the orthogonal procrustes problem ( opp ) is the least square problem on the stiefel manifold .the opp originates from the factor analysis in psychometrics during 1950s and 1960s .the major purpose is to determine an orthogonal matrix that rotates the factor matrix to best fit some hypothesis matrix .the balanced case of the opp was surveyed in multiple introductory textbooks such as .cccc + & rsr &lsr &sp + order of the & & & + complexity & & & + & lr &eb &gpi ( our ) + order of the & & & + complexity & & & +  recently , due to the wide applications of the orthogonal regression in computer science , see , solving the unbalanced opp ( uopp ) is under increasing concern .multiple approaches are proposed to solve uopp such as the expansion balanced algorithm ( eb ) , the right hand side and the left hand side relaxation ( rsr ) , ( lsr ) , the successive projection ( sp ) and the lagrangian relaxation ( lr ) . in , the eb method employs the expanded balanced opp as its objective function . in and respectively , the rsr and the lsr approaches update the solution row by row or column by column iteratively based on solving the least square regression with a quadratic equality constraint ( lsqe ) . in ,the sp method updates the solution column by column by virtue of the projection method combined with correction techniques ( pmct ) discussed by , which is efficient to solve lsqe . in ,the lr method solves uopp by selecting different lagrangian multipliers .all the approaches mentioned above could converge to the solution of uopp successfully , whereas they deal with more complex procedures , which represent high orders of complexity .furthermore , all these methods initialize the parameters deliberately to optimize their proposed algorithms .last but not least , all these approaches are unable to deal with a more general problem known as the quadratic problem on the stiefel manifold ( qpsm ) .  to address the referred deficiencies , we derive a novel generalized power iteration method ( gpi ) for qpsm in order to efficiently solve the orthogonal least square regression ( olsr ) and uopp with a random initial guess and concise computational steps . in sum , the proposed gpi method can deal with a more general problem known as qpsm than other approaches .furthermore , the experimental results show that the proposed gpi method not only takes much less cpu time for the convergence but becomes more efficient dealing with the data matrix of large dimension as well .* notations : * for any matrix , frobenius norm is defined as , where is the trace operator . for any positive integer , denotes a identity matrix .the power iteration method is an iterative algorithm to seek the dominant eigenvalue and the related eigenvector of any given symmetric matrix , where the dominant eigenvalue is defined as the greatest eigenvalue in magnitude .the power iteration can be performed as the following steps :  \1 .random initialize a vector , which has a nonzero  component in the direction of the dominant eigenvector .update .calculate .update .iteratively perform the step 2 - 4 until convergence .+ the power iteration could be further extended to the orthogonal iteration ( also called subspace iteration or simultaneous iteration ) method to find the first dominant eigenvalues and their associated eigenvectors for the given matrix .the orthogonal iteration method could be described as the following iterative algorithm :  \1 . initialization .random initialize .update .calculate via the compact qr factorization of   , where and .update .iteratively perform the step 2 - 4 until convergence .+ apparently , the orthogonal iteration method above indicates a normalization process , which is similar as the normalization in the power iteration method . when the matrix is positive semi - definite ( psd ) , the orthogonal iteration method is equivalent to solving the following optimization problem therefore , the orthogonal iteration method is equivalent to the following steps under the psd matrix :  \1 .random initialize .update .calculate via the compact svd method of ,  where , and .update .iteratively perform the step 2 - 4 until convergence .+ from the observation , the solution of the above algorithm as differs from the solution of the orthogonal iteration method as by the form , where .however , the difference between the solutions of these two algorithms does nt affect the objective value of the problem ( [ e000 ] ) due to the following derivation stiefel manifold is a set of the matrices , which have orthonormal columns as .  in this section , a novel approachis derived to unravel the following quadratic problem on the stiefel manifold ( qpsm ) as where , and the symmetric matrix . in order to solve the problem ( [ e01 ] ) , qpsm in ( [ e01 ] ) can be further relaxed into where .the relaxation parameter is an arbitrary constant such that is a positive definite ( pd ) matrix . instead of the method of the lagrangian multipliers to deal with an optimization problem with orthogonal constraints, one may use a geometric optimization algorithm tailored to the stiefel manifold , such as , for example , the one surveyed in .accordingly , the lagrangian function for the problem ( [ e02 ] ) can be written as   from eq .( [ e09 ] ) , we could obtain the kkt condition for the problem ( [ e02 ] ) as  + which is difficult to solve directly .thus , motivated by and the power iteration method mentioned in section 2 , we could propose the following iterative algorithm :  \1 .random initialize such that .update .calculate by solving the following problem   \4 .update .iteratively perform the step 2 - 4 until convergence .+ besides , a closed form solution of the problem ( [ e05 ] ) can be achieved by the following derivation .suppose the full svd of is with , and , then we have where with and being the -th elements of the matrix and , respectively .note that , thus . on the other hand , since is a singular value of the matrix .therefore , we have   apparently , the equality holds when . that is to say , reaches the maximum when the matrix \in\mathbb{r}^ { k\times m}$ ] . recall that , thus the optimal solution to the problem ( [ e05 ] ) can be represented as \mathbb{v}^t.\label{eq1}\ ] ] since eq .( [ eq1 ] ) is based upon the full svd of the matrix , eq .( [ eq1 ] ) can be rewritten as via the compact svd of the matrix , where with , and .m0.32m0.32m0.4 & & +  based on the above analysis , the generalized power iteration method ( gpi ) can be summarized in the algorithm [ alg1 ] .we will prove that the proposed algorithm [ alg1 ] converges monotonically to the local minimum of qpsm ( [ e01 ] ) .* input : * the symmetric matrix and the matrix . *initialize * a random satisfying and such that is a positive definite matrix .update .calculate via the compact svd method of where , and .update .iteratively perform the step 3 - 5 until the algorithm converges .step 5 of algorithm [ alg1 ] is an instance of a class of methods , called manifold retractions , to update a matrix on the stiefel manifold , that were discussed in details in .if the symmetric matrix is positive definite ( pd ) , then where and are arbitrary matrices.[lem1 ]  proof : since the matrix is positive definite ( pd ) , we could rewrite via cholesky factorization .therefore , we have the following proof for lemma [ lem1 ] as   the algorithm [ alg1 ] decreases the value of the objective function in ( [ e01 ] ) monotonically in each iteration until it converges .[ thm2 ]  proof : suppose the updated is in the algorithm [ alg1 ] , then we have since is the optimal solution of the problem ( [ e05 ] ) .based on the fact that , eq . ( [ e06 ] ) can be further illustrated as   based on lemma [ lem1 ] and eq .( [ e07 ] ) , we could infer that which indicates that the algorithm [ alg1 ] decreases the objective value of qpsm in ( [ e01 ] ) in each iteration until the algorithm converges .  the algorithm [ alg1 ] converges to a local minimum of the qpsm problem ( [ e01 ] ) .  proof : since the algorithm [ alg1 ] performs based on solving the problem ( [ e05 ] ) in each iteration , the lagrangian function for the solution of the algorithm [ alg1 ] can be represented as therefore , the solution of the algorithm [ alg1 ] satisfies the following kkt condition   generally speaking , the matrix will be updated by in each iteration under the algorithm [ alg1 ] . since the algorithm [ alg1 ] converges to the optimal solution i.e. due to theorem [ thm2 ] , eq .( [ e012 ] ) can be further formulated by substituting as by comparing eq .( [ e010 ] ) and ( [ e013 ] ) , we could draw the conclusion that the solution of the algorithm [ alg1 ] and the problem ( [ e02 ] ) satisfy the same kkt condition .therefore , the algorithm [ alg1 ] converges to a local minimum of qpsm ( [ e01 ] ) since the problems ( [ e01 ] ) and ( [ e02 ] ) are equivalent .  m0.32m0.32m0.4 & & +  besides , the problem ( [ e05 ] ) has an unique solution under full column - rank matrix due to the uniqueness of the svd method .on the other hand , the experimental results in section 5 represent that the proposed gpi method uniformly converges to the same objective value with a large amount of random initial guesses .based on the unique solution of the problem ( [ e05 ] ) and the associated experimental results , it is rational to conjecture that the proposed gpi method converges to the global minimum of qpsm .the orthogonal least square regression ( olsr ) can be written as where the data matrix and the hypothesis matrix with .moreover , is the regression matrix and is the bias vector .obviously , is free from any constraint . by virtue of the extreme value condition w.r.t . , we can derive as   by substituting the above result as , eq .( [ e00 ] ) can be simplified to the following form as where .accordingly , the problem ( [ eee ] ) can be further reformulated into in which apparently , eq . ( [ eeee ] ) is in the exact same form as qpsm in ( [ e01 ] ) .therefore , olsr in ( [ e00 ] ) can be solved via the algorithm [ alg1 ] .   with , and , we name the optimization problem    \1 .balanced orthogonal procrustes problem ( opp ) if and only if .  \2 . unbalanced orthogonal procrustes problem ( uopp ) if and only if .[ 1 ] especially when serves as a column vector , the problem ( [ e1 ] ) degenerates to which is known as the least square problem with a quadratic equality constraint ( lsqe ) .   to solve the balanced opp ( ) ,we could expand eq .( [ e1 ] ) into  + which is same as the problem ( [ e05 ] ) with treating .thus , the balanced opp has the analytical solution of the closed form ( [ eq1 ] ) .   when , uopp ( [ e1 ] ) can be expanded into   denote and , then eq .( [ e3 ] ) is in the exact same form as qpsm ( [ e01 ] ) .based on the algorithm [ alg1 ] , the algorithm [ alg2 ] can be proposed to converge to a local minimum of uopp monotonically due to the theoretical supports proved in section 3 .* input : * the matrix and the matrix where . *initialize * and such that and the matrix is positive definite , respectively .* while * not converge * do * update matrix .calculate and via the compact svd of as .update .* end while * * return * .[ alg2 ]  m0.45m0.5 & +  cccccccc +  & rsr&lsr&sp&lr&eb&gpi(our ) + &cpu time&64.940s&23.426s&2.386s&0.541s&0.337s&*0.228s * + &cpu time&136.020s&21.635s&3.221s&1.134s & 0.347s&*0.226s * + &cpu time&229.851s&20.560s&5.054s&1.806s & 0.445s&*0.273s * + & rsr&lsr&sp&lr&eb&gpi(our ) + &cpu time&-&842.849s&132.232s&3.869s&11.440s&*1.290s * + &cpu time&-&851.231s&196.761s&5.180s&12.534s&*1.434s * + &cpu time&-&860.746s&260.132s&7.700s&12.625s&*1.575s * +  generally speaking , qpsm can not be reformulated into uopp while uopp could always be rewritten into qpsm .therefore , the gpi method is more general than other approaches , which can only cope with uopp .based on the experimental results involved in the next section , the proposed gpi method takes much less time to converge to the solution of uopp .in this section , we analyze and report the numerical results of the generalized power iteration method ( gpi ) represented by both the algorithm [ alg1 ] and the algorithm [ alg2 ] .we randomly choose the test data matrix with normally distributed singular values .m0.5  besides , the computer we use is macbook air , whose cpu is 1.4 ghz intel core i5 , ram is 4 gb 1600 mhz ddr3 and operating system is os x yosemite 10.10.5 ._ case 1 : ( parameter dependence ) _ firstly , we try to investigate the gpi method in the algorithm [ alg2 ] via varying the relaxation parameter .suppose is the largest eigenvalue of , then we can let such that is a positive definite matrix , where is an arbitrary constant .\1 ) from the figure [ fig3 ] , we can further notice that although the convergence rate for the algorithm [ alg2 ] is inversely proportional to the value of , the relaxation parameter does not affect the uniform convergence of the gpi method .cccccccc + dimension & & & & & & & + & & & & & & & + & cpu & & & & & & + & time & & & & & & + & cpu & & & & & & + & time & & & & & & + & cpu & & & & & & + & time & & & & & & + & cpu & & & & & & + & time & & & & & & + & cpu & & & & & & + & time & & & & & & + & cpu & & & & & & + & time & & & & & & +  [ tab4 ]  _ case 2:(cpu time comparison for solving uopp ) _ secondly , we further investigate the proposed gpi method in the algorithm [ alg2 ] by comparing it with five existing approaches mentioned in section 1 as eb , rsr , lsr , sp and lr .  based on solving lsqe problem , rsr and lsr respectivelyupdate the solution row by row and column by column iteratively .eb utilizes the expanded balanced opp as the objective function .sp employs the projection method combined with correction techniques ( pmct ) .lr solves uopp by fixing different lagrangian multipliers .the proposed gpi method includes two terms as outside the loop and within the loop , whose orders of complexity are and , respectively . besides , these two terms have the highest orders of complexity for the proposed gpi method . besides, the order of the complexity for each method is shown in the table [ tab2.5 ] .the comparative results are based on fixing as the square matrix at first hand ( table [ tab3 ] ) and then extend to a more general case ( table [ tab4 ] ) afterwards .( mark in the table [ tab3 ] and the table [ tab4 ] represents that it takes too much time to record in the tables . )\1 ) from the figure [ fig4 ] , we notice that the existing methods as eb , rsr , lsr , sp lr and the proposed gpi method converge to the same objective value under the same input data .besides , our proposed gpi method converges faster than other approaches during iteration .\2 ) from the table [ tab2.5 ] , the proposed gpi method has the lowest order of complexity due to its succinct computational process to obtain the optimal solution . during the experiments , we observe that the iteration number for the lr method is usually very large for the convergence .thus , the time consumption for lr method is much larger than that for the proposed gpi method though orders of complexity for these two approaches seem close . besides, the gpi method becomes more efficient when ( the number of data ) is large .\3 ) from the table [ tab3 ] , the proposed algorithm [ alg2 ] ( gpi ) serves as the most efficient method under the square matrix case .\4 ) from the table [ tab4 ] , we can observe that lsr , sp and rsr are unable to compete with lr , eb and gpi due to the complex updating procedures including the expanded opp and solving lsqe .especially when the dimension increases , the superiority of our proposed gpi method would be more obvious ._ case 3:(cpu time comparison for solving lsqe ) _ finally , the projection method combined with correction techniques ( pmct ) is compared to the gpi method in the algorithm [ alg2 ] targeting at solving the least square regression with a quadratic equality constraint ( lsqe ) in ( [ ls ] ) .actually , solving lsqe ( [ ls ] ) is no different from solving uopp ( [ e1 ] ) under .\1 ) from the figure [ fig9 ] , we can notice that pmct and the algorithm [ alg2 ] ( gpi ) converge to the same objective value though in terms of the different patterns .\2 ) from the figure [ fig10 ] , the algorithm [ alg2 ] ( gpi ) takes much less time for convergence than pmct does .in this paper , we analyze the quadratic problem on the stiefel manifold ( qpsm ) by deriving a novel generalized power iteration ( gpi ) method . based on the proposed gpi method , two special and significant cases of qpsm known as the orthogonal least square regression and the unbalanced orthogonal procrustes problem are under further investigation . with the theoretical supports ,the gpi method decreases the objective value of the qpsm problem monotonically to a local minimum until convergence .eventually , the effectiveness and the superiority of the proposed gpi method are verified empirically . in sum, the proposed gpi method not only takes less cpu time to converge to the optimal solution with a random initial guess but becomes much more efficient especially for the data matrix of large dimension as well .green , b. : the orthogonal approximation of an oblique simple structure in factor analysis . psychometrika . * 17 * ( 1952 ) 429 - 440 .hurley , j. , cattell , r. : the procrustes program : producing direct rotation to test a hypothesized factor structure .behavioural science .* 6 * ( 1962 ) 258 - 262 .golub , g. h. , van loan , c. f. : matrix computations .the johns hopkins university press ( 1989 ) .thomas , v. : algorithms for the weighted orthogonal procrustes problem and other least squares problems : [ d ] .( 2006 ) ume university , sweden .souza , p. , leite , c. , borges , h. , fonseca , r. : online algorithm based on support vectors for orthogonal regression .pattern recognition letters * 34 * ( 2013 ) 1394 - 1404 .chu , m. , trendafilov , n. : the orthogonally constrained regression revisted .j. comput . graph. stat . * 10 * ( 2001 ) 746 - 771 .green , b. , goers , j. : a problem with congruence .the annual meeting of the psychometric society .monterey , california ( 1979 ) .park , h. : a parallel algorithm for the unbalanced orthogonal procrustes problem .parallel computing .* 17 * ( 1991 ) 913 - 923 .bojanczyk , a. , lutoborski , a. : the procrustes problem for orthogonal stiefel matrices .* 21 * ( 1999 ) 1291 - 1304 .zhang , z. , du , k. : successive projection method for solving the unbalanced procrustes problem .science in china : series a mathematics .* 49 * ( 2006 ) 971 - 986 .xia , y. , han , y. : partial lagrangian relaxation for the unbalanced orthogonal procrustes problem .* 79 * ( 2014 ) 225 - 237 .zhang , z. , huang , y. : a projection method for least square problems with quadratic equality constraint .* 25 * ( 2003)188 - 212 .journe , m. , nesterov , y. , richtrik , p. , sepulchre , r. : generalized power method for sparse principal component analysis .journal of machine learning research .* 11 * ( 2008 ) 517 - 553 .fiori , s. : formulation and integration of learning differential equations on the stiefel manifold .ieee transactions on neural networks .* 16 * ( 2005 ) 1697 - 1701 .kaneko , t. , fiori , s. , tanaka , t. : empirical arithmetic averaging over the compact stiefel manifold .ieee transactions on signal processing .* 61 * ( 2013 ) 883 - 894 .nie , f. , yuan , j. , huang , h. : optimal mean robust principal component analysis . in proc .icml . 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Abstract: in this paper , we first propose a novel generalized power iteration method ( gpi ) to solve the quadratic problem on the stiefel manifold ( qpsm ) as  along with the theoretical analysis . accordingly , its special case known as the orthogonal least square regression ( olsr ) is under further investigation . based on the aforementioned studies , we then cast major focus on solving the unbalanced orthogonal procrustes problem ( uopp ) . as a result , not only a general convergent algorithm is derived theoretically but the efficiency of the proposed approach is verified empirically as well .   quadratic problem , stiefel manifold , power iteration , procrustes problem , orthogonal least square regression .


