The Complex Network Phenomena and Their Origin An Annotated Bibliography ESL 33C 003180159 Instructor: Gerriet Janssen Match 18, 2004
Introduction A coupled system can be described as a complex network, in which the vertices represent the elements of a system, and the edges stand for the physical or logical interaction between different elements. Recently, with the development of modern statistical physics theory and computing technology, it has become possible to quantitatively study the topological and dynamical properties of huge networks that can be mapped into various real systems. Moreover, it has been proved that research in this field has both theoretical and practical significance. These studies not only offer us a great chance to further our understanding about the mechanisms underlying the real world, it also allow us to investigate some crucial problems in the fields other than physics, such as computer science, epidemiology, economics, and sociology, from the physicists aspects. A variety of real networks have recently been studied in detail; these networks include World Wide Web, communication networks, Protein interaction networks, citation networks, and epidemic spreading networks. Many physicists have reported on the different features they observed in these kinds of networks. However, most of recent work only focus on the phenomena appearing in the networks, and does not pay much attention to the mechanism underlying these phenomena. Therefore, the answers to two key questions are still little known: what is the connection between topological characteristics of one network and its unique dynamical properties? What is the origin why a special topology is chosen by a given real networks? This 1
bibliography will review some major research work about complex networks. The achievements in these articles will not only show the wide existence of complex network phenomena in this world, but also provide a step in the direction of discovering the answers to those two questions as well. Annotated Bibliography A. Barabási, R. Albert, & H. Jeong. Scale-free characteristics of random networks: the topology of the world-wide web. Physica A, 281, 69-77(2000). In his article Scale-free characteristics of random networks: the topology of the world-wide web, A. Barabási claims that the topology of the world-wide web (WWW) network presents some characteristics that belong to the special kind of networks named scale-free. If we consider all the HTML web pages as nodes connected by the hyperlinks pointing from one to another, the whole web documents then form a huge network. There are two most important topological features for this network. First, both P out (k) and P in (k) (defined as the probability that a node have k outgoing and incoming links respectively) follow power law relation with respect to the parameter k over several orders of magnitude, and such a property is called scale-free. Second, the diameter (defined as the average number of hyperlinks that we need to follow to go from one randomly chosen web page to another) of this network is only 19, which is remarkably small comparing with 2
the huge size of this WWW network. To understand what kind of the mechanism leads to these special properties, Barabási goes on to develop a numerical model to simulate the formation process of the real WWW network. In his model, the creative point is that there exist some popular nodes like www.yahoo.com, and these nodes are so popular that most new added web pages would like to provide a hyperlink pointing to them. By simulation based on this rule, the power law distribution, as well as the rather small diameter, is obtained. The results obtained from the numerical model and the real network match with each other on a certain extent; however, there is a slight difference on power law exponent. Barabási claimed that this is probably because the numerical model can not include all the factors underlying the real WWW network, but their model does take some major ones into account. R. Guimerá, A. Arenas, A. Díaz-Guilera, & F. Giralt. Dynamical properties of model communication networks. Physical Review E, 66, 026704(2002). In his article Dynamical properties of model communication networks, Roger Guimera states that the studies about the congestion phenomena in the communication networks information delivering process, such as the internet, has a very important practical meaning. Real communication networks can be simulated as hierarchical networks, 3
in which information packets are travelling from their starting nodes to their destinations. There exist a phase transition between the free regime, in which all the information packets can reach their destinations, to the congested regime, in which some packets are blocked in some key nodes. By a control parameter ξ, which represents the relation between the number of received packets for one node and its ability to deliver packets, Guimera characterized the situation of the whole networks as three typical cases. In the critical case, in which ξ = 1, a continuous phase transition is discovered. Moreover, the block phenomena in this case are investigated in detail in three typical network topologies, and the results indicate that congestion processes in 1 D and Cayley tree networks are nearly the same; but they are different from that in 2 D networks. The author further studies the noncritical cases. For ξ < 1, the phase transition from free regime to congested regime does not occur any more; however, for ξ > 1, a discontinuous transition happens, which means the information delivering process sharply changes into such a bad situation that information even can not flow, and at last the whole network collapses. S. Lehmann, B. Lautrup, & A. Jackson. Citation networks in high energy physics. Physical Review E, 68, 026113(2003). S. Lehmann s recent work, Citation network in high energy physics is another good example of complex networks in our world. In this article, 4
