Complex Systems IMSc

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Complex Systems _at_ IMSc
Sitabhra Sinha The Institute of Mathematical
Sciences (IMSc) Chennai, India
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• IMSc faculty
• Physics
• Maths
• Theoretical Computer Science

Complex Systems related research
Biological Physics Ramesh Anishetty Gautam
Menon Rahul Siddharthan Sitabhra Sinha
Networks Ramesh Anishetty Sudeshna Sinha C R
Subramanian Sitabhra Sinha
Nonlinear Dynamics Sudeshna Sinha
Econophysics Financial Analysis Agent based
models M Krishna Sitabhra Sinha
Statistical Mechanics Purusattam Ray R Rajesh
Game Theory R Ramanujam Kamal Lodaya
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The IMSc Collaboration graph
Physics Dept IIT Madras
Rahul Siddharthan
Sudeshna Sinha
R Ramanujam
Kamal Lodaya
Sitabhra Sinha
Gautam Menon
Purusattam Ray
M Krishna
C R Subramanian
Ramesh Anishetty
R Rajesh
Madras School of Economics
Inst of Financial Mgmt Research
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Promoting Complex Systems
Schools, Meetings and Workshops on topics related
to complex systems Dec 2002 Meeting on
Robustness, Emergent Behavior and Pattern
Formation in Biological Systems March 2003
Discussion Meeting on Statistical Mechanics of
Threshold Activated Systems April 2004
International Conference on Perspectives in
Nonlinear Dynamics Summer 2004 SERC School on
Disordered Systems Dec 2004 Workshop on the
Economy as a Complex System Jan 2006 IMSc
Complex Systems School (in cooperation with the
Santa Fe Institute)
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The IMSc Complex Systems School, Jan 2-27, 2006

• Duration 4 weeks
• 1st week general techniques
• 2nd week biological complexity
• 3rd week socioeconomic complexity
• 4th week student projects
• 31 students from physics, biosciences, economics
  and engineering.

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• Lecturers included
• V Balakrishnan (IIT Madras) Nonlinear Dynamics
• B K Chakrabarti (SINP Kolkata) Complexity in
  earthquake models asset exchange and income
  distribution
• C Dasgupta (IISc Bangalore) Spin Glasses
  Neural Network models
• S Jain (Delhi U) Networks
• J Lobo (Arizona State U, Tempe) Scaling in
  Urban Systems
• G I Menon (IMSc) Statistical Mechanics
• J-P Onnela (HUT, Helsinki) Socioeconomic
  networks
• R Ramanujam (IMSc) Game Theory
• R Ramaswamy (JNU, Delhi) R Siddharthan (IMSc)
  Bioinformatics
• K VS Rao (ICGEB, Delhi) Immune system
• Somdatta Sinha (CCMB, Hyderabad) Biological
  networks
• Sudeshna Sinha (IMSc) Spatiotemporal chaos
• V Savage (SFI) Scaling in ecology
• E Smith (SFI) Biochemical networks
• G West (SFI) Scaling in biology and society

• Student Projects on
• Agent-based models of financial bubbles and
  crashes
• Statistical distribution analysis of Indian
  elections
• Symbolic dynamics of classical Indian music
• Clustering of economic agents
• Infection spreading dynamics of malaria
• Intracellular signaling network modeling
• etc.

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Computational Resources

• IMSc is home of KABRU, 128-node supercomputing
  cluster till last year fastest Indian computer
  in an academic institution
• Originally built exclusively for Lattice Gauge
  Theory calculations now to be open to complex
  systems related research
• In the process of obtaining 16 processor Cray for
  biocomputing group

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Existing European Collaboration in Complex Systems
Indo-French projects 1 existing (in cardiac
dynamics, collaborators Alain Pumir SS) 1
applied for (in soft matter, IMSc collaborator G
I Menon) In addition inter-personal
collaborations with scientists in Germany,
Finland, Norway, Hungary,
3-dimensional Panfilov Model 128x128x128 with
64x64x100 obstacle snapsots every 1000 itrns
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Complex Systems RelatedResearch Snapshots

• Networks
• Econophysics

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Are Complex Networks Unstable ?

• Sitabhra Sinha

In collaboration with Raj Kumar Pan
(IMSc) Sudeshna Sinha (IMSc) Chris Wilmers
(Berkeley) Markus Brede (Leipzig, Canberra)

• S. Sinha and S. Sinha, Phys Rev E, 2005, 71,
  020902 (R).
• S. Sinha, Physica A, 2005, 346, 147.
• C. C. Wilmers, S. Sinha and M. Brede, Oikos,
  2002, 99, 3.

