94,727 research outputs found
Efficiently Generating Geometric Inhomogeneous and Hyperbolic Random Graphs
Hyperbolic random graphs (HRG) and geometric inhomogeneous random graphs (GIRG) are two similar generative network models that were designed to resemble complex real world networks. In particular, they have a power-law degree distribution with controllable exponent beta, and high clustering that can be controlled via the temperature T.
We present the first implementation of an efficient GIRG generator running in expected linear time. Besides varying temperatures, it also supports underlying geometries of higher dimensions. It is capable of generating graphs with ten million edges in under a second on commodity hardware. The algorithm can be adapted to HRGs. Our resulting implementation is the fastest sequential HRG generator, despite the fact that we support non-zero temperatures. Though non-zero temperatures are crucial for many applications, most existing generators are restricted to T = 0. We also support parallelization, although this is not the focus of this paper. Moreover, we note that our generators draw from the correct probability distribution, i.e., they involve no approximation.
Besides the generators themselves, we also provide an efficient algorithm to determine the non-trivial dependency between the average degree of the resulting graph and the input parameters of the GIRG model. This makes it possible to specify the desired expected average degree as input.
Moreover, we investigate the differences between HRGs and GIRGs, shedding new light on the nature of the relation between the two models. Although HRGs represent, in a certain sense, a special case of the GIRG model, we find that a straight-forward inclusion does not hold in practice. However, the difference is negligible for most use cases
Efficiently generating geometric inhomogeneous and hyperbolic random graphs
Hyperbolic random graphs (HRGs) and geometric inhomogeneous random graphs (GIRGs) are two similar generative network models that were designed to resemble complex real-world networks. In particular, they have a power-law degree distribution with controllable exponent and high clustering that can be controlled via the temperature .
We present the first implementation of an efficient GIRG generator running in expected linear time. Besides varying temperatures, it also supports underlying geometries of higher dimensions. It is capable of generating graphs with ten million edges in under a second on commodity hardware. The algorithm can be adapted to HRGs. Our resulting implementation is the fastest sequential HRG generator, despite the fact that we support non-zero temperatures. Though non-zero temperatures are crucial for many applications, most existing generators are restricted to . We also support parallelization, although this is not the focus of this paper. Moreover, we note that our generators draw from the correct probability distribution, that is, they involve no approximation.
Besides the generators themselves, we also provide an efficient algorithm to determine the non-trivial dependency between the average degree of the resulting graph and the input parameters of the GIRG model. This makes it possible to specify the desired expected average degree as input.
Moreover, we investigate the differences between HRGs and GIRGs, shedding new light on the nature of the relation between the two models. Although HRGs represent, in a certain sense, a special case of the GIRG model, we find that a straightforward inclusion does not hold in practice. However, the difference is negligible for most use cases
An invariant of scale-free graphs
In many applications (including biology and the study of computer networks) graphs are found to be scale-free. It has been argued that this property alone does not tell us much about the structure of the graph. In this paper, we introduce a numerical characteristic of a graph, which we call the astral index, and which can be calculated efficiently. We demonstrate that the Barab�si-Albert algorithm for generating scale-free graphs produces not just scale-free graphs, but only scale-free graphs with a constant astral index. On some examples of biological graphs, we see that they not only are scale-free, but also share the value of the astral index with Barab�si-Albert graphs. For comparison, we demonstrate that the Erd?s?R�nyi model for generating random graphs also generates only graphs with a constant astral index, whose value significantly differs from that of graphs generated by the Barab�si-Albert algorithm
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The Unified Theory of Pseudorandomness
Pseudorandomness is the theory of efficiently generating objects that look "random" despite being constructed with little or no randomness. One of the achievements of this research area has been the realization that a number of fundamental and widely studied "pseudorandom" objects are all almost equivalent when viewed appropriately. These objects include pseudorandom generators, expander graphs, list-decodable error-correcting codes, averaging samplers, and hardness amplifi ers. In this survey, we describe the connections between all of these objects, showing how they can all be cast within a single "list-decoding framework" that brings out both their similarities and differences.Engineering and Applied Science
A quantum Monte Carlo algorithm for Bose-Hubbard models on arbitrary graphs
We propose a quantum Monte Carlo algorithm capable of simulating the
Bose-Hubbard model on arbitrary graphs, obviating the need for devising
lattice-specific updates for different input graphs. We show that with our
method, which is based on the recently introduced Permutation Matrix
Representation Quantum Monte Carlo [Gupta, Albash and Hen, J. Stat. Mech.
(2020) 073105], the problem of adapting the simulation to a given geometry
amounts to generating a cycle basis for the graph on which the model is
defined, a procedure that can be carried out efficiently and and in an
automated manner. To showcase the versatility of our approach, we provide
simulation results for Bose-Hubbard models defined on two-dimensional lattices
as well as on a number of random graphs.Comment: 10 pages, 6 figure
Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm
Many practical problems in almost all scientific and technological
disciplines have been classified as computationally hard (NP-hard or even
NP-complete). In life sciences, combinatorial optimization problems frequently
arise in molecular biology, e.g., genome sequencing; global alignment of
multiple genomes; identifying siblings or discovery of dysregulated pathways.In
almost all of these problems, there is the need for proving a hypothesis about
certain property of an object that can be present only when it adopts some
particular admissible structure (an NP-certificate) or be absent (no admissible
structure), however, none of the standard approaches can discard the hypothesis
when no solution can be found, since none can provide a proof that there is no
admissible structure. This article presents an algorithm that introduces a
novel type of solution method to "efficiently" solve the graph 3-coloring
problem; an NP-complete problem. The proposed method provides certificates
(proofs) in both cases: present or absent, so it is possible to accept or
reject the hypothesis on the basis of a rigorous proof. It provides exact
solutions and is polynomial-time (i.e., efficient) however parametric. The only
requirement is sufficient computational power, which is controlled by the
parameter . Nevertheless, here it is proved that the
probability of requiring a value of to obtain a solution for a
random graph decreases exponentially: , making
tractable almost all problem instances. Thorough experimental analyses were
performed. The algorithm was tested on random graphs, planar graphs and
4-regular planar graphs. The obtained experimental results are in accordance
with the theoretical expected results.Comment: Working pape
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