Latent structure blockmodels for Bayesian spectral graph clustering

Spectral embedding of network adjacency matrices often produces node representations living approximately around low-dimensional submanifold structures. In particular, hidden substructure is expected to arise when the graph is generated from a latent position model. Furthermore, the presence of communities within the network might generate community-specific submanifold structures in the embedding, but this is not explicitly accounted for in most statistical models for networks. In this article, a class of models called latent structure block models (LSBM) is proposed to address such scenarios, allowing for graph clustering when community-specific one dimensional manifold structure is present. LSBMs focus on a specific class of latent space model, the random dot product graph (RDPG), and assign a latent submanifold to the latent positions of each community. A Bayesian model for the embeddings arising from LSBMs is discussed, and shown to have a good performance on simulated and real world network data. The model is able to correctly recover the underlying communities living in a one-dimensional manifold, even when the parametric form of the underlying curves is unknown, achieving remarkable results on a variety of real data

Mutually exciting point process graphs for modelling dynamic networks

A new class of models for dynamic networks is proposed, called mutually exciting point process graphs (MEG). MEG is a scalable network-wide statistical model for point processes with dyadic marks, which can be used for anomaly detection when assessing the significance of future events, including previously unobserved connections between nodes. The model combines mutually exciting point processes to estimate dependencies between events and latent space models to infer relationships between the nodes. The intensity functions for each network edge are characterized exclusively by node-specific parameters, which allows information to be shared across the network. This construction enables estimation of intensities even for unobserved edges, which is particularly important in real world applications, such as computer networks arising in cyber-security. A recursive form of the log-likelihood function for MEG is obtained, which is used to derive fast inferential procedures via modern gradient ascent algorithms. An alternative EM algorithm is also derived. The model and algorithms are tested on simulated graphs and real world datasets, demonstrating excellent performance. Supplementary materials for this article are available online

Bayesian estimation of the latent dimension and communities in stochastic blockmodels

Spectral embedding of adjacency or Laplacian matrices of undirected graphs is a common technique for representing a network in a lower dimensional latent space, with optimal theoretical guarantees. The embedding can be used to estimate the community structure of the network, with strong consistency results in the stochastic blockmodel framework. One of the main practical limitations of standard algorithms for community detection from spectral embeddings is that the number of communities and the latent dimension of the embedding must be specified in advance. In this article, a novel Bayesian model for simultaneous and automatic selection of the appropriate dimension of the latent space and the number of blocks is proposed. Extensions to directed and bipartite graphs are discussed. The model is tested on simulated and real world network data, showing promising performance for recovering latent community structure

Modelling dynamic network evolution as a Pitman-Yor process

Dynamic interaction networks frequently arise in biology, communications technology and the social sciences, representing, for example, neuronal connectivity in the brain, internet connections between computers and human interactions within social networks. The evolution and strengthening of the links in such networks can be observed through sequences of connection events occurring between network nodes over time. In some of these applications, the identity and size of the network may be unknown a priori and may change over time. In this article, a model for the evolution of dynamic networks based on the Pitman-Yor process is proposed. This model explicitly admits power-laws in the number of connections on each edge, often present in real world networks, and, for careful choices of the parameters, power-laws for the degree distribution of the nodes. A novel empirical method for the estimation of the hyperparameters of the Pitman-Yor process is proposed, and some necessary corrections for uniform discrete base distributions are carefully addressed. The methodology is tested on synthetic data and in an anomaly detection study on the enterprise computer network of the Los Alamos National Laboratory, and successfully detects connections from a red-team penetration test

Classification of periodic arrivals in event time data for filtering computer network traffic

Periodic patterns can often be observed in real-world event time data, possibly mixed with non-periodic arrival times. For modelling purposes, it is necessary to correctly distinguish the two types of events. This task has particularly important implications in computer network security; there, separating automated polling traffic and human-generated activity in a computer network is important for building realistic statistical models for normal activity, which in turn can be used for anomaly detection. Since automated events commonly occur at a fixed periodicity, statistical tests using Fourier analysis can efficiently detect whether the arrival times present an automated component. In this article, sequences of arrival times which contain automated events are further examined, to separate polling and non-periodic activity. This is first achieved using a simple mixture model on the unit circle based on the angular positions of each event time on the p-clock, where p represents the main periodicity associated with the automated activity; this model is then extended by combining a second source of information, the time of day of each event. Efficient implementations exploiting conjugate Bayesian models are discussed, and performance is assessed on real network flow data collected at Imperial College London

Link prediction in dynamic networks using random dot product graphs

The problem of predicting links in large networks is a crucial task in a variety of practical applications, including social sciences, biology and computer security. In this paper, statistical techniques for link prediction based on the popular random dot product graph model are carefully presented, analysed and extended to dynamic settings. Motivated by a practical application in cyber-security, this paper demonstrates that random dot product graphs not only represent a powerful tool for inferring differences between multiple networks, but are also efficient for prediction purposes and for understanding the temporal evolution of the network. The probabilities of links are obtained by fusing information at multiple levels of resolution: time series models are used to score connections at the edge level, and spectral methods provide estimates of latent positions for each node. In this way, traditional link prediction methods, usually based on decompositions of the entire network adjacency matrix, are extended using edge-specific information. The methods presented in this article are applied to a number of simulated and real-world computer network graphs, showing promising results

Graph link prediction in computer networks using Poisson matrix factorisation

Graph link prediction is an important task in cyber-security: relationships between entities within a computer network, such as users interacting with computers, or system libraries and the corresponding processes that use them, can provide key insights into adversary behaviour. Poisson matrix factorisation (PMF) is a popular model for link prediction in large networks, particularly useful for its scalability. In this article, PMF is extended to include scenarios that are commonly encountered in cyber-security applications. Specifically, an extension is proposed to explicitly handle binary adjacency matrices and include known covariates associated with the graph nodes. A seasonal PMF model is also presented to handle dynamic networks. To allow the methods to scale to large graphs, variational methods are discussed for performing fast inference. The results show an improved performance over the standard PMF model and other common link prediction techniques

Spectral clustering on spherical coordinates under the degree-corrected stochastic blockmodel

Spectral clustering is a popular method for community detection in networks under the assumption of the standard stochastic blockmodel. Taking a matrix representation of the graph such as the adjacency matrix, the nodes are clustered on a low dimensional projection obtained from a truncated spectral decomposition of the matrix. Estimating the number of communities and the dimension of the reduced latent space well is crucial for good performance of spectral clustering algorithms. Real-world networks, such as computer networks studied in cyber-security applications, often present heterogeneous within-community degree distributions which are better addressed by the degree-corrected stochastic blockmodel. A novel, model-based method is proposed in this article for simultaneous and automated selection of the number of communities and latent dimension for spectral clustering under the degree-corrected stochastic blockmodel. The method is based on a transformation to spherical coordinates of the spectral embedding, and on a novel modelling assumption in the transformed space, which is then embedded into an existing model selection framework for estimating the number of communities and the latent dimension. Results show improved performance over competing methods on simulated and real-world computer network data