Combinatorial presentation of multidimensional persistent homology

A multifiltration is a functor indexed by $\mathbb{N}^r$ that maps any morphism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural $\mathbb{N}^r$-graded $R[x_1,\ldots, x_r]$-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and vector spaces. We prove in particular that the $\mathbb{N}^r$-graded $R[x_1,\ldots, x_r]$-modules that can occur as $R$-spans of multifiltrations of sets are the direct sums of monomial ideals.Comment: 21 pages, 3 figure

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

Homotopical decompositions of simplicial and Vietoris Rips complexes

Motivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced by a cover of its vertices. We study how the homotopy type of such decompositions approximates the homotopy of the simplicial complex itself. The difference between the simplicial complex and such an approximation is quantitatively measured by means of the so called obstruction complexes. Our general machinery is then specialized to clique complexes, Vietoris-Rips complexes and Vietoris-Rips complexes of metric gluings. For the latter we give metric conditions which allow to recover the first and zero-th homology of the gluing from the respective homologies of the components

Topological Strata of Weighted Complex Networks

The statistical mechanical approach to complex networks is the dominant paradigm in describing natural and societal complex systems. The study of network properties, and their implications on dynamical processes, mostly focus on locally defined quantities of nodes and edges, such as node degrees, edge weights and --more recently-- correlations between neighboring nodes. However, statistical methods quickly become cumbersome when dealing with many-body properties and do not capture the precise mesoscopic structure of complex networks. Here we introduce a novel method, based on persistent homology, to detect particular non-local structures, akin to weighted 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 hierarchically nested holes, while the second displays larger and longer living inhomogeneities. These classes cannot be reduced to known local or quasilocal network properties, because of the intrinsic non-locality of homological properties, 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. Moreover, this new method creates the first bridge between network theory and algebraic topology, which will allow to import the toolset of algebraic methods to complex systems.Comment: 26 pages, 19 figures, 1 tabl

Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function

A recent publication provides the network graph for a neocortical microcircuit comprising 8 million connections between 31,000 neurons (H. Markram, et al., Reconstruction and simulation of neocortical microcircuitry, Cell, 163 (2015) no. 2, 456-492). Since traditional graph-theoretical methods may not be sufficient to understand the immense complexity of such a biological network, we explored whether methods from algebraic topology could provide a new perspective on its structural and functional organization. Structural topological analysis revealed that directed graphs representing connectivity among neurons in the microcircuit deviated significantly from different varieties of randomized graph. In particular, the directed graphs contained in the order of $10^7$ simplices {\DH} groups of neurons with all-to-all directed connectivity. Some of these simplices contained up to 8 neurons, making them the most extreme neuronal clustering motif ever reported. Functional topological analysis of simulated neuronal activity in the microcircuit revealed novel spatio-temporal metrics that provide an effective classification of functional responses to qualitatively different stimuli. This study represents the first algebraic topological analysis of structural connectomics and connectomics-based spatio-temporal activity in a biologically realistic neural microcircuit. The methods used in the study show promise for more general applications in network science

A Topological Representation of Branching Neuronal Morphologies

The online version of this article (https://doi.org/10.1007/s12021-017-9341-1) contains supplementary material, which is available to authorized users. Among others, we thank Athanassia Chalimourda and Katherine Turner for helpful conversations in various stages of this research and Jay Coggan for a critical reading of the manuscript. We also thank Hanchuan Peng and Xiaoxiao Liu for providing and curating the BigNeuron datasets. This work was supported by funding for the Blue Brain Project (BBP) from the ETH Domain. P.D. and R.L. were supported part by the Blue Brain Project and by the start-up grant of KH. Partial support for P.D. has been provided by the Advanced Grant of the European Research Council GUDHI (Geometric Understanding in Higher Dimensions). MS was supported by the SNF NCCR “Synapsy”.Peer reviewedPublisher PD

Guidelines for diagnosis and management of the cobalamin-related remethylation disorders cblC, cblD, cblE, cblF, cblG, cblJ and MTHFR deficiency

BACKGROUND: Remethylation defects are rare inherited disorders in which impaired remethylation of homocysteine to methionine leads to accumulation of homocysteine and perturbation of numerous methylation reactions. OBJECTIVE: To summarise clinical and biochemical characteristics of these severe disorders and to provide guidelines on diagnosis and management. DATA SOURCES: Review, evaluation and discussion of the medical literature (Medline, Cochrane databases) by a panel of experts on these rare diseases following the GRADE approach. KEY RECOMMENDATIONS: We strongly recommend measuring plasma total homocysteine in any patient presenting with the combination of neurological and/or visual and/or haematological symptoms, subacute spinal cord degeneration, atypical haemolytic uraemic syndrome or unexplained vascular thrombosis. We strongly recommend to initiate treatment with parenteral hydroxocobalamin without delay in any suspected remethylation disorder; it significantly improves survival and incidence of severe complications. We strongly recommend betaine treatment in individuals with MTHFR deficiency; it improves the outcome and prevents disease when given early