many human systems are organized within pyramidal hierarchies including large corporations , universities , armies , trade unions or political parties . while many structural organizational frames are possible in practice , two extreme and opposite cases can be formally singled out as follows . on the one sidestands the dictatorship pyramid initiated at the top of the hierarchy .decisions go down towards the base level by level .it is totally directed and deterministic with a top - down splitting dynamics . on the other sideis the perfectly democratic pyramid built up also level by level , but now from the bottom upwards to the top .the dynamics is bottom - up aggregating with the use of local majority rules . in this paperwe investigate the second case . for a review on voting systems see .more precisely we study a generic model , which exemplifies a dynamical paradox induced by the use of bottom - up majority rules .considering hierarchical organizations , local democratic votes are thus shown to produce a self - neutralization of a growing internal opposition while climbing up the hierarchy from bottom to the top . within a very simple probabilistic frame we show how a top leadership can hamper democratically very large amplitude changes at the hierarchy bottom from a wide support to a huge opposition .although we are illustrating the dynamical paradox in terms of voting process , the model is quite generic and may apply to many different situations where bottom - up majority rules aggregation are used .the frame may be a political group , a firm , a social organization .using tools and concepts from statistical physics , the voting process is found to obey a threshold dynamics with a deterministic top outcome .accordingly as long as the opposition weight is below some critical threshold at the bottom , it is found to fall off quickly to reach zero after only a few hierarchical levels .it thus makes the current leadership very stable . in some cases ,the opposition can grow steadily from few percents up to seventy seven percents with not a single change at the top level .however and in contrast to such a stability , in the vicinity of the threshold , from one election to another less than a one percent additional shift at the bottom level in favor of the opposition can drive a drastic and brutal change at the top .seventy percents increase in bottom opposition has no effect on the elected organization president while less than a one percent change topples it democratically at once .  in this paper, we consider the situation where members of an organization may support either one of two policies denoted respectively a and b. at the bottom , once each member has decided on which policy , a or b it supports , elections are set via the formation of small groups of people whose members are chosen randomly .it means that members of a voting group are not asked about their choice before the voting .then , each group elects a representative either a or b according to the majority of its members .all these elected people constitute the first hierarchical level of the organization .the voting process is then repeated now from the first level to build the second one . at each step up , stands a smaller number of representatives with a single one at the last level , the president .one illustration of the process is shown in figure ( 1 ) in the case of groups of size 3 .  accordingly , having an initial support above the critical threshold guarantees to win the top level provided there exists some minimum number of hierarchical levels .the value of the critical threshold to power is a function of the size of the voting groups . for all groups of odd sizes .however in case of groups of even sizes , .this asymmetry arises from the existence of well kwon inertia of power and status quo . within our model , in the case of a local tie in an even size group , it is the former ruler who is reelected .  for groups of size 4it makes the threshold to jump at for the opposition .it means that if b is already in power , the opposition a needs to win more than support at the bottom level to reach the top of the hierarchy .simultaneously , the threshold to stay in power shrinks to to the ruling party .in addition , in cases of even size groups , the mechanism of self - elimination is even faster than in the case of odd groups .for instance , a proportion of a supporters is self - eliminated within 4 levels . moreover, the number of levels to reach a deterministic outcome decreases with increasing the size group .the structure of the paper is as follows . in the next section we present the main frame of the voting model in the case of 3-person groups .a critical threshold to deterministic power at the top is found at of initial support at the bottom level .the self - elimination of the minority occurs within only a few voting levels .section 3 introduces some natural bias in the voting rule with a tip to the ruling party in terms of inertia of power .illustrated in the case of even 4-person groups it yields an election in favor of the ruling party in case of a local tie 2a-2b .such a bias is found to shift the value of the critical threshold to power from to for the opposition . in parallel , it shrinks it from to for the ruling party .group size effects are then analyzed in section 4 .analytic formulas are derived in section 5 .given an a initial support , the number of voting levels necessary to their self - elimination is calculated .the results are then turned in a more practical perspective for hierarchical organizations in section 6 .some visualization of a numerical simulation is shown in section 7 extension to 3 competing group is sketched in section 8 .the last section discusses possible applications of our generic model to describe in part , the historical last century collapse of eastern european communist parties .the results may also shed a new light on management architectures as well as on alert systems .we consider the simplest case with a population divided between individuals supporting either one of two policies a and b. the respective proportions at the organization bottom denoted level-0 are respectively and .each member does have an opinion .neither strategy nor interactions are explicitly included .it means that we do not address the question of why policy a has a support at the bottom .people do not neither choose their respective voting group to optimize their weight since the groups are formed randomly at each level of the hierarchy .it is worth to stress that although we are illustrating the model in terms of voting process , it is quite generic and may apply to many different situations where bottom - up majority rules aggregation are used .the frame may be a political group , a firm , a social organization .we start studying the case of groups built of 3 persons randomly selected from the population .it could correspond to home localization or working place .each group then elects a representative using a local majority rule .groups with either 3 a or 2 a elect an a. otherwise it is a b who is elected .therefore the probability to have an a elected from the bottom level is , where denotes the voting function , here a simple majority rule .the same process of group forming is repeated within level-1 .the elected persons ( from level-0 ) form groups which in turn elect new higher representatives .the new elected persons constitute level-2 .the process can then be repeated again and again .the probability to have an a elected at level ( n+1 ) from level - n is , where is the proportion of a at level - n .the analysis of the voting function exhibits the existence of 3 fixed points , and .the first one corresponds to not one a elected .the last one represents the totalitarian situation where only a are elected .both and are stable fixed points . in contrast is unstable .it determines indeed the threshold for flowing towards either full power ( with ) or to total disappearance ( with ) .starting from leads to the first case while yields the second .see figure ( 2 ) .  on this basiswe see that majority rule voting produces the self - elimination of any a proportion as long as the initial support is less than ( ) .however this democratic self - elimination requires a sufficient number of voting levels to be completed .  at this stagethe instrumental question is to determine the number of levels required to ensure full leadership to the initial larger party . to make sensethe level number must be small enough , most organizations having only a few levels ( less than 10 ) .  to illustrate the voting dynamics above ,let us calculate the representativity flow starting , for instance , from .we get successively , , , , , down to and . within 8 levels of the populationis self - eliminated .nevertheless the overall process preserves the democratic character of majority rule voting .it is the bottom leading party ( more than ) which eventually gets for sure the full leadership of the organization top level .it is worth to notice the symmetry of situation with respect to a and b parties .the threshold to full power is the same ( ) for both of them .from the above analysis , to turn down a top leadership requires to have more than at the bottom which is a fair constraint . however from real life situations , to produce a change of leadership is much more difficult .a simple majority appears often not to be enough .many institutions are indeed build to strengthen some stability .frequent political changes are not perceived as good for the organization . on this basis ,a tip is often given to the ruling party . to be in charge ,gives some additional power which breaks the symmetry between the two parties .for instance , giving one additional vote to the committee president or allowing the president do designate some committee members .  using the democratic statement to change things ,you need a majority " produces indeed a strong bias in favor of current rulers . to exemplify it , we consider even size groups .again restricting to the simplest case , it means 4 people groups . assuming the b are the ruling party , the a need either 4 or 3 a in a given group to take over . the tie case 2a-2b votes for a b ( see figure ( 3 ) ) .  in going from 3 to 4 person size groups ,the salient new feature is indeed the existence of 2a-2b configurations for which there exists no majority . in most social situations no decision implies de facto , no change .there exists a bias in favor of the rulers which now makes the voting function asymmetric .the probability to get an a elected at level is , where is as before the proportion of a elected persons at level - n .in contrast , for a b to be elected the probability is , where the last term embodies the bias in favor of b. from eqs ( 3 ) and ( 4 ) the stable fixed points are still 0 and 1 .however , the unstable one is now drastically shifted to , for the a. it makes the a threshold to power at about as shown in figure ( 4 ) .simultaneously the b threshold to stay in power is about making both situations drastically different .now , to take over power , the a need to go over of initial bottom support while to stick to power the b only need to keep their support above .in addition to the asymmetry the bias makes the number of levels to democratic self - elimination even smaller than in the precedent case ( 3-group size ) . starting again from we now get , and . instead of 8 levels , 3 are enough to make the a disappear .  