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

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

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

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

5. 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
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.

6. 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

7. Network security __ dynamic breakdown due to non-local coupling
Traffic network congestion 1)
Bi  betweenness
M.E.J. Newman, Phys. Rev. E 64, (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

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

9. 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

10. 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)

11. 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.

12. Why networks are oscillating?
Internal randomness and external forces
M.A. de Menezes and A.-L. Barabasi,
PRL, 93, (2004), ibid, 92, (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

13. 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

14. 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

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

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

17. 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

18. 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

19. 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

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

21. 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

22. Strategy? Attack VS cost: a realistic question4 2 1 3
Attacking important node: large damage but high cost
Attacking non-important node: small damage but low cost
Strategy?

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

24. The security of power-gridB1 B2 ……
[1] A.E. Motter and Y.-C. Lai, Phys. Rev. E 66, (2002). [2] X.G. Wang, Y.-C. Lai, and C.-H. Lai, Phys. Rev. E 74, (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]

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

26. 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

27. The influences of the network parametersbo
<k> a g
Relationship

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

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

30. 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,
Prof. Xingang Wang,

31. Thank You