The impossibility of low rank representations for triangle-rich complex networks

The study of complex networks is a significant development in modern science, and has enriched the social sciences, biology, physics, and computer science. Models and algorithms for such networks are pervasive in our society, and impact human behavior via social networks, search engines, and recommender systems to name a few. A widely used algorithmic technique for modeling such complex networks is to construct a low-dimensional Euclidean embedding of the vertices of the network, where proximity of vertices is interpreted as the likelihood of an edge. Contrary to the common view, we argue that such graph embeddings do not}capture salient properties of complex networks. The two properties we focus on are low degree and large clustering coefficients, which have been widely established to be empirically true for real-world networks. We mathematically prove that any embedding (that uses dot products to measure similarity) that can successfully create these two properties must have rank nearly linear in the number of vertices. Among other implications, this establishes that popular embedding techniques such as Singular Value Decomposition and node2vec fail to capture significant structural aspects of real-world complex networks. Furthermore, we empirically study a number of different embedding techniques based on dot product, and show that they all fail to capture the triangle structure

Foundations of Node Representation Learning

Low-dimensional node representations, also called node embeddings, are a cornerstone in the modeling and analysis of complex networks. In recent years, advances in deep learning have spurred development of novel neural network-inspired methods for learning node representations which have largely surpassed classical \u27spectral\u27 embeddings in performance. Yet little work asks the central questions of this thesis: Why do these novel deep methods outperform their classical predecessors, and what are their limitations? We pursue several paths to answering these questions. To further our understanding of deep embedding methods, we explore their relationship with spectral methods, which are better understood, and show that some popular deep methods are equivalent to spectral methods in a certain natural limit. We also introduce the problem of inverting node embeddings in order to probe what information they contain. Further, we propose a simple, non-deep method for node representation learning, and find it to often be competitive with modern deep graph networks in downstream performance. To better understand the limitations of node embeddings, we prove some upper and lower bounds on their capabilities. Most notably, we prove that node embeddings are capable of exact low-dimensional representation of networks with bounded max degree or arboricity, and we further show that a simple algorithm can find such exact embeddings for real-world networks. By contrast, we also prove inherent bounds on random graph models, including those derived from node embeddings, to capture key structural properties of networks without simply memorizing a given graph

Implications of sparsity and high triangle density for graph representation learning

Recent work has shown that sparse graphs containing many triangles cannot be reproduced using a finite-dimensional representation of the nodes, in which link probabilities are inner products. Here, we show that such graphs can be reproduced using an infinite-dimensional inner product model, where the node representations lie on a low-dimensional manifold. Recovering a global representation of the manifold is impossible in a sparse regime. However, we can zoom in on local neighbourhoods, where a lower-dimensional representation is possible. As our constructions allow the points to be uniformly distributed on the manifold, we find evidence against the common perception that triangles imply community structure

Extending adjacency matrices to 3D with triangles

Social networks are the fabric of society and the subject of frequent visual analysis. Closed triads represent triangular relationships between three people in a social network and are significant for understanding inherent interconnections and influence within the network. The most common methods for representing social networks (node-link diagrams and adjacency matrices) are not optimal for understanding triangles. We propose extending the adjacency matrix form to 3D for better visualization of network triads. We design a 3D matrix reordering technique and implement an immersive interactive system to assist in visualizing and analyzing closed triads in social networks. A user study and usage scenarios demonstrate that our method provides substantial added value over node-link diagrams in improving the efficiency and accuracy of manipulating and understanding the social network triads.Comment: 10 pages, 8 figures and 3 table

Exact Representation of Sparse Networks with Symmetric Nonnegative Embeddings

Many models for undirected graphs are based on factorizing the graph's adjacency matrix; these models find a vector representation of each node such that the predicted probability of a link between two nodes increases with the similarity (dot product) of their associated vectors. Recent work has shown that these models are unable to capture key structures in real-world graphs, particularly heterophilous structures, wherein links occur between dissimilar nodes. In contrast, a factorization with two vectors per node, based on logistic principal components analysis (LPCA), has been proven not only to represent such structures, but also to provide exact low-rank factorization of any graph with bounded max degree. However, this bound has limited applicability to real-world networks, which often have power law degree distributions with high max degree. Further, the LPCA model lacks interpretability since its asymmetric factorization does not reflect the undirectedness of the graph. We address these issues in two ways. First, we prove a new bound for the LPCA model in terms of arboricity rather than max degree; this greatly increases the bound's applicability to many sparse real-world networks. Second, we propose an alternative graph model whose factorization is symmetric and nonnegative, which allows for link predictions to be interpreted in terms of node clusters. We show that the bounds for exact representation in the LPCA model extend to our new model. On the empirical side, our model is optimized effectively on real-world graphs with gradient descent on a cross-entropy loss. We demonstrate its effectiveness on a variety of foundational tasks, such as community detection and link prediction

BOOL-AN: A method for comparative sequence analysis and phylogenetic reconstruction

A novel discrete mathematical approach is proposed as an additional tool for molecular systematics which does not require prior statistical assumptions concerning the evolutionary process. The method is based on algorithms generating mathematical representations directly from DNA/RNA or protein sequences, followed by the output of numerical (scalar or vector) and visual characteristics (graphs). The binary encoded sequence information is transformed into a compact analytical form, called the Iterative Canonical Form (or ICF) of Boolean functions, which can then be used as a generalized molecular descriptor. The method provides raw vector data for calculating different distance matrices, which in turn can be analyzed by neighbor-joining or UPGMA to derive a phylogenetic tree, or by principal coordinates analysis to get an ordination scattergram. The new method and the associated software for inferring phylogenetic trees are called the Boolean analysis or BOOL-AN

Uncertainty in Natural Language Generation: From Theory to Applications

Recent advances of powerful Language Models have allowed Natural Language Generation (NLG) to emerge as an important technology that can not only perform traditional tasks like summarisation or translation, but also serve as a natural language interface to a variety of applications. As such, it is crucial that NLG systems are trustworthy and reliable, for example by indicating when they are likely to be wrong; and supporting multiple views, backgrounds and writing styles -- reflecting diverse human sub-populations. In this paper, we argue that a principled treatment of uncertainty can assist in creating systems and evaluation protocols better aligned with these goals. We first present the fundamental theory, frameworks and vocabulary required to represent uncertainty. We then characterise the main sources of uncertainty in NLG from a linguistic perspective, and propose a two-dimensional taxonomy that is more informative and faithful than the popular aleatoric/epistemic dichotomy. Finally, we move from theory to applications and highlight exciting research directions that exploit uncertainty to power decoding, controllable generation, self-assessment, selective answering, active learning and more