Speaker: Ginestra Bianconi

Abstract:
The dynamics of a network is not only encoded in node variables but also in edge variables, like currents and fluxes. The graph Laplacian has been extensively used in Machine Learning and Artificial Intelligence to process node signals,  however the graph Laplacian and its higher-order versions (the Hodge Laplacians) can only process separately nodes and edge signals. The Topological Dirac operator [1] allows cross-talk of the node and edge signals, encoded in the topological spinor, and is able to jointly treat and process them. The Topological Dirac operator is currently attracting growing attention in AI  and Network Science applications. In this talk we will provide an overview of the theory beyond the Topological Dirac operator and their various applications in AI including Dirac signal processing [2], Dirac neural networks [3,4] and Dirac Gaussian kernels [5].

[1] Bianconi, G., 2021. The topological Dirac equation of networks and simplicial complexes. Journal of Physics: Complexity, 2(3), p.035022.

[2] Calmon, L., Schaub, M.T. and Bianconi, G., 2023. Dirac signal processing of higher-order topological signals. New Journal of Physics, 25(9), p.093013. 

[3] Nauck, C., Gorantla, R., Lindner, M., Schürholt, K., Mey, A.S. and Hellmann, F., 2024. Dirac--Bianconi Graph Neural Networks--Enabling Non-Diffusive Long-Range Graph Predictions. arXiv preprint arXiv:2407.12419.

[4] Battiloro, C., Testa, L., Giusti, L., Sardellitti, S., Di Lorenzo, P. and Barbarossa, S., 2023. Generalized simplicial attention neural networks. arXiv preprint arXiv:2309.02138.

[5] Alain, M., Takao, S., Paige, B. and Deisenroth, M.P., 2023. Gaussian Processes on Cellular Complexes. arXiv preprint arXiv:2311.01198.