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Degree Distribution, and Scale-Free Networks Prof. Ralucca Gera

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Degree Distribution, and Scale-Free Networks Prof. Ralucca Gera Degree Distribution, and Scale-free networks Prof. Ralucca Gera, Applied Mathematics Dept. Naval Postgraduate School Monterey, California [email protected] Excellence Through Knowledge Learning Outcomes Understand the degree distributions of observed networks Understand Scale Free networks Power law degree distribution Versus exponential, log-normal, … 2 Degree Distribution Excellence Through Knowledge Degree distributions • The degree distribution is a frequency distribution of the degree sequence • We define pk to be the fraction of vertices in a network that have degree k • That is the same as saying: pk is the probability that a randomly selected node of the network will have degree k 1 2 4 2 1 p , p , p , p , p , p 0k 5 0 10 1 10 2 10 3 10 4 10 k • A well connected vertex is called a hub 4 Plot of the degree distribution The Internet Commonly seen plots of real networks Right-skewed • Many nodes with small degrees, few with extremely high – Largest degree is 2407 (not shown). Since 19956 this node is adjacent to 12% of the network 5 Degree distribution in directed networks • For directed networks we have both in- and out-degree distributions. The Web 6 Observed networks are SF (power law) • Power laws appear frequently in sciences: • Pareto : income distribution, 1897 • Zipf-Auerbach: city sizes, 1913/1940’s • Zipf-Estouf: word frequency, 1916/1940’s • Lotka: bibliometrics, 1926 • Yule: species and genera, 1924. • Mandelbrot: economics/information theory, 1950’s+ • Research claims that observed real networks are scale free, not consistent defn., [ref. 1–7], i.e. – The fraction of nodes of degree follows a power law distribution , where 1, – Generally being observed ∈ 2, 3 , see [ref. 8, 9] – Or at least the upper tail of the distribution is power law) [ref. 10] – Or it is more plausible than other thin-tailed distributions [ref. 11] Broido and Clauset “Scale-free networks are rare”, Jan 2018, https://arxiv.org/abs/1801.03400 [ref. 13] https://www.eecs.harvard.edu/~michaelm/TALKS/Radcliffe.ppt Power law distributions • Power law distribution: (Infinite mean and variance possible) • Identifying power-law from non power-law distributions is not trivial – Simplest (but not very accurate) strategy visual inspection plots – In 2006 - fit a distribution over the observed data, and test the goodness of the fit. This analysis revealed that none of them fits the theoretical distribution • There are alternatives as we will see 8 Egypt IP-layer data And its degree distribution Notice the long tail. It looks like a power law distribution , 0 A bit hard to differentiate the count values can this visualization be improved to be useful? How? We’ll see in a few slides 10 Statistics for real networks α = power in the power law distribution 11 The powerlaw exponent • Claim: , 0 • Analysis of 1000 networks: observed networks have the power low exponents plotted on the right • Code at: https://github.com/adbroido/ SFAnalysis Broido and Clauset “Scale-free networks are rare”, Jan 2018, https://arxiv.org/abs/1801.03400 [ref. 13] 12 Exponent by network type The values of power law exponents for several types of networks 13 Eikmeier and Gleich, “Revisiting Power-law Distributions in Spectra of Real World Networks”, Aug 2017 https://dl.acm.org/citation.cfm?id=3098128 Binning Excellence Through Knowledge BA Example (n=1000) 15 Binning A binned degree histogram may give poor statistics at the tail of the distribution •As k gets large, in every bin there will be only a few samples large statistical fluctuations in the number of samples from bin to bin which makes the tail end noisy and difficult to decide if it 16 follows a power-law Detecting and visualizing power laws Alternatives: • We could use larger bins to reduce the noise at the tail since more samples fall in the same bin – this reduces the detail captured from the histogram since we end up with fewer bins 17 Logarithmic Binning Excellence Through Knowledge Visualizing power laws Alternatives: • Try to get the best of both worlds different bin sizes in different parts of the histogram – Careful at normalizing the bins correctly: a bin of width 10 will get 10 times more samples compared to the previous bin of width 1 divide the count number by 10 logarithmic view (log-log plots) 19 Logarithmic binning • Plot the degree distribution for the Internet in log-log scale: here the width of the bins is Ck fixed, the display is changed 2 100 0 1 degree k 1 10 M.E.J. Newman, “Power laws, Pareto distributions and Zipf’s law” Logarithmic binning In this scheme each bin is made wider than its predecessor by a constant factor a=10 –The1-st bin will cover the range: ≤ k< –The2-nd bin will cover the range: ≤ k< –The3-rd bin will cover the range: ≤ k< The most common choices for a are 2 and 10, since larger values of a give ranges that are too big, and smaller a values give non-integer ranges limits n-1 n 21 –Then-th bin will cover the range: a ≤ k < a Power Law and Scale Free Excellence Through Knowledge Power laws and scale free networks • Networks that follow power law degree distribution are referred to as scale-free networks • Scale-free network: means that the ration of very connected nodes to the number of nodes in the rest of the network remains constant as the network changes in size (supports research based on some sampling methods) • Real networks do not follow power law degree distribution over the whole range of degree k – That is the degree distribution is not monotonically decreasing over the whole range of degrees – Many times we refer to its tail only (high degrees) • Deviations from power law can appear for high values of k as well http://networksciencebook.com/chapter/4 Power laws and scale free networks • It is important to mention what property of the network is scale-free. – Generally it is the degree distribution, and we say that the “network is scale-free” which in reality says “the degree distribution is scale-free” – Sometimes the exponent of the degree distribution captures this. Why? The exponent is the slope of the line that fits the data on log-log scale. – This has been tested with random subnetworks of scale free models that indeed show scale free degree distribution (but not always the same exponent) 24 Power law for WWW data from 1999 The Degree Distribution of the WWW The incoming (a) and outgoing (b) degree distribution of the WWW (1999). The degree distribution is shown on log-log plot, in which a power law follows a straight line. The symbols correspond to the empirical data and the line corresponds to the power-law fit, with degree exponents γin= 2.1 and γout = 2.45. We also show as a green line the degree distribution predicted by a Poisson function with the 25 average degree 〈kin〉 = 〈kout〉 = 4.60 of the WWW sample. http://networksciencebook.com/chapter/4#power-laws Hubs are Large in Scale-free Networks The estimated degree of the largest node (natural cutoff) in scale-free and random networks with the same average degree 〈k〉= 3. For the scale-free network we chose = 2.5. For comparison, we also show the linear behavior, kmax ~ N − 1, expected for a complete network. Overall, hubs in a scale-free network are several orders of magnitude larger than the biggest node in a random network with the same N and 〈k〉 26 http://networksciencebook.com/chapter/4#hubs A video - scale free property 27 http://networksciencebook.com/chapter/4#introduction4 Scale free Conclusion: in a random network hubs are effectively forbidden, while in scale-free networks they are naturally present. General belief: Scale free networks grow because of preferential attachment • Moreover: preferential attachment scale free But preferential attachment scale free counterexamples: the configuration model, random geometric model 28 New research Excellence Through Knowledge New research (Jan 2018) shows: Example: Power law “exponential distribution, which exhibits a thin tail and relatively low variance, is favored over the power law (Power law is a specific instance of this power law with cuttoff) (36%) nearly as often as the power law is favored over the exponential (37%).” “over half of the data sets favor the cutoff model over the pure power-law model, which suggests that, to the degree that scale-free Broido and Clauset “Scale-free networks are rare”, Jan 2018, https://arxiv.org/abs/1801.03400 [ref. 13] networks are universal, finite-size effects in the extreme upper tail are quite common” Takeaways • Networks have heavy-tailed degree distributions: – scale-free networks are rare: 4% strong SF, and – 33% weak (as likely as other models) SF evidence • Weak correlation of SF across different types: – Stronger correlation shows that SF structures occur in biological and technological networks • “…there is likely no single universal mechanism, or even a handful of mechanisms, that can explain the wide diversity of degree structures found in real-world networks.” Broido and Clauset “Scale-free networks are rare”, Jan 2018, https://arxiv.org/abs/1801.03400 [ref. 13] 31 References [1] R. Albert, H. Jeong, and A. L. Barab´asi, Nature 401, 130 (1999). [2] N. Prˇzulj, Bioinformatics 23, 177 (2007). [3] G. Lima-Mendez and J. van Helden, Molecular BioSystems 5, 1482 (2009). [4] A. Mislove, M. Marcon, K. P. Gummadi, P. Druschel, and B. Bhattacharjee, in Proc. 7th ACM SIGCOMM Conference on Internet Measurement (IMC) (2007) pp. 29–42. [5] M. T. Agler, J. Ruhe, S. Kroll, C. Morhenn, S. T. Kim, D. Weigel, and E. M. Kemen, PLoS Biology 14, 1 (2016).
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