More technically, the joint distribution of set of random variables (networks in this case) could be expressed as something proportional to an arbitrary exponential function on these variables. This function can be expanded as the sum of functions defined on the all possible sets of these variables. According to Hammersley-Clifford theorem (Besag, 1974), this sum can be re-expressed as the sum of terms only associated with cliques (mutually dependent subsets) of variables. Thus, probability of a network depends only on those cliques.