the study of earth - related systems has become even more important with the growing concerns about environmental changes and the awareness of sustainable development . as a paradigm of complex systems ,this research topic relies on multidisciplinary efforts and has also been addressed by physicists via methods of statistical physics .earthquakes  , geomagnetic activities  , climate  and weather - related systems  are just a few examples of systems that researchers have tackled in these pages . in particular , as pointed out by dove and kammen  , one of the foremost global environmental challenges is the climate change . in a broader sense ,the complexity of climate systems is related to the complex interactions between atmosphere , biosphere , cryosphere , lithosphere and hydrosphere .the latter one is the part of the climate system that comprises oceans , lakes and rivers , that is , the liquid water at the earth s surface and underground  and it is well known the extremely important role of water in global environmental change  .  in this context ,important systems are the rivers and their discharges , which have a large impact on human activities , and that may also suffer huge influence from these activities .a river flow results from complicated interactions between the weather - related systems ( such as rainfall , temperature and evaporation ) , the landscape ( such as basin area and land relief ) and human activity ( such as pollution and power generation ) .these many features make river flow rates ( river discharges ) a complex process that has attracted the attention of scholars over the last six decades .for instance , the seminal work of hurst about the long - range dependence of runoff records from several rivers  has fostered several discussions on the fractal / multifractal and scaling properties of the temporal evolution of river flows  .the correlations between river flows and other systems have been also studied , for instance , climate systems such as rain fall  and sunspots  ; and economic systems such as the growth of companies  .moreover , chaos theory  , stochastic models  and permutation entropy  are examples of approaches used to probe the complexity of runoff time series .  despite the considerable attention towards the investigation of river flows ,several works are still based on small datasets and a large - scale characterization of time series related to river flows is rarely reported .furthermore , previous efforts have been mainly focused on well established / traditional methods of times series analysis ( such as fractal / multifractal analysis ) and , for instance , the interesting advances in mapping time series into networks have just recently attracted the attention of researchers working on this topic  . here, we further fill this gap by studying the flows in 141 different measuring stations that cover 53 brazilian rivers via daily time series obtained from the period of 1931 to 2012 .specifically , we have employed the network framework of the horizontal visibility graph  for mapping these river flow series into networks . by tracking the evolution of topological properties of these networks, the horizontal visibility approach reveals that the flow in several stations are becoming more or less correlated ( and displaying more or less complex internal network structures ) over the years , a behavior that could be related to changes in the climate system and other man - made phenomena .this work is organized as follows .we first describe our database and review some properties of the horizontal visibility approach .we next employ the horizontal visibility graph to our time series .then , we characterize these networks by investigating the degree distribution and the clustering coefficient as well as evolutive features of these two measures .we also find that these measures display a kind of coupling .finally , we present a summary of our findings and some concluding remarks .the data we have accessed consist of time series of the natural river flow rates ( river discharges ) with daily resolution measured in 141 different stations . these time series cover 53 brazilian rivers , span the period from 1931 to 2012 [ as described in fig .[ fig:0](a ) and ( b ) ] and are made freely available by the operador nacional do sistema eltrico ons ( a federal institution that controls the power system in brazil )  .let s denote the flow rates by , where is a discrete time variable indexing the days of the year and stands for the year associated with the flow ; thus , represents the flow rate in jan . 10 1986 in a given station . for the matter of convenience, we have removed the datapoint associated to feb .29 from all time series of leap years .we have focused our analysis on a normalized version of the flow rates defined as where ^ 2\ ] ] are respectively the average and the variance flow profile along the days of the year for a given station ( is the number of years available for that station ) .the upper panel of fig .[ fig:1](a ) illustrates the definition of the normalized flow rates by showing a concrete example of its construction . by doing this procedure, we ensure that ( at least ) the main seasonal trend is removed from the original flow rates .  once having the normalized flow , we have applied the horizontal visibility graph approach for mapping each year of the time series in a corresponding complex network .this procedure was proposed by luque et al . as a more restrictive version of the visibility graph approach  with the advantage of providing analytical expressions for fully random time series  .the visibility approach has been applied to several contexts  , enabling researchers from time series analysis to employ tools from network science .several properties of the horizontal visibility graph can be found in refs . ; here , we briefly present its algorithm and some results related to the degree distribution .the horizontal visibility algorithm is a map that assigns each datapoint of a time series to a node / vertex in a complex network .two nodes , and , will be connected whenever one can draw a horizontal line in the time series space that join the datapoints and without intersecting any intermediate datapoint height , that is , > f_t(l)~\forall\,l\ , | ( i < l < j)\,.\ ] ] notice that , in our case , the horizontal visibility algorithm will produce one network per year of the time series of a given station .the bottom panel of fig .[ fig:1](a ) shows an example of a network constructed from the previous procedure by considering data from the paran river collected at the itaipu station in the year .also , fig .[ fig:1](b ) illustrate the horizontal visibility approach applied to a simple time series .thus , our approach enables the investigation of possible evolutive features of the river flow rates by tracking the evolution of network measures .we first evaluated the degree distribution of the networks , where stands for the node degree or its number of connections ( notice that the networks are undirected ) . from the works of luque  and lacasa  , we know that the degree distribution takes the form for fully random time series , regardless the probability distribution underlying the values of the time series .furthermore , time series arising from more complex dynamics are also usually described by asymptotic exponential distributions . however , the value of is usually different from .actually , is associated with chaotic processes ( the smaller the , the smaller the system dimensionality ) and is associated with stochastic processes ( the larger the , the longer the system correlations )  . in our case , fig .[ fig:2](a ) shows the cumulative degree distributions for all the 82 networks ( one for each year ) constructed with data from the paran river at the itaipu station .we observe that all distributions are asymptotically well described by exponential decays , which on log lin scale are represented by straight lines whose slopes match the values of .when comparing the asymptotic behavior of these distributions to the one expected for a random time series [ , we note that the empirical distributions display a faster decay . in order to numerically estimate the empirical value of for each distribution ,we have fitted a linear model to each cumulative distribution ( on log lin scale ) after removing the initial non - exponential behavior ( ) .we have further obtained the average value of after shuffling the times series over 100 realizations .figure [ fig:2](b ) shows the values of for each year of the original series as well as the average ( and confidence intervals ) of for the shuffled versions of these series .we observe that the empirical values of ( for this particular station ) are always larger than and that they are outside the confidence bounds related to random versions of these series .figures [ fig:2](c ) and ( d ) provide another representative example ( for the tocantins river at the tucuru station ) of this analysis , where again we have found values of larger than .  