to illustrate how strong this effect is , let us start far above with for instance .the associated voting dynamics becomes , , , , , and . within only 6 levels , of a population is thus self - eliminated .using an a priori reasonable bias in favor of the b turns a majority rule democratic voting to a dictatorship outcome . to get to power the a must pass over of overall support which is almost out of reach in any normal democratic two party situationup to now we have considered very small groups .but many organizations have larger groups .extending the above cases to groups of any size r is indeed straightforward with the equations getting a bit more complicated .however the main features remain unchanged under size changes .  for a r - size groupthe voting function which accounts for all configurations with a majority of a supporters becomes , where for odd r and for even r which thus accounts for the bias in favor of b.  the two stable fixed points and are unchanged .they are size independent . in the case of odd sizes ,the unstable fixed point is also unchanged with . on the opposite , for even sizes, the asymmetry between the threshold values for respectively rulers and non rulers weakens with increasing sizes . for the a threshold it is for size 4 and it decreases asymptotically towards for r .it is for size 6 , and stays always larger than making hard yet to pass the barrier for the opponents .it is known that in democratic countries a few percent difference between two candidates is seen as huge .  in parallel increasing group sizesreduces the number of levels necessary to get to the stable fixed points as shown in figure ( 5 ) . for any application to real social structure ,combinations of different sizes should be considered since it is not the same organization which prevails at each level .it is left for future work .given an initial support , we want to calculate the corresponding value for a support after n voting levels as a function of .accordingly we expand the voting function around the unstable fixed point , where with . rewriting the last equation as we can then iterate the process to get , from which we get ,   from eq .( 10 ) two different critical numbers of levels and can be obtained .the first one corresponds to and the second one to .putting in eq .( 10 ) gives , which is defined only for showing that only below can the proportion decrease to zero . on the other hand putting in the same eq .( 10 ) gives , which is now defined only for since , showing that only above can the proportion increase to one .though above expansions are a priori valid only in the vicinity of , they turn out to be rather good estimates even in getting close to the two stable fixed points 0 and 1 . while taking the integer part of eqs .( 11 ) and ( 12 ) the rounding is always to the larger value .however once organizations are set , they are usually not modified .therefore within a given organization the number of hierarchical levels is a fixed quantity . along this line , to make a practical use of above analysis the question of how many level are needed to eliminate a party ? "must be turned to ,  given n levels what is the minimum overall support + to get full power for sure ? " .  or alternatively , for the ruling group ,  given n levels what is the critical overall support of the competing party below which it always self - eliminates totally ? " ,  which means no worry for the ruling party about its current policy .  to implement this operative question , we rewrite eq .( 10 ) as , it yields two critical thresholds .first one is the disappearance threshold which gives the value of support under which the a disappear for sure at the top level of the n - level hierarchy .it is given by eq .( 13 ) with , in parallel gives the second threshold above which the a get full and total power . from eq .( 13 ) , there exists now a new regime for .in there , a neither disappears totally nor gets to full power .it is a coexistence region where some democracy is prevailing since results of the election process are only probabilistic .no party is sure of winning making alternate leadership a reality .however as seen from eq .( 15 ) , this democratic region shrinks as a power law of the number n of hierarchical levels .a small number of levels puts higher the threshold to a total reversal of power but simultaneously lowers the threshold for non - existence .above formulas are approximates since we have neglected corrections in the vicinity of the stable fixed points .however we found they give the right quantitative values by adding 1 to and 2 to  .  to get a practical feeling of what eqs .( 14 ) and ( 15 ) mean , let us illustrate the case where and . considering 3 , 4 , 5 , 6 and 7 level organizations , is equal to respectively , , , and . in parallel equals , , , and .these series emphasize drastically the totalitarian character of the voting process .to exhibit the strength of the phenomena , we show some snapshots of a numerical simulation .the two a and b parties are represented respectively in white and black squares , with the bias in favor of the black ones , i. e. , a tie 2 - 2 votes for a black square .a structure with 8 levels is shown .we can see on each picture , how a huge white square majority is self - eliminated .written percentages are for the white representation at each level . the time " and generations" indicators should be discarded .figure ( 6 ) shows a case where of the people at the bottom are in support of the b ( white ) , thus a bit over the expected democratic threshold to take over. however , 3 levels higher in the hierarchy no white squares appear .the bottom majority has self - evaporated .figure ( 7 ) shows the same population with now a substantial increase in b(white ) support with a proportion of , now rather far than . and yet , after 4 levels no more white ( b ) square is found .figure ( 8) shows the situation where the b ( white ) support has climbed up to the huge value of . but again 6 levels higher a a ( black ) is elected with certainty though its bottom support is as low as support ( black ) . finally figure ( 9 ) shows an additional very weak increase in b ( white ) support , putting the actual support at , which in turn prompts an elected b ( white ) president .some insight about the often observed blindness of leaderships towards huge and drastic calls for changement from the botton level of an organization can be gained from the simulation .indeed it shows how and why a president who would get some information about the possible disagreement with a policy can not account for it since it has to check it from what reaches it from the bottom of the hierarchy . as seen from figure ( 7 ) while the opposition is at already at a pick of the president gets of totally satisfied votes from the two levels below it and overwhelming satisfied in the two more lower levels , so why should it make any policy change ?up to now we have treated very simple cases to single out some main trends produced by democratic voting while repeated over several hierarchical levels . in particular the existence of critical thresholdswas found to be instrumental in predicting the voting outcome. moreover these thresholds are not necessarily symmetric with respect to both competing parties . in the 4-group caseit can be for the opposition and for the rulers .such asymmetries are indeed always present one way or another .  in addition , most real situations involve more than two competing groups .let us consider for instance the case of three competing groups a , b and c. assuming a 3-group case , the abc configuration is now unsolved since it has no majority the same way as in the case of the two party configuration aabb with groups of size 4 . for the aabb case we made the bias in favor of the ruling group . for multi - group competition typically the biasis decided in each configuration and from the various parties agreement .usually the two largest parties , say a and b are hostile to each other , while the smallest one c would compromise with either one of them . along this line ,the abc configuration elects a c in a coalition with either a or b. in such a case , we need 2a or 2b to elect respectively an a or a b. otherwise a c is elected.  therefore the elective functions for a and b are the same as for the a in the two party 3-group model .it means that the critical threshold to full power to a and b is . in other words for initial support to both a and b , less than the c get full power .the required number of levels is obtained from the 3-group formula ( equations ( 11 ) and ( 13 ) with ) .let us illustrate what happens for initial support , for instance of for both a and b. the c are thus left with .we will have the series for respectively a , b and c : , , at the first level ; , , at the second level ; , , at the third level ; , , at the forth level ; , , at the fifth level ; and , , at the sixth level giving total power to the c minority within only 6 levels .it is possible to generalize the present approach to as many parties as wanted .the analysis becomes much more heavy but the mean features of voting flows towards fixed point are preserved .moreover power will go even more rarely to the largest groups as seen with the above abc case .to conclude we emphasize on the very generic character of our model which allows to consider many different applications .one could be related to last century s auto - collapse of eastern european communist parties . up to this historical and drastic event , communist parties seemed eternal .once they collapsed many explanations were based on some hierarchical opportunistic change within the various organizations , with in addition for the non russian ones , the end of the soviet army threat .maybe our hierachical model can provide some different new insight at such a unique event .communist organizations are indeed based , at least in principle , on the concept of democratic centralism which is a tree - like hierarchy .suppose that the critical threshold to power was of the order of like in our size 4 case .we could then consider that the internal opposition to the orthodox leadership did grow a lot and massively over several decades to eventually reach and pass the critical threshold .once the threshold was passed , then we had the associated sudden rise to the leadership levels of the internal opposition .therefore , what looked at a sudden and punctual collapse of eastern european communist parties could have been indeed the result of a very long and solid phenomenon inside the communist parties. such an explanation does not oppose to the very many additional features which were instrumental in these collapses .it only singles out some trend within the internal mechanism of these organizations which makes them extremely stable .the results may also shed a new light on management architectures as well as on alert systems . however in order to avoid any confusion between political issues and the management of firms and organizations , it should be noticed that if there exist some similarities between a government as an organization while some constraints are inevitable in management itself , both realities are quite different in nature and objectives .

