Embedding Metadata-Enriched Graphs

This paper presents an on-going research where we studythe problem of embedding meta-data enriched graphs, with a focus onknowledge graphs in a vector space with transformer based deep neuralnetworks. Experimentally, we compare ceteris paribus the performance ofa transformer-based model with other non-transformer approaches. Dueto their recent success in natural language processing we hypothesizethat the former is superior in performance. We test this hypothesizesby comparing the performance of transformer embeddings with non-transformer embeddings on different downstream tasks. Our researchmight contribute to a better understanding of how random walks in-fluence the learning of features, which might be useful in the design ofdeep learning architectures for graphs when the input is generated withrandom walks.

The second Vassiliev measure of uniform random walks and polygons in confined space

Biopolymers, like chromatin, are often confined in small volumes. Confinement has a great effect on polymer conformations, including polymer entanglement. Polymer chains and other filamentous structures can be represented by polygonal curves in 3-space. In this manuscript, we examine the topological complexity of polygonal chains in 3-space and in confinement as a function of their length. We model polygonal chains by equilateral random walks in 3-space and by uniform random walks in confinement. For the topological characterization, we use the second Vassiliev measure. This is an integer topological invariant for polygons and a continuous functions over the real numbers, as a function of the chain coordinates for open polygonal chains. For uniform random walks in confined space, we prove that the average value of the Vassiliev measure in the space of configurations increases as $O(n^2)$ with the length of the walks or polygons. We verify this result numerically and our numerical results also show that the mean value of the second Vassiliev measure of equilateral random walks in 3-space increases as $O(n)$. These results reveal the rate at which knotting of open curves and not simply entanglement are affected by confinement.

Subdiffusive search with home returns via stochastic resetting: A subordination scheme approach

Stochastic resetting with home returns is widely found in various manifestations in life and nature. Using the solution to the home return problem in terms of the solution to the corresponding problem without home returns [Pal et al. Phys. Rev. Research 2, 043174 (2020)], we develop a theoretical framework for search with home returns in the case of subdiffusion. This makes a realistic description of restart by accounting for random walks with random stops. The model considers stochastic processes, arising from Brownian motion subordinated by an inverse infinitely divisible process (subordinator).

Synwalk: community detection via random walk modelling

Complex systems, abstractly represented as networks, are ubiquitous in everyday life. Analyzing and understanding these systems requires, among others, tools for community detection. As no single best community detection algorithm can exist, robustness across a wide variety of problem settings is desirable. In this work, we present Synwalk, a random walk-based community detection method. Synwalk builds upon a solid theoretical basis and detects communities by synthesizing the random walk induced by the given network from a class of candidate random walks. We thoroughly validate the effectiveness of our approach on synthetic and empirical networks, respectively, and compare Synwalk’s performance with the performance of Infomap and Walktrap (also random walk-based), Louvain (based on modularity maximization) and stochastic block model inference. Our results indicate that Synwalk performs robustly on networks with varying mixing parameters and degree distributions. We outperform Infomap on networks with high mixing parameter, and Infomap and Walktrap on networks with many small communities and low average degree. Our work has a potential to inspire further development of community detection via synthesis of random walks and we provide concrete ideas for future research.