🍩 Database of Original & Non-Theoretical Uses of Topology

(found 2 matches in 0.00081s)

  1. Contagion Dynamics for Manifold Learning (2020)

    Barbara I. Mahler
    Abstract Contagion maps exploit activation times in threshold contagions to assign vectors in high-dimensional Euclidean space to the nodes of a network. A point cloud that is the image of a contagion map reflects both the structure underlying the network and the spreading behaviour of the contagion on it. Intuitively, such a point cloud exhibits features of the network's underlying structure if the contagion spreads along that structure, an observation which suggests contagion maps as a viable manifold-learning technique. We test contagion maps as a manifold-learning tool on a number of different real-world and synthetic data sets, and we compare their performance to that of Isomap, one of the most well-known manifold-learning algorithms. We find that, under certain conditions, contagion maps are able to reliably detect underlying manifold structure in noisy data, while Isomap fails due to noise-induced error. This consolidates contagion maps as a technique for manifold learning.

  2. Extended Persistent Homology Distinguishes Simple and Complex Contagions With High Accuracy (2025)

    Vahid Shamsaddini, M. Amin Rahimian
    Abstract The social contagion literature makes a distinction between simple (independent cascade or bond percolation processes that pass infections through edges) and complex contagions (bootstrap percolation or threshold processes that require local reinforcement to spread). However, distinguishing simple and complex contagions using observational data poses a significant challenge in practice. Estimating population-level activation functions from observed contagion dynamics is hindered by confounding factors that influence adoptions (other than neighborhood interactions), as well as heterogeneity in individual behaviors and modeling variations that make it difficult to design appropriate null models for inferring contagion types. Here, we show that a new tool from topological data analysis (TDA), called extended persistent homology (EPH), when applied to contagion processes over networks, can effectively detect simple and complex contagion processes, as well as predict their parameters. We train classification and regression models using EPH-based topological summaries computed on simulated simple and complex contagion dynamics on three real-world network datasets and obtain high predictive performance over a wide range of contagion parameters and under a variety of informational constraints, including uncertainty in model parameters, noise, and partial observability of contagion dynamics. EPH captures the role of cycles of varying lengths in the observed contagion dynamics and offers a useful metric to classify contagion models and predict their parameters. Analyzing geometrical features of network contagion using TDA tools such as EPH can find applications in other network problems such as seeding, vaccination, and quarantine optimization, as well as network inference and reconstruction problems.