a central problem in the search for earth - like planets is to suppress the light from a star so as to reveal a planet which might be 10 to 12 orders of magnitude fainter than the star and might be in quite close to the star . in earlier papers ( nisenson and papliolios , 2001 ; nisenson et al , 2003 ) we showed that apodizing a telescope pupil reduced the diffraction of a telescope that would allow exo - planet detection .in addition to diffraction suppression , extremely low scattering optical surfaces at frequencies of 3 to 30 cycles / mirror are necessary . obtaining such low scattering will require a combination of super - polishing and adaptive optic correction . in this paper, we describe a technique for designing an apodizer that minimizes diffraction in a chosen region of the focal plane while maximizing the throughput of the apodizer .the image of a star ( unaberrated by the atmosphere ) is the point spread function ( psf ) of the telescope .we want to design a real , non - negative , continuous tone mask , i.e. , an apodizer , to place in the aperture ( or a relay plane of the aperture ) such that the psf has a compact , central peak with minimal spread beyond the core of the psf .this problem was first studied by slepian & pollack ( 1961 ) in the context of radar , communications , and superresolution .the authors show that the band - limited signal , , which has the most energy in time t satisfies the integral equation :    where is a zero - one function which time - limits to t seconds and the fourier transform , , of is another zero - one function which band - limits .the solutions ( eigenfunctions ) of this integral equation are prolate spheroidal ( ps ) functions and is the eigenfunction with the largest eigenvalue . the authors show that , inside t , is identical to its fourier transform , , suitably scaled in f. kaiser ( 1966 ) presented an approximation to and subsequent authors ( oppenheim and schafer , 1989 ; aime et al , 2002 ) have used the kaiser window because is difficult to compute .papoulis ( 1968 ) studied the same problem in the optical context and showed that , the fourier transform of , is the optimal one - dimensional apodizer .he showed that the optimal apodizer for a square aperture is the product , which is the form proposed by nisenson and papaliolios ( 2001 ) .he also shows that the optimal apodizer for a circularly symmetric aperture is a sum of the first and second order bessel functions .other authors ( kasdin et al , 2002 and aime et al , 2001 ) , have shown the optimality of a ps apodizer .kasdin et al ( 2002 ) suggest a ps - shaped , binary aperture with the key advantage that the binary aperture may be easier to manufacture .in this paper we present an iterative calculation of an optimal apodizer .the method does not require the kaiser approximation and it can accommodate any special features of the measurement process , such as the likely position of a planet in reference to its star .we use a very simple iterative procedure to calculate the apodizer , an iterative algorithm that has its roots in linear system theory and is closely related to the well - known gerchberg - saxton algorithm ( gerchberg and saxton , 1972 ) .we use the method of alternating projections which was introduced by von neumann ( 1950 ) .we write equation ( 1 ) in the compact form    where r is an operator , sometimes called the kernel .r space - limits the two - dimensional function with a specified spatial mask , , then band - limits it with a specified spatial frequency mask , . the latter is determined by the shape of the aperture .to solve eq .( 2 ) we repeatedly impose r on an initial until eq .( 2 ) is satisfied .more specifically , we calculate    then we calculate the fourier transform of f(x , y ) , namely f(u , v ) , and calculate    the inverse fourier transform of is , which completes one iteration of the algorithm .we rely on the fact that n iterations will produce an expansion of in eigenfunctions of eq .2 with coefficients which are multiplied by eigenvalues raised to the power n. eventually , only the eigenfunction with the largest eigenvalue will emerge , a property which is well - known .the algorithm is described by the figures and text in papoulis 1968 book .because the operation r uses a forward fourier transform , a masking operation , an inverse fourier transform , and another masking operation , the algorithm is similar to the iterative transform algorithm generally known as the gerchberg - saxton algorithm ( gerchberg and saxton , 1972 ) .we emphasize that the functions and can be any desired shape , which gives us the opportunity to shape both the aperture and the search region .a one - dimensional , unapodized aperture of width d has inverse fourier transform , , a sinc function with its first zero at .we call the diffraction - limited spot size or , more simply , a dl element . the magnitude squared of is , the one - dimensional point spread function ( psf ) .a square , two - dimensional aperture has . on the left in figure 1is the logarithm of the psf for such an unapodized aperture .the diagonal line in the image shows where we will sample the psf for comparison purposes . in all of our exampleswe assume that the available light is broadband with a bandwidth which is 20% of the central wavelength .thus , the deep nulls in a conventional sinc function are blurred out when we combine the psf s at various wavelengths .  to form this first apodized aperturewe space - limit a one - dimensional to the interval , say , , then we band - limit the result to d. ( for these examples , we used fft software with a calculation size of n = 256 and we set the aperture size to d = 41 . ) these space and bandwidth limiting operations are defined by equations ( 3 ) and ( 4 ) .we repeat them until the iterations converge .what emerges is the prolate spheroidal function , .its fourier transform , , is the apodizer . following previous authors ( i.e. aime et al , 2002 ) , we form the product aperture , which we call a product prolate spheroidal ( pps ) aperture , and its inverse fourier transform , p(x , y ) , is the telescope s psf . in the center of figure 1is the logarithm of the psf for this apodizer .note that the psf is concentrated along the x and y axes .the aperture has a transmission of .papoulis ( 1968 ) showed that this aperture best concentrates the psf into a central square of size by .figure 1 shows a third psf .this was formed by defining a one - d spatial mask , , which is 1 in the range from 0 to , zero from to and 1 for all larger x ; and is symmetric about .the mask forces the psf to be narrow , as above , but also allows it to rise outside a restricted search region from to .we apply the same iterative algorithm , starting with the pps apodizer and stopping the iterations when the transmission is ( ) .the result is the psf shown on the right of figure 1 .figure 2 shows a cut through three psf s .the horizontal axis plots diagonal distance in units of and the vertical axis plots the logarithm of , scaled such that .figure 2 also shows the psf for the unapodized aperture .note that the pps plot is about seven orders of magnitude below the no - apodizer plot , beyond about .the modified pps plot matches the pps plot from 0 to , drops about 2 db below it to about , then rises almost to the no - apodizer plot .figure 3 shows horizontal cuts through the pps apodizer and the modified pps apodizer .  for our next set of examples we used a circular aperture of diameter .first we calculated the psf for an unapodized aperture .it is the well - known airy disc .the logarithm of this psf is shown on the left of figure 4 .( as in figure 1 the psf s are for wideband light so the rings are somewhat indistinct .the horizontal and vertical bands at the edge of the figures are aliasing , caused by our finite sampling . )next we formed a circular mask of radius 3.5z and exercised a two - dimensional version of the iterative algorithm .this results in the 2-d psf in the center of figure 4 .this is the psf which papoulis ( 1968 ) reports but does not sketch .we call the apodizer a circular bessel ( cb ) apodizer. then we allowed the mask to rise at a radius of , so that the mask looks like a bulls eye , and exercised the two - d algorithm .the psf is on the right of figure 4 .cuts through the psf s are shown in figure 5 . inside the search regionthe bulls eye psf is about two orders of magnitude below the psf for the cb aperture .the apodizers are shown in figure 6 . both have transmission of about .  finally , we formed an apodizer for a square aperture with a two - dimensional mask which is a cross of width , imbedded in a circular mask of radius , a hybrid apodizer .it has a restricted search area as shown on the left in figure 7 .we exercised the iterative algorithm , starting with the cb apodizer .figure 7 shows the mask ( the search area ) and the log of the psf .figure 8 shows a slice through the psf and , for comparison , a slice through the psf for the cb apodizer . despite what appear to be only subtle differences in the shapes of the two apodizers , the hybrid apodizer has a transmission of , almost double that of the cb apodizer ; it also has a narrower central peak and better sidelobe supression .the reduced search area of this hybrid apodizer provides significant advantages over our earlier examples .figure 9 shows cuts through the apodizer at 0 and at , which shows that the hybrid apodizer is not circularly symmetric .we have described an approach to designing an apodization mask that is optimized for minimizing diffraction in a selected local area of the focal plane .this would allow use of selectable apodization masks in a tpf telescope that could , for example , reduce diffraction over a wide area in the focal plane for a survey , and then produce deeper nulls in local areas for spectroscopy of an already detected planet . or one could select an apodizer for the minimum width of the central peak , trading off the size of the searchable region for minimum detectable separation of star and planet. we also investigated use of apodizers that vary only in one direction ( constant in the other ) .these apodizers result in maximizing throughput ( close to 50% ) while allowing detection of luminosity ratios of .this is insufficient for tpf - like missions ( luminosity ratios will be of order for an earth - like planet orbiting a star at 10 pc ) but hybrid apodizers , as described in the paper , allow a continuous trade - off between maximizing throughput and minimizing diffraction .the speckle due to optical surface errors and the diffraction will produce a psf that is highly centro - symmetric when ( as required for a tpf telescope ) the phase errors are extremely small . for very small phases ,the exponential phase can be accurately approximated by only the linear term of the power series expansion of the exponential .so , for example , a telescope with 1/1000 wave rms surface errors will have a centro - symmetric psf to a precision of about 0.1% and 1/10000 wave results in centro - symmetry of 0.01% .one can then use this asymmetry to calibrate the residual speckle and diffraction in the half - plane of the psf that contains a planet using the opposite half - plane .subtraction of the calibration region then increases the dynamic range to an accuracy limited only by photon statistics .accurate intensity transmission of the apodizers is critical for their success - this is evident from the apparently small differences in apodizer shape , as illustrated in figure 3 , producing substantial effects on the psf .we have estimated the precision required in the transmission of the apodizer to be 0.3% or better for it not to affect the dynamic range .this is the same requirement as one for the uniformity of the reflective coatings of the telescope mirrors .numerical simulations that substantiate this result are included in nisenson et al .( 2003 )  the usual problems with iterative algorithms are choosing an initial function and choosing a stopping point in the iterations .for the examples which yielded the ps and cb solutions , we started with a gaussian shape and we stopped the iterations when there was no significant change in the solution , typically a few hundred iterations . for the other examples we started with a ps or cb aperture. continued iterations result in deeper and deeper suppression of diffraction in the chosen region but also resulted in decreasing throughput for the apodizer .we stop the iterations when the diffraction is reduced to a selected level while leaving the throughput at a maximum .we have shown an approach to calculating an optimum apodizer that maximizes the detectable luminosity ratio for exo - planet detection in the focal plane of the tpf telescope .the next step in this analysis would be to include realistic specifications on wavefront errors from the telescope optics and also to factor in practical limits on the accuracy in the manufacture of apodizing masks .how well continuous tone apodizers can be manufactured is an open question .many of the techniques developed for photolithography allow very precise shaped transmission to be generated in special glass .such a mask was used in a laboratory demonstration of imaging two close point sources with a luminosity ratio of nearly ( melnick et al , 2001 ) .melnick , g. j. , fischer , d. , geary , j. c. , gezari , d. y. , hardesty , c. , harwit , m. , holman , m. , korzennik , s. g. , lyon , r. g. , marley , m. s. , mcelroy , m. b. , neufeld , d. a. , nisenson , p. , noyes , r. w. , papaliolios , c. , ridgway , s. t. , sasselov , d. d. 2001 , american astronomical society meeting 199 , # 09.10 .nisenson , p. , melnick , g.j . ,geary , j. , holman , m. korzennik , s.g ., noyes , r.w . ,papaliolios , c. , sasselov , d.d . ,gonsalves , r. , hardesty , c , gezari , d. , lyon , r.g . , fischer , d. , harwit , m. , marley , m.s . ,neufeld , d.a . ,ridgway , s.t .2003 , to appear in asp conf ., scientific frontiers on research of extrasolar planets , eds .d. deming & s. seager , ( san francisco : asp ) .

Abstract: one of two approaches to implementing nasa s terrestrial planet finder is to build a space telescope that utilizes the techniques of coronagraphy and apodization to suppress diffraction and image exo - planets . we present a method for calculation of a telescope s apodizer which suppresses the side lobes of the image of a star so as to optimally detect an earth - like planet . given the shape of a telescope s aperture and given a search region for a detector , we solve an integral equation to determine an amplitude modulation ( an apodizer ) which suppresses the star s energy in the focal plane search region . the method is quite general and yields as special cases the product apodizer reported by nisenson and papaliolios ( 2001 ) and the prolate spheroidal apodizer of kasdin et al ( 2002 ) , and aime et al ( 2002 ) . we show computer simulations of the apodizers and the corresponding point spread functions for various aperture - detector configurations .


