos , chow and hwa  succeeded in calculating , from data on local chaos only , the parameters of the effective kpz equation .it is not clear how such a program could be carried out for the cgl equation in any of its disordered regimes even for phase turbulence , but a detailed analysis of the elementary processes at work in each case appears as a necessary step deserving further work .the general picture of the two - dimensional cgl equation presented here , even though it should be completed along the lines mentioned above , already provides a good starting point to people wanting to study various aspects of spatiotemporal chaos in this system .in particular , our study should help choose specific parameter values .it should also help experimentalists recognize whether the physical or numerical problems they study are typical of the cgl equation and , if so , of what particular regime .finally , natural extensions of this work include a similar study of the three - dimensional case , and of the various modifications of cgl usually considered in the literature .y. kuramoto , _ chemical oscillations , waves and turbulence _ , ( springer , tokyo , 1984 ) ; j. lega , `` dfauts topologiques associs la brisure de linvariance de translation dans le temps , '' thse de doctorat , universit de nice ( 1989 ) ; w. van saarloos , `` the complex ginzburg - landau equation for beginners '' , in _ spatiotemporal patterns in nonequilibrium systems _ ,cladis and p. palffy - muhoray eds ., ( addison - wesley , reading , 1994 ) .see , e.g. : a.c .newell , _ rocky mountains j. math . _ * 8 * ( 1978 ) 25 ; a.c .scott , f.y.f .chu and d.w .mclaughlin , _ proc .ieee _ ( 1973 ) 1443 ; y.s .kivshar and b.a .malomed , _ rev .phys . _ * 61 * ( 1989 ) 762 ; a.c .newell , d.a .rand and d. russell , _ phys .lett . a _ * 132 * ( 1988 ) 112 ; s. popp , o. stiller , i. aranson and l. kramer , _ physica d _ * 84 * ( 1995 ) 424 .shraiman , a. pumir , w. van saarloos , p.c .hohenberg , h. chat , and m. holen , _physica d _ * 57 * ( 1992 ) 241 ; a. pumir , b.i .shraiman , w. van saarloos , p.c .hohenberg , h. chat , and m. holen , `` phase vs. defect turbulence in the one - dimensional complex ginzburg - landau equation , '' in : _ ordered and turbulent patterns in taylor - couette flows _ , c.d. andereck ed .( new york : plenum press , 1992 ) .h. chat , `` disordered regimes of the one - dimensional complex ginzburg - landau equation '' , in _ spatiotemporal patterns in nonequilibrium systems _ , p.e .cladis and p. palffy - muhoray eds ., ( addison - wesley , reading , 1994 ) .a previous study ( see ) , using a discretized version of the cgl equation , attempted such a task . in view of the results presented here, it appears that the effects of the discretization scheme on the phase diagram are rather drastic . in particular , no phase turbulence regime was observed by these authors , an artifact , we believe , due to their numerical scheme .other discrepancies will be discussed elsewhere  .siam j. appl ._ * 42 * ( 1982 ) 762 ; i. aranson , l. kramer and a. weber , `` the theory of motion of spiral waves in oscillatory media '' , in _ spatiotemporal patterns in nonequilibrium systems _cladis and p. palffy - muhoray eds . , ( addison - wesley , reading , 1994 ) ; _ phys .e _ * 47 * ( 1993 ) 3231 .h. chat and p. manneville , `` defect turbulence and frozen states in the two - dimensional complex ginzburg - landau equation '' and `` phase turbulence in the complex ginzburg - landau equation '' , to be published .g. huber , p. alstrm and t. bohr , _ phys .* 69 * ( 1992 ) 2380 ; g. huber , `` vortex solids and vortex liquids in a complex ginzburg - landau equation '' , in _ spatiotemporal patterns in nonequilibrium systems _ ,cladis and p. palffy - muhoray eds . , ( addison - wesley , reading , 1994 ) .m. kardar , g. parisi and y .- c .zhang , _ phys ._ * 56 * ( 1986 ) 889 ; see also the reviews : j. krug and h. spohn , kinetic roughening of growing surfaces , in : c. godrche ed ., _ solids far from equilibrium _ , ( cambridge university press , 1991 ) ; t. halpin - healy and y.c .zhang , _ phys . rep ._ , to appear .v. yakhot , _ phys .a _ * 24 * ( 1981 ) 642 ; s. zaleski , _ physica d _ * 34 * ( 1989 ) 427 ; k. sneppen , j. krug , m.h .jensen , c. jayaprakash , and t. bohr , _ phys .a _ * 46 * ( 1992 ) 7351 ; i. procaccia , m.h .jensen , v.s .lvov , k. sneppen , and r. zeitak , _ phys .a _ * 46 * ( 1992 ) 3220 ; v.s . lvov and i. procaccia , _ phys .lett . _ * 69*(1992 ) 3543 ; c. jayaprakash , f. hayot , and r. pandit , _ phys ._ * 71 * ( 1993 ) 12 .

Abstract: after a brief introduction to the complex ginzburg - landau equation , some of its important features in two space dimensions are reviewed . a comprehensive study of the various phases observed numerically in large systems over the whole parameter space is then presented . the nature of the transitions between these phases is investigated and some theoretical problems linked to the phase diagram are discussed .   this plain latex version of the article , originally published as : +  physica a * 224 * ( 1996 ) 348368   + is placed on arxiv because the publisher s link : +  http://www.sciencedirect.com/science/article/pii/0378437195003614   + only provides badly resolved black - and - white scans of the figures in place of the initial more readable color pictures .   the complex ginzburg - landau equation ( cgl ) is one of the most important simple nonlinear partial differential equations for two main reasons . first , as we will briefly recall below , it arises as the natural description of many physical situations , or at least is the `` kernel '' of many systems of amplitude equations . second , its solutions display a very rich spectrum of dynamical behavior when its parameters are varied , reflecting the interplay of dissipation , dispersion and nonlinearity .  here , we give a brief and mostly qualitative report of the various regimes observed in the two - dimensional case , and discuss several theoretical aspects of these numerical findings .


