The giant component of complex hypergraphs: automated generating function calculations
Abstract
Complex hypergraphs (chygraphs) contain graphs, hypergraphs, multiplex and interacting networks as special cases, and the percolation threshold of all of them follows from one symbolic calculation: the spectrum of a tensor A built from four matrices of first moments. That calculation stops at the threshold. Here I show that A is the Jacobian, at its trivial fixed point, of a non-linear self-consistency map whose non-trivial fixed point is the giant component fraction, so threshold and order parameter are two orders of one object on one index set. Carrying the expansion one order further gives the critical amplitude B in S=BΛ+O(Λ2) in closed form, and exposes a hierarchy: first moments fix the threshold, second moments the amplitude, and only the generating functions themselves the order parameter away from it. Dropping the assumption that a complex's participation in different layers is independent generalises the tensor again, replacing unconditional first moments by inclusion-biased second moments: distributions with identical marginals have different thresholds and different order parameters. Six constructions from the literature are then solved by substitution, among them AND- against OR-logic hypergraph percolation, which share a chygraph and differ in one generating function, and SIR epidemics with two levels of mixing, whose household reproduction number falls out of the tensor. A reported failure of mean-field theory on strongly clustered graphs is shown to be a failure of applying it to the wrong object: mapped onto the treelike backbone rather than the clustered graph, the same calculation reproduces the exact threshold, order parameter and critical exponents. Everything is implemented in one class, in which a chygraph is specified once by its generating functions and then returns all three, and validated against Monte Carlo simulation.
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BibTeX
@misc{vazquez2026giant,
title = {{The giant component of complex hypergraphs: automated generating function calculations}},
author = {Alexei Vazquez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29558},
eprint = {2609.29558},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29558},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - The giant component of complex hypergraphs: automated generating function calculations
AU - Alexei Vazquez
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29558
UR - https://arxiv.org/abs/2609.29558
AB - Complex hypergraphs (chygraphs) contain graphs, hypergraphs, multiplex and interacting networks as special cases, and the percolation threshold of all of them follows from one symbolic calculation: the spectrum of a tensor A built from four matrices of first moments. That calculation stops at the threshold. Here I show that A is the Jacobian, at its trivial fixed point, of a non-linear self-consistency map whose non-trivial fixed point is the giant component fraction, so threshold and order parameter are two orders of one object on one index set. Carrying the expansion one order further gives the critical amplitude B in S=BΛ+O(Λ^2) in closed form, and exposes a hierarchy: first moments fix the threshold, second moments the amplitude, and only the generating functions themselves the order parameter away from it. Dropping the assumption that a complex's participation in different layers is independent generalises the tensor again, replacing unconditional first moments by inclusion-biased second moments: distributions with identical marginals have different thresholds and different order parameters. Six constructions from the literature are then solved by substitution, among them AND- against OR-logic hypergraph percolation, which share a chygraph and differ in one generating function, and SIR epidemics with two levels of mixing, whose household reproduction number falls out of the tensor. A reported failure of mean-field theory on strongly clustered graphs is shown to be a failure of applying it to the wrong object: mapped onto the treelike backbone rather than the clustered graph, the same calculation reproduces the exact threshold, order parameter and critical exponents. Everything is implemented in one class, in which a chygraph is specified once by its generating functions and then returns all three, and validated against Monte Carlo simulation.
ER -