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Detectability threshold in weighted modular networks
Filippo Radicchi1, Filipi N Silva1, Alessandro Flammini1
1Indiana University, Center for Complex Networks and Systems Research, Luddy School of Informatics, Computing, and Engineering, Bloomington, Indiana 47408, USA.
Physical Review. E
|February 20, 2026
Summary
Detecting community structure in networks is possible up to a certain mixing threshold. This threshold depends on node degree and edge weight distributions, with higher variability in weights hindering detection.
Area of Science:
- Network science
- Statistical physics
- Data analysis
Background:
- Community detection algorithms aim to identify groups of nodes in networks.
- The weighted planted-partition model is a standard benchmark for evaluating community detection methods.
- Spectral modularity optimization is a common technique for community detection.
Purpose of the Study:
- To determine the necessary conditions for detecting ground-truth partitions in weighted networks.
- To analytically derive the maximum mixing level tolerated by spectral modularity optimization.
- To investigate the impact of different edge-weight distributions on community detectability.
Main Methods:
- Analytical derivation of the detectability threshold.
- Analysis of the weighted planted-partition model with two equally sized communities.
- Comparison of five edge-weight distributions (Dirac, Poisson, exponential, geometric, signed Bernoulli) under Poisson-distributed node degrees.
Main Results:
- The detectability threshold depends on the first two moments of node degree and edge weight distributions.
- Dirac distributed weights result in the smallest detectability threshold.
- Exponentially distributed weights increase the threshold by a factor of sqrt[2]; higher weight variability can decrease detectability.
Conclusions:
- Edge weight variability significantly influences the detectability of community structure.
- Incorporating edge weights is detrimental when they do not carry information about the community structure.
- The findings provide insights into the limitations and performance of spectral modularity optimization in weighted networks.
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