Oracle inequalities for network models and sparse graphon estimation

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

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.

Motivated by the prevalent data science applications of processing large-scale graph data such as social networks and biological networks, this paper investigates lossless compression of data in the form of a labeled graph. Particularly, we consider a widely used random graph model, stochastic block model (SBM), which captures the clustering effects in social networks. An information-theoretic universal compression framework is applied, in which one aims to design a single compressor that achieves the asymptotically optimal compression rate, for every SBM distribution, without knowing the parameters of the SBM. Such a graph compressor is proposed in this paper, which universally achieves the optimal compression rate with polynomial time complexity for a wide class of SBMs. Existing universal compression techniques are developed mostly for stationary ergodic one-dimensional sequences. However, the adjacency matrix of SBM has complex two-dimensional correlations. The challenge is alleviated through a carefully designed transform that converts two-dimensional correlated data into almost i.i.d. submatrices. The sequence of submatrices is then compressed by a Krichevsky-Trofimov compressor, whose length analysis is generalized to identically distributed but arbitrarily correlated sequences. In four benchmark graph datasets, the compressed files from competing algorithms take 2.4 to 27 times the space needed by the proposed scheme.

The pair-matching problem appears in many applications where one wants to discover matches between pairs of entities or individuals. Formally, the set of individuals is represented by the nodes of a graph where the edges, unobserved at first, represent the matches. The algorithm queries pairs of nodes and observes the presence/absence of edges. Its goal is to discover as many edges as possible with a fixed budget of queries. Pair-matching is a particular instance of multi-armed bandit problem in which the arms are pairs of individuals and the rewards are edges linking these pairs. This bandit problem is non-standard though, as each arm can only be played once.Given this last constraint, sublinear regret can be expected only if the graph presents some underlying structure. This paper shows that sublinear regret is achievable in the case where the graph is generated according to a stochastic block model (SBM) with two communities. Optimal regret bounds are computed for this pair-matching problem. They exhibit a phase transition related to the Kesten–Stigum threshold for community detection in SBM. The pair-matching problem is considered in the case where each node is constrained to be sampled less than a given amount of times, for example for ensuring individual fairness. We show how optimal regret rates depend on this constraint. The paper is concluded by a conjecture regarding the optimal regret when the number of communities is larger than 2 . Contrary to the two communities case, we argue that a statistical-computational gap would appear in this problem.

Block graphons (also called stochastic block models) are an important and widely studied class of models for random networks. We provide a lower bound on the accuracy of estimators for block graphons with a large number of blocks. We show that, given only the number $k$ of blocks and an upper bound $\rho$ on the values (connection probabilities) of the graphon, every estimator incurs error ${\it{\Omega}}\left(\min\left(\rho, \sqrt{\frac{\rho k^2}{n^2}}\right)\right)$ in the $\delta_2$ metric with constant probability for at least some graphons. In particular, our bound rules out any non-trivial estimation (that is, with $\delta_2$ error substantially less than $\rho$) when $k\geq n\sqrt{\rho}$. Combined with previous upper and lower bounds, our results characterize, up to logarithmic terms, the accuracy of graphon estimation in the $\delta_2$ metric. A similar lower bound to ours was obtained independently by Klopp et al.

We investigate how homophily in adherence to anti-epidemic measures affects the final size of epidemics in social networks. Using a modified SIR model, we divide agents into two behavioral groups-compliant and non-compliant-and introduce transmission probabilities that depend asymmetrically on the behavior of both the infected and susceptible individuals. We simulate epidemic dynamics on two types of synthetic networks with tunable inter-group connection probability: stochastic block models (SBM) and networks with triadic closure (TC) that better capture local clustering. Our main result reveals a counterintuitive effect: under conditions where compliant infected agents significantly reduce transmission, increasing the separation between groups may lead to a higher fraction of infections in the compliant population. This paradoxical outcome emerges only in networks with clustering (TC), not in SBM, suggesting that local network structure plays a crucial role. These findings highlight that increasing group separation does not always confer protection, especially when behavioral traits amplify within-group transmission.

