Poisson degree corrected dynamic stochastic block model

It is now widely accepted that knowledge can be acquired from networks by clustering their vertices according to the connection profiles. Many methods have been proposed and in this paper we concentrate on the Stochastic Block Model (SBM). The clustering of vertices and the estimation of SBM model parameters have been subject to previous work, and numerous inference strategies such as variational expectation maximization (EM) and classification EM have been proposed. However, SBM still suffers from a lack of criteria to estimate the number of components in the mixture. To our knowledge, only one model-based criterion, Integrated Complete-data Likelihood (ICL), has been derived for SBM in the literature. It relies on an asymptotic approximation of the integrated complete-data likelihood and recent studies have shown that it tends to be too conservative in the case of small networks. To tackle this issue, we propose a new criterion that we call Integrated Likelihood Variational Bayes (ILvb), based on a non-asymptotic approximation of the marginal likelihood. We describe how the criterion can be computed through a variational Bayes EM algorithm.

We use the Stein‒Chen method to obtain compound Poisson approximations for the distribution of the number of subgraphs in a generalised stochastic block model which are isomorphic to some fixed graph. This model generalises the classical stochastic block model to allow for the possibility of multiple edges between vertices. We treat the case that the fixed graph is a simple graph and that it has multiple edges. The former results apply when the fixed graph is a member of the class of strictly balanced graphs and the latter results apply to a suitable generalisation of this class to graphs with multiple edges. We also consider a further generalisation of the model to pseudo-graphs, which may include self-loops as well as multiple edges, and establish a parameter regime in the multiple edge stochastic block model in which Poisson approximations are valid. The results are applied to obtain Poisson and compound Poisson approximations (in different regimes) for subgraph counts in the Poisson stochastic block model and degree corrected stochastic block model of Karrer and Newman (2011).

Degree-corrected stochastic block models and reliability in networks

Complex networks of material flow in manufacturing and logistics: Modeling, analysis, and prediction using stochastic block models

The proliferation of models for networks raises challenging problems of model selection: the data are sparse and globally dependent, and models are typically high-dimensional and have large numbers of latent variables. Together, these issues mean that the usual model-selection criteria do not work properly for networks. We illustrate these challenges, and show one way to resolve them, by considering the key network-analysis problem of dividing a graph into communities or blocks of nodes with homogeneous patterns of links to the rest of the network. The standard tool for undertaking this is the stochastic block model, under which the probability of a link between two nodes is a function solely of the blocks to which they belong. This imposes a homogeneous degree distribution within each block; this can be unrealistic, so degree-corrected block models add a parameter for each node, modulating its overall degree. The choice between ordinary and degree-corrected block models matters because they make very different inferences about communities. We present the first principled and tractable approach to model selection between standard and degree-corrected block models, based on new large-graph asymptotics for the distribution of log-likelihood ratios under the stochastic block model, finding substantial departures from classical results for sparse graphs. We also develop linear-time approximations for log-likelihoods under both the stochastic block model and the degree-corrected model, using belief propagation. Applications to simulated and real networks show excellent agreement with our approximations. Our results thus both solve the practical problem of deciding on degree correction and point to a general approach to model selection in network analysis.

The stochastic block model is widely used for detecting community structures in network data. However, the research interest in much of the literature focuses on the study of one sample of stochastic block models. Detecting the difference between the two community structures is a less studied issue for stochastic block models. In this article, we propose a novel test statistic based on the largest singular value of a residual matrix obtained by subtracting the geometric mean of two estimated block mean effects from the sum of two observed adjacency matrices. We prove that the null distribution of the proposed test statistic converges in distribution to the Tracy-Widom distribution with index 1, and we show the difference between the two samples for stochastic block models can be tested via the proposed method. We show that the proposed test is asymptotically powerful against alternative models. Further, we extend the proposed method to the degree-corrected stochastic block model. Both simulation studies and real-world data examples indicate that the proposed method works well.

