816 research outputs found
Zero-Temperature Complex Replica Zeros of the Ising Spin Glass on Mean-Field Systems and Beyond
Zeros of the moment of the partition function with respect
to complex are investigated in the zero temperature limit , keeping . We numerically investigate
the zeros of the Ising spin glass models on several Cayley trees and
hierarchical lattices and compare those results. In both lattices, the
calculations are carried out with feasible computational costs by using
recursion relations originated from the structures of those lattices. The
results for Cayley trees show that a sequence of the zeros approaches the real
axis of implying that a certain type of analyticity breaking actually
occurs, although it is irrelevant for any known replica symmetry breaking. The
result of hierarchical lattices also shows the presence of analyticity
breaking, even in the two dimensional case in which there is no
finite-temperature spin-glass transition, which implies the existence of the
zero-temperature phase transition in the system. A notable tendency of
hierarchical lattices is that the zeros spread in a wide region of the complex
plane in comparison with the case of Cayley trees, which may reflect the
difference between the mean-field and finite-dimensional systems.Comment: 4 pages, 4 figure
Growing Cayley trees described by Fermi distribution
We introduce a model for growing Cayley trees with thermal noise. The
evolution of these hierarchical networks reduces to the Eden model and the
Invasion Percolation model in the limit , respectively.
We show that the distribution of the bond strengths (energies) is described by
the Fermi statistics. We discuss the relation of the present results with the
scale-free networks described by Bose statistics
Analytical controllability of deterministic scale-free networks and Cayley trees
According to the exact controllability theory, the controllability is
investigated analytically for two typical types of self-similar bipartite
networks, i.e., the classic deterministic scale-free networks and Cayley trees.
Due to their self-similarity, the analytical results of the exact
controllability are obtained, and the minimum sets of driver nodes (drivers)
are also identified by elementary transformations on adjacency matrices. For
these two types of undirected networks, no matter their links are unweighted or
(nonzero) weighted, the controllability of networks and the configuration of
drivers remain the same, showing a robustness to the link weights. These
results have implications for the control of real networked systems with
self-similarity.Comment: 7 pages, 4 figures, 1 table; revised manuscript; added discussion
about the general case of DSFN; added 3 reference
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