Coupling without Connections Xingang Wang Physics Department, Zhejiang University Physics Department, Shannxi Normal University April 24-26, 2013, Beijing

Coupling through connection A B Examples: synchronization, disease propagation, cascading failure, neural synapse, game theory, sandpile model, etc.

Coupling without connection ? Partial synchronization A B C Traffic detour (betweenness) B A

Oscillation in complex networks X. Wang, Y.-C. Lai, and C.-H. Lai, PRE, 74, 066104 (2006).

The largest blackout in U.S.A Cascading: Cleveland New England  upper Midwest  parts of Canada Within three minutes, starting at 3:06 p.m. Thursday, 21 power plants in the United States shut down, according to Genscape, which monitors power transmissions. (Also said all things happened within 10 seconds) Time: Thursday, August 14, 2003, 3:06 p.m. Effected areas: 50 million people on both sides of US and Canada (7 sates in US plus Ontario in Canada), more than 24 hours. within 10 seconds http://www.thecanadianencyclopedia.com) Reason: Outage probe looks to single power line, of 345,000 volt, in Ohio. Gent told CNN it appears that the first known problem was the loss of a power line in Cleveland, Ohio, at 3:06 p.m.

Network security __ dynamical breakdown due to load re-distribution Y. Moreno, J.B. Gomez and A.F. Pacheco, Europhys. Lett., 58, 630 (2002). 1 4 2 3 1) Node capacity fi =2 2) Set node load d (follow some distribution) 3) Breakdown node of d > fi 4) The load of the broken node distributed to its neighbors equally. 5) Process go on till reaching an stationary state Local coupling: only neighboring nodes are affected

Network security __ dynamic breakdown due to non-local coupling Traffic network congestion 1) Bi  betweenness M.E.J. Newman, Phys. Rev. E 64, 016132 (2001); ibid, PNAS, 98, 404 (2001). B1,2,4,5 = 0; B3 = 4 Ci  capacity Ci = Bi x a a  tolerance parameter 1 2 3 4 5 Bi < Ci : Steady flow Bi > Ci : Congestion B1+B2 2) If B1 > C1 3) If B1+B2 > C2

Cascade-based attacks__ the damage of dynamical breakdown Node capacity Lj  load (betweenness) a  tolerance parameter Dynamics make the damage much catastrophic and unpredictable

Recovery ? Oscillation ? Breakdown ? Former studies  nodes/links removal (attack, epidemic control, cascading, etc.) jamming 1) Function is temporally effected, recoverable e.g, traffic/information/technology networks Jamming  Queuing  Recovery 2) Network response to perturbations Recovery ? Oscillation ? Breakdown ? 3) Jamming Recovery time ? i.e., how long How large an area it effects ? i.e. how big Time evolution of network performance ? Perturbation size/position dependence ? How to reduce the damage once it happens ? Other issues 4) Jamming

World Wide Web Internet fi(t) = number of visits to website i at day t 3000 web sites. Daily visitation for a 30 day period Internet fi(t) = number of bytes passing through router i at time t. 347 routers tmax=2 days (5 min. resolution)

Highways Computer chip fi(t) =traffic at a given point of a road i at day t. Daily traffic on 127 Colorado roads from 1998 to 2001. Computer chip fi(t) =state of a given logic component i at clock cycle t. 462 signal carriers 8,862 clock cycles.

Why networks are oscillating? Internal randomness and external forces M.A. de Menezes and A.-L. Barabasi, PRL, 93, 068701 (2004), ibid, 92, 028701 (2004). Internal randomness  random selection of starting/ending point External forces  the number of connections are randomly time varying Model (Internet) W connections with random starting/ending point W  [w-Dw,w+Dw] Random activities  Oscillation Fixed the connections, W Fixed starting/ending points Oscillation ? Problem: Finding: Oscillation is an intrinsic property for complex networks

The model (e.g., traffic network) 1) Generating SFN (A. Barabasi and R. Albert, science 286, 509 (1999).) Define initial weight at each node Wj (0) = 1 The weight of links djk (t) = ( Wj (t) + Wk (t) ) / 2 Define node capacity Cj = ( 1 + a ) Lj(0) One message to be exchanged between any pair of nodes P = N(N-1)/2 Wj Wk djk Wj (t) = 1 + mj(t)/cj, if mj(t)> Cj 1, if mj(t) < Cj 2) Congest node I, add m packets Wi (0)= 1 + m/ci Recalculate the optimal paths Update the load Li and accumulations mi Update node weight 6. Go to step 3 and so on a > = 0, without perturbation: Steady flow state No congestion Weighted, evolutionary complex network

