ABSTRACT
Real-world heavy-tailed networks are claimed to be scale-free, meaning that the degree distributions follow the classical power-law. But it is evident from a closer observation that there exists a clearly identifiable non-linear pattern in the entire degree distribution in a log-log scale. Thus, the classical power-law distribution is often inadequate to fit the large-scale complex network data sets. The presence of this non-linearity can also be linked to the recent debate on scarcity versus the universality of scale-free networks. The search, therefore, continues to develop probabilistic models that can efficiently capture the crucial aspect of heavy-tailed and long-tailed behavior of the entire degree distribution of real-world complex networks. This paper proposes a new variant of the popular Lomax distribution, termed as modified Lomax (MLM) distribution, which can efficiently fit the entire degree distribution of real-world networks. The newly introduced MLM distribution arises from a hierarchical family of Lomax distributions and belongs to the basin of attraction of Frechet distribution. Some interesting statistical properties of MLM including characteristics of the maximum likelihood estimates have been studied. Finally, the proposed MLM model is applied over several real-world complex networks to showcase its excellent performance in uncovering the patterns of these heavy-tailed networks.
- Ibrahim B Abdul-Moniem. 2012. Recurrence relations for moments of lower generalized order statistics from exponentiated Lomax distribution and its characterization. Journal of Mathematical and Computational Science 2, 4 (2012), 999–1011.Google Scholar
- M Ahsanullah. 1991. Record values of the Lomax distribution. Statistica Neerlandica 45, 1 (1991), 21–29.Google ScholarCross Ref
- SA Al-Awadhi and ME Ghitany. 2001. Statistical properties of Poisson-Lomax distribution and its application to repeated accidents data. Journal of Applied Statistical Science 10, 4 (2001), 365–372.Google Scholar
- Réka Albert and Albert-László Barabási. 2002. Statistical mechanics of complex networks. Reviews of modern physics 74, 1 (2002), 47.Google Scholar
- Réka Albert, Hawoong Jeong, and Albert-László Barabási. 1999. Diameter of the world-wide web. nature 401, 6749 (1999), 130–131.Google Scholar
- Réka Albert, Hawoong Jeong, and Albert-László Barabási. 2000. Error and attack tolerance of complex networks. nature 406, 6794 (2000), 378–382.Google Scholar
- Barry C Arnold. 2015. Pareto distributions. Chapman and Hall/CRC.Google Scholar
- Anthony Barnes Atkinson and Allan James Harrison. 1978. Distribution of personal wealth in Britain. Cambridge Univ Pr.Google Scholar
- N Balakrishnan and M Ahsanullah. 1994. Relations for single and product moments of record values from Lomax distribution. Sankhyā: The Indian Journal of Statistics, Series B (1994), 140–146.Google Scholar
- Albert-Laszlo Barabasi. 2005. The origin of bursts and heavy tails in human dynamics. Nature 435, 7039 (2005), 207–211.Google ScholarCross Ref
- Albert-László Barabási and Réka Albert. 1999. Emergence of scaling in random networks. science 286, 5439 (1999), 509–512.Google Scholar
- Anna D Broido and Aaron Clauset. 2019. Scale-free networks are rare. Nature communications 10, 1 (2019), 1–10.Google Scholar
- Maurice C Bryson. 1974. Heavy-tailed distributions: properties and tests. Technometrics 16, 1 (1974), 61–68.Google ScholarCross Ref
- Majid Chahkandi and Mojtaba Ganjali. 2009. On some lifetime distributions with decreasing failure rate. Computational Statistics & Data Analysis 53, 12 (2009), 4433–4440.Google ScholarDigital Library
- Swarup Chattopadhyay, Asit K Das, and Kuntal Ghosh. 2019. Finding patterns in the degree distribution of real-world complex networks: Going beyond power law. Pattern Analysis and Applications(2019), 1–20.Google Scholar
- Swarup Chattopadhyay, CA Murthy, and Sankar K Pal. 2014. Fitting truncated geometric distributions in large scale real world networks. Theoretical Computer Science 551 (2014), 22–38.Google ScholarDigital Library
- Aaron Childs, N Balakrishnan, and Mohamed Moshref. 2001. Order statistics from non-identical right-truncated Lomax random variables with applications. Statistical Papers 42, 2 (2001), 187–206.Google ScholarCross Ref
- Aaron Clauset, Cosma Rohilla Shalizi, and Mark EJ Newman. 2009. Power-law distributions in empirical data. SIAM review 51, 4 (2009), 661–703.Google ScholarDigital Library
- Gauss M Cordeiro, Edwin MM Ortega, and Božidar V Popović. 2015. The gamma-Lomax distribution. Journal of Statistical computation and Simulation 85, 2(2015), 305–319.Google ScholarCross Ref
- AH El-Bassiouny, NF Abdo, and HS Shahen. 2015. Exponential lomax distribution. International Journal of Computer Applications 121, 13(2015).Google ScholarCross Ref
- Paul Embrechts, Claudia Klüppelberg, and Thomas Mikosch. 2013. Modelling extremal events: for insurance and finance. Vol. 33. Springer Science & Business Media.Google Scholar
