# Topology of complex networks: models and analysis

Carstens, J 2016, Topology of complex networks: models and analysis, Doctor of Philosophy (PhD), Mathematical and Geospatial Sciences, RMIT University.

 Document type: Thesis Theses

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Title Topology of complex networks: models and analysis Carstens, J 2016 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. Doctor of Philosophy (PhD) RMIT University Mathematical and Geospatial Sciences network science random network models topological data analysis motifs neighbourhood analysis
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