1. Scale-free and Hierarchical Structures in Complex Networks L. Barabasi, Z. Dezso, E. Ravasz, S.H. Yook and Z. Oltvai Presented by Arzucan Özgür

2. 21.10.2003 CMPE 588 2 Outline Network Models Network Models Random NetworksRandom Networks Scale-Free NetworksScale-Free Networks Scale-Free ModelScale-Free Model Hierarchical Organization in Complex Networks Hierarchical Organization in Complex Networks Hierarchical Network ModelHierarchical Network Model Hierarchical Organization in Real NetworksHierarchical Organization in Real Networks Halting Viruses in Scale-Free Networks Halting Viruses in Scale-Free Networks Outlook Outlook

3. 21.10.2003 CMPE 588 3 Introduction Behavior of natural and social systems depends on the web through which the system’s constituents interact with each other. Behavior of natural and social systems depends on the web through which the system’s constituents interact with each other. Cell’s metabolizm is maintained by a cellular networkCell’s metabolizm is maintained by a cellular network Nodes  substrates Nodes  substrates Links  chemical reactions Links  chemical reactions Complex networks describe human societiesComplex networks describe human societies Nodes  individuals Nodes  individuals Links  social interactions Links  social interactions WWWWWW Nodes  Web documents Nodes  Web documents Links  URL Links  URL Scientific literatureScientific literature Nodes  publications Nodes  publications Links  citations Links  citations LanguageLanguage Nodes  words Nodes  words Links  syntaxical or grammatical relationships Links  syntaxical or grammatical relationships Networks describing these real life systems constantly evolve by the addition and removal of new nodes and links. Networks describing these real life systems constantly evolve by the addition and removal of new nodes and links. Due to the diversity and large number of nodes and interactions  until recently topology of these complex evolving networks was largely unknown and unexpected. Due to the diversity and large number of nodes and interactions  until recently topology of these complex evolving networks was largely unknown and unexpected. Aim is  review some advances in the area in order to convey the potential for understanding complex systems through the evolution of the networks behind them. Aim is  review some advances in the area in order to convey the potential for understanding complex systems through the evolution of the networks behind them.

4. 21.10.2003 CMPE 588 4 Network Models Random Networks Random Networks Scale-Free Networks Scale-Free Networks Hierarchical Network Model Hierarchical Network Model

5. 21.10.2003 CMPE 588 5 Random Networks Random graphs  since the 1950’s described as large networks with no apparent design principles Random graphs  since the 1950’s described as large networks with no apparent design principles Erdos-Renyi (ER) model of random graphs Erdos-Renyi (ER) model of random graphs start with N nodes and connect every pair of nodes with probability p start with N nodes and connect every pair of nodes with probability p A graph is created with approximately pN(N-1)/2 edges distributed randomly.A graph is created with approximately pN(N-1)/2 edges distributed randomly.

6. 21.10.2003 CMPE 588 6 Scale-Free Networks P(k)  probability that a randomly selected node has exactly k edges. P(k)  probability that a randomly selected node has exactly k edges. In random graphs edges are placed at random  the majority of nodes have approximately the same degree close to the average degree of the network. In random graphs edges are placed at random  the majority of nodes have approximately the same degree close to the average degree of the network. Degrees in random graph follow a Poisson Distribution with a peak at. Degrees in random graph follow a Poisson Distribution with a peak at. It has been shown that most complex networks such as the WWW, Internet, protein networks, language or sexual networks have Power Law degree distribution.  scale-free networks. It has been shown that most complex networks such as the WWW, Internet, protein networks, language or sexual networks have Power Law degree distribution.  scale-free networks. In random networks, the exponential decay of P(k) guarantees the absance of nodes with significantly more links than. In random networks, the exponential decay of P(k) guarantees the absance of nodes with significantly more links than. In scale-free networks, power low distribution implies that nodes with only a few links are numerous, but a few nodes have a very large number of links. In scale-free networks, power low distribution implies that nodes with only a few links are numerous, but a few nodes have a very large number of links.

7. 21.10.2003 CMPE 588 7 Some Scale-Free Networks

8. 21.10.2003 CMPE 588 8 Scale-Free Model Two mechanisms, not present in classical random network models played role in the development of scale-free network model that leads to a network with power-law degree distribution: Two mechanisms, not present in classical random network models played role in the development of scale-free network model that leads to a network with power-law degree distribution: Growth  start with a small number of nodes (m0), at every timestep we add a new node with m edges (m<=m0) that link the new node to m different nodes already present in the network.Growth  start with a small number of nodes (m0), at every timestep we add a new node with m edges (m<=m0) that link the new node to m different nodes already present in the network. Preferential attachment  When choosing the nodes to which the new node connects,we assume that the probability Π that a new node will be connected to node i depends on the degree ki of node i, such thatPreferential attachment  When choosing the nodes to which the new node connects,we assume that the probability Π that a new node will be connected to node i depends on the degree ki of node i, such that

9. 21.10.2003 CMPE 588 9 Scale-Free Model Simulations show that this network evolves into a scale-invariant state with the probability that a node has k edges follows a power-law with an exponent γ=3 Simulations show that this network evolves into a scale-invariant state with the probability that a node has k edges follows a power-law with an exponent γ=3 Scaling exponent is independent of m, the only parameter in the model. Scaling exponent is independent of m, the only parameter in the model. Degree distribution of the scale-free model, with N =m0+t =300,000 and m0 =m=1 (circles), m0 = m = 3 (squares), m0 = m = 5 (diamonds) and m0 = m = 7 (triangles). The slope of the dashed line is γ =2.9, providing the best fit to the data. The inset shows the rescaled distribution P(k)/2m2 for the same values of m, the slope of the dashed line being γ = 3. (b) P(k) for m0 = m = 5 and system sizes N = 100,000 (circles), N = 150,000 (squares) and N = 200,000 (diamonds). The inset shows the time- evolution for the degree of two vertices, added to the system at t 1 = 5 and t2 = 95. Here m0 = m = 5, and the dashed line has slope 0.5 Degree distribution of the scale-free model, with N =m0+t =300,000 and m0 =m=1 (circles), m0 = m = 3 (squares), m0 = m = 5 (diamonds) and m0 = m = 7 (triangles). The slope of the dashed line is γ =2.9, providing the best fit to the data. The inset shows the rescaled distribution P(k)/2m2 for the same values of m, the slope of the dashed line being γ = 3. (b) P(k) for m0 = m = 5 and system sizes N = 100,000 (circles), N = 150,000 (squares) and N = 200,000 (diamonds). The inset shows the time- evolution for the degree of two vertices, added to the system at t 1 = 5 and t2 = 95. Here m0 = m = 5, and the dashed line has slope 0.5

10. 21.10.2003 CMPE 588 10 Continuum Theory The dynamical properties of the scale-free model can be addressed using analytical approaches. The dynamical properties of the scale-free model can be addressed using analytical approaches. Continuum theory is such an approached focusing on the dynamics of node degrees. Continuum theory is such an approached focusing on the dynamics of node degrees. Continuum approach calculates the time dependence of the degree ki of a given node i. Continuum approach calculates the time dependence of the degree ki of a given node i. This degree will increase every time a new node enters the system and links to node i. This degree will increase every time a new node enters the system and links to node i. The probability of this process is Π(ki). The probability of this process is Π(ki).

11. 21.10.2003 CMPE 588 11 Continuum Theory ki is a continuous real variable ki is a continuous real variable The rate at which ki changes is proportional to Π(ki). The rate at which ki changes is proportional to Π(ki). So, ki satisfies the dynamical equation: So, ki satisfies the dynamical equation:

12. 21.10.2003 CMPE 588 12 Continuum Theory

13. 21.10.2003 CMPE 588 13 Hierarchical Organization in Complex Networks In addition of being scale-free, measurements indicate that most networks show a high degree of clustering. In addition of being scale-free, measurements indicate that most networks show a high degree of clustering. Clustering coefficient for node i with ki links is displayed below. Here ni is the number of links between the ki neighbours of i. Clustering coefficient for node i with ki links is displayed below. Here ni is the number of links between the ki neighbours of i.