- Sergey Foss, Dmitry Korshunov, Stan Zachary, 2011. An introduction to heavy-tailed and subexponential distributions. Vol. 6. Springer.Google Scholar
- Amal S Hassan and Amani S Al-Ghamdi. 2009. Optimum step stress accelerated life testing for Lomax distribution. Journal of Applied Sciences Research 5, 12 (2009), 2153–2164.Google Scholar
- Amal S Hassan, Salwa M Assar, and A Shelbaia. 2016. Optimum step-stress accelerated life test plan for Lomax distribution with an adaptive type-II Progressive hybrid censoring. Journal of Advances in Mathematics and Computer Science (2016), 1–19.Google Scholar
- Petter Holme. 2019. Rare and everywhere: Perspectives on scale-free networks. Nature communications 10, 1 (2019), 1–3.Google Scholar
- James Holland Jones and Mark S Handcock. 2003. Sexual contacts and epidemic thresholds. Nature 423, 6940 (2003), 605–606.Google Scholar
- Claudia Klüppelberg. 1988. Subexponential distributions and integrated tails. Journal of Applied Probability 25, 1 (1988), 132–141.Google ScholarCross Ref
- Jure Leskovec and Andrej Krevl. 2014. SNAP Datasets: Stanford Large Network Dataset Collection. http://snap.stanford.edu/data.Google Scholar
- Fredrik Liljeros, Christofer R Edling, Luis A Nunes Amaral, H Eugene Stanley, and Yvonne Åberg. 2001. The web of human sexual contacts. Nature 411, 6840 (2001), 907–908.Google Scholar
- KS Lomax. 1954. Business failures: Another example of the analysis of failure data. J. Amer. Statist. Assoc. 49, 268 (1954), 847–852.Google ScholarCross Ref
- Lev Muchnik, Sen Pei, Lucas C Parra, Saulo DS Reis, José S Andrade Jr, Shlomo Havlin, and Hernán A Makse. 2013. Origins of power-law degree distribution in the heterogeneity of human activity in social networks. Scientific reports 3, 1 (2013), 1–8.Google Scholar
- Mark EJ Newman. 2001. The structure of scientific collaboration networks. Proceedings of the national academy of sciences 98, 2 (2001), 404–409.Google ScholarCross Ref
- Mark EJ Newman. 2003. The structure and function of complex networks. SIAM review 45, 2 (2003), 167–256.Google Scholar
- Mark EJ Newman. 2005. Power laws, Pareto distributions and Zipf’s law. Contemporary physics 46, 5 (2005), 323–351.Google Scholar
- Muhammad Rajab, Muhammad Aleem, Tahir Nawaz, and Muhammad Daniyal. 2013. On five parameter beta Lomax distribution. Journal of Statistics 20, 1 (2013).Google Scholar
- Ryan Rossi and Nesreen Ahmed. 2015. The network data repository with interactive graph analytics and visualization. In Twenty-Ninth AAAI Conference on Artificial Intelligence.Google ScholarCross Ref
- Alessandra Sala, Haitao Zheng, Ben Y Zhao, Sabrina Gaito, and Gian Paolo Rossi. 2010. Brief announcement: revisiting the power-law degree distribution for social graph analysis. In Proceedings of the 29th ACM SIGACT-SIGOPS symposium on Principles of distributed computing. 400–401.Google ScholarDigital Library
- Mukund Seshadri, Sridhar Machiraju, Ashwin Sridharan, Jean Bolot, Christos Faloutsos, and Jure Leskove. 2008. Mobile call graphs: beyond power-law and lognormal distributions. In Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining. 596–604.Google ScholarDigital Library
- Michael PH Stumpf and Mason A Porter. 2012. Critical truths about power laws. Science 335, 6069 (2012), 665–666.Google ScholarCross Ref
- MH Tahir, M Adnan Hussain, Gauss M Cordeiro, GG Hamedani, Muhammad Mansoor, and Muhammad Zubair. 2016. The Gumbel-Lomax distribution: properties and applications. Journal of Statistical Theory and Applications 15, 1(2016), 61–79.Google ScholarCross Ref
- Ivan Voitalov, Pim van der Hoorn, Remco van der Hofstad, and Dmitri Krioukov. 2019. Scale-free networks well done. Physical Review Research 1, 3 (2019), 033034.Google ScholarCross Ref
Recommendations
Asymptotic ruin probabilities in a discrete-time risk model with heavy-tailed random sums
LOPAL '18: Proceedings of the International Conference on Learning and Optimization Algorithms: Theory and ApplicationsIn this paper, we consider a discrete time insurance risk model with some large classes of heavy-tailed claim distributions. we show that the probabilities of the maxima of the partial sums asymptotically equal to the sum of the tail probabilities of ...
How heavy-tailed distributions affect simulation-generated time averages
For statistical inference based on telecommunications network simulation, we examine the effect of a heavy-tailed file-size distribution whose corresponding density follows an inverse power law with exponent α + 1, where the shape parameter α is ...
The Generalized Johnson Quantile-Parameterized Distribution System
Johnson quantile-parameterized distributions (J-QPDs) are parameterized by any symmetric percentile triplet (SPT) (e.g., the 10th–50th–90th) and support bounds. J-QPDs are smooth, highly flexible, and amenable to Monte Carlo simulation via inverse ...
Comments