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Complex gene interactions play a significant role in cancer progression, driving cellular behaviors that contribute to tumor growth, invasion, and metastasis. Gene co-expression networks model the functional connectivity between genes under various biological conditions. Understanding the system-level evolution of these networks in cancer is critical for elucidating disease mechanisms and informing the development of targeted therapies. While previous studies have primarily focused on structural differences between cancer and normal cell co-expression networks, this study applies graph frequency analysis to cancer transcriptomic signals defined on gene co-expression networks, highlighting the graph spectral characteristics of cancer systems. Using a range of graph frequency filters, we showed that cancer cells display distinctive patterns in the graph frequency content of their gene transcriptomic signals, effectively distinguishing between cancer types and stages. The transformation of the original gene feature space into the graph spectral space captured more intricate cancer properties, as validated by significantly higher F-statistic scores for graph frequency-filtered gene features compared to those in the original space.

In this paper, we introduce an adaptive graph normalized least mean pth power (GNLMP) algorithm for graph signal processing (GSP) that utilizes GSP techniques, including bandlimited filtering and node sampling, to estimate sampled graph signals under impulsive noise. Different from least-squares-based algorithms, such as the adaptive GSP Least Mean Squares (GLMS) algorithm and the normalized GLMS (GNLMS) algorithm, the GNLMP algorithm has the ability to reconstruct a graph signal that is corrupted by non-Gaussian noise with heavy-tailed characteristics. Compared to the recently introduced adaptive GSP least mean pth power (GLMP) algorithm, the GNLMP algorithm reduces the number of iterations to converge to a steady graph signal. The convergence condition of the GNLMP algorithm is derived, and the ability of the GNLMP algorithm to process multidimensional time-varying graph signals with multiple features is demonstrated as well. Simulations show the performance of the GNLMP algorithm in estimating steady-state and time-varying graph signals is faster than GLMP and more robust in comparison to GLMS and GNLMS.