extreme values play a crucial role in science , technology , and engineering .they are linked to rare events , large deviations , and optimization : minimizing a lagrangian , for instance , is a frequent task in physics .extreme values also provide an important characterization of random processes , and the study of extreme values is a significant area in statistics and probability theory .  for a scalar random process , there are two extreme values the maximum and the minimum .it suffices to consider the maximum .take the most basic random process , brownian motion . by brownian motion we always mean standard brownian motion , namely the one - dimensional brownian process that starts at the origin : with .for the brownian motion , the maximum process is defined by   this maximum process is non - trivial as demonstrated by the absence of stationarity the increment depends on both and .in contrast , the position increment , , depends only on ; this stationarity is a crucial simplifying feature of brownian motion .still , the maximum process has a number of simple properties , e.g. the probability density of the maximum is a one - sided gaussian the joint probability distribution of the position and maximum , and , also admits an elegant representation \ ] ] this lesser - known formula , which was discovered by lvy , proves useful in many situations ( see ) ; it can be derived using the reflection property of brownian motion .    in this article, we study the leapfrogging of the maxima of two brownian motions ( see fig . [ fig - process ] ) . the probability that one maximum exceeds another during the time interval investigated only recently . herewe explore additional features of the interplay between two brownian maxima .when appropriate , we emphasize the universality of our results , e.g. some of our findings apply to rather general markov processes , e.g. to arbitrary symmetric lvy flights .one of our main results is that the average number of lead changes between the two brownian maxima exhibits a universal logarithmic growth in the long - time limit .this behavior holds for random walks , and more generally for identical symmetric lvy flights .we also study the probability to have exactly lead changes during the time interval , see fig .[ fig - process ] .we show that in the case of identical symmetric lvy flights , the probability decays according to the persistence exponent , that characterizes the probability that the lead does not change , depends only on the lvy index .we evaluate numerically the quantity for .this paper is organized as follows . in sect .[ sec : anlc ] we establish the growth law .first , we provide a heuristic derivation ( sect . [sec : ha ] ) relying on the probability densities . we also mention the generalization to two random walks with different diffusion coefficients . in sect .[ sec : fpa ] we establish a link with first - passage time densities and the arcsine law , and we employ it to provide a more rigorous and more general derivation of . in sect .[ sec : lf ] we study the statistics of lead changes for symmetric lvy processes , particularly symmetric lvy flights with index .the average number of lead changes is shown to exhibit the same universal leading asymptotic behavior independent of the lvy index . in sect .[ sec : nlc ] we show that for identical symmetric random walks the probability to observe exactly lead changes between the maxima behaves as .we conclude with a discussion ( sect .[ sec : disc ] ) .the question about the distribution of the number of lead changes of brownian particles is ill - defined if one considers the positions there is either no changes or infinitely many changes .the situation is different for the maxima : the probability density to have exactly changes during the time interval is a well - defined quantity if the initial positions differ .discretization is still necessary in simulations .the discrete - time framework also simplifies the heuristic reasoning which we employ below .the corresponding results for the continuous - time case follows from the central - limit theorem : a random walk with a symmetric jump distribution that has a finite variance converges to a brownian motion .thus , we consider two discrete time random walks on a one - dimensional line .the position of each walk evolves according to the displacements are drawn independently from the same probability density which is assumed to be symmetric , and hence .we set the variance to unity , .further , we always assume that the walk starts at the origin .the probability distribution of its position becomes gaussian in the long - time limit .the maximal position of the walk during the time interval is the probability distribution of the maximum is given by , and the joint position - maximum distribution is given by .  in the following ,we consider two identical random walkers .we assume that both walkers start at the origin , .denote by and the positions of the walkers and by and the corresponding maxima . if the first walker is considered to be the leader . if the notion of the leader is ambiguous .this happens at , and we postulate that the first walker is the original leader .there are no lead changes as long as both walkers remain in the half - line .once the walkers spend time in the half - line , the identity of the leader can be ambiguous if the probability density is discrete , e.g. , in the following , we tacitly assume that the probability density does not contain delta functions , so once the two maxima become positive they remain distinct .eventually the maximum of the second walker will overtake that of the first and the second walker turns into the leader . yet , at some later moment , the leadership will change again .this leapfrogging proceeds indefinitely .how does the average number of lead changes vary with time ? what is the distribution of the number of lead changes ? in this sectionwe answer the first question .first , we provide a derivation of the asymptotic growth law using heuristic arguments .we now argue that the average number of lead changes exhibits the logarithmic growth .let and the lead changes soon after that .we use the shorthand notation . the probability density for this quantityis given by .we are interested in the asymptotic behavior ( ) and in this regime , the random walker is generally far behind the maximum .this behavior is intuitively obvious , and it can be made quantitative using we compute the average size of the gap between the maximum and the position the second maximum can overtake the first if is close to . since , both and are close to and the probability of this event is once the second particle reaches the leading maximum it will overtake it given that the first particle is typically far behind , see .therefore , the lead changes with rate integrating over all possible yields which gives the announced growth law .the average already demonstrates that the lead changing process is not poissonian . for a poisson process, the probability to have exactly lead changes is fully characterized by the average : this distribution would imply that the probability of having no lead change decays as \sim t^{-1/\pi} ] .one thus obtains by combining with , one recovers the heuristic prediction and confirms .moreover , the derivation is now sufficiently general as it applies to all random walks with symmetric jump distributions .hence the leading asymptotic behavior does not depend on the details of the ( symmetric ) jump distribution , in full agreement with the universality of the amplitude observed in numerical simulations .the first - passage equation and path transformation procedure can be applied to random walks that do not converge to brownian motion , as discussed in the next section .brownian motion belongs to a family of ( symmetric ) lvy processes .these processes are stationary , homogeneous and stable . in a discrete time realization ,a lvy process becomes a lvy flight .lvy flights are ubiquitous in nature , see . here, we examine lead changes of the maxima of two identical lvy flights . for lvy flights ,the jump distribution has a broad tail , and the lvy index quantifies this tail , as . for simplicity , we choose the purely algebraic jump distribution , when , lvy flights are equivalent to an ordinary random walk . for true lvy flights ; in this range the variance diverges . for even the average jump length scales as .we now show that the average number of lead changes exhibits logarithmic growth .denote by the probability density of the maximum process and by the density of the first passage time at level . following the same reasoning as in sect .