c.2.3 [ * * computer - communication networks * * ] : network operationsmeasurement , experimentation  network - wide traffic analysis , random correlations , time - lagged correlations , long range dependencethe cross - correlation matrix is one of the tools of time series analysis , used in studies of the underlying interactions between network structural constituents .the network traffic system is often seen as a collection of times series with traffic bytes per time interval or number of packets per time interval at a single link . within this collection, temporal correlations between network structural constituents and consequently patterns of collective behavior might be present .  to study such patterns one can employ the equal - time cross - correlation matrix of traffic time series at all network links ( see , for example ) . further statistical description of the awareness is commonly done via eigen - decomposition of cross - correlation matrix .detailed studies showed that the major portion of eigenvalues of cross - correlation matrix fall into the theoretically predicted boundaries of eigenvalues spectrum of random matrices .this portion satisfies the so called universal properties described by the random matrix theory ( rmt ) . it also recognizes the uncongested state of the traffic , where each router is able to communicate with any other router and with each subnet under its service .the part of the spectrum deviating from the rmt boundaries represents stable in time non - random correlations between network traffic time series .even though , such distinction provides a valuable insight into the meaning of empirical data , the equal - time cross - correlation matrix on its own may not be sufficient for understanding of the effect of inter - domain correlations at different times .in fact , the analysis of correlation as a function of time lags has already been used in econometric time series systems .for example , the analysis of stock returns portfolio showed the asymmetric lead - lag relationship between stock returns ; high - volume stocks lead the low - volume stocks .this finding is attributed to information adjustment asymmetry .the uncongested balanced state of the network traffic implies the " symmetric " information / traffic flow exchange .thus , to control the congestion level of network traffic systems , it is crucial to recognize the collective behavior or correlation patterns between network traffic time series , and to observe their evolution in time .  in order to trace the evolution of correlation pattern between traffic time series the equal - time cross - correlation matrix is replaced with time - lagged correlation matrix .the obtained matrix of delayed correlations can be subjected to eigen - analysis just like its equal - time counterpart .in contrast to equal - time correlation matrices which have a real eigenvalue spectrum , the spectrum of is complex since matrices of these types are asymmetric . while the general properties of complex spectrum of is unknown so far ,results of symmetrized version of lagged correlation matrices have been reported recently .  in this paper , we concentrated on time - lagged correlations between time series generated by simple network manage protocol ( snmp ) traffic counters of university of louisville backbone routers system .we established that time - lagged correlations between traffic time series sustain for up to , where sec .moreover , the largest and smallest eigenvalues and of and the inverse participation ratio ( ipr ) of the eigenvector corresponding to were found to oscillate with two characteristic periods of and .in addition , within the content of the eigenvector corresponding to we determined the index of a single time series which is driving the oscillation .the ipr of all eigenvectors of shows the single time series contributor in the eigenvector which at the equal - time correlations belonged to the random part of the spectrum .since the theoretical prediction for the spectrum of time - lagged correlation matrices is unknown , we keep the terminology of the spectrum derived from eigen - decomposition of equal - time correlation matrices .random _ in our text refers to the part of the spectrum within the boundaries predicted by the rmt for equal - time correlation matrix ._ non - random _ denotes the part of the spectrum outside of the rmt predictions .finally , we found that the injection of random traffic counts between time series which interact _ non - randomly _ destroys the oscillatory picture of and and that of the corresponding iprs .meanwhile , the injection of traffic counts with smooth ( periodic ) time dependence between _ randomly _ interacting traffic time series with respect to characteristic periods of , and ipr reveals new periodicity ( of ) .the findings of eigenvalue spectrum analysis and experimental results suggest the asymmetric long lasting relationship between traffic time series .network traffic analysis had undergone the evolution from considering the network traffic time series as an outcome of poisson and memory - less processes to recognizing the long range dependencies and self - similarity of the traffic .the statistics of eigenvalue spectrum and ipr of eigenvectors of time - lagged correlation matrices provide essential dimensional reduction in the investigation of long - ranged dependence ( lrd ) of the network traffic . before starting the expensive process of hurstparameter estimation , one can attempt to find just a few indicators of lrd , self - similarity or deviations from thereof .the rest of the paper is organized as follows .we introduce time - lagged correlation matrix and its eigensystem in section ii .section iii is devoted to numerical analysis of time - lagged eigensystem .then , in section iv by experimenting with the content of the original time series we addressed lrd and self - similarity of lagged correlations through proposed indicators . conclusions and discussion are given in section v.the starting point of our discussion is averaged traffic count data collected from router - router and router - vlan subnet connections of the university of louisville backbone routers system . the same data set was used in in the context of equal - time correlation matrix analysis . below we recap the details relevant to the construction of lagged correlation matrix , relegating the information on the network and the way data was processed to .we will be dealing with total of records of time series averaged over seconds , where incoming and outgoing traffic generate independent time series .  in order to define traffic rate change we used the logarithm of the ratio of two successive counts upon calculating the traffic rate change of time series , , over time , .then we introduce normalization , according to and built the time - lagged correlation matrix as follows here the sole purpose of symmetrization is the restriction of the eigenvalues and eigenvectors to real values . in principle , the numerical experiments we run below can be repeated for the eigensystem of non - symmetric correlation matrix .studies of the latter are already in progress in a different setting ( see , for example , ) .   