Quantum mechanics

Electrical resistivity and conductivity

Uncorrelated

Linguistics

Philosophy

Neuroscience

Biology

Statistics

Percolation critical exponents

Crossover

Exponent

Computer science

Percolation (cognitive psychology)

Complex network

Scaling

Statistical physics

Physics

Mathematics

Critical exponent

Mathematical analysis

Artificial intelligence

Geometry

Percolation threshold

Combinatorics

Directed percolation

Generalization

Physical Review X

We consider cumulative merging percolation (CMP), a long-range percolation process describing the iterative merging of clusters in networks, depending on their mass and mutual distance. For a specific class of CMP processes, which represents a generalization of degree-ordered percolation, we derive a scaling solution on uncorrelated complex networks, unveiling the existence of diverse mechanisms leading to the formation of a percolating cluster. The scaling solution accurately reproduces universal properties of the transition. This finding is used to infer the critical properties of the susceptible-infected-susceptible model for epidemics in infinite and finite power-law distributed networks. Here, discrepancies between analytical approaches and numerical results regarding the finite-size scaling of the epidemic threshold are a crucial open issue in the literature. We find that the scaling exponent assumes a nontrivial value during a long preasymptotic regime. We calculate this value, finding good agreement with numerical evidence. We also show that the crossover to the true asymptotic regime occurs for sizes much beyond currently feasible simulations. Our findings allow us to rationalize and reconcile all previously published results (both analytical and numerical), thus ending a long-standing debate.2 MoreReceived 18 June 2019Revised 26 September 2019Accepted 6 February 2020DOI:https://doi.org/10.1103/PhysRevX.10.011070Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasContinuous percolation transitionEpidemicEpidemic prevalencePercolationPercolation phase transitionPercolation theoryInterdisciplinary PhysicsNetworksStatistical Physics

Cumulative Merging Percolation and the Epidemic Transition of the Susceptible-Infected-Susceptible Model in Networks