what-when-how
In Depth Tutorials and Information
≤
, − ,
||
E
|| min{
2
k n
1
}
(4.9)
2
andfor
Rand Switch,
{
}
(
)
≤
,
, − −
||
E
||
2
2
min max min{
k
d n
1
d
}
.
(4.10)
i
i
i
Proof. Equation 4.7 and Equation 4.8 can be easily derived from Weyl's heorem.
Notice that the diagonal elements of
E
are always 0. Hence,
{
} {
}
C E
(
)
=
z
: −
z
e
≤
R
=
z
:
z
≤
R
.
i
ii
i
i
All these circles are concentric, and all the eigenvalues of
A
are thus in the circle of
the largest radius: ||
E
||
2
Y max
i
{
Ri
}, and
Ri
=
Σ
j
≠
i
|
eij
| is actually the total number
of added and deleted edges of vertex
i
.
Hence, for
RandAdd/Del
, when
k
<
n
/2, the worst case is that all the perturba-
tions involve the same vertex; when
k
<
n
/210490_
χο
µµ
Χ
004
ξ
015.
ρτφ
n
/2, the
worst case happens when a certain vertex is removed from all original edges to its
neighbors and adds new edges to all the rest of the vertices. In this case, max
i
{
Ri
}
Y min{2
k
,
n
- 1}, and Equation 4.10 follows.
For
RandSwitch
, if one edge is deleted, there must be an edge added to the same
vertex. herefore,
1 2
≤
, − −
,
through which we immediately get
/
R min d n
{
1
d
}
i
i
i
{
}
,
(
)
≤
,
, − −
max
R
2
min max min{
k
d n
1
d
}
i
i
i
i
i
and Equation 4.10 follows. “
▫
”
he graph spectrum has been well investigated in the graph analysis field. It has
been shown that the eigenvectors of the Laplacian matrix and the normal matrix
are good indicators of community clusters [10, 28, 32, 38]. he diference between
our nonrandomness framework and those traditional spectral clustering methods
is two-folded. First, spectral clustering methods aim to minimize the cut between
communities, while our randomness framework is based on maximizing the densi-
ties of communities. Second, in traditional spectral clustering methods, communi-
ties are represented by dense clusters in the spectral space of the Laplacian or normal
matrix, while communities in our framework are represented by quasi-orthogonal
lines in the spectral space of the adjacency matrix. Our proposed framework can
quantify randomness at all edge, node, and overall graph levels using the spectra of
the adjacency matrix. It is interesting to explore whether similar frameworks can
also be derived using the spectra of the Laplacian or normal matrix. We will study
this issue in our future work.
