Higher-order Link Prediction using Node and Subgraph Embeddings

Social media, academia collaborations, e-commerce websites, biological structures, and other real-world networks are modeled as graphs to represent their entities and relationships in an abstract way. Such graphs are becoming more complex and informative, and by analyzing them we can solve various problems and find hidden insights. Some applications include predicting relationships and potential links between nodes, classifying nodes, and finding the most influential nodes in the graph, etc. A large amount of research is being done in the field of predicting links between two nodes. However, predicting a future relationship among three or more nodes in a graph is a more recent active research topic. Relationships that involve more than two nodes is called a higher-order link. One of the approaches, that we follow in this work, is that of mapping the graph entities, such as nodes, edges, and triangles, into a low dimensional space by generating embeddings vectors. In that way, we work with vectors and reduce the higher-order link prediction to a classification problem. The primary objective of this project is to utilize the GloVeNoR node embedding technique, as well as Simplex2Vec triangle embedding technique, to perform higher- order link prediction, i.e., to predict the possibility of interaction. Additionally, we evaluate the predictions generated by our methods and compare them with existing higher-order link prediction approaches using benchmark datasets. Based on our experiments, we show that the triangle embeddings generated using the techniques discussed in the report increase the average performance over the five datasets evaluated using the AUC-PR relative to random baseline as a metric for higher-order link prediction by 48%

Higher-order Link Prediction Using Graph Embeddings

Link prediction is an emerging field that predicts if two nodes in a network are likely to be connected or not in the near future. Networks model real-world systems using pairwise interactions of nodes. However, many of these interactions may involve more than two nodes or entities simultaneously. For example, social interactions often occur in groups of people, research collaborations are among more than two authors, and biological networks describe interactions of a group of proteins. An interaction that consists of more than two entities is called a higher-order structure. Predicting the occurrence of such higher-order structures helps us solve problems on various disciplines, such as social network analysis, drug combinations research, and news topic connections. Moreover, we can use our methods to get more knowledge about news topics during the COVID-19 pandemic. Higher-order link prediction can be accomplished using neural networks and other machine learning techniques. The primary focus of this project is to explore repre- sentations of three-node interactions, called triangles (a special case of higher-order structure). We propose new methods to embed triangles: by generalizing node2vec algorithm using different operators to learn an embedding for a triangle, and by using 1-hop subgraphs of the triangles to learn embeddings using graph2vec algorithm and graph neural networks. The performance of these techniques is evaluated against the benchmark scores on various datasets used in the bibliography. From the results, it is observed that the node2vec based triangle embedding algorithm performs better or similar on most of the datasets compared to benchmark models

A Growing Self-Organizing Network for Reconstructing Curves and Surfaces

Self-organizing networks such as Neural Gas, Growing Neural Gas and many others have been adopted in actual applications for both dimensionality reduction and manifold learning. Typically, in these applications, the structure of the adapted network yields a good estimate of the topology of the unknown subspace from where the input data points are sampled. The approach presented here takes a different perspective, namely by assuming that the input space is a manifold of known dimension. In return, the new type of growing self-organizing network presented gains the ability to adapt itself in way that may guarantee the effective and stable recovery of the exact topological structure of the input manifold

Flow Smoothing and Denoising: Graph Signal Processing in the Edge-Space

This paper focuses on devising graph signal processing tools for the treatment of data defined on the edges of a graph. We first show that conventional tools from graph signal processing may not be suitable for the analysis of such signals. More specifically, we discuss how the underlying notion of a `smooth signal' inherited from (the typically considered variants of) the graph Laplacian are not suitable when dealing with edge signals that encode a notion of flow. To overcome this limitation we introduce a class of filters based on the Edge-Laplacian, a special case of the Hodge-Laplacian for simplicial complexes of order one. We demonstrate how this Edge-Laplacian leads to low-pass filters that enforce (approximate) flow-conservation in the processed signals. Moreover, we show how these new filters can be combined with more classical Laplacian-based processing methods on the line-graph. Finally, we illustrate the developed tools by denoising synthetic traffic flows on the London street network.Comment: 5 pages, 2 figur