recent years , the network science , a multidiscipline research area , is concentrated by researchers from different backgrounds such as computer science , biology and physics . in these studies , betweenness centrality ( bc )is always applied as a critical metric to measure nodes or edges significance  .for example , girvan and newman developed a community detection algorithm based on edge betweenness centrality  , leydesdorff applied centrality as an indicator of the interdisciplinarity of scientific journals  and motter and lai established a model of cascading failures with node load being its betweenness  .however , the extremely high time and space complexity of calculating betweenness centrality greatly limits its applying on large networks . before the landmark work of brandes  ,the algorithm for computing betweenness centrality requires time and space . while brandes reduced the complexity to on space and and on time for unweighted and weighted networks , respectively , where is the number of vertices and is the number of edges  .however , this improved algorithm still can not satisfy scientific computation requirements in the present information explosion era as more and more unexpected large networks emerge , such as online social networks , gene networks and collaboration networks .for example , twitter possesses hundreds of millions active users which construct a huge online social network .however , a weighted network with one million nodes may take about one year to calculate its betweenness centrality using brandes algorithm , which is an unbearable cost .because of this , there is a pressing need to develop faster bc algorithm for explorations of diverse domains .gpu general computing , which provides excellent parallelization , achieves higher performance compared to traditional cpu sequential algorithms in many issues including network science  .cuda is the most popular gpu - computing framework developed by nvidia corporation and some researchers have even parallelized brandes s algorithm by using it  .however , previous works concentrated on unweighted networks for simplification , but to our best knowledge , most realistic networks are weighted ones .the most significant difference of bc algorithm on unweighted and weighted networks is the shortest path segment . in weighted networks ,dijkstra algorithm should be used to solve the single source shortest path ( sssp ) problem rather than depth first search ( dfs ) algorithm .many efforts in previous work have been devoted to the gpu version of sssp problem using the well - known dijkstra algorithm  .although these algorithms have been presented and developed , establishing a parallel version of betweenness centrality algorithm on weighted networks is nontrivial because the original sssp algorithm have to be modified in many critical points for this task and to our best knowledge , a proper and fast solution is still missing .aiming at filling this vital gap , we propose a fast solution using cuda to calculate bc on large weighted networks based on previous gpu bc algorithms and sssp algorithms in this paper .  to make our algorithm more efficient , we make efforts to optimize it by employing several novel techniques to conquer the influence of irregular network structures .real - world networks have many characters which could deteriorate the performance of gpu parallelization algorithms .for example , the frontier set of nodes is always small compared to the total number of vertices , especially for networks with great diameters . in the meantime, the majority of nodes do not need to be inspected in each step , hence processing all vertices simultaneously in traditional algorithms is wasteful .mclaughlin and bader proposed a work - efficient strategy to overcome this problem  .another well - known issue is that the power - law degree distribution in realistic networks brings in serious load - imbalance .several methods were proposed in previous study to conquer this problem , e.g. , merrill et al . employed edge parallel strategy to avoid load - imbalance  and hong et al . dealt with this problem by using warp technique  . in this paper, we systematically investigate the advantages and drawbacks of these previous methods and implement them in our algorithm to solve the above two problems .experiments on both real - world and synthetic networks demonstrate that our algorithm outperforms the baseline gpu algorithm significantly .our main contributions are listed as follows :  * based on previous gpu parallel sssp and betweenness centrality algorithms , we propose an efficient algorithm to calculate betweenness centrality on weighted networks , which achieves 30 to 150 speedup over the best existing cpu algorithm on realistic networks .* we compare the traditional node - parallel method to the work - efficient version and the warp - centric method .experiments on realistic networks and synthetic networks demonstrate that the combination of the two strategies works better than others , which achieves 2.65 speedup over the baseline method on realistic networks .* we package our algorithm to a useful tool which can be used to calculate both node and edge betweenness centrality on weighted networks .researchers could apply this tool to conveniently calculate bc on weighted networks fast , especially on large networks .the source code is publicly available through https://dx.doi.org/10.6084/m9.figshare.4542405 .first we briefly introduce the well - know brandes s algorithm and dijkstra algorithm based on the preliminary definitions of network and betweenness centrality .a graph can be defined as , where is the set of vertices , and is the set of edges .an edge is a node pair , which means that there is a link connecting nodes and , and its weight is . if the edge exists , it can be traversed from to and from to because we only focus on undirected graphs in this paper .however , our algorithm can be expanded to directed graph version easily .a path is defined as a sequence of vertices connected by edges , where is the starting node and is the end node .the length of is the sum of the weights of the edges involved in . is the distance between and , which represents the minimum length of all paths connecting and . denotes the number of shortest paths from to . according to the definition, we have  , and for undirected graph . denotes the number of shortest paths from to where lies on .based on these definitions , the betweenness centrality can be defined as    from the above definitions , the calculation of betweenness centrality can be naturally separated into the following two steps :  1 .compute and for all node pairs 2 .sum all pair - dependencies ,  in which pair - dependency is defined as .the first step consumes and time for unweighted and weighted graph respectively , therefore the bottleneck of this algorithm is the second step , which requires time .brandse s developed a more efficient bc algorithm which requires time for unweighted graph , and time for weighted graph .the critical point is that the dependency of a node when the source node is is . applying this equation, we can accumulate the dependencies after computing the distance and number of shortest paths from a source vertex to all other vertices , rather than after computing all pair shortest paths .we can develop a parallel version based on brandes s algorithm for unweighted graph because the graph is always traversed as a tree by using dfs algorithm .given a source node , the root of the tree is and the tree produced by dfs method in the first step . in the second step , dependencies related to source node are calculated from the bottom to the root of the tree and the nodes at the same level are isolated and have no influence to each other . as a result , the parallel version can explore nodes at the same level simultaneously in both of the two steps , which will essentially boost the calculation .dijkstra algorithm  and floyd - warshall algorithm  are commonly employed to solve shortest path problems . while dijkstra algorithm is more adaptable to betweenness centrality problem because brandes s algorithm accumulates dependencies after computing single source shortest paths ( sssp ) , rather than finding and storing all pair shortest paths .dijkstra algorithm applies greedy strategy to solve sssp . in this algorithm ,the source node is and if the shortest path from and another node is achieved , will be settled .according to be settled or not , all nodes in graph could be separated into two sets , which are settled vertices and unsettled vertices .an array is used to store tentative distances from to all nodes . at first , and for all other nodes . and the source node settled and considered as the frontier node to be explored . in the second step , for every node and the adjacent frontier node ,if +w(f , u)<d[u] ] will be updated to +w(f , u) ] and is frontier node if =1 ] is the number of shortest paths from to . ] + \leftarrow 0 ] + \leftarrow 0 ] + \leftarrow 0 ] + \leftarrow 0 ] + \leftarrow 0 ] + \leftarrow s;s_{len } \leftarrow 1 ] +   +  +  +  +  +  +   the parallel version of dfs procedure , which is used in bc algorithm for unweighted network , could be modified naturally from its sequential version because vertices located at the same level of the dfs tree can be inspected simultaneously .while for dijkstra algorithm , picking one frontier node each time makes its parallelization a difficult task .however , this restriction can be relaxed , which means that several nodes could be settled becoming frontier set and be inspected simultaneously in the next step . in this paper, we apply the method described in  . in this method, is precomputed .then we define as    where is the tentative distance of node , is the unsettled nodes set in iteration .all nodes that satisfy the following condition    are settled and become frontier nodes . when applying dijkstra algorithm in betweenness centrality calculation , the number of shortest paths should be counted . to achieve this goal ,the above condition should be modified to   fig .[ fig : shortest](a ) demonstrates an example , in which vertex is the source node . if applying eq .[ eq : condition1 ] , and will be frontier nodes after inspecting in the first iteration , and the number of shortest paths will be 1 for both and . then and will be inspected simultaneously in next step .if processing first , the number of shortest paths for will be set to 1 , while the the correct value of shortest paths number for should be 2 .this mistake comes from the overambitious condition and should not be settled after the first iteration .although the distance will be correct for all nodes using eq .[ eq : condition1 ] , but the number of shortest paths will be wrong .however , eq . [ eq : condition2 ] will lead to correct shortest paths number for by only settling after first iteration. this condition could be found at line 22 in algorithm [ alg : shortest ] .algorithm [ alg : shortest ] depicts our parallel dijkstra algorithm in detail . the tentative distance and number of shortest pathsare calculated which can be seen from line 2 to line 12 . in this part , there will be a subtle parallel problem that several nodes in the frontier set may connect to the same node , as can be seen in fig .[ fig : shortest](b ) . in this example , both and are in frontier set and connect to , which results in the classical race condition problem .to avoid this situation , we define a lock for each node . the first thread focus on will achieve the lock and other threadswill not be permitted to change  ] .note that other threads must not wait because in cuda framework , a group of threads in a warp performs as a simd ( single instruction multiple data ) unit . after computing and for all nodes, we can achieve based on the above analysis , as can be seen from line 13 to line 18 . in the end , , , and are updated for next iteration .will make the number of shortest paths of incorrect .( b ) an example of race condition . and are frontier nodes in one iteration , and both of which are connected with .,scaledwidth=80.0% ]   as can be seen on line 2 in algorithm [ alg : shortest ] , threads will be assigned to all nodes but only nodes that in the frontier set will perform the calculation job , which may be inefficient .mclaughlin et al . figured out an excellent work - efficient technique to solve this problem  .here we develop our work - efficient version by employing this technique . will be changed to a _queue _ that stores all frontier nodes and a variable is defined to recode the length of , as can be seen in algorithm [ alg : we : init ] . then on line 2 in algorithm [ alg : we : shortest ] , threads can be assigned to \sim f[f_{len}-1] ] + // inspect // calculate +  +  +  +  +  +  +   real world networks always have scale - free character , which means their degree distributions follow power law .when implementing parallel graph algorithms through node parallel strategy , this feature brings in serious load - imbalance problem .most nodes have low degrees while some nodes have extremely high degrees .threads that assigned to high degree nodes will run slowly and other threads have to wait .edge parallel strategy can solve this problem  but bring in other under - utilizations at the same time . in this paper, we apply the novel warp - centric method  , which allocates a warp to one node rather than a thread . then threads within a warp focus on part of edgesconnected the specific node . as a result, each thread does less job for high degree nodes and the waiting time will be sharply decreased .moreover , memory access patterns can be more coalesced than the conventional thread - level task allocation and because of this , the efficiency of memory access can be essentially improved .nevertheless , the warp - centric method also has some drawbacks .firstly , node degree may be smaller than the warp size , which is always 32 in modern gpu . to solve this problem ,virtual warps are proposed in  .secondly , the number of required threads will be raised as each node needs _warp_size _ threads rather than one thread in this situation .but the number of threads in one block is fixed , hence each thread will be assigned to more nodes iteratively , which may results in low performance .we find that work - efficient method can relieve this problem because it requires less threads compared to the conventional node - parallel method , as can be seen in fig .[ fig : warp ] . in this paper, we apply the warp - centric method for both node - parallel and work - efficient method . as a result , we get four algorithms ( see fig . [fig : real ] ) that using different threads allocation strategies and we compare them on both real - world and synthetic networks .we collect six real - world networks from the internet , which have broad types including collaboration network , epinions trust network , email communication network , wiki vote network and two biological networks .they are publicly available in the internet and have been analyzed extensively by previous literatures  .the details of these networks are listed in table [ tab : real ] . to further understand the effect of network structures to algorithms performance , we generate two types of networks , which are erds rnyi ( er ) random graphs  and kronecker graphs  .the degree distribution of er random graph is poisson , indicating its nodes degrees are relatively balanced . while kronecker graph possesses scale - free and small - world characters , which make it more similar to the realistic network .the two biological networks and the cond - mat-2005 collaboration network are weighted networks and for other networks , we uniformly assign random edge weights ranging from 1 to 10 .we run the four gpu methods on geforce gtx 980 ( only entry - level for scientific computing ) using cuda 7.5 toolkit .we also develop the sequential algorithm using c++ and optimize it by applying binary heap in dijkstra algorithm due to fibonacci heap s inefficiency in practical use , making our cpu version bc algorithm performs better than most of the existing implementations .and we run the cpu version algorithm on competent intel xeon e5620 with 2.40ghz .-0.6in0 in  .*details of networks from public dataset * [ cols="<,<,<,<,<,<",options="header " , ]   [ tab : real ]   from fig .