Heat Kernel of Networks with Long-Range Interactions

Figure 15

Plots of the heat content (using the Laplacian matrix ) over time for 5 Barabási–Albert (blue lines) (a) and 5 Erdös–Rényi graphs (green lines), each comprising 1000 nodes with an average degree ranging from 5 to 20. The plot of the first derivative at the origin of the heat content with time is shown in (b). The curves for the first derivative at zero are already distinct compared to the curves of the heat content at the origin, indicating that the first derivative at zero can be useful for characterizing random graphs.

(a)

(b)