14. 21.10.2003 CMPE 588 14 Hierarchical Organization in Complex Networks Empirical results show that Ci, averaged over all nodes is significantly higher for most real networks that for a random network of similar size. Empirical results show that Ci, averaged over all nodes is significantly higher for most real networks that for a random network of similar size. Clustering coefficient of real networks is to a high degree independent of the number of nodes in the network. Clustering coefficient of real networks is to a high degree independent of the number of nodes in the network. In order to combine modularity, high degree of clustering and scale free topology  it is assumed that modules combine into each other in a hierarchical manner  generating hierarchical network. In order to combine modularity, high degree of clustering and scale free topology  it is assumed that modules combine into each other in a hierarchical manner  generating hierarchical network. Scaling-Law: Scaling-Law:

15. 21.10.2003 CMPE 588 15 Hierarchical Network Model

16. 21.10.2003 CMPE 588 16 Scaling Properties of Hierarchical Model (N = 5 7). (a) The numerically determined degree distribution. The assymptotic scaling, with slope γ=1+ln5/ln4, is shown as a dashedline. (b) The C(k) curve for the model. The open circles show C(k) for a scale-free model of the same size, illustrating that it does not have a hierarchical architecture. (c) The dependence of the clustering coefficient, C, on the size of the network N. While for the hierarchical model C is independent of N (diamond), for the scale-free model C(N) decreases rapidly (circle).

17. 21.10.2003 CMPE 588 17 Hierarchical Organization in Real Networks The scaling of C(k) with k for six large networks: (a) Actor network, two actors being connected if they acted in the same movie according to the www.IMDB.com database. (b) The semantic web, connecting two English words if they are listed as synonyms in the MerriamWebster dictionary. (c) TheWorldWideWeb. (d) Internet at the Autonomous System level, each node representing a domain, connected if there is a communication link between them. (e) The metabolic networks of 43 organisms with their averaged C(k) curves. (f) The protein-protein physical interaction networks using four different databases. The dashed line in each figure has slope -1.

18. 21.10.2003 CMPE 588 18 Summary Measurements indicate that some real networks lack a hierarchical architecture, and do not obey the scaling law. In particular, the power grid and the router level Internet topology have a k independent C(k). In summary, it is shown that for several large networks C(k) is well approximated by C(k) ~ 1/k, in contrast to the k-independent C(k) predicted by both the scale-free and random networks. This indicates that these networks have an inherently hierarchical organization. In contrast, hierarchy is absent in networks with strong geographical contraints, possibly because limitation on the link length strongly constraints the network topology.

19. 21.10.2003 CMPE 588 19 Halting Viruses in Scale-Free Networks Classical epidemiological models predict that infectious diseases with transmission probability under an epidemic threshold will inevitably die out. Classical epidemiological models predict that infectious diseases with transmission probability under an epidemic threshold will inevitably die out. Thus, lowering transmission probability by universally available cure seems an effective action agains virus spreading. Thus, lowering transmission probability by universally available cure seems an effective action agains virus spreading. However, However, It has been shown that in scale-free networks the epidemic threshold is zero.  even extremely weakly infectious viruses spread and prevail.It has been shown that in scale-free networks the epidemic threshold is zero.  even extremely weakly infectious viruses spread and prevail. Network of human sexual contacts has a scale-free topology.Network of human sexual contacts has a scale-free topology. So, infected hubs increase the transmission probability of the epidemics (HIV) by reaching an unusually high percentage of other nodes.So, infected hubs increase the transmission probability of the epidemics (HIV) by reaching an unusually high percentage of other nodes. Given the high cost of cure and immunization there are two approaches that can be taken Given the high cost of cure and immunization there are two approaches that can be taken Random immunization  not very effective as the scale-free nature of the network is not altered.Random immunization  not very effective as the scale-free nature of the network is not altered. Immunizing hubs with higher degree of connectivity  the optimum approach.Immunizing hubs with higher degree of connectivity  the optimum approach.

20. 21.10.2003 CMPE 588 20 Thank You.