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Figure 3. Load distributions for the two classes: A) The PIN of the yeast (ii) and the metabolic network of a eukaryote Emericella nidulans (iii), belonging to the class I. B) WWW within www.nd.edu domain (xi) and the Internet ASes (xiii) which belong to the class II. From Goh KI et al, Proc Nad Acad Sci USA 99:12583-8, ©2002 National Academy of Sciences, USA, with permission.40

The networks that we find to belong to the class II with 8 = 2.0 include:

vi. The Internet at the autonomous systems (AS) level as of October, 2001.43

vii. The metabolic networks for 6 species of archaea in reference 21.

viii.The WWW within www.nd.edu domain.12

x. The deterministic model by Jung et al.44

In particular, the networks (ix) and (x) are of tree structure, where the edge load distribution can be solved analytically. The load distributions for real-world networks (vi) and (viii) are shown in Figure 3B.

Topology of the Shortest Pathways

To understand the generic topological features of the networks in each class, we particularly focus on the topology of the shortest pathways between two vertices separated by a distance d. We define the mass-distance relation M(d) as the mean number of vertices on the shortest pathways between a given pair of vertices, averaged over all pairs separated by the same distance d. If the shortest pathway topology is simple and resembles a fractal with the fractal dimension Dp, M(d) would behave like -dp for large d, while if is tree-like, one would expect M(d) - d. We find that the mass-distance relation behaves differently for each class; for the class I, M(d) behaves nonlinearly (Fig. 4A-B), while for the class II, it is roughly linear (Fig. 4C-D).

For the networks belonging to the class I such as the PIN2 (iii) and the metabolic network for eukaryotes (iv), M(d) exhibits a nonmonotonic behavior (Fig. 4A,B), viz., it exhibits a hump at <4 ~ 10 for (iii) or ¿4 ~ 14 for (iv). To understand why such a hump arises, we visualize the topology of the shortest pathways between a pair of vertices, taken from the metabolic network of a eukaryote organism, Emericella nidulans (EN), as a prototypical example for the class I. Figure 5A shows such a graph with linear size 26 edges {d= 26), where an edge between

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