12,348 research outputs found
Embeddings of 3-connected 3-regular planar graphs on surfaces of non-negative Euler characteristic
Whitney's theorem states that every 3-connected planar graph is uniquely
embeddable on the sphere. On the other hand, it has many inequivalent
embeddings on another surface. We shall characterize structures of a
-connected -regular planar graph embedded on the projective-plane,
the torus and the Klein bottle, and give a one-to-one correspondence between
inequivalent embeddings of on each surface and some subgraphs of the dual
of embedded on the sphere. These results enable us to give explicit bounds
for the number of inequivalent embeddings of on each surface, and propose
effective algorithms for enumerating and counting these embeddings.Comment: 19 pages, 12 figure
LINE: Large-scale Information Network Embedding
This paper studies the problem of embedding very large information networks
into low-dimensional vector spaces, which is useful in many tasks such as
visualization, node classification, and link prediction. Most existing graph
embedding methods do not scale for real world information networks which
usually contain millions of nodes. In this paper, we propose a novel network
embedding method called the "LINE," which is suitable for arbitrary types of
information networks: undirected, directed, and/or weighted. The method
optimizes a carefully designed objective function that preserves both the local
and global network structures. An edge-sampling algorithm is proposed that
addresses the limitation of the classical stochastic gradient descent and
improves both the effectiveness and the efficiency of the inference. Empirical
experiments prove the effectiveness of the LINE on a variety of real-world
information networks, including language networks, social networks, and
citation networks. The algorithm is very efficient, which is able to learn the
embedding of a network with millions of vertices and billions of edges in a few
hours on a typical single machine. The source code of the LINE is available
online.Comment: WWW 201
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