Discrete Curvature and Machine Learning on Graphs
Complex Networks are popular means for studying a wide variety of systems across the social and natural sciences. In a series of articles, we developed geometric tools to describe the structure and evolution of such networks. The core component of our theory, a discrete Ricci curvature, gives rise to two geometric flows that allow for an edge-based network analysis. Thus we extend the commonly used node-based approach to include edge-based information such as edge weights and directionality for a more comprehensive and computationally efficient characterization of networks.
The analysis of a wide range of complex networks suggests connections between curvature and higher order network structure. As a proxy for local assortativity, curvature identifies long-range connections that act as bridges between major network components. By identifying higher order structural features we characterize and classify the network’s geometry.
Publications
- Y. Tian, Z. Lubberts, M. Weber (2022): Mixed-Membership Community Detection via Line Graph Curvature. NeurIPS Symmetry and Geometry in Neural Representations
- M. Weber (2020): Neighborhood Growth Determines Geometric Priors for Relational Representation Learning. AISTATS
- E. Saucan∗, M. Weber∗ (2019): Forman’s Ricci curvature – From networks to hypernetworks. 7th Conference on Complex Networks and Their Applications. (*: co-first authors)
- M. Weber, J. Jost, E. Saucan (2018): Detecting the Coarse Geometry of Networks. NeurIPS Relational Representation Learning
- E. Saucan, A. Samal, M. Weber and J. Jost (2018): Discrete Curvatures and Network Analysis. MATCH, vol. 80 (3), pp. 605–622.
- M. Weber, J. Stelzer, E. Saucan, A. Naitsat, G. Lohmann and J. Jost (2017): Curvature-based Methods for Brain Network Analysis. arXiv: 1707.00180. Technical Report.
- M. Weber, E. Saucan and J. Jost (2017): Coarse Geometry of Evolving Networks. Journal of Complex Networks, vol. 6(5), pp. 706-732.
- M. Weber, E. Saucan and J. Jost (2017): Characterizing Complex Networks with Forman-Ricci Curvature and Associated Geometric Flows. Journal of Complex Networks, vol. 5 (4), 527-550.
- M. Weber, J. Jost and E. Saucan (2016): Forman-Ricci flow for change detection in large dynamic data sets. Axioms, vol. 5 (4).
