Temporal profiles of avalanches on networks

An avalanche or cascade occurs when one event causes one or more subsequent events, which in turn may cause further events in a chain reaction. Avalanching dynamics are studied in many disciplines, with a recent focus on average avalanche shapes, i.e., the temporal profiles of avalanches of fixed duration. At the critical point of the dynamics, the rescaled average avalanche shapes for different durations collapse onto a single universal curve. We apply Markov branching process theory to derive an equation governing the average avalanche shape for cascade dynamics on networks. Analysis of the equation at criticality demonstrates that nonsymmetric average avalanche shapes (as observed in some experiments) occur for certain combinations of dynamics and network topology. We give examples using numerical simulations of models for information spreading, neural dynamics, and behavior adoption and we propose simple experimental tests to quantify whether cascading systems are in the critical state.Cascade propagation models represent a range of processes on networks, such as power-grid blackouts and epidemic outbreaks. Here the authors investigate temporal profiles of avalanches and show how nonsymmetric average avalanche shapes can occur at criticality.

[1]  Jure Leskovec,et al.  Learning to Discover Social Circles in Ego Networks , 2012, NIPS.

[2]  S. Bornholdt,et al.  Avalanches in Self-Organized Critical Neural Networks: A Minimal Model for the Neural SOC Universality Class , 2012, PloS one.

[3]  Esteban Moro Egido,et al.  Branching Dynamics of Viral Information Spreading , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Kristina Lerman,et al.  Information is not a Virus, and Other Consequences of Human Cognitive Limits , 2016, Future Internet.

[5]  Hosung Park,et al.  What is Twitter, a social network or a news media? , 2010, WWW '10.

[6]  James P. Sethna,et al.  Noise in disordered systems: The power spectrum and dynamic exponents in avalanche models , 2000 .

[7]  Piet Van Mieghem,et al.  Epidemic processes in complex networks , 2014, ArXiv.

[8]  Duncan J Watts,et al.  A simple model of global cascades on random networks , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[9]  Duncan J. Watts,et al.  The Structural Virality of Online Diffusion , 2015, Manag. Sci..

[10]  Charles S Peskin,et al.  Synchrony and Asynchrony in a Fully Stochastic Neural Network , 2008, Bulletin of mathematical biology.

[11]  M. A. Muñoz,et al.  Quasi-Neutral Theory of Epidemic Outbreaks , 2011, PloS one.

[12]  Esteban Moro,et al.  Impact of human activity patterns on the dynamics of information diffusion. , 2009, Physical review letters.

[13]  Mark Newman,et al.  Dynamical systems on networks , 2018, Oxford Scholarship Online.

[14]  P. Mucha,et al.  The unreasonable effectiveness of tree-based theory for networks with clustering. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Francesca Colaiori,et al.  Average shape of a fluctuation: universality in excursions of stochastic processes. , 2003, Physical review letters.

[16]  John M. Beggs,et al.  Universal critical dynamics in high resolution neuronal avalanche data. , 2012, Physical review letters.

[17]  Filippo Radicchi,et al.  Breaking of the site-bond percolation universality in networks , 2015, Nature Communications.

[18]  Stefano Zapperi,et al.  Power spectra of self-organized critical sandpiles , 2005 .

[19]  Francesca Colaiori,et al.  Signature of effective mass in crackling-noise asymmetry , 2005, cond-mat/0507607.

[20]  Éva Tardos,et al.  Maximizing the Spread of Influence through a Social Network , 2015, Theory Comput..

[21]  John M. Beggs,et al.  Being Critical of Criticality in the Brain , 2012, Front. Physio..

[22]  J. Sethna,et al.  Crackling noise , 2001, Nature.

[23]  E. Ott,et al.  Statistical properties of avalanches in networks. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Yamir Moreno,et al.  Effects of Network Structure, Competition and Memory Time on Social Spreading Phenomena , 2015, Physical Review. X.

[25]  James P. Gleeson,et al.  Competition-induced criticality in a model of meme popularity , 2013, Physical review letters.

[26]  F. Colaiori.  Exactly solvable model of avalanches dynamics for Barkhausen crackling noise , 2008, 0902.3173.

[27]  Thilo Gross,et al.  Self-organized criticality as a fundamental property of neural systems , 2014, Front. Syst. Neurosci..

[28]  I. Dobson,et al.  Estimating the Propagation and Extent of Cascading Line Outages From Utility Data With a Branching Process , 2012, IEEE Transactions on Power Systems.

[29]  James P Sethna,et al.  Universal pulse shape scaling function and exponents: critical test for avalanche models applied to Barkhausen noise. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  M. Porter,et al.  Critical Truths About Power Laws , 2012, Science.

[31]  R. Durrett.  Random Graph Dynamics: References , 2006 .

[32]  Mikko Alava,et al.  Branching Processes , 2009, Encyclopedia of Complexity and Systems Science.

[33]  B Kahng,et al.  Sandpile on scale-free networks. , 2003, Physical review letters.

[34]  Michael Breakspear,et al.  Scale-Free Bursting in Human Cortex following Hypoxia at Birth , 2014, The Journal of Neuroscience.

[35]  J. Sethna,et al.  Crackling noise : Complex systems , 2001 .

[36]  J. Spencer,et al.  Dynamics of Stochastic Neuronal Networks and the Connections to Random Graph Theory , 2010 .

[37]  John K. McSweeney,et al.  Single-seed cascades on clustered networks , 2015, Network Science.

[38]  Mark Newman,et al.  Networks: An Introduction , 2010 .

[39]  Z. Wang,et al.  The structure and dynamics of multilayer networks , 2014, Physics Reports.

[40]  Peter Nijkamp,et al.  Accessibility of Cities in the Digital Economy , 2004, cond-mat/0412004.

[41]  M. Macy,et al.  Complex Contagions and the Weakness of Long Ties1 , 2007, American Journal of Sociology.

[42]  Sergio Martinoia,et al.  Self-organized criticality in cortical assemblies occurs in concurrent scale-free and small-world networks , 2015, Scientific Reports.

[43]  Woodrow L. Shew,et al.  Information Capacity and Transmission Are Maximized in Balanced Cortical Networks with Neuronal Avalanches , 2010, The Journal of Neuroscience.

[44]  Mason A. Porter,et al.  Multilayer networks , 2013, J. Complex Networks.

[45]  M. Newman.  Power laws, Pareto distributions and Zipf's law , 2005 .

[46]  Rediet Abebe.  Can Cascades be Predicted? , 2014 .

[47]  Kristina Lerman,et al.  Social Contagion: An Empirical Study of Information Spread on Digg and Twitter Follower Graphs , 2012, ArXiv.

[48]  Jari Saramäki,et al.  Small But Slow World: How Network Topology and Burstiness Slow Down Spreading , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[49]  A. Vespignani,et al.  Competition among memes in a world with limited attention , 2012, Scientific Reports.

[50]  J. Sethna,et al.  Universality beyond power laws and the average avalanche shape , 2011 .

[51]  Kimmo Kaski,et al.  Analytically Solvable Model of Spreading Dynamics with Non-Poissonian Processes , 2013, 1309.0701.

[52]  Yamir Moreno,et al.  Cascading behaviour in complex socio-technical networks , 2013, J. Complex Networks.

[53]  Mason A. Porter,et al.  Dynamical Systems on Networks: A Tutorial , 2014, ArXiv.