next we proceed with defining the indicators we focus on in what follows .first of all , the eigenproblem for our cross - correlation matrix is time dependent where is -th eigenvalue , corresponding to eigenvector . in other words ,the eigensystem is defined for each increment of delay time . as opposed to same time eigensystem , our eigensystem does not characterize presence or lack of organization ( localization ) in the system at a given time .instead it can serve as a measure of back - in - time ( or forward in time , depending on prospective ) correlation within the network structure .furthermore , the rmt picture of the eigenvalues and eigenvectors , in which spectrum is split into three parts is no longer valid . by three parts we understand the rmt part , which behave universally at the center of the eigenvalue spectrum , and `` left '' and `` right '' parts ( lying to the left and to the right from predicted rmt bounds ) which exhibit non - universal features .although , for very small this subdivision is clearly still accurate , we can expect , transient behavior of to reveal new , otherwise undetectable correlations within the network .hence , we found it convenient to keep track of quantitative and qualitative changes in eigensystem using left / random / right terminological distinction .  on the other hand , observing the entire system of eigenvalues andthe same amount of eigenvectors can be quite tedious task .one needs to come up with more concise indicators of network behavior .selecting efficient indicators can help in defining `` normal '' state of the system , a task quite challenging on its own , and in predicting structural reaction to the external or internal disruption .indicators can be chosen based on the experiments described below , or on their advanced variations .the candidates are those eigenvalues , that are most receptive to a particular probe .as far as eigenvectors are concerned we decided to test corresponding iprs .given the eigenvector the ipr is computed according to^{4},\label{eq18}\ ] ] with , stands for components of the eigenvector .the ipr is quite indicative in terms of signaling the number of significant , i.e. `` contributors '' to the eigenvector of interest .for example , if we have reasons to expect absence of correlations between routers input into the experimental data , should have its value around .indeed , the eigenvector is normalized , thus ^{2}=1 $ ] .it has components , and they are all roughly the same in magnitude ( otherwise correlations must be present ) .therefore , , and .note , that since is typically much greater than , any finite value of ipr signals _ localization _ in inter - vlan traffic . for non - zero values of , iprs acquire more general meaning in a sense that routers which interact heavily at time may loose their `` bond '' at time , while these not knowing about one another at time may have significant mutual contribution at time .other more complex possibilities can be perceived via as well .upon building the cross - correlation matrix with the help of eq .( [ deeoftau ] ) we perform eigen - decomposition ( see eq .( [ eigenproblem ] ) ) numerically , using standard _matlab _ routine .we look at the result of calculation of eigenvalues for all delay times . a noticeable spike for very small values of delay timeis expected , notwithstanding the position in spectrum .however , our increments in ( ) may not be small enough to observe it .for the remainder of observation the result has to uncover the way system constituents communicate with themselves and their neighbors on a long run .      in fig .[ 1](a)-(c ) we illustrate how left / random / right structure of the spectrum evolves with . as it turns out , `` randomness '' and `` regularity '' find their new interpretations in the context of system remembering itself after has elapsed . with the exception of a few located at the right and left edges of the spectrum , most eigenvalues fall very close to each other numerically . to make it more transparent we plotted their -dependence using different offset values ( these values are the same within each part ) .only ten eigenvalues are offseted in each case and plotted versus time .the lowest eigenvalue was excluded from consideration here and throughout the paper due to it secular behavior in .  at a glance , non - edge eigenvalues fig .[ 1](b ) with the exception of an expected spike at small does not seem to represent any process .such a lack of forward in time correlation is not completely surprising , as the eigenvalues from middle part of the spectrum were referred as rmt - like .it follows , that the random interactions between traffic time series are time delay invariant .in other words , random spectrum of eigenvalues is an indicator of self - similarity .meantime , the eigenvalues at the edges ( figs .[ 1](a ) and ( c ) ) represent a quasiperiodic process , distinguishing themselves from their " peers " , the eigenvalues belonging to regular part of the spectrum for quantitatively , and both qualitatively and quantitatively from the eigenvalues belonging to the rmt - part for .they scale with delay time and clearly exhibit long time dependence .therefore , it makes sense to look further into the properties of edge eigenvalues , especially into the properties of those with relatively high absolute values .the actual values might be used as a measure of delayed time correlations .having located potential indicators we move ahead with the search for others .  a remarkable property of the ipr for equal time cross - correlation matrix ( fig .[ 2](a ) ) is its consistently low , order , value for the major part of the spectrum .this segment in fig .[ 2](a ) is known to obey the rmt . to the left and to the right from this segmentthere is a strong evidence of regular , non - random behavior .now , if the first instances are considered as in figs . [ 2](b ) and ( c ) , where iprs offseted by an arbitrary amount for transparency and plotted versus the eigenvalue position , the situation looks drastically different .the peak located close to the center of the spectrum signifies the presence of previously undetected correlations , and the lead - lag relationship between time series .close examination of figs .[ 2](b ) and ( c ) shows , that initially , the high ipr has changing support in the spectrum .furthermore , as explained in section 2.2 , peak value tells us , that about four time series drive the correlation pattern .later on , the peak " settles down " and establishes itself around median eigenvalue position ( fig .