[ fig : real ] , we can see that all the four gpu programs achieve much better performance than the cpu version on all the six real - world networks .the algorithm that applies work - efficient coupled with warp - centric technique is the best one for achieving 30 to 150 speedup and its performance could be essentially improved on more sophisticated gpu devices .work - efficient method is more efficient than node - parallel in all networks , while warp - centric method is better on large degree networks , such as the two biological networks . for networks with low average degrees , applying warp - centric method aloneis always inefficient because nodes degrees are always smaller than _ warp_size_. using smaller virtual _warp_size _ could be better on these networks and we will demonstrate this hypothesis later .however , combining warp - centric method and work - efficient method is always better than using work - efficient method alone because it needs less threads in each step , which accordingly relieves the influence of the second drawback of warp - centric method .the combination algorithm achieves 2.65 speedup over the baseline node - parallel strategy on average .to deeply mining the relationship of the network structure and the performance of the four gpu implementations , we further run them on two types of synthetic graphs , as can be seen in fig .[ fig : scale ] . from fig .[ fig : scale](a ) , ( b ) , ( c ) and ( d ) , we find that work - efficient algorithm works better than node - parallel algorithm in all networks since it always reduces the required number of threads .as can be seen in fig .[ fig : scale](a ) and ( b ) , warp - centric method works well on networks with large degrees , which is consistent with the conclusion in realistic networks .note that for kronecker graphs , warp - centric method works better than that for random graphs as kronecker graphs have serious load - imbalance problem and warp - centric technique appropriately solves it .while for er random graphs in [ fig : scale](a ) , the advantage of warp - centric method is only the efficient memory access . for low degree graphs ,warp - centric method works even worse than node - parallel strategy because the degrees are always smaller than _ warp_size _ , as can be seen in fig .[ fig : scale](c ) and ( d ) . for random graphs ,the performance of warp - centric method is extremely poor when the average degree is smaller than 8 and fig .[ fig : scale](e ) explains the reason .the small average degree brings in large average depth , which means that the average size of the frontier sets is small . in this case , warp - centric method assigns more useless threads to nodes that do not need inspections . on the contrary ,as the degree grows , it is closer to _ warp_size _ and the depths drop down sharply , which make the warp - centric method performs much better . while low - degree kronecker graphs have power - law degree distributions and small average depths , which make warp - centric method works not as bad as on random graphs .however , the combination of the two methods always runs faster than applying work - efficient method alone because it avoids the second drawback of warp - centric method , which is discussed in the previous section . in conclusion , work - efficient method always achieves better performance while the performance of warp - centric method relies on networks structures but the joint version always achieves the best performance .to for er random and kronecker graphs , respectively . andthe average degrees are fixed to 32 for both of the two types of networks .( c ) and ( d ) separately tune the average degrees for random and kronecker networks , in which the random networks have 20000 vertices and the kronecker networks have nodes .( e ) illustrates the average depths of search trees for random graphs used in ( c ) and kronecker graphs used in ( d ) .networks with larger depths have smaller average frontier sets , indicating the poor performance with parallelism.,scaledwidth=90.0% ]  from the above analysis , applying smaller _warp_size _ may accelerate the two implementations which using warp - centric method when the netowrks average degree is small . andthis hypothesis can be verified in fig .[ fig : warp_adjust ] .we apply smaller _ warp_size _ on email enron network , soc_epinions1 network and other two synthetic graphs whose average degrees are both four . from fig .[ fig : warp_adjust](a ) and ( b ) , we find that implementations with smaller _ warp_size _ do perform better than both of the baseline node - parallel algorithms and the large _ warp_size _ algorithm on both of the low - degree realistic networks . andwhen coupled with work - efficient method , algorithms with smaller _ warp_size _ also perform better than both of the work - efficient strategy alone and the combination of work - efficient and large _warp_size_. the reason is that small _ warp_size _ reduces the required number of threads and then eliminates the waste of assigning more threads to a node than its degree .the implementations which have small _ warp_size _ and coupled with work - efficient method achieve the best performance because they avoid both drawbacks of warp - centric method but utilize the advantages of this technique .the results on low - degree kronecker graph is similar as on realistic networks since kronecker graph is similar with real - world network . for er random graphs , the algorithm with small _ warp_size _does not achieve better performance compared to node - parallel version because the large average depth , which is analyzed in previous section .however , when coupled with work - efficient method , the implementations with small _ warp_size _ perform slightly better than the work - efficient algorithm , which further proves the excellence and stability of the joint algorithm . in summary ,the joint algorithm are most efficient and insensitive to network structure . and if we choose an appropriate _ warp_size _ , its performance could be even better .nodes whose average degree is four .warp - centric method can not accelerate the speed when combining node - parallel strategy .but when combining small _ warp_size _ with work - efficient method , the performance will be slightly better than applying work - efficient method alone .( d ) is kronecker graph with nodes and the average degree is four , on which smaller _ warp_size _ achieves better performance.,scaledwidth=90.0% ]existing gpu version of betweenness centrality algorithms only concentrate on unweighted networks for simplification .our work that computing betweenness centrality on large weighted networks bridges this gap and achieves prominent efficiency enhancement compared to the cpu implementation .moreover , we apply two excellent techniques which are work - efficient and warp - centric methods in our algorithm .work - efficient method allocates threads more efficiently and warp - centric method solves the load imbalance problem and simultaneously optimizes the memory access .we compare these implementations with cpu algorithm and the basic gpu algorithm in realistic networks .the results show that gpu parallel algorithms perform much better than the sequential algorithm and the algorithm which integrates the two techniques is the best , achieving 30 to 150 speedup over the cpu version .results on generated random graphs and kronecher graphs further justify the outperformance of our solution .  for future work, we will consider implementing gpu algorithm to process dynamic networks . when networks changes a little ( like few new nodes come in or several links vanish ) , calculating betweenness centrality for all nodes is unnecessary because betweenness centrality of most nodes and edges will not be changed .some previous works have explored the sequential algorithm on this issue  .we plan to develop gpu version of these algorithms to achieve better performance .this work was supported by nsfc ( grant no . 