Degree Distribution, and Scale-free networks. Ralucca Gera, Applied Mathematics Dept. Naval Postgraduate School Monterey, California

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1 Degree Distribution, and Scale-free networks Ralucca Gera, Applied Mathematics Dept. Naval Postgraduate School Monterey, California

2 Overview Degree distribution of observed networks Scale Free networks (Power law degree distribution) New Observations 2

3 Degree distributions The degree distribution is a frequency distribution of the degree sequence We define p k to be the fraction of vertices in a network that have degree k That is the same as saying: p k is the probability that a randomly selected node of the network will have degree k p , p , p , p , p , p k 0 k 5 A well connected vertex is called a hub 3

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 4 Largest degree is 2407 (not shown). Since 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 5

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, Mandelbrot: economics/information theory, 1950 s+ Research claims that observed real networks are scale free, n consistent defn., [ref. 1 7], i.e. That 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 just that it is more plausible than other thin-tailed distributions [ref Broido and Clauset Scale-free networks are rare, Jan 2018, [ref. 13] 6

7 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 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 7

8 Egypt IP-layer data

9 And its degree distribution Notice the long tail. It looks like a power law distribution, 0 Notice how hard it is to differentiate the count values can this visualization be improved to be useful? How? 9

10 BINNING 10

11 BA Example (n=1000) 11

12 Binning A binned degree histogram will 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 follows a 12 power-law

13 8.4.1 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 However, this reduces the detail captured from the histogram since we end up with fewer bins 13

14 LOGARITHMIC BINNING 14

15 8.4.1 Detecting and 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) 15

16 Logarithmic binning Plot the degree distribution for the Internet in log-log scale: here the width of the bins is fixed, the display is changed Ck degree k

17 Logarithmic binning In this scheme each bin is made wider than its predecessor by a constant factor a 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 Then-th bin will cover the range: a n-1 k < a n 17

18 Statistics for real networks α = power in the power law distribution 18

19 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 19 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

20 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) 20

21 Scale free General belief: Scale free networks grow because of preferential attachment However: preferential attachment scale free But preferential attachment scale free counterexample: the configuration model 21

22 New Research update 22

23 New research (Jan 2018) data: Applying state-of-the-art statistical methods to a diverse corpus of 927 real-world networks: estimate the best-fitting power-law model, test its statistical plausibility, and compare it via a likelihood ratio test to alternative non-scale-free distributions. Broido and Clauset Scale-free networks are rare, Jan 2018, [ref. 13] 23

24 Best fitting model 1. For each degree sequence, estimate the bestfitting power-law distribution model: the power-law holds above some minimum value, chosen before fitting the distribution Test use is Kolmogrov-Smirnov (KS) minimization approach (which actually chooses the optimal value of ) 2. Used goodness-of-fit to obtain a standard p-value: if p 0.1, then the degree sequence is scale free, while if p < 0.1, reject the scale-free hypothesis. failing to reject does not imply that the model is correct, only that it is a plausible data generating process. 24 Broido and Clauset Scale-free networks are rare, Jan 2018, [ref. 13]

25 Testing the goodness-of-fit Alternative heavy tailed distrib. (starting at Exponential (a thin tail and relatively low variance), ): stretched exponential or Weibull distribution (can produce both thin or heavy tails, and is a generalization of the exponential distribution), log-normal (a broad distribution that can exhibit very heavy tails, finite mean and variance), power-law with exponential cutoff in the upper tail (power law is a special case of this). 25 Broido and Clauset Scale-free networks are rare, Jan 2018, [ref. 13]

26 New research (Jan 2018) shows: Example: exponential distribution, which exhibits a thin tail and relatively low variance, is favored over the power law (36%) nearly as often as the power law is favored over the exponential (37%). (Power law is a specific instance of this power law with cuttoff) Broido and Clauset Scale-free networks are rare, Jan 2018, [ref. 13] 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 networks are universal, finite-size effects in the extreme upper tail are quite common

27 The powerlaw exponent Claim:, 0 Analysis of 1000 networks: observed networks have the power low exponents plotted on the right Code at: SFAnalysis Broido and Clauset Scale-free networks are rare, Jan 2018, [ref. 13] 27

28 Exponent by network type The values of power law exponents for several types of networks Eikmeier and Gleich, Revisiting Power-law Distributions in Spectra of Real World Networks, Aug

29 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, [ref. 13] 29

30 Takeaways If real-world networks are not universally or even typically scale-free, the status of a unifying theme in network science over nearly 20 years would be significantly diminished. Such an outcome would require a careful reassessment of the large literature that has grown up around the idea. Hence, an important direction of future work in network theory will be development and validation of novel mechanisms for generating more realistic degree structure in networks Broido and Clauset Scale-free networks are rare, Jan 2018, [ref. 13] 30

31 And then there are the monkeys Miller (psychologist, 1957) suggests following: monkeys type randomly at a keyboard. Hit each of n characters with probability p. Hit space bar with probability 1 - np > 0. A word is sequence of characters separated by a space. Resulting distribution of word frequencies follows a power law. 31

32 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 [5] M. T. Agler, J. Ruhe, S. Kroll, C. Morhenn, S. T. Kim, D. Weigel, and E. M. Kemen, PLoS Biology 14, 1 (2016). [6] G. Ichinose and H. Sayama, Artificial Life 23, 25 (2017). [7] L. Zhang, M. Small, and K. Judd, Physica A 433, 182 (2015). [8] S. N. Dorogovtsev and J. F. F. Mendes, Advances in Physics 51, 1079 (2002). [9] A. L. Barab asi and R. Albert, Science 286, 509 (1999). [10] W. Willinger, D. Alderson, and J. C. Doyle, Notices of the AMS 56, 586 (2009). [11] R. Albert, H. Jeong, and A.-L. Barab asi, Nature 406, 378 (2000). [12] N. Eikmeier, and D. Gleich. "Revisiting Power-law Distributions in Spectra of Real World Networks." Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. ACM, [13] A. Broido and A. Clauset, Scale-free networks are rare,

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