246,262 research outputs found
Exploring Quantum Annealing Architectures: A Spin Glass Perspective
We study the spin-glass transition in several Ising models of relevance for
quantum annealers. We extract the spin-glass critical temperature by
extrapolating the pseudo-critical properties obtained with Replica-Exchange
Monte-Carlo for finite-size systems. We find a spin-glass phase for some random
lattices (random-regular and small-world graphs) in good agreement with
previous results. However, our results for the quasi-two-dimensional graphs
implemented in the D-Wave annealers (Chimera, Zephyr, and Pegasus) indicate
only a zero-temperature spin-glass state, as their pseudo-critical temperature
drifts towards smaller values. This implies that the asymptotic runtime to find
the low-energy configuration of those graphs is likely to be polynomial in
system size, nevertheless, this scaling may only be reached for very large
system sizes -- much larger than existing annealers -- as we observe an abrupt
increase in the computational cost of the simulations around the
pseudo-critical temperatures. Thus, two-dimensional systems with local
crossings can display enough complexity to make unfeasible the search with
classical methods of low-energy configurations.Comment: 13 page
Circular Coloring of Random Graphs: Statistical Physics Investigation
Circular coloring is a constraints satisfaction problem where colors are
assigned to nodes in a graph in such a way that every pair of connected nodes
has two consecutive colors (the first color being consecutive to the last). We
study circular coloring of random graphs using the cavity method. We identify
two very interesting properties of this problem. For sufficiently many color
and sufficiently low temperature there is a spontaneous breaking of the
circular symmetry between colors and a phase transition forwards a
ferromagnet-like phase. Our second main result concerns 5-circular coloring of
random 3-regular graphs. While this case is found colorable, we conclude that
the description via one-step replica symmetry breaking is not sufficient. We
observe that simulated annealing is very efficient to find proper colorings for
this case. The 5-circular coloring of 3-regular random graphs thus provides a
first known example of a problem where the ground state energy is known to be
exactly zero yet the space of solutions probably requires a full-step replica
symmetry breaking treatment.Comment: 19 pages, 8 figures, 3 table
Correlation of Automorphism Group Size and Topological Properties with Program-size Complexity Evaluations of Graphs and Complex Networks
We show that numerical approximations of Kolmogorov complexity (K) applied to
graph adjacency matrices capture some group-theoretic and topological
properties of graphs and empirical networks ranging from metabolic to social
networks. That K and the size of the group of automorphisms of a graph are
correlated opens up interesting connections to problems in computational
geometry, and thus connects several measures and concepts from complexity
science. We show that approximations of K characterise synthetic and natural
networks by their generating mechanisms, assigning lower algorithmic randomness
to complex network models (Watts-Strogatz and Barabasi-Albert networks) and
high Kolmogorov complexity to (random) Erdos-Renyi graphs. We derive these
results via two different Kolmogorov complexity approximation methods applied
to the adjacency matrices of the graphs and networks. The methods used are the
traditional lossless compression approach to Kolmogorov complexity, and a
normalised version of a Block Decomposition Method (BDM) measure, based on
algorithmic probability theory.Comment: 15 2-column pages, 20 figures. Forthcoming in Physica A: Statistical
Mechanics and its Application
The Thermodynamics of Network Coding, and an Algorithmic Refinement of the Principle of Maximum Entropy
The principle of maximum entropy (Maxent) is often used to obtain prior
probability distributions as a method to obtain a Gibbs measure under some
restriction giving the probability that a system will be in a certain state
compared to the rest of the elements in the distribution. Because classical
entropy-based Maxent collapses cases confounding all distinct degrees of
randomness and pseudo-randomness, here we take into consideration the
generative mechanism of the systems considered in the ensemble to separate
objects that may comply with the principle under some restriction and whose
entropy is maximal but may be generated recursively from those that are
actually algorithmically random offering a refinement to classical Maxent. We
take advantage of a causal algorithmic calculus to derive a thermodynamic-like
result based on how difficult it is to reprogram a computer code. Using the
distinction between computable and algorithmic randomness we quantify the cost
