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All your comments are useful for me. Thank you very much.
Let me summary many comments.
1. from Ulrik Brandes
Line Graph are a useful tool for [structural importance of edges]. Apply
a node centrality measure on the line graph. (But these are not appropriate
in all cases.)
2. from Hyo Kim
The way that I understand is that it shows the [strength of the lines]
(With useful example. Thank you.)
3. from Carter T. Butts
In the case of hypergraphs, line graphs can have a very direct
interpretation: [where hyperedges represent groups, adjacent hyperedges
within the line graph represent groups whose memberships overlap.] This
logic can be taken further by considering edges within the line graph as
(in a certain restricted sense) synonymous with the individuals who form
the intersections between the groups in question.
4. from Jim Moody
Mirjam Kretzschmar and Martina Morris, for example, show that the
[concurrency levels in a network] are a [straight-forward function of the
mean and variance of the degree of the line graph].
5. from Martina Morris
We've used the mean [degree of the line graph] to represent the
[epidemic potential] in a network.
I want to add one idea. Maybe we can analyze "roles" of "lines" in line
graph (similary to centrality) and "structural or regular equivalences" of
"lines. Can it be meaningful? (It seems so in my humble opinion.)
In fact, I want to expand "line graph" to [triad graph or tetrad graph].
etc. We can regard "line" as "dyad".
Triad graph is constructed by making "triad (group of three nodes, dyad in
case of line graph)" into node. Tetrad graph is similarly constructed (four
in place of three nodes). Then we can analyze [structural importance] or
[strength] of triads, tetrads. This expanding is not restricted triad,
tetrad. It can be expanded to n-ad.
How about this expanding? (In my humble opinion, these seems to can be used
measuing fixed size's subgroup's centrality.)
In another approach, we can use [total graph] in place of "line graph".
Total graph is original graph + line graph roughly (node -> node, line ->
node, line 1 and line 2 are adjacent if those share node, noe and line are
adjacent if those are incident). In total graph, we can consider node-node
interrelationships, node-edge interrelationships, edge-edge
interrelationships. How about this expanding?
I want to research about this idea, but have only rough idea. If there was
some similar examples, please let me know.
If there wasn't, please advice this idea.
Again thank you for any comments.
Se Kwon, Kim
KAIST (Korean Advanced Institute of Science & Technology)
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