Modularity and Extreme Edges of the Internet

We study the spectral properties of a diffusion process taking place on the Internet network focusing on the slowest decaying modes. These modes identify an underlying modular structure roughly corresponding to individual countries. For instance, in the slowest decaying mode the diffusion current flows from Russia to U.S. military sites. Quantitatively the modular structure manifests itself in a 10 times larger participation ratio of its slow decaying modes compared to a random scale-free network. We propose to use the fraction of nodes participating in slow decaying modes as a general measure of the modularity of a network. For the 100 slowest decaying modes of the Internet this fraction is $\sim 30\%$. Finally, we suggest that the degree of isolation of an individual module can be assessed by comparing its participation in different diffusion modes.

Figure 1

The participation ratio ${\mathrm{P}\mathrm{R}}^{(\alpha )}$ (a) and the eigenvalue density (b) as a function of the eigenvalue $-1<{\lambda }^{(\alpha )}<1$ measured in the Internet (filled circles) and in its randomized counterpart (open squares)—a random scale-free network (RSFN). The participation ratio was averaged over $\lambda$ bins of size 0.05 excluding eigenmodes ${\lambda }^{(\alpha )}=0$ [12] and ${\lambda }^{(1)}=1$. Notice that for $|\lambda |\simeq 1$ participation ratios in the Internet significantly exceed those in an RSFN indicating the modular character of the former network.

Figure 2

Components $c_i^{(2)}$ of the slowest decaying diffusion mode in the Internet (eigenvalue ${\lambda }^{(2)}=0.9626$) as a function of the AS number. The AS known to be geographically located in Russia are marked with circles. The scale of the negative part of the $y$ axis is increased for clarity. Out of 100 autonomous systems with the most negative components $c_i^{(2)}$, those 23 for which we were able to find the description are associated with the U.S. military.

Figure 3

The Internet clustering: Coordinates of the $i$th AS in this plot are its components $(c_i^{(2)},c_i^{(3)})$ in the two slowest decaying nonoscillatory diffusion modes. The color code reveals the geographical location of the AS (Russia: red squares; France: green circles; U.S.: blue crosses; Korea: orange triangles). Note the straight lines corresponding to good country modules.