Multiresolution Graph Transformers and Wavelet Positional Encoding for Learning Hierarchical Structures

Contemporary graph learning algorithms are not well-defined for large molecules since they do not consider the hierarchical interactions among the atoms, which are essential to determine the molecular properties of macromolecules. In this work, we propose Multiresolution Graph Transformers (MGT), the first graph transformer architecture that can learn to represent large molecules at multiple scales. MGT can learn to produce representations for the atoms and group them into meaningful functional groups or repeating units. We also introduce Wavelet Positional Encoding (WavePE), a new positional encoding method that can guarantee localization in both spectral and spatial domains. Our proposed model achieves competitive results on two macromolecule datasets consisting of polymers and peptides, and one drug-like molecule dataset. Importantly, our model outperforms other state-of-the-art methods and achieves chemical accuracy in estimating molecular properties (e.g., GAP, HOMO and LUMO) calculated by Density Functional Theory (DFT) in the polymers dataset. Furthermore, the visualizations, including clustering results on macromolecules and low-dimensional spaces of their representations, demonstrate the capability of our methodology in learning to represent long-range and hierarchical structures. Our PyTorch implementation is publicly available at https://github.com/HySonLab/Multires-Graph-Transforme

Eigenvector localization as a tool to study small communities in online social networks

We present and discuss a mathematical procedure for identification of small "communities" or segments within large bipartite networks. The procedure is based on spectral analysis of the matrix encoding network structure. The principal tool here is localization of eigenvectors of the matrix, by means of which the relevant network segments become visible. We exemplified our approach by analyzing the data related to product reviewing on Amazon.com. We found several segments, a kind of hybrid communities of densely interlinked reviewers and products, which we were able to meaningfully interpret in terms of the type and thematic categorization of reviewed items. The method provides a complementary approach to other ways of community detection, typically aiming at identification of large network modules

Quiver Gauge Theories: Finitude and Trichotomoty

D-brane probes, Hanany-Witten setups and geometrical engineering stand as a trichotomy of standard techniques of constructing gauge theories from string theory. Meanwhile, asymptotic freedom, conformality and IR freedom pose as a trichotomy of the beta-function behaviour in quantum field theories. Parallel thereto is a trichotomy in set theory of finite, tame and wild representation types. At the intersection of the above lies the theory of quivers. We briefly review some of the terminology standard to the physics and to the mathematics. Then, we utilise certain results from graph theory and axiomatic representation theory of path algebras to address physical issues such as the implication of graph additivity to finiteness of gauge theories, the impossibility of constructing completely IR free string orbifold theories and the unclassifiability of N < 2 Yang-Mills theories in four dimensions

Lessons from Love-Locks: The archaeology of a contemporary assemblage

This document is the Accepted Manuscript version. The final, definitive version of this paper has been published in Journal of Material Culture, November 2017, published by SAGE Publishing, All rights reserved.Loss of context is a challenge, if not the bane, of the ritual archaeologist’s craft. Those who research ritual frequently encounter difficulties in the interpretation of its often tantalisingly incomplete material record. Careful analysis of material remains may afford us glimpses into past ritual activity, but our often vast chronological separation from the ritual practitioners themselves prevent us from seeing the whole picture. The archaeologist engaging with structured deposits, for instance, is often forced to study ritual assemblages post-accumulation. Many nuances of its formation, therefore, may be lost in interpretation. This paper considers what insights an archaeologist could gain into the place, people, pace, and purpose of deposition by recording an accumulation of structured deposits during its formation, rather than after. To answer this, the paper will focus on a contemporary depositional practice: the love-lock. This custom involves the inscribing of names/initials onto a padlock, its attachment to a bridge or other public structure, and the deposition of the corresponding key into the water below; a ritual often enacted by a couple as a statement of their romantic commitment. Drawing on empirical data from a three-year diachronic site-specific investigation into a love-lock bridge in Manchester, UK, the author demonstrates the value of contemporary archaeology in engaging with the often enigmatic material culture of ritual accumulations.Peer reviewe

Comparison of Simple Graphical Process Models

Comparing the structure of graphical process models can reveal a number of process variations. Since most contemporary norms for process modelling rely on directed connectivity of objects in the model, connections between objects form sequences which can be translated into performing scenarios. Whereas sequences can be tested for completeness in performing process activities using simulation methods, the similarity or difference in static characteristics of sequences in different model variants are difficult to explore. The goal of the paper is to test the application of a method for comparison of graphical models by analyzing and comparing static characteristics of process models. Consequently, a metamodel for process models is developed followed by a comparison procedure conducted using a graphical model comparison algorithm

On occurrence of spectral edges for periodic operators inside the Brillouin zone

The article discusses the following frequently arising question on the spectral structure of periodic operators of mathematical physics (e.g., Schroedinger, Maxwell, waveguide operators, etc.). Is it true that one can obtain the correct spectrum by using the values of the quasimomentum running over the boundary of the (reduced) Brillouin zone only, rather than the whole zone? Or, do the edges of the spectrum occur necessarily at the set of ``corner'' high symmetry points? This is known to be true in 1D, while no apparent reasons exist for this to be happening in higher dimensions. In many practical cases, though, this appears to be correct, which sometimes leads to the claims that this is always true. There seems to be no definite answer in the literature, and one encounters different opinions about this problem in the community. In this paper, starting with simple discrete graph operators, we construct a variety of convincing multiply-periodic examples showing that the spectral edges might occur deeply inside the Brillouin zone. On the other hand, it is also shown that in a ``generic'' case, the situation of spectral edges appearing at high symmetry points is stable under small perturbations. This explains to some degree why in many (maybe even most) practical cases the statement still holds.Comment: 25 pages, 10 EPS figures. Typos corrected and a reference added in the new versio