29,904 research outputs found
Isomorphism Checking for Symmetry Reduction
In this paper, we show how isomorphism checking can be used as an effective technique for symmetry reduction. Reduced state spaces are equivalent to the original ones under a strong notion of bisimilarity which preserves the multiplicity of outgoing transitions, and therefore also preserves stochastic temporal logics. We have implemented this in a setting where states are arbitrary graphs. Since no efficiently computable canonical representation is known for arbitrary graphs modulo isomorphism, we define an isomorphism-predicting hash function on the basis of an existing partition refinement algorithm. As an example, we report a factorial state space reduction on a model of an ad-hoc network connectivity protocol
The statistical mechanics of networks
We study the family of network models derived by requiring the expected
properties of a graph ensemble to match a given set of measurements of a
real-world network, while maximizing the entropy of the ensemble. Models of
this type play the same role in the study of networks as is played by the
Boltzmann distribution in classical statistical mechanics; they offer the best
prediction of network properties subject to the constraints imposed by a given
set of observations. We give exact solutions of models within this class that
incorporate arbitrary degree distributions and arbitrary but independent edge
probabilities. We also discuss some more complex examples with correlated edges
that can be solved approximately or exactly by adapting various familiar
methods, including mean-field theory, perturbation theory, and saddle-point
expansions.Comment: 15 pages, 4 figure
Notes on Melonic Tensor Models
It has recently been demonstrated that the large N limit of a model of
fermions charged under the global/gauge symmetry group agrees with
the large limit of the SYK model. In these notes we investigate aspects of
the dynamics of the theories that differ from their SYK
counterparts. We argue that the spectrum of fluctuations about the finite
temperature saddle point in these theories has new light
modes in addition to the light Schwarzian mode that exists even in the SYK
model, suggesting that the bulk dual description of theories differ
significantly if they both exist. We also study the thermal partition function
of a mass deformed version of the SYK model. At large mass we show that the
effective entropy of this theory grows with energy like (i.e. faster
than Hagedorn) up to energies of order . The canonical partition function
of the model displays a deconfinement or Hawking Page type phase transition at
temperatures of order . We derive these results in the large mass
limit but argue that they are qualitatively robust to small corrections in
.Comment: 60 pages, 7 figure
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