Mathematics

It is commonly believed that real networks are scale-free and the fraction of nodes P (k) with degree k satisfies the power-law P (k) ∝ k −γ for k > kmin > 0. Preferential attachment is the mechanism that has been considered responsible for such organization of these networks. In many real networks, degree distribution before the kmin varies very slowly to the extent of being uniform as compared to the degree distribution for k > kmin. In this paper, we propose a model that describes this particular degree distribution for the whole range of k > 0. We adopt a two step approach. In the first step, at every time stamp we add a new node to the network and attach it to an existing node using preferential attachment method. In the second step, we add edges between existing pairs of nodes with the node selection based on the uniform probability distribution. Our approach generates weakly scale-free networks that closely follow the degree distribution of real-world networks. We perform a comprehensive mathematical analysis of the model in the discrete domain and compare the degree distribution generated by this model with that of real-world networks.

Graph Sampling provides an efficient yet inexpensive solution for analyzing large graphs. While extracting small representative subgraphs from large graphs, the challenge is to capture the properties of the original graph. Several sampling algorithms have been proposed in previous studies, but they lack in extracting good samples. In this paper, we propose a new sampling method called Weighted Edge Sampling. In this method, we give equal weight to all the edges in the beginning. During the sampling process, we sample an edge with the probability proportional to its weight. When an edge is sampled, we increase the weight of its neighboring edges and this increases their probability to be sampled. Our method extracts the neighborhood of a sampled edge more efficiently than previous approaches. We evaluate the efficacy of our sampling approach empirically using several real-world data sets and compare it with some of the previous approaches. We find that our method produces samples tha...

It is commonly believed that real networks are scale-free and fraction of nodes $P(k)$ with degree $k$ satisfies the power law $P(k) \propto k^{-\gamma} \text{ for } k > k_{min} > 0$. Preferential attachment is the mechanism that has been considered responsible for such organization of these networks. In many real networks, degree distribution before the $k_{min}$ varies very slowly to the extent of being uniform as compared with the degree distribution for $k > k_{min}$ . In this paper, we proposed a model that describe this particular degree distribution for the whole range of $k>0$. We adopt a two step approach. In the first step, at every time stamp we add a new node to the network and attach it with an existing node using preferential attachment method. In the second step, we add edges between existing pairs of nodes with the node selection based on the uniform probability distribution. Our approach generates weakly scale-free networks that closely follow the degree...

Graph sampling allows mining a small representative subgraph from a big graph. Sampling algorithms deploy different strategies to replicate the properties of a given graph in the sampled graph. In this study, we provide a comprehensive empirical characterization of five graph sampling algorithms on six properties of a graph including degree, clustering coefficient, path length, global clustering coefficient, assorta-tivity, and modularity. We extract samples from fifteen graphs grouped into five categories including collaboration, social, citation, technological , and synthetic graphs. We provide both qualitative and quantitative results. We find that there is no single method that extracts true samples from a given graph with respect to the properties tested in this work. Our results show that the sampling algorithm that aggressively explores the neighborhood of a sampled node performs better than the others.