[ sec : fpa ] , we write for the rate at which lead changes take place : we now invoke the scaling properties of lvy flights these scaling forms reflect that the maximum process is -stable and the first - passage time process is -stable . combining and we get consequently , the average number of lead changes grows logarithmically with time ,   versus time . shown ( top to bottom ) are simulation results for identical lvy flights with index and for random walks ( ) .the inset shows versus time , demonstrating that the amplitude in is universal and equal to .,scaledwidth=42.0% ]  our numerical simulations confirm that for identical random walks on a line , and even for random walks on the lattice with the discrete jump distribution , the amplitude is independent of the details of the jump distribution .furthermore , the universality continues to hold for the lvy flights , so independent of the lvy index .this is confirmed by numerical simulations ( see fig . [ fig - nt ] ) .   versus time for lvy flights .shown are ( from top to bottom ) simulation results for the values , , and .,scaledwidth=44.0% ]  the universality can be understood by noting that the path transformation procedure described at the end of the previous section , as well as the arcsine law , are valid for lvy flights .the key requirements are to have cyclic exchangeability of the jumps ( which is guaranteed when these are independent and identically distributed ) , a continuous and symmetric jump distribution , and a certain type of regularity for the supremum , see .the validity of the arcsine law implies that we can use .thus the leading asymptotic holds for a wide class of symmetric lvy processes , and in particular , symmetric lvy flights .next , we examine the probability that there are no lead changes till time .our simulations show that this probability decays algebraically ( see fig .[ fig - f0 ] ) : the persistence exponent varies continuously with , namely is a monotonically decreasing function of , see fig .[ fig - bm ] . for ,lvy flights are equivalent to ordinary random walks , so for .   versus the index characterizing ( symmetric ) lvy flights.,scaledwidth=44.0% ]  in the marginal case ,the mean - square displacement has a logarithmic enhancement over the classic diffusive growth , .consequently , the convergence toward the ultimate asymptotic behavior is very slow near .our numerical simulations suggest that in the marginal case , there may be a logarithmic correction to the algebraic decay .we now focus on the probability to have exactly lead changes until time . for two identical random walks , the probability that there are no lead changes decays as in the long - time limit .since is the probability that the -st lead change takes place after time , the probability density for the -st lead change to occur at time is given by .in particular , from , one can write down the conditional probability density of the time of the next lead change , given that it takes place after time : \sim s^{1/4}\tau ^{-5/4}.\ ] ]  furthermore , the quantity is the probability that the first lead change occurs at some time and that the next one occurs at some time . the probability density associated with sucha configuration is simply proportional to \times \int_t^{\infty}\ , d\tau\ , \rho_s(\tau )  \sim s ^{-5/4}\times  s^{1/4 } t ^{-1/4 } \sim s^{-1}t^{-1/4}.\ ] ] integrating over [ we could introduce a cut - off , but this is akin to a discretization of time , so in the large limit we might as well set , we find   versus time for and .the data corresponds to simulations of two ordinary random walks in one dimension.,scaledwidth=44.0% ]  hence , there is a logarithmic enhancement of the probability to have one lead change compared with having none .the above argument can be generalized to arbitrary to yield one can establish this general behavior by induction : \int_t^{\infty}\ , d\tau\ , \rho_s(\tau ) \\    & \sim & \int_1^{t-1 } ds\ , ( \ln s)^n\,s^{-5/4 } s^{1/4}t^{-1/4 } \\& \sim & t^{-1/4}\int_{1}^{t-1 } ds ( \ln s)^n\,s^{-1}\\    & \sim & t^{-1/4}(\ln t)^{n+1}.\end{aligned}\ ] ] thus , there are logarithmic corrections for all .we probed the behavior of the cumulative distribution using numerical simulations .the quantity is the probability that the number of lead changes in time interval does not exceed .the dominant contribution to is provided by and hence we verified eq .  numerically for and ,see fig .[ fig - fn ] .  for lvy flights , the distribution of the number of lead changes can be established in a similar manner . using and following the derivation of weget where is the aforementioned persistence exponent . using numerical simulations, we verified that holds for and for a few representative values of the lvy index in the range .the simplest model of correlated random variables is a one - dimensional discrete - time random walk .its maximum evolves by a random process that exhibits a number of remarkable features .some of these properties are classical , while others were discovered only recently .for instance , the probability distribution for the total number of distinct maximal values ( records ) achieved by the walk is universal , i.e. , independent of the details of the jump distribution , as long as the jump distribution is symmetric and not discrete .this universality also holds for symmetric lvy flights , and in fact , it is rooted in the sparre andersen theorem . in the case of multiple identical random walks, the total number of records has been investigated in .  in this article, we studied a sort of `` competition '' between maxima of two identical random walks .in particular , we examined the average number of lead changes and the probability to have exactly changes .we found that the average number of lead changes exhibits a universal logarithmic growth .this asymptotic behavior also holds for symmetric lvy flights with arbitrary index .in contrast , the probability distribution is not universal : the persistence exponent in depends on the lvy index . the most interesting challenge for futurework is to determine analytically the continuously varying exponent in the range .one natural generalization of the competing maxima problem is to an arbitrary number of identical random walks or lvy flights .we expect that the average number of lead changes among the maxima of identical random walkers still exhibits a universal logarithmic growth , albeit with a -dependent prefactor .another quantity of interest is the survival probability that lvy maxima remain ordered , , for .based on , we anticipate that the probability decays algebraically , similar algebraic decays with -dependent exponents describe the probability that the positions of random walks remain perfectly ordered , the probability that one random walk remains the leader , etc . .  for , the exponent appearing in is our basic persistence exponent : .the exponents do not depend on when , i.e. , for random walks .for ordinary random walks , the exponents were studied numerically in , while approximate values for these exponents were computed in .  generally one would like to explore the statistics of ordering and lead changes for a collection of random variables . even the case of markovian random variables is far from understood .non - markovian random variables appear intractable , but sometimes the progress is feasible due to hidden connection to markovian processes .for instance , in the case of brownian maxima the first - passage processes are markovian .this allows one to use results from classical fluctuation theory to derive the persistence exponent , see , and yields another derivation of the logarithmic growth of the average number of lead changes . for lvy flights, however , the existence of leapovers at the first passage time of certain points leads to a breakdown of this approach .new results in this direction represent challenges for future work .we benefited from discussions with s. n. majumdar .two of us ( plk and jrf ) thank the galileo galilei institute for theoretical physics for hospitality during the program on `` statistical mechanics , integrability and combinatorics '' and the infn for partial support . the work of ebn was supported through us - doegrant de - ac52 - 06na25396 .k.l . chung and m. kac ,* 6 * , 11 ( 1951 ) ; k.l .chung and m. kac , ann . math . * 57 * , 604 ( 1953 ) ; h. kesten , indiana univ .j. * 12 * , 391 ( 1963 ) .details of the derivation of using the methods developed in these references will be published elsewhere .

Abstract: we investigate statistics of lead changes of the maxima of two discrete - time random walks in one dimension . we show that the average number of lead changes grows as in the long - time limit . we present theoretical and numerical evidence that this asymptotic behavior is universal . specifically , this behavior is independent of the jump distribution : the same asymptotic underlies standard brownian motion and symmetric lvy flights . we also show that the probability to have at most lead changes behaves as for brownian motion and as for symmetric lvy flights with index . the decay exponent varies continuously with the lvy index for , while when .


