StatGraphs -- a package for graphs used in statistical models

Description

This package contains the types of graphs that are used in algebraic statistics: Bigraph and MixedGraph.

A bigraph is graph with bidirected edges and no multiple edges. A mixed graph is a graph with undirected, directed and bidirected edges.

This is an example of a bigraph on 4 vertices. It is created using the method bigraph.

i1 : G = bigraph {{3,4},{1,2},{2,4}}

o1 = Bigraph{1 => {2}   }
             2 => {4, 1}
             3 => {4}
             4 => {3, 2}

o1 : Bigraph

Next is an example of a mixed graph on 4 vertices with undirected, directed and bidirected edges. It is created using the method mixedGraph.

i2 : G = mixedGraph(graph{{1,2}},digraph {{1,3},{2,3}},bigraph {{3,4}})

o2 = MixedGraph{Bigraph => Bigraph{3 => {4}}}
                                   4 => {3}
                Digraph => Digraph{1 => {3}}
                                   2 => {3}
                                   3 => {}
                Graph => Graph{1 => {2}}
                               2 => {1}

o2 : MixedGraph

One can extract key information about mixed graphs using: undirectedGraph(MixedGraph), bigraph(MixedGraph), digraph(MixedGraph), vertices(MixedGraph),

or convert a mixed graph into a more convenient form using: collateVertices(MixedGraph), indexLabelGraph(MixedGraph).

There are several methods that allow to check the properties of mixed graphs: isCyclic(MixedGraph), isLoopless(MixedGraph), isSimple(MixedGraph).

One can also study the properties of vertices using: children(MixedGraph,Thing), parents(MixedGraph,Thing), descendants(MixedGraph,Thing), nondescendants(MixedGraph,Thing), forefathers(MixedGraph,Thing), neighbors(MixedGraph,Thing), nonneighbors(MixedGraph,Thing).

This package allows to construct loopless mixed graphs (LMG) as introduced in Kayvan Sadeghi and Steffen Lauritzen, Markov properties for mixed graphs, Bernoulli 20.2 (2014): 676-696 and to check whether a mixedGraph is an LMG using partitionLMG.

Caveat

StatGraphs requires Graphs version 0.3.3 or later.

Authors

Version

This documentation describes version 0.1 of StatGraphs.

Citation

If you have used this package in your research, please cite it as follows:

@misc{StatGraphsSource,
  title = {{StatGraphs: Graphs specific for algebraic statistics. Version~0.1}},
  author = {Carlos Amendola and Luis David Garcia Puente and Roser Homs Pons and Olga Kuznetsova and Harshit J Motwani},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Types
- Bigraph -- a graph with bidirected edges and no multiple edges
- MixedGraph -- a graph that has undirected, directed and bidirected edges
Functions and commands
- bigraph -- create a bigraph
- collateVertices -- convert a mixed graph into a new mixed graph where each component subgraph has the same set of vertices
- isLoopless -- check whether a graph contains a loop
- mixedGraph -- create a mixed graph from a combination of undirected graph, digraph and bigraph
- partitionLMG -- partition the vertices of a loopless mixed graph into adjacent to undirected edges and adjacent to bidirected edges
- undirectedGraph -- extract the undirected graph component of a mixed graph
Methods
- bigraph(HashTable) -- see bigraph -- create a bigraph
- bigraph(List) -- see bigraph -- create a bigraph
- bigraph(List,List) -- see bigraph -- create a bigraph
- bigraph(List,Matrix) -- see bigraph -- create a bigraph
- bigraph(Matrix) -- see bigraph -- create a bigraph
- bigraph(MixedGraph) -- extract the bigraph component of a mixed graph
- children(MixedGraph,Thing) -- return the children of a vertex of a mixed graph
- collateVertices(MixedGraph) -- see collateVertices -- convert a mixed graph into a new mixed graph where each component subgraph has the same set of vertices
- descendants(MixedGraph,Thing) -- return the descendants of a vertex of a mixed graph
- digraph(MixedGraph) -- extract the digraph component of a mixed graph
- forefathers(MixedGraph,Thing) -- return the forefathers of a vertex of a mixed graph
- graph(MixedGraph) -- convert mixed graph to a hash table
- indexLabelGraph(MixedGraph) -- relabel the vertices of a mixed graph according to their indices, indexed from 0
- isCyclic(MixedGraph) -- check whether a mixed graph contains a directed cycle
- isLoopless(Bigraph) -- see isLoopless -- check whether a graph contains a loop
- isLoopless(Digraph) -- see isLoopless -- check whether a graph contains a loop
- isLoopless(Graph) -- see isLoopless -- check whether a graph contains a loop
- isLoopless(MixedGraph) -- see isLoopless -- check whether a graph contains a loop
- isSimple(MixedGraph) -- check whether a mixed graph is simple
- mixedGraph(Bigraph) -- see mixedGraph -- create a mixed graph from a combination of undirected graph, digraph and bigraph
- mixedGraph(Bigraph,Digraph) -- see mixedGraph -- create a mixed graph from a combination of undirected graph, digraph and bigraph
- mixedGraph(Bigraph,Digraph,Graph) -- see mixedGraph -- create a mixed graph from a combination of undirected graph, digraph and bigraph
- mixedGraph(Bigraph,Graph) -- see mixedGraph -- create a mixed graph from a combination of undirected graph, digraph and bigraph
- mixedGraph(Bigraph,Graph,Digraph) -- see mixedGraph -- create a mixed graph from a combination of undirected graph, digraph and bigraph
- mixedGraph(Digraph) -- see mixedGraph -- create a mixed graph from a combination of undirected graph, digraph and bigraph
- mixedGraph(Digraph,Bigraph) -- see mixedGraph -- create a mixed graph from a combination of undirected graph, digraph and bigraph
- mixedGraph(Digraph,Bigraph,Graph) -- see mixedGraph -- create a mixed graph from a combination of undirected graph, digraph and bigraph
- mixedGraph(Digraph,Graph) -- see mixedGraph -- create a mixed graph from a combination of undirected graph, digraph and bigraph
- mixedGraph(Digraph,Graph,Bigraph) -- see mixedGraph -- create a mixed graph from a combination of undirected graph, digraph and bigraph
- mixedGraph(Graph) -- see mixedGraph -- create a mixed graph from a combination of undirected graph, digraph and bigraph
- mixedGraph(Graph,Bigraph) -- see mixedGraph -- create a mixed graph from a combination of undirected graph, digraph and bigraph
- mixedGraph(Graph,Bigraph,Digraph) -- see mixedGraph -- create a mixed graph from a combination of undirected graph, digraph and bigraph
- mixedGraph(Graph,Digraph) -- see mixedGraph -- create a mixed graph from a combination of undirected graph, digraph and bigraph
- mixedGraph(Graph,Digraph,Bigraph) -- see mixedGraph -- create a mixed graph from a combination of undirected graph, digraph and bigraph
- neighbors(MixedGraph,Thing) -- return the neighbors of a vertex of a mixed graph
- net(MixedGraph) -- print a mixed graph as a net
- nondescendants(MixedGraph,Thing) -- return the nondescendents of a vertex of a mixed graph
- nonneighbors(MixedGraph,Thing) -- return the neighbors of a vertex of a mixed graph
- parents(MixedGraph,Thing) -- return the parents of a vertex of a mixed graph
- partitionLMG(MixedGraph) -- see partitionLMG -- partition the vertices of a loopless mixed graph into adjacent to undirected edges and adjacent to bidirected edges
- toString(MixedGraph) -- print a mixed graph as a string
- undirectedGraph(MixedGraph) -- see undirectedGraph -- extract the undirected graph component of a mixed graph
- vertexSet(MixedGraph) -- see vertices(MixedGraph) -- create a union of all vertices of a mixed graph
- vertices(MixedGraph) -- create a union of all vertices of a mixed graph

For the programmer

The object StatGraphs is a package, defined in StatGraphs.m2.

The source of this document is in StatGraphs.m2:344:0.