Heat Kernel of Networks with Long-Range Interactions

Figure 6

Trace of the generalized heat kernel against time for the simple graph in Figure 7. LRI are accounted for by the Laplace (a) and Mellin (b) transforms of the k-path Laplacian matrix of the graph for different values of and . By increasing and , we can see that the traces tend to the trace of the standard Laplacian without no LRI. This is happening much faster when the Laplace transform is used (a) compared to the Mellin transform (b).

(a)

(b)