solutions to the three dimensional inhomogeneous linear partial differential equations of mathematical physics are expressible in terms their appropriate green s functions ( duffy 2001 ) .single , double , and triple eigenfunction integration and summation expressions for green s functions are obtainable in the coordinate systems that allow for separation of variables ( morse & feshbach 1953 ) .separation of variables maps the linear three dimensional homogeneous partial differential equations into three decoupled ordinary differential equations ( miller 1977 ) .important linear 3d partial differential equations of mathematical physics include the laplace equation , the helmholtz equation , the biharmonic equation , the wave equation , the diffusion equation , and the schroedinger equation . the properties of r - separability for these partial differential equations are reflected in the green s function expansions . in the rotationally invariant coordinate systems that r - separably solve these equations , new discrete fourier representations exist for the green s functions for the laplace equation ( cohl et al .discrete fourier expansions , given in terms of the azimuthal and separation angles , must exist for the rest of the linear partial differential equations of mathematical physics .the coefficients of these discrete fourier representations will be given in terms of identifiable transcendental functions obtained by reversing and collapsing green s function expansions for these partial differential equations .the fundamental mathematical tools required in order to complete this investigation are commonly available in the mathematics and physics literature .the green s function for laplace s equation is   where and are given by the cartesian coordinates and .traditionally , the green s function is expanded in terms of spherical harmonics in spherical coordinates , where ( ) is the smaller ( larger ) of the distances and , and is the legendre polynomial of the first kind .may be viewed either as defining two points relative to the origin or as the coordinates of a three - body system once the center of mass has been separated . in the _ body frame ,_ three out of the six coordinates are dynamical variables , the potential energy depending on them . ''( cohl et al . 2001 ) ]  by utilizing heine s toroidal identity ( cohl & tohline 1999 , cohl et al . 2000 , cohl et al . 2001 ): where and , one can derive two highly compact fourier series representations for the green s function of laplace s equation     in spherical and cylindrical coordinates respectively .similar discrete fourier transform expressions must exist for the linear partial differential equations of mathematical physics .algorithmic implementation of compact fourier representations for the green s functions of mathematical physics significantly improves the accuracy of the solution for these partial differential equations .one may also express potential problems in terms of an inhomogeneous biharmonic equation whose source is proportional to the laplacian of the density .the solution is expressible in terms of an integral of it s green s function , , convolved with source . the resulting inhomogeneous biharmonic equation   where is the laplacian , represents the solution to the potential problem over the infinite three - dimensional domain .the green s function for the biharmonic equation ( vautherin 1972 )   is a much better behaved integration kernel than for laplace s equation because the singularity has been moved to infinity through one can generalize heine s reciprocal square root identity using ( abramowitz & stegun , 1965 ; gradshteyn & ryzhik 1994to within a sign )    and by integrating both sides of eq .( 2 ) with respect to performing the integrations over we obtain the following general result for _ any _ integer - generalizing the last expression at the bottom of p. 182 in magnus , oberhettinger , & soni ( 1966 ) ( see also eqs . ( 26 ) & ( 30 ) in cohl et al . 2000 )   which reduces for to    using this expression , we can rewrite the green s function for the biharmonic equation in two different ways    and    where  ^{\frac{1}{2 } } } { 2\pi\sqrt{rr'}}.\ ] ]this new formulation is amenable in analytic and computational physics applications on high - performance - computing architectures , since laplacians can be readily computed on computational fluid dynamics meshes .higher order harmonic green s function expansions can be generated using this method .boundary values can now be computed effectively along an arbitrarily chosen z - axis and at values chosen to lie within the outer - most extent of a chosen volumetric region .physical solution of the interior inhomogeneous problem can be obtained by proper boundary value treatment as described above .accelerations can be obtained from the potential by performing a precise numerical gradient . in the case of cylindrical and spherical coordinates, the three dimensional poisson solve can be further facilitated through the use of a discrete azimuthal fourier transform .this decouples the three dimensional poisson problem into a set of decoupled 2d problems that can be solved with second order accurate finite differencing using either direct or iterative methods .using highly compact representations of infinite domain green s functions , solutions to the linear partial differential equations of mathematical physics can be more easily obtained . in this paper , and in recent papers , we treat the green s function for the laplace equation . herewe have extended this result to the three dimensional biharmonic and higher harmonic equations .further variants are possible for the 3d helmholtz , wave , and diffusion equations . in the future, we intend to investigate and more precisely describe these new green s function expansions . many areas of theoretical physics will benefit greatly from precise numerical implementations of these compact expansions .azimuthal fourier identities for potentials such as coulomb ( ) or yukawa ( ) lead to new classical and quantum energy theorems .these allow for rapid and precise evaluation of the coulomb or yukawa direct and exchange interactions ( cohl et al .classical yukawa eigenfunctions are obtained through transcendental function identification of the lamb - sommerfeld integral ( see magnus et al . 1966 ) . in quantum physicsthis is accomplished through the use of the azimuthal selection rule for the self - energies , namely for the direct and exchange hamiltonian elements only the and terms survive respectively .the application of the selection rule allows for exact evaluation of the hamiltonian matrix elements for two - electron interactions in atomic physics , molecular physics , condensed matter physics , physical chemistry , and biology .two - electron interactions are critically important in obtaining opacities and correct equations of state in astrophysically dense atomic and molecular fluid media .magnetohydrodynamic problems can now be easily handled with heine expansions for the green s functions of potential theory . by expressing the equations of fluid dynamics of a compressible media in terms of a velocity potential and a vector poisson equation, one may compute precise velocity boundary values in vortex and shock flow regions ( lamb 1932 ) by solving the appropriate poisson problem .radiation transport , and classical and quantum scattering , will be greatly facilitated through compact poisson formulations of the green s function for the 3d diffusion and helmholtz equations .precise coulomb and yukawa energies in the nuclear hamiltonian allow for a higher degree of precision in obtaining nuclear structure .the ultimate resolution of these analytical investigations will be efficient algorithmic implementations of these new schemes .we propose these methods to the three dimensional star community in hope that you may continue to enjoy significantly improved economical and precise boundary values for studies of analytical and numerical three dimensional stellar astrophysics .i would like to thank ernie kalnins , department of mathematics , university of waikato , new zealand , for pointing out the integration formula for legendre functions .i would also like to thank volker oberacker , department of physics and astronomy , vanderbilt university , for pointing out the green s function for the axisymmetric biharmonic equation in terms of the complete elliptic integral of the second kind .i especially wish to thank dr .rau , department of physics and astronomy , louisiana state university , for many helpful comments and suggestions .i would also like to express my gratitude to everyone i had the honor of meeting at the 3d stars workshop , the scientific organizing committee and great appreciation to the editors of this proceedings , rob cavallo , stefan keller , and sylvain turcotte .i would also like to thank richard ward , david dearborn , john castor , john bradley , kem cook , douglas peters , pete eltgroth , and peter eggleton .this work was performed under the auspices of the u.s .department of energy , national nuclear security administration by the university of california , lawrence livermore national laboratory under contract no . w-7405-eng-48 .the unit order negative one half degree legendre function of the second kind can be expressed as

Abstract: precise efforts in theoretical astrophysics are needed to fully understand the mechanisms that govern the structure , stability , dynamics , formation , and evolution of differentially rotating stars . direct computation of the physical attributes of a star can be facilitated by the use of highly compact azimuthal and separation angle fourier formulations of the green s functions for the linear partial differential equations of mathematical physics .   # 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in


