# Michael SchaubRWTH Aachen University · Department of Computer Science

Michael Schaub

PhD

## About

87

Publications

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Introduction

## Publications

Publications (87)

We study the task of node classification for graph neural networks (GNNs) and establish a connection between group fairness, as measured by statistical parity and equal opportunity, and local assortativity, i.e., the tendency of linked nodes to have similar attributes. Such assortativity is often induced by homophily, the tendency for nodes of simi...

The cardiac vascular and perivascular niche are of major importance in homeostasis and during disease, but we lack a complete understanding of its cellular heterogeneity and alteration in response to injury as a major driver of heart failure. Using combined genetic fate tracing with confocal imaging and single-cell RNA sequencing of this niche in h...

In this chapter, we derive and analyse models for consensus dynamics on hypergraphs. As we discuss, unless there are nonlinear node interaction functions, it is always possible to rewrite the system in terms of a new network of effective pairwise node interactions, regardless of the initially underlying multi-way interaction structure. We thus focu...

Higher-order networks have so far been considered primarily in the context of studying the structure of complex systems, i.e., the higher-order or multi-way relations connecting the constituent entities. More recently, a number of studies have considered dynamical processes that explicitly account for such higher-order dependencies, e.g., in the co...

Complex networks are typically not homogeneous, as they tend to display an array of structures at different scales. A feature that has attracted a lot of research is their modular organisation, i.e., networks may often be considered as being composed of certain building blocks, or modules. In this Element, the authors discuss a number of ways in wh...

We study linear filters for processing signals supported on abstract topological spaces modeled as simplicial complexes, which may be interpreted as generalizations of graphs that account for nodes, edges, triangular faces etc. To process such signals, we develop simplicial convolutional filters defined as matrix polynomials of the lower and upper...

We investigate consensus dynamics on temporal hypergraphs that encode network systems with time-dependent, multiway interactions. We compare these consensus processes with dynamics evolving on projections that remove the temporal and/or the multiway interactions of the higher-order network representation. For linear average consensus dynamics, we f...

We propose a method to detect outliers in empirically observed trajectories on a discrete or discretized manifold modeled by a simplicial complex. Our approach is similar to spectral embeddings such as diffusion-maps and Laplacian eigenmaps, that construct vertex embeddings from the eigenvectors of the graph Laplacian associated with low eigenvalue...

Network analysis provides powerful tools to learn about a variety of social systems. However, most analyses implicitly assume that the considered relational data is error-free, and reliable and accurately reflects the system to be analysed. Especially if the network consists of multiple groups (e.g., genders, races), this assumption conflicts with...

The processing of signals supported on non-Euclidean domains has attracted large interest in the last years. Thus far, such non-Euclidean domains have been abstracted primarily as graphs with signals supported on the nodes, though recently the processing of signals on more general structures such as simplicial complexes has also been considered. In...

The processing of signals supported on non-Euclidean domains has attracted large interest in the last years. Thus far, such non-Euclidean domains have been abstracted primarily as graphs with signals supported on the nodes, though recently the processing of signals on more general structures such as simplicial complexes has also been considered. In...

We develop wavelet representations for edge-flows on simplicial complexes, using ideas rooted in combinatorial Hodge theory and spectral graph wavelets. We first show that the Hodge Laplacian can be used in lieu of the graph Laplacian to construct a family of wavelets for higher-order signals on simplicial complexes. Then, we refine this idea to co...

We introduce and analyse a three-body consensus model (3CM) for non-linear consensus dynamics on hypergraphs. Our model incorporates reinforcing group effects, which can cause shifts in the average state of the system even in if the underlying graph is complete (corresponding to a mean-field interaction), a phenomena that may be interpreted as a ty...

We investigate consensus dynamics on temporal hypergraphs that encode network systems with time-dependent, multi-way interactions. We compare this dynamics with that on appropriate projections of this higher-order network representation that flatten the temporal, the multi-way component, or both. For linear average consensus dynamics, we find that...

