Applied Topology and Complex Networks

5–9 February, 2024

About the workshop

This workshop will feature the following two mini courses:

• Exploring the Borsuk-Ulam Theorem: Applications in Geometry and Combinatorics
  by Arijit Ghosh (ISI Kolkata, India)

  The Borsuk-Ulam Theorem, a fundamental result in algebraic topology, establishes a fundamental connection between symmetry and topology. It states that any continuous mapping from an n-dimensional sphere to n-dimensional Euclidean space must necessarily map a pair of diametrically opposite points to the same point. This theorem provides profound insights into the inherent relationship between geometric and algebraic properties. Throughout this lecture series, our primary objective is to understand the foundational principles of the Borsuk-Ulam Theorem and explore its many applications in addressing important questions across the diverse domains of geometry and combinatorics.

  Plan: A series of 5 lectures.

  Course Material

• A tour of methods and algorithms for the manipulation and analysis of temporal networks
  by Esteban Bautista-Ruiz (Universite du Littoral Cote d'Opale, France)

  Computer networks, social networks, the web, or the electric grid are examples of systems that play a crucial role in modern society. In the last decade, the field of complex networks has studied these systems through their interrelated structure, revealing extremely valuable insights about their behavior. As our understanding deepens, it becomes evident that these systems not only possess interrelated characteristics but also undergo evolution, rendering the temporal dimension a powerful source of information. The field of temporal networks emerges as the key to unlocking this temporal wealth. However, leveraging temporal information is not straightforward and it necessitates sophisticated frameworks and ideas. Currently, there is no unified theory, with numerous concepts and methodologies being proposed and tested. The objective of our mini-course is to provide a general tour of current ideas, methods and algorithms in the realm of temporal networks.

  Plan (Tentative):

  • Day 1: Introduction

    A reminder of graph theory and introduction to temporal networks (basic definitions, challenges, overview of course).

  • Day 2: Updating algorithms for static graph properties

    We begin by studying some algorithms that do not care about temporal information. They aim to compute classical graph properties but in a setting where the graph can evolve. Thus, instead of recomputing the graph properties from scratch every time the network changes, they try to update a previously known result in a much more efficient manner.

  • Day 3: Temporal networks as a generalization of graphs

    We study two formalisms that aim to extend graph theory into a theory of temporal networks: the TVG and Link Stream approaches.

  • Day 4: Applications to node centrality and community detection

    We study algorithms that leverage the formalisms presented in Day 3 to do the concrete data mining applications. In particular, that aim to estimate the importance of vertices and the presence of communities in the data.

  • Day 5: Temporal networks as a generalization of time series

    We study a formalism that aims to extend signal processing into a theory of temporal networks.

Schedule