CSE 222 Systems Programming Graph Theory Basics Dr

CSE 222 Systems Programming Graph Theory Basics Dr. Jim Holten

Introduction • Basic Definitions • Concepts and Algorithms • Applications 01/21/09 CSE 222 2

Basic Definitions • Nodes aka Vertices • Links aka Edges aka Arcs 01/21/09 CSE 222 3

Basic Definitions • Subgraphs • Undirected vs Directed Graphs • Paths, Distances, and Graph Diameter • Connected Graphs • Components of a Graph • Fully connected graph or subgraph 01/21/09 CSE 222 4

Undirected Graph 01/21/09 CSE 222 5

Directed Graph 01/21/09 CSE 222 6

Basic Definitions • Node Attributes aka Labels, Indices, Weights, Categories, etc. • Link Attributes aka Labels, Indices, Weights, Categories, etc. • A Semantic Graph is a graph partitioned based on node or link attribute values 01/21/09 CSE 222 7

Semantic Graph Authors 01/21/09 Documents Concepts CSE 222 8

More Definitions • A Hypernode replaces a collection of nodes and their included links. • A Hypergraph is any graph that includes one or more hypernodes. 01/21/09 CSE 222 9

Example of Hypergraph Authors 01/21/09 CSE 222 Documents Concepts 10

Hypergraph of Social Network 01/21/09 CSE 222 11

Some Concepts • A cluster is a subgraph made up of set of highly interconnected nodes. • A critical node is a node on a unique path between two nodes or subgraphs of interest. • A critical link is a link on a unique path between two nodes or subgraphs of interest. 01/21/09 CSE 222 12

Related Algorithms • Finding clusters is defined many ways – Fully connected subgraph or just better than some threshold on connectedness? – Is a node only connected to a node in the cluster to be included in the cluster? • Finding critical paths, nodes, or links 01/21/09 CSE 222 13

Related Concepts • Projections of nodes or subgraphs are mappings via directed links. • The domain subgraph includes the projection's "from" nodes. • The range subgraph includes the projection's "to" nodes. • Basis Node Set is a minimal set of nodes that have a path to every other node 01/21/09 CSE 222 14

Projection Mapping 01/21/09 CSE 222 15

Subset Projections 01/21/09 CSE 222 16

Related Concepts • There are multiple ways of looking at a single collection of nodes and links. • Are multiple graphs just a single multicomponent graph? • Interdependencies among multiple graphs can be useful. 01/21/09 CSE 222 17

Interdependencies 01/21/09 CSE 222 18

Applications • Software Organization • Activity Sequence Organization • Interactions between processes 01/21/09 CSE 222 19

Software Organization • Flow charts • Data interdependencies • Interactions between processes 01/21/09 CSE 222 20

Data Organization Chart 01/21/09 CSE 222 21

Processes Organization Chart 01/21/09 CSE 222 22

Activity Sequence Organization • Gantt and Pert charts • Project resource planning charts • Storyboarding 01/21/09 CSE 222 23

Storyboard of the Software Development Process 01/21/09 CSE 222 24

Network Modeling Applications • Communications (Internet, etc. ) • Energy distribution (electric power, natural gas, etc. ) • Transportation (highways, railways, waterways, etc) • Social associations • Financial and product flows 01/21/09 CSE 222 25

Infrastructure Networks 01/21/09 CSE 222 26

Critical Infrastructure Interdependencies 01/21/09 CSE 222 27

Physics Models • Mesh Models – Space segmented into volumes – Volumes surrounded by faces and edges – Edges meet at vertices • Model element type dependencies form a graph of interdependent element sets 01/21/09 CSE 222 28

Mesh Models 01/21/09 CSE 222 29

Element Type Dependencies 01/21/09 CSE 222 30

Summary Graphs are a significant factor in the present and the future of computer science! 01/21/09 CSE 222 31