15 research outputs found
Sparse random matrices: the eigenvalue spectrum revisited
We revisit the derivation of the density of states of sparse random matrices.
We derive a recursion relation that allows one to compute the spectrum of the
matrix of incidence for finite trees that determines completely the low
concentration limit. Using the iterative scheme introduced by Biroli and
Monasson [J. Phys. A 32, L255 (1999)] we find an approximate expression for the
density of states expected to hold exactly in the opposite limit of large but
finite concentration. The combination of the two methods yields a very simple
simple geometric interpretation of the tails of the spectrum. We test the
analytic results with numerical simulations and we suggest an indirect
numerical method to explore the tails of the spectrum.Comment: 18 pages, 7 figures. Accepted version, minor corrections, references
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Determining the Solution Space of Vertex-Cover by Interactions and Backbones
To solve the combinatorial optimization problems especially the minimal
Vertex-cover problem with high efficiency, is a significant task in theoretical
computer science and many other subjects. Aiming at detecting the solution
space of Vertex-cover, a new structure named interaction between nodes is
defined and discovered for random graph, which results in the emergence of the
frustration and long-range correlation phenomenon. Based on the backbones and
interactions with a node adding process, we propose an Interaction and Backbone
Evolution Algorithm to achieve the reduced solution graph, which has a direct
correspondence to the solution space of Vertex-cover. By this algorithm, the
whole solution space can be obtained strictly when there is no leaf-removal
core on the graph and the odd cycles of unfrozen nodes bring great obstacles to
its efficiency. Besides, this algorithm possesses favorable exactness and has
good performance on random instances even with high average degrees. The
interaction with the algorithm provides a new viewpoint to solve Vertex-cover,
which will have a wide range of applications to different types of graphs,
better usage of which can lower the computational complexity for solving
Vertex-cover
Spectra of Modular and Small-World Matrices
We compute spectra of symmetric random matrices describing graphs with
general modular structure and arbitrary inter- and intra-module degree
distributions, subject only to the constraint of finite mean connectivities. We
also evaluate spectra of a certain class of small-world matrices generated from
random graphs by introducing short-cuts via additional random connectivity
components. Both adjacency matrices and the associated graph Laplacians are
investigated. For the Laplacians, we find Lifshitz type singular behaviour of
the spectral density in a localised region of small values. In the
case of modular networks, we can identify contributions local densities of
state from individual modules. For small-world networks, we find that the
introduction of short cuts can lead to the creation of satellite bands outside
the central band of extended states, exhibiting only localised states in the
band-gaps. Results for the ensemble in the thermodynamic limit are in excellent
agreement with those obtained via a cavity approach for large finite single
instances, and with direct diagonalisation results.Comment: 18 pages, 5 figure
The random K-satisfiability problem: from an analytic solution to an efficient algorithm
We study the problem of satisfiability of randomly chosen clauses, each with
K Boolean variables. Using the cavity method at zero temperature, we find the
phase diagram for the K=3 case. We show the existence of an intermediate phase
in the satisfiable region, where the proliferation of metastable states is at
the origin of the slowdown of search algorithms. The fundamental order
parameter introduced in the cavity method, which consists of surveys of local
magnetic fields in the various possible states of the system, can be computed
for one given sample. These surveys can be used to invent new types of
algorithms for solving hard combinatorial optimizations problems. One such
algorithm is shown here for the 3-sat problem, with very good performances.Comment: 38 pages, 13 figures; corrected typo
Exact solutions for diluted spin glasses and optimization problems
We study the low temperature properties of p-spin glass models with finite
connectivity and of some optimization problems. Using a one-step functional
replica symmetry breaking Ansatz we can solve exactly the saddle-point
equations for graphs with uniform connectivity. The resulting ground state
energy is in perfect agreement with numerical simulations. For fluctuating
connectivity graphs, the same Ansatz can be used in a variational way: For
p-spin models (known as p-XOR-SAT in computer science) it provides the exact
configurational entropy together with the dynamical and static critical
connectivities (for p=3, \gamma_d=0.818 and \gamma_s=0.918 resp.), whereas for
hard optimization problems like 3-SAT or Bicoloring it provides new upper
bounds for their critical thresholds (\gamma_c^{var}=4.396 and
\gamma_c^{var}=2.149 resp.).Comment: 4 pages, 1 figure, accepted for publication in PR
Assigning Codes in a Random Wireless Network
In this paper we present an algorithm that can assign codes in the code division multiple access (CDMA) framework for multihop ad hoc wireless networks