71501005 ) and the fund of the state key lab of software development environment ( grant nos . sklsde-2015zx-05 and sklsde-2015zx-28 ) .r. f. also thanks the innovation foundation of buaa for phd graduates .wang y , davidson a , pan y , wu y , riffel a , owens jd .gunrock : a high - performance graph processing library on the gpu . in : proceedings of the 20th acm sigplan symposium on principles and practice of parallel programming .ppopp 2015 .new york , ny , usa : acm ; 2015 .p. 265266 .harish p , narayanan pj .accelerating large graph algorithms on the gpu using cuda . in : proceedings of the 14th international conference on high performance computing .berlin , heidelberg : springer - verlag ; 2007 .p. 197208 .cong g , bader da .an experimental study of parallel biconnected components algorithms on symmetric multiprocessors ( smps ) . in : proceedings of the 19th ieee international parallel and distributed processing symposium .ipdps 05 .washington , dc , usa : ieee computer society ; 2005 .p. 45b45b .sariyce ae , kaya k , saule e , atalyrek v .betweenness centrality on gpus and heterogeneous architectures . in : proceedings of the 6th workshop on general purpose processor using graphics processing units ; 2013 .p. 7685 .mclaughlin a , bader da . scalable and high performance betweenness centrality on the gpu . in : proceedings of the international conference for high performance computing , networking , storage and analysis ; 2014 .p. 572583 .delling d , goldberg av , nowatzyk a , werneck rf .phast : hardware - accelerated shortest path trees . in : proceedings of the 2011 ieee international parallel & distributed processing symposium .ieee computer society ; 2011 .p. 921931 .davidson a , baxter s , garland m , owens jd .work - efficient parallel gpu methods for single - source shortest paths . in : proceedings of the 2014 ieee 28th international parallel and distributed processing symposium .ieee computer society ; 2014 .p. 349359 .hong s , kim sk , oguntebi t , olukotun k. accelerating cuda graph algorithms at maximum warp . in : proceedings of the 16th acm symposium on principles and practice of parallel programming ; 2011 .p. 267276 .bell n , garland m. implementing sparse matrix - vector multiplication on throughput - oriented processors . in : proceedings of the conference on high performance computing networking , storage and analysis ; 2009 .p. 18:118:11 .crauser a , mehlhorn k , meyer u , sanders p. a parallelization of dijkstra s shortest path algorithm . in : proceedings of the 23rd international symposium on mathematical foundations of computer science ; 1998 .p. 722731 .rossi ra , ahmed nk . the network data repository with interactive graph analytics and visualization . in : proceedings of the twenty - ninth aaai conference on artificial intelligence ; 2015.available from : http://networkrepository.com .

Abstract: recent decades have witnessed the tremendous development of network science , which indeed brings a new and insightful language to model real systems of different domains . betweenness , a widely employed centrality in network science , is a decent proxy in investigating network loads and rankings . however , the extremely high computational cost greatly prevents its applying on large networks . though several parallel algorithms have been presented to reduce its calculation cost on unweighted networks , a fast solution for weighted networks , which are in fact more ubiquitous than unweighted ones in reality , is still missing . in this study , we develop an efficient parallel gpu - based approach to boost the calculation of betweenness centrality on very large and weighted networks . comprehensive and systematic evaluations on both synthetic and real - world networks demonstrate that our solution can arrive the performance of 30x to 150x speedup over the cpu implementation by integrating the work - efficient and warp - centric strategies . our algorithm is completely open - sourced and free to the community and it is public available through https://dx.doi.org/10.6084/m9.figshare.4542405 . considering the pervasive deployment and declining price of gpu on personal computers and servers , our solution will indeed offer unprecedented opportunities for exploring the betweenness related problems in network science .   + rui fan , ke xu and jichang zhao + key lab of software development environment , beihang university + of economics and management , beihang university + author : jichang.edu.cn


the analysis of networks in financial markets has received a growing attention in recent years . however , most studies to date have focused on closed systems , such as inter - bank markets , ownership networks or networks of directors .recently network analysis has been applied to cross - border portfolio investment flows showing that network measurements can be used as indicators of structural robustness of financial systems .research has also been further developed to couple financial networks with real economic networks ( cross - border trades , or , the international trading network ) into an all - inclusive approach .departing from this latter line of research , the present analysis aims to develop a comparative analysis of the international financial system with the environmental loads carried by goods in the international trade network(itn ) within oecd countries .  setting off from this latter line of research, the present analysis aims to develop a comparative analysis of the international financial system in conjunction with environmental loads carried by goods in the itn within oecd countries .the research conducted in this paper relates to two main strands of the literature : the debate on the interactions between financial and bilateral trade flows and that of the nexus between environmental and investment flows , such as foreign direct investment ( fdi) .our analysis concentrates on correlation and reciprocity structure of relations between finance and the environmental content of trades .the existence of reciprocal interaction between financial flows ( particularly long term ones , such as fdi ) and trade between countries is well established in the literature . according to the model proposed by coeurdacier , trade openness ,namely the exposure of domestic firms to increasing international competition , might foster the acquisition of foreign firms equities as a hedging strategy .consequently , bilateral equity holdings and commercial imports are expected to be positively correlated .trade relations could also lead to a reduction in borrowing costs which , in turn , stimulates investments .information asymmetries might provide another explanation : transactions in international trade facilitate information flows between trading partners which , in turn , lowers uncertainties for international financial transactions and vice versa .a growing body of empirical evidence corroborates the hypothesis of the complementarity between trade and fdi .the empirical literature has also investigated the correlation between trade and portfolio investments .aviat and coeurdacie , exploring the geography of trade in goods and asset holdings , for instance , find that the causality between bilateral asset holdings and commercial trade strongly progresses in both ways .lane and milesi - ferretti , using data on international portfolio positions , show that there is a strong correlation between bilateral equity holdings and bilateral trade in goods and services , however the question should be asked if this connection between trade and financial flows has environmental implications and whether highly financialised countries have incentives to re - allocate industrial production to less developed countries to reduce their domestic pollution .recent contributions on the `` pollution haven effect '' suggest that stringency of environmental regulation affects country competitiveness reducing net exports , increasing net imports and affecting firms location choice , and consequently fdi .additionally analyses by aichele and felbermayr reveal that the kyoto protocol affects trade flows by significantly increasing committed countries embodied carbon imports from non - committed countries and the emission intensity of their imports .less attention is paid to the role for portfolio investment as a channel for high financialised countries to re - allocate the industrial production into other countries to reduce their domestic pollution . in our paper, we investigate not only the link