[ 2](b ) ) .the meaning of this and other two peaks differs from that of the ipr peaks in fig .the increase in ipr computed from the time delayed matrix indicate correlations between system s behavior at a given time and its stroboscopic image after , rather than correlations within the spectrum .in addition , it provides reasonable way of tracking down the sources of lead - lag behavior .thus , the observed features make ipr a good candidate for an indicator of the network congestion state .note also significant change in height of the central peak .the next step in getting more quantitative on long memory processes in network traffic is to analyze transient behavior of eigenvalues and iprs of matrix in detail . since quasiperiodic behavior is present in the majority of quantities of interest we focus on their frequency content .the standard way to proceed is to transform into frequency domain using fast fourier transform . in a sense, we construct a spectra of the spectrum .the same operation is performed on respective iprs .we take fast fourier transforms for all of the functions at hand , and then , take the square of their absolute value .the result is referred to as a power spectrum .there should be no confusion , as graphs of power always accompany the corresponding time domain quantity .  in fig .3 we display representative eigenvalue dynamics . once again , random ( fig .3(b ) ) does not exhibit anything remarkable , compared to its regular counterparts .the latter resemble each other , reflecting a symmetry of the spectra induced by symmetrizing procedure ( eq .( [ deeoftau ] ) ) .for now , we can talk about them in parallel .  aside from a substantial low and high frequency contribution , which could have already been guessed from figs .3(a ) and ( c ) , we discover two strong contributions from frequencies corresponding to oscillations with time periods and minutes respectively ( cf . figs. 3(d ) and ( f ) ) .this is in evident contrast to the situation with power of a random eigenvalue .such an eigenvalue has equal ( negligibly small ) contribution from the entire range of frequencies .the existence of these two characteristic frequencies suggests a natural way assessing the current state of the inter - domain network traffic .in fact , it might be possible to use these as the lrd quantifier estimators in the future .next , we investigated consequences of modifying the time - lagged correlations between time series .we have already known the time series contributing the most to the correlation pattern .all of them can be linked to eigenvalues which fall in what we term here as the right segment of eigenvalue spectrum . in these serieswe replaced the original traffic counts with counts obtained by random number generator for a certain period of time .then , we constructed matrix for all hundred increments and repeated manipulations described in previous section 3.1 .the results are shown in fig . [ 1](d ) through ( f ) .the eigenvalues belonging to middle of the spectrum are completely unaffected , i.e. they are still time delay invariant . clearly , our manipulations with the traffic are not disturbing the self - similar nature of delayed correlations .however , edge eigenvalues loose time scales present in their original transient behavior ( see fig . [ 1 ] ( d ) through ( f ) ) .in other words , the lrd gets destroyed .effect on the ipr ( figs .[ 2](d ) and ( f ) . ) is less noticeable but is still there , while for the random segment it is absent .the result of random counts injections can be summarized as presence of randomly positioned of small peaks superimposed on the original ipr picture .indeed , in figs .[ 2](a ) and ( c ) small peaks are very infrequent and unstable in time , unlike these in figs .[ 2](d ) and ( f ) .the above outcome calls for a more close look into eigenvalues and iprs for the system experienced the noisy injections into the time series , which are believed to have major contribution to the overall traffic in router network .we present three eigenvalues considered in section 3.2 as the functions of time delay together with their respected power spectra . as can be concluded from fig .5(a ) and ( c ) , the time dependence looses its lrd structure .it is backed up by the fact that a lot more frequencies contribute to power spectra upon random injection .middle part of the spectra also undergoes certain transformation , but is still scale - free fig .5(b ) , as actual values of power are small relative to the power corresponding to edge eigenvalues .the quantitative changes are also in place for both edge eigenvalues .the effect can be judged based on comparison of the tallest peaks in figs .5(d ) and ( f ) to their counterparts in figs .3(d ) and ( f ) .similar conclusions can be derived for the ipr as we take a look at fig .6(d ) and ( f ) and compare the outcome of our experiment with the graphs in fig .4(d ) and ( f ) .       a logical continuation of the above experiment is the injection of an artificial traffic counts which possess regularity into actual experimental data .this time , however , we perform the replacements for the time series which can be traced back to the eigenvalues falling into the random segment .time series for this replacement were chosen at random .other possibilities can also be considered , but since random segment was much less susceptible to the previous experiment , the above choice seems natural .we choose four injections to be cosinusoidal , having periods of ; ; and and repeating the same manipulations as in the first experiment discussed in section 4.1 .the results are fairly sound .although the random part of the eigenvalue spectrum is again unaltered , the `` reaction '' of left and right parts is both qualitative and quantitative .when cosinusoidal sample with period much smaller than both characteristic periods ( and ) , power spectra in figs .7(d ) and ( f ) are not significantly changed .two characteristic periods are still present , and yet certain narrow frequency range gets suppressed ( note anti - peak between the main two ) .notice slight asymmetry in the way smallest and largest eigenvalues react to the injection .we should add , that we observed essentially the same picture for the injections with period of and minutes ( both not matching , but commensurate with characteristic periods ) .now , we turn to fig . 8 , where the cosinusoidal replacement with period of actual traffic counts leads to the dramatic change in power spectrum .we observe enhancement of the peak corresponding to period of , which can be termed as resonance phenomenon ( figs .8(d ) and ( f ) ) .the very same plots show the suppression of peaks corresponding to the other characteristic period of .similar resonant effect is achieved when the period of injection is changed to ( see fig .this time , both peaks are gone , while the new characteristic period is detected in fig .it is approximately equal to the period of injection .finally , for the experiment , in which period of the injection was chosen to be , i.e. matching to another characteristic period , we obtained yet another result supporting previous conclusions .  