Topology of complex networks: models and analysis

There is a large variety of real-world phenomena that can be modelled and analysed as networks. Part of this variety is reflected in the diversity of network classes that are used to model these phenomena. However, the differences between network classes are not always taken into account in their analysis. This thesis carefully addresses how to deal with distinct classes of networks in two different contexts. First, the well-known switching model has been used to randomise different classes of networks, and is typically referred to as the switching model. We argue that really we should be talking about a family of switching models. Ignoring the distinction between the switching model with respect to different network classes has lead to biased sampling. Given that the most common use of the switching model is as a null-model, it is critical that it samples without bias. We provide a comprehensive analysis of the switching model with respect to nine classes of networks and prove under which conditions sampling is unbiased for each class. Recently the Curveball algorithm was introduced as a faster approach to network randomisation. We prove that the Curveball algorithm samples without bias; a position that was previously implied, but unproven. Furthermore, we show that the Curveball algorithm provides a flexible framework for network randomisation by introducing five variations with respect to different network classes. We compare the switching models and Curveball algorithms to several other random network models. As a result of our findings, we recommend using the configuration model for multi-graphs with self-loops, the Curveball algorithm for networks without multiple edges or without self-loops and the ordered switching model for directed acyclic networks. Second, we extend the theory of motif analysis to directed acyclic networks. We establish experimentally that there is no difference in the motifs detected by existing motif analysis methods and our customised method. However, we show that there are differences in the detected anti-motifs. Hence, we recommend taking into account the acyclic nature of directed acyclic networks. Network science is a young and active field of research. Most existing network measures originate in statistical mechanics and focus on statistics of local network properties. Such statistics have proven very useful. However, they do not capture the complete structure of a network. In this thesis we present experimental results on two novel network analysis techniques. First, at the local level, we show that the neighbourhood of a node is highly distinctive and has the potential to match unidentified entities across networks. Our motivation is the identification of individuals across dark social networks hidden in recorded networks. Second, we present results of the application of persistent homology to network analysis. This recently introduced technique from topological data analysis offers a new perspective on networks: it describes the mesoscopic structure of a network. Finally, we used persistent homology for a classification problem in pharmaceutical science. This is a novel application of persistent homology. Our analysis shows that this is a promising approach for the classification of lipid formulations

Motifs in directed acyclic networks

Finding motifs is important for understanding the structure of a network in terms of its building blocks. A network motif is a sub graph that appears significantly more often in a real network than expected in randomised networks. This paper looks at motif detection for a special class of directed networks: directed a cyclic networks. Normally, randomised networks are obtained using the switching algorithm. This algorithm preserves the in-degree and out-degree of each node. However, it does not preserve the directed a cyclic nature of directed a cyclic networks. Karrer and Newman introduced an algorithm that does preserve the directed a cyclic property but which may create multiple edges. This paper introduces alternative null-models that maintain the degree sequences, directed a cyclic property and do not introduce multiple edges. It is shown that there are explicit formulas for the number of occurrences of each possible 3-node pattern in such random networks. Even though the different random network models result in networks with different properties, the patterns that are keyed as network motifs in three real-world directed a cyclic networks do not depend on the choice of null-model. However, when using the switching model as a null-model, sometimes anti-motifs are found that contain directed cycles