Epidemic Spreading in Scale-Free Networks

Epidemic Spreading in Scale-Free Networks Romualdo Pastor-Satorras Alessandro Vespignani

Tag layer of the DHSS Meta-Tagging Project. The article text is reproduced from the staged copy named in sourceDesc and is not altered; the tags are held separately in standOff so that overlapping and nested spans are preserved.

Epidemic Spreading in Scale-Free Networks 2001 Physical Review Letters / arXiv 10.1103/PhysRevLett.86.3200 cond-mat/0010317 pastorsatorras2001-epidemic-scalefree pastorsatorras2001-epidemic-scalefree.txt, the project's staged copy 030556a9e1e1c83314f5a25fd4bd76ce91153dff arxiv-open epidemiology

the paper's own words

the tag layer; a sentence written by the tagger about the paper, always anchored to a quotation but not itself a quotation

0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them The spread of computer viruses on the Internet, treated as an epidemic process running on a scale-free network. Quotation withheld because this record is not cleared for redistribution. Why do computer viruses persist for years at very low prevalence, when standard epidemic models say an infection either dies out exponentially or takes off? Quotation withheld because this record is not cleared for redistribution. Shows that on a scale-free network the epidemic threshold vanishes, so an infection persists at any spreading rate - which explains the observed long-lived, low-prevalence computer viruses without tuning anything. Quotation withheld because this record is not cleared for redistribution. mechanism Quotation withheld because this record is not cleared for redistribution. epidemic spreading on scale-free networks Quotation withheld because this record is not cleared for redistribution. simulation simulation Quotation withheld because this record is not cleared for redistribution. mechanisms the scale-free network diverging connectivity fluctuations remove the epidemic threshold, so infections persist for any positive transmission rate Quotation withheld because this record is not cleared for redistribution. mechanisms diverging connectivity fluctuations act as an infinite connectivity, and the threshold falls to zero Quotation withheld because this record is not cleared for redistribution. mechanisms a link is more likely to land on a highly connected node, so hubs are infected first and keep re-infecting the rest Quotation withheld because this record is not cleared for redistribution. mechanisms e-mail attachments make the contact graph the Internet's topology rather than physical proximity Quotation withheld because this record is not cleared for redistribution. the SIS model reduces individuals to two discrete states (healthy/infected) what this limits: every result of the SIS model: INDIVIDUALS EXIST ONLY IN TWO DISCRETE STATES Quotation withheld because this record is not cleared for redistribution. The model is run at one exponent, γ = 3, and the generalisation to other exponents is asserted qualitatively rather than shown. what this limits: the range of exponents the conclusion covers: THE PRESENT FRAMEWORK CAN BE GENERALIZED TO NETWORKS WITH 2 < gamma <= 3, RECOVERING QUALITATIVELY THE SAME RESULTS - asserted qualitatively, by citation Quotation withheld because this record is not cleared for redistribution. Individuals are reduced to two states, healthy or infected, which the paper flags as a coarse description. what this limits: the same scope as limitations[0], in the paper's own words: THIS MODEL RELIES ON A COARSE GRAINED DESCRIPTION OF INDIVIDUALS IN THE POPULATION Quotation withheld because this record is not cleared for redistribution. The empirical half rests on one industry data source over a four-year window. what this limits: the empirical half only: WE HAVE ANALYZED THE PREVALENCE DATA REPORTED BY THE VIRUS BULLETIN FROM FEBRUARY 1996 TO MARCH 2000 - one industry source, four years Quotation withheld because this record is not cleared for redistribution. modelling epidemic spreading on scale-free contact networks Quotation withheld because this record is not cleared for redistribution. the susceptible-infected-susceptible model and the notion of an epidemic threshold, applied to computer virus outbreaks on the Internet Quotation withheld because this record is not cleared for redistribution. testing the model's prediction against independent real-world prevalence data reported by the Virus Bulletin, February 1996 to March 2000 eliminates: that the absence of an epidemic threshold is an artefact of the scale-free model rather than a property of how real computer viruses spread · outcome: held Quotation withheld because this record is not cleared for redistribution. epidemic threshold epidemic threshold Quotation withheld because this record is not cleared for redistribution. scale-free networks scale-free networks Quotation withheld