between fdi and trade of goods but also the correlation between other financial flows , such as equity and bond securities , which are characterized by a shorter time horizon than fdis .the research question we want to address is : what role do the long and short term financial flows plays with respect to the direction and intensity of international trade and their expected environmental impact , in terms of different pollution variables ?we find that embodied environmental content of imports have , on average , stronger and more stable correlation and reciprocity with financial outflows , either as fdi and cross - borders portfolio flows , than with inward financial flows .moreover , this pattern is more accentuated for highly financialised countries .in particular , in the multiplex network of cross - border financial investments and of environmental flows embodied in trade movements , four ( france , germany , usa and uk ) out of five hubs are net importers of environmental load .these results can not definitively detect a causal link but are consistent with the notion that financial markets help the most financialised countries export capitals in exchange of environment load s displacement .the rest of the paper is organized as follows .we first present the used data sources and data content and we briefly present the indexes of correlation and reciprocity calculated in the analysis .then , we discuss the main results and their economic implications . the last section concludes .supplementary materials and all details of the analyses are reported in the sm .our network analysis combines various data sources for measures of trade and bilateral financial flows and positions between oecd countries over the 2002 - 2010 period .more precisely , we consider ten networks where nodes are given by 33 oecd countries and where edges are represented by aggregated / effective cross - border flows of financial investments or goods ( see the si for a detailed list of nodes and flows ) .financial flows comprise bilateral foreign direct investment flows ( fdi ) and portfolio capital flows .fdi data are reported in current usd millions and come from oecd statistics .portfolio flows are also measured in current usd millions and data are drawn from the coordinated portfolio investment survey ( cpis ) carried out by imf .the cpis records data on the bilateral composition of year - end portfolio holdings ( long - term debt , short - term debt and equity portfolio assets ) for over seventy reporting / source countries _ vis - a - vis _ over 200 destination countries , however it does not provide information on portfolio flows .for this reason , all portfolio flows in this paper are measured as the difference between consecutive positions : an increase ( decrease ) in debt securities issued by country and held by residents in country is recorded as a capital movement from country j to country ( from to ) .earlier studies have already underscored the main limitations of the cpis in terms of incompleteness , lack of strong data consistence , risk of under - reporting and problematic treatment of intermediated holdings in financial centers. however , the inclusion of portfolio flows , in addition to fdi , allows us to track trends and features of financial investment decisions that are based on different time - horizons and driven by different economic motives .moreover , we try to contain some of these problems by focusing the analysis on oecd countries which are more likely to report comprehensive information .we therefore select a set of countries which are expected to ensure the best _ trade - off _ between data quality and geographical and time coverage .this , however , implies that the interpretation of our results should take into account that large players , such as china , and important resource - rich countries , such as the gulf states , are excluded from the analysis . as for trade networks , we concentrate on environmental flows which are , directly and indirectly , generated by trade flows .the estimation of the environmental footprint of trade flows draws from the eora global multi - region , input output ( mrio ) database .eora elaborates a time series of environmentally extended input - output ( io ) tables for 187 countries based on the un system of national accounts ( sna ) , un comtrade , eurostat , ide / jetro , and several national io tables with matching environmental and social satellite accounts .we refer to this database to obtain footprint embodied in bilateral import and export flows in terms of five environmental dimensions : emissions of  ( in gigagrams , gg ) ,  ( gg ) , (gg ) , total  emissions ( gg ) and water footprint ( in m3 ) .the five financial networks analysed in this study comprise the networks of bilateral fdi , portfolio investments in short and long term security debt ( sd and ld ) , in equities as well as in total security debts ( sd+ld ) .. environmental and financial networks share the same set of nodes , namely the 33 oecd countries .the entanglement of their relationships gives rise to a _multiplex _ of interacting networks .henceforth , every network will be defined as a _ layer _ of this financial - environmental multiplex .  to start with, we investigate the _ spatial _ correlation among layers by applying a previous method based on the pearson correlation coefficient .figure [ fig1 ] shows the temperature map ( colors from white to dark red indicates an increasing , positive correlation ) for the five financial layers ( 1 to 5 ) and the five environmental layers ( 6 to 10 ) .correlations are averaged over the period under investigation ( 2002 - 2010 ) .the first matrix shows correlations between flows going in the same direction ( synergic flows ) and the second shows correlations between flows going in opposite direction ( reverse flows ) .interestingly , the most correlated financial layers to the environmental ones are _ equities _( layer2 ) and _ total debts _ ( layer 5 ) .all the environmental layers , are more correlated to ( with the exception of reverse flows of water , layer 10 , that is correlated more with sd , layer2 ) .the second most correlated financial layer to the environment is _td_. among the environmental layers , ( layer 8) is always the most correlated to any financial layer .this may suggest that financial layers tend to be more correlated to manufacturing , metallurgy and industry in general that relies heavily on the combustion of raw fossil fuels , but more research is needed on this aspect .the second most correlated is ( layer 6 ) , further suggesting a link to combustion and , foremost , to low - efficiency combustion .this may hint at a specific topology in correlation that will be addressed further in a subsequent section of this paper .: fdi , equity , short - term debts , long - term debts and total debts .layers : , ,  ,  and water footprint . according to the pearson index equityis the financial layer most correlated to the environmental layers , followed by td ( see si , table [ tab3 ] ) . , and water are the most correlated layers with the equity layer and most of the other financial layers .the first two polluters are linked to combustion and may hint to a nexus with the heavy industry .the footprint of water is high in the energy intensive and agricultural sectors , and in some manufactures ., title="fig : " ] : fdi , equity , short - term debts , long - term debts and total debts .layers : , ,  ,  and water footprint . according to the pearson index equityis the financial layer most correlated to the environmental layers , followed by td ( see si , table [ tab3 ] ) . , and water are the most correlated layers with the equity layer and most of the other financial layers .the first two polluters are linked to combustion and may hint to a nexus with the heavy industry .the footprint of water is high in the energy intensive and agricultural sectors , and in some manufactures ., title="fig : " ]  a second notable result of our analysis is that , in most cases ( 15 out of 25 , see table [ tab3 ] and [ tab4 ] in si ) and particularly for the most correlated layers ( equity , td , and ) , _ reverse _ flows display higher correlation to the environment