The dynamics of many social, technological and economic phenomena are driven by individual human actions, turning the quantitative understanding of human behavior into a central question of modern science. Current models of human dynamics, used from risk assessment to communications, assume that human actions are randomly distributed in time and thus well approximated by Poisson processes. Here we provide direct evidence that for five human activity patterns, such as email and letter based communications, web browsing, library visits and stock trading, the timing of individual human actions follow non-Poisson statistics, characterized by bursts of rapidly occurring events separated by long periods of inactivity. We show that the bursty nature of human behavior is a consequence of a decision based queuing process: when individuals execute tasks based on some perceived priority, the timing of the tasks will be heavy tailed, most tasks being rapidly executed, while a few experiencing very long waiting times. In contrast, priority blind execution is well approximated by uniform interevent statistics. We discuss two queuing models that capture human activity. The first model assumes that there are no limitations on the number of tasks an individual can hadle at any time, predicting that the waiting time of the individual tasks follow a heavy tailed distribution P͑ w ͒ϳ w −␣ with ␣ =3/2. The second model imposes limitations on the queue length, resulting in a heavy tailed waiting time distribution characterized by ␣ = 1. We provide empirical evidence supporting the relevance of these two models to human activity patterns, showing that while emails, web browsing and library visitation display ␣ = 1, the surface mail based communication belongs to the ␣ =3/2 universality class. Finally, we discuss possible extension of the proposed queuing models and outline some future challenges in exploring the statistical mechanics of human dynamics.

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We study the sensitivity to estimation error of portfolios optimized under various risk measures, including variance, absolute deviation, expected shortfall and maximal loss. We introduce a measure of portfolio sensitivity and test the various risk measures by considering simulated portfolios of varying sizes N and for different lengths T of the time series. We find that the effect of noise is very strong in all the investigated cases, asymptotically it only depends on the ratio N/T , and diverges at a critical value of N/T , that depends on the risk measure in question. This divergence is the manifestation of a phase transition, analogous to the algorithmic phase transitions recently discovered in a number of hard computational problems. The transition is accompanied by a number of critical phenomena, including the divergent sample to sample fluctuations of portfolio weights. While the optimization under variance and mean absolute deviation is always feasible below the critical value of N/T , expected shortfall and maximal loss display a probabilistic feasibility problem, in that they can become unbounded from below already for small values of the ratio N/T , and then no solution exists to the optimization problem under these risk measures. Although powerful filtering techniques exist for the mitigation of the above instability in the case of variance, our findings point to the necessity of developing similar filtering procedures adapted to the other risk measures where they are much less developed or nonexistent. Another important message of this study is that the requirement of robustness (noise-tolerance) should be given special attention when considering the theoretical and practical criteria to be imposed on a risk measure.

Recent studies inspired by results from random matrix theory [1,2,3] found that covariance matrices determined from empirical financial time series appear to contain such a high amount of noise that their structure can essentially be regarded as random. This seems, however, to be in contradiction with the fundamental role played by covariance matrices in finance, which constitute the pillars of modern investment theory and have also gained industry-wide applications in risk management. Our paper is an attempt to resolve this embarrassing paradox. The key observation is that the effect of noise strongly depends on the ratio r = n/T , where n is the size of the portfolio and T the length of the available time series. On the basis of numerical experiments and analytic results for some toy portfolio models we show that for relatively large values of r (e.g. 0.6) noise does, indeed, have the pronounced effect suggested by [1,2,3] and illustrated later by [4,5] in a portfolio optimization context, while for smaller r (around 0.2 or below), the error due to noise drops to acceptable levels. Since the length of available time series is for obvious reasons limited in any practical application, any bound imposed on the noise-induced error translates into a bound on the size of the portfolio. In a related set of experiments we find that the effect of noise depends also on whether the problem arises in asset allocation or in a risk measurement context: if covariance matrices are used simply for measuring the risk of portfolios with a fixed composition rather than as inputs to optimization, the effect of noise on the measured risk may become very small.

We address the problem of portfolio optimization under the simplest coherent risk measure, i.e. the expected shortfall. As it is well known, one can map this problem into a linear programming setting. For some values of the external parameters, when the available time series is too short, the portfolio optimization is ill posed because it leads to unbounded positions, infinitely short on some assets and infinitely long on some others. As first observed by Kondor and coworkers, this phenomenon is actually a phase transition. We investigate the nature of this transition by means of a replica approach.

