Structural sparseness and complex networks

The field of complex networks has seen a steady growth in the last decade, fuelled by an ever-growing collection of relational data that our life in the information age generates. While several structural commonalities of complex networks have been observed—e.g. low density, heavily skewed degree-distributions, or the small world property—so far no property has been discovered that is algorithmically exploitable on a broad scale. Concurrently, the theory of structurally sparse graphs has been revolutionised by Robertson and Seymour’s graph minors programme. Many tools and techniques, developed as ‘by-products’ in the programme, have had a tremendous impact on the research of parametrised and approximation algorithms. They in particular enabled the development of several algorithmic meta-theorems, that is, algorithms that work for a large spectrum of problems on sparse inputs. In this thesis, we work towards bringing the field of structural sparse graphs and the field of complex networks closer together. We identify two notions of structural sparseness based on the density of shallow minors as keys for this endeavour: classes of bounded expansion and nowhere dense classes as introduced by Nešetřil and Ossona de Mendez in their seminal work on a robust theory of sparseness. In the following, we demonstrate that these sparse classes admit efficient algorithms for a huge number of problems, some of which have applications in domain-specific areas of network science. We further prove that several fundamental network models exhibit these properties and demonstrate empirically that this also holds true for a selection of realworld networks from various domains. As a result, we can state that the theory of structurally sparse graphs is applicable to complex networks and, as a corollary, so is the rich algorithmic toolkit it provides. This connection offers researchers from both the field of algorithmic graph theory and network science new approaches, insights, and productive questions.

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