Scale-free and Hierarchical Structures in Complex Networks

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. Outline • Network Models • Random Networks • Scale-Free Networks • Scale-Free Model • Hierarchical Organization in Complex Networks • Hierarchical Network Model • Hierarchical Organization in Real Networks • Halting Viruses in Scale-Free Networks • Outlook CMPE 588

3. Introduction • 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 network • Nodessubstrates • Linkschemical reactions • Complex networks describe human societies • Nodes individuals • Links social interactions • WWW • Nodes Web documents • Links URL • Scientific literature • Nodes publications • Links citations • Language • Nodes words • Links syntaxical or grammatical relationships • 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. • 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. CMPE 588

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

5. Random Networks • Random graphs since the 1950’s described as large networks with no apparent design principles • Erdos-Renyi (ER) model of random graphs • 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. CMPE 588

6. Scale-Free Networks • 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 <k> of the network. • Degrees in random graph follow a Poisson Distribution with a peak at <k>. • 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 <k>. • 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. CMPE 588

7. Some Scale-Free Networks 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: • 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 idepends on the degree ki of node i, such that 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 • 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 CMPE 588

10. Continuum Theory • 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 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. • The probability of this process is Π(ki). CMPE 588

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

12. Continuum Theory 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. • Clustering coefficient for node i with ki links is displayed below. Here ni is the number of links between the ki neighbours of i. 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. • 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. • Scaling-Law: CMPE 588

15. Hierarchical Network Model 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). 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. CMPE 588

18. Summary • Measurements indicate thatsome 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. 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. • Thus, lowering transmission probability by universally available cure seems an effective action agains virus spreading. • However, • 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. • 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 • 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. CMPE 588

20. Thank You. CMPE 588