Epidemic Spreading in Scale-Free Networks — TEI-tagged view

Romualdo Pastor-Satorras, Alessandro Vespignani · 2001 · epidemiology · doi:10.1103/PhysRevLett.86.3200 · full text at the source · TEI xml · tagged edition · index · concepts · authors
Where this record's text came from. faceted 2026-08 by reading the locally held text (pdftext); every evidence string verified as an exact substring of that text before merging. limitations left empty: the paper states no limits on its own results; the nearest thing is a claim of robustness (finite-size effects excluded, generalisable to 2 < gamma <= 3), which is the opposite of a limitation. Only one claim_relation recorded: the paper takes its scale-free graph from 'the algorithm devised in Refs. [6]' and its data from 'the Virus Bulletin [16]', but these targets appear as bare reference numbers with no author or work named in the running text, so per the rules they were not turned into relations. Method is really twofold (large-scale simulation plus a mean-field analytic solution); the facet records the simulation. The extraction garbles all mathematics (e.g. 'hk2i', 'ki/ Pj kj') and splices page numbers into the middle of sentences ('In a SF network the unbounded 5 fluctuations in connectivity'), so quotes were taken only from unbroken prose stretches. 2026-08-12 and 2026-08-15 schema-0.3 passes: every quote verified as an exact substring of the staged text and anchored at merge time; concept terms verified present; every edge endpoint an existing node. Update, 16 September 2026: EXCEPTION TO THE DATED PASS ABOVE, on this record only: this record carries 1 composite quote field - a quote assembled from two or more passages joined by an ellipsis. A COMPOSITE IS NOT AN EXACT CONTIGUOUS SUBSTRING of the staged text and is deliberately not anchored: each of its parts is verified separately at the offsets in evidence composite spans, and a part occurring more than once carries span null rather than a guessed offset. So the 'every quote verified as an exact substring' clause of the dated pass above DOES NOT HOLD for those fields, and never did - the parts were verified, the whole was not. Nothing is deleted, re-grounded or withdrawn: this writes down an exception that was always true of composites and had not been recorded, so a reader was told more than the pass had checked. 2026-09-16 the project CORRECTION TO THE EXEMPTION ABOVE, SAME DAY: THE EXEMPTION DOES NOT APPLY TO THIS RECORD. Its evidence composite spans row(s) are FOSSILS of a 2026-08-25 composite marking that was superseded by a 2026-08-26 re-slice; the quote is an exact contiguous substring of the staged text, so the dated substring claim above holds for it WITHOUT exception. The clause is left in place with this correction beside it rather than removed, so that the error and its repair are both on the record.
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Other fields define these differently the same word, defined another way in another discipline
threshold 2 fields · connectivity 2 fields · simulation 2 fields · statistical-significance 2 fields
Tags on this paper every tag the record carries, whether or not the words occur in the text below — a number is how many papers in the corpus carry that tag; a tag with no number is unique to this paper
scale-free network4 epidemic threshold contact network degree distribution3 basic reproduction number scale-free networks node4 individual link4 susceptible-infected-susceptible contact process Euclidean lattice random networks2 Internet2 surviving probability prevalence6 simulation51

