\r\nand a “periphery.” The existence of a core suggests that some

\r\nvertices are central and form the skeleton of the network, to which

\r\nall other vertices are connected. An alternative view of graphs is

\r\nthrough communities. Multiple measures have been proposed for

\r\ndense communities in graphs, the most classical being k-cliques,

\r\nk-cores, and k-plexes, all presenting groups of tightly connected

\r\nvertices. We here show that the edge number thresholds for such

\r\ncommunities to emerge and for their percolation into a single dense

\r\nconnectivity component are very close, in all networks studied. These

\r\npercolating cliques produce a natural core and periphery structure.

\r\nThis result is generic and is tested in configuration models and in

\r\nreal-world networks. This is also true for k-cores and k-plexes. Thus,

\r\nthe emergence of this connectedness among communities leading to

\r\na core is not dependent on some specific mechanism but a direct

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