Eigenvalues of a graph are of high interest in graph analytics for Big Data due to their relevance to many important properties of the graph including network resilience, community detection and the speed of viral propagation. Accurate computation of eigenvalues of extremely large graphs is usually not feasible due to the prohibitive computational and storage costs and also because full access to many social network graphs is often restricted to most researchers. In this paper, we present a series of new sampling algorithms which solve both of the above-mentioned problems and estimate the two largest eigenvalues of a large graph efficiently and with high accuracy. Unlike previous methods which try to extract a subgraph with the most influential nodes, our algorithms sample only a small portion of the large graph via a simple random walk, and arrive at estimates of the two largest eigenvalues by estimating the number of closed walks of a certain length. Our experimental results using real graphs show that our algorithms are substantially faster while also achieving significantly better accuracy on most graphs than the current state-of-the-art algorithms.

Monitoring key properties of dynamic graphs in real time can help detect spam accounts in social networks, identify anomalies and intrusions, and inform resource management. Traditional frameworks for dynamic graphs have relied on processing only the stream of edges added into or deleted from an evolving graph, but not any additional related information such as the degrees or neighbor lists of nodes associated with the edges. In this paper, we propose a new edge sampling framework for big-graph analytics in dynamic graphs which enhances the traditional model and offers significantly improved trade-off between accuracy and computational/memory costs in the running estimates of a graph's properties. To demonstrate this higher accuracy using the framework, we present a new sampling algorithm, called Edge Sample and Discard, which generates an unbiased estimate of the total number of triangles in a fully dynamic graph. The experimental results show that, with the help of the neighborhood information of the sampled edges, the accuracy achieved by our algorithm is substantially better than the accuracy achieved by current state-of-the-art algorithms.

The spectral radius of a graph, i.e., the largest eigenvalue of its adjacency matrix, is of high interest in graph analytics for Big Data due to its relevance to many important properties of the graph including network resilience, community detection and the speed of viral propagation. Accurate computation of the largest eigenvalue of extremely large graphs is infeasible due to the prohibitive computational and storage costs, and also because full access to many social network graphs is often restricted to most researchers. In this paper, we present a new and efficient algorithm called Closed Walk Sampler (cWalker) which solves both of the above-mentioned problems. Unlike previous methods which try to extract a subgraph with the most influential nodes, the cWalker samples only a small portion of the large graph via a simple random walk, and arrives at an estimate of the largest eigenvalue by estimating the number of closed walks of a certain length. Our experimental results using real graphs show that the cWalker is substantially faster while also achieving significantly better accuracy on most graphs than the current state-of-the-art algorithms.

Algorithms for mining very large graphs, such as those representing online social networks, to discover the relative frequency of small subgraphs within them are of high interest to sociologists, computer scientists and marketeers alike. However, the computation of these network motif statistics via naive enumeration is infeasible for either its prohibitive computational costs or access restrictions on the full graph data. Methods to estimate the motif statistics based on random walks by sampling only a small fraction of the subgraphs in the large graph address both of these challenges. In this paper, we present a new algorithm, called the Waddling Random Walk (WRW), which estimates the concentration of motifs of any size. It derives its name from the fact that it sways a little to the left and to the right, thus also sampling nodes not directly on the path of the random walk. The WRW algorithm achieves its computational efficiency by not trying to enumerate subgraphs around the random walk but instead using a randomized protocol to sample subgraphs in the neighborhood of the nodes visited by the walk. In addition, WRW achieves significantly higher accuracy (measured by the closeness of its estimate to the correct value) and higher precision (measured by the low variance in its estimations) than the current state-of-the-art algorithms for mining subgraph statistics. We illustrate these advantages in speed, accuracy and precision using simulations on well-known and widely used graph datasets representing real networks.