Related papers: Information topology identifies emergent model cla…
Large knowledge graphs combine human knowledge garnered from projects ranging from academia and institutions to enterprises and crowdsourcing. Within such graphs, each relationship between two nodes represents a basic fact involving these…
There is currently growing interest in modeling the information diffusion on social networks across multi-disciplines. The majority of the corresponding research has focused on information diffusion independently, ignoring the network…
We propose a novel method for topological analysis of unweighted graphs which is based on \textit{persistent homology}. The proposed method maps the input graph to a complete weighted graph where the weighting function maps each edge to a…
Topological data analysis asks when balls in a metric space $(X,d)$ intersect. Geometric data analysis asks how much balls have to be enlarged to intersect. We connect this principle to the traditional core geometric concept of curvature.…
Topological invariants have proved useful for analyzing emergent function as they characterize a property of the entire system, and are insensitive to local details, disorder, and noise. They support boundary states, which reduce the system…
Knowledge is a network of interconnected concepts. Yet, precisely how the topological structure of knowledge constrains its acquisition remains unknown, hampering the development of learning enhancement strategies. Here we study the…
We apply modern methods in computational topology to the task of discovering and characterizing phase transitions. As illustrations, we apply our method to four two-dimensional lattice spin models: the Ising, square ice, XY, and…
Singular statistical models arise whenever different parameter values induce the same distribution, leading to non-identifiability and a breakdown of classical asymptotic theory. While existing approaches analyze these phenomena in…
The advent of Internet and World Wide Web has led to unprecedent growth of the information available. People usually face the information overload by following a limited number of sources which best fit their interests. It has thus become…
We introduce a new information-geometric structure associated with the dynamics on discrete objects such as graphs and hypergraphs. The presented setup consists of two dually flat structures built on the vertex and edge spaces,…
For safety and robustness of AI systems, we introduce topological parallax as a theoretical and computational tool that compares a trained model to a reference dataset to determine whether they have similar multiscale geometric structure.…
Topological data analysis is a powerful tool for describing topological signatures in real world data. An important challenge in topological data analysis is matching significant topological signals across distinct systems. In geometry and…
A standard problem in applied topology is how to discover topological invariants of data from a noisy point cloud that approximates it. We consider the case where a sample is drawn from a properly embedded C1-submanifold without boundary in…
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field.…
Topological data analysis provides a multiscale description of the geometry and topology of quantitative data. The persistence landscape is a topological summary that can be easily combined with tools from statistics and machine learning.…
Emergent patterns in complex systems are related to many intriguing phenomena in modern science and philosophy. Several conceptions such as weak, strong and robust emergence have been proposed to emphasize different epistemological and…
Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such…
Understanding the decision-making processes of large language models is critical given their widespread applications. To achieve this, we aim to connect a formal mathematical framework - zigzag persistence from topological data analysis -…
We study a class models of correlated random networks in which vertices are characterized by \textit{hidden variables} controlling the establishment of edges between pairs of vertices. We find analytical expressions for the main topological…
Modeling Internet growth is important both for understanding the current network and to predict and improve its future. To date, Internet models have typically attempted to explain a subset of the following characteristics: network…