Exponential generalised network descriptors

Abstract

In communication networks theory the concepts of networkness and network surplus have recently been defined. Together with transmission and betweenness centrality, they were based on the assumption of equal communication between vertices. Generalised versions of these four descriptors were presented, taking into account that communication between vertices u and v is decreasing as the distance between them is increasing. Therefore, we weight the quantity of communication by λ d (u,v) where λ ∈ 0, 1 . Extremal values of these descriptors are analysed.

One-alpha weighted network descriptors

“…It can be easily seen that i) is obtained for the sequence (n − 1, 0, ..., 0), ii) for the sequence (2, ..., 2) and iii) for the sequence (2, ..., 2, 1). Moreover, it can be easily seen that T n (2, ..., 2) > T n (n − 1, 0, ..., 0) if and only if α ≥ 4 3+n , and that T n (2, ..., 2, 1) > T n (n − 1, 0, ..., 0) if and only if α ≥ 4n−12 n 2 −8 . Hence, it is sufficient to prove that for all n:…”

“…As this is not a case in real life networks we searched for a descriptor which amends this problem and gives a more realistic representation of the actual communication dynamic in the network. In the paper [3] we presented one way of fixing this by defining a generalized version of the observed descriptors, where the amount of communication was weighted by d(u, v) λ , for λ < 0, and in paper [4] by weighting the amount of communication by λ d (u,v) , for λ ∈ 0, 1 .…”

“…Here we consider network descriptors where the amount of communication between two non-neighbor vertices decreases proportional to α, compared to the amount of communication between neighboring vertices, for some α ∈ 0, 1 . This modification, unlike the ones in [3] and [4], takes in the account that neighboring vertices communicate much more than non-neighboring ones, which is of course true in vast majority of networks.…”