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Aim

• To understand the stability of network dynamics
  with many interacting components
• Species ( Ecology ) predator-prey, competitive
  cooperative interactions.
• Markets/economic agents ( Economics )
  producer-consumer, competitive and cooperative
  interactions.
• Human groups ( Social networks )

The Question

• How does the stability (response to
  perturbations) of networks change as a function
  of
• network size (number of interacting components
  in the network)
• degree of connectivity between the nodes of the
  network
• the strength of interaction between the nodes of
  the network

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The Empiricists View
The Physicists View
complexity ? network instability
Complexity is essential for network stability

• Robert May (1972)
• Randomly constructed networks become less stable
  with complexity

• Charles Elton (1958)
• Simple ecosystems less stable than complex ones

Construct randomly generated matrices
representing interaction strengths in a network,
whose individual nodes are stable (Jii -1)

• Field observations
• Violent fluctuations in population density more
  common in simpler communities.
• Simple communities more likely to experience
  species extinctions.
• Insect outbreaks rare in diverse tropical
  forests common in less diverse sub-tropical
  forests.

Obtain the eigenvalues ? of J and use the
criterion that if ?maxgt 0, the system is unstable.
May-Wigner Theorem Stability of a network
decreases as its size, connectivity and
interaction strength increases
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A Fresh look at Complexity ? Instability

• Consider networks which have structures in the
  arrangement of their interactions
• Complexity ? Instability
• Small-world connectivity S. Sinha, Physica A,
  2005, 346, 147.
• Assortativity M. Brede and S. Sinha, submitted.
• Hierarchical modular connectivity R. K. Pan and
  S. Sinha, forthcoming

• Consider networks with full dynamics (fixed
  point, oscillatory, chaotic) at each
  nodeComplexity ? InstabilityS. Sinha and S.
  Sinha, Phys Rev E, 2005, 71, 020902 (R).S. Sinha
  and S. Sinha, submitted.

• Consider networks which evolve over time through
  addition (migration) or deletion (extinction) of
  nodes
• C. C. Wilmers, S. Sinha and M. Brede, Oikos,
  2002, 99, 3.

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In nature, networks are not random they have
structure.
Example small-world networks
Regular Network
Random Network
Small-world Network
p 1
p 0
0 lt p lt 1
Increasing Randomness
Watts and Strogatz (1998) Many biological,
technological and social networks have connection
topologies that lie between the two extremes of
completely regular and completely random.
Question Does small-world topology affect the
stability of a network ?
The stability-instability transition occurs at
the same critical value as random network but
transition gets sharper with randomness (Sinha
2005)
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Hierarchical Modular Networks (Pan Sinha,
forthcoming)
C. Elegans neural network 302 neurons
C. Elegans synaptic connectivity matrix
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Hierarchical Modular Networks (Pan Sinha,
forthcoming)
Increasing modularity
r 1
r 0

• r 1 randomly coupled network.
• r 0 isolated sub-networks (modules)
• 0 lt r lt 1 hierarchically structured network.

Increasing levels of hierarchy
l 0
l 3
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Networks with structure
Introducing certain structures in the network
topology does not change the Complexity ?
Instability result!
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Network dynamics
Nodes may have non-trivial dynamics. (Sinha
Sinha, 2005)
Introduce explicit dynamics at the nodes X
(n1) F( X (n)) What happens when such nodes
are coupled together to form a sparsely connected
network ? Activity at a node may stop as a result
of interactions
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(Sinha Sinha, 2005)
Dynamics of network nodes X (n1) F( X (n))
Example Discrete exponential logistic growth
model X n1 X n exp r( 1 X n )
Network dynamics Xi (n1) F( Xi (n) 1S Jij
Xj (n) )
Fixed-point, periodic and chaotic dynamics
extinction
A node is extinct if S Jij Xj lt -1 Question How
many nodes survive asymptotically ?
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The size of the set of active nodes is extremely
robust !
initial
final
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Global stability of network Probability of
persistence of active nodes
C -1
C -1
N -1
r -3
s -2
s -1.4
Scaling of S Jij Xj distribution confirms the
May-Wigner results
Probability of stability depends not on details
of map dynamics but on the extent of the
attractor as xrmax-3
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Network dynamics
No change in results for global stability with
dynamics at nodes Complexity ? Instability
Universal feature of network dynamics!
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Puzzle ! How can complex networks be robust at
all ?Possible solution Network
EvolutionNetworks do not occur fully formed
but gradually evolve over time
Example Assembling ecological communities How
are ecological networks gradually organized over
time by species introduction and/or extinction
? Community Assembly rules decide which species
can coexist in a system, and the sequence in
which species are able to colonize a habitat.
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WSB Network Assembly Model Algorithm(Wilmers-Si
nha-Brede, 2002)
Network Evolution

• Start with one node.

a12 ? ? a21
1
2

• Add another node with Poisson distributed number
  of links, and Normal (0,?2 ) distributed
  interaction strengths aij .