in order to fully characterize the degree distributions of our entire dataset , we have proceeded as in the two previous - discussed examples for estimating the values of for every time series .after , we calculate the probability distribution for the values of obtained from the original time series as well as for one shuffled version of each time series .figure [ fig:2](e ) shows both distributions , where it is evident that the values of for the original series are larger than for practically all time series in our dataset , only of the time series display slightly smaller values ( but close to ) .it is worth noting that the average value of is 0.65 , a value that can be related to long - range correlations in the river flows .in particular , if consider that our time series are described by a power - law correlation function , an average corresponds to ( see figure 3 of ref . ) and consequently to a hurst exponent around 0.75 , a result that is compatible with the existence of long - range persistent correlations in the river flows .another question that our data pose is regarding possible evolutive features in the river flow dynamics . to address this question ,we have investigated whether the values of of a given station display a time dependence over the years .specifically , we have tested the hypothesis of the relationship versus showing a linear trend by fitting the linear model ( via the ordinary least squares method ) where and are the linear coefficients .thus , the value of and its statistical significance provide clues of whether is changing over the years or not . figure [ fig:3 ] shows the value of for each station . in order to check the statistical significance of ,we have employed the -test for examining the null hypothesis of being different from zero via the two - tail -value . for a confidence level of 95% ( that is , -value smaller than 0.05 ), we find that the null hypothesis can not be rejected in 46 among 141 measuring stations ( fig .[ fig:3 ] ) . also , among the stations where is statistically significant , we find that the majority ( % ) displays an increasing trend .this result thus indicates that some stations are presenting a more correlated dynamics over the years .it is important to mention that the -test assumes that the residual distributions are normal and it can be considered not ideal in this context  . however , we have further applied the bootstrap regression approach  and the results obtained are quite similar , that is , the linear trends considered significant by the -test , were also considered statistically significant by the bootstrap regression .in addition to the degree distribution , another common - studied network property is the clustering coefficient  .this quantity measures the likelihood of nodes to create tight - knit groups with a relatively high density of ties .specifically , we have employed the global clustering coefficient , which is the number of closed paths of length two in the network over the number of all paths of length two . similarly to the analysis of the degree distribution , we have compared the values of obtained from the networks emerging from the original time series with the networks obtained from shuffled versions of these series ( average over 100 realizations ) .figures [ fig:4](a ) and [ fig:4](b ) show representative cases for the values of evaluated from two measuring stations .for both stations , we note that the values of evaluated for the original time series ( black lines ) are considerably larger than the values obtained from the randomized series ( red lines ) .similar behaviors are observed for almost all time series .in fact , we note from fig . [fig:4](c ) that the values of from the original series are distributed around , whereas the values of related to the shuffled series present a more acute distribution around .this result thus indicates that the underlying fluctuations of the river flows produce networks with more complex internal structures , which are visually observed in fig .[ fig:1 ] by noticing the formation of cliques among closer days .we have also investigated the evolutive features in the values of over the years for all stations . as we did previously for the degree distribution, we have verified the hypothesis of versus displaying a linear trend by least squares fitting the model where and are now the model parameters .again , the value of and its significance provide clues of whether is changing over the years or not .figure [ fig:5 ] shows the values of for each station , where the asterisk markers indicate the values that are statistically significant ( confidence level of 95% ) . among the 141 measuring stations , we find out that 68 stations exhibit a statistically significant linear trend and that the majority of these stations ( % ) present an increasing trend in (similar results are obtained with the bootstrap regression approach ) .we further show in fig .[ fig:5 ] some representative cases of these evolutive trends .    another intriguing aspect of the fig .[ fig:5 ] is that most of the stations displaying evolutive features in have also presented a similar dynamics in .this behavior suggests that the values of and may be somehow coupled . in order to check this possibility ,we have plotted the values of against the values of evaluated for each station and year of the time series .figure [ fig:6 ] shows this relationship where ( despite the scatter ) we note that the increasing of is ( on average ) followed by an increasing of .in order to overcome the noise and to focus on the main tendency of these data , we have calculated the window average values of this relationship .our results suggest that an exponential function describes quite well the average relationship , which confirms a coupling ( on average ) between the values of and .we have studied the river discharges from a relatively large dataset composed of almost one hundred and fifty stations covering more than fifty brazilian rivers in time periods of more than eighty years . in our investigations , we have proposed to employ the network framework of the horizontal visibility method for characterizing river flow fluctuations . our approach has pointed some intriguing evolutive features of the river discharges .we have shown that river discharges in certain measuring stations are becoming more or less correlated as well as displaying more or less complex internal network structures . despite the difficult in testing this hypothesis from our data , we believe that these evolutive features may be related to changes in the climate system , in particular regarding the rainfall system . other man - made phenomena such as the large - scale use of water in agricultural activities may also have contributed to these evolutive features . perhaps , other investigations at more local level could help to elucidate the mechanisms underlying the evolutive aspects of the river discharges presented here . finally , we believe that our work sheds new light on river flow dynamics and open the possibility for other direct investigations based on the horizontal visibility approach .this work has been supported by the cnpq , capes and fundao araucria ( brazilian agencies ) .hvr thanks the financial support of the cnpq under grant no . 440650/2014 - 3 .aat thanks the financial support of the cnpq under grant no . 462067/2014 - 9 .the authors are also grateful to dr .francesco serinaldi for helpful discussions during the review process of this manuscript .ribeiro , l.s .costa , l.g.a .alves , p.a .santoro , s. picoli , e.k .lenzi , r.s .mendes , analogies between the cracking noise of ethanol - dampened charcoal and earthquakes , phys .( 2015 ) 025503 .kantelhardt , d. rybski , s.a .zschiegner , p. braun , e. koscielny - bunde , v. livina , s. havlin , a. bunde , multifractality of river runoff and precipitation : comparison of fluctuation analysis and wavelet methods , physica a 330 ( 2003 ) 240 .y. zou , r.v .donner , n. marwan , m. small , j. kurths , long - term changes in the north - south asymmetry of solar activity : a nonlinear dynamics characterization using visibility graphs , nonlin .processes geophys . 21 ( 2014 )

Abstract: we report on a large - scale characterization of river discharges by employing the network framework of the horizontal visibility graph . by mapping daily time series from 141 different stations of 53 brazilian rivers into complex networks , we present an useful approach for investigating the dynamics of river flows . we verified that the degree distributions of these networks were well described by exponential functions , where the characteristic exponents are almost always larger than the value obtained for random time series . the faster - than - random decay of the degree distributions is an another evidence that the fluctuation dynamics underlying the river discharges has a long - range correlated nature . we further investigated the evolution of the river discharges by tracking the values of the characteristic exponents ( of the degree distribution ) and the global clustering coefficients of the networks over the years . we show that the river discharges in several stations have evolved to become more or less correlated ( and displaying more or less complex internal network structures ) over the years , a behavior that could be related to changes in the climate system and other man - made phenomena .