because this record is not cleared for redistribution. node node Quotation withheld because this record is not cleared for redistribution. individual individual Quotation withheld because this record is not cleared for redistribution. link link Quotation withheld because this record is not cleared for redistribution. susceptible-infected-susceptible susceptible-infected-susceptible Quotation withheld because this record is not cleared for redistribution. contact process contact process Quotation withheld because this record is not cleared for redistribution. Euclidean lattice Euclidean lattice Quotation withheld because this record is not cleared for redistribution. random networks random networks Quotation withheld because this record is not cleared for redistribution. Internet Internet Quotation withheld because this record is not cleared for redistribution. surviving probability surviving probability Quotation withheld because this record is not cleared for redistribution. prevalence prevalence Quotation withheld because this record is not cleared for redistribution. Euclidean lattice → epidemic threshold A NONZERO EPIDEMIC THRESHOLD IS A PROPERTY OF THE LOCAL-CONNECTIVITY REGIME, NOT SOMETHING THE LATTICE PRODUCES. The lattice does not increase, reduce or eliminate a threshold; models with local connectivity simply HAVE one, and that is the standard result this paper is written against. None of the eight types says 'X holds in regime Y', and none of the four effect directions says a quantity EXISTS - all four describe a change in it. Quotation withheld because this record is not cleared for redistribution. network - social, biological and communication systems alike, whose nodes are individuals or organizations and whose links are the interactions among them Quotation withheld because this record is not cleared for redistribution. network - the Internet as the contact graph along which an e-mail-borne infection actually travels, as opposed to physical proximity Quotation withheld because this record is not cleared for redistribution. epidemic - a computer virus outbreak, analysed with the models of biological epidemiology Quotation withheld because this record is not cleared for redistribution. claim_type: mechanism a second reading reaching the same value. Recorded: mechanism — note: a second reading reaching the same value from a different sentence. Not applied - a person decides. Quotation withheld because this record is not cleared for redistribution. method: simulation a second reading reaching the same value. Recorded: simulation — note: a second reading reaching the same value from a different sentence. Not applied - a person decides. Quotation withheld because this record is not cleared for redistribution. phenomenon: absence of an epidemic threshold for spreading on scale-free networks a second reading DIFFERING from the recorded value. Recorded: epidemic spreading on scale-free networks — note: a second reading, verbatim-verified, disagreeing with the value on the record. Not applied - a person decides. Quotation withheld because this record is not cleared for redistribution. builds-on → scale-free network model Quotation withheld because this record is not cleared for redistribution. contradicts → the classical nonzero-epidemic-threshold prediction Quotation withheld because this record is not cleared for redistribution. contradicts → the standard nonzero epidemic threshold of mathematical epidemiology [11] Quotation withheld because this record is not cleared for redistribution. builds-on → the scale-free graph model of Barabasi and Albert [6] Quotation withheld because this record is not cleared for redistribution. contradicts → standard epidemic modelling on lattices and mean-field models, which predicts a nonzero threshold differs in: THE AXIS IS THE CONNECTIVITY REGIME. 'Quotation withheld.' The threshold is not wrong for the models it was derived in - it is a property OF THOSE MODELS' CONNECTIVITY, and this paper changes the connectivity. Its unhoused_relations[0] in this corpus records the same point as a relation the vocabulary could not express: a threshold is a property of a regime, not something a lattice produces. Quotation withheld because this record is not cleared for redistribution. extends → the observational literature on computer virus prevalence [10,13] Quotation withheld because this record is not cleared for redistribution. scale-free networks —contrasted-with→ random networks scale-free networks random networks strength of the relation: 1 Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but endpoint by anaphora: 'That' = the scale-free connectivity of the Internet (previous sentence). Quotation withheld because this record is not cleared for redistribution. contact process —is-a→ susceptible-infected-susceptible contact process susceptible-infected-susceptible