Big Ramsey degrees in universal inverse limit structures

We build a collection of topological Ramsey spaces of trees giving rise to universal inverse limit structures, extending Zheng's work for the profinite graph to the setting of Fra\"{\i}ss\'{e} classes of finite ordered binary relational structures with the Ramsey property. This work is based on the Halpern-L\"{a}uchli theorem, but different from the Milliken space of strong subtrees. Based on these topological Ramsey spaces and the work of Huber-Geschke-Kojman on inverse limits of finite ordered graphs, we prove that for each such Fra\"{\i}ss\'{e} class, its universal inverse limit structures has finite big Ramsey degrees under finite Baire-measurable colourings. For finite ordered graphs, finite ordered $k$-clique free graphs ($k\geq 3$), finite ordered oriented graphs, and finite ordered tournaments, we characterize the exact big Ramsey degrees.Comment: 20 pages, 5 figure

Dendrites or lambda-dendroids as generalized inverse limits

This research centers on the study of generalized inverse limits. We show that all members of an infinite family of inverse limit spaces are homeomorphics to one particularly complicated inverse limit space known as The Monster . Further, properties of factor spaces and graphs of bonding functions which are preserved in generalized inverse limit spaces with upper semi-continuous bonding functions with appropriate restrictions are investigated. Some of the properties are locally connectedness, hereditary decomposability, hereditary indecomposability, hereditary unicoherence, arc-likeness, and tree-likeness. The theorems are illustrated by several examples --Abstract, page iv

Convolutional 2D Knowledge Graph Embeddings

Link prediction for knowledge graphs is the task of predicting missing relationships between entities. Previous work on link prediction has focused on shallow, fast models which can scale to large knowledge graphs. However, these models learn less expressive features than deep, multi-layer models -- which potentially limits performance. In this work, we introduce ConvE, a multi-layer convolutional network model for link prediction, and report state-of-the-art results for several established datasets. We also show that the model is highly parameter efficient, yielding the same performance as DistMult and R-GCN with 8x and 17x fewer parameters. Analysis of our model suggests that it is particularly effective at modelling nodes with high indegree -- which are common in highly-connected, complex knowledge graphs such as Freebase and YAGO3. In addition, it has been noted that the WN18 and FB15k datasets suffer from test set leakage, due to inverse relations from the training set being present in the test set -- however, the extent of this issue has so far not been quantified. We find this problem to be severe: a simple rule-based model can achieve state-of-the-art results on both WN18 and FB15k. To ensure that models are evaluated on datasets where simply exploiting inverse relations cannot yield competitive results, we investigate and validate several commonly used datasets -- deriving robust variants where necessary. We then perform experiments on these robust datasets for our own and several previously proposed models and find that ConvE achieves state-of-the-art Mean Reciprocal Rank across most datasets.Comment: Extended AAAI2018 pape

On simplicial maps and chainable continua

AbstractAn operation d on simplicial maps between graphs is introduced and used to characterize simplicial maps which can be factored through an arc. The characterization yields a new technique of showing that some continua are not chainable and allows to prove that span zero is equivalent to chainability for inverse limits of trees with simplicial bonding maps

Δ-related functions and generalized inverse limits

For any continuous single-valued functions (f,g: [0,1] rightarrow [0,1]) we define upper semicontinuous set-valued functions (F,G: [0,1] multimap [0,1]) by their graphs as the unions of the diagonal (Delta) and the graphs of set-valued inverses of (f) and (g) respectively. We introduce when two functions are (Delta)-related and show that if (f) and (g) are (Delta)-related, then the inverse limits (varproj F) and (varproj G) are homeomorphic. We also give conditions under which (varproj G) is a quotient space of (varproj F)