the issue of controlling chaos has attracted great interest in the last two decades ; much work has been done in the case of dissipative systems .ott , grebogi and york ( ogy )  a method by which chaos can be suppressed by making small time - dependent perturbations in order to shadow one of the infinitely many periodic orbits embedded in the chaotic attractor .this method has been extended by many following studies  and successfully applied to experimental systems  .  as a result of the absence of attractors in conservative systems and the complex nature of the phase space combining regular and irregular regions , constructingan effective method has remained a challenging question .lai ding and grebogi ( ldg ) extended the ogy method to hamiltonian systems by incorporating the notion of stable and unstable directions at each periodic point .  in relation to the above question , chandre _ et al . _ have developed a method for constructing barriers in phase space to subdue chaotic behavior for large times .this method is based on the introduction of a specifically designed small control term which changes the dynamics from chaotic to regular behavior .et al . _ have introduced a method for controlling chaos in two - dimensional hamiltonian system which they called adaptive integrable mode coupling based on the separation of the system into two coupled subsystems one of which is stable and the other unstable ; when the unstable system comes into the vicinity of the integrable system the conditions are reset resulting in an effective adaptive control .et al . _ have constructed a method which permits stabilization of kam islands through forward iteration of the orbits , and transforming them into global attractors of the embedded system .et al . _ have used a method of adding a small perturbation to modify systems , such as nonlinear plasmas , to follow more regular motion .additional efficient methods have been proposed in  ; some of these methods require tracking the trajectories while others involve the addition of interacting terms for which the criteria are somewhat sensitive .geometric approaches for the analysis of the stability of a given hamiltonian have been widely discussed  , for which curvature of a manifold is associated with stability .  the pioneering work of oloumi and teychenn  proposes a stabilization method by considering the gaussian curvature of the potential energy as a source of chaos . even if the condition of negative curvature of the potential and instability of the dynamics are not completely equivalent they were able to successfully control the instability by avoidance of negative curvature regions of the potential energy ( ancrp ) .  in this workwe make use of a recently developed geometrical criterion  which is highly sensitive to instability in hamiltonian systems , and has been shown to be in complete agreement with the results of the numerical technique of surface of section ( poincar plot ) and more effective than other geometric methods  .this criterion is based on an equivalence between motions generated by hamiltonian in the standard form with quadratic kinematic terms and additive potentials ( which we shall call the _ hamilton _description ) and a hamiltonian in which the dynamics is described by a metric - type function of coordinates multiplying the momenta in bilinear form ( which we shall call the _ gutzwiller _ description )  .the mapping between these equivalent descriptions was first introduced by appel  .the criterion for stability developed in  establishes an inverse mapping of the motion described completely geometrically in the gutzwiller space back to the hamilton description , carrying with it the covariance under diffeomorphisms that is a property of the gutwiller dynamics .the resulting orbits in the hamilton description , with the special choice of coordinates serving as the basis for the appel type relation , reduce to the standard description of the hamilton orbits under the hamilton equations . in general geometric form , as geodesic equations , however , they are subject to analysis in terms of geodesic deviation , and the resulting formula can be reduced to a computation in the special coordinates of a new ( symmetric ) matrix valued criterion for stability  which has been shown to be very effective in a wide range of examples  .  in the analysis of these systems, it was shown that the presence of negative eigenvalues for the stability matrix in the admissible physical region ( for which ) results in a chaotic type poincar plot .since the condition is local in coordinate space , the result necessarily implies some degree of ergodicity .there has , however , been no proof that the existence of negative eigenvalues is not strongly associated with the structure of the dynamics more globally , as is often the case , for example , for analytic function theory , where the presence of a complex pole implies a distortion of a relatively large region of the complex plane . the results of our application to the control of dynamical systems , however , indicates that the regions of negative eigenvalue correspond to a very localized phenomena .the observed instability they induce appears to be due to the passage of orbits through these regions , and they are not strongly correlated with the more global structure of the dynamics .what we have done here is to identify the regions of negative eigenvalue in the coordinate space , and modify the hamiltonian in these regions locally , removing the source of instability either by removing the instability inducing terms or varying the coupling to these terms to bring the hamiltonian closer to an integrable form in these regions .the remaining part of the space is left to develop according to the full , nonlinear , symmetry breaking , evolution . the effect on the poincar plots , as we show here , is dramatic .this result provides strong evidence that chaotic hamiltonian systems derive their properties from sources that may be thought of as highly localized in the regions of negative eigenvalues for the stability matrix .in addition to giving an interesting insight into the nature of hamiltonian chaos , it also provides an effective control method in which the hamiltonian dynamics is left to exercise its full function in a large domain of configurations , but is only constrained in prescribed local regions of the configuration space .  in the next section ,we review the basic ideas underlying the geometrical criterion , and in the following section we give numerical results for the application of our control procedure for the cases of an oscillator with broken symmetry and a potential which may be derived from the toda form .  +  +  +     +  +  +  +  has been shown  that a hamiltonian system of the form ( we use the summation convention ) where is a function of space variables alone , can be put into the equivalent form where is conformal and is a function of the coordinates alone .one can easily see that the orbits described by the hamilton equations for coincide with the geodesics on a riemannian space associated with the metric , i.e. , it follows directly from the hamilton equations associated with that  where the connection form is given by and is the inverse of .  for a metric of conformal form with inverse ,onthe hypersurface defined by , and assuming that the observable momenta are the same at every , the requirement of equivalence implies that    to see that the hamilton equations obtained from can , be put into correspondence with those obtained from the hamiltonian of the potential model , we first note , from the hamilton equations for , that we then define the _ velocity field _ coinciding formally with one of the hamilton equations implied by , for which we label the coordinates .  to complete our correspondence with the dynamics induced by ,consider the hamilton equation for generated by , with this becomes where   eq . has the form of a geodesic equation , with a truncated connection form  .note that performing parallel transport on the local flat tangent space of the gutzwiller manifold ( for which and are compatible ) , the resulting connection , after raising the tensor index to reach the hamilton manifold , is exactly the `` truncated '' connection .  