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