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Learned latent vector representations are key to the success of many recommender systems in recent years. However, traditional approaches like matrix factorization produce vector representations that capture global distributions of a static recommendation scenario only. Such latent user or item representations do not capture background knowledge and are not customized to a concrete situational context and the sequential history of events leading up to it. This is a fundamentally limiting restriction for many tasks and applications, since the latent state can depend on a) abstract background information, b) the current situational context and c) the history of related observations. An illustrating example is a restaurant recommendation scenario, where a user’s assessment of the situation depends a) on taxonomical information regarding the type of cuisine, b) on situational factors like time of day, weather or location and c) on the subjective individual history and experience of this user in preceding situations. This situation-specific internal state of the user is not captured when using a traditional collaborative filtering approach, since background knowledge, the situational context and the sequential nature of an individual’s history cannot easily be represented in the matrix. In this paper, we investigate how well state-of-the-art approaches do exploit those different dimensions relevant to POI recommendation tasks. Naturally, we represent such a scenario as a temporal knowledge graph and compare plain knowledge graph, a taxonomy and a hypergraph embedding approach, as well as a recurrent neural network architecture to exploit the different context-dimensions of such rich information. Our empirical evidence indicates that the situational context is most crucial to the prediction performance, while the taxonomical and sequential information are harder to exploit. However, they still have their specific merits depending on the situation.

We count the number of graph compositions of suspended Y -trees via flats of the cycle matroid of the suspended Y -trees. 1. Introduction. The notion of graph composition was introduced by Knopfmacher and Mays in [3]. In this original work, various formulas, generating functions and recurrence relations for composition counting functions are given for several families of graphs. Graph compositions play an important role in the generalization of both ordinary compositions of positive integers and partitions of finite sets.This work is extended further to bipartite graphs, as well as some operations on graphs, including unions of graphs and the 2-sum of graphs in [1, 2, 7]. In [4], the terminology of graph composition is explored further, but under a new term, comppartition. In [5], it is shown that the number of graph compositions is equal to the total number of flats of the cycle matroid.The number of compositions for the following graphs are given in [3]:(i) T n : a tree on n vertices, (ii) P n : a path with n vertices, (iii) S n : a star graph on n vertices, (iv) K n : a complete graph on n vertices, (v) C n : a cycle graph on n vertices, (vi) W n : a wheel on n vertices, (vii) L n : a ladder graph on 2n vertices, and (viii) K m,n : a complete bipartite graph.In particular, it is shown that C(T n ) = C(P n ) = C(S n ) = 2 n−1 .