We consider state-aggregation schemes for Markov chains from an information-theoretic perspective. Specifically, we consider aggregating the states of a Markov chain such that the mutual information of the aggregated states separated by T time steps is maximized. We show that for T=1 this recovers the maximum-likelihood estimator of the degree-corr...

Higher-order networks have so far been considered primarily in the context of studying the structure of complex systems, i.e., the higher-order or multi-way relations connecting the constituent entities. More recently, a number of studies have considered dynamical processes that explicitly ac- count for such higher-order dependencies, e.g., in the...

Recent studies have exposed that many graph neural networks (GNNs) are sensitive to adversarial attacks, and can suffer from performance loss if the graph structure is intentionally perturbed. A different line of research has shown that many GNN architectures implicitly assume that the underlying graph displays homophily, i.e., connected nodes are...

This paper re-examines the concept of node equivalences like structural equivalence or automorphic equivalence, which have originally emerged in social network analysis to characterize the role an actor plays within a social system, but have since then been of independent interest for graph-based learning tasks. Traditionally, such exact node equiv...

Motivation
Ligand-receptor (LR) network analysis allows the characterization of cellular crosstalk based on single cell RNA-seq data. However, current methods typically provide a list of inferred LR interactions and do not allow the researcher to focus on specific cell types, ligands or receptors. In addition, most of these methods cannot quantify...

In this chapter, we derive and analyse models for consensus dynamics on hypergraphs. As we discuss, unless there are nonlinear node interaction functions, it is always possible to rewrite the system in terms of a new network of effective pairwise node interactions, regardless of the initially underlying multi-way interaction structure. We thus focu...

In this tutorial, we provide a didactic treatment of the emerging topic of signal processing on higher-order networks. Drawing analogies from discrete and graph signal processing, we introduce the building blocks for processing data on simplicial complexes and hypergraphs, two common higher-order network abstractions that can incorporate polyadic r...

Modeling complex systems and data using the language of graphs and networks has become an essential topic across a range of different disciplines. Arguably, this network-based perspective derives is success from the relative simplicity of graphs: A graph consists of nothing more than a set of vertices and a set of edges, describing relationships be...

In this paper, we study linear filters to process signals defined on simplicial complexes, i.e., signals defined on nodes, edges, triangles, etc. of a simplicial complex, thereby generalizing filtering operations for graph signals. We propose a finite impulse response filter based on the Hodge Laplacian, and demonstrate how this filter can be desig...

This tutorial paper presents a didactic treatment of the emerging topic of signal processing on higher-order networks. Drawing analogies from discrete and graph signal processing, we introduce the building blocks for processing data on simplicial complexes and hypergraphs, two common abstractions of higher-order networks that can incorporate polyad...

Networks are a widely-used tool to investigate the large-scale connectivity structure in complex systems and graphons have been proposed as an infinite size limit of dense networks. The detection of communities or other meso-scale structures is a prominent topic in network science as it allows the identification of functional building blocks in com...

Network analysis provides powerful tools to learn about a variety of social systems. However, most analyses implicitly assume that the considered data is error-free and reliable. Especially if the network consists of multiple groups, this assumption conflicts with the range of systematic reporting biases, measurement errors and other inaccuracies t...

We study the problem of recovering a planted hierarchy of partitions in a network. The detectability of a single planted partition has previously been analysed in detail and a phase transition has been identified below which the partition cannot be detected. Here we show that, in the hierarchical setting, there exist additional phases in which the...

Modular and hierarchical structures are pervasive in real-world complex systems. A great deal of effort has gone into trying to detect and study these structures. Important theoretical advances in the detection of modular, or "community", structures have included identifying fundamental limits of detectability by formally defining community structu...

Networks and data supported on graphs have become ubiquitous in the sciences and engineering. This paper studies the 'blind' community detection problem, where we seek to infer the community structure of a graph model given the observation of independent graph signals on a set of nodes whose connections are unknown. We model each observation as fil...

We consider state aggregation schemes for Markov chains from an information-theoretic perspective. Specifically, we consider aggregating the states of a Markov chain such that the mutual information of the aggregated states separated by T time steps is maximized. We show that for T = 1 this approach recovers the maximum-likelihood estimator of the...

