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 sample subgraph induced by a randomly selected set of nodes has been discussed in several works on graph sampling @cite_3 @cite_11 @cite_27 @cite_13 . Selecting nodes randomly ensures that nodes of a given degree are chosen with probability proportional to the number of such nodes in the network. The selected set of nodes have a degree distribution very similar to that of the original graph, but these degrees are the degrees of the nodes in the original graph @math and not the degrees in the induced subgraph @math . There are at least two additional problems with such a sampling strategy: (i) when the desired sample size is as small as 5 is highly likely to be a disconnected graph even if @math is connected and thus, unrepresentative; and (ii) in real networks that have to be crawled, it is usually very hard or infeasible to generate a statistically valid set of uncorrelated random nodes from the full graph @math given that the full graph is not known (even though a few random nodes can always be selected from within the known portion of the graph).
@cite_27 @cite_13 @cite_3 @cite_11
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We discuss two sampling schemes for selecting random subnets from a network, random sampling and connectivity dependent sampling, and investigate how the degree distribution of a node in the network is affected by the two types of sampling. Here we derive a necessary and sufficient condition that guarantees that the degree distributions of the subnet and the true network belong to the same family of probability distributions. For completely random sampling of nodes we find that this condition is satisfied by classical random graphs; for the vast majority of networks this condition will, however, not be met. We furthermore discuss the case where the probability of sampling a node depends on the degree of a node and we find that even classical random graphs are no longer closed under this sampling regime. We conclude by relating the results to real Eschericia coli protein interaction network data. We consider secure and reliable connectivity in wireless sensor networks that utilize the heterogeneous random key predistribution scheme. We model the unreliability of wireless links by an on off channel model that induces an Erds-Renyi graph, while the heterogeneous scheme induces an inhomogeneous random key graph. The overall network can thus be modeled by the intersection of both graphs. We present conditions (in the form of zero-one laws) on how to scale the parameters of the intersection model, so that with high probability: i) all of its nodes are connected to at least @math other nodes, i.e., the minimum node degree of the graph is no less than @math , and ii) the graph is @math -connected, i.e., the graph remains connected even if any @math nodes leave the network. These results are shown to complement and generalize several previous results in the literature. We also present numerical results to support our findings in the finite-node regime. Finally, we demonstrate via simulations that our results are also useful when the on off channel model is replaced with the more realistic disk communication model . We consider the problem of finding a minimum edge cost subgraph of a graph satisfying both given node-connectivity requirements and degree upper bounds on nodes. We present an iterative rounding algorithm of the biset linear programming relaxation for this problem. For directed graphs and @math -out-connectivity requirements from a root, our algorithm computes a solution that is a 2-approximation on the cost, and the degree of each node @math in the solution is at most @math , where @math is the degree upper bound on @math . For undirected graphs and element-connectivity requirements with maximum connectivity requirement @math , our algorithm computes a solution that is a @math -approximation on the cost, and the degree of each node @math in the solution is at most @math . These ratios improve the previous @math -approximation on the cost and @math -approximation on the degrees. Our algorithms can be used to improve approximation ratios for other node-connectivity problems such as undirected $k... Considerable attention has been focused on the properties of graphs derived from Internet measurements. Router-level topologies collected via traceroute-like methods have led some to conclude that the router graph of the Internet is well modeled as a power-law random graph. In such a graph, the degree distribution of nodes follows a distribution with a power-law tail. We argue that the evidence to date for this conclusion is at best insufficient We show that when graphs are sampled using traceroute-like methods, the resulting degree distribution can differ sharply from that of the underlying graph. For example, given a sparse Erdos-Renyi random graph, the subgraph formed by a collection of shortest paths from a small set of random sources to a larger set of random destinations can exhibit a degree distribution remarkably like a power-law. We explore the reasons for how this effect arises, and show that in such a setting, edges are sampled in a highly biased manner. This insight allows us to formulate tests for determining when sampling bias is present. When we apply these tests to a number of well-known datasets, we find strong evidence for sampling bias.