than synergic flows , indicating that among oecd countries financial flows tend to be associated to environmental flows going in the _ opposite direction _ , more often than flows going in the _same direction_. furthermore , reverse correlations are more stable than synergic correlations ( see si for further details ) .reverse correlations among flows suggest that different layers tend to reciprocate .we measure the cross - product reciprocity between layer and and country and , normalized on country s exports ( ) :    figure [ fig2 ] shows the cross - product reciprocity between financial layers on aggregate ( the average of the four financial layers ) and environmental layers on aggregate ( the average of the five environmental layers ) . in the first matrix ( figure [ fig2 ] , left panel ) , entries ( countries ) are reported in alphabetical order , while in the second ( figure [ fig2 ] right panel ) , countries are ordered by increasing financialisation ( measured as value added of the financial sector over the total value added ) . shifting the array - order of countries we can see a gradient of increasing reciprocity emerging from the matrix , when going from less to more financialised countries .this result confirms that there is a specific topology in the financial - environment correlation network that mirrors the level of countries financialisaton. the top - right block of the matrix ( figure [ fig2 ] right panel ) shows an average higher reciprocity than the bottom - left block ( see table [ tab4 ] in si ) .this asymmetry is evident when looking at the different degrees of yellows in the two regions , indicating a dominant pattern of reciprocity .exports of finance from the most financialised countries ( entries 19 - 33 ) towards the less financialised ( entries 1 - 18 ) are more correlated to environmental imports than flows from low financialised to high financialised countries .the countries of the low financialised group show higher correlation with the countries of the high financialised group than within themselves , contrary to the the counties of this latter group , which are instead tightly connected among themselves . in other words, it seems that highly financialised countries tend to exchange financial flows with environmental flows with countries that are less financialised , by means , most prominently in terms of equities and td .given the prevalence of reverse correlations , we can focus our analysis on reciprocal exchanges between the financial layers and environmental layers .it should be noted that a _ multiplex _ is a very complex structure .spurious correlations may rise from overlapping effects driven by two factors : the reciprocity structure within each layer and the topology peculiar to each layer . in what follows ,we want to disentangle the correlation between each pair of layers from the topology specific to every single layer ( with respect to our analysis , reciprocity between layers can be considered as a different form of correlation ) . by utilising a methodology recently developed in ,we are able to provide a measure of correlation between layers for reverse flows , on both a global ( correlations ) and local scale ( equation [ localrho ] ) that incorporates a null model and thereby clearing spurious effects from our analysis : this measure signals when reciprocity exceeds the expected reciprocity trivially produced by the null model .the chosen null model is based on an exponential randomization that preserves export , import and reciprocated trade flows for each country pair , which are set as constrains into the model . on the one hand, this null model enables us to test previous results and , on the other hand , to extract the backbone of significant correlations among countries .according to the cross - product reciprocity and the statistical validation , results of the pearson index hold : equities and td are more correlated to environmental flows than fdi and reverse correlation tends to prevail over synergic correlation .we can further test the structure of the reciprocal relationship among oecd countries between financial outgoing flows and environmental incoming flows with the null model based on exponential random graph ( see si ) .the matrix of figure [ fig3 ] shows the reciprocal exchanges between financial layers and environmental layers exceeding the single - layer reciprocity level set by the null model .the degree of yellow signals positive values of whereas the white dots indicates that there is no significant correlation ( equation [ localrho ] ) .the top - right block shows , on average , stronger correlations than the bottom - left , pointing to a statistically significant connection between financial outgoing flows reciprocated by incoming environmental flows in the region of the highly financialised countries with respect to the lowest financialised countries ( table [ tab3 ] in si ) .  ) ; countries ordered by degree of financialisation , left to right ( years 2002 - 2010 ) .financial flows are outgoing and environmental flows are incoming ; exports are normalized over columns ( example : entry indicates the share of financial export of country that is reciprocated by the share of environmental export of country . on the left the temperature map of the reciprocity values that exceed the significance threshold posed by the null model .wihte dots indicate that there is no significant reciprocity between countries .yellow dots concentrate in two regions : bottom - right and upper - right quadrants .the first indicate significant reciprocity among higly financialized countries .the second that there is a significant reciprocity from higly financialsed countries to less financialised countries .the the right panel shows the backbone network of the reciprocity structure : an arrow indicate a link of an outgoing financial flow reciprocated by an incoming environmental flow .the five hub - countries are shown with red dots.,title="fig : " ] ) ; countries ordered by degree of financialisation , left to right ( years 2002 - 2010 ) .financial flows are outgoing and environmental flows are incoming ; exports are normalized over columns ( example : entry indicates the share of financial export of country that is reciprocated by the share of environmental export of country . on the left the temperature map of the reciprocity values that exceed the significance threshold posed by the null model .wihte dots indicate that there is no significant reciprocity between countries .yellow dots concentrate in two regions : bottom - right and upper - right quadrants .the first indicate significant reciprocity among higly financialized countries .the second that there is a significant reciprocity from higly financialsed countries to less financialised countries .the the right panel shows the backbone network of the reciprocity structure : an arrow indicate a link of an outgoing financial flow reciprocated by an incoming environmental flow .the five hub - countries are shown with red dots.,title="fig : " ]  the yellow dots in figure [ fig3 ] can be converted into directed links in order to depict the topology of reciprocal relations among oecd countries . in what follows ,links map the network backbone of positive correlations exceeding the level of significance posed by the null model .the arrows in the right panel of figure [ fig3 ] stand for a statistically significant outgoing financial flow reciprocated by an incoming environmental flow . for almost all layer pairs and for the financial - environmental layers in aggregate, the backbone of the correlation network highlights the central role of five distinctive nodes : usa , france , germany , bel - lux and uk . figure [ fig4 ] show the case of the equity layer compared to the five environmental layers .those five countries display double the number of links compared to the other oecd countries ( figure [ s3 fig ] ) .this observation mirrors the central role these countries take in both financial markets and international trade . contrary to most of the developed and financialised countries , usa , france , germany , and uk ,are net importers of environmental flows , with the remarkable exception of bel - lux ( figure [ s4 fig ] ) . at the same time , but not surprisingly , these five countries are also net importers of financial flows ( figure [ s5 fig ] ) .it is noteworthy that these five countries are net importers in terms the value and the mass of international trade of goods .industrialized countries transform row materials into composite goods .this explains their negative mass imbalance .goods that are positioned higher in the value chain have arguably a lesser content of mass and an higher environmental footprint by unit of value .  , ,  ,  and water footprint .increasing dark red indicate an increasing out - degree of the node .the hubs are pleaced in the core of the cloud .the reciprocity analysis confirms that equity is mostly reciprocated with and , suggesting a link with the industrial sector . ]in this paper , we have focused on the linkage between financial and environmental movements which are generated by cross - border investment activities and the trade of goods .in contrast to previous studies , we have compared the correlation between environmental and portfolio investment flows , including fdi as well as ipi .we have found that most environmental indicators , apart from water , are highly correlated with short term financial flows ( which might also be more speculative in nature ) than fdis , which are mainly long - term and stable investments in the real economy .this result appears to be coherent with the hypothesis that short - termism is not only linked to the financial system s stability but also to the real economy and environmental sustainability .a second finding is that reverse flows , particularly short term equity trades , show higher correlation with environmental flows than synergic flows , indicating that , among oecd countries , financial flows tend to be associated with environmental flows going in the opposite direction . reverse correlations among flows suggest that different layers tend to reciprocate .furthermore , the dominant pattern in reverse correlations between finance and environment is seen to be directed from more financialised to less financialised countries .the conclusion from this is that agents in net - importer and highly financilised countries tend to take speculative exposures in stocks traded in less industrialized countries .however , not all fiancialised countries exhibit a signicant correlation between financial layers and environmental layers ( see table [ tab_oecd ] of si for the complete ranking of oecd countries ) .the back - bone within this highly complex structure is formed by five countries , which are themselves hubs for international trade and investment , namely : the usa , france , germany , bel - lux and the uk .these five countries are both net importers of finance and net importers of mass. do these results , concerning the topology of correlations , hold even if we would enlarge the scope of the analysis , including emerging economies and developing countries ?what is the causal relationship beyond these correlations ?do financial investments draw environmental load or financial flows follow the channels of trade ? these questions remain unanswered because of limitations of the model and uncertainties in the data .although we can not derive conclusions on causality , we built a first step in contributing to these areas of research in three main ways : by applying network analysis instruments to study the links between financial flows and trade - related environmental movements ( beyond bilateral view ) , by including portfolio investment flows ( beyond fdi ) , and by evaluating trade flows according to their environmental content rather than in monetary or mass terms ( beyond disciplinary separatism ) .what we get is a picture of the world , albeit reduced to only oecd countries , of immense complexity , in which the interacting systems , even those seemingly distinct , can not be seen separately .to say it succinctly , with the stylish and sarcastic words of oscar wilde :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ london is too full of fogs and serious people .whether the fogs produce the serious people or whether the serious people produce the fogs , i do nt know ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _we would like to thank diego garlaschelli , tiziano squartini , francesco picciolo and rossana mastandrea for their help . without their contributionthis work would have not been possible .what is a _ multiplex _ ?a multiplex is a bundle of networks ( layers ) that share the _ same nodes _ but potentially with different flows ( edges ) within each layer .each network , or _ layer _ , can feature relationships of the _ same nature _ and measured in the same unit or can display a broad variety of connections and , thus , links will be measured in different units .metrics of our analysis will be of two natures : monetary units ( usd ) for the financial layers and mass units for the environmental layers ( table [ tab_layers ] ) .the nodes of the multiplex are 33 oecd countries and will be ordered in every layers matrix adjacency according to the increasing financialisation of their economy , expressed by value added accounted by financial intermediation , real estate , renting and business activities as a percentage of total value added in 2000 , namely prior to the period under study ( table [ tab_oecd ] ) .we analyzed the interactions of the resulting financial - environmental multiplex over a time span of 9 years , from 2002 to 2010 .these multiplex interactions over 9 years for 10 layers and 33 nodes can be expressed as a tensor with dimensions ( ) and the entry of the tensor expresses the flow of the layer at time from country to country .the total dataset for this analysis will thus has thus 95040 data points .fdi & foreign direct investment & 1 + equity & & 2 + sd & short term debts & 3 + ld & long term debts & 4 + td & total debts & 5 +  + nox & nitrogen oxide & 6 + pm10 & particulate matter & 7 + so2 & sulfur dioxide & 8 + co2 & carbon dioxide & 9 + water & & 10 +                                                  akoglu l. , vaz de melo p. o. s. & faloutsos c. quantifying reciprocity in large weighted communication networks .. science _* 7302 * , 85 - 96 ( 2012 ) .snijders t. a. b. , pattison p. e. , robins g. l. & handcock m. s. new specifications for exponential random graph models . _sociological methodology _ * 36*(1 ) , 99 - 153 ( 2006 ) .robins g. , pattison p. , kalish y. & lusher d. an introduction to exponential random graph ( p * ) models for social networks ._ social networks _* 29*(2 ) , 173191 ( 2007 ) .park j & newman mej ( 2004 ) the statistical mechanics of networks . _ phys rev e _ * 70 * , 066117 .

Abstract: we analyse a multiplex of networks between oecd countries during the decade 2002 - 2010 , which consists of five financial layers , given by foreign direct investment , equity securities , short - term , long - term and total debt securities , and five environmental layers , given by emissions of ,  ,  and the water footprint associated with international trade . we present a new measure of cross - layer correlations between flows in different layers based on reciprocity . for the assessment of results , we implement a null model for this measure based on the exponential random graph theory . we find that short - term financial flows are more correlated with environmental flows than long - term investments . moreover , the correlations between reverse financial and environmental flows ( i.e. flows of different layers going in opposite directions ) are generally stronger than correlations between synergic flows ( flows going in the same direction ) . this suggests a trade - off between financial and environmental layers , where , more financialised countries display higher correlations between outgoing financial flows and incoming environmental flows from lower financialised countries , which could have important policy implications . five countries are identified as hubs in this finance - environment multiplex : the united states , france , germany , belgium - luxembourg and the united kingdom .   keywords : finance , environment , international trade , network theory , multiplex