HSC Mathematics Standard: Networks

Overview of Networks for standard mathematics.

Table of Contents

What is a Network?

  • A network is a visualisation of objects and how they are connected
  • Networks have 3 key features:
    • The Network is the entire visualisation
    • The vertices (singular: vertex) are the objects in the network, represented as points/dots
    • The Edges are the lines connecting the vertices

What properties can network features have?

  • Vertices have a property known as its “degree”, which is the number of other unique vertices it is connected to
  • Edges have two properties: weight (how strong the connection is) and direction (if a connection is one-way or two-way)
    • Edges with direction have an arrow pointing in the direction they travel
    • Loops are a special type of edge:
      • A loop starts and ends at the same vertex
      • It counts as 1 edge, but adds two to the degree of the vertex

How are Vertices represented?

  • Vertices can be represented by listing them all in braces:

$\color{orange}{V= \{ A,B,C,D,E,F \} }$

How are Edges represented?

  • Edges are represented by writing the start and end points in brackets
  • For example, an edge between $A$ and $B$ would be represented as $(A,B)$
  • A loop is represented the same as an edge, but with the vertex in both places
    • For example, a loop around $X$ would be represented as $(X,X)$
  • In a Network diagram, the weight of an edge is written next to it as a label

Diagram of a weighted edge network
The vertices are A, B, C, D, E, and F, and the weights of the edges are the numbers in red.

How are networks navigated?

  • A walk is a connect set of edges from one vertex to another
  • An example of a walk is a GPS guided drive, in which each street is an edge, and your start and end point are the vertices
  • Usually, your aim is to find the shortest possible walk between any two vertices, or to find a path which includes every edge/vertex

What types of walk are there?

  • A trail is a walk in which no edge is traversed more than once (i.e. no repeated edges)
  • A path is a walk with no repeated vertices or edges
  • A circuit is a walk with no repeated edges which starts at the same vertex it ends on
  • A cycle is a walk with no repeated vertices which starts at the same vertex it ends on
  • This is “neatly” summarised by this flowchart:
graph TD
    B[Are Repeated Vertices Allowed?]
    B -->| Yes | C[Are Repeated Edges Allowed?]
    B -->| No | D[Are Repeated Edges Allowed?]
    C -->|Yes| E[Walk]
    C -->|No| F[Trail]
    D -->|Yes| G[Circuit]
    D -->|No| H[Can the First and Last vertex be the same?]
    H -->|Yes| I[Cycle]
    H -->|No| J[Path]

How are Networks Drawn?

  1. Mark out every vertex IN PENCIL
  2. Connect any connected vertices with edges IN PENCIL
  3. Label each vertex
  4. Label the weight and direction of each edge

What is an isomorphic network?

  • An isomorphic network is a network in which every edge has the same weight, and every vertex has the same degree

What are Eulerian Trails?

  • Eulerian trails are trails which use every edge of a network exactly once
  • By definition, they must start and end at different vertices
  • Eulerian trails will always exist if there are exactly two vertices with an odd degree
    • These two vertices must be the start and end of the trail

What are Eulerian Circuits?

  • Eulerian circuits are a subset of eulerian trails
  • They are identical to Eulerian trails in every way, except that they start and end at the same point (because they are circuits)
  • Eulerian circuits will always exist if every vertex is of even degree

What is a Tree?

  • A tree is a connected graph which has no cycles or loops
  • In a tree, the number of edges is always 1 less than the number of vertices

What is a Spanning Tree?

  • Within a network, there will always be a tree which can be created
  • A tree which connects all of the vertices in a network is known as a spanning tree
  • Minimum spanning trees are spanning trees which have the lowest possible total weight
    • The sum of the weights of each edge should be as low as possible, while still connecting every vertex

What is Prim’s Algorithm?

  • Prim’s algorithm is a set of rules to determine a minimum spanning tree of a graph:
  1. Choose any vertex (vertex 1)
  2. Follow the edge with the lowest weight to the next vertex (vertex 2)
  3. Follow the edge of vertex 2 with the lowest weight, excluding the edge connecting to vertex 1
  4. Repeat step 3, ignoring edges which will result in doubling up on a vertex, until all vertices are connected

What is a Connector Problem?

  • Connector problems use minimum spanning trees to find the lowest cost method to link multiple objects to a network
  • They usually take the form of “Which path will be the lowest cost, while reaching every vertex in the network?”

How can the Shortest Path Between Vertices be Calculated?

  • The shortest path is the path between two vertices with the lowest total edge weight
  • Following Prim’s algorithm between the two vertices is the most efficient way to do this
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