Enterprise Collaboration and
Innovation Support Systems
(GE/IE 498 ECI):
Connectivity Everywhere;
Graph Theory 101
Xavier Llorà
Illinois Genetic Algorithms Lab &
National Center for Supercomputing Applications
University of Illinois at Urbana-Champaign
xllora@uiuc.edu

Where did we leave it?
• Giving structure to information exchange
• Markup languages
• XML basics
• The semantic web (RDF)
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What’s the plan for today?
• Graphs as structure
• Basic concepts
• Types of graphs
• Basic concepts of graph analysis
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Königsberg and Mr. Euler
Leonhard Euler was a Swiss mathematician
•
(April 15, 1707 – September 7, 1783)
His paper in 1736 is considered the first paper on
•
graph theory
People say Mr. Euler spent their Sundays walking
•
around, trying to find a starting point so that he
could walk about Königsberg, cross each bridge
exactly once, and then return to their starting point.
Mr. Euler was a mathematician so he ask himself questions like the following
•
ones:
– Could I find a path that allow me to do so? Moreover, is that possible at all?
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Welcome to Königsberg
http://www-gap.dcs.st-and.ac.uk/%7Ehistory/HistTopics/Topology`_in_mathematics.html
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The formalized Königsberg
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What is a graph?
A graph is a set of vertex and a set of edges
•
G=<V,E>
Vertices are usually labeled
•
– V = {A,B,C,D}
The set of edges
•
– Direction is important (directed graph)
• E={<A,B>,<B,C>,<C,B>}
• <X,Y>: X = head, Y = tail
• <X,Y> ≠ <Y,X> unless X=Y
– Direction is not important (undirected graph)
• E = { (A,B), (B,C), (C,B) }
• (X,Y): X = head, Y = tail
• (X,Y) = (Y,X) (unless we have a multigraph)
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Some basic examples
A A
B D B D
C C
Undirected graph: Directed graph:
G=<V,E> G=<V,E>
V={A,B,C,D} V={A,B,C,D}
E={(A,B),(B,C),(C,D),(A,D),(B,D)} E={<A,B>,<C,B>,<A,D>,<D,B>,<D,C>}
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Königsberg is a multigraph
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Mixed graph = directed + undirected
A E
B D F H
C G
G=<V,E>
V={A,B,C,D,E,F,G,H}
E={(A,B),(B,C),(C,D),(A,D),(B,D), <E,F>,<G,F>,<E,H>,<H,F>,<H,G>,<A,E>,(C,G)}
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Information on the edges
Edges may be labeled
•
Edges may contain weight (weighted graph)
•
A
3
1
2
B D
1.1
3.6
C
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Degree of a vertex
Undirected graphs
•
– The degree of a node is the number of edges involving that node
Directed graphs
•
– Indegree: number of edges that arrive (head) to a node
– Outdegree: number of edges departing (tail) from a node
2 0,2
A A
3 3,0 1,2
3
B D B D
3 1,1
C C
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Other concepts
• A loop: an edge where the to vertex are the same
• A simple graph: a graph that does not contain loops or
parallel edges
• A multigraph: a graph that contains parallel edges
• To edges are adjacent if they connect to the same
vertex
• To vertices are adjacent if they are connected by an
edge
• A complete graph: a graph that every pair of vertices
are adjacent
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A path in the graph
A path through a graph is a traversal of consecutive vertices along
•
a sequence of edges
– The vertex at the end of one edge in the sequence must also be the
vertex at the beginning of the next edge in the sequence
– The vertices that begin and end the path are termed the initial vertex
and terminal vertex, respectively
– With the exception of these initial and terminal vertices, each vertex
within the path has two neighboring vertices that must also be
adjacent to the vertex
– The length of the path is the number of edges that are traversed along
the path
– An elementary path is a path where no vertex appears more than once
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Some path examples
A A
B D B D
C C
P= {A,D,B} is a path
P= {A,B,C} P= {A,D,C,B} is a path
P= {A,D,A} P= {A,B,C} is not a path
P= {D,B,A} P= {A,D,A} is not a path
P= {D,B,A} is not a path
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Cycle
• A cycle is a path in which the initial vertex of the path
is also the terminal vertex of the path
• Removing all the cycles in a path we get an elementary
path
A
P= {A,B,A}
P= {A,B,D,A}
P= {A,B,C,D,A}
P= {{A,B,C,D,A},B,C}
B D
P= {D,C,B,{A,B,C,D,A},B,C}
C
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Connectedness
Undirected graphs
•
– An undirected graph is considered to be connected if a path exists
between all pairs of vertices thus making each of the vertices in a pair
reachable from the other
– An unconnected graph may be subdivided into what are termed
connected subgraphs or connected components of the graph.
Directed graphs
•
– A weakly connected graph is where the direction of the graph is
ignored and the connectedness is defined as if the graph was
undirected
– A unilaterally connected graph is defined as a graph for which at least
one vertex of any pair of vertices is reachable from the other
– A strongly connected graph is one in which for all pairs of vertices,
both vertices are reachable from the other
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Implementation
Graphs can be implemented in several way
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1. Set of nodes, and list of edges
2. Adjacency matrix (triangular matrix for undirected graphs)
A B C D E F
A
A D
C
B
B
C
D F
E
E
F
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What will we do in the next lecture?
• Human communication
• Text as a form of unstructured data
• Basic notions of text mining
• Common operations in text mining
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Something to think about
Further reading
•
http://ocw.mit.edu/NR/rdonlyres/Electrical-Engineering-and-Computer-Science/6-042JSpring-2005/FF95BA8F-6457-4592-B473-349A5C1CA277/0/l8_graphs1.pdf
http://www.math.lsa.umich.edu/mmss/coursesONLINE/graph/
http://www.cs.usask.ca/resources/tutorials/csconcepts/1999_8/
Homework:
•
1. What is the solution to Mr. Euler’s question, and why?
2. Review last lesson’s RDF examples and paint down the underlying graph and write
the adjacency matrix. What else can you tell about this graph?
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