Role of Local Geometry in Resilience and functions of Complex Networks

There is a growing tendency nowadays to study a broad range of man-made, socio-demographic and natural science systems, from brain connectome to interbank transactions to critical infrastructures, under the framework of complex networks. Methods of complex network analysis have provided new insights into the fundamental and intrinsic characteristics of complex system functionality, vulnerability and resilience. The most widely explored characteristics of complex system in the network analysis are node degree distribution, mean degree, small world properties and, to a lesser extent, betweenness centrality measures - that is, primarily lower-order connectivity features that are investigated at the level of individual nodes and edges. However, networks that are similar in terms of global topological properties may differ noticeably at a local level. At the same time, many studies of networks still tend to focus on global topological measures often failing to understand hidden mechanisms of their functionality, vulnerability and dynamic response to malfunctions. Therefore the analysis of the local network structure and underlying geometry are indispensable to unveil the properties of complex networks. In this dissertaion we investigate utility of two approaches to describe network local topology and geometry, network motifs and topological data analysis. First, a study of motif-based analysis of network resilience and reliability under various types of intentional attacks is presented, with the goal of shedding light on local dynamics and vulnerability of networks. These methods are illustrated in application to resilience analysis on electricity transmission networks of four European countries and the results are compared with commonly used resilience and reliability measures. We find that power grids show different degrees of local sensitivity and degradation with respect to the type of attack and the type of motif. Hence, motif characteristics can be potentially used as alternative local metrics of network resilience as well as early warning indicators of complex system degradation and failure. Second, we introduce a novel approach to study robustness of a power grid network employing the tools of topological data analysis (TDA). We represent the networks as a weighted graph, equipping it with a nested simplicial complex structure and extracting topological summaries in the form of the Betti numbers and persistent diagrams. These summaries are then used to characterize network vulnerability under critical conditions such as targeted attacks. We observe that vulnerability properties of grids tend to be linked to the persistence of onedimensional holes. Third, we introduce a novel concept of chainlets, or blockchain motifs,to incorporate information on local topology and geometry of the blockchain transaction graph. Chainlets allow us to evaluate the role of local topological structure of the blockchain on the joint Bitcoin and Litecoin price formation and dynamics. We investigate the Granger causality of chainlets and identify certain types of chainlets that exhibit the highest predictive influence on cryptocurrency price and investment risk. Our findings indicate that chainlets exhibit high predictive utility in Litecoin and Bitcoin price and volatility prediction. Finally, a new method for intentional islanding of power grids is proposed, based on a datadriven and inherently geometric concept of data depth. The utility of the new depth-based islanding is illustrated in application to the Italian power grid. Our results indicate that spectral clustering with data depths outperforms spectral clustering with k-means in terms of k-way expansion. This dissertation aims to open up new horizons for the analysis of the local geometric properties of a complex network and their impact on network functionality. We hope that the findings of this dissertation will enhance our understanding of complex network functionality, reliability, and resilience, at a local level.