<s>
A	O
bigraph	B-Application
can	O
be	O
modelled	O
as	O
the	O
superposition	O
of	O
a	O
graph	O
(	O
the	O
link	B-Application
graph	I-Application
)	O
and	O
a	O
set	O
of	O
trees	O
(	O
the	O
place	B-Application
graph	I-Application
)	O
.	O
</s>
<s>
Each	O
node	O
of	O
the	O
bigraph	B-Application
is	O
part	O
of	O
a	O
graph	O
and	O
also	O
part	O
of	O
some	O
tree	O
that	O
describes	O
how	O
the	O
nodes	O
are	O
nested	O
.	O
</s>
<s>
Bigraphs	B-Application
can	O
be	O
conveniently	O
and	O
formally	O
displayed	O
as	O
diagrams	O
.	O
</s>
<s>
They	O
have	O
applications	O
in	O
the	O
modelling	O
of	O
distributed	O
systems	O
for	O
ubiquitous	B-Architecture
computing	I-Architecture
and	O
can	O
be	O
used	O
to	O
describe	O
mobile	B-Operating_System
interactions	O
.	O
</s>
<s>
Aside	O
from	O
nodes	O
and	O
(	O
hyper	O
-	O
)	O
edges	O
,	O
a	O
bigraph	B-Application
may	O
have	O
associated	O
with	O
it	O
one	O
or	O
more	O
regions	O
which	O
are	O
roots	O
in	O
the	O
place	O
forest	O
,	O
and	O
zero	O
or	O
more	O
holes	O
in	O
the	O
place	B-Application
graph	I-Application
,	O
into	O
which	O
other	O
bigraph	B-Application
regions	O
may	O
be	O
inserted	O
.	O
</s>
<s>
These	O
controls	O
are	O
drawn	O
from	O
a	O
bigraph	B-Application
signature	O
.	O
</s>
<s>
In	O
the	O
link	B-Application
graph	I-Application
we	O
define	O
inner	O
and	O
outer	O
names	O
,	O
which	O
define	O
the	O
connection	O
points	O
at	O
which	O
coincident	O
names	O
may	O
be	O
fused	O
to	O
form	O
a	O
single	O
link	O
.	O
</s>
<s>
A	O
bigraph	B-Application
is	O
a	O
5-tuple	O
:	O
</s>
<s>
The	O
notation	O
indicates	O
that	O
the	O
bigraph	B-Application
has	O
holes	O
(	O
sites	O
)	O
and	O
a	O
set	O
of	O
inner	O
names	O
and	O
regions	O
,	O
with	O
a	O
set	O
of	O
outer	O
names	O
.	O
</s>
<s>
These	O
are	O
respectively	O
known	O
as	O
the	O
inner	O
and	O
outer	O
interfaces	O
of	O
the	O
bigraph	B-Application
.	O
</s>
<s>
Formally	O
speaking	O
,	O
each	O
bigraph	B-Application
is	O
an	O
arrow	O
in	O
a	O
symmetric	O
partial	O
monoidal	O
category	O
(	O
usually	O
abbreviated	O
spm-category	O
)	O
in	O
which	O
the	O
objects	O
are	O
these	O
interfaces	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
the	O
composition	O
of	O
bigraphs	B-Application
is	O
definable	O
in	O
terms	O
of	O
the	O
composition	O
of	O
arrows	O
in	O
the	O
category	O
.	O
</s>
<s>
Directed	O
Bigraphs	B-Application
are	O
a	O
generalisation	O
of	O
bigraphs	B-Application
where	O
hyper-edges	O
of	O
the	O
link-graph	O
are	O
directed	O
.	O
</s>
<s>
Directed	O
bigraphs	B-Application
were	O
introduced	O
as	O
a	O
meta-model	O
for	O
describing	O
computation	O
paradigms	O
dealing	O
with	O
locations	O
and	O
resource	O
communication	O
where	O
a	O
directed	O
link-graph	O
provides	O
a	O
natural	O
description	O
of	O
resource	O
dependencies	O
or	O
information	O
flow	O
.	O
</s>
<s>
Examples	O
of	O
areas	O
of	O
applications	O
are	O
security	B-Protocol
protocols	I-Protocol
,	O
resource	O
access	O
management	O
,	O
and	O
cloud	B-Architecture
computing	I-Architecture
.	O
</s>
<s>
Bigraphs	B-Application
with	O
sharing	O
are	O
a	O
generalisation	O
of	O
Milner	O
's	O
formalisation	O
that	O
allows	O
for	O
a	O
straightforward	O
representation	O
of	O
overlapping	O
or	O
intersecting	O
spatial	O
locations	O
.	O
</s>
<s>
In	O
bigraphs	B-Application
with	O
sharing	O
,	O
the	O
place	B-Application
graph	I-Application
is	O
defined	O
as	O
a	O
directed	O
acyclic	O
graph	O
(	O
DAG	O
)	O
,	O
i.e.	O
</s>
<s>
The	O
definition	O
of	O
link	B-Application
graph	I-Application
is	O
unaffected	O
by	O
the	O
introduction	O
of	O
sharing	O
.	O
</s>
<s>
Note	O
that	O
standard	O
bigraphs	B-Application
are	O
a	O
sub-class	O
of	O
bigraphs	B-Application
with	O
sharing	O
.	O
</s>
<s>
Areas	O
of	O
application	O
of	O
bigraphs	B-Application
with	O
sharing	O
include	O
wireless	O
networking	O
protocols	O
,	O
real-time	O
management	O
of	O
domestic	O
wireless	O
networks	O
and	O
mixed	B-General_Concept
reality	I-General_Concept
systems	O
.	O
</s>
<s>
BigraphER	O
is	O
a	O
modelling	O
and	O
reasoning	O
environment	O
for	O
bigraphs	B-Application
consisting	O
of	O
an	O
OCaml	B-Language
library	O
and	O
a	O
command-line	O
tool	O
providing	O
an	O
efficient	O
implementation	O
of	O
rewriting	O
,	O
simulation	O
,	O
and	O
visualisation	O
for	O
both	O
bigraphs	B-Application
and	O
bigraphs	B-Application
with	O
sharing	O
.	O
</s>
<s>
jLibBig	O
is	O
a	O
Java	B-Language
library	O
providing	O
efficient	O
and	O
extensible	O
implementation	O
of	O
reactive	O
systems	O
for	O
both	O
bigraphs	B-Application
and	O
directed	O
bigraphs	B-Application
.	O
</s>
<s>
BigMC	O
is	O
model	B-Application
checker	I-Application
for	O
bigraphs	B-Application
which	O
includes	O
a	O
command	O
line	O
interface	O
and	O
visualisation	O
.	O
</s>
<s>
Big	O
Red	O
is	O
a	O
graphical	O
editor	O
for	O
bigraphs	B-Application
with	O
easily	O
extensible	O
support	O
for	O
various	O
file	O
formats	O
.	O
</s>
<s>
SBAM	O
is	O
a	O
stochastic	O
simulator	O
for	O
bigraphs	B-Application
,	O
aimed	O
at	O
simulation	O
of	O
biological	O
models	O
.	O
</s>
<s>
DBtk	O
is	O
a	O
toolkit	O
for	O
directed	O
bigraphs	B-Application
that	O
provides	O
calculation	O
of	O
IPOs	O
,	O
matching	O
,	O
and	O
visualisation	O
.	O
</s>