• Check stability of the resultant network
  interaction matrix
• If unstable, remove a node at random and analyze
  the stability again.
• If stable, add another node.

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Network initially grow in size monotonically and
then settles down to a pattern of growth spurts
and collapses.
Communities with overall weaker interactions
support a larger mean number of species ? weak
links are stabilizing (R. May).
The randomness in network connectivity is
quenched ? long-range memory!
Probability of stability of random network
(annealed randomness)
Agrees with May (1972)
gaussian PDF for annealed randomness
observed PDF for quenched randomness
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Surprise! For the evolved networks complexity ?
robustness
Larger networks are less variable (i.e., more
robust) and more resilient (resilience
normalized mean return time to average network
size) Frequency Distribution of Extinction
Cascades
Larger networks have smaller chance of a large
magnitude collapse ? increased resistance
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Bottomline

• Introducing dynamics and/or structure into
  networks does not change stability-instability
  transition at increased complexity.
• Introducing network evolution ?
• synthesis of opposing views in
  stability/diversity debate
• Stronger interactions increased connectivity
  lead to smaller networks, yet,
• given a large, highly connected network it is
  more likely to be robust than its smaller,
  sparsely connected counterpart.
• The results imply that the traditional approach
  of taking snapshot views of networks may be
  inadequate to build an understanding of their
  stability.
• Implications not just for ecology, but from cell
  to society!

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Complexity vs Stability inCellular Signaling
NetworkSS and Kanury V S Rao (ICGEB, New Delhi)

• Question
• How does the signaling network allow the cell
  response
• to be sensitive to various different stimuli,
  and,
• yet robust enough to withstand noise ?

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Cellular Signaling Network
In all forms of life, sensing and responding
appropriately at the cellular level depends on
signal transduction.
The mechanism a sequence of linked biochemical
reactions inside the cell, carried out by
enzymes.
The system A network whose nodes are enzymes,
and links are reactions
Emergence From interaction of these
simultaneous reaction processes, system-level
properties arise that are essential to cell
function.
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The System
B-Cell Response Signal Transduction Network
provides a good experimental model to study the
structure of signaling networks.
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• Aim
• To understand the intracellular communication
  process of the cell.
• Why ?
• Breakdown of communication ? disease.
• Hijacked by intracellular infectious agents for
  proliferating.

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BCR signaling network
Pyk2
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(No Transcript)
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BCR signaling network
Tyr Kinase
Ser/Thr Kinase
Dual specificity Kinase
Other
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The Approach
Step 1. Reconstructing the network (by
correlation analysis)
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Under normal condition measure activation
p38
Lets focus on a specific node How does it
respond when particular nodes in the network are
blocked ?
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Correlation analysis of activity Which nodes
influence which other nodes ? Block a node, and
find out how other nodes behave in its absence
positive effect
negative effect
no effect
Surprise e.g., p38 affects and is affected by
many vertices of the BRC signaling network
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The Approach
Step 2. Understanding network behavior (by
dynamical modeling)
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Dynamics of Kinase Activation (x)
dxi/dt Sj aij xj ßi xi
Addition/removal of even single links can have
remarkable consequences






1
R
1
2
2
3
3
4
4
R

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• Eventual Goal
• To determine the locus and extent of functional
  breakdown in specific situations
• Use this knowledge to apply appropriate remedial
  measures, e .g. targeting the relevant hub with
  specific drug.

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How a hit is bornThe Emergence of
Popularityfrom the Dynamics of Collective Choice

• Sitabhra Sinha
• in collaboration with
• Raj K Pan ( IMSc )
• S Raghavendra ( MSE, NUI )

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The case of movie popularity
as measured in terms of gross income
Empirical analysis of all movies released across
theaters in USA during 1999-2004
Most properties of movie popularity distribution
are a consequence of 3 independent features
log 10 (opening theaters)

• The distribution of opening screens is bimodal
• The opening gross per theater distribution is
  log-normal
• The gross per theater decays with time t as 1/t

US Box Office
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Explaining the emergence of popularity
collective choice through agent-based models

• Interaction between agents driven by
• Personal belief (expectation from a particular
  choice)
• Herding (through interaction with neighbors)
• 2 factors affect the evolution of an agents
  belief
• Adaptation (to previous choice)
• Belief changes with time to make subsequent
  choice of the same alternative less likely
• Learning (by global feedback through media)
• The agent will be affected by how her previous
  choice accorded with the collective choice (M S
  S).

