IRIS Pol. Torinohttps://iris.polito.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Sun, 15 Sep 2019 07:47:43 GMT2019-09-15T07:47:43Z1041Topology of Social and Managerial Networkshttp://hdl.handle.net/11583/2507606Titolo: Topology of Social and Managerial Networks
Abstract: 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.
Fri, 01 Jan 9999 00:00:00 GMThttp://hdl.handle.net/11583/25076069999-01-01T00:00:00ZCombinatorial presentation of multidimensional persistent homologyhttp://hdl.handle.net/11583/2651578Titolo: Combinatorial presentation of multidimensional persistent homology
Abstract: A multifiltration is a functor indexed by N^r that maps any morphism
to a monomorphism. The goal of this paper is to describe in an explicit
and combinatorial way the natural N^r-graded R[x_1,...,x_r]-module structure
on the homology of a multifiltration of simplicial complexes. To do that we
study multifiltrations of sets and R-modules. We prove in particular that the
N^r-graded R[x_1,...,x_r]-modules that can occur as R-spans of multifiltrations
of sets are the direct sums of monomial ideals.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/11583/26515782017-01-01T00:00:00ZNetworks and cycles: a persistent homology
approach to complex networkshttp://hdl.handle.net/11583/2504859Titolo: Networks and cycles: a persistent homology
approach to complex networks
Abstract: Persistent homology is an emerging tool to identify robust
topological features underlying the structure of high-dimensional data
and complex dynamical systems (such as brain dynamics, molecular folding,
distributed sensing).
Its central device, the ltration, embodies this by casting the analysis
of the system in terms of long-lived (persistent) topological properties
under the change of a scale parameter.
In the classical case of data clouds in high-dimensional metric spaces,
such ltration is uniquely dened by the metric structure of the point
space. On networks instead, multiple ways exists to associate a ltration.
Far from being a limit, this allows to tailor the construction to the speci
c analysis, providing multiple perspectives on the same system.
In this work, we introduce and discuss three kinds of network ltrations,
based respectively on the intrinsic network metric structure, the hierarchical
structure of its cliques and - for weighted networks - the topological
properties of the link weights. We show that persistent homology is robust
against dierent choices of network metrics. Moreover, the clique
complex on its own turns out to contain little information content about
the underlying network. For weighted networks we propose a ltration
method based on a progressive thresholding on the link weights, showing
that it uncovers a richer structure than the metrical and clique complex
approaches.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/11583/25048592013-01-01T00:00:00ZTopological Strata of Weighted Complex Networkshttp://hdl.handle.net/11583/2508494Titolo: Topological Strata of Weighted Complex Networks
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/11583/25084942013-01-01T00:00:00Z