The Graphon Fourier Transform

In many network problems, graphs may change by the addition of nodes, or the same problem may need to be solved in multiple similar graphs. This generates inefficiency, as analyses and systems that are not transferable have to be redesigned. To address this, we consider graphons, which are both limit objects of convergent graph sequences and random graph models. We define graphon signals and introduce the Graphon Fourier Transform (WFT), to which the Graph Fourier Transform (GFT) is shown to converge. This result is demonstrated in two numerical experiments where, as expected, the GFT converges, hinting to the possibility of centralizing analysis and design on graphons to leverage transferability.

[1]  R. Morgan.  Computing Interior Eigenvalues of Large Matrices , 1991 .

[2]  José M. F. Moura,et al.  Discrete signal processing on graphs: Graph filters , 2013, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing.

[3]  Bala Rajaratnam,et al.  Model-free consistency of graph partitioning , 2016 .

[4]  P. Wolfe,et al.  Nonparametric graphon estimation , 2013, 1309.5936.

[5]  Santiago Segarra,et al.  Optimal Graph-Filter Design and Applications to Distributed Linear Network Operators , 2017, IEEE Transactions on Signal Processing.

[6]  Alejandro Ribeiro,et al.  Optimal Wireless Resource Allocation With Random Edge Graph Neural Networks , 2019, IEEE Transactions on Signal Processing.

[7]  Santiago Segarra,et al.  Spectral Partitioning of Time-varying Networks with Unobserved Edges , 2019, ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[8]  M. Bálek,et al.  Large Networks and Graph Limits , 2022 .

[9]  José M. F. Moura,et al.  Discrete Signal Processing on Graphs , 2012, IEEE Transactions on Signal Processing.

[10]  Gitta Kutyniok,et al.  On the Transferability of Spectral Graph Filters , 2019, 2019 13th International conference on Sampling Theory and Applications (SampTA).

[11]  Antonio G. Marques,et al.  Rating Prediction via Graph Signal Processing , 2018, IEEE Transactions on Signal Processing.

[12]  Pierre Vandergheynst,et al.  Graph Signal Processing: Overview, Challenges, and Applications , 2017, Proceedings of the IEEE.

[13]  V. Sós,et al.  Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing , 2007, math/0702004.

[14]  Geert Leus,et al.  Signal processing on kernel-based random graphs , 2017, 2017 25th European Signal Processing Conference (EUSIPCO).

[15]  V. Sós,et al.  Convergent Sequences of Dense Graphs II. Multiway Cuts and Statistical Physics , 2012 .

[16]  Graham J. Wills,et al.  NicheWorks - Interactive Visualization of Very Large Graphs , 1997, GD.

[17]  José M. F. Moura,et al.  Discrete Signal Processing on Graphs: Frequency Analysis , 2013, IEEE Transactions on Signal Processing.

[18]  Christopher C. Paige,et al.  The computation of eigenvalues and eigenvectors of very large sparse matrices , 1971 .

[19]  Pascal Frossard,et al.  The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains , 2012, IEEE Signal Processing Magazine.

[20]  F. Maxwell Harper,et al.  The MovieLens Datasets: History and Context , 2016, TIIS.

[21]  S. Arya.  Air Pollution Meteorology and Dispersion , 1998 .

[22]  Edoardo M. Airoldi,et al.  Stochastic blockmodel approximation of a graphon: Theory and consistent estimation , 2013, NIPS.

[23]  Ngai-Man Cheung,et al.  Dimensionality reduction of brain imaging data using graph signal processing , 2016, 2016 IEEE International Conference on Image Processing (ICIP).

[24]  Francesca Parise,et al.  Graphon Games , 2018, EC.

[25]  M. Penrose.  CONNECTIVITY OF SOFT RANDOM GEOMETRIC GRAPHS , 2013, 1311.3897.

[26]  Alejandro Ribeiro,et al.  Invariance-Preserving Localized Activation Functions for Graph Neural Networks , 2019, IEEE Transactions on Signal Processing.

[27]  Albrecht Seelmann.  Notes on the sin 2Θ Theorem , 2014 .

[28]  Santiago Segarra,et al.  Centrality Measures for Graphons: Accounting for Uncertainty in Networks , 2017, IEEE Transactions on Network Science and Engineering.

[29]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .