E-mail network at Universitat Rovira i Virgili
Communities in the e-mail network of URV
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(a) Binary tree showing the
result of applying the GN algorithm to the e-mail network of URV. The
position indicated by the arrow represents the root of the tree and branches
are depicted so that they can be clearly differentiated. In particular, only
the leaves of the tree, that correspond to e-mail addresses, are plotted, as
shown in the detail that is zoomed. (b) Same as before but without showing
the leaves. Branches are now colored according to their Horton-Strahler
index. (c) Binary tree showing the result of applying the GN algorithm to a random
graph with the same size and connectivity than the e-mail network. The lack
of community structure is reflected in the absence of branches in the tree,
which contrasts with the intricate self-similar structure of (a) and (b).
Again, colors correspond to Horton-Strahler indices. |
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Self-similarity in the community
structure. (a) Calculation of the community size distribution for a binary
tree generated by the community identification algorithm. Black nodes
represent the actual nodes of the original graph while white nodes are just
graphical representations of communities that arise as a result of the
splitting procedure. Nodes A and B belong to a community of size 2, and
together with E form a community of size 3. Similarly, C, D and F form another
community of size 3. These two groups together form a higher level community
of size 6. Following up to higher and higher levels, the community structure
can be regarded as the set of nested groups. In this case there are three
communities of size 2, three communities of size 3, one community of size 6,
one community of size 7, and one community of size 10. Note that a single
node belongs to different communities at different levels. (b) Calculation of
the drainage area distribution for a river network. (c) Calculation of the
Horton-Strahler index. In this case, there are 10 branches with index 1, 3
branches with index 2, and 1 branch with index 3. (d) The distribution of
community sizes, P(s). The distribution of community sizes in a random
network is shown with a dotted line for comparison. (e) The number of
branches with HS index i, as a function of i. The community tree of the
e-mail network is topologically self-similar with B=5.76. Topological
self-similarity does not hold for the random case. |
Albert Díaz-Guilera