We explore the problem of inferring the graph Laplacian of a weighted, undirected network from snapshots of a single or multiple discrete-time consensus dynamics, subject to parameter uncertainty, taking place on the network. Specifically, we consider three problems in which we assume different levels of knowledge about the diffusion rates, observa...

We introduce and analyse a three-body consensus model (3CM) for non-linear consensus dynamics on hypergraphs. Our model incorporates reinforcing group effects, which can cause shifts in the average state of the system even in if the underlying graph is complete (corresponding to a mean-field interaction), a phenomena that may be interpreted as a ty...

Networks and data supported on graphs have become ubiquitous in the sciences and engineering. This paper studies the 'blind' community detection problem, where we seek to infer the community structure of a graph model given the observation of independent graph signals on a set of nodes whose connections are unknown. We model each observation as fil...

This chapter focuses on the rich interplay between network structure and a dynamics acting on top of the network as a means of identifying modules in the network or describing the effect that modules can have on the dynamical behavior of a system. One of the main motivations for identifying modular structures in networks is that they provide a simp...

This chapter unfolds different aims underpinning community detection – in a relaxed form that includes assortative as well as disassortative group structures with dense and sparse internal connections, respectively – and discusses how the resulting problem perspectives relate to various applications. It focuses on four broad perspectives that have...

We explore the problem of inferring the graph Laplacian of a weighted, undirected network from snapshots of a single or multiple discrete-time consensus dynamics, subject to parameter uncertainty, taking place on the network. Specifically, we consider three problems in which we assume different levels of knowledge about the diffusion rates, observa...

We present a graph-based semi-supervised learning (SSL) method for learning edge flows defined on a graph. Specifically, given flow measurements on a subset of edges, we want to predict the flows on the remaining edges. To this end, we develop a computational framework that imposes certain constraints on the overall flows, such as (approximate) flo...

Complex systems and relational data are often abstracted as dynamical processes on networks. To understand, predict and control their behavior a crucial step is to extract reduced descriptions of such networks. Inspired by notions from Control Theory, here we propose a time-dependent dynamical similarity measure between nodes, which quantifies the...

We consider a blind identification problem in which we aim to recover a statistical model of a network without knowledge of the network's edges, but based solely on nodal observations of a certain process. More concretely, we focus on observations that consist of snapshots of a diffusive process that evolves over the unknown network. We model the n...

We present a graph-based semi-supervised learning (SSL) method for learning edge flows defined on a graph. Specifically, given flow measurements on a subset of edges, we want to predict the flows on the remaining edges. To this end, we develop a computational framework that imposes certain constraints on the overall flows, such as (approximate) flo...

We discuss a variant of `blind' community detection, in which we aim to partition an unobserved network from the observation of a (dynamical) graph signal defined on the network. We consider a scenario where our observed graph signals are obtained by filtering white noise input, and the underlying network is different for every observation. In this...

As relational datasets modeled as graphs keep increasing in size and their data-acquisition is permeated by uncertainty, graph-based analysis techniques can become computationally and conceptually challenging. In particular, node centrality measures rely on the assumption that the graph is perfectly known --- a premise not necessarily fulfilled for...

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 co...

Modeling complex systems and data with graphs has been a mainstay of the applied mathematics community. The nodes in the graph represent entities and the edges model the relations between them. Simplicial complexes, a mathematical object common in topological data analysis, have emerged as a model for multi-nodal interactions that occur in several...

We develop a mathematical model considering a random walker with long-range hops on arbitrary graphs. The random multi-hopper can jump to any node of the graph from an initial position, with a probability that decays as a function of the shortest-path distance between the two nodes in the graph. We consider here two decaying functions in the form o...

Network-based data mining techniques such as graph mining,
(social) network analysis, link prediction and graph clustering
form an important foundation for data science applications
in computer science, computational social science, and the
life sciences. They help to detect patterns in large data sets
that capture dyadic relations between pairs of...

