A Steepest Descent Method for Time Estimation of Heat Diffusion Over Graph

Heat diffusion over a graph extends the classical heat diffusion problem into the realm of graph theory, where a graph consists of nodes connected by edges. This concept is widely used in various fields, such as network theory, computer science, and data analysis, to model and study the spread of heat through complex networks. Understanding heat diffusion over a graph provides powerful insights into the dynamic processes occurring within networks, offering a robust framework for analyzing and solving complex problems across various domains. In this paper, the heat diffusion process is treated as a graph signal governed by a discrete heat equation. When a noisy graph signal of heat diffusion is measured, the challenge is to estimate the diffusion time from the noisy measurement signal. To tackle this issue, a steepest descent method is proposed to estimate the diffusion time by minimizing the error function between the model signal and the measured signal. The matrix exponential used to compute model signal is calculated by using polynomial graph filter method for reducing the computation complexity. Experiments conducted on complex networks in the USA, with single-point, double-point, and triple-point heat sources, demonstrate the effectiveness of the proposed method.