  We model a social network by a random graph whose nodes represent agents and
links between two of them stand for a reciprocal interaction; each agent is
also associated to a binary variable which represents a dichotomic opinion or
attribute. We consider both the case of pair-wise (p=2) and multiple (p>2)
interactions among agents and we study the behavior of the resulting system by
means of the energy-entropy scheme, typical of statistical mechanics methods.
We show, analytically and numerically, that the connectivity of the social
network plays a non-trivial role: while for pair-wise interactions (p=2) the
connectivity weights linearly, when interactions involve contemporary a number
of agents larger than two (p>2), its weight gets more and more important. As a
result, when p is large, a full consensus within the system, can be reached at
relatively small critical couplings with respect to the p=2 case usually
analyzed, or, otherwise stated, relatively small coupling strengths among
agents are sufficient to orient most of the system.