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Results

• Long-range order when learning is introduced
• Clusters of neighboring agents with similar
  choice behavior
• Self-organized pattern formation
• Multiple ordered domains
• Behavior of agents belonging to each such domain
  is highly correlated Distinct cultural groups
  (Axelrod).
• These domains eventually cover the entire system.
  dislocation lines at the boundary of two
  domains
• Phase transition
• Unimodal to bimodal distribution as learning rate
  increases.

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OK but does it explain reality ?
Rank distribution Compare real data with model
US Movie Opening Gross
Model randomly distributed ?
Model
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Using the model for financial markets
Make the links dynamic No learning Every agent
independent
Learning Interaction strength betn nbrg agents
evolve over time
µ rand 0, 0.1
N 200, T 10000 itrns Hexagonal Lattice (6
neighbors)
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Learning Interaction strength betn nbrg agents
evolve over time
200 x 200 agents
Mt ?i Sit
Return rt log (Mt1/Mt)
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Fat-tailed distribution of returns
obtained through interactions between agents
mediated via an evolving network J
compare with
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Analysis of Financial Market Data from Indian
Markets
About 20 different stock markets in India The two
largest are BSE NSE Bombay Stock Exchange
(BSE) founded in 1875, the oldest stock market
in Asia Index BSE 30, BSE 100, BSE 500
National Stock Exchange (NSE) Index Nifty
50 Nifty stocks represent about 58 of the total
market capitalization (30/12/05) and 47 of
traded value of all stocks in NSE
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(Power) Laws of the Market
Gabaix et al, Nature 423, 267 (2003)
The Inverse Cubic Law of return distribution
valid for ?t 1 min 1 month
Plerou et al, PRE 60, 6519 (1999)
1000 largest companies TAQ Database 94-95

• Universality
• Common distribution for
• Different indices, stocks
• Different markets
• Different countries

P (r gt x) x - a
?t 15 min
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Do Indian Markets Follow a Different Law ?
Some recent studies claim yes
log-log
Return distrn of daily closing price, p t ?t 1
day for 49 largest stocks in NSE (Nov 1994 -
Jun 2002) Return, r t log ( p t /p t ?t
) Normalized return, r r / s (r) Return
distribution decays as an exponential function P
( r ) exp ( - ß r ) ß characteristic scale
Indian Stocks
US Stocks
semi-log
Indian Stocks
US Stocks
Matia et al, EPL 66, 909 (2004)
the stock market of the less highly developed
economy of India belongs to a different class
from that of highly developed countries. (Matia
et al, 2004)
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Breakdown of universality !!!
Or is it ???
We re-examine the question come to the opposite
conclusion Indian markets have a fat-tailed
return distribution, consistent with a power law
having exponent a 3.

• Based on our analysis of trading data from BSE
  NSE with different time resolutions
• Macro-scale (daily closing price)
• Micro-scale (transaction-by-transaction or
  tick-by-tick)

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Do Indian markets follow the inverse-cubic law of
stock price return distribution ? (SS and Raj K
Pan)
Probability density function A single
distribution of all the normalized returns for
479 NSE stocks (belonging to BSE 500 index)
?t 5 min
Power law tail over approximately two decades !
Slope of positive tail -3.2 Slope of negative
tail -2.7
?t 5 min
Cumulative probability distribution
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Price-Impact function (M Krishna S K
Shanthi) How is the price of a stock affected by
the transaction volume ? Are stock price
movements subject to the Principle of
demand-and-supply ?

• Price reaction to a single transaction
• may not be relevant !
• unlikely that human reaction occurs at the time
  scale of less
• than a second
• Rather, look at std dev of prices the total
  volume over small time intervals t

t 5 min
st Vt?, ? 1
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Correlation Matrices
The stocks in the NSE are strongly correlated !
Jan 2003
Jan 2006
Movie 152 frames 1 Jan 2003-30 Jan 2006 Window
size 21 days Window step 5 days
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Results

• Examined the 2 largest Indian markets, BSE NSE
• Focus on stock prices, rather than index
• Price and transaction data at various time
  scales
• tick-by-tick to 1 day
• Return distribution has long tails consistent
  with a power law of exponent a 3
• Volume and number of trades distributions appear
  to be log-normal
• Volume and number of trades are linearly related
• Movement of most stocks appear to be strongly
  correlated