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.
A related strategy is one called Forest Fire , first introduced in @cite_8 . In this method, as in Snowball sampling, we choose a random node from the known part of @math and use a breadth-first approach. With a forward burning probability'' @math , the node burns links attached to it. The nodes at the other end of a burned link are added to the sample subgraph and they now continue spreading the fire'' by burning links attached to them. This continues until the desired size of the sample subgraph is achieved. It has been found in @cite_26 that the Forest Fire sampling strategy works best with @math and this is what we use in all our simulations in this report. In general, it has been found that methods based on BFS search are likely to overestimate node-degrees and underestimate symmetry @cite_5 . As we will show later, the Forest Fire sampling strategy, being based on a scaled-down'' BFS, is not entirely able to reduce the likelihood of adding high-degree nodes to the sample subgraph.
@cite_5 @cite_26 @cite_8
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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. Breadth First Search (BFS) is a widely used approach for sampling large graphs. However, it has been empirically observed that BFS sampling is biased toward high-degree nodes, which may strongly affect the measurement results. In this paper, we quantify and correct the degree bias of BFS. First, we consider a random graph RG(pk) with an arbitrary degree distribution pk. For this model, we calculate the node degree distribution expected to be observed by BFS as a function of the fraction f of covered nodes. We also show that, for RG(pk), all commonly used graph traversal techniques (BFS, DFS, Forest Fire, Snowball Sampling, RDS) have exactly the same bias. Next, we propose a practical BFS-bias correction procedure that takes as input a collected BFS sample together with the fraction f. Our correction technique is exact (i.e., leads to unbiased estimation) for RG(pk). Furthermore, it performs well when applied to a broad range of Internet topologies and to two large BFS samples of Facebook and Orkut networks. To face fire it is crucial to understand its behaviour in order to maximize fighting means. To achieve this task, the development of a metrological tool is necessary for estimating both geometrical and physical parameters involved in forest fire modelling. A key parameter is to estimate fire positions accurately. In this paper an image processing tool especially dedicated to an accurate extraction of fire from an image is presented. In this work, the clustering on several colour spaces is investigated and it appears that the blue chrominance Cb from the YCbCr colour space is the most appropriate. As a consequence, a new segmentation algorithm dedicated to forest fire applications has been built using first an optimized k-means clustering in the Cb-channel and then some properties of fire pixels in the RGB colour space. Next, the performance of the proposed method is evaluated using three supervised evaluation criteria and then compared to other existing segmentation algorithms in the literature. Finally a conclusion is drawn, assessing the good behaviour of the developed algorithm. Recently, there has been a great deal of research focusing on the development of sampling algorithms for networks with small-world and or power-law structure. The peerto-peer research community (e.g., [7]) have used sampling to quickly explore and obtain a good representative sample of the network topology, as these networks are hard to explore completely and have significant amounts of churn in their topology. For collecting data from social networks, researchers often use snowball sampling (e.g., [2]) due to the lack of access to the complete graph. have developed Forest Fire Sampling, which uses a hybrid combination of snowball sampling and random-walk sampling to produce samples that match the temporal evolution of the underlying social network [5]. have developed a Metropolis algorithm which samples in a manner designed to match desired properties in the original network [3]. Although there has been a great deal of research focusing on the the development of sampling algorithms, much of this work is based on empirical study and evaluation (i.e., measuring the similarity between sampled and original network properties). There has been some work (e.g., [4, 8, 6]) that has studied the statistical properties of samples of complex networks produced by traditional sampling algorithms such as node sampling, edge sampling and random walks. However, there has been relatively little attention paid to the development of a theoretical foundation for sampling from networks—including a formal framework for sampling, an understanding of various network characteristics and their dependencies, and an analysis of their impact on the accuracy of sampling algorithms. In this paper, we reconsider the foundations of network sampling and attempt to formalize the goals, and process of, sampling, in order to frame future development and analysis of sampling algorithms.