<s>
The	O
(	O
also	O
known	O
as	O
the	O
HCS	O
algorithm	O
,	O
and	O
other	O
names	O
such	O
as	O
Highly	O
Connected	O
Clusters/Components/Kernels	O
)	O
is	O
an	O
algorithm	O
based	O
on	O
graph	B-Application
connectivity	O
for	O
cluster	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
It	O
works	O
by	O
representing	O
the	O
similarity	O
data	O
in	O
a	O
similarity	O
graph	B-Application
,	O
and	O
then	O
finding	O
all	O
the	O
highly	O
connected	O
subgraphs	O
.	O
</s>
<s>
The	O
goal	O
of	O
cluster	B-Algorithm
analysis	I-Algorithm
is	O
to	O
group	O
elements	O
into	O
disjoint	O
subsets	O
,	O
or	O
clusters	O
,	O
based	O
on	O
similarity	O
between	O
elements	O
,	O
so	O
that	O
elements	O
in	O
the	O
same	O
cluster	O
are	O
highly	O
similar	O
to	O
each	O
other	O
(	O
homogeneity	O
)	O
,	O
while	O
elements	O
from	O
different	O
clusters	O
have	O
low	O
similarity	O
to	O
each	O
other	O
(	O
separation	O
)	O
.	O
</s>
<s>
Similarity	O
graph	B-Application
is	O
one	O
of	O
the	O
models	O
to	O
represent	O
the	O
similarity	O
between	O
elements	O
,	O
and	O
in	O
turn	O
facilitate	O
generating	O
of	O
clusters	O
.	O
</s>
<s>
To	O
construct	O
a	O
similarity	O
graph	B-Application
from	O
similarity	O
data	O
,	O
represent	O
elements	O
as	O
vertices	O
,	O
and	O
elicit	O
edges	O
between	O
vertices	O
when	O
the	O
similarity	O
value	O
between	O
them	O
is	O
above	O
some	O
threshold	O
.	O
</s>
<s>
In	O
the	O
similarity	O
graph	B-Application
,	O
the	O
more	O
edges	O
exist	O
for	O
a	O
given	O
number	O
of	O
vertices	O
,	O
the	O
more	O
similar	O
such	O
a	O
set	O
of	O
vertices	O
are	O
between	O
each	O
other	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
if	O
we	O
try	O
to	O
disconnect	O
a	O
similarity	O
graph	B-Application
by	O
removing	O
edges	O
,	O
the	O
more	O
edges	O
we	O
need	O
to	O
remove	O
before	O
the	O
graph	B-Application
becomes	O
disconnected	O
,	O
the	O
more	O
similar	O
the	O
vertices	O
in	O
this	O
graph	B-Application
.	O
</s>
<s>
Minimum	O
cut	O
is	O
a	O
minimum	O
set	O
of	O
edges	O
without	O
which	O
the	O
graph	B-Application
will	O
become	O
disconnected	O
.	O
</s>
<s>
HCS	B-Algorithm
clustering	I-Algorithm
algorithm	I-Algorithm
finds	O
all	O
the	O
subgraphs	O
with	O
n	O
vertices	O
such	O
that	O
the	O
minimum	O
cut	O
of	O
those	O
subgraphs	O
contain	O
more	O
than	O
n/2	O
edges	O
,	O
and	O
identifies	O
them	O
as	O
clusters	O
.	O
</s>
<s>
Given	O
a	O
similarity	O
graph	B-Application
G(V,E )	O
,	O
HCS	B-Algorithm
clustering	I-Algorithm
algorithm	I-Algorithm
will	O
check	O
if	O
it	O
is	O
already	O
highly	O
connected	O
,	O
if	O
yes	O
,	O
returns	O
G	O
,	O
otherwise	O
uses	O
the	O
