ACARA v9 CONTENT DESCRIPTION “interpret networks and network diagrams used to represent relationships in practical situations and describe connectedness”
Builds on: Deductive Proof: Reasoning from Theorems, Not Measuring (AC9M10SP01). Proof trained the habit of reasoning carefully about figures. Networks turn attention from the exact shape of a figure to its pattern of connections, a different kind of spatial thinking that powers maps of transport, computer systems, and social links.
Stripping a situation down to dots and lines
A network, also called a graph, is one of the most useful pictures in mathematics because it throws away everything except connections. You draw a dot, called a vertex, for each thing you care about, and a line, called an edge, between two dots whenever those things are related in the way you are studying. Vertices might be towns with edges for the roads between them, or people with edges for friendships, or computers with edges for cables. Once drawn, the network answers questions about how things link up without any of the clutter of the real situation. It does not care how far apart the towns are or where the people live; it records only who is joined to whom, and that turns out to be exactly what many practical problems depend on.
Dots for things, lines for relationships
A network represents a situation as vertices (the things) joined by edges (the relationships between them).
This is a network: five dots joined by lines. Reveal what the parts mean to read it as a friendship map.
Only the connections matter, not the picture
Because a network stores nothing but its links, the same network can be drawn in many ways that all mean the same thing. Slide the dots around, stretch the lines, flip the whole thing over, and as long as the same pairs stay joined, it is the very same network. This is a freeing idea and a common source of confusion at first, because two drawings that look quite different on the page can be identical as networks. The test is never how the diagram looks; it is which vertices are connected. When you compare two networks, you check their lists of edges, not their layouts, and when you tidy a messy diagram, you may move vertices freely so long as you keep every connection intact.
Same network, different drawing
A network carries only the connections, so moving the vertices around leaves the network unchanged.
These two pictures are the same network, even though the dots sit in different places. What a network records is purely which pairs are joined; stretching or bending the drawing changes nothing about the relationships.
Connectedness: can you get everywhere?
The central question this topic asks of a network is whether it is connected. A network is connected when you can travel from any vertex to any other by following edges, hopping from dot to dot along the lines. Such a route is called a path. If even one vertex is stranded, with no chain of edges reaching it from the rest, the network is not connected. The idea is intensely practical: a connected transport network means you can get from any station to any other, a connected computer network means every machine can reach every other, and a break in connectedness means someone or something is cut off. Checking connectedness is simply a matter of asking whether a path exists between every pair of vertices, which you can do by tracing routes outward from one vertex and seeing if you reach them all.
Connected means you can reach everyone
A network is connected if there is a path along edges between every pair of vertices; a stranded vertex breaks this.
A network is connected when you can travel from any vertex to any other along its edges. Here every person can be reached from A, so it is connected.
Counting the links at each vertex
A handy measurement on a network is the degree of a vertex, the number of edges that meet at it. A person with three friendships sits at a vertex of degree three; a town at the end of a single road has degree one. Degree gives a quick sense of how connected each vertex is, and it carries a neat fact: if you add up the degrees of every vertex, the total is always twice the number of edges, because each edge has two ends and so gets counted once at each. This double-counting rule is a reliable check when reading a network, and degree itself helps spot important features, such as a hub with very high degree or an isolated vertex of degree zero that threatens connectedness.
Degree: how many links meet at a vertex
The degree of a vertex is the number of edges at it, and all the degrees add to twice the number of edges.
The degree of a vertex counts the edges at it. Reveal the degrees to compare how connected each person is.
Reading paths to describe a network
Putting these ideas together, describing a network well means talking about its paths. To show a network is connected, you exhibit a path between vertices that might look far apart, as when a route from one corner reaches another by passing through the dots between them. To describe how robust a network is, you might notice whether there are several different paths between two vertices or only one fragile link holding a part of it on. Interpreting a network diagram, then, is less about the drawing and more about answering questions of reachability: who can get to whom, by how many routes, and what would happen to connectedness if a single edge were removed. These are the questions that make networks such a powerful tool for practical situations.
A path is a route along the edges
A path links two vertices by a sequence of edges; connectedness is the existence of a path between every pair.
To check connectedness you look for a path between vertices. Trace a route from C to D along the edges.
Quick self-check
1. In a network diagram, the vertices (dots) and edges (lines) represent:
2. Two network diagrams have their dots in completely different positions but join exactly the same pairs. They are:
3. A network is described as connected when:
4. The degree of a vertex is:
5. One vertex in a network has no edges at all, while the rest are linked together. The network is: