what-when-how
In Depth Tutorials and Information
can conclude that
x
ij
reflects the degree of node
j
's attachment to the community
Gi
.“
▫
”
Property1
.
AnodeubelongstoonecommunityGtifthet-thentryof
α
u,x
tu
,is
muchgreaterthantherestoftheentriesandx
iu
≈
0fori
≠
t.Anodeudoesnotbelongto
anycommunityifalltheentriesof
α
uarecloseto0,orequivalently,||
α
||2
≈
0.We call
suchnodesnoisenodes
.
Property2
.
Ifnodesuandvbelongtothesamecommunity,then|cos(
α
u,
α
v)|≈1.
Ifnodesuandvbelongtotwodiferentcommunities,respectively,then|cos(
α
u,
α
v)|≈
0.Otherwise,ifnodeubelongstoonecommunityG
t
,andbridgingnodevlocatesinthe
overlapoftwocommunitiesGtandGw,then|cos(
α
u,
α
v)|isnotclosetoeither0or1
.
Explanation
. Notice that
α α
T
α α
,
=
u
.
cos(
)
u
v
||
α
|| ||
α
||
u
2
v
2
When nodes
u
and
v
are in the same community
G
t
,
x
tu
, we have that
x
tv
is much
greater than the rest of entries in
α
u
and
α
v
. Hence
∑
k
x x
α α
T
x x
iu iv
u
=
i
=
1
≈
|
tu tv
x
|
= ± .
1
||
α
|| ||
α
||
1
2
1
2
x
||
⎛
∑
k
⎞
⎛
∑
k
⎞
u
2
v
2
tu
tv
x
2
x
2
⎜
⎜
⎟
⎟
⎜
⎜
⎟
⎟
iu
iv
⎝
i
=
1
⎠
⎝
i
=
1
⎠
In other words, points
α
u
and
α
v
approximately locate along a straight line that goes
through the origin.
Similarly, when node
u
and
v
are in two different communities
G
t
and
G
w
,
respectively, with
x
wu
≈ 0 and
x
tv
≈ 0, we have
α α
+
T
x x
x
x
u
≈
tu tv
wu wv
≈
0
||
α
|| ||
α
||
|
x
||
x
|
u
2
v
2
tu
wv
which means that
α
u
and
α
v
are approximately orthogonal.
If a bridging node
v
is in the overlap of two communities
St
and
S
w
, both
t
-th
and
w
-th entries in
α
v
are not negligible. Hence,
||
(
)
. For a node
1
2
α
v
||
≈
x
2
+
x
2
2
tv
wv
u
from
Gt
, we have
α α
T
|
|
|
|
x x
x
u
≈
tu tv
=
tv
.
||
α
|| ||
α
||
)
1
2
)
1
2
(
(
|
x
|
x
+
x
x
+
x
2
2
2
2
u
2
v
2
tu
tv
wv
tv
wv
Since neither
xtv
nor
xwv
is close to 0, |cos(
u
,
v
)| is not close to either 1 or 0, which
indicates that bridging nodes locate between the quasi-orthogonal lines formed by
communities, and are also away from the origin. “
▫
”
