Heat Kernel of Networks with Long-Range Interactions

Figure 14

Results of simulations for the generalized heat content over time for the graph in Figure 7. Panel (a) depicts the results for various values of the parameter in the Mellin transform-based generalized normalized Laplacian matrix, while Panel (b) corresponds to the Laplace transform-based generalized normalized Laplacian for different values of . As and increase, we observe that the heat contents in both panels tend to approach the heat content of the standard Laplacian (depicted by the blue dotted curve) without LRI. Notably, this convergence occurs more rapidly when the Laplace transform is utilized (b) compared to the Mellin transform (a).

(a)

(b)