Simulations Interested characters System parameters Perturbations Large a = 0.4 Interested characters Average diameter <d(t)> Network betweenness <B(t)> The jammed nodes <n(t)> The jammed packets <P(t)> System parameters Size: N = 1000 <k> = 4 kmax = 27 Bmax = 2 x 106 <d> = 5.18 Perturbations Observations: Position: the most important node Size: 10 x Bmax Recoverable: congested  free-flow Perturbation  cascading  recovery Cascading time  diameter Recovery time  perturbation size

Small tolerance parameter a a = 0.3, period 2 a = 0.2, chaotic

Medium a a = 0.26, p4 a = 0.26, p4 a = 0.22, p8 a = 0.22, p8

DLk > a Lk (0) ao = DLk/ Lk(0) Three regimes: Find the oscillation condition Set wk = 2 at all nodes Change wk to 1 The necessary condition for oscillation DLk > a Lk (0) ao = DLk/ Lk(0) Then the critical value is: Three regimes: a > ao , free-flow a < ac , chaotic oscillation ac < a < ao, periodic oscillation Numerically: ao = 0.32; ac = 0.23 Findings: ao is an universal constant for SFN (independent of N and <k>) This method of estimation overlooks the details of weight distribution, thus ao is overestimated. ao* = 0.37 > ao

Simulate the real world Internet at AS level: N= 6494 Total links: 13895 <d> = 4.71 Kmax = 1460 Lmax = 2 x 108 Perturbation: 10 x Lmax Computing time: 3 weeks Few remarks: Independent to the network form Small perturbation can also induce network oscillation (also for lower k) The oscillation is permanent, t->104

Understanding network oscillation The reason for oscillation: Competition Cooperation Limited resource Network oscillation Unlimited resource: = infinity  no congestion  free-flow 1 2 3 4 5 Driven by efficiency (optimal path) The existence of loops (choices) 1  5 and 2  5 when 3 jammed Competing the resource in 4 1 wish 2 owns a small weight  cooperation

Protecting Infrastructure Networks from Cost-based Attacks X. Wang, S. Guan, and C.-H. Lai, NJP, 11, 033006 (2009).

Random failure VS intentional attack 1. Moore and Shannon, J. Franklin Inst. 262, 281 (1956) 2. Margulis, Probl. Inf. Transm. 10, 101 (1974) 3. Bollobas, Random Graphs (Academic, London) 1985 1 S Failures f’c Random network of exponential degree-distribution: Scale-free random network (SFN): Failures Attacks f fc 4. R.Cohen, et.al, PRL, 85, 4626 (2000) 5. R.Albert, et.al, Nature 406, 378 (2000) 6. H. Jeong, et.al, Nature 411, 41 (2001) SFN is robust to random failures but vulnerable to intentional attacks

Strategy? Attack VS cost: a realistic question 4 2 1 3 Attacking important node: large damage but high cost Attacking non-important node: small damage but low cost Strategy?

VS Size: L = 4941 Link: <k> = 2.67 Nd units Na units Na << Nd Flexible VS Global VS

The security of power-grid B1 B2 …… [1] A.E. Motter and Y.-C. Lai, Phys. Rev. E 66, 065102 (2002). [2] X.G. Wang, Y.-C. Lai, and C.-H. Lai, Phys. Rev. E 74, 066104 (2006). Load-based cascade model [1,2] Bi(0)  Load of node i Ci = (1+a) Bi (0)  Capacity of node i If Bi(t) > Ci  node failed Important node identification [1]

? Protection resources distribution Assume: while Overall network damage: The defense parameter : Distributed attack Concentrated attack The optimal defense ?

Simulations on BA model Network parameters: N=3000, <k>=4, a = 0.3 Concentrated attack: attack only the largest-degree node Distributed attack: attack a group of small-degree nodes Gc,d: the number of remaining nodes after CA or DA. Bc,d: the total network damage induced by CA or DA bo = 1.25 bo = 1.27

The influences of the network parameters bo <k> a g Relationship

Power grid of U.S.A The Internet bo(a=0.3)=1.35; bo(a=0.5)=1.42

Summary Oscillation in traffic networks Cost-based attacks of infrastructure networks Complex structure Loops Indirect coupling New phenomena

Positions @ SNNU (The center for complex systems) At all levels Research directions 1) Complex network dynamics; 2) Nonlinear dynamics and chaos theory; 3) Neural networks and system biology; 4) Pattern formation and non-equilibrium statistical physics. Contacting information: Prof. Shixian Qu, sxqu@snnu.edu.cn Prof. Xingang Wang, wangxg@snnu.edu.cn

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