\title{Community detection algorithms for GBF-PUM signal approximation on graphs} \author{Roberto Cavoretto$^{*}$, Chiara Comoglio$^{*}$ \underline{Alessandra De Rossi$^{*}$}} \address{ $^{*}$ Department of Mathematics "Giuseppe Peano", University of Torino \\ via Carlo Alberto 10, 10123 Torino, Italy \\ e-mail: \{alessandra.derossi@unito.it\} } \begin{document} \begin{center} \bf ABSTRACT \end{center} Graph signal approximation plays a key role in processing irregularly distributed data on graphs, where achieving smooth and computationally efficient interpolation is a crucial point. In this talk, we introduce a new approach that combines a spectral community detection algorithm with the partition of unity method (PUM) applied to signal approximation on large graphs \cite{cav21}. The PUM provides an effective technique for handling irregularly distributed data by dividing the graph into smaller subgraphs, constructing local interpolants and combining them to produce a global approximation \cite{cav24}. Since the first step in the PUM consists in dividing the graph into disjoint communities, we focus in particular on exploring and testing some community detection algorithms based on the maximization of the modularity \cite{new13}. Then, we integrate the PUM with a local graph basis function approximation scheme, resulting in an accurate and computationally efficient technique for graph signal approximation. All the MATLAB codes implemented and used in this work are available as open-source software \cite{comdata}. \begin{thebibliography}{99} \bibitem{cav21} R. Cavoretto, A. De Rossi and W. Erb, \emph{Partition of unity methods for signal processing on graphs}, J. Fourier Anal. Appl. \textbf{27}(4) (2021), 66. \bibitem{cav24} R. Cavoretto, A. De Rossi, S. Lancellotti and F. Romaniello, \emph{Node-bound communities for partition of unity interpolation on graphs}, Appl. Math. Comput. \textbf{467} (2024), 128502. \bibitem{comdata} C. Comoglio, GitHub repository, https://github.com/chiaracomoglio/CommDet \bibitem{new13} M. E. Newman, \emph{Spectral methods for community detection and graph partitioning}, Phys. Rev. E \textbf{88}(4) (2013), 042822.