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| {{about|the formalism for [[ubiquitous computing]]|graphs whose edges alternate between two kinds of vertices|Bipartite graph}}
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| A '''bigraph''' (often used in the plural '''bigraphs''') can be modelled as the superposition of a [[Graph (mathematics)|graph]] (the ''link graph'') and a set of [[Tree (mathematics)|trees]] (the ''place graph'').<ref name="intro">''[http://www.itu.dk/research/pls/wiki/index.php/A_Brief_Introduction_To_Bigraphs A Brief Introduction To Bigraphs]'', [[IT University of Copenhagen]], Denmark.</ref><ref name="milner">Milner, Robin. ''[http://www.cl.cam.ac.uk/archive/rm135/uam-theme.html The Bigraphical Model]'', [[University of Cambridge Computer Laboratory]], UK.</ref>
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| Each [[Node (mathematics)|node]] of the bigraph is part of a graph and also part of some tree that describes how the nodes are nested. Bigraphs can be conveniently and formally displayed as [[diagram]]s.<ref name="intro" /> They have applications in the modelling of distributed systems for [[ubiquitous computing]] and can be used to describe [[Mobile agent|mobile]] interactions. They have also been used by [[Robin Milner]] in an attempt to subsume [[Calculus of Communicating Systems]] (CCS) and [[Pi calculus|π-calculus]].<ref name="milner" /> They have been studied in the context of [[category theory]].<ref>{{cite journal
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| |first=Robin
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| |last=Milner
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| |title=Bigraphs and Their Algebra
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| |journal=[[Electronic Notes in Theoretical Computer Science]]
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| |volume=209
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| |pages=5–19
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| |year=2008
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| |note=Proceedings of the LIX Colloquium on Emerging Trends in Concurrency Theory (LIX 2006)
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| |doi=10.1016/j.entcs.2008.04.002}}</ref>
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| ==Anatomy of a bigraph==
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| Aside from nodes and (hyper-)edges, a bigraph may have associated with it one or more ''regions'' which are roots in the place forest, and zero or more ''holes'' in the place graph, into which other bigraph regions may be inserted. Similarly, to nodes we may assign ''controls'' that define identities and an arity (the number of ''ports'' for a given node to which link-graph edges may connect). These controls are drawn from a bigraph ''signature''. In the link graph we define ''inner'' and ''outer'' names, which define the connection points at which coincident names may be fused to form a single link.
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| ==Foundations==
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| A bigraph is a 5-tuple:
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| <math>(V,E,ctrl,prnt,link) : \langle k,X \rangle \to \langle m,Y \rangle,</math>
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| where <math>V</math> is a set of nodes, <math>E</math> is a set of edges, <math>ctrl</math> is the ''control map'' that assigns controls to nodes, <math>prnt</math> is the ''parent map'' that defines the nesting of nodes, and <math>link</math> is the ''link map'' that defines the link structure. | |
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| The notation <math>\langle k,X \rangle \to \langle m,Y \rangle</math> indicates that the bigraph has <math>k</math> ''holes'' (sites) and a set of inner names <math>X</math> and <math>m</math> ''regions'', with a set of ''outer names'' <math>Y</math>. These are respectively known as the ''inner'' and ''outer'' interfaces of the bigraph.
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| Formally speaking, each bigraph is an arrow in a symmetric partial [[monoidal category]] (usually abbreviated ''spm-category'') in which the objects are these interfaces.<ref>{{cite conference
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| |first=Robin
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| |last=Milner
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| |title=Bigraphical Categories
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| |series=[[Lecture Notes in Computer Science]]
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| |volume=5710
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| |pages=30–36
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| |year=2009
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| |booktitle=CONCUR 2009 - ''Concurrency Theory''
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| |publisher=Springer-Verlag
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| |doi=10.1007/978-3-642-04081-8_3}}</ref> As a result, the composition of bigraphs is definable in terms of the composition of arrows in the category.
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| ==See also==
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| * [[Bisimulation]]
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| * [[Combinatorial species]]
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| ==Bibliography==
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| * {{cite book
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| |first=Robin
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| |last=Milner
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| |title=The Space and Motion of Communicating Agents
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| |publisher=[[Cambridge University Press]]
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| |year=2009
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| |isbn=0521738334
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| |note=(200 pages)}}
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| * {{cite conference
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| |first=Robin
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| |last=Milner
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| |title=Bigraphical reactive systems, (invited paper)
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| |booktitle=CONCUR 2001 – Concurrency Theory, Proc. 12th International Conference
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| |volume=2154
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| |series=[[Lecture Notes in Computer Science]]
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| |publisher=[[Springer-Verlag]]
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| |year=2001
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| |pages=16–35
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| |doi=10.1007/3-540-44685-0_2}}
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| * {{cite conference
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| |first=Robin
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| |last=Milner
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| |title=Bigraphs as a Model for Mobile Interaction (invited paper)
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| |booktitle=ICGT 2002: First International Conference on Graph Transformation
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| |series=[[Lecture Notes in Computer Science]]
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| |publisher=Springer-Verlag
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| |volume=2505
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| |year=2002
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| |pages=8–13
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| |doi=10.1007/3-540-45832-8_3
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| }}
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| * {{cite book
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| |first1=Søren
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| |last1=Debois
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| |first2=Troels Christoffer
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| |last2=Damgaard
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| |chapter=Bigraphs by Example
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| |id={{citeseerx|10.1.1.73.176}}
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| |title=IT University Technical Report Series TR-2005-61
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| |publisher=[[IT University of Copenhagen]]
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| |location=Denmark
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| |year=2005
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| |isbn=87-7949-090-5}}
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| ==References==
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| {{reflist}}
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| ==External links==
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| * [http://www.itu.dk/~mikkelbu/research/bigraphsbib/ Bibliography on Bigraphs]
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| [[Category:Formal methods]]
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| [[Category:Theoretical computer science]]
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