Dynamical response networks under perturbations

1
Dynamical response networks under perturbations

• Seung-Woo Son, Dong-Hee Kim, Yong-Yeol Ahn, and
  Hawoong Jeong
• Complex System and Statistical Physics Lab.,
• Dept. Physics, KAIST, Daejeon 305-701, Korea

2
Motivation Microarray Data

• Microarray data show the response of each gene to
  an experiment, which is a kind of perturbation to
  the genetic network.
• ex) gene deletion, temperature change etc
• Like building the genetic network from microarray
  data, the secondary network can be constructed
  from the response of primary network under
  perturbation.
• ex) node removal (?)
• Can the secondary network represents the
  primary network correctly ?
• What is the meaning of the response under
  perturbation ?
• Ultimately, can we find out primary network
  from the secondary network ?

3
Introduction Node Removal Perturbations

• When a node is removed, network structure
  changes. The network can break into several
  isolated clusters.
• Giant cluster size decreases gradually and the
  average path length increases.

R. Albert and A.-L.Barabási, Reviews of Modern
Physics, 74, 47 (2002)

• SF network is more tolerant against random
  removal better than random network.
• In SF network, the diameter changes under a node
  removal follow the power-law distributtion.

J.-H. Kim, K.-I. Goh, B. Kahng and D.
Kim, Physical Review Letters, 91, 5 (2003)
4
Introduction Load Betweenness Centrality

• What is the Load ?
• When every pair of nodes in a network exchanges
  data packets along the shortest path, load of a
  node is the total number of data packets passing
  through that node.
• ex) Internet traffic jam

j

• Betweenness Centrality BC ( Freeman, 1977 )
• if is the number of geodesic paths from i to
  j and is the number of paths from i to j
  that pass through k, then is the
  proportion of geodesic paths from i to j that
  pass through k. The sum for
  all i,j pairs is betweenness centrality.

5
Introduction BC Changes . - BA
model
J.-H. Kim, K.-I. Goh, B. Kahng and D.
Kim, Physical Review Letters, 91, 5 (2003)
6
Distribution of . - BA model
7
MST Percolation Network

• How to build the secondary network ?
• Based on correlation
  bewtween node i and j
• MST (minimum spanning tree)
• A graph G (V,E) with weighted edges. The
  subset of E of G of minimum weight which forms a
  tree on V MST .A node is linked to the most
  influential one with constraint such that N
  vertices must be connected only with (N-1) edges.
• Percolation
• After sorting ?bi(j) in descending order, add a
  link between i and j following that order. When
  all nodes make a giant cluster, stop the
  attachment. It means the links with values
  ?bi(j) gt b (percolation threshold) are valid and
  connected.

MST
8
Result Secondary Networks
BA 100
MST
Secondary network construction

• The degree k of secondary networks contain the
  global information of primary network, because it
  is constructed from BC that is calculated from
  the information of whole network.
• More sparse or dense networks which contain the
  information of original network can be
  constructed.
• Secondary networks represent the primary network
  well with significant link matches.

9
Result Minimal Spanning Tree

• In MST network, the degree distribution shows the
  power-law with exponent 2.2 not 3.0 (
  Scale-free )
• The degree of each node in secondary network is
  linearly correlated to that of primary network.

10
Result Percolation Network

• The degree distribution of percolation network
  shows power-law. ( exponent -1.9 )
• Percolation features appear during giant cluster
  fromation.

11
Similarity Measurement between Two Networks

• The links of each node are regarded as vector in
  N dimensional vector space.
• Vector inner product shows the similarity between
  two networks.
• Binary undirected network case It means how
  many links are overlapping each other.

• The network similarity measure between secondary
  and primary networks are significantly higher
  than other network.
• ( MST 90.8 , percolation 76.6 )
• The secondary networks well represent the
  primary network.

BA model ( N 1000 , M 1996 ) BA model ( N 1000 , M 1996 ) BA model ( N 1000 , M 1996 ) BA model ( N 1000 , M 1996 )
links X matches
MST network 999 0.908 907
Percolation net. 3377 0.766 1529
Other BA net. 1996 0.019 39
RG network 1996 0.012 23
Random net. 2041 0.003 5
Random net. 957 0.001 1
12
Conclusions Future Works

• Conclusions
• Two secondary networks, MST percolation
  network, reproduce the scale-free behavior and
  its degree of each node is in proportion to
  degree of primary network. Its degree contains
  the global information of primary network.
• Similarity measurement shows that the secondary
  networks reproduce original network quite well. (
  MST 91 , percolation 77 )
• BC change ?bi(j) values represent the interaction
  between i-node and j-node. And It is related to
  diameter change directly.
• ?bi(j) and b(i) relations might help to explain
  network classification with BC distribution
  exponents.
• Future Works
• BC change calculation for other network models
  and real networks.
• Precise relation between ?bi(j) and b(i) ,
  analytic calculation.
• Finding primary network from secondary network
  information.

13
Distribution of BC Changes .
bi summation of BC after i-th node removed bo
summation of BC over whole network.
b? summation of BC from ?-th node to all.
14
Distribution of BC Changes .
?bi ( i-th node removed ) summation of BC
changes.
1
Network deformation
select alternative shortest path detour
( Contribution to ?bi gt 0 )
2
Lost a source of BC
( Contribution to ?bi lt 0 )
77.4
Contribution of ?bi portion of b(i)
select alternative shortest path detour
Nonlinear!
22.6
15
Distribution of BC Changes .

• Small closeness centrality of A
• Large sum of distance from A
• Large ? contribution and
• small network deformation

A
A
Network
B

B

• Large closeness centrality of B
• small sum of distance from B
• small ? contribution and
• large network deformation

( ? detour length )
16
Introduction Scale-free network

• What is the Scale Free Network?
• SF network is the network with the power-law
  degree distribution.
• Ex) BA model growth and preferential
  attachmentA.-L.Barabási and R. Albert,
  Emergence of scaling in random networks, Science,
  286, 509 (1999)

Ex) Empirical Results of Real Networks
World-Wide Web, Internet, Movie actor
collaboration network, Science collaboration
graph, Cellular network, etc.R. Albert, H.
Jeong, and A.-L.Barabási, Nature(London), 406,
378 (2000)

• SF network shows error and attack tolerance.

17
Introduction Load Classification of networks

• What is the Load ?
• When every pair of nodes in a network exchanges
  data packets along the shortest path, load, or
  betweenness centrality(BC), of a node is the
  total number of data packets passing through that
  node.
• Ex) Internet traffic jam, influential people in
  social network, etc.

d is universal value !

• It is found that the load distribution follows a
  power-law with the exponent d2.2(1) K.-I.
  Goh, B. Kahng, and D. Kim, Universal Behavior of
  Load Distribution in Scale-Free Networks, PRL,
  87, 27 (2001)

• The exponent of load is robust without network
  model dependency. It can be used to classify the
  networks.
• Kwang-Il Goh, et al., Classification of
  scale-free networks, PNAS, 99, 20 (2002)