A phase transition for the strength of weak ties

Abstract

Granovetter's ``strength of weak ties'' argument links local triadic closure to the global importance of weak relationships, but it does not by itself quantify when strong ties can or cannot sustain large-scale connectivity. We study the subgraph formed by strong ties under a uniform, edgewise closure condition. A strong-tie graph is called p-transitive when, for every ordered adjacent pair (A,B), at least a fraction p of the strong neighbors of B other than A are also strong neighbors of A. Under a fixed maximum-degree bound, we prove a sharp structural transition at p = 2/3. If p > 2/3, every strong-tie component has size bounded by an explicit function of p and the degree bounds, independently of the total network size. If p < 2/3, bounded-degree p-transitive graphs can have arbitrarily large connected strong-tie components. We also revisit three unweighted real-world networks by thresholding a Jaccard neighborhood-overlap proxy for tie strength and plotting global transitivity against normalized size-biased mean component size. These empirical curves show fragmentation as low-overlap edges are removed, but their statistic and thresholds are not the theorem's edgewise parameter. The theorem is therefore a conditional structural result, while the data illustrate a related clustering--connectivity tradeoff rather than a direct validation of the constant 2/3.