substituting and into and, the kronecker deltas identify the indices of and ; the resulting square of the velocity cancels a factor of , leaving the hamilton - newton law derived from the hamilton equations directly from .is therefore a geometrically covariant form of the hamilton - newton law , exhibiting what can be considered an underlying geometry of standard hamiltonian motion .since the coefficients constitute a connection form , they can be used to construct a covariant derivative .it is this covariant derivative which must be used to compute the rate of transport of the geodesic deviation along the ( approximately common ) motion of neighboring orbits in the hamilton manifold , since it follows the geometrical structure of the geodesic curves on .the relation obtained from , can be factorized in terms of the covariant derivative one obtains where the index refers to the connection , and corresponds to the curvature associated with the connection form this expression does not coincide with the curvature of the gutzwiller manifold ( given by this formula with in place of ) , but is a _ dynamical curvature _ which is appropriate for geodesic motion in   with the conformal metric in noncovariant form , ( in the coordinate system in which is defined ) , the dynamical curvature can be written in terms of derivatives of the potential , and the geodesic deviation equation becomes where the matrix is given by and with , defining a projection into a direction orthogonal to .  instability should occur if at least one of the eigenvalues of is negative , in terms of the second covariant derivatives of the transverse component of the geodesic deviation .this condition is easily seen to be equivalent to the same condition imposed on the spectrum of , and is thus independent of the direction of the motion on the orbit .the condition implied by the geodesic deviation equation , in terms of covariant derivatives , in which the orbits are viewed geometrically as geodesic motion , is a new condition for instability  , based on the underlying geometry , for a hamiltonian system of the form , providing new insight into the structure of the unstable and chaotic behavior of hamiltonian dynamical systems .the method has been successfully applied to many potential models  .we now apply the method , which is simple and straightforward , since the instability criterion is defined locally in coordinate space , to extract from a chaotic hamiltonian the part of the dynamics in the regions for which has negative eigenvalues which lead to unstable motion .we show by direct simulation that this procedure is remarkably effective .the results confirm the notion of the locality of our criterion , since the ( local ) removal of parts of the hamiltonian inducing instability by this criterion strongly affects the global stability of the motion .in the following we give some examples of hamiltonian motion which are unstable and exhibit chaotic behavior . in two dimensions , the terms in the potential that break rotational symmetryhave a form , in these examples , that induces chaotic behavior .the matrix computed over the physically accessible region exhibits certain regions with negative eigenvalues .we have arranged our simulation to replace the hamiltonian _ in the regions of negative eigenvalues _ by a hamiltonian in which the symmetry breaking terms are either removed or for which the coupling coefficient is decreased , but in regions of positive eigenvalues , the hamiltonian remains in its original form .the changes are carried in relatively small regions of the configuration space in the neighborhood of the regions of negative eigenvalues .we take for illustration here a simple and important case of coupled harmonic oscillators with perturbation and a generalization of the toda potential   the first of these is known to generate chaotic behavior ; as we have shown previously , the transition to chaotic behavior as a function of energy and of the coupling to the perturbation is well described by our geometric criterion . in the present work, we set the coupling to the chaos inducing perturbation to zero in the regions for which the geometric criterion results in negative eigenvalues .  in our previous study of the second example eq .we studied the sensitivity of the dynamical behavior to the choice of energy . in the present work , at the energies for which the system exhibits chaotic behavior we have removed the chaos inducing perturbation in the regions for which the geometric criterion results in negative eigenvalues .1 shows the effects on the dynamical behavior of the change of the coupling to a stable value in the regions of negative eigenvalues for the perturbed oscillator potential .the results for different values of energy are shown ; these values correspond to different sizes of the physically accessible region .note that the radius of the region of positive eigenvalues does not change in this example .the first row shows the physical region and the location of negative eigenvalues .the interior of the circle corresponds to a region in which no negative eigenvalue occur , chosen in this way to simplify the computation .the second row shows the surface of section poincar plot(surface of section ) of the original uncontrolled hamiltonian indicating chaotic dynamics .the third row shows the poincar plot generated by the controlled potential :   where stands for the radius of the region of positive eigenvalues .2 shows the effect of control on second system , using the controlled potential : where stands for the limit ( a horizontal dashed line ) of the region of positive eigenvalues .in both cases , for sufficiently high energies the uncontrolled systems become less stable and a chaotic signature appears .we examine the method for different values of the energy , all corresponding to chaotic motion of the uncontrolled system .one can easily see that the poincar plots in controlled systems present almost completely regular motion .we see from these computations that in regions of positive eigenvalues the hamiltonian with chaos inducing terms do not generate instability .if the chaos inducing terms are removed or decreased in coupling in the regions of negative eigenvalues , the chaotic motion of the system is reduced or disappears .we conclude that the chaotic behavior of the system is associated with local properties of the hamiltonian , and that our local criterion is effective in identifying these regions .furthermore , our procedure provides an effective method for control of chaotic systems of this type .

Abstract: an effective characterization of chaotic conservative hamiltonian systems in terms of the curvature associated with a riemannian metric tensor derived from the structure of the hamiltonian has been extended to a wide class of potential models of standard form through definition of a conformal metric . the geodesic equations reproduce the hamilton equations of the original potential model through an inverse map in the tangent space . the second covariant derivative of the geodesic deviation in this space generates a dynamical curvature , resulting in ( energy dependent ) criteria for unstable behavior different from the usual lyapunov criteria . we show here that this criterion can be constructively used to modify locally the potential of a chaotic hamiltonian model in such a way that stable motion is achieved . since our criterion for instability is local in coordinate space , these results provide a new and minimal method for achieving control of a chaotic system .


human social interactions often lead to complex dynamics .repeated social interactions produce spontaneous variations which are manifested as inequalities at various levels .the availability of huge amount of empirical data for a plethora of measures of human social interactions has made it possible to uncover the patterns , analyze them and look for the reasons behind various socio - economic inequalities . besides using tools of statistical physics , researchers are also bringing in knowledge and techniques from various other disciplines  , e.g. , statistics , applied mathematics , information theory and computer science to better the understanding of the precise nature ( spatio - temporal ) and origin of socio - economic inequalities prevalent in our society .socio - economic inequality  basically concerns the existence of unequal ` wealth ' and ` fortunes ' accumulated due to complex dynamics within the society .it usually contains structured and recurrent patterns of unequal distributions of goods , wealth , opportunities , and even rewards and punishments , and classically measured in terms of _ inequality of conditions _ , and _ inequality of opportunities_. _ inequality of conditions _ refers to the unequal distribution of income , wealth , assets and material goods ._ inequality of opportunities _ refers to the unequal distribution of ` life chances ' .this is reflected in measures like level of education , health status , treatment by the criminal justice system etc .socio - economic inequalities are responsible for conflict , war , crisis , oppression , criminal activities , political instability and unrest , and that indirectly affects economic growth  of a region .traditionally , economic inequalities have been extensively studied in the context of income and wealth  , although it is also measured for many quantities like energy consumption  .the studies of inequality in society  has always been very important , and is also a topic of current focus and immediate global interest , bringing together researchers across various disciplines economics , sociology , mathematics , statistics , demography , geography , graph theory , computer science , and even theoretical physics .socio - economic inequalities are quantified in numerous ways .the most detailed measures are of course given by probability distributions of various quantities .what is usually observed is that most quantities display broad distributions most common are log - normals , power - laws or their combinations . for example, the distribution of income is usually an exponential followed by a power law  .however , such distributions can widely differ in their forms and subtleties , and as such they are not quite convenient to handle .this lead to the introduction of various indices like the gini  , theil  , pietra  and other socio - geometric indices  , which try to characterize various geometric features of these distributions .the degree of inequality is most commonly measured by the gini index .one considers the lorenz curve  , representing the cumulative proportion of ordered ( from poorest to richest ) individuals ( entries ) in terms of the cumulative of their wealth . can of course represent income or wealth of individuals but it can as well represent citation of articles , votes in favor of candidates , population of cities etc . the gini index ( ) , defined as the ratio of the area enclosed between the lorenz curve and the equality line , to that below the equality line , is the most common measure to quantify socio - economic inequality , taking values for absolute equality and for absolute inequality from a given statistical distribution .if the area between ( i ) the lorenz curve and the equality line is represented as , and ( ii ) that below the lorenz curve as ( see fig . [ fig : lorenz ] ) , the gini index is  .ghosh et al . recently introduced the kolkata index ( symbolizing the extreme nature of social inequalities in kolkata ) or ` -index ' , which is defined as the fraction such that poorest fraction of people possess fraction of income  .in fact , another recently proposed measure , the perpendicular width index  can be shown to be equal to .-index is given by area of the shaded region ( normalized by the area of the triangle afb ) .-index is computed from the coordinate of the point of intersection c ( ) of the lorenz curve and the diagonal perpendicular to the equality line .obviously , while -index measures the overall inequality in the system , -index gives the precise fraction of wealth possessed by fraction of richer population ., width=264 ]a large variety of socio - economic data suggest that there exists a simple relation between the two seemingly different inequality measures .we analyzed citations of papers published from academic institutions and journals ( data from isi web of science  and reported in ref . ) , consumption expenditure data of india  , brazil  , italy  , income data from usa  , voting data from open list proportional elections  of italy , netherlands and sweden , _ first past the post _ election data for indian parliamentary elections and legislative assembly elections  , united kingdom  , canada  , bangladesh  , tanzania  , and city population data from ref .  .see tables i - ix in the appendix for details .the relation is perfectly linear for smaller values while the curve becomes non - linear as or approaches unity , the limit of extreme inequality ( fig .[ fig : gk_emp ] ) .the most intriguing part is that the data from a variety of sources hardly depart from this smooth curve .we explore a spectrum of data such as income , expenditure , journal citations and impact factors , votes , city population to arrive at this conclusion .the -index and -index show a linear relationship with .in fig . [ fig : lorenz ] , the thick red line is a typical lorenz curve corresponding to a probability distribution function . denotes the cumulative share of from lowest to highest while denotes the cumulative share of .the lorenz curve cuts the anti - diagonal at point c and thus the -index is defined as the following ( in terms of wealth posessed by individuals , say ) : fraction of total wealth is possessed by fraction of the top wealthiest people . the gini index is simply given by . if is the shaded area enclosed by the lorenz curve ( acb ) and the equality line adb ( ) , then gini coefficient is given by we discusss below three approximate ways to calculate .   from fig .[ fig : lorenz ] , ab= and cd= .thus the area of the triangle cab is .cd .thus , giving it may be noted that the equality in the above relation corresponds to for .   here ,area ab.cd .this gives or in this approximation , the equality in the above relation corresponds to for .analysis of the observed data suggests that line ( eq .[ eq : emp ] ; fig .[ fig : gk_emp ] ) touches line at around .let us now imagine that the lorenz curve is represented by the arc acb of a circle ( fig .[ fig : cvstheta]a ) of radius ( = ae = be ) . de is perpendicular to ab such that . the total area of the sector beac is then .the area of the triangle abe is given by .de.ab = .thus our required area acdb is given by ( difference between the sector and the triangle defined above ) if we write .ab.cd , then referring back to case i , and incorporating the factor , we get the approximate value as . hence ( using eq .[ eq : a1 ] ) .this gives thus the slope of the line is .variation of with is plotted in fig .[ fig : cvstheta]b . the observed approximate value of ( from fig .[ fig : gk_emp ] ) is which corresponds to ( see fig . [fig : cvstheta]b ) .this would imply that the lorenz curve can be approximated as a quadrant arc of a circle with centre at e ( see fig .[ fig : cvstheta]a ) , subtending and angle at e ( compare fig .[ fig : cvstheta]a with fig .[ fig : lorenz ] ) . in that case, the line will touch the line at around ( from eq .[ eq : kg_c ] ) .   at the centre of the circle e. ( b ) plot of from eq .( [ eq : c ] ) for different values of ., title="fig:",height=151 ]  at the centre of the circle e. ( b ) plot of from eq .( [ eq : c ] ) for different values of ., title="fig:",height=173 ]  in fact , this linear relationship ( with the value of the slope ) derived here for a circular ( quadrant ) lorenz curve is more generally valid . if the lorenz curve in fig .[ fig : lorenz ] is taken as a parabola ( for uniform and normalized distribution of income / wealth ; ) , one gets and , giving .these values of and satisfy the above relationship ( eq .[ eq : emp ] ) very well .   vs. in cc and ccm models ( for agents ) .( a ) for cc model , varying parameter .the inset shows the plots of and in the range of .the solid line corresponds to eq .[ eq : gpck ] for vs. which agrees perfectly with the results from simulations . in the plot, the points fit to with .( b ) for ccm model , varying parameter .the inset shows the variation of and in the range of ., title="fig:",width=207 ]  vs. in cc and ccm models ( for agents ) .( a ) for cc model , varying parameter .the inset shows the plots of and in the range of .the solid line corresponds to eq .[ eq : gpck ] for vs. which agrees perfectly with the results from simulations . in the plot, the points fit to with .( b ) for ccm model , varying parameter .the inset shows the variation of and in the range of ., title="fig:",width=207 ]let us now consider some market models developed by econophysicists , in particular the kinetic exchange models  . in the cc model  there, an agent keeps a fraction ( same for all ) of their income or wealth before going for any ( stochastic ) exchange ( trade or scattering ) with another agent .formally , the dynamics is defined by \\m_j(t+1 ) & = \lambda m_j(t)+(1-r)(1-\lambda)\left [ m_i(t)+m_j(t ) \right ] , \end{aligned } \label{eq : cc}\ ] ] where is a random fraction in  ] by definition .fig [ fig : gk_emp1 ] shows this plot with the strict inequality line , which is never exceeded since ( eq . [ eq : a1 ] ) .the transformation to makes the slope of the vs plot for smaller values equal to ..computed and indices for academic institutions from citation data for different years .data retreived from isi web of science  and available from ref . . [ cols="^,^,^,^",options="header " , ]

Abstract: socio - economic inequality is characterized from data using various indices . the gini ( ) index , giving the overall inequality is the most common one , while the recently introduced kolkata ( ) index gives a measure of fraction of population who possess top fraction of wealth in the society . here , we show the relationship between the two indices , using both empirical data and analytical estimates . the significance of their relationship has been discussed .