in this case , however , the resonance phenomenon is slightly more difficult to establish . from the results displayed in figwe see , that relative contribution to power spectrum is now changed for two main peaks .before the experiment was performed , the higher harmonic ( smaller period ) dominated by quite a few orders of magnitude . after running the experiment ,this is still the case for the spectrum of largest eigenvalue , but now the difference is marginal ( see fig .10(d ) ) . at the same time for the left most eigenvalue, we determined , that lower harmonic ( period , matching the period of injection ) now contributes the most , as can be checked in fig .two power spectra for the edge eigenvalues are no longer symmetric and the contributions from some ranges of frequencies are again strongly diminished . as for the random eigenvalue considered fig .10(b ) and ( e ) , no impact has been recorded , just as in all other cases .long range or time dependent processes which show significant correlations across large time scales were first discovered in network traffic over a decade ago . since then , lrd was found and studied intensively in various aspects of network behavior .lrd is a manifestation of self - similarity of the process , meaning that the behavior of the process is space and time scale invariant .leland and colleagues performed first rigorous statistical analysis of self - similar characteristics in local area network ( lan ) traffic .they showed that the aggregated ethernet traffic is not smoothing out with accordance to poisson model , it is time scale invariant . in this frameworkthe traditional poisson or memory - less models of network traffic became inadequate .since high variability across different time scales produces high congestion level , the impact of the self - similar traffic models on queuing performance is considerable .however , the identification of self - similarity origin and estimation of lrd in network traffic are far from straightforward .one of the effective procedures to quantify lrd is to calculate the value of hurst parameter .even though the hurst parameter is a scalar it can not be calculated directly , it can be only estimated .there are several methods to estimate the hurst parameter and sometimes they produce the conflicting results .ineptly , it is not clear which method provides the most accurate estimation .  in this paper , we proposed the lrd and self - similarity indicators of delayed correlations in network traffic .we demonstrated , that the time delay invariant behavior of non - edge eigenvalues of reflects the self - similar nature of delayed correlations .meanwhile , the scaling with time of edge eigenvalues or their lagged - time dependence is an exhibition of self - similarity of delay correlations .in addition , we established that the ipr for eigenvectors of lagged correlations are concise parameters of realistic model for network congestion pattern . as was shown in ,ipr for contains two localization trends in network interactions , i.e. two regions in spectrum , traced back to a small number of time series , which create the bottleneck at the routers .it is noteworthy , that the ipr for where , reveals the third localization trend , which has different origin .the significantly increased and time delay invariant ipr around the median eigenvalue indicates presence of lead - lag relationship between time series .  with experiments altering the original traffic timeseries several distinctive effects has been uncovered .first of all , we demonstrated that tempering with time series has no effect on self - similar transient behavior of eigenvalues and iprs , located in the middle segment of the spectrum .by contrast , both stochastic and periodic injections into the right ( non - random ) and middle ( random ) segments respectively yielded dramatic changes in chosen indicators .in particular , we recorded the destructive effect of random noise on otherwise simplistic double - peaked power spectra .one of the main results we obtained from periodic injection experiments was presence of resonance phenomenon .when the period of injection coincides with one of the characteristic time scales of the network ( i.e. oscillation periods of edge eigenvalues ) the corresponding spectral peak gets enhanced .the fourier transform peak , corresponding to the other scale gets suppressed and sometimes even annihilated . finally , injection with the period much less than both scales has little effect on fourier spectra , while period of the same order in magnitude rearranges the original spectra completely .the above described time - lagged correlational analysis has a broad area of applications , where delayed correlations between system substructures are essential . for instance , it can be applied to electro - physiological time series of brain response , earthquake relocations , financial portfolios , and atmospheric data . to support this assertionwe point out that edge eigenvalues of behave almost identically to these of atmospheric data , while the delay eigenvalues of the stock market data act just like the eigenvalues , we termed random .this research was partially supported by a grant from the us department of treasury through a subcontract from the university of kentucky .10 v. rojkova and m. kantardzic , analysis of inter - domain traffic correlations : random ratrix rheory approach , arxiv.org:cs/0706.2520 ( 2007 - 06 - 21 ) . m. barthelemy , b. gondran and e. guichard , large scale cross - correlations in internet traffic , arxiv : cond0mat/0206185 vol * 2 * 3 dec 2002 .mayya and r.e .amritkar , analysis of delay correlation matrices , oai : arxiv.org : cond - 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223 .m. krunz , on the limitations of the variance - time test for inference of long - range dependence , ieee infocom , 2001 , 1254 - 1260 .s. molnar and t.d .dang , pitfalls in long range dependence testing and estimation .globecom , 2000 .t. karagiannis , m. molle , and m. faloutsos , long - range dependence , ten years of internet traffic modeling , ieee internet computing , oct 2004 .j. kwapien , s. drozdz , and a.a .ioannides , temporal correlations versus noise in the correlation matrix formalism : an example of the brain auditory response , arxiv : cond - mat/0002175 , vol * 1 * , 11 feb 2000 .du , c. h. thurber , and d. eberhart - phillips , earthquake relocation using cross - correlation time delay estimates verified with the bispectrum method , bulletin of the seismological society of america , june 2004 , vol * 94 * , 3 , 856 - 866 .leland et al . , on the self - similar nature of ethernet traffic , ieee / acm trans .networking , vol * 2 * , 1 , 1994 , 1 - 15 .

