Albert-Lszl Barabsi With Emma K. Towlson and Sean P. Cornelius - PowerPoint PPT Presentation

Network Science Chapter 7: Degree Correlations Albert-László Barabási With Emma K. Towlson and Sean P. Cornelius www.BarabasiLab.com

Questions 1. What are degree correlations? Why do we want to study correlations? 2. What is the degree correlation matrix? What do we expect it to look like for random, assortative and disassortative networks? Why? 3. What is the degree correlation function? What do we expect to see for random, assortative and disassortative networks? Why? 4. What is the degree correlation coefficient r ? What values do we expect for random, assortative and disassortative networks? Why? 5. What is structural disassortativity? What kind of network is affected and how can we detect it? 6. What is the impact of the degree correlations? Why do we study them? Why does the threshold for the phase transition in Fig. 7.15 change? 7. Summary 8. Differences between undirected and directed networks.

TOPOLOGY OF THE PROTEIN NETWORK Nodes: proteins Links: physical interactions (binding) Puzzling pattern: Hubs tend to link to small degree nodes. Why is this puzzling? In a random network, the probability that a node with degree k links to a node with degree k’ is: p kk ' = kk ' 2 L k ≅ 50, k’=13, N=1,458, L=1746 p 50,13 = 0.15 Yet, we see many links between degree 2 and 1 nodes, and no p 2,1 = 0.0004 links between the hubs. H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N. Oltvai, Nature 411, 41-42 (2001) Network Science: Degree Correlations

DEGREE CORRELATIONS IN NETWORKS Assortative: Neutral : Disassortative: hubs show a tendency to nodes connect to each Hubs tend to avoid link to each other. other with the expected linking to each other. random probabilities. Quantifying degree correlations (three approaches):  full statistical description ( Maslov and Sneppen, Science 2001 )  degree correlation function ( Pastor Satorras and Vespignani, PRL 2001 )  correlation coefficient ( Newman, PRL 2002 ) Network Science: Degree Correlations

STATISTICAL DESCRIPTION e jk : probability to find a node with degree j and degree k at the two ends of a randomly selected edge å å = 1 e jk = q k e jk j , k j q k : the probability to have a degree k node at the end of a link. q k = kp k Probability to find a node at the end of a link is biased towards the more connected Where: nodes, i.e. q k =Ckp k, where C is a normalization constant . After normalization we k find C=1/<k>, or q k =kp k /<k> If the network has no Deviations from this prediction are a e jk = q j q k signature of degree correlations. degree correlations: M. E. J. Newman, Phys. Rev. Lett. 89, 208701 (2002) Network Science: Degree Correlations

EXAMPLE: e jk FOR A SCALE-FREE NETWORK Assortative: More strength in the diagonal, e jk e jk hubs tend to link to each other. Neutral Disassortative: Hubs tend to connect to small nodes. e jk Each matrix is the average of 100 independent scale-free networks, generated using the static model with N=10 4 , γ=2.5 and <k>=3. Network Science: Degree Correlations

EXAMPLE: e jk FOR A SCALE-FREE NETWORK Assortative: Perfectly assortative More strength in network : the diagonal, e jk hubs tend to link e jk =q k δ jk to each other. Disassortative: Perfectly Hubs tend to disassortative connect to small network : nodes. e jk Each matrix is the average of 100 independent scale-free networks, generated using the static model with N=10 4 , γ=2.5 and <k>=3. Network Science: Degree Correlations

REAL-WORLD EXAMPLES Astrophysics co-authorship network Yeast PPI e jk e jk Assortative : Disassortative : More strength in Hubs tend to the diagonal, connect to small hubs tend to nodes. link to each other. Network Science: Degree Correlations

PROBLEM WITH THE FULL STATISTICAL DESCRIPTION (1) Difficult to extract (2) Based on e jk and hence requires a large information from a visual number of elements to inspect: inspection of a matrix. ( ) Nr. of k max k max - 1 independent - 1 - k max elements 2 å å Undirected network: = q k = 1 e jk e jk k max x k max matrix j = 1, k max j , k Constraints We need to find a way to reduce the information contained in e jk M. E. J. Newman, Phys. Rev. Lett. 89, 208701 (2002) Network Science: Degree Correlations

Section 7.3 Measuring Degree Correlations

Average n ext n eighbor degree k nn (k) : average degree of the first neighbors of nodes with degree k . R. Pastor-Satorras, A. Vázquez, A. Vespignani, Phys. Rev. E 65 , 066130 (2001) Network Science: Degree Correlations

Average n ext n eighbor degree k nn (k) : average degree of the first neighbors of nodes with degree k . Network Science: Degree Correlations

Average n ext n eighbor degree Network Science: Degree Correlations

Degree Correlation Coefficient If there are degree correlations, e jk will differ from q j q k . The magnitude of the correlation is captured by <jk>-<j><k> difference, which is: å jk ( e jk - q j q k ) jk <jk>-<j><k> is expected to be: positive for assortative networks, zero for neutral networks, negative for dissasortative networks To compare different networks, we should normalize it with its maximum value; the maximum is reached for a perfectly assortative network , i.e. e jk =q k δ jk 2 = max å å s r jk ( e jk - q j q k ) = jk ( q k d jk - q j q k ) normalization: jk jk å r £ 0 jk ( e jk - q j q k ) disassortative r = 0 - 1 £ r £ 1 neutral jk r = r ³ 0 assortative s r 2 Network Science: Degree Correlations M. E. J. Newman, Phys. Rev. Lett. 89, 208701 (2002)

REAL NETWORKS Social networks are assortative Biological, technological networks are disassortative r>0: assortative network: r<0: disassortative network: Hubs tend to connect to other hubs. Hubs tend to connect to small nodes. Network Science: Degree Correlations

Section 7.4 Structural cutofgs

Example: Degree sequence introduces disassortatjvity Scale-free network generated with the confjguratjon model (N=300, L=450, γ=2.2). The measured r=-0.19!  Dissasortatjve! Purple hub: 55 neighbors. White hub: 46 neighbors. Calculatjon of the expected number of links between purple node (k=55) and white node (k=46) for uncorrelated networks: p k k ¢ p ¢ k k 55 1 300 × 46 1 300 E 55,46 = k N × e 55,46 = 900 × » 2.8 > 1 3 2 In order for the network to be neutral, we k need 2.8 links between these two hubs.

Section 7.10

Section 7.5 Correlations in Real Networks

Section 7.5 Correlations in Real Networks

Section 7.5 Correlations in Real Networks

Section 7.5 Correlations in Real Networks

Section 7.7 The Impact of Degree Correlations

Section 7.7 The Impact of Degree Correlations

Section 7.7 The Impact of Degree Correlations

Section 7.8 Summary

Section 7.8 Summary

DEGREE CORRELATIONS IN NETWORKS Assortative: Neutral : Disassortative: hubs show a tendency to nodes connect to each Hubs tend to avoid link to each other. other with the expected linking to each other. random probabilities. Quantifying degree correlations (three approaches):  full statistical description ( Maslov and Sneppen, Science 2001 )  degree correlation function ( Pastor Satorras and Vespignani, PRL 2001 )  correlation coefficient ( Newman, PRL 2002 ) Network Science: Degree Correlations

Section 7.8 Directed networks: knn