the annual data traffic generated by mobile devices is expected to surpass exabits by 2020 .this deluge of traffic will significantly exacerbate the spectrum crunch that cellular providers are already experiencing . to address this issue ,it is envisioned that in g cellular systems certain portions of the mmwave band will be used , spanning the spectrum between ghz to ghz with the corresponding wavelengths between - mm .this will substantially increase the spectrum available to cellular providers , which is currently between mhz and ghz with only mhz of bandwidth allocation for all current cellular technologies .however , before mmwave communications can become a reality , there are significant challenges that need to be overcome .firstly , compared with the rf bands ( i.e. , sub-6 ghz ) , the propagation loss in the mmwave band is much higher due to atmospheric absorption and low penetration .while small wavelengths allow deployment of large antenna arrays in relatively small chip areas , it causes several other issues such as high energy consumption by components ( e.g. , analog - to - digital ( a / d ) converters ) .for instance , at a sampling rate of gsamples / sec , an quantizer consumes of power . during active transmissions, this would constitute up to of the overall power consumed for a typical smart phone .moreover , in order to fully utilize the directional antenna arrays , continuous beamforming and signal training at the receiver is needed .digital beamforming is highly efficient in delay , but there is a need for a separate a / d converter for each antenna , which may not be feasible for even a small to mid - sized antenna array due to high energy consumption .in contrast , analog beamforming requires only one a / d , but it can focus on one direction at a time , making the angular search costly in delay .there are also proposals on hybrid digital / analog beamforming , which strikes a balance between analog and digital beamforming , using a few a / d converters rather than one per antenna .secondly , a given mmwave channel can be highly variable with intermittent connectivity since most objects lead to blocking and reflections as opposed to scattering and diffraction in typical rf frequencies .when the users and/or surrounding objects are mobile , different propagation paths become highly variable with intermittent on - off periods , which can potentially result in long outages and poor mmwave delay performance . on the other hand , ultimately , the very - high bandwidth available in the mmwave band should translate into performance guarantees , required by next generation real - time applications that are expected to dominate the traffic in the next generation networks . in order to address the high energy consumption by components and beamforming overhead in mmwave, we consider a hybrid rf / mmwave beamforming scheme .due to high cost and energy consumption by a / d converters in fully - digital beamforming as well as the delay in fully - analog beamforming , we investigate the feasibility of conducting a coarse angle of arrival ( aoa ) estimation on the rf channel and then utilizing the fully - analog beamforming for fine tuning and transmissions . in this case , our a / d follows the beamformer at the receiver , eliminating the need for a separate a / d per antenna in the mmwave antenna array . to this end , we first experimentally verify the correlation between the rf and mmwave aoa , especially in the presence of line - of - sight ( los ) .our measurements taken jointly at different bands and for both _ indoor and outdoor settings _ show that under los conditions and in of all measurements , the identified aoa of signal in the rf band is within accuracy for the aoa of the mmwave signal .based on the estimated rf aoa , the angular range over which we scan for the mmwave transmitter reduces to no more than on average , from in stand - alone mmwave systems .the authors in have also proposed a beamforming method based on out - of - band measurements for ghz wifi and under _static indoor _ conditions .  to mitigate the issue of highly intermittent mmwave connectivity, we consider a hybrid rf / mmwave transceiver model , in which , in addition to beamforming , the rf interface is used for data transfer .the link speed of the mmwave interface ( multi - gbps ) is comparable to the speed at which a typical processor in a smart device operates .this is different from classical wireless interfaces in which data rates are much smaller than the clock speeds of the processors .thus , the mmwave interface can not be assumed to operate at smaller time - scales and the algorithms run at the processor may not be able to respond to variations in real time and execute control decisions ._ this necessitates the use of proactive queue - control solutions along with a reasonably large buffer at the mmwave interface ._ for instance , if the queue size at the mmwave interface gets small , the risk of wasting the abundant capacity from mmwave increases .conversely , if we keep the queue at the mmwave interface large , if the channel goes down , we incur a high delay .  to understand the tradeoff between full exploitation of the mmwave capacity and the delay for mmwave channel access , _ we model the hybrid rf / mmwave transceiver as a communication network , and investigate an optimal scheduling policy using network optimization tools ._ in the equivalent network model , the rf and mmwave interfaces are represented by individual network nodes with dedicated queues .hence , the optimal transmission policy across the rf and mmwave interfaces is transformed into an optimal scheduling policy across the rf and mmwave nodes .we formulate an optimal scheduling problem where the objective is to achieve maximum mmwave channel utilization with bounded delay performance subsequently , in order to determine `` when '' a data packet should be added to the rf or mmwave queues , we prove that the optimal policy is of the _ threshold - type _ such that the scheduler routes the arrival traffic to the mmwave queue if and only if its queue length is smaller than a threshold .we show that the threshold - based scheduling policy efficiently captures the dynamics of the mmwave channel , and indeed maximizes the channel utilization .  in summary ,our main contributions are as follows :  * we have conducted a wide variety of experiments to evaluate the correlation between the measured channel gains for the ghz mmwave and ghz rf interfaces under various indoor and outdoor situations involving existence of los between the transmitter and receiver .* we propose a hybrid rf / mmwave system that exploits the cross - interface correlations for beamforming as well as data transfer .our a / d follows the beamformer at the receiver , and eliminates the need for a separate a / d for each element in the mmwave antenna - array .* we propose a framework to model the hybrid rf / mmwave transceiver as a network and jointly manage the transmission across the rf and mmwave interfaces .our queue management formulation explicitly takes into account the mmwave channel dynamics , and our approach enables full utilization of the available mmwave channel capacity , despite the highly variable nature of the channel .we prove using subadditivity analysis that the optimal scheduling policy is a simple threshold based one , which can be easily adopted despite the high link variations .we should emphasize that the rf / mmwave correlation was studied in , and applied only for beamforming in .however , a coherent design that fully integrates the rf and mmwave interfaces and optimally design the rf / mmwave transceivers is missing .hence , we aim to develop a hybrid architecture for which the rf interface is utilized for both beamforming and data transfer .we use the following notation throughout the paper .bold uppercase and lowercase letters are used for matrices and vectors , respectively , while non - bold letters are used for scalers .in addition , denotes the conjugate transpose , denotes the matrix trace operator , and  ]   1**(average throughput and reneging rate ) * * under the scheduling policy with timeout , and given that packets are completely served by the mmwave queue , while packets renege at time slot , the average throughput and reneging rate of the policy is respectively defined as : ,  \label{throughput - definition}\ ] ] .\label{reneging - rate}\ ] ]  in order for the expectations in and to exist , we assume that and are stationary ergodic . in this model , imposing the service deadline ensures that the average waiting time of packets in the mmwave queue is smaller than or equal to .hence , the reneging mechanism explicitly dictates a constraint on the mmwave waiting time .therefore , our goal is to derive a throughput - optimal policy with bounded reneging rate .  *problem 1 * * ( constrained throughput optimization ) * _ given that there is a timeout for packets in the mmwave queue , we define problem 1 as follows : for a given . _  in , the objective function and the first constraint can be relaxed as : where is a positive lagrange multiplier .for any particular fixed value of , it is straightforward to show that there is no loss of optimality in the relaxed problem . to see this ,let be the optimal policy for the original problem , and be the optimal policy for the relaxed problem .we have : the first inequality holds since is feasible in the original problem , and the second inequality holds because is the optimal solution for the relaxed problem .thus , there is no loss of optimality in the relaxed problem .we note that the relaxed formulation can be interpreted as an optimization over obtained _ rewards _ and paid _costs_. in particular , each packet that receives service from the mmwave link , results in units of reward ( i.e. , in terms of mmwave throughput ) , while a packet reneging incurs a cost of ( i.e. , in terms of wasted waiting time in the mmwave queue ) .this leads to the following problem .