3,530 research outputs found
On the Spectral Gap of a Quantum Graph
We consider the problem of finding universal bounds of "isoperimetric" or
"isodiametric" type on the spectral gap of the Laplacian on a metric graph with
natural boundary conditions at the vertices, in terms of various analytical and
combinatorial properties of the graph: its total length, diameter, number of
vertices and number of edges. We investigate which combinations of parameters
are necessary to obtain non-trivial upper and lower bounds and obtain a number
of sharp estimates in terms of these parameters. We also show that, in contrast
to the Laplacian matrix on a combinatorial graph, no bound depending only on
the diameter is possible. As a special case of our results on metric graphs, we
deduce estimates for the normalised Laplacian matrix on combinatorial graphs
which, surprisingly, are sometimes sharper than the ones obtained by purely
combinatorial methods in the graph theoretical literature
Representation Learning for Scale-free Networks
Network embedding aims to learn the low-dimensional representations of
vertexes in a network, while structure and inherent properties of the network
is preserved. Existing network embedding works primarily focus on preserving
the microscopic structure, such as the first- and second-order proximity of
vertexes, while the macroscopic scale-free property is largely ignored.
Scale-free property depicts the fact that vertex degrees follow a heavy-tailed
distribution (i.e., only a few vertexes have high degrees) and is a critical
property of real-world networks, such as social networks. In this paper, we
study the problem of learning representations for scale-free networks. We first
theoretically analyze the difficulty of embedding and reconstructing a
scale-free network in the Euclidean space, by converting our problem to the
sphere packing problem. Then, we propose the "degree penalty" principle for
designing scale-free property preserving network embedding algorithm: punishing
the proximity between high-degree vertexes. We introduce two implementations of
our principle by utilizing the spectral techniques and a skip-gram model
respectively. Extensive experiments on six datasets show that our algorithms
are able to not only reconstruct heavy-tailed distributed degree distribution,
but also outperform state-of-the-art embedding models in various network mining
tasks, such as vertex classification and link prediction.Comment: 8 figures; accepted by AAAI 201
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