Abstract: to observe the evolution of network traffic correlations we analyze the eigenvalue spectra and eigenvectors statistics of delayed correlation matrices of network traffic counts time series . delayed correlation matrix is composed of the correlations between one variable in the multivariable time series and another at a time delay . we determined , that inverse participation ratio ( ipr ) of eigenvectors of deviates substantially from that of eigenvectors of the equal time correlation matrix . the largest eigenvalue of and the corresponding ipr oscillate with two characteristic periods of and . found delayed correlations between network time series fits well into the long range dependence ( lrd ) property of the network traffic .  injecting the random traffic counts between non - randomly correlated time series , we were able to break the picture of periodicity of . in addition , we investigated influence of the periodic injections on both largest eigenvalue and the ipr , and addressed relevance of these indicators for the lrd and self - similarity of the network traffic .  department of computer engineering and computer science , university of louisville , louisville , ky 40292 email : \{vbrozh01 , mmkant01}.edu


the scientific study of the problems of growth and form occupies an anomalously broad set of disciplines .whether the emergent patterns are physical or biological in origin , their quantitative description presents many challenging and compelling issues in , for example , applied mathematics , biophysics , condensed matter and geophysics wherein the motion of free boundaries is of central interest . in all such settings a principal goal is to predict the evolution of a boundary that is often under the influence of an instability .here we study a novel variant of such a situation that occurs naturally on the frozen surfaces of lakes .lakes commonly freeze during a snowfall .when a hole forms in the ice cover , relatively warm lake water will flow through it and hence through the snow layer .in the process of flowing through and melting the snow this warm water creates dark regions .the pattern so produced looks star - like ( see figure [ fig : photo ] ) and we refer to it as a `` lake star '' .these compelling features have been described qualitatively a number of times ( e.g.  ) but work on the formation process itself has been solely heuristic .knight  outlines a number of the physical ideas relevant to the process , but does not translate them into a predictive framework to model field observations .knight s main idea is that locations with faster flow rates melt preferentially , leading to even faster flow rates and therefore to an instability that results in fingers .this idea has features that resemble those of many other instabilities such as , for example , those observed during the growth of binary alloys , in flow of water through a rigid hot porous media , or in more complex geomorphological settings , and we structure our model accordingly .  katsaros  and woodcock  attribute the holes from which the stars emanate and the patterns themselves to thermal convection patterns within the lake , but do not measure or calculate their nature .however , often the holes do not exhibit a characteristic distance between them but rather form from protrusions ( e.g. sticks that poke through the ice surface )  and stars follow thereby ruling out a convective mechanism as being necessary to explain the phenomena .the paucity of literature on this topic provides little more than speculation regarding the puncturing mechanism but lake stars are observed in all of these circumstances .therefore , while hole formation is necessary for lake star formation , its origin does not control the mechanism of pattern formation , which is the focus of the present work .the water level in the hole is higher than that in the wet snow slush layer and hence we treat this warm waterc . ] region as having a constant height above the ice or equivalently a constant pressure head , which drives flow of water through the slush layer , which we treat as a darcy flow of water at .we model the temperature field within the liquid region with an advection - diffusion equation and impose an appropriate ( _ stefan _ ) condition for energy conservation at the water - slush interface .the water is everywhere incompressible .finally , the model is closed with an outer boundary condition at which the pressure head is assumed known .although we lack _ in - situ _ pressure measurements , circular water - saturated regions ( a few meters in radius ) are observed around the lake stars .hence , we assume that the differential pressure head falls to zero somewhere in the vicinity of this circular boundary .the actual boundary at which the differential pressure head is zero is not likely to be completely uniform ( as in figure 4 of knight  ) but treating it as uniform is a good approximation in the linear regime of our analysis .finally , we treat the flow as two - dimensional .thus , although the water in direct contact with ice must be at , we consider the depth - averaged temperature , which is above freezing . additionally , the decreasing pressure head in the radial direction must be accompanied by a corresponding drop in water level .therefore , although the driving force is more accurately described as deriving from an axisymmetric gravity current , the front whose stability we assess is controlled by the same essential physical processes that we model herein .our analysis could be extended to account for these three - dimensional effects .the system is characterized by the temperature , a darcy fluid velocity , pressure , and an evolving liquid - slush interface .the liquid properties are ( thermal diffusivity ) , ( specific heat at constant pressure ) and ( dynamic viscosity ) and the slush properties are ( permeability ) , ( solid fraction ) and ( latent heat ) .we non - dimensionalized the equations of motion by scaling the length , temperature , pressure and velocity with , , , and , respectively .thus , our model consists of the following system of dimensionless equations :        with boundary conditions    and where ( [ temp1n ] ) describes the temperature evolution in the liquid , ( [ pres2n ] ) and ( [ continuityn ] ) describe mass conservation with a darcy flow ( [ darcyn ] ) in the slush , ( [ stefann ] ) is the stefan condition , and ( [ temp_bcn ] ) and ( [ pres_bcn ] ) are the temperature and pressure boundary conditions , respectively ( see figure [ fig : schematic ] ) .note that ( [ pres1n ] ) and ( [ continuityn ] ) can both be satisfied since the liquid region has an effectively infinite permeability .the dimensionless parameters and of the system are given by which describe an inverse peclet number and a stefan number respectively . because the liquid must be less than or equal to , we make the conservative estimates