Abstract : Complex systems often exhibit emergent phenomena that cannot be understood by studying mere pairwise interactions. Emblematic examples include chemical reaction and molecular biology, where interactions often involve more than just two elements. Nevertheless, most of the mathematical tools of network science have been developed using traditional graphs, which are an explicitly pairwise representation. This mini-symposium will focus on the mathematical objects which are better suited to modelling higher order interactions, such as simplicial complexes and hypergraphs, as well as on the tools that can be used to investigate them.
[02953] Towards mixed volumes of binomial reaction networks
Format : Online Talk on Zoom
Author(s) :
Jane Ivy Coons (University of Oxford)
Mark Curiel (University of Hawai'i)
Elizabeth Gross (University of Hawai'i)
Abstract : Mass action kinetics on a chemical reaction network give rise to a polynomial system of equations. The zeros of this system are the steady states of the network, and its mixed volume is an upper bound on the number of complex steady states. We show that when the steady state ideal is binomial and the conservation laws satisfy a "partitionability" property, the mixed volume can be easily computed as a determinant.
[02968] A Hypergraph Model of Opinion Dynamics
Format : Online Talk on Zoom
Author(s) :
Abigail Hickok (UCLA)
Yacoub Kureh (Industry)
Heather Zinn Brooks (Harvey Mudd College)
Michelle Feng (Caltech)
Mason A. Porter (UCLA)
Abstract : Networks encode only pairwise interactions and cannot take into account polyadic interactions. I’ll discuss how the incorporation of polyadic interactions (encoded by a hypergraph) changes the qualitative behavior of an opinion dynamics model, which is a model of how people’s opinions change with time as they interact. We show that our hypergraph model converges to consensus under a wide range of initial conditions, whereas network models converge to polarization under the same initial conditions.
[02963] Topological techniques for classification of agent-based tumour-immune model
Format : Talk at Waseda University
Author(s) :
Gillian Grindstaff (University of California, Los Angeles)
Jingjie Yang (University of Oxford)
Hai Fang (University of Oxford)
Jagdeep Dhesi (University of Oxford)
Hee Rhang Yoon (Wesleyan University)
Joshua A. Bull (University of Oxford)
Helen M. Byrne (University of Oxford)
Heather A. Harrington (University of Oxford)
Abstract : We address the problem of classifying time series of synthetic 2-d spatial data from an agent-based model of tumour growth that includes tumour cells, macrophages, and blood vessels. We implement and compare the predictive power of four topological vectorizations specialized to such cell data: persistence images of Vietoris-Rips and radial filtrations at static time points, and persistence images for zigzag filtrations and persistence vineyards varying in time.
[02931] Topological methods for spatial data in molecular biology
Format : Talk at Waseda University
Author(s) :
Katherine Benjamin (University of Oxford)
Abstract : Single-parameter persistent homology – the flagship tool in topological data analysis – has witnessed a wide range of successful applications in the biological sciences over the last decade. Multiparameter persistent homology is a natural generalisation allowing for higher-order analysis of more complex phenomena including time-varying data. In this talk, we demonstrate some applications of multiparameter persistent homology to spatial data in biology.
00686 (2/3) : 3D @F412 [Chair: Gillian Grindstaff]
[02955] Clustering and trajectory classification via the Hodge Laplacian
Format : Talk at Waseda University
Author(s) :
Michael Schaub (RWTH Aachen University)
Abstract : We present methods to cluster point cloud data and trajectories based on spectral properties of the Hodge Laplacian. Our approach relies on similar ideas as found in spectral embeddings such as Laplacian eigenmaps. However, rather than constructing a single graph to cluster the data, we consider appropriately constructed simplicial complexes, and (a set of) associated Hodge-Laplacians which allow us to leverage a rich set of topological features for classification.
[02956] Kernel-based independence measures, hypergraphs, and higher-order interactions
Format : Talk at Waseda University
Author(s) :
Mauricio Barahona (Imperial College London)
Abstract : This talk will cover the use of kernel-based methods for the detection of higher order interactions and the relationships with formalizations using hypergraphs.
