In this paper we consider the dyadic effect introduced in complex networks when nodes are distinguished by a binary characteristic. Under these circumstances two independent parameters, namely dyadicity and heterophilicity, are able to measure how much the assigned characteristic affects the network topology. All possible configurations can be represented in a phase diagram lying in a two-dimensional space that represents the feasible region of the dyadic effect, which is bound by two upper bounds on dyadicity and heterophilicity. Using some network's structural arguments, we are able to improve such upper bounds and introduce two new lower bounds, providing a reduction of the feasible region of the dyadic effect as well as constraining dyadicity and heterophilicity within a specific range. Some computational experiences show the bounds' effectiveness and their usefulness with regards to different classes of networks. Keywords: Complex networks, dyadic effect, upper and lower bound. arXiv:1609.04547v1 [math.CO] 15 Sep 2016
Nodes' characteristics and dyadic effectHerein, we refer to a given characteristic c i , which can assume the values 0 or 1, for each i ∈ N . Consequently, N can be divided into two subsets: the set of n 1 nodes with characteristic c i = 1, the set of n 0 nodes with characteristic c i = 0; thus, N = n 1 + n 0 .We distinguish three kinds of dyads, i.e. edges and their two end nodes, in the network:(1 − 1), (1 − 0), and (0 − 0) as depicted in the Figure 1.We label the number of each dyad in the graph as m 11 , m 10 , m 00 , respectively. Hence, M = m 11 +m 10 +m 00 . We consider m 11 and m 10 as independent parameters that represent i∈D T G (n 1 ) d i − n 1 (n 1 − 1). Indeed, every stub that constitutes the residual degree will find its endpoint in a node with c i = 0. Otherwise, i∈D T G (n 1 ) d i < n 1 (n 1 − 1) and m 10 = 1.