that , , and use the fact that from which we see . using , and the field observations of knight  to constrain ( ) and ( ) , we find that .we therefore employ the quasi - stationary ( ) and large peclet number ( ) approximations , and hence equations ( [ temp1n ] ) - ( [ pres_bcn ] ) are easily solved for a purely radial flow with cylindrical symmetry ( no dependence ) and circular liquid - slush interface .this ( boundary layer ) solution is          where equation ( [ bla ] ) has an approximate implicit solution for given by   we perform a linear stability analysis around this quasi - steady cylindrically symmetrical flow .proceeding in the usual way , we allow for scaled perturbations in and with scaled wavenumber , non - dimensional growth rate , and amplitudes and respectively . keeping only terms linear in , and , we solve ( [ pres2n ] ) subject to ( [ pres_bcn ] ) , substitute into ( [ u_matchn ] ) and satisfy ( [ continuityn ] ) and ( [ temp1n ] ) .this gives the non - dimensional growth rate ( ) as a function of scaled wave number ( ) :    equation ( [ stability ] ) can be approximated in as where .the stability curve ( [ stability ] ) and the approximation ( [ stability_2 ] ) are plotted in figure [ fig : stability_curve ] .the essential features of ( [ stability ] ) are a maximum in the range , zero growth rate at and a linear increase in stability with for large .the long - wavelength cut - off is typical of systems with a peclet number , here with the added effect of latent heat embodied in the stefan number .this demonstrates the competition between the advection and diffusion of heat and momentum ( in a harmonic pressure field ) ; the former driving the instability and the latter limiting its extent .the maximum growth rate occurs at approximately with ( non - dimensional ) growth rate   translating ( [ stability_kap ] ) and ( [ stability_sap ] ) back into dimensional quantities , we find that the most unstable mode has angular size given by and has growth rate given by observations of lake stars can not be _ controlled_. a reasonable estmate for is the radius of the wetted ( snow ) region around the lake stars , and observations  bound the value as .this is simply because if there were significant excess pressure at this point then the wetting front would have advanced further .however , it is also possible that the effective value of , say , is less than this either because the wetted radius is smaller earlier in the star formation process or because the ambient pressure level is reached at smaller radii . here , we take to be the radius of the roughly circular liquid - filled region at the center of the lake star ( ) as the best approximation during the initial stages of star formation ( see figure [ fig : obs_schem ] ) .field observations show that ,  and hence .we note that equations ( [ stability_deg ] ) and ( [ stability_ap ] ) are more sensitive to than or independently may also be interpreted as the radius of the lake star ( ) .field observations show  and hence . ] . with this interpretation of find a reasonable estimate of as .using these parameter values , the most unstable mode should have wavelength between and . letting the number of branchesbe , then and we clearly encompass the observed values for lake stars ( ) , but note that values ( ) are never seen in the field .  despite the dearth of field observations ,many qualitative features embolden our interpretation .for example , the stars with larger values of have a larger number of branches .moreover , for any value of , our analysis predicts an increase in with and .indeed , increases with ( higher water height within the slush layer ) and ( less well - packed snow ) .therefore , we ascribe some of the variability among field observations to variations in these quantities ( which have not been measured in the field ) and the remainder to nonlinear effects . because the dendritic arms are observed long after onset and are far from small perturbations to a radially symmetric pattern , as one might see in the initial stages of the saffman - taylor instability , the process involves non - linear cooperative phenomena .hence , our model should only approximately agree with observations . although a rigorous non - linear analysis of the long term star evolution process ( e.g.  ) may more closely mirror field observations , the present state of the latter does not warrant that level of detail . instead , we examine the model physics through simple proof of concept experimentation described presently .a 30 cm diameter circular plate is maintained below freezing ( ) , and on top of this we place a 0.5 to 1 cm deep layer of slush through which we flow water .given the technical difficulties associated with its production , the grain size , and hence the permeability , of the slush layer , is not a controlled variable .this fact influences our results quantitatively . in fourteen runs we varied the initial size of the water - filled central hole ( ) , that of the circular slush layer ( ) , and the flow rate ( ) , which determines .the flow rate is adjusted manually so that the water level ( ) in the central hole remains constant . ) .it is significantly more difficult to prepare a uniform permeability sample with a circular hole initially present ; these runs are therefore more difficult to interpret . ] fingering is observed in every experimental run andhence we conclude that fingers are a robust feature of the system. two distinct types of fingering are observed : small - scale fingering ( see figure [ fig : fingers ] ) that forms early in an experimental run , and larger channel - like fingers ( see figure [ fig : channels ] ) that are ubiquitous at later times and often extend from the central hole to the outer edge of the slush .since the channel - like fingers provide a direct path for water to flow , effectively _ shorting _ darcy flow within the slush , their subsequent dynamics are not directly analogous to those in natural lake stars .however , in _ all _ runs , the initial small - scale fingers have the characteristics of lake stars and hence we focus upon them . we note that because the larger channel - like fingers emerge out of small - scale fingers , they likely represent the non - linear growth of the linear modes of instability , a topic left for future study .finally , we measure the distance between fingers ( ) , so that for each experiment we can calculate , , from equation ( [ stability_deg ] ) , and , and we can thereby compare experiment , theory and field observations .  