As long as a square nonnegative matrix A contains sufficient nonzero elements, then the matrix can be balanced, that is we can find a diagonal scaling of A that is doubly stochastic. A number of algorithms have been proposed to achieve the balancing, the most well known of these being Sinkhorn-Knopp. In this paper we derive new algorithms based on inner-outer iteration schemes. We show that Sinkhorn-Knopp belongs to this family, but other members can converge much more quickly. In particular, we show that while stationary iterative methods offer little or no improvement in many cases, a scheme using a preconditioned conjugate gradient method as the inner iteration can give quadratic convergence at low cost. Introduction.For at least 70 years, scientists in a wide variety of disciplines have attempted to transform square nonnegative matrices into doubly stochastic form by applying diagonal scalings. That is, given A ∈ R n×n , A ≥ 0, find diagonal matrices D 1 and D 2 so that P = D 1 AD 2 is doubly stochastic. Motivations for achieving this balance include interpreting economic data [1], preconditioning sparse matrices [16], understanding traffic circulation [14], assigning seats fairly after elections [3], matching protein samples [4] and ordering nodes in a graph [12]. In all of these applications, one of the main methods considered is SK 1 . This is an iterative process that attempts to find D 1 and D 2 by alternately normalising columns and rows in a sequence of matrices starting with A. Convergence conditions for this algorithm are well known: if A has total support 2 then the algorithm will converge linearly with asymptotic rate equal to the square of the subdominant singular value of P [22,23,12].Clearly, in some cases the convergence will be painfully slow. The principal aim of this paper is to derive some new algorithms for the matrix balancing problem with an eye on speed, especially for large systems. First we look at a simple Newton method for symmetric matrices, closely related to a method proposed by Khachiyan and Kalantari [11] for positive definite (but not necessarily nonnegative) matrices. We will show that as long as Newton's method produces a sequence of positive iterates, the Jacobians we generate will be positive semi-definite and that this is also true when we adapt the method to cope with nonsymmetric matrices.To apply Newton's method exactly we require a linear system solve at each step, and this is usually prohibitively expensive. We therefore look at iterative techniques for approximating the solution at each step. First we look at splitting methods and we see that SK is a member of this family of methods, as is the algorithm proposed by Livne and Golub in [16]. We give an asymptotic bound on the (linear) rate of convergence of these methods. For symmetric matrices we can get significant improvement *

Abstract. As long as a square nonnegative matrix A contains sufficient nonzero elements, then the Sinkhorn-Knopp algorithm can be used to balance the matrix, that is, to find a diagonal scaling of A that is doubly stochastic. It is known that the convergence is linear and an upper bound has been given for the rate of convergence for positive matrices. In this paper we give an explicit expression for the rate of convergence for fully indecomposable matrices.We describe how balancing algorithms can be used to give a measure of web page significance. We compare the measure with some well known alternatives, including PageRank. We show that with an appropriate modification, the Sinkhorn-Knopp algorithm is a natural candidate for computing the measure on enormous data sets.

We study a class of Markovian systems of N elements taking values in [0, 1] that evolve in discrete time t via randomized replacement rules based on the ranks of the elements. These rank-driven processes are inspired by variants of the Bak-Sneppen model of evolution, in which the system represents an evolutionary 'fitness landscape' and which is famous as a simple model displaying self-organized criticality. Our main results are concerned with long-time large-N asymptotics for the general model in which, at each time step, K randomly chosen elements are discarded and replaced by independent U [0, 1] variables, where the ranks of the elements to be replaced are chosen, independently at each time step, according to a distribution κ N on {1, 2, . . . , N } K . Our main results are that, under appropriate conditions on κ N , the system exhibits threshold behaviour at s * ∈ [0, 1], where s * is a function of κ N , and the marginal distribution of a randomly selected element converges to U [s * , 1] as t → ∞ and N → ∞. Of this class of models, results in the literature have previously been given for special cases only, namely the 'mean-field' or 'random neighbour' Bak-Sneppen model. Our proofs avoid the heuristic arguments of some of the previous work and use Foster-Lyapunov ideas. Our results extend existing results and establish their natural, more general context. We derive some more specialized results for the particular case where K = 2. One of our technical tools is a result on convergence of stationary distributions for families of uniformly ergodic Markov chains on increasing state-spaces, which may be of independent interest.

International audienceWe present an iterative algorithm which asymptotically scales the $\infty$-norm of each row and each column of a matrix to one. This scaling algorithm preserves symmetry of the original matrix and shows fast linear convergence with an asymptotic rate of $1/2$. We discuss extensions of the algorithm to the one-norm, and by inference to other norms. For the 1-norm case, we show again that convergence is linear, with the rate dependent on the spectrum of the scaled matrix. We demonstrate experimentally that the scaling algorithm improves the conditioning of the matrix and that it helps direct solvers by reducing the need for pivoting. In particular, for symmetric matrices the theoretical and experimental results highlight the potential of the proposed algorithm over existing alternatives.Nous décrivons un algorithme itératif qui, asymptotiquement, met une matrice à l'échelle de telle sorte que chaque ligne et chaque colonne est de taille 1 dans la norme infini. Cet algorithme préserve la symétrie. De plus, il converge assez rapidement avec un taux asymptotique de 1/2. Nous discutons la généralisation de l'algorithme à la norme 1 et, par inférence, à d'autres normes. Pour le cas de la norme 1, nous établissons que l'algorithme converge avec un taux linéaire. Nous démontrons expérimentalement que notre algorithme améliore le conditionnement de la matrice et qu'il aide les méthodes directes de résolution en réduisant le pivotage. Particulièrement pour des matrices symétriques, nos résultats théoriques et expérimentaux mettent en valeur l'intérêt de notre algorithme par rapport aux algorithmes existants

Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. The Bak-Sneppen model is a well-known stochastic model of evolution that exhibits selforganized criticality; only a few analytical results have been established for it so far. We report a surprising connection between Bak-Sneppen type models and more tractable Markov processes that evolve without any reference to an underlying topology. Specifically, weshow that in the case of a large number of species, the long time behaviour of the fitness profile in the Bak-Sneppen model can be replicated by a model with a purely rank-based update rule whose asymptotics can be studied rigorously.