While reading about community detection and stochastic block model I came up with the article "Community detection and stochastic block models: recent developments" by Emmanuel Abbe. I saw there a definition of graphical channel which connects community detection with information theory. I have difficulty to completely understand the definition especially the phrase "k node variables adjacent to edge I". Can someone explain me in simple words ?

Here is the link of article https://arxiv.org/pdf/1703.10146.pdf

and here is the definition image:



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