* problem 2 * * ( total reward optimization ) * _ we consider the maximization problem over total rewards obtained as a result of serving packets , and costs due to packets reneging , i.e. , : \\  & \ \text{s.t . }  \ \ \  \limsup_{t \rightarrow \infty } \frac{1}{t } \mathds{e } \left [ \sum_{t=0 } ^t \beta^\pi(t ) \right ] \leq \lambda , \end{aligned } \label{payoff - optimization}\ ] ] where the constraint is implicit ._  it is straightforward to show that an optimal solution for the relaxed formulation of problem 1 is the optimal solution for and vice versa . to see this ,assume that the set of feasible solutions for is denoted by .for all , we have : multiplying both sides by , and setting , we conclude that is the optimal solution for as well since their feasible sets are identical . for and , two formulations will be identical . in general , the values of and are set based on the application and performance requirements .for instance , a large value of ensures high throughput , while a large value of prioritizes low - latency performance ( i.e. , a _ conservative policy _ ) .therefore , captures the tradeoff between full exploitation of the mmwave capacity and the delay for mmwave channel access through the control knob : if is set to a very small value for all time slots ( i.e. , a conservative policy ) then would be small as well , and the objective function reduces due to the first term . on the other hand , if is set to a large value ( e.g. , matched to the arrival rate for all time slots ) and the link state fluctuates according to the process , then the objective function could decrease due to the reneging cost that is captured by the second term .therefore , there is an optimal value of within these two extreme cases that results in the maximum return rate .the following example illustrates this point clearly .* example ( probabilistic admission policy ) * : using a probabilistic admission policy , the input arrival rate is mapped to the admitted rate such that a packet is admitted to the mmwave queue with an admission probability .we assume that the arrival process is a batch arrival such that there is a batch of size randomly distributed with normal distribution with mean and standard deviation .the probability of a batch arrival within a time slot is set to .the length of on and off periods of the mmwave channel is set according to log - normal distributions with mean and , respectively .[ probabilistic ] demonstrates behavior of the objective function in as increases .we observe that the objective value increases by admitting more packets into the queue up to a certain threshold , and thereafter the objective value decreases due to the dominant reneging cost . in this context , next we tackle the problem of deriving an optimal admission policy .     from and using the lagrangian relaxation ,we define : in which the lagrange multiplier is positive , and it can be interpreted as a _ subsidy _ for taking the _ passive action_. in our problem , passive action is equivalent to adding packets to the rf queue , while active action corresponds to admitting packets into the mmwave queue .hence , the objective is to maximize the long - term expected reward by balancing the reward for serving and the subsidy for passivity .the solution of partitions the state space into three sets , , and , where , respectively , the optimal action is for , for , or some randomization between and for . from , it is known that in a markov decision process if the state space contains a finite number of states , then the set does not contain more than one state .this case holds in our model since the mmwave queue length is upper - bounded by its buffer size , and the waiting time in the queue can be at most .thus , the set does not contain more than one state .in addition , the following theorem states that problems 1 and 2 are solved by a _ monotone _policy , where a class of policies has monotone structure if for , there exists such that : where denotes the mmwave queue length . in other words , the optimal policy routes a packet to the mmwave queue if and only if the mmwave queue length is smaller than a threshold .( optimality of monotone policy ) the solution for the reward optimization in has a monotone structure .[ monotone ]  let us denote by , the value function corresponding to problem 2 when mmwave is at state , and . from the bellman equation , we have :   in which , is the reneging rate and is the mmwave service rate .we prove that if passive action is optimal in then passive action is optimal in .similar to , let us define :   and .it then suffices to show that for .assuming , we have .let us now prove that has the _ subadditivity _ property .2**(subadditive function ) * * let and be partially ordered sets and a real - valued function on .we say that is subadditive if for in and in we have :   to prove that is a subadditive function , it suffices to show that for all and , the inequality holds . if or , then the inequality is satisfied . if and , then we show that , or equivalently , .this inequality is true due to the fact that is non - decreasing , and thus the theorem statement follows .note that is a function of reward and channel state , and it is proportional to the number of packets in the mmwave queue .intuitively , for a first - in - first - out ( fifo ) queue , the likelihood that an admitted packet reneges before receiving service increases with the number of queued packets .therefore , given that the reneging and moving packets from the mmwave queue to the rf queue incurs a delay cost , it is in the scheduler interest to exercise admission control and deny entry to packets when the mmwave queue grows and becomes larger than a threshold .next we characterize the value of optimal threshold .   in order to characterize the value of optimal threshold , we first need to calculate the limiting distribution of the state of mmwave queue . to this end , the authors in introduced an _ embedding technique _ such that an embedded process is obtained by sampling the process at the beginning of each on period ( see for details ) .our model involves an admission policy that regulates the arrival process , and thus length of the mmwave queue does not exceed an optimal threshold .moreover , we assume that the limiting distribution of the mmwave queue state is denoted by and for and , respectively :   as in , the limiting distribution of all states under the off and on link state is then obtained in a matrix form as follows : }{\mathds{e}\big[t_{\text{on } } + t_{\text{off}}\big ] } ; \nonumber \\\boldsymbol{\xi}_{\text{on } } & =  \frac{\boldsymbol{\nu } \mathds{e}\big[\sum_{k = 1}^{t_{\text{on } } }  ( \boldsymbol{m}_{\text{on}})^{k-1}\big]}{\mathds{e}\big[t_{\text{on } } + t_{\text{off}}\big]},\end{aligned}\ ] ] where is the vector of limiting distribution for the embedded process .moreover ,  ] such that : the proof is similar to .therefore , the limiting distribution vector of the state space is obtained as : .a sufficient condition for existence of the limiting distribution is that the embedded process has finite state space , which holds here due to bounded queue length and waiting time . to denote the limiting distribution at the state , we use the notation .* optimal threshold : * the optimal policy imposes a threshold such that if and only if . under the ergodicity assumption , we rewrite problem 2 as : - \big(w+c\big)\mathds{e}[\beta_h]\bigg).\ ] ]  [ non - decreasing ] given an admission threshold , if - \mathds{e}[\beta_{h-1}]}{\mathds{e}[\alpha_{h } ] - \mathds{e}[\alpha_{h-1 } ] } ,  \label{decreasing - exp}\ ] ] then is non - decreasing in .  in order to prove that is non - decreasing , we note that both  ] are increasing in since a larger threshold causes the admission of more packets ( i.e. , a larger  ] .moreover ,  ] is an increasing and concave , and $ ] is an increasing and affine function of , we conclude that is non - negative , and thus for .  [ optimal - threshold ] given an admission threshold , we define if , then .  from lemma [ non - decreasing ] , we conclude that is non - decreasing in as well , i.e. , . for a given threshold that satisfies , we conclude that : - ( w+c ) \mathds{e}[\beta_{h+1 } ] \leq ( r+c ) \mathds{e}[\alpha_{h } ] - ( w+c ) \mathds{e}[\beta_{h}].\ ] ] therefore , achieves a higher objective value than . now in order to establish this result for , we can show that : - \mathds{e}[\beta_h]}{\mathds{e}[\alpha_{h+2 } ] - \mathds{e}[\alpha_{h}]},\ ] ] from which we conclude that is optimal with respect to as well . by induction, we extend this result for all .similarly , based on the constraint we prove that is optimal with respect to all as well .thus , is the optimal threshold value in general , and we have .note that and , with reneging rate and to be limiting probability of the state .the optimal threshold increases with respect to and decreases with respect to .[ thm : optimality - behavior ]  we note that is increasing in and decreasing in due to the fact that .the condition is necessary in order to avoid the trivial scenario where the subsidy is larger than the reward of successful transmission .the trivial scenario leads to always choosing the passive action , and thus we pose the constraint to avoid the trivial condition . from lemma, we note that is non - decreasing in . from theorem[ optimal - threshold ] and because increases in and decreases in , we conclude that the optimal threshold increases in and decreases in as well .the above theorem shows that if the value of increases , throughput performance will have a higher priority than delay , and thus optimal threshold increases , as expected . on the other hand , by increasing the value of , the optimal threshold decreases to avoid high reneging costs . as a result , based on the performance requirements ,the tradeoff between full exploitation of the mmwave capacity and the delay for mmwave channel access is adjusted through the use of parameters and .   