1. Introduction to Graph & Network Theory Thinking About Networks: From Metabolism to the Genome to Social Conflict Summer Workshop for Teachers June 27 th -July 9 th, 2010

2. Graph Theory ● Graphs are collections of nodes (points) and edges (links), where each edge connects exactly two nodes. ● The focus is on fundamental properties: degree of the nodes (i.e. number of edges connected to a node), connectedness of the graph, subsets of nodes and edges, and distance between nodes as a function of the number of “hops” between them. ● Concepts such as length of edges, capacity limits of nodes or edges, etc. are not generally addressed in graph theory and applications.

3. The Bridges of Königsberg ● Is it possible to take a tour of the city, using only the bridges to get across the river (no swimming or boats), and crossing each bridge exactly once? ● Euler's insight: treat the map as a graph, where each bridge is an edge, and the separate land areas are nodes. ● Euler's solution (which proved that such a tour is not possible in a graph with more than two nodes of odd degree) is considered to be the first theorem of graph theory.

4. The Four-Color Problem ● What's the minimum number of colors required to color a plane divided into regions, so that no two adjacent regions have the same color? ● This may be formulated as a graph theory problem, where each region is a node, and an edge connects each pair of adjacent regions. ● The conjecture that no more than four colors are required was proven in 1976, with the extensive use of computers.

5. Network Theory ● A specialization of graph theory, in which additional properties of graphs are incorporated – e.g. edge weight/cost/distance, edge capacity. ● For example, the numbers next to each edge in the graph to the right could be the cost of traversing that edge. ● The applications of network theory are very broad – including data transmission networks, power grids, transportation networks, and social networks. ● Graph and network are often used interchangeably. 3 5 10 2 1 13 4

6. Shortest Path Problem ● Find the path that connects nodes A and E that has the minimum cost (or length). ● Examples of this problem are found in telecommunications and data networking, as well as travel assist tools (e.g. Google Maps), facilities layout, “six degrees of separation”, etc. ● Best-known solution method is Dijkstra's algorithm, invented by Edsgar Dijkstra. ● Open Shortest Path First (OSPF) is a variation of Dijkstra's algorithm used for internet routing. 3 5 10 2 1 13 A B C D E F 4

7. Other Network Problems ● Minimum Spanning Tree – Find the shortest/least-cost “tree” (component without cycles) that connects all the nodes in an undirected network. ● Traveling Salesman Problem – Find the shortest route that visits every node in the network exactly once, and then returns to the starting node. ● Max-Flow Problem – Find the maximum flow through a network where each edge has a given capacity. ● Critical Path Method – Find the sequence of tasks in a project which have the least “slack” - i.e. those for which any delays in the task completion will delay the entire project.

8. Transportation Theory ● Focused on the optimal transportation and allocation of resources in a network. ● One of the foundational areas of study and application in the multi- disciplinary field of Operations Research. ● Many early applications dealt with the allocation and movement of military personnel and equipment. ● As in network theory, the network itself need not be a physical construct; it can be a collection of entities, and the logical relationships between them. For example, one special case of the transportation problem (see next page) is the assignment problem, where the edges are feasible pairings of resources and activities.

9. The Transportation Problem ● Each node on a network has a stated supply or demand (which may be expressed as a surplus or a shortfall) of a given resource. ● Each edge has an associated cost, incurred when moving one unit of the resource via that edge. ● Often formulated with a very simple network topology, where shortest paths have already been determined, and non- terminal nodes are not shown. ● The problem is to minimize the total cost of moving the resource over the network, to satisfy the demand. A (+10) B (+12) C (+8) D (+15) I (-12) II (-16) III (-17) 9 8 10 11 13 7 9 19 16 13

10. Example: Food Web Caribbean Reef Trophic Web, courtesy of foodwebs.org.

11. Example: AT&T Communications Network

12. Example: Les Miserables Character Relationships