1. Computational Methods for Management and Economics Carla Gomes Module 9a Network Models Terminology (Slides adapted from J.Orlin’s)

2. Network Optimization Problems Many optimization problems can be represented by a graphical network representation. Examples: –Distribution problems –Routing problems –Maximum flow problems –Designing computer / phone / road networks –Equipment replacement

3. Networks are Everywhere Physical Networks –Road Networks –Railway Networks –Airline traffic Networks –Electrical networks, e.g., the power grid Abstract networks –organizational charts –precedence relationships in projects Others – e.g. relationships networks (6 degrees of Kevin Bacon )

4. Networks Network representation also widely used for problems in such diverse areas as: –production –distribution, –project planning –facility location –resource management –financial planning. General framework used to study scientific, social, and economic systems. Many network models are special cases of linear programming (e.g., transportation and assignment problems). (see Network Flows by Ahuja, Magnanti, and Orlin) for lots and lots of models and applications

5. Notation and Terminology Note: Network terminology is not (and never will be) standardized. The same concept may be denoted in many different ways. Called: NETWORK directed graph digraph graph 2 1 4 3 2 1 4 3 Node set N = {1, 2, 3, 4} Arc Set Network G = (N, A) {(1,2), (1,3), (3,2), (3,4), (2,4)} A={1-2,1-3,3-2,3-4,2-4} Graph G = (V,E) Edge set: Also Seen Vertex set V = {1,2,3,4} Network: a set of points (called nodes or vertices) and a set of lines (called arcs, edges, links or branches).

6. Applications Physical analog of nodes Physical analog of arcs Flow Communication systems phone exchanges, computers, transmission facilities, satellites Cables, fiber optic links, microwave relay links Voice messages, Data, Video transmissions Hydraulic systems Pumping stations Reservoirs, Lakes Pipelines Water, Gas, Oil, Hydraulic fluids Integrated computer circuits Gates, registers, processors WiresElectrical current Mechanical systemsJoints Rods, Beams, Springs Heat, Energy Transportation systems Intersections, Airports, Rail yards Highways, Airline routes Railbeds Passengers, freight, vehicles, operators Applications of Network Optimization

7. Examples Flows in networks NodesArcsFlow Intersectionsroadsvehicles Airportsair lanesaircraft Switching pointsWires, channelsmessages Pumping stationspipesfluids Hollywood ActorsCo-staring in moviesJoint movies ScientistsResearch collaborationJoint papers

8. Terminology about flows Maximum amount of flow (possibly infinity) that can be carried on a directed arc  arc capacity Supply (or source) node  nodes that are net generators of flow; the flow out of the node exceeds the flow into the node. Demand (or sink) node  nodes that are net absorbers of flow; the flow into the node exceeds the flow out of the node. Transshipment (or intermediate) node  satisfies the conservation of flow; the flow in equals the flow out.

9. Directed and Undirected Networks 2 34 1 a b c d e An Undirected Graph 2 34 1 a b c d e A Directed Graph  The field of Network Optimization concerns optimization problems on networks  Networks are used to transport commodities physical goods (products, liquids) communication electricity, etc.

10. The Adjacency Matrix (for directed graphs) 2 34 1 a b c d e A Directed Graph Have a row for each node 1 2 3 4 12341234 Have a column for each node Put a 1 in row i- column j if (i,j) is an arc What would happen if (4,2) became (2,4)?

11. The Adjacency Matrix (for undirected graphs) Have a row for each node 1 2 3 4 12341234 Have a column for each node Put a 1 in row i- column j if (i,j) is an arc 2 34 1 a b c d e An Undirected Graph The degree of a node is the number of incident arcs degree 23232323

12. The Node-Arc Incidence Matrix (for directed graphs) 2 34 1 a b c d e A Directed Graph Have a row for each node a b c d e 12341234 Have a column for each arc Put a 1 in row i- column j if arc j starts at node i. Put a -1 in row i- column j if arc j ends at node i. What would happen if arc (4,2) became arc (2,4)?

13. On Incidence Matrices If the constraint matrix for a linear program is a node- arc incidence matrix (at most one 1 and at most one – 1 per column), then the linear program solves in integer optima. Note: we saw this result earlier with the transportation problem

14. Representation of arc lists (for directed graphs) 2 34 1 a b c d e A Directed Graph Create a list arcs for each node 1: (1,2), (1,4) 2: (2,3) 3:  4: (4,2), (4,3) There are lots of very similar variants of this type of representation

15. On network representations Each representation has its advantages –Major purpose of a representation efficiency in algorithms  very important ease of use Node arc incidence matrix shows up in Linear Programs

16. Directed Path. Example: 1, 2, 5, 3, 4 (or 1, a, 2, c, 5, d, 3, e, 4) No node is repeated. Directions are important. Cycle (or circuit or loop) 1, 2, 3, 1. (or 1, a, 2, b, 3, e) A path with 2 or more nodes, except that the first node is the last node. Directions are ignored. Path: Example: 5, 2, 3, 4. (or 5, c, 2, b, 3, e, 4) No node is repeated. Directions are ignored. Directed Cycle: (1, 2, 3, 4, 1) or 1, a, 2, b, 3, c, 4, d, 1 No node is repeated. Directions are important. 2 3 4 ab c 1 5 d e 2 3 4 a b c d 1 e 2 3 4 ab c 1 5 d e 2 3 4 a b c d 1 e

17. Walks 2 34 1 a b c d e 5 2 34 1 a b c d e 5 Walks are paths that can repeat nodes and arcs Example of a directed walk: 1-2-3-5-4-2-3-5 A walk is closed if its first and last nodes are the same. A closed walk is a cycle except that it can repeat nodes and arcs.

18. More terminology An undirected network is connected if every node can be reached from every other node by a path 2 1 4 3 5 2 1 4 3 5 A directed network is connected if it’s undirected version is connected. This directed graph is connected, even though there is no directed path between 2 and 5.

19. More Definitions 2 1 4 3 A network is connected if every node can be reached from every other node by following a sequence of arcs in which direction is ignored. 5 A spanning tree is a connected subset of a network including all nodes, but containing no cycles. 2 1 4 3 5 2 1 4 3 5 2 1 4 3 5

20. More on Trees An out-tree is a spanning tree in which every node has exactly one incoming arc except for the root. Theorem. In an out-tree, there is a directed path from the root to all other nodes. (All paths come out of the root). 2 1 4 3 5 2 1 4 3 5 789 6 1011 1213