Estimating distributions of node characteristics (labels) such as number of connections or citizenship of users in a social network via edge and node sampling is a vital part of the study of complex networks. Due to its low cost, sampling via a random walk (RW) has been proposed as an attractive solution to this task. Most RW methods assume either that the network is undirected or that walkers can traverse edges regardless of their direction. Some RW methods have been designed for directed networks where edges coming into a node are not directly observable. In this work, we propose Directed Unbiased Frontier Sampling (DUFS), a sampling method based on a large number of coordinated walkers, each starting from a node chosen uniformly at random. It is applicable to directed networks with invisible incoming edges because it constructs, in real-time, an undirected graph consistent with the walkers trajectories, and due to the use of random jumps which prevent walkers from being trapped. DUFS generalizes previous RW methods and is suited for undirected networks and to directed networks regardless of in-edges visibility. We also propose an improved estimator of node label distributions that combines information from the initial walker locations with subsequent RW observations. We evaluate DUFS, compare it to other RW methods, investigate the impact of its parameters on estimation accuracy and provide practical guidelines for choosing them. In estimating out-degree distributions, DUFS yields significantly better estimates of the head of the distribution than other methods, while matching or exceeding estimation accuracy of the tail. Last, we show that DUFS outperforms uniform node sampling when estimating distributions of node labels of the top 10 largest degree nodes, even when sampling a node uniformly has the same cost as RW steps.
In the last decade, there has been a growing interest in graph sketching for processing massive networks. A sketch is a compact representation of data. Unlike a sample, a sketch is computed over the entire graph, that is observed as a data stream. For a survey on graph sketching techniques, please refer to @cite_24 .
@cite_24
1979110895 1592346261 2025622191 2016289973
Many dynamic applications are built upon large network infrastructures, such as social networks, communication networks, biological networks and the Web. Such applications create data that can be naturally modeled as graph streams, in which edges of the underlying graph are received and updated sequentially in a form of a stream. It is often necessary and important to summarize the behavior of graph streams in order to enable effective query processing. However, the sheer size and dynamic nature of graph streams present an enormous challenge to existing graph management techniques. In this paper, we propose a new graph sketch method, gSketch, which combines well studied synopses for traditional data streams with a sketch partitioning technique, to estimate and optimize the responses to basic queries on graph streams. We consider two different scenarios for query estimation: (1) A graph stream sample is available; (2) Both a graph stream sample and a query workload sample are available. Algorithms for different scenarios are designed respectively by partitioning a global sketch to a group of localized sketches in order to optimize the query estimation accuracy. We perform extensive experimental studies on both real and synthetic data sets and demonstrate the power and robustness of gSketch in comparison with the state-of-the-art global sketch method. We initiate the study of graph sketching, i.e., algorithms that use a limited number of linear measurements of a graph to determine the properties of the graph. While a graph on n nodes is essentially O(n2)-dimensional, we show the existence of a distribution over random projections into d-dimensional "sketch" space (d<< n2) such that the relevant properties of the original graph can be inferred from the sketch with high probability. Specifically, we show that: 1. d = O(n · polylog n) suffices to evaluate properties including connectivity, k-connectivity, bipartiteness, and to return any constant approximation of the weight of the minimum spanning tree. 2. d = O(n1+γ) suffices to compute graph sparsifiers, the exact MST, and approximate the maximum weighted matchings if we permit O(1 γ)-round adaptive sketches, i.e., a sequence of projections where each projection may be chosen dependent on the outcome of earlier sketches. Our results have two main applications, both of which have the potential to give rise to fruitful lines of further research. First, our results can be thought of as giving the first compressed-sensing style algorithms for graph data. Secondly, our work initiates the study of dynamic graph streams. There is already extensive literature on processing massive graphs in the data-stream model. However, the existing work focuses on graphs defined by a sequence of inserted edges and does not consider edge deletions. We think this is a curious omission given the existing work on both dynamic graphs in the non-streaming setting and dynamic geometric streaming. Our results include the first dynamic graph semi-streaming algorithms for connectivity, spanning trees, sparsification, and matching problems. A growing body of work addresses the challenge of processing dynamic graph streams: a graph is defined by a sequence of edge insertions and deletions and the goal is to construct synopses and compute properties of the graph while using only limited memory. Linear sketches have proved to be a powerful technique in this model and can also be used to minimize communication in distributed graph processing. We present the first linear sketches for estimating vertex connectivity and constructing hypergraph sparsifiers. Vertex connectivity exhibits markedly different combinatorial structure than edge connectivity and appears to be harder to estimate in the dynamic graph stream model. Our hypergraph result generalizes the work of (PODS 2012) on graph sparsification and has the added benefit of significantly simplifying the previous results. One of the main ideas is related to the problem of reconstructing subgraphs that satisfy a specific sparsity property. We introduce a more general notion of graph degeneracy and extend the graph reconstruction result of (IPDPS 2011). Over the last decade, there has been considerable interest in designing algorithms for processing massive graphs in the data stream model. The original motivation was two-fold: a) in many applications, the dynamic graphs that arise are too large to be stored in the main memory of a single machine and b) considering graph problems yields new insights into the complexity of stream computation. However, the techniques developed in this area are now finding applications in other areas including data structures for dynamic graphs, approximation algorithms, and distributed and parallel computation. We survey the state-of-the-art results; identify general techniques; and highlight some simple algorithms that illustrate basic ideas.