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Multilayer networks continue to gain significant attention in many areas of study, particularly due to their high utility in modeling interdependent systems such as critical infrastructures, human brain connectome, and socioenvironmental ecosystems. However, clustering of multilayer networks, especially using the information on higher-order interactions of the system entities, still remains in its infancy. In turn, higher-order connectivity is often the key in such multilayer network applications as developing optimal partitioning of critical infrastructures in order to isolate unhealthy system components under cyber-physical threats and simultaneous identification of multiple brain regions affected by trauma or mental illness. In this paper, we introduce the concepts of topological data analysis to studies of complex multilayer networks and propose a topological approach for network clustering. The key rationale is to group nodes based not on pairwise connectivity patterns or relationships between observations recorded at two individual nodes but based on how similar in shape their local neighborhoods are at various resolution scales. Since shapes of local node neighborhoods are quantified using a topological summary in terms of persistence diagrams, we refer to the approach as clustering using persistence diagrams (CPD). CPD systematically accounts for the important heterogeneous higher-order properties of node interactions within and in-between network layers and integrates information from the node neighbors. We illustrate the utility of CPD by applying it to an emerging problem of societal importance: vulnerability zoning of residential properties to weather- and climate-induced risks in the context of house insurance claim dynamics. 

Network motifs are often called the building blocks of networks. Analysis of motifs has been found to be an indispensable tool for understanding local network structure, in contrast to measures based on node degree distribution and its functions that primarily address a global network topology. As a result, networks that are similar in terms of global topological properties may differ noticeably at a local level. This phenomenon of the impact of local structure has been recently documented in network fragility analysis and classification. At the same time, many studies of networks still tend to focus on global topological measures, often failing to unveil hidden mechanisms behind vulnerability of real networks and their dynamic response to malfunctions. In this paper, 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 demonstrated on electricity transmission networks of 4 European countries, and the results are compared with commonly used resilience and reliability measures. 

A new method for intentional islanding of power grids is proposed, based on a data-driven and inherently geometric concept of data depth. The utility of the new depth-based islanding is illustrated in application to the Italian power grid. It is found that spectral clustering with data depths outperforms spectral clustering with k-means in terms of k-way expansion. Directions on how the k-depths can be extended to multilayer grids in a tensor representation are outlined. 

Network motifs are often called the building blocks of networks. Analysis of motifs is found to be an indispensable tool for understanding local network structure, in contrast to measures based on node degree distribution and its functions that primarily address a global network topology. As a result, networks that are similar in terms of global topological properties may differ noticeably at a local level. In the context of power grids, this phenomenon of the impact of local structure has been recently documented in fragility analysis and power system classification. At the same time, most studies of power system networks still tend to focus on global topo-logical measures of power grids, often failing to unveil hidden mechanisms behind vulnerability of real power systems and their dynamic response to malfunctions. In this paper a pilot study of motif-based analysis of power grid robustness under various types of intentional attacks is presented, with the goal of shedding light on local dynamics and vulnerability of power systems.