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OPTIMIZATION of FLOW and CASCADING EFFECTS in WEIGHTED COMPLEX NETWORKS

OPTIMIZATION of FLOW and CASCADING EFFECTS in WEIGHTED COMPLEX NETWORKS. Andrea Asztalos Sameet Sreenivasan , Boleslaw K. Szymanski, György Korniss Department of Computer Science Department of Physics April 9, 2011. Research funded by DTRA ARL NS CTA. NETWORKS.

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OPTIMIZATION of FLOW and CASCADING EFFECTS in WEIGHTED COMPLEX NETWORKS

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  1. OPTIMIZATION of FLOW and CASCADING EFFECTS in WEIGHTED COMPLEX NETWORKS Andrea Asztalos SameetSreenivasan, Boleslaw K. Szymanski, GyörgyKorniss Department of Computer Science Department of Physics April 9, 2011 Research funded by DTRA ARL NS CTA

  2. NETWORKS Complex Network – composed of many non-identical elements (nodes) connected by diverse interactions (links) Social Network: Friend Wheel on Facebook: network of social connections • Social networks • Technological networks • Biological networks • Information networks S. Wasserman and K. Faust, Social network analysis (1994) R. Albert and A.-L. Barabási, Rev. Mod. Phys. (2002) M. Newman, SIAM Review (2003) Boccaletti S. et. al., Phys. Reports (2006)

  3. Technological network:INTERNET domain2 domain1 router domain3 Router level map of Internet Nodes: IP addresses (computers) Links: physical lines

  4. NETWORK - GRAPH GRAPH - mathematical representation of network’s connectivity graph For simple, connected graphs: Nodes (vertices) Edges (links) Degree of a node - the number of links a node has Connectivity specified by: A - adjacency matrix Many network characteristics: local and global One relevant to this talk: degree (connectivity) distribution P(k)

  5. Poisson degree distribution Connect with probability p p=1/6 N=10k ~ 1.5 Erdős-Rényi RANDOM GRAPH (1960) P. Erdős A. Rényi Homogeneous graph

  6. HOMOGENEOUS versus HETEROGENEOUS networks Power-law deg. distribution Poisson deg. distribution Exponential Network Scale-free Network

  7. There is MORE to a network THAN its STRUCTURE Various dynamical processes on networks: transport, flow, traffic, searching, synchronization, epidemic spreading, etc. Internet data flow (2008) US electric power grid (www.anl.gov) French river network Source: JBA Consulting Worldwide airline routes (www.visualcomplexity.com)

  8. RESISTOR NETWORKS – fundamental model for distributed flows E. Lopez et. al., PRL (2005) I • Each link has a conductance (1/resistance): • The traffic is assumed to flow like an • electrical current • I unit current enters at source (s) node and leaves • at target (t) node • Interested in the currents along the edges I hub avoidance Observed in real-world networks Macdonald et al., EPL (2005), Barratet al., PNAS (2004) unweighted network Target hub preference - Control parameter power, packets, information, carbon, etc. Source

  9. CURRENT FLOW and LOAD Ohm’s law: I current enters at source (s) node and leaves at target (t) node Charge conservation (Kirchoff’s law) and Ohm’s law for node i: Rewritten in the form of a matrix equation: Obtain the voltages Graph Laplacian: FOR: I=1 • Current through link (i,j) • Current through node i

  10. CURRENT FLOW and LOAD • FOR: I=1, N source-target pairs • All nodes are simultaneously sources • One unit current enters at all sources per unit time • For each source node a target node is chosen randomly and uniformly • from among the rest N-1 nodes FOR: I=1, one source-target pair Node/Edge LOAD: given by the net current flowing through a node/edge due to N source-target pairs • Current through link (i,j) • Current through node i

  11. LOAD LANDSCAPES Conductance of an edge: Heterogeneous (scale-free) networks Load distribution on the nodes The maximum load is the lowest when The loads are balanced for an optimal G. Korniss et. al., “Optimizing Synchronization, Flow, and Robustness in Weighted Complex Networks”, in Handbook of Optimization in Complex Networks , edited by My T. Thai and  P. Pardalos (Springer, in press)

  12. MAXIMIZING TRANSPORT CAPACITY • Assume units of current entering the network • Each node has unit processing capability • Congestion-free network if : for all i nodes Transport capacity (network throughput) – the maximum unit of inserted current for which the network is congestion free: Effect of a uniform (unit) bandwidth

  13. NETWORK ROBUSTNESS TO CASCADING FAILURES • Node forwarding capacities are proportional • to the initial load • Node i fails when - tolerance parameter Motter, Lai PRE (2002) The removal (failure) or one or more node might trigger: Cascadeof failing nodes – subsequent removal of failed nodes, till none of the loads on the node exceed their respective capacities Question: After the failure of one or more nodes, how does the redistribution of loads affect the stability of the network? Question: Can the network be saved by varying and ?

  14. Severity of cascades for different control parameter and Load tolerance values N - original network size N’ - size of the largest network component after cascading failure Scale-free network Robustness and network throughput is optimal for

  15. SUMMARY • Simple model for distributed flows – traffic flows like an electrical current • Tunable weights along the edges coupled with the network’s structure • Unweighted scale-free networks have unbalanced load profile • The network throughput is maximum when the loads are balanced E. Katifori et. al. PRL (2010)

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