slice modularity
DESCRIPTION
Presentation at seminar Université catholique de Louvain, March 12, 2010TRANSCRIPT
Slice modularity
Vincent Traag
12 March 2010
Random walk and modularity
Random walk
• Random walk on graph pj(t + 1) =∑
j pi (t)pij where pij =Aij
ki.
• Steady state for undirected connected graphs p∗j =
kj
2m.
Modularity
• Assign a community ci to each node i .
• Probability P1(C ) to stay in community C is∑
ci ,cj=C
p∗i pij .
• Probability P∞(C ) to be in community C is∑
ci ,cj=C
p∗i p
∗j .
• Modularity is Q =∑
C P1(C ) − P∞(C ).
R. Lambiotte, J.-C. Delvenne, M. Barahona, arXiv:0812.1770
Slices
t=0t=1
t=2
Si ,rs Aij ,s
• Define
ki ,s =∑
j Aij ,s
si ,s =∑
r Si ,sr
κi ,s = ki ,s + si ,s
• Random walk with probability
pir ,js =δsrAij ,s + δijCj ,sr
κi ,r
Peter J. Mucha et al. arXiv:0911.1824
Slices
t=0t=1
t=2
Si ,rs Aij ,s
• Steady state
p∗jr =
κjr∑
jr κjr
• Going from slice-node ir to js
p∗ir |js = δrs
kj ,s
2ms
ki ,s
κi ,s
+ δij
Si ,rs
si ,s
si ,s
κi ,s
• Multi slice modularity
Q =∑
ijrs
∑
C
pir ,jsp∗ir − pir |jsp
∗ir
Peter J. Mucha et al. arXiv:0911.1824
Flatten the slices
t=0
t=1
t=2
Interslice and intraslice links considered a ‘layer’
Q =∑
s
sgns
∑
ij
(Aij ,s − pij ,s) δ(ci , cj)
• and interslice links
∑
sr
∑
i
Si ,srδ(cis , cir )
• We split it into intraslice
∑
s
∑
ij
[
Aij ,s −ki ,skj ,s
2ms
]
δ(cis , cjs)
• Multi slice modularity is
∑
ijrs
[
Aij ,s − δrski ,skj ,s
2ms+ δijSjsr
]
δ(cis , cjr )
Results (International Conflicts and Alliances)
1916 – World War I
Results (International Conflicts and Alliances)
1943 – World War II
Results (International Conflicts and Alliances)
1960 – Cold War
Results (International Conflicts and Alliances)
2000
Results (International Conflicts and Alliances)
0
5
10
15
20
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
’./conflict_alliance_change.txt’ u 1:2