Heat Kernel of Networks with Long-Range Interactions

Figure 20

Time evolution of the heat content based on the generalized Laplacian matrix using the Mellin transform for 5 Barabási–Albert (BA) graphs (depicted by blue lines) and 5 Erdös–Rényi (ER) graphs (represented by green lines). These graphs consist of 1000 nodes each, with an average degree varying between 5 and 20. As can be seen, the generalized heat content is able to distinguish random graphs having similar structure. The discriminating power increases with value of for both BA and ER graphs. (a)  = 1.5. (b)  = 2. (c)  = 3. (d)  = 4. (e)  = 5.

(a)

(b)

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(d)

(e)