Emergence of Scaling in Random Networks

Emergence of Scaling in Random Networks Albert-László Barabási Réka Albert

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Emergence of Scaling in Random Networks 1999 Science 10.1126/science.286.5439.509 cond-mat/9910332 barabasi-albert1999-scale-free barabasi-albert1999-scale-free.txt, the project's staged copy 49449de5d5ab9a4757260b5b634c4d6a6c45263d arxiv-open sociology

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0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them The connectivity distribution of large real networks - film-actor collaboration, the world wide web, the western US power grid and scientific citation - and what generates it. Quotation withheld because this record is not cleared for redistribution. Why does the connectivity of large real networks follow a power law, when the established random-graph models predict that highly connected vertices should be practically absent? Quotation withheld because this record is not cleared for redistribution. Identifies growth and preferential attachment as the two mechanisms that generate scale-free connectivity, and gives a model reproducing it. Quotation withheld because this record is not cleared for redistribution. mechanism Quotation withheld because this record is not cleared for redistribution. scale-free degree distribution in large networks Quotation withheld because this record is not cleared for redistribution. simulation simulation Quotation withheld because this record is not cleared for redistribution. mechanisms network growth with preferential attachment (new nodes attach preferentially to already-highly-connected nodes; a rich-gets-richer process) generates the scale-free power law Quotation withheld because this record is not cleared for redistribution. mechanisms growth - the network is open and gains vertices continuously, so N is not fixed Quotation withheld because this record is not cleared for redistribution. mechanisms preferential attachment - a new vertex attaches with probability proportional to a vertex's existing connectivity Quotation withheld because this record is not cleared for redistribution. mechanisms compounding: an early advantage in connectivity widens as the network grows, so old vertices become hubs Quotation withheld because this record is not cleared for redistribution. the model reproduces the scale-free degree distribution but cannot account for all aspects of real networks what this limits: every application of the model beyond the degree distribution: IT CANNOT BE EXPECTED TO ACCOUNT FOR ALL ASPECTS OF THE STUDIED NETWORKS Quotation withheld because this record is not cleared for redistribution. The model is not expected to capture everything about the networks it is fitted to. what this limits: the same scope as limitations[0] Quotation withheld because this record is not cleared for redistribution. Only a few real networks have detailed topological data available to test against. what this limits: the empirical support for the whole claim: DETAILED TOPOLOGICAL DATA IS AVAILABLE ONLY FOR A FEW of the systems that form complex networks Quotation withheld because this record is not cleared for redistribution. The model assumes strictly linear preferential attachment. what this limits: every result derived from the model: IN THE MODEL WE ASSUMED LINEAR PREFERENTIAL ATTACHMENT - an assumption, not a measurement Quotation withheld because this record is not cleared for redistribution. The single model exponent does not match the spread of exponents measured across the real networks. what this limits: the predicted exponent specifically: Quotation withheld, while the model gives a single value Quotation withheld because this record is not cleared for redistribution. The model is not offered as a full account of the systems it is fitted to. duplicate of: limitations[1] · what this limits: the same scope as limitations[0] Quotation withheld because this record is not cleared for redistribution. Attachment is assumed linear; scaling survives only at that exponent, which is a strong assumption rather than a result. what this limits: the scale-free result itself, which survives at one parameter value only: SIMULATIONS INDICATE THAT SCALING IS PRESENT ONLY FOR alpha = 1 Quotation withheld because this record is not cleared for redistribution. The model gives γ = 3 while the measured exponents run from 2.1 to 4, so the central prediction does not match any of the four datasets exactly. what this limits: the same scope as limitations[4] - the central prediction against the measured spread Quotation withheld because this record is not cleared for redistribution. The paper's dynamic prediction about how a vertex accumulates edges over time could not be tested with the data available. what this limits: the dynamic growth prediction, which is untested here: Quotation withheld Quotation withheld because this record is not cleared for redistribution. The power-law region for the power grid is weak, and the paper says so rather than fitting it anyway. what this limits: the power-grid case only: DUE TO THE RELATIVELY MODEST SIZE OF THE NETWORK, CONTAINING ONLY 4941 VERTICES, THE SCALING REGION IS LESS PROMINENT Quotation withheld because this record is not cleared for redistribution. the Erdos-Renyi random graph as the incumbent model of complex network topology, taken up and then shown to be the wrong model because real networks grow and attach preferentially Quotation withheld because this record is not cleared for redistribution. two VARIANTS of the model were built, each dropping one of the two ingredients eliminates: that only one of growth and preferential attachment is doing the work - an ablation, so the scale-free degree distribution cannot be attributed to the wrong mechanism · outcome (stated by the tagger, no quotation): held Quotation withheld because this record is not cleared for redistribution. preferential attachment preferential attachment Quotation withheld because this record is not cleared for redistribution. network network Quotation withheld because this record is not cleared for redistribution. scale-free scale-free Quotation withheld because this record is not cleared for redistribution. vertex vertex Quotation withheld because this record is not cleared for redistribution. random graph random graph Quotation withheld because this record is not cleared for redistribution. small world small world Quotation withheld because this record is not cleared for redistribution. genetic network genetic network Quotation withheld because this record is not cleared for redistribution. world wide web world wide web Quotation withheld because this record is not cleared for redistribution. six degrees of separation six degrees of separation Quotation withheld because this record is not cleared for redistribution. power-law power-law Quotation withheld because this record is not cleared for redistribution. small-world phenomenon small-world phenomenon Quotation withheld because this record is not cleared for redistribution. network - a genetic network, where the vertices are proteins and genes and the edges are chemical interactions Quotation withheld because this record is not cleared for redistribution. network - the nervous system, where the vertices are nerve cells and the edges are axons Quotation withheld because this record is not cleared for redistribution. network - a social network, where the vertices are individuals or organizations and the edges are social interactions Quotation withheld because this record is not cleared for redistribution. network - the world wide web, where the vertices are HTML documents and the edges are links between pages Quotation withheld because this record is not cleared for redistribution. network - a citation network, where the vertices are published papers and the edges are the references between them Quotation withheld because this record is not cleared for redistribution. network - a collaboration graph, where the vertices are film actors and two are joined if they appeared in the same film Quotation withheld because this record is not cleared for redistribution. method: computational-model a second reading. Recorded: simulation — 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. phenomenon: scale-free power-law distribution of vertex connectivity in large networks a second reading. Recorded: scale-free degree distribution in large 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. contradicts → existing random-network models Quotation withheld because this record is not cleared for redistribution. contradicts → the Erdos-Renyi random graph model and the Watts-Strogatz small-world model (refs 7 and 10) differs in: the functional form of P(k): THE PROBABILITY OF FINDING A HIGHLY CONNECTED VERTEX (THAT IS, A LARGE k) DECREASES EXPONENTIALLY WITH k in the ER and WS models, against the power-law tail measured here Quotation withheld because this record is not cleared for redistribution. builds-on → Redner (11), on the power-law distribution of citation counts Quotation withheld because this record is not cleared for redistribution. contradicts → the Erdős-Rényi random graph model (7) differs in: the functional form of P(k) - exponential decay in ER, so VERTICES WITH LARGE CONNECTIVITY ARE PRACTICALLY ABSENT, against a power law in which they dominate Quotation withheld because this record is not cleared for redistribution. builds-on → Redner's measurement of citation counts (11) Quotation withheld because this record is not cleared for redistribution. contradicts → the Watts-Strogatz small world model (10) differs in: exponential against power law: IN CONTRAST, THE POWER-LAW TAIL CHARACTERIZING P(k) FOR THE STUDIED NETWORKS INDICATES THAT HIGHLY CONNECTED (LARGE k) VERTICES HAVE A LARGE CHANCE OF OCCURRING, DOMINATING THE CONNECTIVITY Quotation withheld because this record is not cleared for redistribution. genetic network —is-a→ network genetic network network strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. world wide web —is-a→ network world wide web network strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. small world —same-as→ six degrees of separation small world six degrees of separation strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. preferential attachment —causes→ scale-free preferential attachment scale-free strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. scale-free —operationalised-as→ power-law scale-free power-law strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. vertex —part-of→ network vertex network strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. γ_www = 2.1 ± 0.1 — power-law exponent of the World Wide Web link distribution Quotation withheld because this record is not cleared for redistribution. γ_model = 2.9 ± 0.1 — power-law exponent produced by the preferential-attachment model Quotation withheld because this record is not cleared for redistribution. γ_cite = 3 — power-law exponent of the scientific citation network Quotation withheld because this record is not cleared for redistribution. γ_power ≃ 4 — approximate power-law exponent of the power-grid network Quotation withheld because this record is not cleared for redistribution. γactor = 2.3 ± 0.1 — power-law exponent of the film-actor collaboration graph Quotation withheld because this record is not cleared for redistribution. γwww = 2.1 ± 0.1 — power-law exponent of the world wide web Quotation withheld because this record is not cleared for redistribution. over 800 million vertices — size of the world wide web as the paper takes it Quotation withheld because this record is not cleared for redistribution. 4941 vertices — size of the western US power grid Quotation withheld because this record is not cleared for redistribution. γmodel = 2.9±0.1 — exponent produced by the simulated model Quotation withheld because this record is not cleared for redistribution. The probability P(k) that a vertex interacts with k other vertices decays as a power law, P(k) ~ k^-γ. Quotation withheld because this record is not cleared for redistribution. Scale-free structure emerges from two generic mechanisms: continuous growth by adding vertices, and preferential attachment to already well-connected sites. Quotation withheld because this record is not cleared for redistribution. These systems share a scale-free distribution of local connectivity, following a power law for large k with an exponent γ between 2.1 and 4. Quotation withheld because this record is not cleared for redistribution. Both ingredients, growth and preferential attachment, are needed for the stationary power-law distribution to develop. Quotation withheld because this record is not cleared for redistribution. The model's connectivity distribution P(k) is independent of time, and so of system size. Quotation withheld because this record is not cleared for redistribution. Solved analytically, the model gives an exponent of 3 that does not depend on how many edges each new vertex brings. Quotation withheld because this record is not cleared for redistribution. growth growth Quotation withheld because this record is not cleared for redistribution. preferential attachment preferential attachment Quotation withheld because this record is not cleared for redistribution. scale-free power-law distribution scale-free power-law distribution Quotation withheld because this record is not cleared for redistribution. preferential attachment preferential attachment Quotation withheld because this record is not cleared for redistribution. network network Quotation withheld because this record is not cleared for redistribution. preferential attachment preferential attachment Quotation withheld because this record is not cleared for redistribution. random graph model random graph model Quotation withheld because this record is not cleared for redistribution. small-world model small-world model Quotation withheld because this record is not cleared for redistribution. power-law power-law Quotation withheld because this record is not cleared for redistribution. poisson-distribution poisson-distribution Quotation withheld because this record is not cleared for redistribution. probability-density probability-density Quotation withheld because this record is not cleared for redistribution. exponential-distribution exponential-distribution Quotation withheld because this record is not cleared for redistribution.

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