We introduce random directed acyclic graph and use it to model the information diffusion network. Subsequently, we analyze the cascade generation model (CGM) introduced by [19]. Until now only empirical studies of this model were done. In this paper, we present the first theoretical proof that the sizes of cascades generated by the CGM follow the power-law distribution, which is consistent with multiple empirical analysis of the large social networks. We compared the assumptions of our model with the Twitter social network and tested the goodness of approximation.
@cite_5: Due to its fast, dynamic, and distributed growth process, it is hard to obtain an accurate map of the Internet. In many cases, such a map-representing the structure of the Internet as a graph with nodes and links-is a prerequisite when investigating properties of the Internet. A common way to obtain such maps is to make certain local measurements at a small subset of the nodes, and then to combine these in order to "discover" (an approximation of) the actual graph. Each of these measurements is potentially quite costly. It is thus a natural objective to minimize the number of measurements which still discover the whole graph. We formalize this problem as a combinatorial optimization problem and consider it for two different models characterized by different types of measurements. We give several upper and lower bounds on the competitive ratio (for the online network discovery problem) and the approximation ratio (for the offline network verification problem) in both models. Furthermore, for one of the two models, we compare four simple greedy strategies in an experimental analysis
@cite_18: Breadth First Search (BFS) and other graph traversal techniques are widely used for measuring large unknown graphs, such as online social networks. It has been empirically observed that incomplete BFS is biased toward high degree nodes. In contrast to more studied sampling techniques, such as random walks, the bias of BFS has not been characterized to date. In this paper, we quantify the degree bias of BFS sampling. In particular, we calculate the node degree distribution expected to be observed by BFS as a function of the fraction of covered nodes, in a random graph RG(p k ) with a given (and arbitrary) degree distribution p k . Furthermore, we also show that, for RG(p k ), all commonly used graph traversal techniques (BFS, DFS, Forest Fire, and Snowball Sampling) lead to the same bias, and we show how to correct for this bias. To give a broader perspective, we compare this class of exploration techniques to random walks that are well-studied and easier to analyze. Next, we study by simulation the effect of graph properties not captured directly by our model. We find that the bias gets amplified in graphs with strong positive assortativity. Finally, we demonstrate the above results by sampling the Facebook social network, and we provide some practical guidelines for graph sampling in practice.
@cite_11: This paper presents a detailed examination of how the dynamic and heterogeneous nature of real-world peer-to-peer systems can introduce bias into the selection of representative samples of peer properties (e.g., degree, link bandwidth, number of files shared). We propose the Metropolized Random Walk with Backtracking (MRWB) as a viable and promising technique for collecting nearly unbiased samples and conduct an extensive simulation study to demonstrate that our technique works well for a wide variety of commonly-encountered peer-to-peer network conditions. We have implemented the MRWB algorithm for selecting peer addresses uniformly at random into a tool called ion-sampler. Using the Gnutella network, we empirically show that ion-sampler. yields more accurate samples than tools that rely on commonly-used sampling techniques and results in dramatic improvements in efficiency and scalability compared to performing a full crawl.
Results similar to ours were also obtained in the study on the bias of traceroute sampling. Achlioptas traceroute1 characterize the degree distribution of a BFS tree for a random graph with a given degree distribution. Their explanation why the degree distribution under traceroute sampling exhibits power-law motivated researchers to study bias in P2P systems @cite_11 and network discovery @cite_5 . Their research also resulted in the development of the new tools in the social networks sampling @cite_18 .