This research addresses the limitations of traditional network models in capturing the complexity and dynamics of real-world social networks. Motivated by the need for a more comprehensive and flexible framework, the study introduces the Hybrid Modern Network Model (HMNM). The HMNM integrates foundational models like the Stochastic Block Model (SBM) and Preferential Attachment with advanced machine learning techniques, including Graph Neural Networks (GNNs), Reinforcement Learning (RL), Hierarchical Random Graphs (HRGs), Generative Adversarial Networks (GANs), and Variational Autoencoders (VAEs). The methods employed involve constructing initial network structures using SBM, simulating network growth through preferential Attachment, learning node embeddings with GNNs, dynamically optimizing network properties using RL, capturing hierarchical community structures with HRGs, controlling degree distributions using GANs, and uncovering latent patterns with VAEs. The empirical illustration of HMNM highlights its effectiveness in providing a more realistic, scalable, and comprehensive analysis of social networks compared to traditional models. Integrating diverse methodologies allows for accurately modeling of network structures, dynamic processes, and latent patterns. In conclusion, the HMNM offers significant advancements in network modeling, providing a robust and flexible framework for analyzing social networks. This model overcomes the limitations of traditional models and delivers deeper insights into the complexities and dynamics of social structures. Future research will optimize the HMNM and explore its applications across various domains. The R programming code used for the network simulations and visualizations is conceptual and demonstrates the HMNM framework. The results and metrics are illustrative placeholders, emphasizing the methodology rather than empirical validation.

Clustering and deformation are important for dynamics of nuclear structure, and cluster structures couple to deformed states. However, its mechanism is unclear. Cluster structures that couple to each deformed state and the mechanism are investigated in $^{35}$Cl. The antisymmetrized molecular dynamics (AMD) and the generator coordinate method (GCM) are used. An AMD function is a Slater determinant of Gaussian wave packets. By energy variational calculations with constraints for deformation and clustering, wave functions that have deformed and $\alpha$-$^{31}$P and $t$-$^{32}$S cluster structures are obtained. Adopting those wave functions as GCM basis, wave functions of ground and excited states are calculated. Various cluster deformed bands are obtained and predicted. The $K^\pi = \frac{1}{2}^-$ deformed bands contains large amount of $\alpha$-$^{31}$P and $t$-$^{32}$S cluster structure components. Particle-hole configurations of both cluster structures are same as dominant components of the $K^\pi = \frac{1}{2}^-$ bands in small intercluster distance. Particle-hole configurations of cluster structure in small intercluster distance are important for coupling of cluster structure to deformed states.

Many statistical inference problems correspond to recovering the values of a set of hidden variables from sparse observations on them. For instance, in a planted constraint satisfaction problem such as planted 3-SAT, the clauses are sparse observations from which the hidden assignment is to be recovered. In the problem of community detection in a stochastic block model, the community labels are hidden variables that are to be recovered from the edges of the graph. Inspired by ideas from statistical physics, the presence of a stable fixed point for belief propogation has been widely conjectured to characterize the computational tractability of these problems. For community detection in stochastic block models, many of these predictions have been rigorously confirmed. In this work, we consider a general model of statistical inference problems that includes both community detection in stochastic block models, and all planted constraint satisfaction problems as special cases. We carry out the cavity method calculations from statistical physics to compute the regime of parameters where detection and recovery should be algorithmically tractable. At precisely the predicted tractable regime, we give: (i) a general polynomial-time algorithm for the problem of detection: distinguishing an input with a planted signal from one without; (ii) a general polynomial-time algorithm for the problem of recovery: outputting a vector that correlates with the hidden assignment significantly better than a random guess would. Analogous to the spectral algorithm for community detection [1], [2], the detection and recovery algorithms are based on the spectra of a matrix that arises as the derivatives of the belief propagation update rule. To devise a spectral algorithm in our general model, we obtain bounds on the spectral norms of certain families of random matrices with correlated and matrix valued entries. We then demonstrate how eigenvectors of various powers of the matrix can be used to partially recover the hidden variables.

