TU Graz: Course: 707.000 Web Science and Web Technology: Lecture 2: Small World Problem

We will discuss several examples and research efforts related to the small world problem and set the ground for our discussion of network theory and social network analysis. Readings: An Experimental Study of the Small World Problem, J. Travers and S. Milgram Sociometry 32 425-443 (1969) [Protected Access] Optional: The Strength of Weak Ties, M.S. Granovetter The American Journal of Sociology 78 1360--1380 (1973) [Protected Access] Optional: Worldwide Buzz: Planetary-Scale Views on an Instant-Messaging Network, J. Leskovec and E. Horvitz MSR-TR-2006-186. Microsoft Research, June 2007. [Web Link, the most recent and comprehensive study on the subject!] Originally from: http://kmi.tugraz.at/staff/markus/courses/SS2008/707.000_web-science

Exploring the topology of small-world networks

Complex networks have been studied for a long time in order to understand various real-world complex systems around us. Complex systems, such as the WWW, the movie-actor network, social networks and neural networks, are systems made of many non-identical elements connected by diverse interactions. The study of the network topology is one of important issues on the way of exploring such systems, because the structure always affects the system function. Traditionally, these systems have been modeled as either completely ordered graphs or completely random graphs. Until recently, some surprising empirical results in the field of complex networks, like 19 clicks of the web s diameter and 6 degrees of separation in social networks, show us the small-world phenomena existing in some large sparse networks. This finding motivates the interest in small-world networks. The objective of the project is to study the properties of small-world networks and the network evolution over time via experiments on a movie actor collaboration network; to find their different characteristics by comparing small-world networks with random networks; and to analyze the factors that result in such differences. The properties of small-world networks discussed here include small diameter, sparseness, clustering, giant component, power-law degree distribution and short path discovery. Also, four existing network models are studied in this project: Watts-Strogatz Small-world model, Erd s R nyi Random-graph model, A.-L. Barab si Scale-free model and Jon Kleinberg Small-world model

Self-similarity of complex networks

Complex networks have been studied extensively due to their relevance to many real systems as diverse as the World-Wide-Web (WWW), the Internet, energy landscapes, biological and social networks \cite{ab-review,mendes,vespignani,newman,amaral}. A large number of real networks are called ``scale-free'' because they show a power-law distribution of the number of links per node \cite{ab-review,barabasi1999,faloutsos}. However, it is widely believed that complex networks are not {\it length-scale} invariant or self-similar. This conclusion originates from the ``small-world'' property of these networks, which implies that the number of nodes increases exponentially with the ``diameter'' of the network \cite{erdos,bollobas,milgram,watts}, rather than the power-law relation expected for a self-similar structure. Nevertheless, here we present a novel approach to the analysis of such networks, revealing that their structure is indeed self-similar. This result is achieved by the application of a renormalization procedure which coarse-grains the system into boxes containing nodes within a given "size". Concurrently, we identify a power-law relation between the number of boxes needed to cover the network and the size of the box defining a finite self-similar exponent. These fundamental properties, which are shown for the WWW, social, cellular and protein-protein interaction networks, help to understand the emergence of the scale-free property in complex networks. They suggest a common self-organization dynamics of diverse networks at different scales into a critical state and in turn bring together previously unrelated fields: the statistical physics of complex networks with renormalization group, fractals and critical phenomena.Comment: 28 pages, 12 figures, more informations at http://www.jamlab.or

Multifractal analysis of complex networks

Complex networks have recently attracted much attention in diverse areas of science and technology. Many networks such as the WWW and biological networks are known to display spatial heterogeneity which can be characterized by their fractal dimensions. Multifractal analysis is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. In this paper, we introduce a new box covering algorithm for multifractal analysis of complex networks. This algorithm is used to calculate the generalized fractal dimensions $D_{q}$ of some theoretical networks, namely scale-free networks, small world networks and random networks, and one kind of real networks, namely protein-protein interaction networks of different species. Our numerical results indicate the existence of multifractality in scale-free networks and protein-protein interaction networks, while the multifractal behavior is not clear-cut for small world networks and random networks. The possible variation of $D_{q}$ due to changes in the parameters of the theoretical network models is also discussed.Comment: 18 pages, 7 figures, 4 table

Fractal Boundaries of Complex Networks

We introduce the concept of boundaries of a complex network as the set of nodes at distance larger than the mean distance from a given node in the network. We study the statistical properties of the boundaries nodes of complex networks. We find that for both Erd\"{o}s-R\'{e}nyi and scale-free model networks, as well as for several real networks, the boundaries have fractal properties. In particular, the number of boundaries nodes {\it B} follows a power-law probability density function which scales as $B^{-2}$. The clusters formed by the boundary nodes are fractals with a fractal dimension $d_{f} \approx 2$. We present analytical and numerical evidence supporting these results for a broad class of networks. Our findings imply potential applications for epidemic spreading

The World Trade Network

      This paper uses the tools of network analysis and graph theory to graphically and analytically represent the characteristics of world trade. The structure of the World Trade Network is compared over time, detecting and interpreting patterns of trade ties among countries. In particular, we assess whether the entrance of a number of new important players into the world trading system in recent years has changed the main characteristics of the existing structure of world trade, or whether the existing network was simply extended to a new group of countries. We also analyze whether the observed changes in international trade flow patterns are related to the multilateral or the regional liberalization policies. The results show that trade integration at the world level has been increasing but it is still far from being complete, with the exception of some areas, that there is a strong heterogeneity in the countries’ choice of partners, and that the WTO plays an important role in trade integration. The role of the extensive and the intensive margin of trade is also highlighted.Network analysis,International Trade,WTO,Extensive and Intensive Margins of Trade,Gravity

Link prediction methods and their accuracy for different social networks and network metrics

Currently, we are experiencing a rapid growth of the number of social–based online systems. The availability of the vast amounts of data gathered in those systems brings new challenges that we face when trying to analyse it. One of the intensively researched topics is the prediction of social connections between users. Although a lot of effort has been made to develop new prediction approaches that could provide a better prediction accuracy in social networked structures extracted from large–scale data about people and their activities and interactions, the existing methods are not comprehensively analysed. Presented in this paper, research focuses on the link prediction problem in which in a systematic way, we investigate the correlation between network metrics and accuracy of different prediction methods. For this study we selected six time–stamped real world social networks and ten most widely used link prediction methods. The results of our experiments show that the performance of some methods have a strong correlation with certain network metrics. We managed to distinguish ’prediction friendly’ networks, for which most of the prediction methods give good performance, as well as ’prediction unfriendly’ networks, for which most of the methods result in high prediction error. The results of the study are a valuable input for development of a new prediction approach which may be for example based on combination of several existing methods. Correlation analysis between network metrics and prediction accuracy of different methods may form the basis of a metalearning system where based on network characteristics and prior knowledge will be able to recommend the right prediction method for a given network at hand