What is the difference between weakly strongly and giant connected components?

This post hopes to make a clear distinction between weak, strong and giant components, and I use the open-source code of Gephi to illustrate.

The definition of a giant component is simply the connected component which has the largest number of nodes.

You can see it in the code. The giant component of a graph (or network) is simply identified by:

  • considering all the connected components,
  • selecting the one that has the largest size (meaning, that has the most nodes)

So, what is a “connected component”?

It depends whether your graph is directed or not. A graph is said to be directed when the links (or connections, or edges) have a direction (A –> B : A is connected “in the direction of” B). If the direction of the connection has no relevance (as in: A-B, there is no “direction” or “arrow” between A and B), the graph is said to be “undirected”.

if your graph is undirected:

A connected component is used to mean a “weakly connected component”. In non technical terms, this is a group of nodes where no node or group of nodes is isolated, there is at least one path connecting a node to another. See this part of the code which shows that weak components are computed when the graph is undirected.

if your graph is directed:

A connected component is used to mean a “strongly connected component”, according to this definition: https://en.wikipedia.org/wiki/Strongly_connected_component. In non technical term, this is a group of nodes where it is possible to go from one node to any other node (there is at least one “path”), even with taking into account that some links can be followed in just one direction.

In the Gephi software, the method used to find these strongly connected components is Tarjan’s algorithm. The code for the tarjan’s algorithm is visible here.

Written on December 3, 2021