Metric characterization of cluster dynamics on the Sierpinski gasket

Agliari, Elena; Casartelli, Mario; Vivo, Edoardo

doi:10.1088/1742-5468/2010/09/p09002

Cited by 4 publications

(9 citation statements)

References 29 publications

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“…The following investigation is just aimed at the study of its topological features, which, as well known, are intimately connected with the dynamical properties of phenomena occurring on the network itself (e.g. diffusion [26,2,5], transport [9,7], critical properties [31,8], coherent propagation [6], relaxation [44], just to cite a few). We first focus on the topology neglecting the role of weights and we say that two nodes i and j are connected whenever J ij is strictly positive; disorder on couplings will be addressed in Sec.…”

Section: The Emergent Networkmentioning

confidence: 99%

Equilibrium statistical mechanics on correlated random graphs

Barra¹,

Agliari²

2011

J. Stat. Mech.

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Biological and social networks have recently attracted enormous attention between physicists. Among several, two main aspects may be stressed: A non trivial topology of the graph describing the mutual interactions between agents exists and/or, typically, such interactions are essentially (weighted) imitative. Despite such aspects are widely accepted and empirically confirmed, the schemes currently exploited in order to generate the expected topology are based on a-priori assumptions and in most cases still implement constant intensities for links. Here we propose a simple shift [−1, +1] → [0, +1] in the definition of patterns in an Hopfield model to convert frustration into dilution: By varying the bias of the pattern distribution, the network topology -which is generated by the reciprocal affinities among agents (the Hebbian kernel)-crosses various well known regimes (fully connected, linearly diverging connectivity, extreme dilution scenario, no network), coupled with small world properties, which, in this context, are emergent and no longer imposed a-priori. The model is investigated at first focusing on these topological properties of the emergent network, then its thermodynamics is analytically solved (at a replica symmetric level) by extending the double stochastic stability technique, and presented together with its fluctuation theory for a picture of criticality: both a statistical mechanics and a topological phase diagrams are obtained. Overall the picture depicted from statistical mechanics is quite intuitive: at least at equilibrium, dilution (of whatever kind) simply decreases the strength of the coupling felt by the spins, but leaves the paramagnetic/ferromagnetic flavors unchanged. The main difference with respect to previous investigations and a naive picture is that within our approach replicas do not appear: instead of (multi)-overlaps as order parameters, we introduce a class of magnetizations on all the possible sub-graphs belonging to the main one investigated: As a consequence, for these objects a closure for a self-consistent relation is achieved. 1As we will see, small values of a give rise to highly correlated, diluted networks, while, as a gets larger the network gets more and more connected and correlation among links vanishes.Even though the theory is defined at each finite V and L, as standard in statistical mechanics, we are interested in the large V behavior (such that, under central limit theorem permissions, deviations from averaged values become negligible and the theory predictive). To this task we find meaningful to let even L diverge linearly with the system size (to bridge conceptually to high storage neural networks), such that lim V →∞ L/V = α defines α as another control parameter. Finally, since we are interested in the regime of large V and large L we will often confuse V with V − 1 and L with L − 1. 4

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Section: The Emergent Networkmentioning

confidence: 99%

Equilibrium statistical mechanics on correlated random graphs

Barra¹,

Agliari²

2011

J. Stat. Mech.

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“…The non-similarity between two partitions could be confused and weakened by the presence of a tight common factor, that we would eliminate as far as possible, in order to amplify the Rohlin distance giving evidence to the real emerging novelty. However, such a “reduction” operation (analogous to the reduction to minimal terms for fractions) is not uniquely defined because partitions do not admit a unique factorization into primes [33] , [34] . The role of prime (i.e.…”

Section: Methodsmentioning

confidence: 99%

“…The process, therefore, essentially depends on the family of elementary factors, a choice which a priori can be implemented in many ways, reflecting the kind of interest the observer has in the experiment. Details and procedures in abstract probability spaces may be found in [33] , [34] . Here we sketch an algorithmically easy recipe, fitting the very special case of character strings.…”

Section: Methodsmentioning

confidence: 99%

Rohlin Distance and the Evolution of Influenza A Virus: Weak Attractors and Precursors

2011

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The evolution of the hemagglutinin amino acids sequences of Influenza A virus is studied by a method based on an informational metrics, originally introduced by Rohlin for partitions in abstract probability spaces. This metrics does not require any previous functional or syntactic knowledge about the sequences and it is sensitive to the correlated variations in the characters disposition. Its efficiency is improved by algorithmic tools, designed to enhance the detection of the novelty and to reduce the noise of useless mutations. We focus on the USA data from 1993/94 to 2010/2011 for A/H3N2 and on USA data from 2006/07 to 2010/2011 for A/H1N1. We show that the clusterization of the distance matrix gives strong evidence to a structure of domains in the sequence space, acting as weak attractors for the evolution, in very good agreement with the epidemiological history of the virus. The structure proves very robust with respect to the variations of the clusterization parameters, and extremely coherent when restricting the observation window. The results suggest an efficient strategy in the vaccine forecast, based on the presence of “precursors” (or “buds”) populating the most recent attractor.

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“…an efficient behavior in the whole range of temperatures from 0 to ∞. We remark, that previous results for such a dynamics on homogeneous or disordered systems, are limited to the stationary regime, which is established after a long-time interaction between the systems and the thermostats [19,20,21]. On the other hand, as already mentioned, here we focus on genuine non-stationary phenomena emerging when the system is suddenly put in contact with a thermostat at different temperature.…”

Section: Introductionmentioning

confidence: 80%

“…Therefore, the presence of very long transients described by a power law seems to be typical of infinite systems, while finite systems thermalize in a rapid (possibly exponential) way. This differences ensure that the dynamics may be used for the study of the stationary properties of finite systems as it has been done in [19,20,21]. Moreover, Fig.…”

Section: Finite-size Effectsmentioning

confidence: 99%

Slow relaxation in microcanonical warming of a Ising lattice

2011

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Abstract. We study the warming process of a semi-infinite cylindrical Ising lattice initially ordered and coupled at the boundary to a heat reservoir. The adoption of a proper microcanonical dynamics allows a detailed study of the time evolution of the system. As expected, thermal propagation displays a diffusive character and the spatial correlations decay exponentially in the direction orthogonal to the heat flow. However, we show that the approach to equilibrium presents an unexpected slow behavior. In particular, when the thermostat is at infinite temperature, correlations decay to their asymptotic values by a power law. This can be rephrased in terms of a correlation length vanishing logarithmically with time. At finite temperature, the approach to equilibrium is also a power law, but the exponents depend on the temperature in a non-trivial way. This complex behavior could be explained in terms of two dynamical regimes characterizing finite and infinite temperatures, respectively. When finite sizes are considered, we evidence the emergence of a much more rapid equilibration, and this confirms that the microcanonical dynamics can be successfully applied on finite structures. Indeed, the slowness exhibited by correlations in approaching the asymptotic values are expected to be related to the presence of an unsteady heat flow in an infinite system.

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Metric characterization of cluster dynamics on the Sierpinski gasket

Cited by 4 publications

References 29 publications

Equilibrium statistical mechanics on correlated random graphs

Equilibrium statistical mechanics on correlated random graphs

Rohlin Distance and the Evolution of Influenza A Virus: Weak Attractors and Precursors

Slow relaxation in microcanonical warming of a Ising lattice

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