Abstract: the stability of a leadership against a growing internal opposition is studied in bottom - up hierarchical organizations . using a very simple model with bottom - up majority rule voting , the dynamics of power distribution at the various hierarchical levels is calculated within a probabilistic framework . given a leadership at the top , the opposition weight from the hierarchy bottom is shown to fall off quickly while climbing up the hierarchy . it reaches zero after only a few hierarchical levels . indeed the voting process is found to obey a threshold dynamics with a deterministic top outcome . accordingly the leadership may stay stable against very large amplitude increases in the opposition at the bottom level . an opposition can thus grow steadily from few percent up to seventy seven percent with not one a single change at the elected top level . however and in contrast , from one election to another , in the vicinity of the threshold , less than a one percent additional shift at the bottom level can drive a drastic and brutal change at the top . the opposition topples the current leadership at once . in addition to analytical formulas , results from a large scale simulation are presented . the results may shed a new light on management architectures as well as on alert systems . they could also provide some different insight on last century s eastern european communist collapse . +  + keywords : local majority rules , bottom - up hierarchies , statistical physics


the integration by fractional expansion technique ( ibfe ) constitutes a rather simple method that allows to evaluate feynman diagrams starting from the corresponding schwinger s parametric representation , and whose main advantage is to transfer the complexity of the direct calculation of multidimensional integrals to the evaluation of linear systems of equations that the method generates .this technique is particularly useful when the propagator exponents are arbitrary , and the general solutions obtained are sums of generalized hypergeometric functions , whose number of variables depends on the number of invariants in the diagram and also on the topological family to which the diagram belongs .as is the case with any integration technique , although ibfe can be applied to any diagram , it is not always possible to obtain simple solutions due to the number and multivariability of the resulting hypergeometric series .nevertheless , for certain families of diagrams its application is particularly simple and optimal . in a previous work we showed that a modular application of ibfe is a tool that simplifies considerably the complexity of the solutions : specifically we considered bubble - type modules or subgraphs . herewe extend this modular application to cases in which the modules are triangular loops , which then provides a rather simple procedure to find the solutions to a large number of feynman diagrams .this work is divided in the following way . in sec ._ we apply the ibfe technique to the massless triangular module , and find its representing multiregion expansion for the case of off - shell external lines . herewe define the one - loop function , which allows to describe the multiregion expansion ( mre ) of this module as the sum over this loop function and a vertex , thus effectively reducing the loop to a node .one of the main characteristic features of the feynman diagram general solutions , when the propagators have arbitrary powers , is that they are given by sums of multivariable hypergeometric functions .the arguments or variables in these functions are ratios between invariants associated to the graph , and given the structure of the resulting hypergeometric function , the convergence condition requires that at least this ratio of invariants be less than unity .therefore from a particular diagram mre different solutions can be extracted , each of them associated to a particular kinematical regime , where the differentiating element is determined by the ratio of two invariants .this form of differentiating the solutions is direct for certain families of diagrams .once all terms associated to a specific region of interest are found , summing them will give us the desired solution .nevertheless , for topologies that do not belong to the previously mentioned family , and which is the most general case , this criteria for finding the correct terms of the solutions extracted from the mre actually fails , because in the general case the application of ibfe can generate solutions which contain hypergeometric functions with a particular argument , the unity .given the structure of the diagram solutions , this value corresponds to the ratio of two invariants or , when .nevertheless , both ratios are associated to different regions ( different solutions ) and the fact that they are equal does not allow to differentiate them . herewe present a method that solves this problem , and apply it to the specific case of diagrams that contain triangle subdiagrams .  in sec ._ iii _ we evaluate two different two - loop diagrams which contain the triangle subgraph in their topology .the first corresponds to a correction to a three - point function , with six propagators .we find the general solution for arbitrary propagator powers , and then particularize it to unity . in order to validate this technique, we compare the solution with the one obtained from a conventional method ibp ( integration by parts ) tka .the second diagram is a self - energy correction with five propagators , for which we present a new solution for the case or arbitrary values of the propagator powers . finally in sec ._ iv _ we describe some cases in which this massless triangle function can be used and the generalization to massive loops .in this section we will find the mre associated to the one - loop triangular subgraph or module , considering a case in which the propagators are massless and the external lines are off - shell . from the mre of this graphwe will define the one - loop function , which will allow to represent this module as a particular sum of three line vertices .this formulation will then be applied as examples to a self - energy and a three - point function diagram , both with two loops .the self - energy example will show quite clearly the advantages of this modular procedure .we start from the momentum integral representation for this graph , which is given in the massless case and in dimensions by :    when the propagator powers are integers , this expression can be easily reduced to simpler topologies using the integration by parts technique .the situation is more difficult for arbitrary indices , which we are considering in this work . graphically the integralcan be represented by the following diagram :   { t1.eps } \end{minipage}\ ] ]  the ibfe integration technique is applied to schwinger s parametric representation of the diagram , which can be obtained from the matrix of parameters  associated to eq . . in the present casethis matrix can be easily obtained and it is given by :    which allows to express schwinger s parametric representation as :    where and .the coefficients can be obtained from the determinants of the submatrices of the matrix of parameters :    replacing in and remembering that , we get after a little algebra schwinger s parametric representation of :    now we will deduce the mre of the parametric integral . for this purpose we expand the exponentials that are present in the integrand , which leads us to the following multiple series :    and this allows us to obtain the mre of the polynomial . for the denominator in the integral we have that :  and then , integrals already separated in factors of the form , can be replaced by the equivalent constraints .this is the final step of the fractional expansion process , and we have finally obtained the mre of the diagram from its parametric representation :    where we have summarized the constraints in the following identities :    from expression we see that the composition of sums and kronecker deltas ( ) of the mre of this module or graph is .   in order to generate a systematic reduction procedure of interior triangle subgraphs of a more complex topology , we define from eq . loop function denoted , such that can be written as :    where the one - loop function by comparison is given by the expression :    let us look at the propagator structure of the eq .it is clear that it can be represented pictorially as a sum whose argument contains the original diagram with the loop reduced to an effective vertex where the external lines reach , as inverse propagators .the cost of this topological minimization is a multiple sum .graphically the representation of the mre of the triangular module is given by :   { t1.eps } \end{minipage}=\sum% \limits_{n_{1}, .. ,n_{3}}g_{t}(a_{1},a_{2},a_{3};n_{1},n_{2},n_{3})% \begin{minipage}{4.6 cm } \includegraphics[scale=.7 ] { t2.eps } \end{minipage} }  \label{t14}\ ] ]the topological reduction formula , which corresponds to an mre of the triangular module , will be useful for finding the mre of generic graphs that contain in their geometry one or more of these triangular modules .we will now explain how to use formula in specific cases , showing the advantages of this procedure .as the first example of the modular reduction technique and of the topological formula , let us consider a diagram with three external lines , two loops and six propagators .we take a massless theory , with mass shell external lines and ( ) .the diagram is :   { 2loopstriangle.eps } \end{minipage }  \label{t13}\ ] ]   the first step in order to find the diagram mre with the modular procedure is to reduce the subgraph associated to the indices , using formula . in this waywe obtain an expression where we have eliminated one of the loops of the diagram , which has been replaced by an effective vertex where the three propagators associated to the indices meet . in detail :   { 2loops1.eps } \end{minipage}% \end{array } \label{t10}\ ] ]  in eq . we can see that one of the external lines , the one associated with the momentum , is affected by the modular reduction on .it is possible to actually extract this propagator from the diagram , since it is not involved in the loop integration , and write it as a factor that multiplies this graph .looking at eq . , and its pictorial equivalent , we conclude that it is possible to eliminate from the graph the information related to the index of summation . rewriting thus have :   { 2loops2.eps } \end{minipage}% \end{array}%\ ] ]  the resulting triangle allows us to apply the reduction formula .we then get the mre of the topology as :   { 2loops3.eps } \end{minipage}% \end{array}%\ ] ]  since each function is associated to six summation indices , in order to use the same notation for these indices , the second function starts with the summation index .therefore we get :    where after replacing the functions by their respective mre , we obtain the expression :    we have defined the factor :    the constraints are given by the following identities :    finally , applying the conditions of the problem , the summation indices , and must vanish in order to have a nonzero solution . replacing these values and eliminating the sums in ,we have obtain the mre of the diagram .after renaming the indexes so that they become consecutive , the multiregion expansion is finally given by :    and now the constraints are given by: the multiplicity of the hypergeometric series that can be extracted from the mre is one hypergeometric series of the type  and at most series of this type whose argument for this case is one .   now we proceed to find the general solutions associated with the diagram coming from the mre represents it .we maintain all the indexes arbitrary values , and for notational simplicity make and .the solutions we present here correspond to two regions that can be differentiated according to the ratio of the fictitious invariants used in order to separate them : the region where and the region where .both are solutions of , related by analytical continuation .in particular , in the limit they become identical , as will be seen later on .[ [ analytical - solution - in - the - region - leftvert - dfracabrightvert - leq-1 ] ] analytical solution in the region + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  let us first define the following simplifying notation for the indexes :    moreover , since in practice each constraint  eliminates a sum in the mre , the remaining or free sum at the end of the elimination process generates the corresponding hypergeometric representation  which we will identify as , since it is the contribution obtained when the summation index is free in the multiregion representation of diagram .  