Average path length in random networks

Analytic solution for the average path length in a large class of uncorrelated random networks with hidden variables is found. We apply the approach to classical random graphs of Erdös and Rényi (ER), evolving networks introduced by Barabási and Albert as well as random networks with asymptotic scale-free connectivity distributions characterized by an arbitrary scaling exponent $\alpha >2$. Our result for $2<\alpha <3$ shows that structural properties of asymptotic scale-free networks including numerous examples of real-world systems are even more intriguing than ultra-small world behavior noticed in pure scale-free structures and for large system sizes $N\to \infty$ there is a saturation effect for the average path length.

Figure 1

The average path length $l_{ER}$ versus network size $N$ in ER classical random graphs with $⟨k⟩=4,10,20$. The solid curves represent the numerical prediction of Eq. (19).

Figure 2

Characteristic path length $l_{BA}$ versus network size $N$ in BA networks. Solid lines represent Eq. (25).

Figure 3

Model $C$. Degree distribution $P\left(k\right)$ (the main layer) for random network with an underlying hidden variable distribution given by a power-law (the inset). Scatter data represent results of numerical calculations, whereas solid curves express formulas (6, 29), respectively, for $P\left(k\right)$ and $\rho \left(h\right)$.

Figure 4

Model $C$. The average path length versus network size $N$ for $\alpha =4$ (C1), $\alpha =3$ (C2) and $\alpha =2.5$ (C3). The scatter data represent numerical calculations. Solid curves with open squares in the case of $m=5$ (open circles in the case of $m=20$) express analytical predictions of Eqs. (30, 31, 32), respectively, for (C1), (C2) and (C3).