Let @math be a simplicial complex with a piecewise linear function @math . The Reeb graph @math is the quotient of @math , where we collapse each connected component of @math to a single point. Let the nodes of @math be all homologically critical points where any homology of the corresponding component of the level set @math changes. Then we can label every arc of @math with the Betti numbers @math of the corresponding @math -dimensional component of a level set. The homology labels give more information about the original complex @math than the classical Reeb graph. We describe a canonical embedding of a Reeb graph into a multi-page book (a star cross a line) and give a unique linear code of this book embedding.
Our methods for book embeddings were inspired by basic embeddings of graphs @cite_5 and by a reduction of the topological classification for graphs embedded in @math to word problems in finitely presented semigroups @cite_0 .
@cite_0 @cite_5
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The Web offers rich relational data with different semantics. In this paper, we address the problem of document recommendation in a digital library, where the documents in question are networked by citations and are associated with other entities by various relations. Due to the sparsity of a single graph and noise in graph construction, we propose a new method for combining multiple graphs to measure document similarities, where different factorization strategies are used based on the nature of different graphs. In particular, the new method seeks a single low-dimensional embedding of documents that captures their relative similarities in a latent space. Based on the obtained embedding, a new recommendation framework is developed using semi-supervised learning on graphs. In addition, we address the scalability issue and propose an incremental algorithm. The new incremental method significantly improves the efficiency by calculating the embedding for new incoming documents only. The new batch and incremental methods are evaluated on two real world datasets prepared from CiteSeer. Experiments demonstrate significant quality improvement for our batch method and significant efficiency improvement with tolerable quality loss for our incremental method. The notion of a basic embedding appeared in research motivated by Kolmogorov–Arnold's solution of Hilbert's 13th problem. Let K,X,Y be topological spaces. An embedding K⊂X×Y is called basic if for every continuous function f:K→R there exist continuous functions g:X→R, h:Y→R such that f(x,y)=g(x)+h(y) for any point (x,y)∈K. Let Ti be an i-od. Theorem. There exists only a finite number of prohibited' subgraphs for basic embeddings into R×Tn. Consequently, for a finite graph K there is an algorithm for checking whether K is basically embeddable into R×Tn. Our theorem is a generalization of Skopenkov's description of graphs basically embeddable into R2, and our proofs is a (non-trivial) extension of that one. Abstract An algorithm is presented which embeds any planar graph in a book of four pages. The algorithm runs in linear time. This paper presents a novel, graph embedding based recommendation technique. The method operates on the knowledge graph, an information representation technique alloying content-based and collaborative information. To generate recommendations, a two dimensional embedding is developed for the knowledge graph. As the embedding maps the users and the items to the same vector space, the recommendations are then calculated on a spatial basis. Regarding to the number of cold start cases, precision, recall, normalized Cumulative Discounted Gain and computational resource need, the evaluation shows that the introduced technique delivers a higher performance compared to collaborative filtering on top-n recommendation lists. Our further finding is that graph embedding based methods show a more stable performance in the case of an increasing amount of user preference information compared to the benchmark method.