network provides a powerful tool for representing the structure of complex systems .these networks include social networks  , information networks  , and biological networks  .much of recent research on networks actually aims to understand the structural regularities and further to reveal the relationship between such structural regularities and the function of networks .for example , as a widely - studied structural characteristic of network , community structure is of high interest because communities often correspond to functional units such as pathways for metabolic networks and collections of pages on a similar topic on the web .  community structure is a kind of assortative structure , in which nodes are divided into groups such that the members within each group are mostly connected with each other .contrary to community structure , multipartite structure is another important kind of structural regularities observed in real world networks .multipartite structure means that nodes of network can be divided into groups such that most of edges are across groups . beside these salient structural characteristics , other types of structureare also observed in real world networks , such as hierarchical structure and core - periphery structure .however , existing methods mostly presume that certain type of structure exists in the target network and then devote to detect such structure .this raises concerns to the reliability of the resulted structure . on one hand, the assumed structure may not match the intrinsic structure of the target network and thus these methods are not applicable to these situations . on the other hand ,several real world networks contain multiple types of structure simultaneously .most existing methods are designed for certain type of structure and thus can not detect the broad types of structure .in addition , several unknown types of structure may also exist in networks and a desired method should be able to detect such structure as well .thus , it is the time to explore multiple types of structural regularities in networks .  in the last decade , the identification of community structure has attracted much attention in various scientific fields .many methods have been proposed and applied successfully to some specific complex networks  . for review , the reader can refer to ref .these methods are from different perspectives , such as the centrality measures , modularity , link density , percolation theory , network compression , and spectral analysis .recently , several generative models for network data are proposed to detect community structure  .these models view network structure as observed quantities and take communities as hidden groups of nodes .the communities are then identified by fitting the model to the observed network structure .for example , ren _et al . _ proposed a probabilistic model to uncover the overlapping community structure .this model assumes that the two end nodes of each edge are from the same community and this assumption is satisfied by the fuzzy membership of nodes .et al . _ applied the latent dirichlet allocation ( lda , a well - known generative model ) to social network analysis and gave a method to detect community structure .the common drawback of these two models is that they can only uncover the community structure and fail to reveal other types of structural regularities , e.g. , multipartite structure .  to characterize the hierarchical organization of networks ,clauset _ et al ._ proposed the hierarchical random graph model , which is capable of expressing both assortative and disassortative structure  . to explore more broad types of structure , newman _proposed a mixture model for exploratory analysis of network structure  . in this model , the nodes with similar connection preference rather than the highly connected nodes are classified into the same group . in such a general way, this model can reveal several other kinds of structural regularities beyond community structure .however , this model fails to tell us which kind of structural regularities has been identified .more importantly , this model may produce a result which is a mixture of several types of structure , and thus the identified structure may not provide clear information about the structural regularities .the shortcoming of this model is attributed to that it only models the relationship between groups and nodes rather than the relationship among groups .stochastic blockmodel provides an appropriate alternative to the mixture model for exploring broad range of structural regularities .utilized a degree - corrected stochastic blockmodel  to investigate community structure of network ._ gave a mixed membership stochastic blockmodel  to model network data .these work has demonstrated that stochastic blockmodel is a good choice for exploring regularities of network .however , the effectiveness of these models are limited by their inflexible model assumptions , e.g. , the hard partition assumption or neglecting the directionality of edges .  in this paper , we focus on exploring the intrinsic structural regularities in network by dividing network nodes into groups such that the members of each group have similar patterns of connections to other groups .we propose a general stochastic blockmodel ( referred to as gsb model in this paper ) to model the network structure . in this model ,node groups are represented by unobserved or hidden quantities and the relationship among groups are explicitly modeled by a block matrix as the traditional blockmodels .then , using the expectation - maximization algorithm , we fit the model to specific network data and thus detect intrinsic structural regularities of the network without prior knowledge of what type of regularity exists in the network . compared with existing models ,the most prominent strength of our model is the high flexibility .this strength enables it to possess the advantages of existing models and overcomes their shortcomings in a unified way . as a result ,not only broad types of structure can be detected , but also the type of identified structure can be indicated by the block matrix . in addition , our model can tell us the centrality of the node in each group and the mixed membership of nodes as well .tests on a number of artificial and real world networks demonstrate that our model outperforms the state - of - the - art models at shedding light on the structural features of networks , including the overlapping community structure , multipartite structure and several other types of structure which is beyond the capability of existing models .generally , a network with nodes can be represented mathematically by an adjacency matrix with elements if there is an edge from node to node and otherwise . for weighted networks , is generalized to represent the weight of the edge from to .  to investigate the structural regularities in network, we suppose that the nodes of the network fall into groups whose memberships are unknown , i.e. , we can not observe or measure them directly . in this paper , we propose a statistical model to infer the group membership from the observed network structure .the model we used is a kind of stochastic blockmodel .blockmodel is a generative model and has a long tradition of study in the social science and computer science . for a standard blockmodel , a matrix is generally adopted such that the matrix element denotes the probability that a randomly selected edge connects group to group , i.e., the tail node of the edge is from group and the head node is from .the advantage of blockmodel lies in that the matrix explicitly characterizes various types of connecting patterns among groups .  in the standard blockmodel ,the nodes in the same group are identical , i.e. , each node in a group has equal probability to be the end node of an edge adjacent to the group .this constraint is relaxed in our model .specifically , for an edge with its tail node being from group and its head node being from group , we use to denote the probability that the tail node is and to denote the probability that the head node is respectively .in addition , we use and to denote respectively the group membership of the tail node and head node of the edge .  up to now , we have given all the quantities in our model .they can be classified into three classes : observed quantities , hidden quantities , and model parameters . to simplify the notations ,we henceforth denote by the entire set and similarly , , , , for , , , and .  with our model , an edge generated in the following process :  1 .select two groups and respectively for the tail node and head node of the edge with probability ; 2 .draw the tail node from the group with probability ; 3 .draw the head node from the group with probability .summing over the latent quantities and , the probability that we observe an edge can be written as   then , the likelihood of the observed network with respect to our model is note that the self - loop edges are allowed and the weight is taken as the number of multi - edges connecting node to node as done in many existing models including , for instance , the widely studied configuration model  .intuitively , the parameter characterizes the centrality of node in the group from the perspective of outgoing edges while describes the centrality of node in the group from the perspective of incoming edges . differently from traditional blockmodels , by differentiating these two kinds of centrality , our model can provide more flexibility to explore broad types of intrinsic structural regularities in network .note that the parameters , , satisfy the normalization conditions   now our task is to estimate the model parameters and to infer the unobserved quantities by fitting the model to the observed network data .the standard framework for such a task is likelihood maximization .generally , one works not with the likelihood [ eq . ( [ eq2 ] ) ] itself but with its logarithm ( log - likelihood ) the maximums of the likelihood and its logarithm are in the same place since the logarithm is a monotonically increasing function .  directly maximizingthe log - likelihood is difficult because of the inner sum over the unobserved quantities and . using jensen s inequality ,the maximization of the log - likelihood can be transformed to the maximization of the expected log - likelihood \nonumber \\ & = & \sum_{ijrs}{q_{ijrs}a_{ij}\left(\text{ln}\omega_{rs}+\text{ln}\theta_{ri}+\text{ln}\phi_{sj}\right)},\label{eq5}\end{aligned}\ ] ] where to simplify the notation we have defined , which denotes the probability that one observes an edge with its tail node from group and its head node from group given the observed network and the model parameters .  