1. Self-Similarity of Complex Networks Maksim Kitsak Advisor: H. Eugene Stanley Collaborators: Shlomo Havlin Gerald Paul Zhenhua Wu Yiping Chen Guanliang Li Kitsak, Havlin, Paul, Pammolli, Stanley, (submitted), Phys. Rev. E. (2006)

2. Motivation and Objectives: Many real networks are fractals. Fractal real networks are shown to have a topology distinct from non-fractal networks. Do fractal and non-fractal networks have different properties? (Transport properties) What are the possible applications of these properties?

3. Networks: Definitions 1) Network is a set of nodes (objects) connected with edges (relations). 2) Degree (k) of a node is a number of edges connected to it. 3) Degree Distribution P(k) is the probability that a randomly chosen node has degree k NodesEdges k=3 k=2 k=1

4. Networks: Properties 1)Small World property. Despite the large size, the shortest path between any two nodes is small. (WWW, Internet, Biological) Number of nodes accessible from a random node (seed) grows exponentially with the distance measured from the seed.

5. Networks: Properties 2) Degree Distribution Many real networks have Poisson or power-law degree distribution. Edges connect randomlyPreferential Attachment

6. Networks: Properties 3) Self-Similarity Self-similar network is approximately similar to a part of itself and is fractal. Fractal typically has fractional dimension and doesn’t possess translational symmetry. f d

7. Networks: Properties 2) Degree Distribution Many real networks have Poisson or power-law degree distribution. 3) Self-Similarity Self-similar network is approximately similar to a part of itself. Self-similar networks are fractals and have fractal dimension. It turns out that many real networks possess all three properties (Small World, Scale-Free, Fractal)!!! 1) Small World property. Despite the large size, the shortest path between any two nodes is small. (WWW, Internet, Biological)

8. Dimension Calculation: Box Covering Algorithm A box of size is an imaginary ‘container’ that can hold a part of the network, so that the shortest path between any 2 nodes What is the minimal number of boxes with size needed to cover the entire network? (Non-Fractal)(Fractal) Strategy to calculate dimension of the network: 1.Calculate minimal number of boxes needed to cover the network as a function of their size. 2.Analyze obtained function

9. Fractal analysis with box-covering algorithm Song, Havlin, Makse, 2005

10. Origin of fractals in scale-free networks: Repulsion between hubs In fractal networks large degree nodes (hubs) tend to connect to small degree nodes and not to each other! Song, Havlin, Makse, 2005 Fractal networkNon-fractal Network Probability of having a node of degree connected to node of degree.

11. Transport on networks: Betweenness Centrality Most of the transport on the network flows along the shortest paths. Central nodes are critical: if they are blocked – transport becomes inefficient Betweenness centrality of node : Sociology - L.C. Freeman, 1979 Number of shortest paths between nodes and that pass node. Total number of shortest paths between nodes and. C=0 C=2

12. Transport on networks: Betweenness Centrality How do we identify nodes with high Centrality? Is it true that high centrality nodes also have large degree? Centrality is weakly correlated with degree in fractal scale-free networks!

13. Transport on networks: Betweenness Centrality Why is centrality weakly correlated with degree in fractal scale-free networks? Non-Fractal TopologyFractal topology Due to ‘repulsion between hubs’ small degree nodes appear at all parts of the fractal network. Thus, their centralities can have both small and large values.

14. Centrality-degree correlation in real networks One can’t compare centralities of networks directly due to uniqueness of real networks. The network can be compared to its random counterpart ! Rewire 10000 times Preserve degrees of nodes Rewired network has degree distribution identical to the original network. Repulsion between hubs is broken by random rewiring. The random network is always non-fractal.

15. Centrality-degree correlation. Centrality – degree correlation in non-fractal scale-free networks is much stronger than that in SF fractal networks. Average centrality of small degree nodes in scale-free fractal networks is significantly larger due to repulsion between hubs. Fractal networks should be more stable to conventional degree attacks. Immunization/Attack strategies should be optimized for fractal networks. Kitsak, Havlin, Paul, Pammolli, Stanley, 2006

16. What is the overall Centrality distribution in scale-free networks? Centrality distribution obeys power-law for both types of networks We show both analytically and numerically that Nodes of fractal networks generally have larger centrality than nodes of non-fractal networks Kitsak, Havlin, Paul, Pammolli, Stanley, 2006

17. Box covering algorithm applied to a fractal network with added random edges Transition from Fractal to Non-Fractal Behavior. Real networks are neither pure fractals nor non- fractals due to statistical effects. What happens if we add random edges to a scale-free fractal network? Fractal behaviorNon-fractal behavior

18. Scaling Ansatz: How does the crossover length depend on the density of random edges ? Under rescaling all plots should collapse onto a single curve Crossover length

19. How does the crossover length depend on the density of random edges ? Rescale All plots collapse onto a single curve Rescaling parameter as a function of. We show analytically that

20. Summary and Conclusions 1.Centrality-degree correlation is much weaker in fractal networks than in non-fractal. Fractal networks should be more stable to conventional degree attacks. Immunization/Attack strategies should be optimized for fractal networks. 2.Power-law centrality distribution Centralities of nodes are larger in fractal scale-free networks. fractal networks have different transport properties. 3.Transition from fractal to non-fractal networks. A crossover is observed from fractal to non-fractal networks. Relatively small percent of edges is needed to turn fractal network into non-fractal. Findings of present work have been submitted to Phys. Rev. E.

23. Centrality-degree correlation.

24. Transition from Fractal to Non-Fractal Behavior: Analytical Consideration Consider a fractal network of dimension with nodes and edges. Suppose random edges are added to the network. The probability that a random edge is connected to a given node: A cluster of radius centered on a randomly chosen seed has nodes. The probability that a cluster of size contains a random edge. The crossover length corresponds to Seed

25. Centrality distribution: Analytical Consideration Consider a fractal tree network of dimension with nodes and edges. A small region with nodes will have a typical diameter. The region will be connected to the rest of the network via approximately nodes with centrality. The total number of such nodes in the network: The number of nodes with centrality n

26. Contents: 1.Introduction. (Definitions, Properties…) 2.Self-Similarity and Fractality. (Fractal Networks: General Results) 3.Betweenness Centrality in Fractal and Non-Fractal Networks. (Centrality distribution, Centrality-Degree correlation…) 4.Transition from Fractal to Non-Fractal networks. (Crossover phenomenon, Scaling…) 5. Summary and Conclusions

27. Betweenness Centrality Betweenness Centrality is a measure of Importance of a node in the network. C=0 C=2 Is it true that larger degree nodes generally have larger centrality? L.C. Freeman, 1979 Fractal topology Non-Fractal Topology

28. Transition from Fractal to Non-Fractal Behavior. What happens when random edges are added to a fractal network? Fractal behavior Non-Fractal behavior All plots collapse onto a single curve Scaling Ansatz:

29. Edges connect randomlyPreferential Attachment Networks: Properties 2) Degree Distribution Many real networks have Poisson or power-law degree distribution. 3) Self-Similarity Self-similar network is approximately similar to a part of itself. Self-similar networks are fractals and have fractal dimension. It turns out that many real networks possess all three properties (Small World, Scale-Free, Fractal)!!! 1) Small World property. Despite the large size, the shortest path between any two nodes is small. (WWW, Internet, Biological)