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BIMSA-HSE Joint Seminar on Data Analytics and Topology
Geometry of Electric Networks and data analysis in phylogenetic
Geometry of Electric Networks and data analysis in phylogenetic
Organizers
Speaker
Time
Monday, December 23, 2024 8:00 PM - 9:00 PM
Venue
A6-101
Online
Zoom 468 248 1222
(BIMSA)
Abstract
A classic problem in data analysis is studying the systems of subsets defined by either a similarity or a dissimilarity function which is either observed directly or derived from a data set X.
For an electrical network there are two functions defined by the resistance matrix and the response matrix either of which defines the network completely. We argue that these functions should be viewed as a similarity and a dissimilarity function moreover as such they are related via the covariance mapping also known as the Farris transform or the Gromov product. We will explore the properties of electrical networks from this point of view.
It is known that the resistance matrix defines a metric on the nodes of the electrical networks. Moreover for a circular planar electrical network this metric obeys the Kalmanson property as it was shown recently. We will call such a metric an electrical Kalmanson metric. The main results in the talk will be a complete description of the electrical Kalmanson metrics in the set of all Kalmanson metrics in terms of geometry of the positive Isotropic Grassmannian whose connection to the theory of electrical networks was discovered earlier.
One important area of applications where the Kalmanson metrics are actively used is the theory of phylogenetic networks which are a generalisation of phylogenetic trees. Our results allow to use the powerful methods of reconstruction of the minimal graphs of electrical networks in phylogenetics.
This is a joint work with Anton Kazakov.
For an electrical network there are two functions defined by the resistance matrix and the response matrix either of which defines the network completely. We argue that these functions should be viewed as a similarity and a dissimilarity function moreover as such they are related via the covariance mapping also known as the Farris transform or the Gromov product. We will explore the properties of electrical networks from this point of view.
It is known that the resistance matrix defines a metric on the nodes of the electrical networks. Moreover for a circular planar electrical network this metric obeys the Kalmanson property as it was shown recently. We will call such a metric an electrical Kalmanson metric. The main results in the talk will be a complete description of the electrical Kalmanson metrics in the set of all Kalmanson metrics in terms of geometry of the positive Isotropic Grassmannian whose connection to the theory of electrical networks was discovered earlier.
One important area of applications where the Kalmanson metrics are actively used is the theory of phylogenetic networks which are a generalisation of phylogenetic trees. Our results allow to use the powerful methods of reconstruction of the minimal graphs of electrical networks in phylogenetics.
This is a joint work with Anton Kazakov.
Speaker Intro
Vasilii Gorbunov obtained a PhD in topology in 1987 from Novosibirsk State University. He held postdoc positions at University of Manchester, UK and Northwestern University, USA. After that he held Professor positions in University of Kentucky, USA and Aberdeen University, UK until 2020. Currently he is Professor at the Higher School of Economics in Moscow, Russia.