Lecture 2: Graph Preliminaries January 20, 1999 1Lecture 2: Graph PreliminariesMark PeotRead: [CGH] Chapter 4.1 to 4.6.Comments during class:The moral graph definition for chain graphs in the notes is correct.There was a comment on the definition of parent. The definition in the notes is correct.There was a question on the uniqueness of paths in a tree. Given any two nodes, there is only one path from one node to another. Thus the path definition works.1.0 Why do we care about graphs?Graphs will be used to model the independence properties of distributions.The idea of graph separation will be used to capture a subset of the independence proper-ties implied by a distribution. Our general goal (approximately): graph separation will imply probabilistic independence that, in turn, will imply that the graph can be factored. If a graph can factored, then that will imply the structure of a graph, such that graphical sep-aration will imply independence.The idea of graph decomposition will be used in inference. If a graph is decomposable, we can use the distributed law of multiplication to simplify the computation of marginals in the distribution represented by that graph.A brief note on notation. I like using big roman capitals for sets and small roman capitals for the elements of sets.Note: Much of this section is adapted from [Lauritzen].2.0 GraphsA graph is a pair , where is a set of vertices or nodes and is a set of edges or links.1 Each edge is an ordered pair of nodes2.The graphs that we will consider are simple: In a simple graph, each edge is an ordered pair of distinct vertices (no self loops are permitted).1. In the belief net literature, both sets of terminology (nodes/links and vertices/edges) are used. However, the ‘cool’ people use the terms vertices and edges when referring to graphs.2. The book uses for edge (i,j). I think that the is redundant.GXL,()=XLab,()XX×∈LijLLecture 2: Graph Preliminaries January 20, 1999 2Edge is called undirected if both and are in . An edge is called directed if and . will be used to denote a directed edge . will be used to denote an undi-rected edge between and .Graphically, vertices are represented by ellipses, undirected edges by lines and directed edges by arrows. The figure below represents the graph or, alter-natively, .2.1 Relationships in GraphsGiven a graph :If , then is a parent of and is a child of .A path from to is an ordered sequence of two or more distinct nodes such that , and for . The length of the path is the number of links that it contains. If there is a path from to , we will write .The adjacency set for is , that is the set of nodes that can be reached by following all one edge paths. The neighbors of are .A closed path1 (or cycle) is similar to a path save that it starts and ends with the same node ( ). A cycle is called an n-cycle if it contains n edges.A cycle is directed if one of the edges on the path is directed, otherwise it is undirected ([CGH] calls this a loop).1. The book calls this a closed path, but the cool people call this a cycle.ab,() ab,() ba,() Lab,()L∈ba,()L∉ab→ ab,()ab–abG abc,,{}ab,()bc,()cb,(),,{},()=G abc,,{}abbc–,→{},()=abcGXL,()=ab→ abbaab x0… xn,,()ax0= bxn= xi 1–xi,()L∈ i 1 … n,,{}=ab ab⇒a adj a() bX∈ ab,()L∈{}=ane a() bX∈ ab– G∈{}=a … a,,Lecture 2: Graph Preliminaries January 20, 1999 3Given graph and a set of nodes , the induced subgraph of is the graph . That is, the induced subgraph consists only of nodes in and those links that pass between those nodes in .A node is a descendent of if , but not . Given graph , the descendents of are . The nondescendents of , , is the set . A node is an ancestor of if , but not . The ancestors of are .Two nodes and are connected if and . We will write this . A component is a maximal set of nodes with the property that every node is connected to every other node. Let denote the set of nodes that are connected to in induced sub-graph .The boundary of is .The expressions and refer the collections of parents of, children of, and nodes in the boundary of the nodes in that are not themselves in . For example, .3.0 Kinds of GraphsAn undirected graph is a graph that contains only undirected edges.A directed graph is a graph that contains only directed edges.A directed acyclic graph (or DAG) is a directed graph that contains no directed cycles.A chain graph is a graph with the property that the nodes can be partitioned1 into a sequence of numbered subsets such that1. all edges in each are undirected.2. if , , and then . 1. A partition of is a set of subsets of such that if and .GXL,()= AX∈ GAGGAAL A A×∩,()=AAbaab⇒ ba⇒ GXL,()=a de a() ca c⇒()ca⇒()¬∧{}= and a() X\dea() a{}∪()baba⇒ ab⇒ aan a() cc a⇒()ac⇒()¬∧{}=ab ab⇒ ba⇒ ab⇔a[]AaAbd a() a pa a() nea()∪pa A()chA(),bd A()AAbd A() bda()aA∈∪\A=GXAA1… An,,AAiAj∩∅= ij≠AA1…∪ An∪=XX1() … Xn()∪∪=GXi()ab→aXi()∈bXj()∈ ij>Lecture 2: Graph Preliminaries January 20, 1999 4The subsets are called the chain components of .A concise definition of chain graph: a graph with no directed cycles.A directed acyclic graph is a chain graph where each chain component consists of a single node. An undirected graph is a chain graph with only one chain component.3.1 Undirected GraphsThe undirected version of graph is the graph derived by replacing all of the edges of with undirected edges.A graph is connected if there is a path between any two distinct nodes in its correspond-ing undirected graph . A tree is an undirected graph with the property that there is exactly one path between any two distinct nodes.A forest is an undirected graph with the property that there is at most one path between any two distinct nodes.A graph is complete if there is an edge between every distinct pair of nodes in . A subset of the nodes of is complete if the induced subgraph is com-plete. A complete subgraph will also be called a clique.1If a clique A is not a subgraph of another clique in the graph, we will say that is maxi-mal.The complement of an undirected graph if iff .A set is a cover for the edges in if every edge in contains a vertex in . If no sub-set of is a cover, then is minimal.Thm. 1: If is a cover for the complement of , then is a clique in