243,525 research outputs found
Brief Announcement: Node Sampling Using Centrifugal Random Walks.
We propose distributed algorithms for sampling networks based on a new class of random walks that we call Centrifugal Random Walks (CRW). A CRW is a random walk that starts at a source and always moves away from it. We propose CRW algorithms for connected networks with arbitrary probability distributions, and for grids and networks with regular concentric connectivity with distance based distributions. All CRW sampling algorithms select a node with the exact probability distribution, do not need warm-up, and end in a number of hops bounded by the network diameter
Random graphs with arbitrary degree distributions and their applications
Recent work on the structure of social networks and the internet has focussed
attention on graphs with distributions of vertex degree that are significantly
different from the Poisson degree distributions that have been widely studied
in the past. In this paper we develop in detail the theory of random graphs
with arbitrary degree distributions. In addition to simple undirected,
unipartite graphs, we examine the properties of directed and bipartite graphs.
Among other results, we derive exact expressions for the position of the phase
transition at which a giant component first forms, the mean component size, the
size of the giant component if there is one, the mean number of vertices a
certain distance away from a randomly chosen vertex, and the average
vertex-vertex distance within a graph. We apply our theory to some real-world
graphs, including the world-wide web and collaboration graphs of scientists and
Fortune 1000 company directors. We demonstrate that in some cases random graphs
with appropriate distributions of vertex degree predict with surprising
accuracy the behavior of the real world, while in others there is a measurable
discrepancy between theory and reality, perhaps indicating the presence of
additional social structure in the network that is not captured by the random
graph.Comment: 19 pages, 11 figures, some new material added in this version along
with minor updates and correction
Self-avoiding walks and connective constants in small-world networks
Long-distance characteristics of small-world networks have been studied by
means of self-avoiding walks (SAW's). We consider networks generated by
rewiring links in one- and two-dimensional regular lattices. The number of
SAW's was obtained from numerical simulations as a function of the number
of steps on the considered networks. The so-called connective constant,
, which characterizes the long-distance
behavior of the walks, increases continuously with disorder strength (or
rewiring probability, ). For small , one has a linear relation , and being constants dependent on the underlying
lattice. Close to one finds the behavior expected for random graphs. An
analytical approach is given to account for the results derived from numerical
simulations. Both methods yield results agreeing with each other for small ,
and differ for close to 1, because of the different connectivity
distributions resulting in both cases.Comment: 7 pages, 5 figure
Sliced Wasserstein Generative Models
In generative modeling, the Wasserstein distance (WD) has emerged as a useful
metric to measure the discrepancy between generated and real data
distributions. Unfortunately, it is challenging to approximate the WD of
high-dimensional distributions. In contrast, the sliced Wasserstein distance
(SWD) factorizes high-dimensional distributions into their multiple
one-dimensional marginal distributions and is thus easier to approximate. In
this paper, we introduce novel approximations of the primal and dual SWD.
Instead of using a large number of random projections, as it is done by
conventional SWD approximation methods, we propose to approximate SWDs with a
small number of parameterized orthogonal projections in an end-to-end deep
learning fashion. As concrete applications of our SWD approximations, we design
two types of differentiable SWD blocks to equip modern generative
frameworks---Auto-Encoders (AE) and Generative Adversarial Networks (GAN). In
the experiments, we not only show the superiority of the proposed generative
models on standard image synthesis benchmarks, but also demonstrate the
state-of-the-art performance on challenging high resolution image and video
generation in an unsupervised manner.Comment: This paper is accepted by CVPR 2019, accidentally uploaded as a new
submission (arXiv:1904.05408, which has been withdrawn). The code is
available at this https URL https:// github.com/musikisomorphie/swd.gi
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