Robust Distributed Routing in Dynamical Networks–Part II: Strong Resilience, Equilibrium Selection and Cascaded Failures

Strong resilience properties of dynamical networks are analyzed for distributed routing policies. The latter are characterized by the property that the way the outflow at a non-destination node gets split among its outgoing links is allowed to depend only on local information about the current particle densities on the outgoing links. The strong resilience of the network is defined as the infimum sum of link-wise flow capacity reductions making the asymptotic total inflow to the destination node strictly less than the total outflow at the origin. A class of distributed routing policies that are responsive to local information is shown to yield the maximum possible strong resilience under such local information constraints for an acyclic dynamical network with a single origin-destination pair. The maximal achievable strong resilience is shown to be equal to the minimum node residual capacity of the network. The latter depends on the limit flow of the unperturbed network and is defined as the minimum, among all the non-destination nodes, of the sum, over all the links outgoing from the node, of the differences between the maximum flow capacity and the limit flow of the unperturbed network. We propose a simple convex optimization problem to solve for equilibrium flows of the unperturbed network that minimize average delay subject to strong resilience guarantees, and discuss the use of tolls to induce such an equilibrium flow in traffic networks. Finally, we present illustrative simulations to discuss the connection between cascaded failures and the resilience properties of the network.

[1]  Leandros Tassiulas,et al.  Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks , 1992 .

[2]  Dirk Helbing,et al.  Transient dynamics increasing network vulnerability to cascading failures. , 2007, Physical review letters.

[3]  Tim Roughgarden,et al.  Selfish routing and the price of anarchy , 2005 .

[4]  Massimo Marchiori,et al.  Model for cascading failures in complex networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  W. Marsden.  I and J , 2012 .

[6]  Munther A. Dahleh,et al.  Stability analysis of transportation networks with multiscale driver decisions , 2011, Proceedings of the 2011 American Control Conference.

[7]  Masakazu Sengoku,et al.  On a function for the vulnerability of a directed flow network , 1988, Networks.

[8]  Munther A. Dahleh,et al.  Robust distributed routing in dynamical flow networks , 2011, IEEE Conference on Decision and Control and European Control Conference.

[9]  Munther A. Dahleh,et al.  On Robustness Analysis of Large-scale Transportation Networks , 2010 .

[10]  Mauro Garavello,et al.  Traffic Flow on Networks , 2006 .

[11]  Vivek S. Borkar,et al.  Dynamic Cesaro-Wardrop equilibration in networks , 2003, IEEE Trans. Autom. Control..

[12]  Xin-She Yang,et al.  Introduction to Algorithms , 2021, Nature-Inspired Optimization Algorithms.

[13]  Munther A. Dahleh,et al.  Robust Distributed Routing in Dynamical Networks - Part I: Locally Responsive Policies and Weak Resilience , 2013, IEEE Trans. Autom. Control..

[14]  C. B. Mcguire,et al.  Studies in the Economics of Transportation , 1958 .

[15]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[16]  J. G. Wardrop,et al.  Some Theoretical Aspects of Road Traffic Research , 1952 .

[17]  Pradeep Dubey,et al.  Inefficiency of Nash Equilibria , 1986, Math. Oper. Res..

[18]  Adilson E Motter,et al.  Cascade-based attacks on complex networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Fernando Paganini,et al.  Internet congestion control , 2002 .

[20]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[21]  T. Koopmans,et al.  Studies in the Economics of Transportation. , 1956 .

[22]  Thomas H. Cormen,et al.  Introduction to algorithms [2nd ed.] , 2001 .

[23]  Michael Patriksson,et al.  The Traffic Assignment Problem: Models and Methods , 2015 .

[24]  Munther A. Dahleh,et al.  Robust distributed routing in dynamical networks with cascading failures , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[25]  J. Wardrop.  ROAD PAPER. SOME THEORETICAL ASPECTS OF ROAD TRAFFIC RESEARCH. , 1952 .