minimum	O
cut	O
of	O
G	O
to	O
partition	O
G	O
into	O
two	O
subgraphs	O
H	O
and	O
H	O
 '	O
,	O
and	O
recursively	O
run	O
HCS	B-Algorithm
clustering	I-Algorithm
algorithm	I-Algorithm
on	O
H	O
and	O
H	O
 '	O
.	O
</s>
<s>
The	O
following	O
animation	O
shows	O
how	O
the	O
HCS	B-Algorithm
clustering	I-Algorithm
algorithm	I-Algorithm
partitions	O
a	O
similarity	O
graph	B-Application
into	O
three	O
clusters	O
.	O
</s>
<s>
The	O
step	O
of	O
finding	O
the	O
minimum	O
cut	O
on	O
graph	B-Application
is	O
a	O
subroutine	O
that	O
can	O
be	O
implemented	O
using	O
different	O
algorithms	O
for	O
this	O
problem	O
.	O
</s>
<s>
The	O
running	O
time	O
of	O
the	O
HCS	B-Algorithm
clustering	I-Algorithm
algorithm	I-Algorithm
is	O
bounded	O
by	O
×	O
f(n, m )	O
.	O
</s>
<s>
f(n, m )	O
is	O
the	O
time	O
complexity	O
of	O
computing	O
a	O
minimum	O
cut	O
in	O
a	O
graph	B-Application
with	O
n	O
vertices	O
and	O
m	O
edges	O
,	O
and	O
is	O
the	O
number	O
of	O
clusters	O
found	O
.	O
</s>
<s>
For	O
fast	O
algorithms	O
for	O
finding	O
a	O
minimum	O
cut	O
in	O
an	O
unweighted	O
graph	B-Application
:	O
</s>
<s>
The	O
clusters	O
produced	O
by	O
the	O
HCS	B-Algorithm
clustering	I-Algorithm
algorithm	I-Algorithm
possess	O
several	O
properties	O
,	O
which	O
can	O
demonstrate	O
the	O
homogeneity	O
and	O
separation	O
of	O
the	O
solution	O
.	O
</s>
<s>
Theorem	O
1	O
The	O
diameter	O
of	O
every	O
highly	O
connected	O
graph	B-Application
is	O
at	O
most	O
two	O
.	O
</s>
<s>
There	O
is	O
a	O
famous	O
theorem	O
in	O
graph	B-Application
theory	O
that	O
says	O
that	O
if	O
every	O
vertex	O
has	O
degree	O
>=	O
n/2	O
,	O
then	O
the	O
diameter	O
of	O
G	O
(	O
the	O
longest	O
path	O
between	O
any	O
two	O
nodes	O
)	O
<=	O
2	O
.	O
</s>
<s>
Theorem	O
2	O
(	O
a	O
)	O
The	O
number	O
of	O
edges	O
in	O
a	O
highly	O
connected	O
graph	B-Application
is	O
quadratic	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
number	O
of	O
edges	O
in	O
a	O
highly	O
connected	O
graph	B-Application
must	O
be	O
at	O
least	O
(	O
n	O
×	O
n/2	O
)	O
/2	O
,	O
where	O
we	O
sum	O
the	O
degrees	O
of	O
each	O
vertex	O
and	O
divide	O
by	O
2	O
.	O
</s>
<s>
Removing	O
Low	O
Degree	O
Vertices	O
:	O
When	O
the	O
input	O
graph	B-Application
has	O
vertices	O
with	O
low	O
degrees	O
,	O
it	O
is	O
not	O
worthy	O
to	O
run	O
the	O
algorithm	O
since	O
it	O
is	O
computationally	O
expensive	O
and	O
not	O
informative	O
.	O
</s>
<s>
The	O
CLICK	O
clustering	B-Algorithm
algorithm	I-Algorithm
is	O
an	O
adaptation	O
of	O
HCS	O
algorithm	O
on	O
weighted	O
similarity	O
graphs	O
,	O
where	O
the	O
weight	O
is	O
assigned	O
with	O
a	O
probability	O
flavor	O
.	O
</s>
