129 research outputs found
Spectra of Sparse Random Matrices
We compute the spectral density for ensembles of of sparse symmetric random
matrices using replica, managing to circumvent difficulties that have been
encountered in earlier approaches along the lines first suggested in a seminal
paper by Rodgers and Bray. Due attention is payed to the issue of localization.
Our approach is not restricted to matrices defined on graphs with Poissonian
degree distribution. Matrices defined on regular random graphs or on scale-free
graphs, are easily handled. We also look at matrices with row constraints such
as discrete graph Laplacians. Our approach naturally allows to unfold the total
density of states into contributions coming from vertices of different local
coordination.Comment: 22 papges, 8 figures (one on graph-Laplacians added), one reference
added, some typos eliminate
Disentangling Giant Component and Finite Cluster Contributions in Sparse Matrix Spectra
We describe a method for disentangling giant component and finite cluster
contributions to sparse random matrix spectra, using sparse symmetric random
matrices defined on Erdos-Renyi graphs as an example and test-bed.Comment: 7 pages, 2 multi-part figure
A Random Walk Perspective on Hide-and-Seek Games
We investigate hide-and-seek games on complex networks using a random walk
framework. Specifically, we investigate the efficiency of various degree-biased
random walk search strategies to locate items that are randomly hidden on a
subset of vertices of a random graph. Vertices at which items are hidden in the
network are chosen at random as well, though with probabilities that may depend
on degree. We pitch various hide and seek strategies against each other, and
determine the efficiency of search strategies by computing the average number
of hidden items that a searcher will uncover in a random walk of steps. Our
analysis is based on the cavity method for finite single instances of the
problem, and generalises previous work of De Bacco et al. [1] so as to cover
degree-biased random walks. We also extend the analysis to deal with the
thermodynamic limit of infinite system size. We study a broad spectrum of
functional forms for the degree bias of both the hiding and the search strategy
and investigate the efficiency of families of search strategies for cases where
their functional form is either matched or unmatched to that of the hiding
strategy. Our results are in excellent agreement with those of numerical
simulations. We propose two simple approximations for predicting efficient
search strategies. One is based on an equilibrium analysis of the random walk
search strategy. While not exact, it produces correct orders of magnitude for
parameters characterising optimal search strategies. The second exploits the
existence of an effective drift in random walks on networks, and is expected to
be efficient in systems with low concentration of small degree nodes.Comment: 31 pages, 10 (multi-part) figure
Optimal trading strategies - a time series approach
Motivated by recent advances in the spectral theory of auto-covariance
matrices, we are led to revisit a reformulation of Markowitz' mean-variance
portfolio optimization approach in the time domain. In its simplest incarnation
it applies to a single traded asset and allows to find an optimal trading
strategy which - for a given return - is minimally exposed to market price
fluctuations. The model is initially investigated for a range of synthetic
price processes, taken to be either second order stationary, or to exhibit
second order stationary increments. Attention is paid to consequences of
estimating auto-covariance matrices from small finite samples, and
auto-covariance matrix cleaning strategies to mitigate against these are
investigated. Finally we apply our framework to real world data
A Structural Model for Fluctuations in Financial Markets
In this paper we provide a comprehensive analysis of a structural model for
the dynamics of prices of assets traded in a market originally proposed in [1].
The model takes the form of an interacting generalization of the geometric
Brownian motion model. It is formally equivalent to a model describing the
stochastic dynamics of a system of analogue neurons, which is expected to
exhibit glassy properties and thus many meta-stable states in a large portion
of its parameter space. We perform a generating functional analysis,
introducing a slow driving of the dynamics to mimic the effect of slowly
varying macro-economic conditions. Distributions of asset returns over various
time separations are evaluated analytically and are found to be fat-tailed in a
manner broadly in line with empirical observations. Our model also allows to
identify collective, interaction mediated properties of pricing distributions
and it predicts pricing distributions which are significantly broader than
their non-interacting counterparts, if interactions between prices in the model
contain a ferro-magnetic bias. Using simulations, we are able to substantiate
one of the main hypotheses underlying the original modelling, viz. that the
phenomenon of volatility clustering can be rationalised in terms of an
interplay between the dynamics within meta-stable states and the dynamics of
occasional transitions between them.Comment: 16 pages, 8 (multi-part) figure
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