Measuring Preferential Attachment in Evolving Networks

1. Measuring Preferential Attachment in Evolving Networks H.JEONG[1,2], Z.NEDA[1], A.L.BARABASI[1] 1-Department of Physics, University of Nore Dame,USA 2- Department of Physics, Korea Advanced Institute of Science and Technology accepted in 4 December 2002 (last revision July 9,2004)

2. Abstract • A key ingredient of many current models proposed to capture the topological evolution of complex networks is the hypothesis that highly connected nodes increase their connectivity faster than their less connected peers, a phenomenon called preferential attachment. • Measurements on four networks; • Science citation network • Internet • Actor collaboration • Science coauthorship network indicate that the rate at which nodes acquire links depends on the node’s degree, offering direct quantitative support for the presence of preferential attachment.

3. Introduction • Many networks as social, biological and communication systems were seen as random networks. However recent studies show that they have scale-free property. • The studies resulted that these networks are evolving dynamical systems rather than static graphs. • Evolving network models are based on two ingredients; • Growth • Preferential attachment Growth: Networks continuously expand through the addition of new nodes and new links between nodes Preferential attachment: rate (k) with which a node with k links acquire new links is a monotonically increasing function of k.

4. Preferential attachment • Most results show that (k) is linear in k, however recently several authors proposed it could follow a power law. • Time evolution of the degree ki of node i can be obtained from; • where m is constant and(k) has the form • with α > 0 an unknown scaling exponent. For α=1 these models reduce to the scale-free model , for which the degree distribution P(k), giving the probability that a node has k links, follows P(k) ∝ k*exp(-γ)with γ = 3.

5. Preferential attachment (cont.) • for α <1 the degree distribution follows a stretched exponential, while for α >1 a gelation-like phenomenon is expected.

6. Questions about Preferential Attachment • There are fundamental questions that are not yet supported by experimental data. • Is preferential attachment indeed present in real networks? • If (k) doesindeed depend on k, what is its functional form? Is it linear or does it follow a power law? • Could (k) follow some unknown and yet unexplored functional form? In the paper, a numerical method is proposed that allows us to extract functional form of (k) and its characteristic (power law). This paper also shows that for Internet and citation networks, the value of α is 1 while for science collaboration and actor network α<1

7. Methods • To measure (k) we need to monitor to which old node, the new nodes link, as a function of degree of the old node. • However there is an important problem with this approach, normalization constant, C(t), depends on the time at which a given node joins the system. C(t) creates unwanted biases in measurement. • Solution is: to collect data in very tiny time intervals such that nodes in the network in time T0 will be “T0 nodes”. Call “T1 nodes” as the nodes added between [T1,T1+T] where T<<T1 and T1>T0. • When a T1 node joins, we record the degree of T0 node to which the new node links. • The histogram providing the number of links acquired by the T0 nodes with exactly k degree, after normalization, gives the (k,T0,T1)

8. Methods(cont.) • If the growing network develops a stationery state then (k,T0,T1) independent of T0and T1. • Large networks with hundreds of thousands of nodes, (k) has significant fluctations for large k. • To reduce the noise level, instead of (k) we study the cumulative function; • If (k) follows the previous definition,we expect;

9. Measurements 4 networks are analyzed: • In the coauthorship network of neuro-science (NS) the nodes are scientists, two nodes being linked if they coauthored a paper. The database consist of journals published between 1991-98. Papers published between 1991-9x are used to reveal the network topology so that papers published in 199x+1 are used to measure (k). • In the citation network the nodes are papers published in 1988 in Physical Review Letters, and links represent the citations these articles received. T0chosen as the year 1989. • In the actor network nodes are actors which are linked if they acted together in a movie. The (k) values for actors that debuted between 1920 and 1940 are chosen. T0=1940. And evolution of new links between 1942 and 1993 is followed. • For the Internet data the investigated nodes represent Autonomous Systems (AS) and links are direct connections between them. The network structure data contains nodes from 1997 to present. The (k) was determined for nodes existing in 2000.

10. Results • The κ(k) functions are obtained for the discussed databases. If preferential attachment is absent i.e. (k) is independent of k, we expect κ(k)∝ k • However in the figures below, the increase of κ(k) is faster than linear, offering direct evidence that preferential attachment is present in each system. For internet, measurement was performed for only one year, while for citation network the κ(k) values are observed for eight different years. Citation Network Internet

11. Results(cont.) Collaboration • Furthermore, curves follow a straight line on a log-log plot, indicating that with a good approximation power law hypothesis at the beginning is valid. • In Citation network and Internet, we obtain =1.05 and =0.95 ± 0.1. For these two networks, linear preferential attachment hypothesis offers a good approximation. • For scientific collaboration and actor networks, we find <1, on the average 0.81 ±0.1 and 0.79 ±0.1 respectively. • The observed sub-linear behavior in the scientific collaboration network predicts that P(k) should follow a stretched exponential. However the measured P(k) indicates that a power law offers a better fit. This is a contradiction. Actor network

12. Internal and external links • The links in these networks don’t occur by addition of new nodes, existing nodes can link to each other. For this reason, the measurement contained both external and internal links for the latter 2 network. • For science collaboration and actor networks, there exist internal links. When determining (k) the measurement is limited first only to external links, and then only to internal links. • Note that, for the citation network, the internal links are not allowed and data resolution for the Internet does not allow us to perform the same experiment.

13. Internal and external links • A new measurement is performed for external links and internal links separately in the actor network. • Probability that a new internal link appears between two nodes with degree k1 and k2 scales with k1k2 product. • The results shows that, internal links are also governed by preferential attachment, which scales linearly with k. • The P(k) distribution in the network is believed to be majorly driven by the characteristic of the internal attachment. Preferential attachment of new internal nodes Preferential attachment of new nodes

14. Initial attractiveness • Dorogovtsev, Mendes and Samukhin have suggested that nodes with no links can acquire links so (k) should have an additive term, k0 , called initial attactiveness, so that (k) ∝ k0+k*exp(α) • According to available statistics, k0 has a small value in the 10*exp(-6). • Thus it has no effect on the scaling of K(k) at large k.

15. Results (cont.) • Summary of the investigated databases

16. Conclusion • Measurements shows preferential attachment exists in real evolving networks. • (k)follows a power law distribution, however  is system dependent. For scale-free network =1. • For Internet and citation network a linear (k) offers a reasonable fit, for actor and collaboration network attachment rate is sublinear.

17. Further Work • What is the microscopic origin of the preferential attachment? • What determines the exponent  in general?

18. Thanks for listening QUESTIONS are WELCOME