1. Response network emerging from simple perturbation Seung-Woo Son 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 i j k 11 start target i k 11 start target 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 cf) diameter changes J.-H. Kim, K.-I. Goh, B. Kahng and D. Kim, Physical Review Letters, 91, 5 (2003)

6. Distribution of. - BA model distribution is power law distribution with exponent 2.1 Summation of BC changes after i-th node removal is linearly proportional to BC of i-th node in 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 Δb i (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 Δb i (j) > b * (percolation threshold) are valid and connected. a c b d e f 3 4 5 4 2 6 3 1 2 7 MST a c b d e f 3 4 5 4 2 6 3 1 2 7 Percolation

8. Result : Secondary Networks 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. BA 100 MST Secondary network construction

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. 1 2 30 45 6 1 2 30 45 6 compare BA model ( N = 1000, M = 1996 ) linksXmatches MST network9990.908907 Percolation net.33770.7661529 Other BA net.19960.01939 RG network19960.01223 Random net. 20410.0035 9570.0011 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.

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 Δb i (j) values represent the interaction between i-node and j-node. And It is related to diameter change directly. – Δb i (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 Δb i (j) and b(i), analytic calculation. – Finding primary network from secondary network information.

13. Distribution of BC Changes. b i : summation of BC after i-th node removed b o : summation of BC over whole network. b κ : summation of BC from κ-th node to all. A B C F E D G start

14. Distribution of BC Changes. Δb i : ( i-th node removed ) summation of BC changes. Network deformation = Lost a source of BC = select alternative shortest path + detour ( Contribution to Δb i < 0 ) ( Contribution to Δb i > 0 ) select alternative shortest path + detour Nonlinear! Contribution of Δb i = portion of b(i) 77.4% 22.6% 1 2

15. Distribution of BC Changes. A B A Small closeness centrality of A  Large sum of distance from A  Large ② contribution and small network deformation B : : Large closeness centrality of B  small sum of distance from B  small ② contribution and large network deformation Network ( λ : 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 attachment A.-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) 0 2 1 3 4 5 6 Next One? –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. A B C F E D G start –“It is found that the load distribution follows a power-law with the exponent δ~2.2(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) δ is universal value !