Mathematical Modelling Lecture 8 Networks

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1 Lecture 8 Networks

2 Overview of Course Model construction dimensional analysis Experimental input fitting Finding a best answer optimisation Tools for constructing and manipulating models networks, differential equations, integration Tools for constructing and simulating models randomness Real world difficulties chaos and fractals The material in these two lectures is not in the course textbook.

3 What is a network? A network is any system of interconnected locations Locations usually less important than the connections Not restricted to computer networks! Using networks we can answer questions like What is the shortest route between two points? What is the cheapest route between two points? Are there any bottlenecks?

4 What s in a name? Locations are called nodes or vertices Links are called connections or edges

5 Classified according to the nature of the connections: May contain cycles May contain directed (one-way) or undirected edges May be complete (every node directly connected to every other) or incomplete

6 : Trees A tree is a special type of network. Unique path from any one node to any other Not cyclic Incomplete May have directed and/or undirected edges

7 : Trees A tree is a special type of network. Unique path from any one node to any other Not cyclic Incomplete May have directed and/or undirected edges

8 : Bipartite A bipartite network has two distinct groups of nodes Paths only exist between nodes in different groups No paths between nodes in same group May have directed and/or undirected edges

9 : Bipartite A bipartite network has two distinct groups of nodes Paths only exist between nodes in different groups No paths between nodes in same group May have directed and/or undirected edges

10 : Complete A complete network has: Paths from each node to all other nodes Lots of cycles

11 We are interested in the cheapest path from a particular node, called the root node, to another node.

12 Our method is to build up a linked list. We need to define three things: P j is the previous node in the cheapest path found so far from the root node to node j K j is the cost of the cheapest path found so far from the root node to node j C ij is the cost of the connection from neighbouring node i to node j

13 We start by initialising P and K for all the nodes: [P root, K root ] = 0 [P i, K i ] = [0, ] for non-root nodes Label the root node with a slash

14 1 From the slashed node i, look at each non-slashed neighbour j 2 Calculate K i + C ij 3 If this is less than the current K j : K j = K i + C ij and P j = i Sketch the new [P j, K j ] by the node 4 Find the neighbour with the lowest K j 5 Label this node with a slash and repeat from step 1 until all nodes are slashed

15 Example 1 Introduction Let s look at the cheapest path from node 1 to nodes 9 and 10.

16 Example 1 Introduction

17 Example 1 Introduction

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24 Example 1 Introduction

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27 Example 1 Introduction

28 Example 1 Introduction

29 At the end, K target is the cost of the cheapest path to the target node We can find the cheapest path by working back along the links of previous nodes P j : from node j move to node P j and repeat until we reach the root node.

30 Example 1 Introduction

31 Example 1 Introduction

32 Example 2 Introduction

33 Example 3 - Transportation network Using the network from the first example, but 1 3 and 3 4 are directional (one-way). There is a further restriction on 3 4, that the flow must not exceed 50 units a day.

34 Example 3 - Transportation network Using the cheapest routes: Flow due to path Edge A B C D Total flow

35 Example 3 - Transportation network Route Cost Flow x x x y y y z z w w 2

36 Example 3 - Transportation network Total cost K : K = 6x 1 +8x 2 +7x 3 +8y 1 +10y 2 +9y 3 +10z 1 +12z 2 +11w 1 +13w 2 Subject to: x 1 + x 2 + x 3 = 35 y 1 + y 2 + y 3 = 20 z 1 + z 2 = 15 w 1 + w 2 = 30 x 1 + y 1 + w 1 + z 1 50 i.e. minimise K subject to constraints a linear programming problem!

37 Summary Introduction Networks consist of nodes joined by connections Examples include trees, bipartite networks, complete networks May have cycles, be directed/undirected, complete/incomplete Dijkstra s algorithm computes the cheapest path from a particular node to all other nodes

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