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Scale-free networks Péter Kómár Statistical physics seminar 07/10/2008

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2 Elements of graph theory I. A graph consists: vertices edges Edges can be: directed/undirected weighted/non-weighted self loops multiple edges Non-regular graph

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3 Elements of graph theory II. Degree of a vertex: the number of edges going in and/or out Diameter of a graph: distance between the farthest vertices Density of a graph: sparse dense

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4 Networks around us I. Internet: routers cables WWW: HTML pages hyperlinks Social networks: people social relationship

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5 Networks around us II. Transportation systems: stations / routes routes / stations Nervous system: neurons axons and dendrites Biochemical pathways: chemical substances reactions

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6 Real networks Properties: Self-organized structure Evolution in time (growing and varying) Large number of vertices Moderate density Relatively small diameter (Small World phenomenon) Highly centralized subnetworks

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7 Random networks Measuring real networks: Relevant state-parameters Evolution in time Creating models: Analytical formulas Growing phenomenon Checking: ‘Raising’ random networks Measuring

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8 Scale-free property 1999. A.-L. Barabási, R. Albert measured the vertex degree distribution → power-law tail: movie actors: www: US power grid: A.-L. Barabási, R. Albert (1999) ‘Emergence of Scaling in Random Networks’, Science Vol. 286 actors www

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9 Small diameter 2000. A.-L. Barabási, R. Albert measured the diameter of a HTML graph 325 729 documents, 1 469 680 links found logarithmic dependence: ‘small world’ A.-L. Barabási, R. Albert, H. Jeong (2000) ‘Scale-free characteristics of random networks: the topology of the wold-wide web’, Physica A, Vol. 281 p. 69-77

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10 Erdős-Rényi graph (ER) Construction: N vertices probability of each edge: p ER Properties: p ER ≥ 1/N → → Asympt. connected degree distribution: Poisson (short tail) not centralized small diameter A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187 (1960. P. Erdős, A. Rényi) N=10 4 p ER = 6∙10 -4 10 -3 1.5 ∙ 10 -3

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11 ER graph example

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12 Small World graph (WS) Construction: N vertices in sequence 1 st and 2 nd neighbor edges rewiring probability: p WS Properties: p WS = 0 → clustered, 0 < p WS < 0.01 → clustered → small-world propery p WS = 1 → not clustered, A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187 (1998. D.J. Watts, S.H. Strogatz)

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13 WS graph example

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14 ER graph - WS graph WSER

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15 Barabási-Albert graph (BA) New aspects: Continuous growing Preferential attachment Construction: m 0 initial vertices in every step: +1 vertex with m edges P (edge to vertex i ) ~ degree of i A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187 m 0 = 3 m = 2

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16 Barabási-Albert graph II. Properties: Power-law distribution of degrees: Stationary scale-free state Very high clustering Small diameter A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187 1 = m 0 = m 3 5 7 N = 300 000

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17 BA graph example

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18 ER graph – BA graph

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19 Mean-field approximation I. Time dependence of k i (continuous): solution: probability of an edge to i th vertex time of occurrence of the i th vertex A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187 k i (t) t titititi ~ t 1/2

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20 Mean-field approximation II. Distribution of degrees: Distribution of t i : Probability density: A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187

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21 Without preferential attachment Uniform growth: Exponential degree distribution: A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187 p(k)p(k)p(k)p(k) k scale-free exponential

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22 Without growth Construction: Constant # of vertices + new edges with preferential attachment Properties: At early stages → power-law scaling After t ≈ N 2 steps → dense graph A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187 t = N 5N 40N N=10 000

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23 Conclusion Power-law = Growth + Pref. Attach. Varieties Non-linear attachment probability: → affects the power-law scaling Parallel adding of new edges → Continuously adding edges (eg. actors) → may result complete graph Continuous reconnecting (preferentially) → may result ripened state A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187

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24 Network research today A.-L. Barabási, R. Albert, ‘Statistical Mechanics of Complex Networks’, arXiv:cond-mat/0106096 v1 6 Jun 2001 Centrality Adjacency matrix Spectral density Attack tolerance

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Thank you for the attention!

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27 ER – WS – BA

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