Help with the definition of clique percolation

So I can find the Wiki article okay, as I think I understand that this is a definition of a special community in a graph. However their image in Fig. 1 confuses me from the description.

As I understand it, the definition of a clique percolation within a graph is:

1. You have a set of k-maximal cliques, which contain k nodes.
2. You are said to have k-adjacent cliques if each k-maximal cliques are connected by (k-1)-nodes.

So a graph that was (1)<-->(2)<-->(3) could be considered a community because it has 2 2-maximal cliques, [[1,2], [2,3]], and each of them are connected by 1 vertex.

Technically yes, the graph you describe is one 2-clique percolated community. However, this is a reduced case, since a 2-clique is no more than an edge, and so a 2-clique percolated community is just a connected component.

For k > 2, clique percolation results in communities that are only detectable using this method. The example in Wiki uses 4-cliques.