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 set of sampling strategies is based on selecting random edges instead of random nodes or a combination of node and edge sampling @cite_11 . In general, however, edge sampling does not overcome the problems of node sampling mentioned above. Sampling strategies based on random deletion @cite_2 @cite_28 instead of selection also suffer the same problems and are not suitable as solutions to the problem statement expressed in Section . Node or edge sampling is useful in contexts where the goal is to infer properties of nodes but not necessarily the topological properties of the graph. The choice of nodes guided by simulated annealing can target a specific set of topological properties @cite_27 , but this method also relies on randomly choosing nodes from the entire network.
@cite_28 @cite_27 @cite_2 @cite_11
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Given a huge real graph, how can we derive a representative sample? There are many known algorithms to compute interesting measures (shortest paths, centrality, betweenness, etc.), but several of them become impractical for large graphs. Thus graph sampling is essential.The natural questions to ask are (a) which sampling method to use, (b) how small can the sample size be, and (c) how to scale up the measurements of the sample (e.g., the diameter), to get estimates for the large graph. The deeper, underlying question is subtle: how do we measure success?.We answer the above questions, and test our answers by thorough experiments on several, diverse datasets, spanning thousands nodes and edges. We consider several sampling methods, propose novel methods to check the goodness of sampling, and develop a set of scaling laws that describe relations between the properties of the original and the sample.In addition to the theoretical contributions, the practical conclusions from our work are: Sampling strategies based on edge selection do not perform well; simple uniform random node selection performs surprisingly well. Overall, best performing methods are the ones based on random-walks and "forest fire"; they match very accurately both static as well as evolutionary graph patterns, with sample sizes down to about 15 of the original graph. Mapping the Internet generally consists in sampling the network from a limited set of sources by using "traceroute"-like probes. This methodology, akin to the merging of different spanning trees to a set of destinations, has been argued to introduce uncontrolled sampling biases that might produce statistical properties of the sampled graph which sharply differ from the original ones. Here we explore these biases and provide a statistical analysis of their origin. We derive a mean-field analytical approximation for the probability of edge and vertex detection that exploits the role of the number of sources and targets and allows us to relate the global topological properties of the underlying network with the statistical accuracy of the sampled graph. In particular we find that the edge and vertex detection probability is depending on the betweenness centrality of each element. This allows us to show that shortest path routed sampling provides a better characterization of underlying graphs with scale-free topology. We complement the analytical discussion with a throughout numerical investigation of simulated mapping strategies in different network models. We show that sampled graphs provide a fair qualitative characterization of the statistical properties of the original networks in a fair range of different strategies and exploration parameters. The numerical study also allows the identification of intervals of the exploration parameters that optimize the fraction of nodes and edges discovered in the sampled graph. This finding might hint the steps toward more efficient mapping strategies. Networks are characterized by nodes and edges. While there has been a spate of recent work on estimating the number of nodes in a network, the edge-estimation question appears to be largely unaddressed. In this work we consider the problem of estimating the average degree of a large network using efficient random sampling, where the number of nodes is not known to the algorithm. We propose a new estimator for this problem that relies on access to node samples under a prescribed distribution. Next, we show how to efficiently realize this ideal estimator in a random walk setting. Our estimator has a natural and simple implementation using random walks; we bound its performance in terms of the mixing time of the underlying graph. We then show that our estimators are both provably and practically better than many natural estimators for the problem. Our work contrasts with existing theoretical work on estimating average degree, which assume that a uniform random sample of nodes is available and the number of nodes is known. The graph embedding paradigm projects nodes of a graph into a vector space, which can facilitate various downstream graph analysis tasks such as node classification and clustering. To efficiently learn node embeddings from a graph, graph embedding techniques usually preserve the proximity between node pairs sampled from the graph using random walks. In the context of a heterogeneous graph, which contains nodes from different domains, classical random walks are biased towards highly visible domains where nodes are associated with a dominant number of paths. To overcome this bias, existing heterogeneous graph embedding techniques typically rely on meta-paths (i.e., fixed sequences of node types) to guide random walks. However, using these meta-paths either requires prior knowledge from domain experts for optimal meta-path selection, or requires extended computations to combine all meta-paths shorter than a predefined length. In this paper, we propose an alternative solution that does not involve any meta-path. Specifically, we propose JUST, a heterogeneous graph embedding technique using random walks with JUmp and STay strategies to overcome the aforementioned bias in an more efficient manner. JUST can not only gracefully balance between homogeneous and heterogeneous edges, it can also balance the node distribution over different domains (i.e., node types). By conducting a thorough empirical evaluation of our method on three heterogeneous graph datasets, we show the superiority of our proposed technique. In particular, compared to a state-of-the-art heterogeneous graph embedding technique Hin2vec, which tries to optimally combine all meta-paths shorter than a predefined length, our techn