the shape of collaborations

Abstract The structure of scientific collaborations has been the object of intense study both for its importance for innovation and scientific advancement, and as a model system for social group coordination and formation thanks to the availability of authorship data. Over the last years, complex networks approach to this problem have yielded important insights and shaped our understanding of scientific communities. In this paper we propose to complement the picture provided by network tools with that coming from using simplicial descriptions of publications and the corresponding topological methods. We show that it is natural to extend the concept of triadic closure to simplicial complexes and show the presence of strong simplicial closure. Focusing on the differences between scientific fields, we find that, while categories are characterized by different collaboration size distributions, the distributions of how many collaborations to which an author is able to participate is conserved across fields pointing to underlying attentional and temporal constraints. We then show that homological cycles, that can intuitively be thought as hole in the network fabric, are an important part of the underlying community linking structure

Weak Form of Stokes-Dirac Structures and Geometric Discretization of Port-Hamiltonian Systems

We present the mixed Galerkin discretization of distributed parameter port-Hamiltonian systems. On the prototypical example of hyperbolic systems of two conservation laws in arbitrary spatial dimension, we derive the main contributions: (i) A weak formulation of the underlying geometric (Stokes-Dirac) structure with a segmented boundary according to the causality of the boundary ports. (ii) The geometric approximation of the Stokes-Dirac structure by a finite-dimensional Dirac structure is realized using a mixed Galerkin approach and power-preserving linear maps, which define minimal discrete power variables. (iii) With a consistent approximation of the Hamiltonian, we obtain finite-dimensional port-Hamiltonian state space models. By the degrees of freedom in the power-preserving maps, the resulting family of structure-preserving schemes allows for trade-offs between centered approximations and upwinding. We illustrate the method on the example of Whitney finite elements on a 2D simplicial triangulation and compare the eigenvalue approximation in 1D with a related approach.Comment: Copyright 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0

Profinite complexes of curves, their automorphisms and anabelian properties of moduli stacks of curves

Let ${\cal M}_{g,[n]}$, for $2g-2+n>0$, be the D-M moduli stack of smooth curves of genus $g$ labeled by $n$ unordered distinct points. The main result of the paper is that a finite, connected \'etale cover {\cal M}^\l of ${\cal M}_{g,[n]}$, defined over a sub-$p$-adic field $k$, is "almost" anabelian in the sense conjectured by Grothendieck for curves and their moduli spaces. The precise result is the following. Let \pi_1({\cal M}^\l_{\ol{k}}) be the geometric algebraic fundamental group of {\cal M}^\l and let {Out}^*(\pi_1({\cal M}^\l_{\ol{k}})) be the group of its exterior automorphisms which preserve the conjugacy classes of elements corresponding to simple loops around the Deligne-Mumford boundary of {\cal M}^\l (this is the "$\ast$-condition" motivating the "almost" above). Let us denote by {Out}^*_{G_k}(\pi_1({\cal M}^\l_{\ol{k}})) the subgroup consisting of elements which commute with the natural action of the absolute Galois group $G_k$ of $k$. Let us assume, moreover, that the generic point of the D-M stack {\cal M}^\l has a trivial automorphisms group. Then, there is a natural isomorphism: {Aut}_k({\cal M}^\l)\cong{Out}^*_{G_k}(\pi_1({\cal M}^\l_{\ol{k}})). This partially extends to moduli spaces of curves the anabelian properties proved by Mochizuki for hyperbolic curves over sub-$p$-adic fields.Comment: This paper has been withdrawn because of a flaw in the paper "Profinite Teichm\"uller theory" of the first author, on which this paper built o

Sampling and Recovery of Signals on a Simplicial Complex using Neighbourhood Aggregation

In this work, we focus on sampling and recovery of signals over simplicial complexes. In particular, we subsample a simplicial signal of a certain order and focus on recovering multi-order bandlimited simplicial signals of one order higher and one order lower. To do so, we assume that the simplicial signal admits the Helmholtz decomposition that relates simplicial signals of these different orders. Next, we propose an aggregation sampling scheme for simplicial signals based on the Hodge Laplacian matrix and a simple least squares estimator for recovery. We also provide theoretical conditions on the number of aggregations and size of the sampling set required for faithful reconstruction as a function of the bandwidth of simplicial signals to be recovered. Numerical experiments are provided to show the effectiveness of the proposed method