Emergence of ScaleFree Network with Chaotic Units Pulin Gong, Cees van Leeuwen

1
Emergence of Scale-Free Network with Chaotic
Units Pulin Gong, Cees van Leeuwen

• by
• Oya Ünlü
• Instructor Haluk Bingöl

2
Outline

• Introduction
• Random Scale-Free Networks
• The Model for Network with Dynamic Units
• Algorithm
• Evolution of the Network
• Robust feature of Spatiotemporal Dynamics

3
Introduction

• A diversity of the systems such as world-wide
  web, social networks, protein interactions
  network and metabolic reactions in a cell, belong
  to complex systems.
• Distinctive feature of complex systems is
  emergent order resulting from their many
  interacting elements.
• The structure emerges spontaneously rather than
  by design.
• For many years science treated all complex
  networks as being completely random.
• However, Barabasi and Albert presented a
  scale-free model and showed that most of these
  networks are scale-free.

4
Random vs. Scale-Free
5
Random vs. Scale-Free (contd)

• In random networks
• Most of the nodes have approximately equal number
  of links.
• Distribution of node linkages will follow a
  bell-shaped curve.
• In scale-free networks
• Most nodes have just a few connections where some
  have a tremendous number of links and system has
  no scale.
• Distribution of node linkages follows a power law.

6
Scale-Free Networks

• Scale-free network model based on two mechanisms
• Growth Cumulative addition of new units.
• Preferential Attachment Probability of
  attachment of a newly created node is
  proportional to the connected degree of target
  nodes. (Richly connected nodes tend to get
  richer).
• More recent models allows nodes to age so they no
  longer accept new links or a node acquire new
  links with other monotonically increasing
  function not limited to linear preferential
  attachment.

7
Oscillatory Activity

• Many of the realistic features are included in
  the researches. However, in the studies dynamics
  of the system havent been used to explain the
  emergence of the network structure.
• In real biological systems such as neuronal,
  genetic, metabolic network models etc., units
  with oscillatory activity is very common.
• Subject of the study Growth and adaptive
  rewiring according to dynamical coherence is
  studied in order to explain the emergence of
  scale-free network structure.

8
The Model for Network with Dynamic Units

• Proposed a model for growth combined with
  adaptive rewiring according to the oscillatory
  dynamics of the network units.
• Studies showed that scale-free network with a
  high clustering coefficient is obtained if the
  oscillatory behavior is chaotic.
• Chaos
• The disordered formless matter supposed to have
  existed before the ordered universe.
• Stochastic (random) behavior occurring in a
  deterministic system.

9
The Model for Network with Dynamic Units (contd)

• Spatiotemporal Dynamics dynamics observed both
  in space and time in systems.
• Chaotic logistic maps are used to model the
  dynamics of the network

10
The Model for Network with Dynamic Units (contd)

• Parameters
• xi(n) Activity of the i th unit at the n th
  time step.
• N Total number of units in the current network.
• B(i) Set of all neighbors of the unit i.
• Mi Number of units in the current set B(i)
• a System parameter which controls the dynamics
  of each unit.
• dij Coherence between unit i and j.
• e Coupling strength

11
Algorithm

• Starting from a sparsely, fully connected small
  random network with M0 total number of units and
  L0 number of connections.
• Add a new node in with m connections to m
  different nodes in the current network randomly.
• Choose random initial activation values in the
  range(-1,1) for all units of the new network.
  Calculate the state of the system according to
  second equation and discard an initial transient
  time T.
• Calculate dinj(T1) for the newly added node in.
  Obtain the value jj1 where the value dinj1(T1)
  is minimum among all other units.

12
Algorithm

• Obtain the value jj2 where the dinj2(T1) value
  is maximum among the neighbors of the new unit
  in.
• If j1 is a neighbor of unit in then no change in
  the system. Otherwise connetion between in and j2
  is replaced by a connection between in and j1.
• Go to step 2 and repeat the algorithm for K0
  times.
• Go to step 1 and repeat adding a new node.