the author studies the citation networks, in which the vertices are scientific publications, and an edge represents a citation from one paper pointing to another. Through basic statistics about the network composed by the data from SPIRES HEP database (a database about high-energy physics related articles), the authors demonstrates us that the citations in high energy physics and its different subfields, namely theory, experiment, phenomenology, reviews and instrumentation, have the same pattern, that is, the probability that a given article has been cited for k times by other articles follows the power law distribution P (k) k α, which means that highly cited, thus relatively important articles constitute only few percentage of all papers. At the second part of this article, Lehmann utilizes the result above to give a practical application of this research work. It is well known that how many times an article has been cited is a good reflection its quality. Similarly, the parameter r defined in this article is a criterion for evaluating the research achievements of an institute or an individual. This parameter takes many factors into account, thus the authors considers it to be an objective criterion for judging of the research work s value from many aspects. Moreover, it is especially suitable to make comparison between two research institutes, both of which publish papers in various fields. However, at the end of this article, the author emphasizes that any single parameter can not include all the meaning of this complex citation network, thus the role of a database that is entirely available to the users will be far more important than we have ever expected. 5
N. Mathias, & V. Gopal. Small worlds: how and why. Physical Review E, 63, 021117(2001). In her article Small worlds: how and why, the author Nisha Mathias states that it is the tradeoff between the highest efficiency of connectivity and the lowest cost of wiring that drives real networks to choose the topological structure of small world networks. In this article, Mathias first defines an objective function which includes the effects of both the efficiency of transportation and the cost of network s topology, and her aim is to find the special network topological structure for which this objective function reaches its minimum value, which meets the requirements of maximizing connectivity and minimizing wiring cost simultaneously. Numerical simulations then present a clear picture of how the structure of one network evolves from random to small world aiming this goal; in addition, it also demonstrates that the hubs will emerge in the networks during this optimizing process. In real networks, such as neural networks, a higher connectivity always means a more efficient way for transporting information or energy between nodes, however, the more links exist in networks, the more wasteful of space or materials they are. The simulation results in this article indicate that there exists a balance between these two contradictory requirements, and the balance is nothing but small world networks. After simulation, a comparison between the small world networks ob- 6
tained from this optimization model and the previous WS model is made in detail with respect to the small world characteristics, and this comparison reveals that the small world networks got from the optimization model are more efficient than that obtained from the WS model. Finally, the author ends her article with the conclusion that that any efficient transportation networks will have the structure of small world. D. Volchenkov, L. Volchenkova, & Ph. Blanchard. Epidemic spreading in a variety of scale free networks. Physical Review E, 66, 046137(2002). In the article Epidemic spreading in a variety of scale free networks, the author D. Volchenkov claims that epidemic spreading properties are very sensitive to the characteristics of communication networks among people. It has been known for a long time that scale free networks can represent the communication network in our daily life, in which each node stands for a person. In this article, the author developed a epidemic spreading model based on the scale free network. In the model, only a few ratio of population are infected initially, and the virus can spread via contact between people. Based on the evolution equation that describes how the fraction of infected people varies, the author first discusses the stationary case, namely the final fraction of infected people. Results indicate that the stationary fraction increases with the effective spreading rate of the virus, but the effective ways for 7
preventing epidemic spread in two typical kinds of societies are divergent. In the unstructured society, the more popular an individual is, the more probable s/he is chosen to be partner by others. Calculations imply that the effective program in this society is to decentralize the network, say, decrease the contact between the popular star and others. However, things are quite different for the structured society, in which one prefer to choose partner in his same class, say, s/he would like to make friends within the equivalent social class. The efficient way for preventing the epidemic spread in the structured society should be to vaccinate the hubs and increase their communication with others. Following the stationary case, the author continues discussing the dynamical properties of the evolution equation, and finally finds an expression for relaxation time, in which the initial distribution of infected people in this network underlies the actual disease spreading. This parameter is very important since it indicates about how long the epidemic spreading will reach its stationary solution, perhaps in which a large number of the population are infected. Conclusion Though it has been only less than ten years since researchers began to study the phenomena in the field of complex networks, a variety of interesting phenomena have been reported in different kinds of real networks, and obviously, this implies the broad application of this new born subject. However, as pointed out in the 8
Introduction, two basic questions about the complex networks still require further studies: what are the connections between one network s topological and dynamical properties? What is the origin why a given real network chooses a special topological structure? In the articles cited above, the authors Barabási, Lehmann, Volchenkov, Guimera investigate the special features in four different kinds of real networks, and their studies indicate the relation between the topological structure and dynamical properties in WWW network, citation networks, epidemic spreading networks, and communication networks respectively. Moreover, the studies of Barabási and Mathias give the possible explanation of the origin of scale-free and small-world networks. In conclusion, the studies aimed to those two key questions have been carried in some typical networks; however, the general theories that can be applied into each kind of networks are still not discovered. The field of complex networks is rapidly developing. Due to its young age and importance, we can foresee this prolific subject has great potential to discover new theories that have very practical applications. So far we lack some important knowledge that would help us to overcome those two central questions; however, recent research work, such as articles cited in this bibliography, does present a good path towards this final destination. 9