Our tag layer for this paper

Cross-disciplinary analysis
content-tags & facets: phenomenon · claim-type · cross-domain-transfer · mechanism · claim-relation · replication-status · limitation — grounded in the full text
phenomenonepidemic spreading on scale-free networks— “To address the effects of scale-free connectivity in epidemic spreading we study the SIS model on the SF network.”
claim-typemechanism— “We find the absence of an epidemic threshold and its associated critical behavior, which implies that SF networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents possess.”
method — evidence— “With this aim, we study by large scale simulations and analytical methods the susceptible-infected-susceptible [11] model on SF graphs.”
object of study — evidence— “we have analyzed the prevalence data reported by the Virus Bulletin [16] from February 1996 to March 2000”
research question — evidence— “all viruses that are able to pervade, spread much slower than exponentially, and saturate to a very low level of persistence, affecting just a tiny fraction of the total number of computers [10]”
contribution — evidence— “We find the absence of an epidemic threshold and its associated critical behavior, which implies that SF networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents possess.”
transferfrom network science / statistical physics: modelling epidemic spreading on scale-free contact networks— “We define a dynamical model for the spreading of infections on scale-free networks”transfers[0]
transferfrom biological epidemiology: the susceptible-infected-susceptible model and the notion of an epidemic threshold, applied to computer virus outbreaks on the Internet— “The epidemiological analysis of computer viruses has been the subject of a continuous interest in the computer science community [10,13–15], following mainly approaches borrowed from biological epidemiology [11].”transfers[1]
mechanismthe scale-free network diverging connectivity fluctuations remove the epidemic threshold, so infections persist for any positive transmission rate— “finding the absence of an epidemic threshold and its associated critical behavior”mechanisms[0]
mechanismdiverging connectivity fluctuations act as an infinite connectivity, and the threshold falls to zeromechanisms[1]
mechanisma link is more likely to land on a highly connected node, so hubs are infected first and keep re-infecting the restmechanisms[2]
mechanisme-mail attachments make the contact graph the Internet's topology rather than physical proximity
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mechanisms[3]
builds-on scale-free network model
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claim_relations[0]
contradicts the classical nonzero-epidemic-threshold prediction
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claim_relations[1]
contradicts the standard nonzero epidemic threshold of mathematical epidemiology [11]
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claim_relations[2]
builds-on the scale-free graph model of Barabasi and Albert [6]
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claim_relations[3]
contradicts standard epidemic modelling on lattices and mean-field models, which predicts a nonzero threshold
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claim_relations[4]
extends the observational literature on computer virus prevalence [10,13]
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claim_relations[5]
sensenetwork - social, biological and communication systems alike, whose nodes are individuals or organizations and whose links are the interactions among them
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senses[0]
sensenetwork - the Internet as the contact graph along which an e-mail-borne infection actually travels, as opposed to physical proximity
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senses[1]
senseepidemic - a computer virus outbreak, analysed with the models of biological epidemiology
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senses[2]
controltesting the model's prediction against independent real-world prevalence data reported by the Virus Bulletin, February 1996 to March 2000 — held
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— rules out 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
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controls[0]
differs instandard epidemic modelling on lattices and mean-field models, which predicts a nonzero threshold — THE AXIS IS THE CONNECTIVITY REGIME. 'In models with local connectivity (Euclidean lattices and mean-field models), the most significant result is the general prediction of a nonzero epidemic threshold.' 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.
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claim_relations[4]
auditscale-free networks —contrasted-with→ random networks — 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).
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concept_edges[2]
auditcontact process —is-a→ susceptible-infected-susceptible — 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.
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concept_edges[3]
weightcontact process —is-a→ susceptible-infected-susceptible — 0.9
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concept_edges[3]
effect directionEVIDENCE CORRECTED (RE-GROUNDED) · scale-free networks —causes→ epidemic threshold — eliminates
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concept_edges[4]
retractedEVIDENCE CORRECTED (RE-GROUNDED) · scale-free networks —causes→ epidemic threshold — 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.
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concept_edges[4]
outcomecontrol — held
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controls[0]
for which uselimitation — every result of the SIS model: INDIVIDUALS EXIST ONLY IN TWO DISCRETE STATES
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limitations[0]
for which uselimitation — 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
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limitations[1]
for which uselimitation — 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
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limitations[2]
for which uselimitation — 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
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limitations[3]
replicationuntested
no type for thisEuclidean 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.
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unhoused_relations[0]
limitationthe SIS model reduces individuals to two discrete states (healthy/infected)
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limitations[0]
limitationThe model is run at one exponent, γ = 3, and the generalisation to other exponents is asserted qualitatively rather than shown.
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limitations[1]
limitationIndividuals are reduced to two states, healthy or infected, which the paper flags as a coarse description.
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limitations[2]
limitationThe empirical half rests on one industry data source over a four-year window.
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limitations[3]
Content tags
content-observations: finding · quantity · variable · definition · key-term — grounded in the full text
findingan epidemic threshold and its associated critical behavior are absent
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content_tags.findings[0]
findingReal computer-virus data are analysed to give the average lifetime and prevalence of viral strains on the Internet.
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content_tags.findings[1]
findingOn a scale-free network the epidemic threshold is zero: any spreading rate, however small, lets the infection persist.
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content_tags.findings[2]
findingSteady-state virus prevalence decays with decreasing spreading rate as ρ ~ exp(−C/λ).
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content_tags.findings[3]
findingFitting the data gives a lifetime of about 14 months for boot and macro viruses and 6-9 months for file viruses.
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content_tags.findings[4]
findingThe higher a node's connectivity, the higher its probability of being infected.
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content_tags.findings[5]
quantityγ between 2 and 3 — power-law degree exponent of the scale-free networks (Internet/WWW)
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content_tags.quantities[0]
quantity0 (no threshold) — the epidemic threshold vanishes for scale-free networks
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content_tags.quantities[1]
quantityτ ≃ 14 months — characteristic lifetime of boot and macro virus strains
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content_tags.quantities[2]
quantityλc = 0 — the epidemic threshold on a scale-free network
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content_tags.quantities[3]
quantityN = 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)
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content_tags.quantities[4]
evidence status notequantity — 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.content_tags.quantities[4]
quantityγ ≃ 2.5 — measured scale-free exponent of the Internet and the world-wide-web
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content_tags.quantities[5]
quantity100 starting configurations x 10 network realisations — averaging used for every prevalence figure
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content_tags.quantities[6]
variablespreading rate λ (independent)
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content_tags.variables[0]
variableeffective spreading rate λ (independent)
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content_tags.variables[2]
variablevirus prevalence (dependent)
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content_tags.variables[3]
definitionepidemic threshold λc
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content_tags.definitions[0]
definitionscale-free network
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content_tags.definitions[1]
definitionsusceptible-infected-susceptible
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content_tags.definitions[2]
definitioneffective spreading rate λ
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content_tags.definitions[3]
definitionepidemic threshold
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content_tags.definitions[4]
key-termsepidemic, network, scale-free, spreading, threshold, virus, power-law, infection, connectivity, scale-free networks, epidemic threshold, node, link, computer viruses, prevalence, susceptible-infected-susceptible, Internet, Euclidean lattice, mean-field model, surviving probability, contact process, random networks, world-wide-web
Concepts
terms this paper carries, each with the sentence it was cut from — the layer that joins one paper to another. Terms with no quote yet are on CONCEPTS.html.
conceptepidemic threshold
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concepts[1]
conceptscale-free networks — not defined in this paper's own words; a network whose connectivity follows a power law with exponent between 2 and 3, so that connectivity fluctuations diverge
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concepts[5]
conceptnode — not defined in this paper's own words; an element of the network; in this paper an individual computer, which is either healthy or infected
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concepts[6]
conceptindividual — not defined in this paper's own words; the epidemiological name for what graph theory calls a node; in this paper, one computer
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concepts[7]
conceptlink — not defined in this paper's own words; a connection between two nodes, along which the infection can travel
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concepts[8]
conceptsusceptible-infected-susceptible — not defined in this paper's own words; the epidemic model in which a node is infected at rate ν by infected neighbours and cured back to susceptible at rate δ
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concepts[9]
conceptcontact process — not defined in this paper's own words; the absorbing-state model of which SIS is a generalisation
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concepts[10]
conceptEuclidean lattice — not defined in this paper's own words; a regular local topology in which every node has the same small number of neighbours - the standard setting of classical epidemic modelling
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concepts[11]
conceptrandom networks — not defined in this paper's own words; a network in which every node has approximately the average number of links
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concepts[12]
conceptInternet — not defined in this paper's own words; the communication network through which electronic mail travels, and therefore the real contact graph for e-mail-borne viruses
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concepts[13]
conceptsurviving probability — not defined in this paper's own words; the fraction of virus strains still alive at time t after their birth - the measure by which strain lifetime is estimated
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concepts[14]
conceptprevalence — not defined in this paper's own words; the steady-state density of infected nodes in the network
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concepts[15]
Statistics — cross-field join keys
field-invariant statistical terms, used in their statistical sense (gated on a number / symbol / test). A p-value or a standard deviation means the SAME thing in physics, biology, psychology and sociology — so these are the strongest cross-field join keys. Grounded verbatim.
statistical-significance as written: statistically significant
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content_tags.statistics[0]
power-law P(k) ~ k^-γ
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content_tags.statistics[1]
mean <k>
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content_tags.statistics[2]
exponential-distribution exp(-t/τ)
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content_tags.statistics[3]
Second readings
a facet read a second time, independently, from its own sentence — and whether the two readings agree
claim typemechanism — a second reading reaching the same value; recorded: mechanism
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alternatives.claim_type
methodsimulation — a second reading reaching the same value; recorded: simulation
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alternatives.method
phenomenonabsence 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
Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher.
alternatives.phenomenon

