Programming for Geographical Information Analysis: Advanced Skills

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Programming for Geographical Information
AnalysisAdvanced Skills

• Online mini-lecture Introduction to Networks
• Dr Andy Evans

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This lecture

• Introduction.
• Graph theory.
• Types of Networks.
• Path analysis.
• Flow analysis.

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Networks?

• Vehicle / Distribution
• Motorways, rail, electricity grid, water, nerves,
  circuits.
• Socio-economic
• Trade, politics, friendship.
• Spatial-temporal
• Timetables, zonal maps, chemical structures.
• Almost any set of complex relationships between
  variables can be represented as a network in
  variable space.

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Graph Theory

• The study of networks as graphs.
• How do we efficiently move around networks?
• How do networks break down?
• Can we add links to improve networks?
• How do things spread around networks?

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What is a graph?

• Combination of (potentially disconnected)
  Vertices and Edges.
• Can be Directional (sometimes called digraphs)

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What is a graph?

• Vertices are the points on a network.
• Edges are the relationships.
• The degree of a vertex is the number of edges it
  has connected to it.

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Representing networks

• Note that it is only the connections that matter,
  not how theyre drawn.
• E.g. contiguousness in maps.

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Types of graph
Complete each vertex is connected to all the
others. Regular each vertex has the same
number of edges. Simple no loops or multiple
paths.
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Walks, Trails and Paths

• A Walk any sequence of adjacent edges.
• A Trail any sequence of unrepeated adjacent
  edges.
• A Path any sequence of adjacent edges that dont
  visit the same vertex more than once.

A trail, but not a path.
A path that ends where it begins is called a
Cycle.
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Eulerian networks

• A graph is Eulerian if there is a trail
  containing every edge in a graph.
• Euler solved the Königsberg Bridges problem can
  you cross all the bridges and return to your
  starting point, crossing each once and once only

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Eulerian networks

• A graph will contain a cycle if each vertex has a
  degree of gt2.
• A connected graph is Eulerian if, and only if,
  the degree of each vertex is even.
• A semi-Eulerian network is the same but doesnt
  end up at its start.
• A connected graph is semi-Eulerian when only two
  of its vertices are odd.
• Uses
• Designing one-way systems.
• Designing diversions / flow alterations.

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Fleurys Algorithm

• How to construct a Eulerian trail in a Eulerian
  graph.
• Start with any vertex, and travel through the
  edges, following these rules
• As you go through each edge, mark it as taboo.
• If you create isolated vertices, mark them as
  taboo.
• Only use a bridge as a last resort.
• A bridge is an edge connecting two otherwise
  disconnected graphs

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Hamiltonian networks

• A graph is Hamiltonian if theres a path taking
  in every vertex once only, that ends where it
  began.
• A graph is semi-Hamiltonian if theres a trail
  taking in every vertex once only that doesnt end
  where it began.
• Diracs Theorem
• A simple graph with n vertices (more than two) is
  Hamiltonian if the degree of each vertex is
  greater than n/2.
• Note that there may be more Hamiltonian
  situations.

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Weighted graphs

• So far weve only been interested in connections,
  but we can add strength, capacity, or length of
  connections.
• This gives much more realistic networks, and
  allows us to study real path optimisation
  problems.

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Path optimisation problems

• Three major problems
• Shortest Path Problem
• What is the shortest path between two points?
• Chinese Postman Problem
• What is the shortest way around a network back to
  the start taking in each edge (imagining the
  edges to be roads of houses)?
• Travelling Salesman Problem
• What is the shortest way around a network back to
  the start taking in each vertex (imagining each
  to be a city)?

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Shortest Path Problem

• Find the shortest path across a weighted graph.

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Shortest Path Problem algorithm

• Move from A to B, treating all the vertices at
  each distance from the start in turn.
• Attribute the value of the shortest path to each
  vertex to that vertex, then reassess each of its
  neighbours to see if the path is now shorter to
  them.
• Continue until at the destination and follow back
  shortest path.

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Chinese Postman Problem

• Suggested by Mei-Ku Kwan
• Find the shortest walk taking in all the weighted
  edges and coming back to the start.
• If the graph is Eulerian we can use Fleurys
  method.
• If not we find a semi-Eulerian trail using
  Fleurys to take in all the edges.
• We then use the Shortest Path algorithm to find
  our way back to the start.

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Travelling Salesman Problem

• Find the shortest weighted path needed to visit a
  number of cities once (i.e. the shortest
  Hamiltonian path).
• No efficient accurate algorithm.
• Could calculate all the possible Hamiltonians and
  pick the shortest.
• However, for 20 cities on a Hamiltonian graph,
  there are about 6 x 1016 such possible paths.
• One solution is to use a Minimum Connector.

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The Minimum Connector Problem

• Work out the smallest sub-network to connect a
  set of vertices on a weighted network a
  spanning tree.
• Choose the smallest weighted edge.
• Continue picking the next smallest connected
  edges but not using edges that form cycles with
  the ones already picked.

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The Travelling Salesman revisited

• If we remove a vertex from a Hamiltonian path, we
  end up with a spanning tree.
• To get an good approximation for the Travelling
  Salesman path, we
• Remove a vertex from the graph, and its
  associated edges.
• Form the minimum connector for the rest of the
  graph.
• Add in the vertex and the two of its edges with
  the minimum weights.
• This will give us a semi-Hamiltonian which the
  solution cant be better than, and which the
  solution will be close to.
• Alternatively use AI to search the solution space.

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Network flows strong connection

• Strongly connected graphs are those digraphs
  where every vertex can be reach from every other.
• A graph can be made into a strongly connected one
  if each edge is in a cycle.
• I.e. you cant make a purely one-way system in a
  town split by a single bridge.

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Maximum Flow Minimum Cut

• Given each part of a network has a capacity, what
  is the maximum capacity of the network?
• It equals the minimum capacity cut of the
  graph.
• A cut is a set of edges which a flow must pass
  through one of.

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Other things Graph Theory can help with

• Finding the distribution of flow in
  interconnected networks with resistances and
  inputs/outputs.
• Calculating whether graph lines, e.g. railways,
  need to cross.
• Calculating the number of rearrangements possible
  between vertices on a graph.
• The maximum number of different paths between two
  places.
• Timetabling (using rooms and people as
  components).
• How many colours it takes to colour a zonal map.
• How long it takes to do a number of parallel
  tasks.
• Magic squares.
• Arranged marriages.

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Summary

• Graph theory encapsulates the mathematics of
  networks.
• Eulerian graphs are those where you can visit all
  the edges one and end up at the start.
• Hamiltonian graphs are those where you can visit
  all the vertices once and end up at the start.
• The rules for these graphs help us design network
  systems.

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Summary

• By giving weights to edges we can model real
  dynamic systems.
• We can find out optimal paths.
• We can work out transport loads.