1,921 research outputs found
A Hebbian approach to complex network generation
Through a redefinition of patterns in an Hopfield-like model, we introduce
and develop an approach to model discrete systems made up of many, interacting
components with inner degrees of freedom. Our approach clarifies the intrinsic
connection between the kind of interactions among components and the emergent
topology describing the system itself; also, it allows to effectively address
the statistical mechanics on the resulting networks. Indeed, a wide class of
analytically treatable, weighted random graphs with a tunable level of
correlation can be recovered and controlled. We especially focus on the case of
imitative couplings among components endowed with similar patterns (i.e.
attributes), which, as we show, naturally and without any a-priori assumption,
gives rise to small-world effects. We also solve the thermodynamics (at a
replica symmetric level) by extending the double stochastic stability
technique: free energy, self consistency relations and fluctuation analysis for
a picture of criticality are obtained
Quasi-Static Brittle Fracture in Inhomogeneous Media and Iterated Conformal Maps: Modes I, II and III
The method of iterated conformal maps is developed for quasi-static fracture
of brittle materials, for all modes of fracture. Previous theory, that was
relevant for mode III only, is extended here to mode I and II. The latter
require solution of the bi-Laplace rather than the Laplace equation. For all
cases we can consider quenched randomness in the brittle material itself, as
well as randomness in the succession of fracture events. While mode III calls
for the advance (in time) of one analytic function, mode I and II call for the
advance of two analytic functions. This fundamental difference creates
different stress distribution around the cracks. As a result the geometric
characteristics of the cracks differ, putting mode III in a different class
compared to modes I and II.Comment: submitted to PRE For a version with qualitatively better figures see:
http://www.weizmann.ac.il/chemphys/ander
Equilibrium statistical mechanics on correlated random graphs
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 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 - 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. At least at
equilibrium, dilution 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.Comment: 30 pages, 4 figure
Ferromagnetic models for cooperative behavior: Revisiting Universality in complex phenomena
Ferromagnetic models are harmonic oscillators in statistical mechanics.
Beyond their original scope in tackling phase transition and symmetry breaking
in theoretical physics, they are nowadays experiencing a renewal applicative
interest as they capture the main features of disparate complex phenomena,
whose quantitative investigation in the past were forbidden due to data
lacking. After a streamlined introduction to these models, suitably embedded on
random graphs, aim of the present paper is to show their importance in a
plethora of widespread research fields, so to highlight the unifying framework
reached by using statistical mechanics as a tool for their investigation.
Specifically we will deal with examples stemmed from sociology, chemistry,
cybernetics (electronics) and biology (immunology).Comment: Contributing to the proceedings of the Conference "Mathematical
models and methods for Planet Heart", INdAM, Rome 201
Inertial terms to magnetization dynamics in ferromagnetic thin films
Inertial magnetization dynamics have been predicted at ultrahigh speeds, or
frequencies approaching the energy relaxation scale of electrons, in
ferromagnetic metals. Here we identify inertial terms to magnetization dynamics
in thin NiFe and Co films near room temperature. Effective
magnetic fields measured in high-frequency ferromagnetic resonance (115-345
GHz) show an additional stiffening term which is quadratic in frequency and
80 mT at the high frequency limit of our experiment. Our results extend
understanding of magnetization dynamics at sub-picosecond time scales.Comment: 11 pages, 3 figure
Analogue neural networks on correlated random graphs
We consider a generalization of the Hopfield model, where the entries of
patterns are Gaussian and diluted. We focus on the high-storage regime and we
investigate analytically the topological properties of the emergent network, as
well as the thermodynamic properties of the model. We find that, by properly
tuning the dilution in the pattern entries, the network can recover different
topological regimes characterized by peculiar scalings of the average
coordination number with respect to the system size. The structure is also
shown to exhibit a large degree of cliquishness, even when very sparse.
Moreover, we obtain explicitly the replica symmetric free energy and the
self-consistency equations for the overlaps (order parameters of the theory),
which turn out to be classical weighted sums of 'sub-overlaps' defined on all
possible sub-graphs. Finally, a study of criticality is performed through a
small-overlap expansion of the self-consistencies and through a whole
fluctuation theory developed for their rescaled correlations: Both approaches
show that the net effect of dilution in pattern entries is to rescale the
critical noise level at which ergodicity breaks down.Comment: 34 pages, 3 figure
Transport and dynamics on open quantum graphs
We study the classical limit of quantum mechanics on graphs by introducing a
Wigner function for graphs. The classical dynamics is compared to the quantum
dynamics obtained from the propagator. In particular we consider extended open
graphs whose classical dynamics generate a diffusion process. The transport
properties of the classical system are revealed in the scattering resonances
and in the time evolution of the quantum system.Comment: 42 pages, 13 figures, submitted to PR
A Two-populations Ising model on diluted Random Graphs
We consider the Ising model for two interacting groups of spins embedded in
an Erd\"{o}s-R\'{e}nyi random graph. The critical properties of the system are
investigated by means of extensive Monte Carlo simulations. Our results
evidence the existence of a phase transition at a value of the inter-groups
interaction coupling which depends algebraically on the dilution of
the graph and on the relative width of the two populations, as explained by
means of scaling arguments. We also measure the critical exponents, which are
consistent with those of the Curie-Weiss model, hence suggesting a wide
robustness of the universality class.Comment: 11 pages, 4 figure
Criticality in diluted ferromagnet
We perform a detailed study of the critical behavior of the mean field
diluted Ising ferromagnet by analytical and numerical tools. We obtain
self-averaging for the magnetization and write down an expansion for the free
energy close to the critical line. The scaling of the magnetization is also
rigorously obtained and compared with extensive Monte Carlo simulations. We
explain the transition from an ergodic region to a non trivial phase by
commutativity breaking of the infinite volume limit and a suitable vanishing
field. We find full agreement among theory, simulations and previous results.Comment: 23 pages, 3 figure
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