When we have changed our mind. Some entries in the table begin with a word in capitals and the reason for it. EVIDENCE CORRECTED — the finding still stands, but we had quoted the wrong sentence from the paper, and the quotation has been replaced with the right one. RETRACTED — the entry itself was wrong; it stays on the page, marked, with the reason, rather than being deleted. WITHDRAWN AND REPLACED — the entry was wrong, and a corrected version of it appears in the entries below. Each mark carries its own reason, in brackets, because the reasons differ from one entry to the next. An entry we have WITHDRAWN is removed from this page and kept in the record, with its reason. A line reading Audit and a date is a later re-reading of the entry against its whole sentence: DOUBTFUL means it was kept, and the reason says what is loose about it. Quotation marks inside the tagger's sentence. Where a tag is the tagger's own sentence, a phrase in quotation marks inside it is borrowed from the paper — the tagger's wording around it is not. The paper's own sentence is the evidence quote for that row; on this page it is withheld with the article, and the article is at its publisher. Nothing inside quotation marks anywhere on this page is ours: if it is quoted, it is the paper's. Whose words a concept carries. A concept entry records a term the paper itself uses, so the term, the sense given for it, and its evidence are the paper's own words, quoted where the paper fixes them. This is the opposite of a sense entry in the tag table, which is the tagger's sentence about how the paper uses a term. Where a concept entry could not be grounded in the paper's words, the entry says so on its own row.

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