in information loss associated with reprogramming. To illustrate this we apply
the algorithmic refinement to Maxent on graphs and introduce a Maximal
Algorithmic Randomness Preferential Attachment (MARPA) Algorithm, a
generalisation over previous approaches. We discuss practical implications of
evaluation of network randomness. Our analysis provides insight in that the
reprogrammability asymmetry appears to originate from a non-monotonic
relationship to algorithmic probability. Our analysis motivates further
analysis of the origin and consequences of the aforementioned asymmetries,
reprogrammability, and computation.Comment: 30 page
Parameterized Complexity of Graph Constraint Logic
Graph constraint logic is a framework introduced by Hearn and Demaine, which
provides several problems that are often a convenient starting point for
reductions. We study the parameterized complexity of Constraint Graph
Satisfiability and both bounded and unbounded versions of Nondeterministic
Constraint Logic (NCL) with respect to solution length, treewidth and maximum
degree of the underlying constraint graph as parameters. As a main result we
show that restricted NCL remains PSPACE-complete on graphs of bounded
bandwidth, strengthening Hearn and Demaine's framework. This allows us to
improve upon existing results obtained by reduction from NCL. We show that
reconfiguration versions of several classical graph problems (including
independent set, feedback vertex set and dominating set) are PSPACE-complete on
planar graphs of bounded bandwidth and that Rush Hour, generalized to boards, is PSPACE-complete even when is at most a constant
Next nearest neighbour Ising models on random graphs
This paper develops results for the next nearest neighbour Ising model on
random graphs. Besides being an essential ingredient in classic models for
frustrated systems, second neighbour interactions interactions arise naturally
in several applications such as the colour diversity problem and graphical
games. We demonstrate ensembles of random graphs, including regular
connectivity graphs, that have a periodic variation of free energy, with either
the ratio of nearest to next nearest couplings, or the mean number of nearest
neighbours. When the coupling ratio is integer paramagnetic phases can be found
at zero temperature. This is shown to be related to the locked or unlocked
nature of the interactions. For anti-ferromagnetic couplings, spin glass phases
are demonstrated at low temperature. The interaction structure is formulated as
a factor graph, the solution on a tree is developed. The replica symmetric and
energetic one-step replica symmetry breaking solution is developed using the
cavity method. We calculate within these frameworks the phase diagram and
demonstrate the existence of dynamical transitions at zero temperature for
cases of anti-ferromagnetic coupling on regular and inhomogeneous random
graphs.Comment: 55 pages, 15 figures, version 2 with minor revisions, to be published
J. Stat. Mec
Evolutionary constraints on the complexity of genetic regulatory networks allow predictions of the total number of genetic interactions
Genetic regulatory networks (GRNs) have been widely studied, yet there is a
lack of understanding with regards to the final size and properties of these
networks, mainly due to no network currently being complete. In this study, we
analyzed the distribution of GRN structural properties across a large set of
distinct prokaryotic organisms and found a set of constrained characteristics
such as network density and number of regulators. Our results allowed us to
estimate the number of interactions that complete networks would have, a
valuable insight that could aid in the daunting task of network curation,
prediction, and validation. Using state-of-the-art statistical approaches, we
also provided new evidence to settle a previously stated controversy that
raised the possibility of complete biological networks being random and
therefore attributing the observed scale-free properties to an artifact
emerging from the sampling process during network discovery. Furthermore, we
identified a set of properties that enabled us to assess the consistency of the
connectivity distribution for various GRNs against different alternative
statistical distributions. Our results favor the hypothesis that highly
connected nodes (hubs) are not a consequence of network incompleteness.
Finally, an interaction coverage computed for the GRNs as a proxy for
completeness revealed that high-throughput based reconstructions of GRNs could
yield biased networks with a low average clustering coefficient, showing that
classical targeted discovery of interactions is still needed.Comment: 28 pages, 5 figures, 12 pages supplementary informatio
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