in figure [fig : comp ] we plot versus for the various field observations for which we have estimates of parameters , the laboratory experiments described above , and the model [ equation ( [ stability_deg ] ) ] .there is a large amount of scatter in both the experimental and observational data and the data does not lie on the one - to - one curve predicted by the model .however , the experiments are meant to demonstrate the features of the model predictions , and the results have the correct qualitative trend ( having a best - fit slope of 0.34 ). we also attempt to find trends in the experimental data not represented by the model by comparing vs. various combinations of control parameters ( ) including , , , , and .for all plots of vs. , our model predicts a zero slope ( and y - intercept of 1 ) . a non - random dependence of on would point to failure of some part of our model .thus , to test the validity of our model , we perform significance tests on all non - flagged data with the null hypothesis being a non - zero slope . in all cases ,the null hypothesis is accepted ( not rejected ) at the 95% confidence level .thus , although the agreement is far from perfect , the simple model captures all of the significant trends in the experimental data .by generalizing and quantifying the heuristic ideas of knight  , we have constructed a theory that is able to explain the radiating finger - like patterns on lake ice that we call lake stars .the model yields a prediction for the wavelength of the most unstable mode as a function of various physical parameters that agrees with field observations .proof of concept experiments revealed the robustness of the fingering pattern , and to leading order the results also agree with the model .there is substantial scatter in the data , and the overall comparison between field observations , model and experiment demonstrates the need for a comprehensive measurement program and a fully nonlinear theory which will yield better quantitative comparisons .however , the general predictions of our theory capture the leading order features of the system .we thank k. bradley and j. a. whitehead for laboratory and facilities support .this research , which began at the geophysical fluid dynamics summer program at the woods hole oceanographic institution , was partially funded by national science foundation ( nsf ) grant oce0325296 , nsf graduate fellowship ( vct ) , nsf grant opp0440841 ( jsw ) , and department of energy grant de - fg02 - 05er15741 ( jsw ) .

Abstract: star patterns , reminiscent of a wide range of diffusively controlled growth forms from snowflakes to saffman - taylor fingers , are ubiquitous features of ice covered lakes . despite the commonality and beauty of these `` lake stars '' the underlying physical processes that produce them have not been explained in a coherent theoretical framework . here we describe a simple mathematical model that captures the principal features of lake - star formation ; radial fingers of ( relatively warm ) water - rich regions grow from a central source and evolve through a competition between thermal and porous media flow effects in a saturated snow layer covering the lake . the number of star arms emerges from a stability analysis of this competition and the qualitative features of this meter - scale natural phenomena are captured in laboratory experiments .


many practical and scientific problems require the collaboration of groups of experts , with different expertise and background .remarkably , it turns out that large groups of cooperative agents are extremely adept at finding efficient strategies for solving such problems  ; the development of the scientific method within the physical science or the development of entire suites of computer software by the open source movement  being perhaps two of the most notable instances  .indeed , even loosely structured groups have demonstrated an ability to coordinate and find innovative solutions to complex problems .  in the corporate world , several companies including ibm , hp and various consulting companies have used `` the wisdom of the crowd '' principle as the justification for the creation of knowledge communities , the so - called `` communities of practice , '' which have enabled organizations to spawn new ideas for products and services  .other companies , such as intel , eli lilly , and procter & gamble , which created company - sponsored closed knowledge networks , are now opening them to outsiders .  recognizing that knowledge exists not merely in the members of the network but in the networks themselves that is , in the members _ and _ in their interactions naturally leads to the question of what characterizes successful communities and what measures could be taken to improve the ability of groups and organizations to innovate .previous work investigated the importance of group diversity  , team formation  , and the structure of the interaction network  . here , we focus instead on the effect of micro - level strategies on macro - level performance .recent work demonstrates that under quite general conditions , well - intentioned and completely trusting agents can efficiently solve information aggregation and coordination tasks .we investigate how changes in the intention and trust level of agents affects the efficiency of solving such tasks by considering three types of agents : naives , conservatives and partisans .naive agents are well - intentioned and completely trusting .conservative agents are well - intentioned but not completely trusting .partisan agents are neither well - intentioned nor completely trusting .  remarkably , we find that conservative agents , despite slowing the information aggregation and coordination process , actually enhance the populations ability to efficiently solve these tasks under large levels of noise .in contrast , we find that even a small fraction of partisans holding the minority position will result in deadlock or a consensus on an incorrect answer. significantly , only by completely disregarding partisan opinions , can the population recover its original ability to solve the task .we use here the density classification task , a model of decentralized collective problem - solving  , to quantitatively investigate information aggregation and coordination . the density classification taskis completed successfully if ( i ) all agents converge to the same state within a determined time period , and ( ii ) that consensual state was the majority state in the initial configuration .  before proceeding ,let us explain the reasons why the density classification task is a good paradigm for the type of problems into which we aim to gain insight . consider a population of agents tackling a problem in which there is a large uncertainty and for which no agent will be able , by herself , to demonstrate that a particular solution is correct .if one assumes that all agents are well - intentioned , that is , that they want to find a good solution to the problem , then it is plausible to assume that the answer reached by a specific agent `` contains '' a good answer distorted by some noise . under these conditions ,an efficient strategy is to aggregate the answers from all agents , as information aggregation cancels the distorting component of the individual answers .however , in many situations a centralized structure may not be practical or desirable because it is too inefficient , too costly , or because it would be difficult to secure an unbiased central authority . for these reasons , decentralized strategiesmay be preferable or even the only ones feasible .