with these definitions we can write the solutions for this region as :    where the terms inside the sum correspond to the following functions :    and :    [ [ analytical - solution - in - the - region - leftvert - dfracbarightvert - leq-1 ] ] analytical solution in the region + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  analogously we have that in this fictitious region the solution is given by :    with :      and finally :     in the actual feynman diagram evaluation , in general the corresponding momentum integral has indexes or propagator powers that are equal to unity . from the previous results , making , we obtain the solution :  [ [ analytic - solution - in - the - region - leftvert - dfracabrightvert - leq-1 ] ] analytic solution in the region + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  the solutions in this region are given by :    where now :    and :    [ [ analytical - solution - in - the - region - leftvert - dfracbarightvert - leq-1 - 1 ] ] analytical solution in the region + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  in this region the solution is :    where the hypergeometric representations  are now given by :    or equivalently :      and finally :     it is possible to show that in the limit the solutions  and  are equal , and therefore both are related by analytic continuation . for this purpose we use the following identity :    using this formula we easily find that the summed series have the following form :    y :    on the other hand we have that :    and :    the hypergeometric function contained in the term and which has the form , can be transformed in another one of the type , and then we can apply the identity ,    and then we get that    after a bit of algebra , and using known gamma function properties , it is straightforward to show that the solutions and are equal in the limit , giving    therefore , defining as the solution for the graph in , the final solution in general can be written as :     when the propagator powers are integers , it is possible to evaluate such topologies with the integration by parts method . in the particular case of diagrams that contain subgraphs , this technique can be quite useful in order to simplify the integration problem , since it allows to simplify the geometry of the diagram .  in the particular case of the diagram which contain triangle subgraphs , this technique can be in order to simplify the integration problem .one very useful formula , which can be deduced applying ibp to a triangle graph , and which is known as the triangle identity , is the following :   { rtriangle.eps } \end{minipage}= & \dfrac{1}{\left ( d - a_{1}-a_{2}-2a_{3}\right ) } \left [ \;a_{1}\left ( \begin{minipage}{3.0 cm } \includegraphics[scale=.6 ] { triangle1.eps } \end{minipage}-% \begin{minipage}{3.0 cm } \includegraphics[scale=.6 ] { triangle2.eps } \end{minipage}\;\right ) \right .\\ &  \\ & \left .+ \;a_{2}\left ( \begin{minipage}{3.0 cm } \includegraphics[scale=.6 ] { triangle3.eps } \end{minipage}-% \begin{minipage}{3.0 cm } \includegraphics[scale=.6 ] { triangle4.eps } \end{minipage}\;\right ) \;\right ] .%\end{array } \label{rt}\ ] ]  where it has been assumed that the indexes and have arbitrary values .we now apply directly this identity to the diagram depicted in , but taking all the propagator powers to be unity .the result that we obtain is :   { tibp1.eps } \end{minipage } \\ &  \\ & = \dfrac{1}{d-4}\left [ 2\left ( \begin{minipage}{3.0 cm } \includegraphics[scale=.6 ] { tibp2.eps } \end{minipage}\right ) -%\begin{minipage}{3.0 cm } \includegraphics[scale=.6 ] { tibp3.eps } \end{minipage}-% \begin{minipage}{3.0 cm } \includegraphics[scale=.6 ] { tibp4.eps } \end{minipage}\right ].% \end{array}%\ ] ]  the third diagram in the left - hand side of the equation vanishes ( external on - shell line condition ) .the second diagram can be further simplified using again , giving :   { tibp3.eps } \end{minipage}=\frac{1}{d-5}\left [ \begin{minipage}{2.8 cm } \includegraphics[scale=.65 ] { tibp5.eps } \end{minipage}+2\left ( \begin{minipage}{3.0 cm } \includegraphics[scale=.6 ] { tibp6.eps } \end{minipage}-% \begin{minipage}{3.0 cm } \includegraphics[scale=.6 ] { tibp7.eps } \end{minipage}\right ) \right ] .\ ] ]  now each term of the solution can be easily rewritten evaluating the bubble diagram and then the triangle . for thiswe need the general formulae for these graphs , given by :   { form_b.eps } \end{minipage } & = \;\mathbf{g}(a_{1},a_{2})\;\dfrac{1}{\left ( p^{2}\right ) ^{a_{1}+a_{2}-\frac{d}{2 } } } \\ a_{2 } & \end{array } \label{t19}\ ] ]  where :    in the case of the triangle with conditions one has the fundamental formula :   { form_t.eps } \end{minipage}=\mathbf{g}(a_{1},a_{2},a_{3})\;\left ( p_{3}^{2}\right ) ^{\frac{d}{2}-a_{1}-a_{3}-a_{2 } }  \label{t20}\ ] ]  where the factor is given by :    applying formulae  and , the solution for the diagram is then :   \right]% \end{array } \label{sol_ibp}\ ] ]  choosing one of the solutions for  , it can be easily shown that the solution of this diagram evaluated using the ibfe technique gives the same result as with ibp: is then a concrete confirmation of the ibfe technique , applied modularly to a diagram with triangle subgraphs .the important point is that this has allowed to obtain the mre of the graph in a noticeably more direct and systematic way than finding it by taking schwinger 's parametric representation of the whole diagram .a more relevant example is the two - loop propagator correction , with five internal lines. { i25_1.eps } \end{minipage}\ ] ]   in order to find the diagram mre we apply formula on the left - hand side of the diagram , obtaining the following graphical expression :   { i25_2.eps } \end{minipage}\ ] ]  using then the massless bubble one - loop functions  , we quickly obtain the diagram mre as :   { propagator.eps } \end{minipage } \\ \end{array}% \end{array}%\ ] ]  or equivalently :    the one - loop functions and are given by the expressions :    whose constraints are :    and for we have :    where the constraints become :    with this information we finally are able to write the diagram mre as :    this expansion contains and , and therefore the solution of this diagram corresponds to a double series and since we are dealing with a propagator , the kinematical variable that is present does not appear explicitly as an argument in this series .we expect that for this type of diagrams the solution will not be a single term , since it does not correspond to an optimal topology ( in the sense that the multiplicity of the resulting series does not depend on the number of loops or equivalently only depends on the invariants of the physical process ) .  in order to simplify the notation in  we take  and . with these invariantssix regions of interest can be generated , each one with a different solution , but in the limit they are all identical .these regions are given by :    we arbitrarily take the region in which .then when we extract the different terms of the mre ( bivalued hypergeometric functions ) we only sum those that have as one of their arguments the following simple combinations :    or equivalently all the double combinations generated when combining these simple combinations :  etc . given these conditions ,the general solution that we get starting from  is the following :    where we have that :            and finally :    the previous results have been written in terms of the functions and ( for more information see ref . ) , which are defined as : similarly we also have :    this last series is called kamp de friet function .finally the solution is obtained by simply taking .we have solved this diagram in a modular or loop by loop form . in order to appreciate the advantages of this way of applying ibfe , it is important to compare the mre  with the mre of the same diagram obtained by the application of ibfe to the parametric integral of the complete diagram . for this purposelet us consider first schwinger s parametric representation of the complete diagram , which is given by :   } { \left [ \left ( x_{1}+x_{2}\right ) \left ( x_{3}+x_{4}+x_{5}\right ) + x_{3}\left ( x_{4}+x_{5}\right ) \right ] ^{\frac{d}{2% } } } ,  \label{t11}\ ] ]  where the polynomial is given by :   \;p^{2}.\ ] ]  if we now obtain the diagram mre using can compare ( _ table i _ ) this mre with mre :    we find that the mre obtained in terms of the one - loop functions is considerably reduced with respect to the one that we get when ibfe is applied to both loops of the diagram simultaneously , not only in the number of sums and kronecker deltas , but also the difference between them is reduced , which implies that the multiplicity of the resulting series is less , a nontrivial fact that simplifies considerably the complexity of the solutions .the calculation technology that we have presented here can be useful in order to find minimal expressions for the mre of diagrams that contain triangular subgraphs , such as those that we see in the following figure :   { comentario.eps } \end{minipage}\;\text{etc.}\ ] ]  the loop function is very useful since it opens up the possibility of evaluating other families of graphs and to obtain solutions with less complexity and less number of terms when compared with the application of ibfe to a parametric integral of the complete diagram .although we have presented here the one - loop function for the massless case , it is certainly possible to consider also massive propagators and define new loop functions for these cases . it will be always possible to use here the same reduction procedure for diagrams that contain bubbles or triangles that we have used for the massless cases .the same idea can be extended to consider one - loop functions for modules that contain four or more propagators .nevertheless , the reductions are no longer so simple as for the bubble and triangle subgraphs .the important point is that the modular ibfe application , particularly when it is done loop by loop , allows to obtain a minimal representation for the diagram mre , from which the analytical solution follows .the general solutions that can be obtained in the evaluation of feynman diagrams always correspond to multivariable hypergeometric series , whose multiplicity can be deduced directly from the mre that represents the diagram .one of the characteristics of the ibfe technique is that it has a lower bound that fixes the multiplicity of these hypergeometric series , and which is related to the minimal number of invariants that characterizes a specific feynman diagram , described through the formula :   the greater - equal sign is here included since besides the number of invariants one must consider also the topological family to which the diagram belongs . if this is not the optimal for ibfe application , in the sense described in , then the multiplicity is also increased according to the number of loops in the diagram .this multiplicity can be directly recognized from the mre of the diagram , and it is equal to the difference between the number of sums and constraints ( kronecker deltas ) that are present . solutions with 0 multiplicity ( which in reality is not a series , and contains only one term ) , or multiplicity 1 or 2 , correspond to series whose properties are extensively discussed in the literature wba , lsl , lgr , hex .nevertheless , for triple or higher multiplicity series , the information is rather limited and it is therefore important to have methods that allow to reduce the diagram mre , not only in the sense of lowering the number of sums and deltas , but also to reduce their difference , which is something that we have shown here can be done modularly ( see example ii ) .the idea of the modular application of ibfe can be easily extended to one - loop modules or subgraphs that contain four or more propagators. the result will be in general a reduced mre , that will also contain simpler solution , i.e. hypergeometric functions of lower multiplicity . in the same way , the idea of using fictitious invariants for selecting the correct solution associated to a diagram can be also generalized to all the cases in which hypergeometric functions with unit arguments are generated .the modular application of the integration technique ibfe has shown to be a powerful calculational tool , not only for its simplicity but also because it allows to calculate many families of graphs .here we have seen that the systematical graphical procedure that we have implemented for triangular modules , makes the reduction just as simple as in the case of bubbles contained in a diagram .

