Measuring the Sensitivity of Graph Metrics to Missing Data

EPL (Europhysics Letters), 2014

Identifying the nodes of small sub-graphs with no a priori information is a hard problem. In this work, we want to find each node of a sparse sub-graph embedded in both dynamic and static background graphs, of larger average degree. We show that exploiting the summability over several background realizations of the Estrada-Benzi communicability and the Krylov approximation of the matrix exponential, it is possible to recover the sub-graph with a fast algorithm with computational complexity O(N n). Relaxing the problem to complete sub-graphs, the same performance is obtained with a single background. The worst case complexity for the single background is O( n log(n)).

Graph sampling allows mining a small representative subgraph from a big graph. Sampling algorithms deploy different strategies to replicate the properties of a given graph in the sampled graph. In this study, we provide a comprehensive empirical characterization of five graph sampling algorithms on six properties of a graph including degree, clustering coefficient, path length, global clustering coefficient, assorta-tivity, and modularity. We extract samples from fifteen graphs grouped into five categories including collaboration, social, citation, technological , and synthetic graphs. We provide both qualitative and quantitative results. We find that there is no single method that extracts true samples from a given graph with respect to the properties tested in this work. Our results show that the sampling algorithm that aggressively explores the neighborhood of a sampled node performs better than the others.

Proceedings of the 9th International Conference on Applied Informatics, Volume 2, 2015

In this paper we consider three different properties of social networks, namely the number of the different triangles, the sizes of the strongly connected components (SCC) and Linerank, which is offered as a substitute for betweenness and can be efficiently calculated on graphs with more than 10 9 nodes [7]. We implemented the computation of these measures in three different ways: sequentially, using the widely spread MapReduce model-Hadoop in our case-and its newly emerging rival Giraph, which is based on the Pregel system [11]. MapReduce and Pregel use distributed computation enabling high parallelization of the calculations. The algorithms calculating the above measures can be grouped into three different families. In the case of the triangles for a node only its close neighbourhood should be taken into consideration. In the case of the SCC's at least in the sequential approach the graph should be traversed in a depth search manner, which can be prohibitively expensive for the distributive systems. Finally, Linerank can be computed by using an iterative matrix-vector multiplication. We ran our implementations on several generated and real-world networks of different sizes firstly in order to assess the degree of the benefit of parallelization offered by the last two technologies for the algorithm families. Secondly, to see for which magnitude of the size of the networks it is more beneficial to choose one the aforementioned models instead of the sequential approach.

The study of a large real world network in terms of graph sample representation constitutes a very powerful and useful tool in several domains of network analysis. This is the motivation that has led the work of this paper towards the development of a new graph sampling algorithm. Previous research in this area proposed simple processes such as the classic Random Walk algorithm, Random node and Random edge sampling and has evolved during the last decade to more advanced graph exploration approaches such as Forest Fire and Frontier sampling. In this paper, we propose a new graph sampling method based on edge selection. In addition, we crawled Facebook collecting a large dataset consisting of 10 million users and 80 million users’ relations, which we have also used to evaluate our sampling algorithm. The experimental evaluation on several datasets proves that our approach preserves several properties of the initial graphs, leading to representative samples and outperforms all the other approaches.

Data Mining and Knowledge Discovery, 2020

Large real-world graphs claim lots of resources in terms of memory and computational power to study them and this makes their full analysis extremely challenging. In order to understand the structure and properties of these graphs, we intend to extract a small representative subgraph from a big graph while preserving its topology and characteristics. In this work, we aim at producing good samples with sample size as low as 0.1% while maintaining the structure and some of the key properties of a network. We exploit the fact that average values of degree and clustering coefficient of a graph can be estimated accurately and efficiently. We use the estimated values to guide the sampling process and extract tiny samples that preserve the properties of the graph and closely approximate their distributions in the original graph. The distinguishing feature of our work is that we apply traversal based sampling that utilizes only the local information of nodes as opposed to the global information of the network and this makes our approach a practical choice for crawling online networks. We evaluate the effectiveness of our sampling technique using real-world datasets and show that it surpasses the existing methods.

2015

Using the replica method, we develop an analytical approach to compute the characteristic function for the probability PN(K,λ) that a large N × N adjacency matrix of sparse random graphs has K eigenvalues below a threshold λ. The method allows to determine, in principle, all moments of PN(K,λ), from which the typical sample to sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with N ≫ 1 for |λ| > 0, with a model-dependent prefactor that can be exactly calculated. Explicit results are discussed for Erdös-Rényi and regular random graphs, both exhibiting a prefactor with a non-monotonic behavior as a function of λ. These results contrast with rotationally invariant random matrices, where the index variance scales only as lnN , with an universal prefactor that is independent of λ. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support ...

2006

Abstract 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 (eg, the diameter), to get estimates for the large graph.

SIAM Journal on Scientific Computing, 2010

We address the problem of obtaining by local computation a sample of a graph such that the vertices are sampled uniformly. The presented solution uses a Markov chain to combine a rapidly mixing random walk that does not reach a uniform distribution with a slow-mixing walk that does. The resulting chain mixes at a notably faster rate than the standard uniform random walk and can be simulated exactly using only local information.

Physical Review E, 2015

Using the replica method, we develop an analytical approach to compute the characteristic function for the probability PN (K, λ) that a large N × N adjacency matrix of sparse random graphs has K eigenvalues below a threshold λ. The method allows to determine, in principle, all moments of PN (K, λ), from which the typical sample to sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with N ≫ 1 for |λ| > 0, with a model-dependent prefactor that can be exactly calculated. Explicit results are discussed for Erdös-Rényi and regular random graphs, both exhibiting a prefactor with a non-monotonic behavior as a function of λ. These results contrast with rotationally invariant random matrices, where the index variance scales only as ln N , with an universal prefactor that is independent of λ. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support the Gaussian nature of the index fluctuations.