How is this going to make us 100K

Applications of Graph Theory

Konigsberg BridgeKonigsberg Bridge

Konigsberg is a city which was the capital of East Prussia but now is known as Kaliningrad in Russia. There are seven bridges that join the different parts of the city.

People wondered if they could travel the city crossing all bridges, but never crossing any bridge twice.

Leonhard Euler (1707-1783) In 1735, Euler realized that the size of the land In 1735, Euler realized that the size of the land

masses and the widths of the bridges were masses and the widths of the bridges were irrelevant. Land masses could be expressed as irrelevant. Land masses could be expressed as

points, and the bridges as lines or arcs.points, and the bridges as lines or arcs.

This is considered to be the start of This is considered to be the start of

Graph TheoryGraph Theory

Graphs Represent RelationshipsGraphs Represent Relationships Graphs may be used in many ways . . . . . .

This graph represents friendships.

Assume a set of people:Assume a set of people: Each person is a node.Each person is a node.

If two people know each If two people know each other, then there is a link, other, then there is a link, or an edge between them.or an edge between them.

Mathematicians study the Mathematicians study the relationship between relationship between nodes and the number of nodes and the number of edges to find patterns.edges to find patterns.

NodeNode – a point of interest also known as a – a point of interest also known as a vertex.vertex.Edge Edge – a line connecting two nodes– a line connecting two nodesDegreeDegree – The number of edges attached – The number of edges attached to a node, also know as a valenceto a node, also know as a valencePathPath – A connected sequence of edges in – A connected sequence of edges in a graph a graph LengthLength - is the number of edges - is the number of edges transversed.transversed.DiameterDiameter – The shortest length between – The shortest length between the two most distant nodes.the two most distant nodes.

Simple Graph Complete Graph A complete graph is a graph in which each pair of vertices is connected by an edge.

Multi-Graph A simple graph is nothing more than a set of vertices and sets of edges between them.

A multi-graph has at least one pair of vertices with multiple edges between them.

Circuit A circuit is a graph that has it’s starting point and end point on the same node.

We need 5 volunteers

Construct a simple graph where each one of you is a node.

Hint: three of you should be of degree 2 and two of you will be of degree one.

How many edges have you constructed ? ________

What is the relationship between the number of edges and the number of nodes ?

E = __________Where . . .

E represents the # of edges. N represents the # of nodes.

The relationship between nodes and edges for a complete graph is . . .E = n – n2

2

To illustrate this, our volunteers will now form a complete graph.

Remember: a complete graph is a graph where each node is connected to each of the other nodes.

So how many edges are in a

complete graph of 5 nodes ? _________

Graph Theory studies structure The graph’s

diameter.The degree of

each nodeThe paths in the

graph.The graph’s

connectivity.

Graph Theory studies structure The graph’s

diameter.The degree of

each nodeThe paths in the

graph.The graph’s

connectivity.

Large scale structure

Applied mathematicians study large sets of data, such as telephone calls and web sites, to deduce their structure. Each telephone or web page is a node. The edges are formed by the nodes that call or link to each other.

Mathematicians find the degree of each node, the degree can be tabulated to construct a model.

For example, Google looks at the incoming links for each web page and ranks the pages based on those links.

Emergent StructureEmergent Structure• Modeling the expected number of links pointing to a web page

– Find the degree for all web pages– Count the number of web pages with a certain degree– Calculate the regression equation that models the

relationship.• This equation has the form

– Where k represent the degree of the node– This function falls slower than an exponential, allowing for

nodes with many links.– The exponent value has been found to be around 2.4.

k

Current Research

• Dr. Butenko studies relations between stocks.• Each stock is a node.• For each pair of stocks, a correlation coefficient is

calculated.• The coefficient measures how closely each stock

fluctuates.• The nodes are connected if the coefficient is above a

certain value.• Count the number of links for each stock

• This number indicates the quantity of stocks with similar fluctuations.

• This graph is not connected• Has subsets of stocks that seem to be related. • These subsets are called cliques.

Results

• He creates a model that predicts global behavior of the stock market

• Has shown that the distribution of the fluctuations of the nodes is nearly a normal distribution, with a nearly central mean, indicating the stocks move substantially independently.

• Has shown that the number of edges for each stock follows a power law over long periods, indicating that the global structure of the market is stable over time.

• The edge density, or the number of edges divided by the number of possible edges, is increasing over time.

– This indicates that more stocks appear to be changing in the same way.

– This indicates that the randomness of the global market is decreasing over time, which means the stocks move less independently.

Why are these problems interesting ?

• Very hard to solve exactly– No known algorithm

to find a general exact solution quickly

– Sometimes no known method to check guessed solution.

• Optimization– Create heuristics to

systematically find best guess solution

– Create tests to estimate confidence in solutions

• Examples– Traveling Salesman

– Prize Collection

Our Project

• Investigate the flight schedule of airlines• Each city served by the airline is a node• Each flight to another city is an edge• Tabulate the degree of each city

• Models• Create a model relating cities to the number of

edges.• Predict the diameter of the airline graph• Calculate edge density

• Optimization• Tabulate other possible edges between cities• Calculate parameters for each model and optimize.

What about the bridge ! !