Figure 1

Mean distance $⟨l_{ij}⟩$ between pairs of nodes $i$ and $j$ as a function of a product of their degrees $k_i{k}_j$: (a) Erdős-Rényi random graphs: $⟨k⟩=20$ and $N=10000$ (squares) $N=50000$ (circles); (b) Barabási-Albert networks: $⟨k⟩=8$ and $N=1000$ (squares) $N=10000$ (circles); (c) biological networks: Silwood (squares), Yeast (triangles), Ythan (diamonds); (d) coauthorship networks: Astro (squares), Cond-mat (circles); (e) Internet Autonomous Systems: Year 1997 (squares), Year 1998 (circles) Year 1999 (triangles), Year 2001 (diamonds); (f) public transport networks in Polish cities: Gorzów Wlkp. (squares), Łódź (triangles), Zielona Góra (circles) In (b), (d), and (e) data are logarithmically binned with the power of 2, in the case of (c) and (a) with the power of 1.25 and in case of (f) data are not binned.Reuse & Permissions

Figure 2

Dependence of average path length on $k_i$, for fixed $k_i{k}_j$ product. The lines connecting the symbols are there for clarity. The bars show point weight, meaning relative numbers of pairs $ij$. The horizontal lines are weighted averages over $k_i$, and are just average path lengths for given $k_i{k}_j$. The top shows data for the Internet Autonomous Systems, while bottom for Astro coauthorship network. Note: The very small shifts on $k_i$ axis between data for different $k_i{k}_j$ are artificially introduced to make the weight bars not overlap.Reuse & Permissions

Figure 3

Tree formed by a random process, starting from the node $i$ and approaching the node $j$.Reuse & Permissions