A phase transition for the strength of weak ties
- 1Office of Educational Assessment, Santa Clara University , 500 El Camino Real, Santa Clara, 95053, CA ,
- 2Department of Statistics, Stanford University , 390 Jane Stanford Way, Stanford, 94305, CA ,
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.
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BibTeX
@article{chatterjee2026phase,
title = {{A phase transition for the strength of weak ties}},
author = {Esha Chatterjee and Sourav Chatterjee},
journal = {PNAS Nexus},
year = {2026},
month = oct,
doi = {10.1093/pnasnexus/pgag364},
url = {https://academic.oup.com/pnasnexus/advance-article/doi/10.1093/pnasnexus/pgag364/8979534},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - A phase transition for the strength of weak ties
AU - Esha Chatterjee
AU - Sourav Chatterjee
PY - 2026
DA - 2026/10/22
JO - PNAS Nexus
DO - 10.1093/pnasnexus/pgag364
UR - https://academic.oup.com/pnasnexus/advance-article/doi/10.1093/pnasnexus/pgag364/8979534
AB - 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.
SN - 2752-6542
ER -