with the expected log - likelihood, we can give the best estimate of the value and the position of its maximum represents the best estimate of the most likely values of the model parameters .specifically , if the value of is known , we can find the values of the model parameters , , where reaches its maximum .however , the calculation of requires the values of these model parameters . to address such a problem , an expectation - maximization ( em )algorithm is adopted .  under the framework of em algorithm, we first calculate the value of by once we have the values of the , we can use them to evaluate the expected log - likelihood and hence to find the values of , , that maximize it .  introducing the lagrange multipliers , and to incorporate the normalization conditions in eq .( [ eq3 ] ) , the expected log - likelihood expression to be maximized becomes   by letting the derivative of to be , the maximum of the expected log - likelihood occurs at the places where    eqs .( [ eq6 ] ) and ( [ eq8 ] ) constitute our expectation - maximization algorithm . in the expectation step, the expected value of log - likelihood is calculated through evaluating the values of with eq .( [ eq6 ] ) . in the maximization step ,the expected value of log - likelihood is maximized when the values of model parameters , , are evaluated with eq .( [ eq8 ] ) .implementation of the algorithm consists merely of iterating eqs .( [ eq6 ] ) and ( [ eq8 ] ) until convergence .when the algorithm converges , we obtain a set of values for hidden quantity and model parameters , , .this set of values is self - consistent with respect to eqs .( [ eq6 ] ) and ( [ eq8 ] ) . however , it is not always the place where the log - likelihood reaches its maximum . in other words ,the expectation - maximization algorithm may converge to local maxima of the log - likelihood . with different starting values ,the algorithm will give rise to different solutions . to obtain a satisfactory solution ,it is necessary to perform several runs with different initial conditions and take the solution giving the highest log - likelihood over all the runs performed .  by fitting the model to the observed network structure with the expectation - maximization algorithm ,the estimated model parameters provide us vital information for structural regularities of the network .specifically , and describe the centrality of a node in groups containing it respectively from the perspective of outgoing edges and incoming edges .the parameter characterizes the connecting patterns among different groups , i.e. , the type of structural regularities .more importantly , according to the model parameters , we can define two kinds of group memberships and respectively from the perspective of outgoing edges and incoming edges .specifically , is the probability that node is from group when it acts as the tail node of edges while is the probability that node is from group when it acts as the head node of edges . for , it can be calculated by actually , provides a soft or fuzzy membership , i.e. , node can belong to more than one groups simultaneously .when the identified structural regularity corresponds to community structure , we actually obtain the overlapping community structure which has attracted much research attention ever since it is proposed .if one wants to get a hard partition , we can simply assign each node to the group satisfying .these statements for also apply to defined as   finally , the model described above so far is based on directed networks .actually , the model can be easily generalized to undirected networks by letting the parameter be identical to .the derivation follows the case of directed networks and the results are the same to eqs .( [ eq6 ] ) and ( [ eq8 ] ) .now we discuss the computational cost of the expectation - maximization algorithm for the fitting of our model . for each iteration in this algorithm ,the cost consists in two parts .the first part is from the calculation of using eq .( [ eq6 ] ) , whose time - complexity is . here is the edges in the network and is the number of groups .the second part is from the estimation of the model parameters using eq .( [ eq8 ] ) , whose time - complexity is also .we use to denote the number of iterations before the iteration process converges .then , the total cost of the expectation - maximization algorithm for our model is .it is difficult to give a theoretical estimation to the number of iterations . generally speaking , is determined by the network structure and the initial condition .a well - designed initial condition may result in good convergence rate .the computational cost limits our model to deal with networks with moderate scale networks .we look forward to seeing more efficient implementation for our model .note that the method proposed in  provides a promising way to improve the computational efficiency and decrease the memory space required .in this section , we illustrate the difference and connections between our model with several existing models . fig .[ fig1 ] gives the schematic for our model and two existing generative models , namely newman s mixture model and ren s model .   and indicates the existence of one ( directed ) edge connecting them .the dashed - line connecting two circles indicates that the relation between the corresponding quantities is unobserved and requires being learned from the observed network data .arrows represent the directions of relation.,scaledwidth=44.0% ]  for newman s model , as shown in fig .[ fig1](a ) , each group is characterized by the connecting preference to node , no matter the node is contained by the group or not .the nodes belonging to the same group have similar connecting preference . as a result ,both assortative and disassortative structural regularities can be detected by this model .however , this model has no parameter to explicitly characterize the type of the identified structure .more importantly , this model may produce a result which is a mixture of several types of structure and thus in these cases the identified structure may provide confused information about the structural regularities .for example , for the network shown in fig .[ fig2 ] , nodes , , , , , are identified by this model as overlapped nodes shared by the two groups , denoted by circles and squares , although these nodes only have connections to one of the two groups .  for ren s model , as shown in fig .[ fig1](b ) , the two end nodes of each edge is assumed to be from the same group . as a result ,only the assortative structure ( community ) can be detected using this model .note that , for this model , no edge is allowed to connect different groups .the relationship between communities is instead reflected by the overlapped nodes .   ranging from for the smallest nodes to for the largest.,scaledwidth=44.0% ]  for our model , it essentially is a kind of stochastic blockmodel , in which the relationships among different node groups are explicitly modeled by the block matrix . in this way, our model possesses the advantages of both newman s model and ren s model and overcomes the shortcoming of these two models .  on one hand , through learning the matrix according to observed network data , various types of structural regularities can be explored by our model . the type of the identified structure is indicated by the matrix .specifically , when the matrix is an identity matrix , the identified structural regularity corresponds to an obvious community structure .meanwhile , multipartite or anti - community structure is revealed when the estimated model parameter is an anti - diagonal matrix with all the anti - diagonal elements being .for other types of structure such as core - periphery structure and hierarchical structure , the form of is the same to the block matrix in traditional block models  .  on the other hand , using the matrix , our model discards the assumption of ren s model that two end nodes of one edge are required to be from the same community . in this sense , ren s model is a special case of our model . in addition , our model also provides several other flexibility . by representing the centrality of nodes in group from two different perspective respectively according to the outgoing edges and incoming edges ,our model can detect more broad range of structural regularities which is out of the capability of other models .this will be shown later in the subsequent section . moreover , our model can be further generalized by not requiring the matrix be a square matrix .finally , we compare our model to two recently proposed stochastic models for community detection  .firstly , both our model and karrer s model  are stochastic block model where a block matrix is adopted to characterize the connecting patterns among groups .the main difference between these two models lies in that karrer s model is designed to detect disjoint structural regularities while our model is for fuzzy structural regularities .this difference is reflected by the definition of the model parameters and in our model and the definition of the model parameter in karrer s model .in addition , our model differentiates the out - edges from in - edges of nodes while karrer s model does not .secondly , similar to ren s model , ball s model  focused on the community structure while our model can uncover multiple types of structural regularities .in this section , we demonstrate the effectiveness of our model at exploring the structural regularities of networks by experiments on several real world or artificial networks with various types of intrinsic structural regularities . then we discuss the model selection issue , i.e. , how to determine the optimal number of groups .the test network is the famous karate club network constructed by zachary .this network characterizes the acquaintance relationship between members of a karate club in an american university .a dispute arose between the club s administrator and its principal karate teacher , and as a result the club eventually split into two smaller clubs , centered around the administrator and the teacher respectively .the network and its fission are depicted in figure [ fig2 ] .the administrator and the teacher are represented by nodes and respectively .  by setting the group number , we fit our model to the karate club network data .the resulted matrix is a identity matrix , indicating that the obtained structure is community structure .