The problem of diagonalizing a class of complicated matrices, to be called ultrametric matrices, is investigated. These matrices appear at various stages in the description of disordered systems with many equilibrium phases by the technique of replica symmetry breaking. The residual symmetry, remaining after the breaking of permutation symmetry between replicas, allows us to bring all ultrametric matrices to a block diagonal form by a common similarity transformation. A large number of these blocks are, in fact, of size 1 × 1, i.e. in a vast sector the transformation actually diagonalizes the matrix. In the other sectors we end up with blocks of size (R + 1) × (R + 1) where R is the number of replica symmetry breaking steps. These blocks cannot be further reduced without giving more information, in addition to ultrametric symmetry, about the matrix. Similar results for the inverse of a generic ultrametric matrix are also derived. PACS classification numbers: 75.10.Nr, 05.50.+q. * E-mail:temtam@hal9000.elte.hu IntroductionLow temperature disordered systems often possess many equilibrium phases. The technique of replica symmetry breaking (RSB) provides a theoretical framework in which these systems can be described analytically , starting from a microscopic basis. Discovered and developed in the theory of spin glasses [1], RSB has recently penetrated into a number of other problems, including the theory of random manifolds [2][3][4], random field problems [5,6], protein folding [7][8][9], vortex pinning [10], etc. In each of these theories randomness is handled via the replica trick, and the multitude of equilibrium phases is captured by breaking the permutation symmetry between the replicas. As always, symmetry breaking means that the low temperature solutions realize a particular subgroup of the underlying symmetry group of the theory, here of the permutation group. The proper choice of the subgroup proved to be a highly nontrivial task in the case of RSB. The succesful Ansatz for the symmetry breaking pattern, proposed by Parisi originally in the context of spin glasses, turned out to embody a particular, hierarchical organization of the equilibrium phases, usually referred to as ultrametricity [1].The corresponding subgroup determines the structure not only of the order parameter, but also of all other quantities in the theory, like self-energies, propagators, etc. The structure imposed by this residual symmetry on quantities depending on two replica indices is by now widely known. The algebra of such quantities has been worked out by Parisi [11] with further results, most notably on the inversion problem, added by Mézard and Parisi [2]. At a certain stage of the development of RSB theories, however, one has to face also more complicated objects, depending on three or four replica indices. The structure of these is much harder to grasp and their algebra is much more involved than that of the two-index quantities. Our purpose here is to analyse and exploit the structure imposed by ultametricity on fou...

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The state of art in spin glass field theory is reviewed. We start from an Edwards-Anderson-type model in finite dimensions, with finite but long range forces, construct the effective field theory that allows one to extract the long wavelength behaviour of the model, and set up an expansion scheme (the loop expansion) in the inverse range of the interaction. At the zeroth order we recover mean field theory. We evaluate systematic corrections to this around Parisi's replica symmetry broken solution. At the level of quadratic fluctuations we derive a set of coupled integral equations for the free propagators of the theory and show how they can be solved for short, intermediate and extreme long distances. To reveal the physical meaning of these results, we relate the various propagator components to overlaps of spin-spin correlation functions inside a single phase space valley resp. between different valleys. Next we calculate the first loop corrections to the theory above 8 dimensions, where we find that it maps back onto mean field theory, with basically temperature independent renormalization of the coupling constants, thereby demonstrating that Parisi's mean field theory is, at least perturbatively, stable against finite range corrections. In the range between six and eight dimensions various physical quantities pick up nontrivial temperature dependences which can, however, still be determined exactly. Upon approaching the upper critical dimension (d = 6) of the model, scaling which is badly violated in Parisi's mean field theory is gradually restored. Below 6 dimensions one should apply renormalization group methods. Unfortunately, the structure of RG is not completely understood in spin glass theory. Nevertheless, the first corrections in 6 − d to e.g. the exponent of the order parameter can still be calculated, moreover exponentiation to this power can be checked at the next order. The theory is, however, plagued by infrared divergences due to the presence of zero modes and soft modes. Systematic methods (like those developed in the O(n) model) to handle these infrared singularities are not yet available in spin glass theory.