strength of the relation: 0.9 Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but the span says the SIS model is a GENERALIZATION of the CP model; is-a is loose, endpoints abbreviated. Quotation withheld because this record is not cleared for redistribution. EVIDENCE CORRECTED (RE-GROUNDED) · scale-free networks —causes→ epidemic threshold scale-free networks epidemic threshold EVIDENCE CORRECTED: RE-GROUNDED. The old span - 'annulling thus the threshold' - states the mechanism and names neither endpoint. This sentence names both AND states the finding, and the sign travels with it. — direction on the measure: eliminates · strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. γ between 2 and 3 — power-law degree exponent of the scale-free networks (Internet/WWW) Quotation withheld because this record is not cleared for redistribution. 0 (no threshold) — the epidemic threshold vanishes for scale-free networks Quotation withheld because this record is not cleared for redistribution. τ ≃ 14 months — characteristic lifetime of boot and macro virus strains Quotation withheld because this record is not cleared for redistribution. λc = 0 — the epidemic threshold on a scale-free network Quotation withheld because this record is not cleared for redistribution. N = 103 to N = 8.5 × 106 — range of simulated network sizes (the exponents are lost in extraction; the paper means 10^3 to 8.5x10^6) about this quote: FLATTENED EXPONENT IN THIS QUOTE - THE NUMBER IS WRONG BY ORDERS OF MAGNITUDE AS IT STANDS. The staged text reads '8.5 × 106', where the paper writes 10 to the power 6. The superscript was pushed onto the line by the PDF extractor. The quote is verbatim to the staged text and its anchor is sound; the STAGED TEXT is what is damaged. NOT repaired - repair means re-staging the body and re-anchoring the record. Quotation withheld because this record is not cleared for redistribution. γ ≃ 2.5 — measured scale-free exponent of the Internet and the world-wide-web Quotation withheld because this record is not cleared for redistribution. 100 starting configurations x 10 network realisations — averaging used for every prevalence figure Quotation withheld because this record is not cleared for redistribution. an epidemic threshold and its associated critical behavior are absent Quotation withheld because this record is not cleared for redistribution. Real computer-virus data are analysed to give the average lifetime and prevalence of viral strains on the Internet. Quotation withheld because this record is not cleared for redistribution. On a scale-free network the epidemic threshold is zero: any spreading rate, however small, lets the infection persist. Quotation withheld because this record is not cleared for redistribution. Steady-state virus prevalence decays with decreasing spreading rate as ρ ~ exp(−C/λ). Quotation withheld because this record is not cleared for redistribution. Fitting the data gives a lifetime of about 14 months for boot and macro viruses and 6-9 months for file viruses. Quotation withheld because this record is not cleared for redistribution. The higher a node's connectivity, the higher its probability of being infected. Quotation withheld because this record is not cleared for redistribution. spreading rate λ spreading rate λ Quotation withheld because this record is not cleared for redistribution. effective spreading rate λ effective spreading rate λ Quotation withheld because this record is not cleared for redistribution. virus prevalence virus prevalence Quotation withheld because this record is not cleared for redistribution. epidemic threshold λc epidemic threshold λc Quotation withheld because this record is not cleared for redistribution. scale-free network scale-free network Quotation withheld because this record is not cleared for redistribution. susceptible-infected-susceptible susceptible-infected-susceptible Quotation withheld because this record is not cleared for redistribution. effective spreading rate λ effective spreading rate λ Quotation withheld because this record is not cleared for redistribution. epidemic threshold epidemic threshold Quotation withheld because this record is not cleared for redistribution. statistical-significance statistical-significance Quotation withheld because this record is not cleared for redistribution. power-law power-law Quotation withheld because this record is not cleared for redistribution. mean mean Quotation withheld because this record is not cleared for redistribution. exponential-distribution exponential-distribution Quotation withheld because this record is not cleared for redistribution.

The full text of this article is not redistributable, so it is not included here. The evidence sentences are withheld with it; the tag layer and its grounding status remain. The article itself is at https://doi.org/10.1103/PhysRevLett.86.3200.