[02966] Higher-Order Phase Oscillator Networks from Phase Reductions
Format : Talk at Waseda University
Author(s) :
Christian Bick (Vrije Universiteit Amsterdam)
Abstract : Synchronization is a fascinating effect of the interaction between coupled oscillatory units and is ubiquitous in biological systems. If the coupling between units is sufficiently weak, phase reductions provide an adequate description of the dynamics. We discuss phase reductions beyond first order that yield phase oscillator networks with higher-order interactions. Specifically, we discuss how the nonpairwise higher-order phase interactions depend on the shape of the limit cycles and the underlying network structure.
[03193] When do two networks have the same steady state ideal?
Format : Talk at Waseda University
Author(s) :
Mark Curiel (University of Hawaii at Manoa)
Elizabeth Gross (University of Hawaii at Manoa)
Carlos Munoz (San Jose State University)
Abstract : Chemical reaction networks are often used to model biological processes, e.g. cell signaling. Assuming mass action kinetics, a reaction network gives rise to a polynomial system. We consider the ideal generated by these polynomials, called steady-state ideals. Our main results describe three combinatorial operations on the reaction graph that preserve the steady-state ideal. Furthermore, we give combinatorial conditions to identify monomials in a steady-state ideal.
00686 (3/3) : 3E @F412 [Chair: Gillian Grindstaff]
[02960] Hypergraph representation of topological features in complex systems
Format : Talk at Waseda University
Author(s) :
Agnese Barbensi (The University of Melbourne)
Abstract : Understanding how a system's behaviour emerges from its shape and structure is a critical question across modern science. Topological data analysis provides a powerful computational window on this problem. I will present some recent work in which we develop a framework to analyse the behaviour of complex systems via their structures. The proposed method is based on an interpretation of persistent homology summaries with network theoretical tools, combined with statistical and computational techniques.
[02964] Spectral theory of graphs and hypergraphs
Format : Online Talk on Zoom
Author(s) :
Raffaella Mulas (Vrije Universiteit Amsterdam)
Abstract : Spectral graph theory studies the qualitative properties of a graph that can be inferred from the eigenvalues and the eigenvectors of an operator associated with it. It has a long history, and it is widely used in applications. In this talk, we recall the key properties of the graph normalized Laplacian and we generalize it to the case of hypergraphs.
[02989] Role analysis for higher-order social systems
Format : Talk at Waseda University
Author(s) :
Nina Otter (Queen Mary University of London)
Abstract : Social scientists have been using networks to model social systems since at least the 1970s. A social network model is typically a collection of simple graphs, which can be thought of as a multi-relational graph. Studying the roles of actors in the network amounts to studying the semigroup structure of compound relations that arise from the multi-relational graph. These graph models assume a pairwise interaction between social actors. We study ways to extend the analysis of social roles to models that take into account not only pairwise but also higher-order relationships between social actors, such as social simplicial complexes and hypergraphs. A challenge in this process is to define compound higher-order relations with desirable algebraic and topological properties. In this talk I will discuss several ways to address this problem, drawing inspiration from various areas to find an appropriate generalization of composition, including q-analysis, PROPs, and chemical hypergraphs. I will then illustrate how different composition operations capture different types of information in real-world social systems.
The talk is based on joint work in progress, started at the AMS MRC 2022 on Applied Category Theory, jointly with Daniel Cicala, Rachel Hardeman Morrill, Abigail Hickok, Elise McMahon, Nikola Milicevic, Nima Motamed, Emily Roff.
[02951] Topological Information Retrieval with Dilation-Invariant Bottleneck Comparative Measures
Abstract : Representing elements in a database so that queries may be accurately matched is a central task in information retrieval. This is achieved by embedding the graph of the database into a manifold using a variety of metrics. Persistent homology is able to characterize a database in terms of hierarchy and connectivity. We show that embeddings retaining the database topology coincide topologically with dilation-invariant comparisons, which we propose to address metric distortion on manifolds.
