Topology of Social and Managerial Networks

With the explosion of innovative technologies in recent years, organizational and man- agerial networks have reached high levels of intricacy. These are one of the many complex systems consisting of a large number of highly interconnected heterogeneous agents. The dominant paradigm in the representation of intricate relations between agents and their evolution is a network (graph). The study of network properties, and their implications on dynamical processes, up to now mostly focused on locally defined quantities of nodes and edges. These methods grounded in statistical mechanics gave deep insight and explanations on real world phenomena; however there is a strong need for a more versatile approach which would rely on new topological methods either separately or in combination with the classical techniques. In this thesis we approach this problem introducing new topological methods for network analysis relying on persistent homology. The results gained by the new methods apply both to weighted and unweighted networks; showing that classi- cal connectivity measures on managerial and societal networks can be very imprecise and extending them to weighted networks with the aim of uncovering regions of weak connectivity. In the first two chapters of the thesis we introduce the main instruments that will be used in the subsequent chapters, namely basic techniques from network theory and persistent homology from the field of computational algebraic topology. The third chapter of the thesis approaches social and organizational networks studying their con- nectivity in relation to the concept of social capital. Many sociological theories such as the theory of structural holes and of weak ties relate social capital, in terms of profitable managerial strategies and the chance of rewarding opportunities, to the topology of the underlying social structure. We review the known connectivity measures for social networks, stressing the fact that they are all local measures, calculated on a node’s Ego network, i.e considering a nodes direct contacts. By analyzing real cases it, nevertheless, turns out that the above measures can be very imprecise for strategical individuals in social networks, revealing fake brokerage opportunities. We, therefore, propose a new set of measures, complementary to the existing ones and focused on detecting the position of links, rather than their density, therefore extending the standard approach to a mesoscopic one. Widening the view from considering direct neighbors to considering also non-direct ones, using the “neighbor filtration”, we give a measure of height and weight for structural holes, obtaining a more accurate description of a node’s strategical position within its contacts. We also provide a refined version of the network efficiency measure, which collects in a compact form the height of all structural holes. The methods are implemented and have been tested on real world organizational and managerial networks. In pursuing the objective of improving the existing methods we faced some technical difficulties which obliged us to develop new mathematical tools. The fourth chapter of the thesis deals with the general problem of detecting structural holes in weighted networks. We introduce thereby the weight clique rank filtration, to detect particular non-local structures, akin to weighted structural holes within the link-weight network fabric, which are invisible to existing methods. Their properties divide weighted networks in two broad classes: one is characterized by small hierarchi- cally nested holes, while the second displays larger and longer living inhomogeneities. These classes cannot be reduced to known local or quasi local network properties, because of the intrinsic non-locality of homology, and thus yield a new classification built on high order coordination patterns. Our results show that topology can provide novel insights relevant for many-body interactions in social and spatial networks. In the fifth chapter of the thesis, we develop new insights in the mathematical setting underlying multipersistent homology. More specifically we calculate combinatorial resolutions and efficient Gro ̈bner bases for multipersistence homology modules. In this new frontier of persistent homology, filtrations are parametrized by multiple elements. Using multipersistent homology temporal networks can be studied and the weight filtration and neighbor filtration can be combined.