Abstract: a modular application of the integration by fractional expansion method for evaluating feynman diagrams is extended to diagrams that contain loop triangle subdiagrams in their geometry . the technique is based in the replacement of this module or subdiagram by its corresponding multiregion expansion ( mre ) , which in turn is obtained from schwinger s parametric representation of the diagram . the result is a topological reduction , transforming the triangular loop into an equivalent vertex , which simplifies the search for the mre of the complete diagram . this procedure has important advantages with respect to considering the parametric representation of the whole diagram : the obtained mre is reduced and the resulting hypergeometric series tend to have smaller multiplicity .  * pacs * : 11.25.db ; 12.38.bx  * keywords * : perturbation theory ; scalar integrals ; multiloop feynman diagrams ; schwinger parameters ; negative dimension integration method ( ndim ) , integration by fractional expansion ( ibfe ) .


[ sec1 ] natural selection is a fundamental evolutionary process and finding genomic regions experiencing selective pressure has important applications , including identifying the genetic basis of diseases and understanding the molecular basis of adaptation .there has been a long line of theoretical and experimental research devoted to modeling and detecting selection acting at a given locus .several earlier works have considered modeling the stationary distribution of allele frequencies in a population undergoing nonneutral evolution [ ( ) , ] .more recently , there has been growing interest to utilize time series genetic variation data to enhance our ability to infer allele frequency trajectories , thereby enabling better estimates of selection parameters .for example , the sequencing of samples over several generations in experimental evolution of a population ( e.g. , bacteria [ ] , yeast [ ] and _ drosophila _ [ ] ) under controlled laboratory environments , or direct measurements in fast evolving populations such as hiv [ ] , has allowed us to better understand the genetic basis of adaptation to changes in the environment .also , recent technological advances have given us the unprecedented ability to acquire ancient dna samples ( e.g. , for humans [ ] , ancient hominids [ ] and horses [ ] ) , providing useful information about allele frequency trajectories over long evolutionary timescales .most methods for analyzing times series dna data model the underlying population - wide allele frequency as an unobserved latent variable in a hidden markov model ( hmm ) framework , in which the sample of alleles drawn from the population at a given time is treated as a noisy observation of the hidden population allele frequency . in this framework , computing the probability of observing time series genetic variation data involves integrating over all possible hidden trajectories of the population allele frequency . for short evolutionary timescales , a discrete - time wright fisher model of random matingis often used to describe the dynamics of the population allele frequency in the underlying hmm .this approach has been used to estimate the effective population size from temporal allele frequency variation , assuming a neutral model of evolution [ ] .more recently , temporal and spatial variations of advantageous alleles have been investigated through an hmm framework that can incorporate migration between multiple subpopulations [ ] .if the evolutionary timescale between consecutive sampling times is large , it can become computationally cumbersome to work with discrete - time models of reproduction .however , by a suitable rescaling of time , population size and population genetic parameters , one can obtain a continuous - time process ( the wright fisher diffusion ) which accurately approximates the population allele frequency of the discrete - time wright fisher model . the key quantity needed when applyingthe diffusion process is the transition density function , which describes the probability density of the allele frequency changing from value to value in time .this transition density function satisfies a certain partial differential equation ( pde ) with coefficients that depend on the mutation and selection parameters . used a finite - difference numerical method to approximate the solution to the pde and incorporated the results into the aforementioned hmm framework to infer the strength of selection from time series data .recently , an alternative approach [ ] based on a one - step markov process has been proposed to compute the necessary transition densities . in both of these approaches ,the allele frequency space has to be discretized finely enough in order to reliably approximate various numerical integrals that are needed for computing the hmm likelihood .the efficiency and accuracy of these grid - based numerical methods depend critically on the spacing and distribution of the discrete grid points .furthermore , an appropriate choice of this discretization scheme could be strongly dependent on the underlying population genetic parameters .another limitation of these previous works is that only a few restricted models of selection have been considered . recently developed a likelihood - ratio test for identifying signatures of selection from time series data in which they combined a deterministic model and a gaussian noise process .this approximation is less accurate than the diffusion approximation , but it facilitates computation and seems sufficiently accurate provided that the allele frequency does not get too close to the boundaries during the period of observation .  in this paper, we develop a novel algorithm based on the spectral method to circumvent the limitations mentioned above . specifically ,instead of approximating the solution to the pde numerically , we utilize a method recently developed by which finds an explicit spectral representation of the transition density as a function of , and .we show that the probability of observing a given time series data set can be computed analytically by combining the spectral representation with the forward algorithm for hmms to _ efficiently _ and _ analytically _ integrate over all population allele frequency trajectories .the key idea in our work is to represent the intermediate densities in the forward algorithm in the basis of eigenfunctions of the infinitesimal generator of the wright fisher diffusion process . exploiting the spectral representation of the transition density, we can then efficiently compute the coefficients in this basis representation . furthermore , since this spectral representation applies to general diploid models of selection , we are able to leverage this representation to consider more complex models of selection than previously possible .we first demonstrate the accuracy of our method on simulated data .we then apply the method to analyze time series ancient dna data from genetic loci ( asip and mc1r ) that are associated with horse coat coloration .in contrast to the conclusions of previous studies which considered only a few special models of selection [ ] , our exploration of the full parameter space of general diploid selection reveals that a heterozygote advantage form of balancing selection may have been acting on these loci .we implemented the algorithms described in this paper in a publicly available software package called _spectralhmm_.  the remainder of this paper is organized as follows . in section [ secmodel ]we formally introduce the hmm framework and describe the details of our spectral algorithm .the proofs of the theoretical results underlying our algorithm are provided in the supplemental article [ ] . in section [ secresults ]we use simulated data to investigate the statistical properties of our maximum likelihood estimator and also apply our method to analyze the aforementioned ancient dna data for the loci associated with horse coat coloration [ ] .we conclude in section [ secdiscussion ] with a discussion of future extensions of our model .here we provide a formal description of the time series data considered in this paper and present our inference method for analyzing such data .   the data we analyze consist of genotype samples obtained from individuals at distinct times in the past ( given in years ) . the present time is denoted by . at each time point , a sample of individuals is randomly drawn from the population .we assume that the locus under consideration is biallelic , and that the identities of the ancestral allele and the derived allele are known .we also assume that the allele became selected at some time .we use to denote the number of derived alleles in the sample of alleles drawn at time , where . for notational convenience, we use to denote the tuple and } ] . ]   to analyze the time series data described earlier , we employ a hidden markov model ( hmm ) framework as in . in this approach ,the population - wide frequency of the allele at time is modeled as an unobserved hidden variable ( see figure [ figdata ] ) .we denote a realization of the frequencies at the sampling times by . the initial frequency at time is distributed according to the density function , that is , .for example , the density function models the case where the selected allele arose as a _ de novo _ mutation in one individual of the population at time .the probability of transitioning from frequency at time to frequency at time is described by the transition density function of the wright fisher diffusion process , where and are population - scaled parameters as given in equations ( [ eqdifftimepointsidxdef])([eqmutbetadef ] ) .the observations in the hmm are the number of copies of the allele among the alleles in the sample drawn at time .the probability of such an observation at time  with population allele frequency is given by the probability mass function of a binomial distribution   to compute the probability } \} ] under the model parameters , we introduce the forward density functions , given by } , { \mathbf{y}}({{\tau}}_{k } ) \in d{y } \bigr\ } , \qquad{k}\in\{0,1,\ldots,{{k}}\}. \label{eqforwarddensity}\ ] ] the function is the joint density of the observed data up to time and the hidden population allele frequency at time .we also find it convenient to consider a second auxiliary density function , , given by } , { \mathbf{y}}({{\tau}}_{k } ) \in d{y } \bigr\ } , \qquad{k } \in\{1,\ldots,{{k}}\}. \label{eqauxiliarydensity}\ ] ] this function is the joint density of the observed data up to time and the hidden frequency at .the forward density function is given by the density function for the initial allele frequency as since we approximate the time evolution of the hidden population allele frequency by the wright fisher diffusion , we can get a recurrence relation between the density functions and by integrating over all possible allele frequencies at  : where . using the binomial distribution for sampling derived alleles out of individuals at time , we get another recurrence relation between the density functions and as follows : finally , the probability } \} ] , where the intermediate density functions have a natural interpretation .  