+   we consider a population of agents . for simplicity , and without loss of generality .the state of each agent is a binary variable that represents the answer to a stated problem .updating occurs using boolean functions . as we indicated earlier, agents can hold various types of intentions .specifically , well - intentioned agents have a bias toward their present state , while `` partisan '' agents have a bias toward a particular state , for example , .naive and conservative agents are thus not biased for or against either state _ per se _ , they merely prefer whatever answer they currently hold . while all three types of agents may change their state in response to peer pressure, a partisan agent will defect back to his preferred state if peer pressure decreases below a threshold value .both conservative and partisan agents can have different levels of trust on their neighbors , that is , different thresholds for responding to peer pressure .we define the `` strength ''  ] with which non - partisans weigh the information held by partisan agents . as demonstrated by a recent study , we must also enable partisan agents to discount the signals of both non - partisan and opposing partisan agents with _ at least _ the same discount rate .surprisingly , we find that the increased distrust among agents actually has a deleterious effect on the efficiency of the system in solving the density classification task ( fig .the reason for this apparently counter - intuitive result is that for partisans , so that when , partisan agents discount the answer of non - partisan agents to such an extent that they will never abandon their preferred answer . in fact , only for is a system containing partisan agents able to efficiently complete the density classification task .common experience demonstrates the existence of partisanship within groups of any kind .one possible interpretation for partisanship is a strong _ a priori _ belief that a certain answer is correct .alternatively , one may consider the case where agents have personal interests that may in fact differ from the `` common good . '' in this case individual decision rules , such as the degree of partisanship , can be interpreted as the solution to a maximization problem at the individual level .we model this question by assigning to a set of rational agents an idiosyncratic utility function that each agent tries to maximize , while the rest of the agents use the naive strategy .the interesting case is the one in which the agent s self - interest comes from answer `` '' being adopted while the `` common good '' comes from `` '' being adopted .a utility function for the rational agents is : where is the degree to which the agent values his own `` self - interest over the common good . ''if , the agent s utility is maximized by minimizing .thus , if an agent has the optimal individual strategy is partisanship with .if , then the optimal strategy depends on the strength of the noise ( fig .4 ) ; the stronger the noise , the more conservative the agents should be .these findings thus provide a possible explanation for the emergence of conservatism and partisanship as mechanisms to maximize individual rather than collective advantage .the question then arises of how one can reconcile the advantage of self - interest with the evolution toward cooperative societies .the answer is that for individual decisions to serve common good , societies must develop and adopt norms that regulate self - interest  .importantly , only in the presence of norms and the `` metanorms '' that support their enforcement  , will individuals adopt strategies that lead toward cooperation and better social outcomes .  our findings for the effect of partisan agents on the efficiency with which the system completes the density classification task are striking .even a small fraction of partisans can completely erase the efficiency of the system .it is not difficult to envision the consequences of this result on our daily lives .democratic societies face many situations in which `` difficult '' decisions must be made  .moreover , the ability of policy makers to reach timely decisions on difficult matters clearly increases when a strategy has broad support . reaching such broad consensus ,unfortunately , is unlikely to occur if partisan agents are present . sadly , partisanship is the rational individual strategy if there are no norms against it .we thank roger guimer for discussions . m.s .-p . acknowledges the support of ctsa grant 1 ul1 rr025741 from ncrr / nihnih and of nsf scisip 0830338 award .l.a.n.a . gratefully acknowledges the support of nsf awards sbe 0624318 and scisip 0830338 .for concreteness , and without loss of generality , we assign a probability to state `` '' in the initial configuration of the system . by setting , we avoid finite size effects ( supporting online material ) .we then define the instantaneous efficiency of the coordination process as where is the number of agents that are in state `` '' and is the number of agents that are in state `` '' . for each realization, we allow the system to evolve for time steps . in order to ensure that the strategies used by the agents are scalable, we let the system evolve for a number of time steps proportional to the number of agents in the system .we define the efficiency of a single realization as setting .the efficiency for a given set of parameter values is the average of over 1000 realizations .crutchfield and mitchell  allowed the system to converge to a point where all the agents have the same state , and a realization is considered to be successful provided the converged state is the same as the majority state . instead of requiring that the system reaches consensus , we focus instead on the steady state configuration reached by the system .

Abstract: we use agent - based modeling to investigate the effect of conservatism and partisanship on the efficiency with which large populations solve the density classification task a paradigmatic problem for information aggregation and consensus building . we find that conservative agents enhance the populations ability to efficiently solve the density classification task despite large levels of noise in the system . in contrast , we find that the presence of even a small fraction of partisans holding the minority position will result in deadlock or a consensus on an incorrect answer . our results provide a possible explanation for the emergence of conservatism and suggest that even low levels of partisanship can lead to significant social costs .