13
Evolution of the Network

• By choosing the parameter a, system parameter
  that controls the dynamics of the units, within a
  chaotic range a network with 501700 (M0t) nodes
  is formed.

14
Evolution of the Network

• The network has evolved into a scale-free state
  with the probability a node has k connections,
  following a power-law p(k)k-? with ?3.090.17.
• The clustering coefficient is calculated as 0.15
  and average path length is 2.70.
• It is known that in real networks clustering
  coefficient is much larger than random networks.
• To compare the corresponding random graph with
  the model of the self-organized scale-free
  network, a random graph is formed by connecting
  nodes randomly. (Same number of nodes and
  connections).
• Clustering coefficient1.710-2 and avg. shortest
  path length2.56.

15
Evolution of the Network

• When we compare the random graph with the model
• Shortest path length is closer to random graph.
• Clustering coefficient is much larger than random
  graph.
• So A growing network with chaotic units
  according to the model, produces a scale-free
  network with the characteristics of small-world
  networks.

16
Evolution of the Network

• To see the preferential attachment Ki , the
  number of connections for the set of nodes at
  time t is calculated. Also at time t?t, ?tltltt,
  the number of connections ,Ji, is measured. So
  increasing number of connections ?ki Ji-Ki is
  found.

17
Chaotic vs. Stochastic vs. Periodic

• In order to investigate the mechanisms
  responsible for the emergence of the scale-free
  network, some variants of the model is studied.
• The first variant Dynamic of units Periodic.
• To make the model periodic, parameter a is chosen
  0.51 for equation 2, making the units period-1
  state.
• Second variant Dynamic of units Stochastic.
• To make the model stochastic, a random generator
  is used instead of the logistic functions.
• For both of the models, same small random network
  is chosen and the algortihm is used for the same
  number of times as in chaotic model.

18
Chaotic vs. Stochastic vs. Periodic
19
Chaotic vs. Stochastic vs. Periodic

• For periodic units, distribution doesnt have
  scale-free characteristics. Clustering
  coefficient is close to random network.
• For stochastic units, the clustering coefficient
  is close to random network. However, the
  distribution of connections has at least a
  scale-free part with a cut-off.

20
Robust feature of Spatiotemporal Dynamics

• The dynamics of the scale-free network of coupled
  maps is investigated for its robustness in terms
  of the change of average shortest path length
  under lesioning.
• The spatiotemporal dynamics characteristics of
  the networks are of central importance because
  they provide the functional significance of the
  networks.
• For the scale-free network the size of each
  dynamical cluster (the number of participating
  units) is calculated for every iteration.
• Distribution of the sizes of the dynamical
  clusters are calculated over a longtime period.

21
Robust feature of Spatiotemporal Dynamics
22
Robust feature of Spatiotemporal Dynamics

• The distribution has a power-law part followed by
  an exponential cut.
• For the scale-free network with dynamical units,
  the spatiotemporal dynamics of the complex
  network is robust.
• Also a small number of nodes with large numbers
  of connections certainly play a very important
  role in the spatiotemporal dynamics.

23
Conclusion

• The development of networks with chaotic units is
  investigated.
• Adding new nodes one by one and adaptive rewiring
  of the connections according to the dynamical
  coherence of the nodes, the network
  self-organizes into a scale-free network.
• Self-organized network has a high clustering
  coefficient and small characteristic path length.
• Adaptive rewiring without growth does not produce
  a scale-free network but a small-world network.

24
Conclusion

• Chaotic dynamical behaviors, growth, and adaptive
  rewiring according to the dynamical coherence of
  the nodes is needed for scale-free networks.
• The model can help to understand the real systems
  with dynamic units such as neural systems, or
  some complex systems such as the brain and visual
  system.

25
Questions

• ?