text: Many real-world networks are prohibitively large for data retrieval, storage and analysis of all of its nodes and links. Understanding the structure and dynamics of these networks entails creating a smaller representative sample of the full graph while preserving its relevant topological properties. In this report, we show that graph sampling algorithms currently proposed in the literature are not able to preserve network properties even with sample sizes containing as many as 20 of the nodes from the original graph. We present a new sampling algorithm, called Tiny Sample Extractor, with a new goal of a sample size smaller than 5 of the original graph while preserving two key properties of a network, the degree distribution and its clustering co- efficient. Our approach is based on a new empirical method of estimating measurement biases in crawling algorithms and compensating for them accordingly. We present a detailed comparison of best known graph sampling algorithms, focusing in particular on how the properties of the sample subgraphs converge to those of the original graph as they grow. These results show that our sampling algorithm extracts a smaller subgraph than other algorithms while also achieving a closer convergence to the degree distribution, measured by the degree exponent, of the original graph. The subgraph generated by the Tiny Sample Extractor, however, is not necessarily representative of the full graph with regard to other proper ties such as assortativity. This indicates that the problem of extracting a truly representative small subgraph from a large graph remains unsolved.
  @cite_6: We study the algorithmic and structural properties of very large, realistic social contact networks. We consider the social network for the city of Portland, Oregon, USA, developed as a part of the TRANSIMS EpiSims project at the Los Alamos National Laboratory. The most expressive social contact network is a bipartite graph, with two types of nodes: people and locations; edges represent people visiting locations on a typical day. Three types of results are presented. (i) Our empirical results show that many basic characteristics of the dataset are well-modeled by a random graph approach suggested by Fan Chung Graham and Lincoln Lu (the CL-model), with a power-law degree distribution. (ii) We obtain fast approximation algorithms for computing basic structural properties such as clustering coefficients and shortest paths distribution. We also study the dominating set problem for such networks; this problem arose in connection with optimal sensor-placement for disease-detection. We present a fast approximation algorithm for computing near-optimal dominating sets. (iii) Given the close approximations provided by the CL-model to our original dataset and the large data-volume, we investigate fast methods for generating such random graphs. We present methods that can generate such a random network in near-linear time, and show that these variants asymptotically share many key features of the CL-model, and also match the Portland social network.The structural results have been used to study the impact of policy decisions for controlling large-scale epidemics in urban environments.
  @cite_12: In a time of information glut, observations about complex systems and phenomena of interest are available in several applications areas, such as biology and text. As a consequence, scientists have started searching for patterns that involve interactions among the objects of analysis, to the effect that research on models and algorithms for network analysis has become a central theme for knowledge discovery and data mining (KDD). The intuitions behind the plethora of approaches rely upon few basic types of networks, identified by specific local and global topological properties, which we term "pure" topology types.In this paper, (1) we survey pure topology types along with existing sampling algorithms that generate them, (2) we introduce novel algorithms that enhance the diversity of samples, and address the case of cellular topologies, (3) we perform statistical studies of the stability of the properties of pure types to alternative generative algorithms, and a joint study of the separability of pure types, in terms of their embedding in a space of metrics for network analysis, widely adopted in the social and physical sciences.We conclude with a word of caution to the practitioners, who sample pure topology types to assess the "statistical significance" of their findings, e.g., the p-value of the clustering coefficient is sensitive to the sampling algorithm used. We find that different pure types share similar topological properties. Further, real world networks hardly present the variability profile of a single pure type. We suggest the assumption of "mixtures of types" as an alternative starting point for developing models and algorithms for network analysis.
summary: A variety of strategies have been used to shrink'' a graph for purposes of analysis. Shrinking a graph by extracting an actual subgraph allows one to discover patterns in the subgraph that can be validated later in the original graph because of the one-to-one correspondence between the nodes in the two graphs. The generation of synthetic topologies which have a specific set of properties in them @cite_12 @cite_6 , though useful in many contexts, is not considered in this report.