while ( [ eqtransitionintegral ] ) , ( [ eqbinomialemission ] ) and ( [ eqfinalintegral ] ) succinctly describe the sampling probability of the data } ] of real - valued functions on  ] .this algorithm recursively computes the density functions and given in ( [ eqforwarddensity ] ) and ( [ eqauxiliarydensity ] ) , respectively . to update these density functions efficiently ,we represent them in the basis of scaled eigenfunctions of the diffusion generator .more precisely , we express and as where we employ the vector notation   we now describe how the coefficient vectors and the probability }\} ] can be computed using the following proposition :  [ propfinalstep ] the probability } \ } ] given the population genetic parameters is } \ } = \frac { { { c}}_0 } { { b}_0(0 ) } { { b}}_{{{k}},0},\ ] ] where is given by this section we perform parametric inference via the maximum likelihood framework using a finite grid in the parameter space .we first test the accuracy on simulated data and then apply it to analyze an ancient dna data set related to coat coloration in domesticated horses [ ] .  since ancient dna data are often collected from only those loci which are segregating at the present time , in our empirical study we condition on observing at least one copy of the derived allele at the last sampling time .in particular , the likelihood of the parameters is given by } { |}{d}_{{k } } > 0 \} ] . for each bootstrapped data set , we resampled alleles at each time .the number of derived alleles for data set was obtained by binomial sampling from the empirical frequency of derived alleles in the original asip data set , that is , we then reported the empirical maximum of the likelihood surface for each of these resampled data sets .figure [ figbootstrapasip](b ) shows the empirical maximum likelihood estimates and marginal histograms of the maxima for the 100 resampled data sets .the marginal 2.5% and 97.5% quantiles of the empirical distribution are  ] for the derived allele homozygote fitness , thus providing further evidence that the data are significantly better explained by a selection model where a heterozygous individual is selectively advantageous over the homozygous individuals . as figure[ figmultipleasip ] shows , changing from 1000 to 10,000 , or changing from 19,000 bce to 15,000 , bce has only a minimal effect on the shape of the likelihood surface and maximum likelihood estimate , again supporting that a selective scheme of heterozygote advantage best explains the data .   for temporal samples from the asip locus wherethe likelihood is computed over a grid .the maximum is attained at , indicated by the `` . '' a joint density plot and marginal histograms of the maximum likelihood estimates for 100 bootstrap resampled data sets of the temporal data at the asip locus .the circles are centered on the grid points at which the likelihood function is evaluated , and the sizes of the circles indicate the proportion of maximum likelihood estimates that occupy the same grid point . the marginal empirical 2.5% and97.5%-quantiles are  ] for the derived allele homozygote fitness , as indicated by the dashed box . ][ figsingleasip][figbootstrapasip ]  \{1000 , 2500 , 10,000 } and \{15,000 bce , 17,000 bce , 19,000 bce}. the respective maxima are indicated by an `` . '' ]  a similar analysis of the mc1r locus can be found in figures [ figmc1r ] and [ figmultiplemc1r ] . for this data set ,the maximum of the likelihood surface is attained at , and the empirical marginal 2.5% and 97.5%-quantiles are  ] for the derived allele homozygote fitness . together with the results shown in figure [ figmultiplemc1r ], this suggests that the data at the mc1r locus is also best explained by a selection model of heterozygote advantage .however , although the marginal quantiles for the homozygote fitness cover , they are rather far apart , so the evidence of heterozygote advantage for the mc1r locus is weaker than that for the asip locus .   and bce .likelihood surface for the mc1r locus .the maximum likelihood estimate is at and is indicated by the `` . ''a joint density plot and marginal histograms of the maximum likelihood estimates for 100 bootstrap resampled data sets obtained from the mc1r data as described in section [ sechorses ] .the marginal 2.5% and 97.5%-quantiles are  ] for the derived allele homozygote fitness , as indicated by the dashed box . ]\{1000 , 2500 , 10,000 } and \{5000 bce , 7000 bce , 9000 bce}. the respective maxima are indicated by an `` . '' ]   the running time of our algorithm for computing the likelihood of a given set of population - scaled parameters is dependent on the dimensions of the truncation of the infinite matrix given in ( [ eqoperatormatrix ] ) . in particular , the time complexity of computing a single likelihood is the cost of computing the eigenvalues and eigenvectors of plus the cost of computing the coefficients in theorem [ thmtransitionemission ] , where . to compute the eigenvalues and eigenvectors of to high precision, we first used lapack to compute them to double precision , and then refine them by using inverse iteration [ , chapter 11.8 ] .each step of the inverse iteration involves solving a linear system with matrix , where is an estimate for an eigenvalue of .since this matrix has bandwidth at most 9 , this linear system can be solved in time , where is the dimension of . by using the repeated squaring algorithm for taking powers of the matrices and and exploiting the fact that and are tridiagonal matrices ,each coefficient can be computed in time , where the first term comes from the matrix - vector multiplications in ( [ eqauxtoforward ] ) .  for the analysis of the asip and mc1r data sets reported in figures [ figrealasip ] and [ figmc1r ] , we approximated the eigenvalues and eigenvectors of defined in ( [ eqoperatormatrix ] ) using a submatrix. furthermore , we used the first terms in ( [ eqdefefun ] ) to approximate the eigenfunctions , and the dimensions of the vectors of coefficients in ( [ eqvecforward ] ) and ( [ eqvecauxiliary ] ) were set to .we empirically verified that these cutoffs produced a stable approximation of the likelihood . using these values , the computation time for a single point of the grid in figure [ figsingleasip](a ) was approximately 95 seconds .we adjusted the cutoffs appropriately for the other analyses reported in section [ secresults ] .in this paper we have developed a novel , efficient spectral algorithm to analyze time series allele frequency data under a general diploid selection model .we have demonstrated that our method can be used to accurately estimate selection parameters on simulated data .we have also applied our method to investigate loci involved in horse coat coloration .our inferred selection coefficients show that the data are best explained by a heterozygote advantage model of balancing selection . as mentioned earlier , provided evidence for slightly positive selection at the asip locus , assuming a model of genic selection ( where ) .more precisely , they obtained a point estimate of and a 95% confidence interval of  ] , thus not rejecting neutrality at the asip locus . in our work , we have allowed our method to explore the two - dimensional parameter space of general diploid selection models and presented evidence for a selection mode where heterozygous individuals are advantageous over homozygous individuals .it is possible that previous analyses have only been able to infer very weak selection acting at the asip locus because they have restricted the model of selection to certain one - dimensional models .indeed , if we restrict our analysis to a model of genic selection , we get results similar to those reported by . our analysis does not conclusively prove that individuals that were heterozygous at the asip locus had a constant evolutionary advantage since 17,000 bce , because we have ignored the interaction of selection and demographic history , epistatic interactions between loci , time - varying models of selection and other factors .however , our results suggest the possibility that some mode of heterozygote advantage balancing selection has maintained polymorphism at the asip locus that is involved in horse coat coloration .although we have focused on time series samples taken at a biallelic locus , the mathematical framework presented here could be readily extended to handle an arbitrary number of alleles using the spectral representation derived by .further , changes in the population size and selection coefficients could be modeled by suitably combining the spectral representations for different population genetic parameters at the change points .it is also possible to extend the method to multiple populations and to incorporate samples taken from extinct ancestral populations . in light of emerging ancient dna sequence data for ancient hominids [ ] , such temporal sequence data and inference methods present novel opportunities to gain insight into adaptation in humans . for a more adequate modeling of biologically relevant scenarios, it is also necessary to incorporate the exchange of migrants into the model [ ] and extend the framework to incorporate variation at linked loci . by taking advantage of genetic hitchhiking at closely linked sites during the course of selective sweeps, one might be able to further improve the inference of selection coefficients .we thank rasmus nielsen , joshua schraiber andmontgomery slatkin for helpful comments and discussions .we also thank karen kafadar and two anonymous referees for suggestions that improved the exposition of this paper .moreover , we thank richard j. mathar [ ] for making his source code available to us .

Abstract: the increased availability of time series genetic variation data from experimental evolution studies and ancient dna samples has created new opportunities to identify genomic regions under selective pressure and to their associated fitness parameters . however , it is a challenging problem to compute the likelihood of nonneutral models for the population allele frequency dynamics , given the observed temporal dna data . here , we develop a novel spectral algorithm to analytically and efficiently integrate over all possible frequency trajectories between consecutive time points . this advance circumvents the limitations of existing methods which require fine - tuning the discretization of the population allele frequency space when numerically approximating requisite integrals . furthermore , our method is enough to handle general diploid models of selection where the heterozygote and homozygote fitness parameters can take any values , while previous methods focused on only a few restricted models of selection . we demonstrate the utility of our method on simulated data and also apply it to analyze ancient dna data from genetic loci associated with coat coloration in horses . in contrast to previous studies , our exploration of the full fitness parameter space reveals that a heterozygote advantage form of balancing selection may have been acting on these loci .  ,