Computing Vertex Centrality Measures in Massive Real Networks with a Neural Learning Model

Vertex centrality measures are a multi-purpose analysis tool, commonly used in many application environments to retrieve information and unveil knowledge from the graphs and network structural properties. However, the algorithms of such metrics are expensive in terms of computational resources when running real-time applications or massive real world networks. Thus, approximation techniques have been developed and used to compute the measures in such scenarios. In this paper, we demonstrate and analyze the use of neural network learning algorithms to tackle such task and compare their performance in terms of solution quality and computation time with other techniques from the literature. Our work offers several contributions. We highlight both the pros and cons of approximating centralities though neural learning. By empirical means and statistics, we then show that the regression model generated with a feedforward neural networks trained by the Levenberg-Marquardt algorithm is not only the best option considering computational resources, but also achieves the best solution quality for relevant applications and large-scale networks. Keywords: Vertex Centrality Measures, Neural Networks, Complex Network Models, Machine Learning, Regression ModelComment: 8 pages, 5 tables, 2 figures, version accepted at IJCNN 2018. arXiv admin note: text overlap with arXiv:1810.1176

HyperANF: Approximating the Neighbourhood Function of Very Large Graphs on a Budget

The neighbourhood function N(t) of a graph G gives, for each t, the number of pairs of nodes such that y is reachable from x in less that t hops. The neighbourhood function provides a wealth of information about the graph (e.g., it easily allows one to compute its diameter), but it is very expensive to compute it exactly. Recently, the ANF algorithm (approximate neighbourhood function) has been proposed with the purpose of approximating NG(t) on large graphs. We describe a breakthrough improvement over ANF in terms of speed and scalability. Our algorithm, called HyperANF, uses the new HyperLogLog counters and combines them efficiently through broadword programming; our implementation uses overdecomposition to exploit multi-core parallelism. With HyperANF, for the first time we can compute in a few hours the neighbourhood function of graphs with billions of nodes with a small error and good confidence using a standard workstation. Then, we turn to the study of the distribution of the shortest paths between reachable nodes (that can be efficiently approximated by means of HyperANF), and discover the surprising fact that its index of dispersion provides a clear-cut characterisation of proper social networks vs. web graphs. We thus propose the spid (Shortest-Paths Index of Dispersion) of a graph as a new, informative statistics that is able to discriminate between the above two types of graphs. We believe this is the first proposal of a significant new non-local structural index for complex networks whose computation is highly scalable

Injecting Uncertainty in Graphs for Identity Obfuscation

Data collected nowadays by social-networking applications create fascinating opportunities for building novel services, as well as expanding our understanding about social structures and their dynamics. Unfortunately, publishing social-network graphs is considered an ill-advised practice due to privacy concerns. To alleviate this problem, several anonymization methods have been proposed, aiming at reducing the risk of a privacy breach on the published data, while still allowing to analyze them and draw relevant conclusions. In this paper we introduce a new anonymization approach that is based on injecting uncertainty in social graphs and publishing the resulting uncertain graphs. While existing approaches obfuscate graph data by adding or removing edges entirely, we propose using a finer-grained perturbation that adds or removes edges partially: this way we can achieve the same desired level of obfuscation with smaller changes in the data, thus maintaining higher utility. Our experiments on real-world networks confirm that at the same level of identity obfuscation our method provides higher usefulness than existing randomized methods that publish standard graphs.Comment: VLDB201

Principal manifolds and graphs in practice: from molecular biology to dynamical systems

We present several applications of non-linear data modeling, using principal manifolds and principal graphs constructed using the metaphor of elasticity (elastic principal graph approach). These approaches are generalizations of the Kohonen's self-organizing maps, a class of artificial neural networks. On several examples we show advantages of using non-linear objects for data approximation in comparison to the linear ones. We propose four numerical criteria for comparing linear and non-linear mappings of datasets into the spaces of lower dimension. The examples are taken from comparative political science, from analysis of high-throughput data in molecular biology, from analysis of dynamical systems.Comment: 12 pages, 9 figure

Correlation of Automorphism Group Size and Topological Properties with Program-size Complexity Evaluations of Graphs and Complex Networks

We show that numerical approximations of Kolmogorov complexity (K) applied to graph adjacency matrices capture some group-theoretic and topological properties of graphs and empirical networks ranging from metabolic to social networks. That K and the size of the group of automorphisms of a graph are correlated opens up interesting connections to problems in computational geometry, and thus connects several measures and concepts from complexity science. We show that approximations of K characterise synthetic and natural networks by their generating mechanisms, assigning lower algorithmic randomness to complex network models (Watts-Strogatz and Barabasi-Albert networks) and high Kolmogorov complexity to (random) Erdos-Renyi graphs. We derive these results via two different Kolmogorov complexity approximation methods applied to the adjacency matrices of the graphs and networks. The methods used are the traditional lossless compression approach to Kolmogorov complexity, and a normalised version of a Block Decomposition Method (BDM) measure, based on algorithmic probability theory.Comment: 15 2-column pages, 20 figures. Forthcoming in Physica A: Statistical Mechanics and its Application