Unravelling the complexity of metabolic networks

Network science provides an invaluable set of tools and techniques for improving our understanding of many important biological processes at the systems level. A network description provides a simplied view of such a system, focusing upon the interactions between a usually large number of similar biological units. At the cellular level, these units are usually interacting genes, proteins or small molecules, resulting in various types of biological networks. Metabolic networks, in particular, play a fundamental role, since they provide the building blocks essential for cellular function, and thus, have recently received a lot of attention. In particular, recent studies have revealed a number of universal topological characteristics, such as a small average path-length, large clustering coecient and a hierarchical modular structure. Relations between structure, function and evolution, however, for even the simplest of organisms is far from understood. In this thesis, we employ network analysis in order to determine important links between an organism's metabolic network structure and the environment under which it evolved. We address this task from two dierent perspectives: (i) a network classication approach; and (ii) a more physiologically realistic modelling approach, namely hypernetwork models. One of the major contributions of this thesis is the development of a novel graph embedding approach, based on low-order network motifs, that compares the structural properties of large numbers of biological networks simultaneously. This method was prototyped on a cohort of 383 bacterial networks, and provides powerful evidence for the role that both environmental variability, and oxygen requirements, play in the forming of these important networked structures. In addition to this, we consider a hypernetwork formalism of metabolism, in an attempt to extend complex network reasoning to this more complicated, yet physiologically more realistic setting. In particular, we extend the concept of network reciprocity to hypernetworks, and again evidence a signicant relationship between bacterial hypernetwork structure and the environment in which these organisms evolved. Moreover, we extend the concept of network percolation to undirected hypernetworks, as a technique for quantifying robustness and fragility within metabolic hypernetworks, and in the process nd yet further evidence of increased topological complexity within organisms inhabiting more uncertain environments. Importantly, many of these relationships are not apparent when considering the standard approach, thus suggesting that a hypernetwork formalism has the potential to reveal biologically relevant information that is beyond the standard network approach

Model-free inference of direct network interactions from nonlinear collective dynamics

The topology of interactions in network dynamical systems fundamentally underlies their function. Accelerating technological progress creates massively available data about collective nonlinear dynamics in physical, biological, and technological systems. Detecting direct interaction patterns from those dynamics still constitutes a major open problem. In particular, current nonlinear dynamics approaches mostly require to know a priori a model of the (often high dimensional) system dynamics. Here we develop a model-independent framework for inferring direct interactions solely from recording the nonlinear collective dynamics generated. Introducing an explicit dependency matrix in combination with a block-orthogonal regression algorithm, the approach works reliably across many dynamical regimes, including transient dynamics toward steady states, periodic and non-periodic dynamics, and chaos. Together with its capabilities to reveal network (two point) as well as hypernetwork (e.g., three point) interactions, this framework may thus open up nonlinear dynamics options of inferring direct interaction patterns across systems where no model is known.Comment: 10 pages, 7 figure

Topological and Geometric Methods with a View Towards Data Analysis

In geometry, various tools have been developed to explore the topology and other features of a manifold from its geometrical structure. Among the two most powerful ones are the analysis of the critical points of a function, or more generally, the closed orbits of a dynamical system defined on the manifold, and the evaluation of curvature inequalities. When any (nondegenerate) function has to have many critical points and with different indices, then the topology must be rich, and when certain curvature inequalities hold throughout the manifold, that constrains the topology. It has been observed that these principles also hold for metric spaces more general than Riemannian manifolds, and for instance also for graphs. This thesis represents a contribution to this program. We study the relation between the closed orbits of a dynamical system and the topology of a manifold or a simplicial complex via the approach of Floer. And we develop notions of Ricci curvature not only for graphs, but more generally for, possibly directed, hypergraphs, and we draw structural consequences from curvature inequalities. It includes methods that besides their theoretical importance can be used as powerful tools for data analysis. This thesis has two main parts; in the first part we have developed topological methods based on the dynamic of vector fields defined on smooth as well as discrete structures. In the second part, we concentrate on some curvature notions which already proved themselves as powerful measures for determining the local (and global) structures of smooth objects. Our main motivation here is to develop methods that are helpful for the analysis of complex networks. Many empirical networks incorporate higher-order relations between elements and therefore are naturally modeled as, possibly directed and/or weighted, hypergraphs, rather than merely as graphs. In order to develop a systematic tool for the statistical analysis of such hypergraphs, we propose a general definition of Ricci curvature on directed hypergraphs and explore the consequences of that definition. The definition generalizes Ollivier’s definition for graphs. It involves a carefully designed optimal transport problem between sets of vertices. We can then characterize various classes of hypergraphs by their curvature. In the last chapter, we show that our curvature notion is a powerful tool for determining complex local structures in a variety of real and random networks modeled as (directed) hypergraphs. Furthermore, it can nicely detect hyperloop structures; hyperloops are fundamental in some real networks such as chemical reactions as catalysts in such reactions are faithfully modeled as vertices of directed hyperloops. We see that the distribution of our curvature notion in real networks deviates from random models

Multilayer Networks

In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such "multilayer" features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize "traditional" network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others. We also survey and discuss existing data sets that can be represented as multilayer networks. We review attempts to generalize single-layer-network diagnostics to multilayer networks. We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks. We conclude with a summary and an outlook.Comment: Working paper; 59 pages, 8 figure