This chapter discusses the interplay between structure and dynamics in complex networks. Given a particular network with an endowed dynamics, our goal is to find partitions aligned with the dynamical process acting on top of the network. We thus aim to gain a reduced description of the system that takes into account both its structure and dynamics....

Networks provide a powerful formalism for modeling complex systems, by representing the underlying set of pairwise interactions. But much of the structure within these systems involves interactions that take place among more than two nodes at once; for example, communication within a group rather than person-to-person, collaboration among a team ra...

Directed information transmission is a paramount requirement for many social, physical, and biological systems. For neural systems, scientists have studied this problem under the paradigm of feedforward networks for decades. In most models of feedforward networks, activity is exclusively driven by excitatory neurons and the wiring patterns between...

A precise definition of what constitutes a community in networks has remained elusive. Consequently, network scientists have compared community detection algorithms on benchmark networks with a particular form of community structure and classified them based on the mathematical techniques they employ. However, this comparison can be misleading beca...

Using an information theoretic point of view, we investigate how a dynamics acting on a network can be coarse grained through the use of graph partitions. Specifically, we are interested in how aggregating the state space of a Markov process according to a partition impacts on the thus obtained lower-dimensional dynamics. We highlight that for a dy...

We consider the problem of identifying the topology of a weighted, undirected network $\mathcal G$ from observing snapshots of multiple independent consensus dynamics. Specifically, we observe the opinion profiles of a group of agents for a set of $M$ independent topics and our goal is to recover the precise relationships between the agents, as spe...

Graphs provide a natural mathematical abstraction for systems with pairwise interactions, and thus have become a prevalent tool for the representation of systems across various scientific domains. However, as the size of relational datasets continues to grow, traditional graph-based approaches are increasingly replaced by other modeling paradigms,...

V.Y.K., T.A., A.Y. and M.H. are supported by Wellcome Trust Grants. K.N.N. is supported by the Wellcome Trust Strategic Award 'Single cell genomics of mouse gastrulation'. M.T.S. acknowledges support from FRS-FNRS; the Belgian Network DYSCO (Dynamical Systems, Control and Optimisation), funded by the Interuniversity Attraction Poles Programme initi...

Single-cell RNA-seq enables the quantitative characterization of cell types based on global transcriptome profiles. We present single-cell consensus clustering (SC3), a user-friendly tool for unsupervised clustering, which achieves high accuracy and robustness by combining multiple clustering solutions through a consensus approach (http://bioconduc...

Community detection, the decomposition of a graph into meaningful building blocks, has been a core research topic in network science over the past years. Since a precise notion of what constitutes a community has remained evasive, community detection algorithms have often been compared on benchmark graphs with a particular form of community structu...

Using single-cell RNA-seq (scRNA-seq), the full transcriptome of individual cells can be acquired, enabling a quantitative characterisation of cell-type based on expression profiles. Due to the large variability in gene expression, assigning cells into groups based on the transcriptome remains challenging. We present Single-Cell Consensus Clusterin...

Allosteric regulation is central to many biochemical processes. Allosteric sites provide a target to fine-tune protein activity, yet we lack computational methods to predict them. Here, we present an efficient graph-theoretical approach for identifying allosteric sites and the mediating interactions that connect them to the active site. Using an at...

Supplementary Figures 1-5, Supplementary Tables 1-6, Supplementary Notes 1-3, Supplementary Methods and Supplementary References.

Synchronization over networks depends strongly on the structure of the coupling between the oscillators.When the coupling presents certain regularities, the dynamics can be coarse-grained into clusters by means of External Equitable Partitions of the network graph and their associated quotient graphs. We exploit this graph-theoretical concept to st...

We exploit flow propagation on the directed neuronal network of the nematode C. elegans to reveal dynamically relevant features of its connectome. We find flow-retaining groupings of neurons at different levels of granularity, which we relate to functional and anatomical constituents of its nervous system. A systematic in silico evaluation of the f...

Allosteric regulation is central to many biochemical processes. Allosteric sites provide a target to fine-tune protein activit