Higher-order architecture of metastatic networks: A statistical mechanics approach via validated hypergraphs

Abstract

Metastasis is a complex, stochastic systemic process characterised by non-random spreadpatterns to secondary organs. Conventional dyadic projections fragment multi-organ interactionsinto independent events, effectively discarding irreducible higher-order topological informationand the synergistic nature of clonal dissemination. This limitation often leads to conflatingfirst-order topological properties with intrinsic biological affinities, particularly in the absenceof validation against structure-aware null models. To address this methodological gap, we present a higher-order hypergraph frameworkthat preserves simultaneous colonisation patterns and accounts for emergent architectures thatcannot be reduced to simple pairwise interactions. Our approach incorporates: (1) a functionalpartition of the network nodes (𝑉𝑆, , 𝑉𝑇) to distinguish organs with documented reseedingcapacity; and (2) a directed association matrix S quantifying the non-linear information fluxand dependencies among source organs.We implement a rigorous validation protocol by exploring the constrained null-space ofthe metastatic hypergraph using a Metropolis–Hastings MCMC algorithm. By contrasting theempirical data against a degree-preserving Canonical Ensemble (Configuration Model), thisframework acts as a structural filter, demonstrating that many observed metastatic hotspots arespurious correlations emergent from nodal degree heterogeneity rather than genuine biologicalco-selection. By de-convoluting effective affinities from degree-driven effects, our methodidentifies a robust structural backbone that remains invariant under topological perturbations.This work shifts the focus from counting isolated events to quantifying the statistical mechanicsof non-linear disease progression in complex biological networks.