Cascading Edge Failures: A Dynamic Network Process

This paper studies a network process that can be used to model cascading failures in networks. The Dynamic Bond Percolation (DBP) process models, through stochastic local rules, the failure or recovery of an edge/link <inline-formula><tex-math notation="LaTeX">$(i,j)$</tex-math><alternatives> <inline-graphic xlink:href="zhang-ieq1-2764921.gif"/></alternatives></inline-formula> in a network. The probability that a working link fails or a failed link recovers may be independent of the state of other links <italic>or</italic> may be dependent locally on the state of neighboring links as described by a cascade function <inline-formula> <tex-math notation="LaTeX">$f$</tex-math><alternatives><inline-graphic xlink:href="zhang-ieq2-2764921.gif"/> </alternatives></inline-formula>. In applications, this means that failures or recovery of links may have a regional preference, or, alternatively, relationships between neighbors in the network can lead to changes in the links between neighbors of neighbors. This paper shows that the dynamic evolution of <inline-formula><tex-math notation="LaTeX"> $P(\mathbf{A},t)$</tex-math><alternatives><inline-graphic xlink:href="zhang-ieq3-2764921.gif"/></alternatives> </inline-formula>, the probability that the network is in some state <inline-formula><tex-math notation="LaTeX"> $\mathbf{A}$</tex-math><alternatives><inline-graphic xlink:href="zhang-ieq4-2764921.gif"/></alternatives> </inline-formula>, describing the collective states of all the links at time <inline-formula><tex-math notation="LaTeX"> $t$</tex-math><alternatives><inline-graphic xlink:href="zhang-ieq5-2764921.gif"/></alternatives></inline-formula>, converges to a stationary distribution. We use this distribution to study the emergence of global behaviors like consensus (i.e., catastrophic failure or full recovery of all the edges) or mixed (i.e., some failed and some working substructures). In particular, we show that, depending on the local dynamical rule, different network substructures, such as hub or triangle subgraphs, are more prone to failure.

[1]  Alain Billionnet,et al.  Maximizing a supermodular pseudoboolean function: A polynomial algorithm for supermodular cubic functions , 1985, Discret. Appl. Math..

[2]  Garry Robins,et al.  An introduction to exponential random graph (p*) models for social networks , 2007, Soc. Networks.

[3]  D. West.  Introduction to Graph Theory , 1995 .

[4]  Alessandro Vespignani,et al.  Dynamical Processes on Complex Networks , 2008 .

[5]  D. Vere-Jones.  Markov Chains , 1972, Nature.

[6]  Sandip Roy,et al.  Network models: growth, dynamics, and failure , 2001, Proceedings of the 34th Annual Hawaii International Conference on System Sciences.

[7]  I. Dobson,et al.  A LOADING-DEPENDENT MODEL OF PROBABILISTIC CASCADING FAILURE , 2005, Probability in the Engineering and Informational Sciences.

[8]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.

[9]  Mahdi Jalili,et al.  Cascading Failure Tolerance of Modular Small-World Networks , 2011, IEEE Transactions on Circuits and Systems II: Express Briefs.

[10]  Roger Th. A. J. Leenders,et al.  MODELS FOR NETWORK DYNAMICS - A MARKOVIAN FRAMEWORK , 1995 .

[11]  S. Borgatti,et al.  Social Networks of Drug Users in High-Risk Sites: Finding the Connections , 2002, AIDS and Behavior.

[12]  Edmund M. Yeh,et al.  Resilience to Degree-Dependent and Cascading Node Failures in Random Geometric Networks , 2011, IEEE Transactions on Information Theory.

[13]  Ulrik Brandes,et al.  Studying Social Networks - A Guide to Empirical Research , 2013 .

[14]  T. Liggett,et al.  Stochastic Interacting Systems: Contact, Voter and Exclusion Processes , 1999 .

[15]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[16]  P. Moran,et al.  Reversibility and Stochastic Networks , 1980 .

[17]  S. Wasserman.  Analyzing Social Networks as Stochastic Processes , 1980 .

[18]  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.

[19]  Marco Aiello,et al.  The Power Grid as a Complex Network: a Survey , 2011, ArXiv.

[20]  D S Callaway,et al.  Network robustness and fragility: percolation on random graphs. , 2000, Physical review letters.

[21]  José M. F. Moura,et al.  Role of subgraphs in epidemics over finite-size networks under the scaled SIS process , 2014, J. Complex Networks.

[22]  Ian Dobson,et al.  A branching process approximation to cascading load-dependent system failure , 2004, 37th Annual Hawaii International Conference on System Sciences, 2004. Proceedings of the.

[23]  José M. F. Moura,et al.  Diffusion in Social Networks as SIS Epidemics: Beyond Full Mixing and Complete Graphs , 2014, IEEE Journal of Selected Topics in Signal Processing.

[24]  Mark S. Granovetter.  The Strength of Weak Ties , 1973, American Journal of Sociology.

[25]  Matt J Keeling,et al.  Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[26]  Silvio Micali,et al.  An O(v|v| c |E|) algoithm for finding maximum matching in general graphs , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

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

[28]  T. Snijders.  Models for longitudinal network datain , 2005 .

[29]  Edmund M. Yeh,et al.  Cascading Link Failure in the Power Grid: A Percolation-Based Analysis , 2011, 2011 IEEE International Conference on Communications Workshops (ICC).

[30]  Endre Boros,et al.  Pseudo-Boolean optimization , 2002, Discret. Appl. Math..

[31]  Nicholas J. A. Harvey.  Algebraic Algorithms for Matching and Matroid Problems , 2009, SIAM J. Comput..

[32]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[33]  J. Steif.  A survey on dynamical percolation , 2009, 0901.4760.

[34]  D. L. Pepyne,et al.  Topology and cascading line outages in power grids , 2007 .