The emergence of the ultra-small world state. Panel a: Sketch of a hypothetical network where nodes v and u are separated by a distance H +1. The neighbors of v are then at either H (the light blue node), or H + 1 (the green node s), or H + 2 (the red nodes) edges from u. For a better visualization, paths of different lengths are marked with the corresponding colors. Panel b: A direct (yellow) link is added between v and u. Our study demonstrates rigorously (see Theorem 3 of the SI) that the network configuration of panel a is incompatible with an equilibrium state.

Contexts in source publication

Context 1

... proof of the Theorem (see SI for details) is given by contradiction i.e., by supposing that there is a node u in the final state of the network whose distance from v is at least H + 1 i.e., l(u, v) ≥ H + 1. To better illustrate the situation, we depicted in panel a of Fig. 4 the case in which nodes v and u are separated by a distance H + 1. In that circumstance, the nodes directly connected to v (the neighbors of v) may be found at either H (the light blue node), or H + 1 (the green node), or H + 2 (the red nodes) edges from u. Looking at the figure, it is easy to understand that all network's shortest ...

Context 2

... one, instead, includes a direct link between v and u [the yellow link in panel b of Fig. 4], then the shortest path between any neighbor of v (denoted by w) and u becomes w − v − u, since H ≥ 3. Calculating then the value of ∆W BC(v) corresponding to the addition of such a link, and recalling that the equilibrium requires ∆W BC(v) to be smaller than the cost c, one easily get to an expression which is in explicit ...