Dominance of weighted nested split graph networks in connections models

Characterization of efficient networks in a generalized connections model with endogenous link strength

“…7 It is straightforward to check that the model described in Sect. 3 satisfies all these conditions, under which the dominance of weighted DNSG-networks is established in Olaizola and Valenciano (2019). We omit the very easy detailed checking here to avoid a trivial digression.…”

“…2 shows an adjacency matrix that follows this pattern. 6 In order to make the paper basically self-contained, we briefly review the family of connections models considered in Olaizola and Valenciano (2019). As in the current model, a link-formation technology (as per Definition 2) rules the formation of costly weighted links among a set of nodes, each of them endowed with value v. The value that one node receives from another is what it receives through the strongest path connecting them, and what it receives through a path is a fraction of v proportional to the "strength" of their connection through that path.…”

“…As in the current model, a link-formation technology (as per Definition 2) rules the formation of costly weighted links among a set of nodes, each of them endowed with value v. The value that one node receives from another is what it receives through the strongest path connecting them, and what it receives through a path is a fraction of v proportional to the "strength" of their connection through that path. Olaizola and Valenciano (2019) make the following assumptions about the "strength" of the connection of any two nodes in a weighted network through a path: (i) the strength of a link is its weight; (ii) the strength of the connection of i with j is the same as that of j with i; (iii) the strength of the connection through a path is non-decreasing w.r.t. the increase of the strength of its links; (iv) the strength of the connection through a path is not stronger than the connection of any two nodes in it through the subpath.…”

“…We omit the very easy detailed checking here to avoid a trivial digression. On the other hand, the constructive proof of the result in Olaizola and Valenciano (2019) consists of an algorithm that generates a DNSG-network that dominates the initial network by rearranging the links of any network whose net value is positive, perhaps disposing of some of them. The procedure consists basically of forming a star with the strongest links and adding the weakest of the remaining available links at each stage only if adding it improves the connection of the two worst connected nodes in the network.…”

“…We prove that even in this general setting, virtually, whatever the technology, the only possibly efficient networks are the empty network, the all-encompassing star and the complete network. Olaizola and Valenciano (2019) prove constructively that, under rather general conditions for models extending the connections model of Jackson and Wolinsky (1996) and for any link-formation technology, any network with positive net value is dominated by a weighted nested split graph network (NSG-network) of particular type called dominant nested split graph network (DNSG-network). 2 The generalized connections model considered here meets those conditions, so that result applies to the current model and provides the basic point of support for the result obtained.…”