Fluctuations in complex systems

1
Fluctuations in complex systems

• Zoltán Eisler
• János Kertész
• Budapest University of Technology and Economics
• Institute of Physics,
• Dept. of Theoretical Physics

2
What makes a system complex?

• Complex systems have many interacting elements
• Interaction strongly affects the behavior
• electrons (superconductivity)

3
What makes a system complex?

• Complex systems have many interacting elements
• Interaction strongly affects the behavior
• electrons
• Internet (data traffic, congestion)

4
What makes a system complex?

• Complex systems have many interacting elements
• Interaction strongly affects the behavior
• electrons
• Internet
• humans (Mexican waves)

5
What makes a system complex?

• Complex systems have many interacting elements
• Interaction strongly affects the behavior
• electrons
• Internet
• humans
• economy (rallies, panic)

6
What makes a system complex?

• Complex systems have many interacting elements
• Interaction strongly affects the behavior
• electrons ? good models, quantum mechanics
• Internet
• humans
• economy

? NO good models Let us analyze real data!
How?
7
A practical method

• Take a complex system with components i1N
• servers
• companies

8
A practical method

• Take a complex system with components i1N
• Monitor the activity of the components fi(t) 0
• servers ? data flow (bytes/minute)
• companies ? trading activity of their stocks
  (USD/min)

9
A practical method

• Take a complex system with components i1N
• Monitor the activity of the components fi(t) 0
• servers ? data flow (bytes/minute)
• companies ? trading activity of their stocks
  (USD/min)
• average activity
• standard deviation (fluctuations) of activity

10
A practical method

• What is the origin of this relation?
• What is the value of a?

11

• any server, n neighbors
• Independent data queries from each neighbor with
  probability p
• mean data traffic
• variance of data traffic

a1/2
12

• fluctuations not determined by internal factors
  (random data queries)
• external factor the total number of drivers
  varies strongly from day to day

a1
13
Stock market

• Internet for larger servers there are more
  independent queries but still the same average
  size
• Stock market for larger companies there are more
  trades and the average value/trade is also larger
• value/trade trades/min0.69

• fi(t) trading activity of stock i at time t
  (traded value in USD)

a0.72

• 5 second resolution at New York Stock Exchange,
  2000-2002, 2000 stocks

14
Take home message

• Empirical method based on a simple
  proportionality
• One can measure a for any system without
  assuming any model
• The value of a gives hints on possible driving
  mechanism

15
Thank you!

• 1 Menezes and Barabási, PRL 92, 28701 (2004)
• 2 Menezes and Barabási, PRL 93, 68701 (2004)
• 3 Z. E. et al. Multiscaling and
  non-universality in fluctuations of driven
  complex systems, EPL 69, 664-670 (2005)
• 4 Z. E., J. Kertész Inhomogeneous dynamics on
  complex networks, arXivcond-mat/0501391 (2005)
• 5 J. Kertész, Z. E. Proceedings of the Nikkei
  Econophysics III Symposium, arXivphysics/0503139
  (2005)
• 6 A.-L. Barabási, ELTE Stat. Phys. Day
  2004.04.07