Heat Kernel of Networks with Long-Range Interactions

Figure 5

Plots of the trace of the generalized heat kernel against time for the star (blue curve), ring (green curve), and path (orange curve) graphs of Figure 4(b). For comparison purposes, subfigure (a) shows the plot of the trace of the standard heat kernel. Subfigures (b) and (c) correspond to plots of the trace function for the generalized heat kernel based on the Mellin transform with and , respectively. Subfigures (d) and (e) are plots of trace function of the generalized heat kernel based on Laplace transform for and , respectively. As can be seen, for the same values of and , the trace function based on the Mellin transform decreases faster than that for the Laplace transform-based function. This fact is justified by the claim of Lemma 5.

(a)

(b)

(c)

(d)

(e)