CSE 222 Systems Programming Graph Theory Basics Dr. Jim Holten.

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Presentation transcript:

CSE 222 Systems Programming Graph Theory Basics Dr. Jim Holten

CSE /21/09 Introduction Basic Definitions Concepts and Algorithms Applications

CSE /21/09 Basic Definitions Nodes aka Vertices Links aka Edges aka Arcs

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

CSE /21/09 Undirected Graph

CSE /21/09 Directed Graph

CSE /21/09 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

CSE /21/09 Semantic Graph Authors Documents Concepts

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

CSE /21/09 Example of Hypergraph Authors Documents Concepts

CSE /21/09 Hypergraph of Social Network

CSE /21/09 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.

CSE /21/09 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

CSE /21/09 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

CSE /21/09 Projection Mapping

CSE /21/09 Subset Projections

CSE /21/09 Related Concepts There are multiple ways of looking at a single collection of nodes and links. Are multiple graphs just a single multi- component graph? Interdependencies among multiple graphs can be useful.

CSE /21/09 Interdependencies

CSE /21/09 Applications Software Organization Activity Sequence Organization Interactions between processes

CSE /21/09 Software Organization Flow charts Data interdependencies Interactions between processes

CSE /21/09 Data Organization Chart

CSE /21/09 Processes Organization Chart

CSE /21/09 Activity Sequence Organization Gantt and Pert charts Project resource planning charts Storyboarding

CSE /21/09 Storyboard of the Software Development Process

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

CSE /21/09 Infrastructure Networks

CSE /21/09 Critical Infrastructure Interdependencies

CSE /21/09 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

CSE /21/09 Mesh Models

CSE /21/09 Element Type Dependencies

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