Networks are represented by the mathematical concept of a “graph”:
“Nodes” are the individual locations / states / items being described
“Edges” are the lines / routes / relationships connecting them.
Edges may have values associated with them, variously called “weights” or “distances,” and nodes often have additional properties as well. Edges may also be “directed” or “undirected”, according to whether they can be traversed in one or two directions.
On the left is a basic graph with nodes A-F. The middle graph adds a weight or distance to each edge. The right graph has unweighted, directional edges.
This concept of a graph is used in computer science to describe databases, and relationships between computers on the Internet. Closely related diagrams can show a program’s class structure or logical flow.
The diagrams can be used to illustrate the “Six Degrees of Separation” concept from pop culture - the estimate that any two people can be connected by a chain of at most six intermediaries. This is enabled by humans’ large number of social contacts and the interconnectedness of global society. (Dunbar’s number means that each “node” or person connects to around 150 other people, and society promotes multiple connections between different groups, so that large clusters are far from isolated.)
Mind maps, family trees, and evolutionary biology also make use of these graphs.
The benefits of all these diagrams is they emphasize connections, relationships between objects.
Nodes within a graph may be characterized by their number of connections, or in a directional graph, by what sources / destinations are possible for a given graph. For example, in the left most graph above, F connects to only one other node, while A connects to all five other nodes. (Connections of a node to itself are also possible, but not shown.) In the middle graph, the shortest path is A-B with a distance of 1. The longest chain is D-F, with a distance of 8. In the right graph, F cannot be reached from any other node, and no other node can be reached from B.
Graphs may be classified based on the number of links to a typical node, or the distance connecting multiple nodes, or the number of nearly-isolated subgraphs which are much more interconnected than the overall network. There are 9 edges and 6 nodes in each of the graphs above, giving a mean connectivity of 3 edges per node, since each edge connects two nodes.
Any time relationships or state changes are important, examination of the appropriate graphs are likely to yield useful insights.