Heat Kernel of Networks with Long-Range Interactions

Figure 13

Plots illustrating the derivative of the zeta function at the origin for the graph in Figure 7. Panel (a) is for the generalized Laplacian matrix, L_G. In comparison, the plot in Panel (b) corresponds to the generalized normalized Laplacian (18). Notably, for small values of and , the Laplace-based case consistently yields higher values than the Mellin-based case. However, as , the derivatives of the zeta function at the origin converge.

(a)

(b)