3,239 research outputs found
Switch chain mixing times through triangle counts
Sampling uniform simple graphs with power-law degree distributions with
degree exponent is a non-trivial problem. We propose a method to
sample uniform simple graphs that uses a constrained version of the
configuration model together with a Markov Chain switching method. We test the
convergence of this algorithm numerically in the context of the presence of
small subgraphs. We then compare the number of triangles in uniform random
graphs with the number of triangles in the erased configuration model. Using
simulations and heuristic arguments, we conjecture that the number of triangles
in the erased configuration model is larger than the number of triangles in the
uniform random graph, provided that the graph is sufficiently large.Comment: 7 pages, 8 figures in the main article. 2 pages, 2 figures in the
supplementary materia
Switch chain mixing times through triangle counts
Sampling uniform simple graphs with power-law degree distributions with degree exponent Ïâ(2,3) is a non-trivial problem. We propose a method to sample uniform simple graphs that uses a constrained version of the configuration model together with a Markov Chain switching method. We test the convergence of this algorithm numerically in the context of the presence of small subgraphs. We then compare the number of triangles in uniform random graphs with the number of triangles in the erased configuration model. Using simulations and heuristic arguments, we conjecture that the number of triangles in the erased configuration model is larger than the number of triangles in the uniform random graph, provided that the graph is sufficiently large
A triangle process on graphs with given degree sequence
The triangle switch Markov chain is designed to generate random graphs with
given degree sequence, but having more triangles than would appear under the
uniform distribution. Transition probabilities of the chain depends on a
parameter, called the activity, which is used to assign higher stationary
probability to graphs with more triangles. In previous work we proved
ergodicity of the triangle switch chain for regular graphs. Here we prove
ergodicity for all sequences with minimum degree at least 3, and show rapid
mixing of the chain when the activity and the maximum degree are not too large.
As far as we are aware, this is the first rigorous analysis of a Markov chain
algorithm for generating graphs from a a known non-uniform distribution.Comment: 35 page
Simulation of quantum walks and fast mixing with classical processes
We compare discrete-time quantum walks on graphs to their natural classical equivalents, which we argue are lifted Markov chains (LMCs), that is, classical Markov chains with added memory. We show that LMCs can simulate the mixing behavior of any quantum walk, under a commonly satisfied invariance condition. This allows us to answer an open question on how the graph topology ultimately bounds a quantum walk's mixing performance, and that of any stochastic local evolution. The results highlight that speedups in mixing and transport phenomena are not necessarily diagnostic of quantum effects, although superdiffusive spreading is more prominent with quantum walks. The general simulating LMC construction may lead to large memory, yet we show that for the main graphs under study (i.e., lattices) this memory can be brought down to the same size employed in the quantum walks proposed in the literature
Parallel Global Edge Switching for the Uniform Sampling of Simple Graphs with Prescribed Degrees
The uniform sampling of simple graphs matching a prescribed degree sequence
is an important tool in network science, e.g. to construct graph generators or
null-models. Here, the Edge Switching Markov Chain (ES-MC) is a common choice.
Given an arbitrary simple graph with the required degree sequence, ES-MC
carries out a large number of small changes, called edge switches, to
eventually obtain a uniform sample. In practice, reasonably short runs
efficiently yield approximate uniform samples.
In this work, we study the problem of executing edge switches in parallel. We
discuss parallelizations of ES-MC, but find that this approach suffers from
complex dependencies between edge switches. For this reason, we propose the
Global Edge Switching Markov Chain (G-ES-MC), an ES-MC variant with simpler
dependencies. We show that G-ES-MC converges to the uniform distribution and
design shared-memory parallel algorithms for ES-MC and G-ES-MC. In an empirical
evaluation, we provide evidence that G-ES-MC requires not more switches than
ES-MC (and often fewer), and demonstrate the efficiency and scalability of our
parallel G-ES-MC implementation
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