Finding communities in complex networks is a challenging task and one promising approach is the Stochastic Block Model (SBM). But the influences from various fields led to a diversity of variants and inference methods. Therefore, a comparison of the existing techniques and an independent analysis of their capabilities and weaknesses is needed. As a first step, we review the development of different SBM variants such as the degree-corrected SBM of Karrer and Newman or Peixoto’s hierarchical SBM. Beside stating all these variants in a uniform notation, we show the reasons for their development. Knowing the variants, we discuss a variety of approaches to infer the optimal partition like the Metropolis-Hastings algorithm. We perform our analysis based on our extension of the Girvan-Newman test and the Lancichinetti-Fortunato-Radicchi benchmark as well as a selection of some real world networks. Using these results, we give some guidance to the challenging task of selecting an inference method and SBM variant. In addition, we give a simple heuristic to determine the number of steps for the Metropolis-Hastings algorithms that lack a usual stop criterion. With our comparison, we hope to guide researches in the field of SBM and highlight the problem of existing techniques to focus future research. Finally, by making our code freely available, we want to promote a faster development, integration and exchange of new ideas.

Leclerc's conjecture on a cluster structure for type A Richardson varieties

Random walk-based node embedding algorithms learn vector representations of nodes by optimizing an objective function of embedding vectors and skip-bigram statistics computed from random walks on the network. They have demonstrated state-of-the-art performance in many supervised learning problems such as link prediction and node classification. Yet, their properties remain poorly understood. This paper studies properties of random walk based node embeddings in the unsupervised setting of discovering latent network block structure, i.e., learning node representations whose cluster structure in Euclidean space reflects node adjacencies within the network. We characterize the ergodic limits of the embedding objective, its generalization, and related convex relaxations to derive corresponding non-randomized versions of the node embedding objectives. We also characterize the optimal node embedding Grammians of the non-randomized objectives for the expected graph of a two-community Stochastic Block Model (SBM). We prove that the solution Grammian has rank 1 for a suitable nuclear norm relaxation of the non-randomized objective. Comprehensive experimental results on SBM random networks reveal that our non-randomized ergodic objectives yield node embeddings whose distribution is Gaussian-like, centered at the node embeddings of the expected network within each community, and concentrate in the linear degree-scaling regime as the number of nodes increases.

In today’s networked systems a massive amount of data is produced every day. These data can be modelled using graphs, where the nodes typically correspond to users or devices and the edges to the connections between them. Almost all networks change over time, with new nodes and edges appearing or disappearing as the system matures. Therefore dynamic graph models are more adequate to analyse such networks than static graphs, and appropriate tools need to be implemented to protect them. In this paper we obtain an edge differentially private version of dynamic stochastic block model. We show experimentally that the trends in the dynamic stochastic block model obtained from the original data are well preserved with the additional privacy guarantees.

We analyze the performance of spectral clustering for community extraction in stochastic block models. We show that, under mild conditions, spectral clustering applied to the adjacency matrix of the network can consistently recover hidden communities even when the order of the maximum expected degree is as small as $\log n$, with $n$ the number of nodes. This result applies to some popular polynomial time spectral clustering algorithms and is further extended to degree corrected stochastic block models using a spherical $k$-median spectral clustering method. A key component of our analysis is a combinatorial bound on the spectrum of binary random matrices, which is sharper than the conventional matrix Bernstein inequality and may be of independent interest.

We study the problem of detecting the community structure from the generalized stochastic block model with two communities (G2-SBM). Based on analysis of the Stieljtes transform of the empirical spectral distribution, we prove a Baik–Ben Arous–Péché (BBP)-type transition for the largest eigenvalue of the G2-SBM. For specific models, such as a hidden community model and an unbalanced stochastic block model, we provide precise formulas for the two largest eigenvalues, establishing the gap in the BBP-type transition.