Two kinds of large connected cores in directed networks are predicted to emerge at the same point during random node or link removal, but the giant strongly biconnected component, or SBC, grows more slowly than the giant strongly connected component, or SCC, after that threshold. The SCC is the large mutually reachable core, while the SBC tracks redundant routes within that core, making it the more demanding measure.
That conclusion comes from a study built around an analytical framework for giant SBC size and robustness under random failures in directed and biological networks. The authors compare the SBC with the giant SCC, then check the framework against directed random-network simulations, a weighted larval-brain connectome and directed networks from the Cell Collective database.
One threshold, different growth
To derive the model, the authors use generating functions to obtain equations for giant SBC size and estimate its percolation threshold. In the failure setup, node removal is treated as site percolation and link removal as bond percolation, with each node or link assigned an independent activation probability f. This lets the analysis compare the point at which the two giant structures emerge with the way their sizes change afterward.
Under either site or bond percolation, the predicted threshold is f_c = ⟨k⟩/⟨k_i k_o⟩, and it is identical for the giant SBC and giant SCC. The analytical predictions agreed well with simulations. Once the common threshold was crossed, the SBC increased more slowly than the SCC.
Near the threshold, the reported scaling laws make the gap sharper. The giant SCC size follows S ∼ (f − f_c)^2, while the giant SBC follows B ∼ (f − f_c)^4. Put simply, the reported SBC size rises with the fourth power of its distance above the threshold, compared with the second power for the SCC. No uncertainty interval for these exponents was reported.
The biological networks tell a similar story
The empirical checks start with a weighted neural connectome of the larval brain containing 2,952 nodes and 110,677 non-zero-weight edges. The authors remove links below a weight cutoff θ and recalculate giant SCC and SBC sizes. The theoretical curves agreed well with the empirical thresholding results, and the giant SBC was consistently smaller than the giant SCC.
They also examine 14 directed biological networks from the Cell Collective, each larger than 50. To create a randomized comparator, they swap edge targets while preserving source out-degrees and target in-degrees.
Those randomized networks had larger giant SBCs than their real-world counterparts, with a mean difference of ⟨B_w − B⟩ = 0.0656 ± 0.0136. The paper reports that uncertainty as a standard error. Giant SBC size correlated with theoretical predictions at r = 0.878 for original networks and r = 0.957 for rewired networks.
The authors interpret the smaller giant SBCs in real biological networks as reflecting optimized structures with reduced or locally concentrated redundant pathways. Their conclusion stays at the level of network structure and modeled robustness under random failures.
The supplied document identifies the work as an arXiv preprint, version 1. The authors report support from the National Research Foundation of Korea, the LAMP Program and IITP/ITRC grants funded by Korean government ministries.
Paper data and sources
Original title: Giant strongly biconnected components of directed networks: a generating function approach
Authors: Minsoo Yang, Reinhard Laubenbacher, Byungjoon Min
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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