[ fig2 ] shows the two groups found by our model with the expectation - maximization method . as shown in fig .[ fig2 ] , the shades of the nodes in the figure represent the values of . ] , where group is the left group .as we can see , our model assigns most of the nodes strongly to one group or the other .actually , all but nodes are assigned to one of the groups ( black and white nodes in the figure ) . if we simply divide the nodes into two disjoint groups by assigning each node to the group according to the belong coefficients , the resulting groups perfectly correspond to the real split of the club .in addition , table [ tab1 ] gives the belonging coefficient of the overlapped nodes which are shared by the two groups .these overlapped nodes are nodes , , , , , .note that these overlapped nodes are often misclassified by traditional partition - based community detection methods . for comparison, we also gives the mixed membership of these six nodes according to newman s mixture model and ren s model .as we can see , our model and ren s model produce the same results , which is attributed to the fact that ren s model is a special case of our model .however , newman s model behave very differently from the other two models .actually , for newman s model , another nodes are also assigned to two groups , e.g. , nodes , .such a result is counterintuitive to the real structure of this network . as a conclusion , our model performs better than newman s model at detecting the overlaps between groups .ren s model can only detect community structure while our model can detect other types of structural regularities as illustrated in the following test ..[tab1 ] mixed membership of overlapped nodes . [ cols="<,<,<,<",options="header " , ]   now we illustrate the detection of multipartite or anti - community structure according to our model .the test network is the adjacency network of english words taken from ref . . in this network ,the nodes represent commonly occurring adjectives and nouns in the novel _ david copperfield _ by charles dickens , with edges connecting any pair of words that appear adjacent to each other at any place in the text .generally , adjectives occur next to nouns in english .thus most edges in the network connect an adjective to a noun and the network is approximately bipartite , i.e. , this network possesses anti - community structure .this can be seen clearly in fig .[ fig3 ] , where the adjectives and nouns are respectively represented by circles and squares .  fitting our model to this network with , the resulted is a transposed identity matrix , indicating that the identified structure is bipartite structure .the obtained two groups and node memberships are shown by the shades of nodes as shown in fig .we can see that most nodes are assigned to only one group , although there are several ambiguous cases corresponding to the nodes with intermediate shades .if we assign each node to its most preferred group , the resulted two disjoint groups well separate the adjectives from the nouns .in fact , of all the nodes are correctly classified .this accuracy is the same to the result given by newman s mixture model .  as a comparison, we also apply ren s model to this network by setting the group number being .only nodes of all the nodes are correctly classified , similar to the accuracy of random assignment .the ineffectiveness of ren s model at this network is attributed to that ren s model presumes the existence of community structure in the network while the intrinsic structural regularity is bipartite structure .     in the previous tests , we have demonstrate that our model can be used to detect both the assortative structure ( i.e. , community structure ) and the disassortative structure ( i.e. , multipartite structure ) without being told that which type of structural regularities exists in the target networks .now we will further show that our model can also detect other type of structure which can not be revealed by competing models .we consider the schematic network depicted in fig .[ fig4](a ) .this network is constructed according to the rules in fig .[ fig4](b ) .intuitively , according to the outgoing edges in this network , the nodes can be divided into two groups : and . meanwhile , according to the incoming edges , the nodes of this network belong to another two groups : and .we apply newman s model , ren s model and our model to this schematic network . limited by the assumptions of models , both newman s model and ren s modelfail to uncover the intrinsic structural regularity according the construction rules . for our model ,the flexibility of model assumption enables it to accurately detect such type of structure .specifically , by fitting our model to this network , the obtained reveal the two groups indicated by the outgoing edges while the reflect the two groups indicated by the incoming edges .   a schematic network .the directed edges are placed according to the rules described the right table.,title="fig:",scaledwidth=90.0% ] +   a schematic network .the directed edges are placed according to the rules described the right table.,title="fig:",scaledwidth=70.0% ] +   in the previous tests , we need to specify the group number before fitting our model to network . however , the group number is unknown a prior for many cases .thus it is helpful to give a criterion to determinate the appropriate group number for given network .this task is known as the model selection issue in statistics .we deal with this problem by using minimum description length principle , which is also used to handle the model selection issue in ren s model .  according to minimum description length principle , the required length to describe the network datais composed of two parts .the first part describes the coding length of the network using our model .this coding length is for directed network and for undirected network .the second part gives the length for coding model parameters .this part is for directed network and for undirected network . in this way ,the optimal is the one which minimizes the total description length .  as tests , we consider two real world networks with prior knowledge of the intrinsic group numbers .these two networks are respectively the journal citation network constructed in ref . and the american football team network described in ref . . in the journal citation network, each node corresponds to a journal and all the journals are from four different fields : multidisciplinary physics , chemistry , biology and ecology .journals from the same field are more likely connected by citation relation . for the football network , nodes represent the teams respectively belonging to conference and generally games are more frequent between members of the same conference than between teams of different conferences .  as shown in fig .[ fig5 ] , the number of intrinsic groups is correctly identified for the journal citation network .however , for the football network , is the optimal number of groups while the intrinsic number is . by checking the found node groups , we find that only node groups have their identities , i.e. , each group contains at least one node after assigning nodes to their most preferred groups according to the obtained belonging coefficient or .this indicates that the appropriate group number is for the football network .in fact , many well - known community detection methods also identify communities .in this paper , we have studied the exploration of intrinsic structural regularities in network using a general stochastic blockmodel . without prior knowledge ,our model can not only detect broad types of intrinsic structural regularities , but also can learn the type of identified structure directly from the network data .tests on a number of artificial and real world networks demonstrate that our model outperforms the state - of - the - art models at shedding light on the structural features of networks .the flexibility enables our model be an effective way to reveal the structural regularities of network and further to help us understand the relationship between structure and function of network . for potential application , our model can be used to predict the emergence or vanishing of edges in network . as future work, we will generalize our model by releasing the requirement that the block matrix is a square matrix and investigate the possible applications of the more flexible model .this work was funded by the national natural science foundation of china under grant number , , and .this work was also partly funded by the national high - tech r&d program of china ( the 863 program ) with grant number 2010aa012500 .the authors thank alex arenas , mark newman , and santo fortunato for providing network and other data used in this paper .

Abstract: in this paper , we consider the problem of exploring structural regularities of networks by dividing the nodes of a network into groups such that the members of each group have similar patterns of connections to other groups . specifically , we propose a general statistical model to describe network structure . in this model , group is viewed as hidden or unobserved quantity and it is learned by fitting the observed network data using the expectation - maximization algorithm . compared with existing models , the most prominent strength of our model is the high flexibility . this strength enables it to possess the advantages of existing models and overcomes their shortcomings in a unified way . as a result , not only broad types of structure can be detected without prior knowledge of what type of intrinsic regularities exist in the network , but also the type of identified structure can be directly learned from data . moreover , by differentiating outgoing edges from incoming edges , our model can detect several types of structural regularities beyond competing models . tests on a number of real world and artificial networks demonstrate that our model outperforms the state - of - the - art model at shedding light on the structural features of networks , including the overlapping community structure , multipartite structure and several other types of structure which are beyond the capability of existing models .