Criminal networks analysis in missing data scenarios through graph distances

Data collected in criminal investigations may suffer from issues like: (i) incompleteness, due to the covert nature of criminal organizations; (ii) incorrectness, caused by either unintentional data collection errors or intentional deception by criminals; (iii) inconsistency, when the same information is collected into law enforcement databases multiple times, or in different formats. In this paper we analyze nine real criminal networks of different nature (i.e., Mafia networks, criminal street gangs and terrorist organizations) in order to quantify the impact of incomplete data, and to determine which network type is most affected by it. The networks are firstly pruned using two specific methods: (i) random edge removal, simulating the scenario in which the Law Enforcement Agencies fail to intercept some calls, or to spot sporadic meetings among suspects; (ii) node removal, modeling the situation in which some suspects cannot be intercepted or investigated. Finally we compute spectral distances (i.e., Adjacency, Laplacian and normalized Laplacian Spectral Distances) and matrix distances (i.e., Root Euclidean Distance) between the complete and pruned networks, which we compare using statistical analysis. Our investigation identifies two main features: first, the overall understanding of the criminal networks remains high even with incomplete data on criminal interactions (i.e., when 10% of edges are removed); second, removing even a small fraction of suspects not investigated (i.e., 2% of nodes are removed) may lead to significant misinterpretation of the overall network.

Shape-aware Stochastic Neighbour Embedding for Robust Data Visualisations

The t-distributed Stochastic Neighbour Embedding (t-SNE) method has emerged as one of the leading methods for visualising High Dimensional (HD) data in a wide variety of fields, especially for revealing cluster structure in HD single cell transcriptomics data. However, several shortcomings of the algorithm have been identified. Specifically, t-SNE is often unable to correctly represent hierarchical relationships between clusters and spurious patterns may arise in the embedding due to incorrect parameter settings, which could lead to misinterpretations of the data. Here we incorporate t-SNE with shape-aware graph distances, a method termed shape-aware stochastic neighbour embedding (SASNE), to mitigate these limitations of the t-SNE. The merits of the SASNE are first demonstrated using synthetic data sets, where we see a significant improvement in embedding imbalanced and nonlinear clusters, as well as preservation of hierarchical structure, based on quantitative validation in clustering and dimensionality reductions. Moreover, we propose a data-driven parameter setting which we find consistently optimal in all test cases. Lastly, we demonstrate the superior performance of SASNE in embedding the MNIST image data and the single cell transcriptomics gene expression data.

Network comparison and the within-ensemble graph distance

Quantifying the differences between networks is a challenging and ever-present problem in network science. In recent years, a multitude of diverse, ad hoc solutions to this problem have been introduced. Here, we propose that simple and well-understood ensembles of random networks—such as Erdős–Rényi graphs, random geometric graphs, Watts–Strogatz graphs, the configuration model and preferential attachment networks—are natural benchmarks for network comparison methods. Moreover, we show that the expected distance between two networks independently sampled from a generative model is a useful property that encapsulates many key features of that model. To illustrate our results, we calculate this within-ensemble graph distance and related quantities for classic network models (and several parameterizations thereof) using 20 distance measures commonly used to compare graphs. The within-ensemble graph distance provides a new framework for developers of graph distances to better understand their creations and for practitioners to better choose an appropriate tool for their particular task.