in the previous section, we presented an optimal threshold - based scheduling policy along with the optimal mmwave admission threshold . in practice ,the mmwave link is highly dynamic such that the mmwave data rate can vary over two orders of magnitude . in this context , it is desirable to be able to adjust the admission threshold on - the - fly , capturing temporal information of the mmwave channel . in this case , the optimal admission threshold is adjusted to accommodate the dynamics of channels . in what follows ,we provide an online scheduling policy that preserve the form of optimal policy , while adjusts the admission threshold on - the - fly . in order to obtain the online algorithm of theorem [ optimal - threshold ], we note that the optimal threshold is a function of the ratio with .moreover , the optimal threshold is expressed in terms of function that is an increasing function of . as an example ,[ threshold_behavior ] demonstrates a sample path of the function introduced in lemma 1 . in order to calculate the optimal threshold at time , we consider the value of function up to time and adjust the optimal threshold accordingly ( as shown in fig .[ threshold_behavior ] ) .algorithm 1 provides an online scheme to calculate the optimal threshold .-2 - 2    set set update , and  = update - threshold   calculate  * return * this section , we investigate the performance of our proposed scheduling policy . to this end , we use the experimental traces to model the on - off mmwave link . in our experiment ,a mobile receiver moves with the speed of m / s over a path characterized by sudden link transitions due to human blockers ( hb ) and reflectors ( ref ) . figure [ fig : channel - mobile ] illustrates the received signal strength as the mobile moves away from the transmitter .we assume a signal reception cutoff threshold ( determined based on the hardware used and environment ) such that if the signal strength is below , the channel is in the off state .moreover , in order to adequately capture the dynamics of the mmwave channels , the timeout value is set on - the - fly such that at time , we set with to be the rf average waiting time .thus , on average , packets would not get stuck in the mmwave queue longer than if they would have joined the rf queue .[ threshold - effect ]   we first investigate the tradeoff between mmwave throughput ( or , conversely , _link wastage _ ) and the average waiting time .link wastage is defined as the fraction of time slots that there are packets in the system , but the mmwave queue is empty and the mmwave link is available ( i.e. , ) . the tradeoff between link wastage and the average waiting time is shown in fig .[ tradeoff ] . from the results, we observe that if there are so many packets added to the mmwave queue and if the mmwave link becomes unavailable , high average delay incurs . on the other hand ,a conservative policy is not desirable either such that due to lack of packets in the mmwave queue , the link wastage increases .figure [ threshold - based ] illustrates the total reward obtained as a function of the admission threshold where the maximum reward is obtained for threshold , which is the same threshold value with zero link wastage and the smallest average waiting in fig .[ tradeoff ] .   in order to optimally design the rf / mmwave hybrid transceiver, we represented the transceiver node as a communication network . in this model ,the rf and mmwave interfaces are modeled as individual network nodes and the objective is to fully exploit the abundant mmwave capacity , while the waiting time is guaranteed to be bounded . in the context of through - optimal scheduling , it is well known that traditional network utility optimization , such as backpressure policy , promises optimal throughput performance for a wide range of networking problems .backpressure policy requires knowledge of channel state ( i.e. , link rate ) , while due to the high data rate of the mmwave interface , real - time tracking of the link state may not be feasible .therefore , the scheduler node ( in fig . [ network - model ] ) obtains information of the data rate of interface with a delay of . under the assumption of delayed network state information, the authors in have shown that the modified version of backpressure policy achieves optimal throughput , i.e. , : \mathds{e } [ r_a(t ) | r_a(t-\tau_a ) ] ,  \label{delayed - information}\ ] ] in which is the optimal interface selected at time , and is the link rate of interface at time . under the assumption of delayed csi ,the network stability region ( and thus maximum achievable throughput ) shrinks as the csi delay increases .[ threshold_backpressure_delayedcsi ]  figure [ throughput - delayedcsi ] and [ delay - delayedcsi ] demonstrate the throughput and delay performance of our threshold - based scheduler compared with the backpressure algorithm applied for the network model in fig .[ network - model ] .we investigate the performance under both real - time csi and delayed csi scenarios . from the results , we observe that the threshold - based scheme achieves a similar throughput performance to the backpressure .however , delay of backpressure policy increases with the arrival traffic , while threshold - based policy provides a bounded delay performance .moreover , delayed csi degrades the performance of both policies , however , the threshold based policy is more robust ( in terms of delay performance ) towards csi delay since it is not _ directly _ expressed in terms of csi for scheduling , while the backpressure policy requires csi for scheduling .[ threshold_reneging_cost ]   from theorem [ thm : optimality - behavior ] , the optimal threshold increases by the reward value and decreases by the reneging cost . as a result ,depending on the application ( throughput vs. latency ) , the value of and are set , and the optimal threshold value is regulated accordingly .figure [ throughput - cost ] and [ delay - cost ] demonstrate the throughput and delay performance of the threshold based policy as the reneging cost increases . from the results ,we note that by increasing the value of , the optimal threshold decreases and thus less packets are admitted to the mmwave queue , as expected . as a result ,mmwave waiting time decreases while due to the lack of backlogged packets , throughput performance degrades as well . on the other hand ,the trade off between the mmwave throughput and waiting time can be balanced by adjusting the value of reward .figure [ throughput - reward ] and [ delay - reward ] illustrate the throughput and delay performance as the reward value increases .similarly , as the reward value increases , the optimal threshold increases and more packets are admitted into the mmwave queue , which results in a higher throughput at the cost of larger waiting time .in this paper , we proposed a hybrid rf / mmwave architecture for g cellular systems .our proposed architecture includes an rf - assisted beamforming that exploits the correlation between the rf and mmwave interfaces in order to enhance energy efficiency of mmwave beamforming .in addition to beamforming , we utilized the rf interface for data transfer , and formulated an optimal scheduling policy in order to maximize the long - term throughput of the mmwave interface provided that the average delay is bounded .we cast the constrained throughput maximization as a reward optimization , and proved that the optimal scheduling policy has a simple monotone structure . as a result , using the rf interface as a secondary data transfer mechanism , the abundant yet intermittent mmwave bandwidth is fully utilizedindeed , we believe that mmwave will most likely be deployed with an overlay of rf in g .

Abstract: we propose a hybrid architecture that integrates rf ( i.e. , sub-6 ghz ) and millimeter wave ( mmwave ) technologies for g cellular systems . in particular , communications in the mmwave band faces significant challenges due to variable channels , intermittent connectivity , and high energy usage . on the other hand , speeds for electronic processing of data is of the same order as typical rates for mmwave interfaces which makes the use of complex algorithms for tracking channel variations and adjusting resources accordingly impractical . our proposed architecture integrates the rf and mmwave interfaces for _ beamforming _ and _ data transfer _ , and exploits the spatio - temporal correlations between the interfaces . based on extensive experimentation in indoor and outdoor settings , we demonstrate that an integrated rf / mmwave signaling and channel estimation scheme can remedy the problem of high energy usage and delay associated with mmwave beamforming . in addition , cooperation between two interfaces at the higher layers effectively addresses the high delays caused by highly intermittent mmwave connectivity . we design a scheduler that fully exploits the mmwave bandwidth , while the rf link acts as a fallback mechanism to prevent high delay . to this end , we formulate an optimal scheduling problem over the rf and mmwave interfaces where the goal is to maximize the delay - constrained throughput of the mmwave interface . we prove using subadditivity analysis that the optimal scheduling policy is based on a single threshold that can be easily adopted despite high link variations .