Heat Kernel of Networks with Long-Range Interactions

Figure 19

Derivative of the generalized heat content at the origin using the Laplace transform against time for 5 Barabási–Albert (BA) graphs (represented by blue lines) and 5 Erdös–Rényi (ER) graphs (depicted by green lines). These graphs consist of 1000 nodes each, with an average degree ranging between 5 and 20. At , the curves are distinctly separated, especially for small values of the Laplace exponent . The introduction of long-range interactions (LRI) further enhances the separation of the curves and contributes to the effective characterization of complex random graphs at . (a)  = 1.5. (b)  = 2. (c)  = 3. (d)  = 4.

(a)

(b)

(c)

(d)