Graphical models facilitate the representation of psychological variables as complex systems of interacting variables structured as a network. However, their current statistical estimation procedures overlook the assumption of clustering, which refers to the grouping of subsets of variables that are more densely connected within the network, despite this assumption playing a central role in many psychological theories. We address this gap by proposing the use of the Stochastic Block Model (SBM) as a prior distribution on the network structure of a graphical model for binary and ordinal data. The SBM assumes that variables belong to latent clusters, where the probability of an edge depends on the cluster membership of the nodes. Embedding this prior in a Bayesian graphical modeling framework allows researchers to formally incorporate theoretical expectations about clustering, test hypotheses about the number of clusters, and estimate cluster assignments from cross-sectional data. We demonstrate the benefits of this approach in a simulation study and reanalyze 30 openly available empirical datasets to test for clustering. This work highlights how the Bayesian framework can embed theoretical assumptions into network models via priors and introduces a new tool for latent cluster inference in psychological network analysis.

Neurons in the brain form thousands of connections, or synapses, with one another, allowing signals to pass from one cell to the next. To activate a neuron, a high enough activating signal or ‘action potential’ must be reached. However, the accepted view of signal transmission assumes that the great majority of synapses are too weak to activate neurons. This means that often simultaneous inputs from many neurons are required to trigger a single neuron’s activation. However, such coordination is likely unreliable as neurons can react differently to the same stimulus depending on the circumstances. An alternative way of transmitting signals has been reported in turtle brains, where impulses from a single neuron can trigger activity across a network of connections. Furthermore, these responses are reliably repeatable, activating the same neurons in the same order. Riquelme et al. set out to understand the mechanism that underlies this type of neuron activation using a mathematical model based on data from the turtle brain. These data showed that the neural network in the turtle’s brain had many weak synapses but also a few, rare, strong synapses. Simulating this neural network showed that those rare, strong synapses promote the signal’s reliability by providing a consistent route for the signal to travel through the network. The numerous weak synapses, on the other hand, have a regulatory role in providing flexibility to how the activation spreads. This combination of strong and weak connections produces a system that can reliably promote or stop the signal flow depending on the context. Riquelme et al.’s work describes a potential mechanism for how signals might travel reliably through neural networks in the brain, based on data from turtles. Experimental work will need to address whether strong connections play a similar role in other animal species, including humans. In the future, these results may be used as the basis to design new systems for artificial intelligence, building on the success of neural networks.

In several sciences but especially in animal and plant breeding, the general mixed model with fixed and random effects plays a great role. Statistical inference on variance components means tests of hypotheses about variance components, constructing confidence intervals for them, estimating them, and using the variance components to get best estimates for fixed effects as well as to predict random effects.Many problems in the statistical inference of variance components already arise in even the most simple mixed model, describing nested designs; they are already present in the balanced nested designs, but become more pronounced in the unbalanced case. To find a guideline for solving the problems for the general mixed model, the study of the nested designs is worthwile.In chapter 1 the outline of the author's research on statistical inference of variance components is given.In chapter 2 the historical development of variance components estimation is described. More than 125 years ago the notion of variance components appears explicitly in astronomy. The expansion began with the development of quantitative genetics in 1918 after the first World War, bull the tremendous increase in research in variance components dates from after the second World War.Just as in the fixed effects model, the notions of vectors, vector spaces and projections of vectors on vector spaces, can be fruitfully used for the mixed effects model. This approach is given in chapter 3.For balanced nested designs exact tests about ratios of variance components and the calculation of their power, are well- known. For the unbalanced three-stage nested designs an exact test exists about the variance component belonging to the second stage, but for the first stage variance component no exact test was available in practice before 1974. in chapter 4 the onset of the author's research on the testing problem is described. It was possible to calculate the critical level or P-value for the exact test.The estimation of variance components was directed towards finding, for several designs, the best quadratic unbiased estimators for the variance components. In chapter 5 the onset of author's research on the estimation problem is described. On introduction of the concept of a permissible estimator it becomes clear which approaches to estimate variance components are unsatisfactory. To use permissible estimators for variance components means to use estimators which give non-negative estimates. A solution for a non-negative estimator has been found using the least squares approach as a unified procedure in variance components estimation.To demonstrate the danger of the use of the best quadratic unbiased estimator in the simplest random model, the balanced one-way classification, the probabilities for negative estimates were calculated in chapter 6.In chapter 7 a new exact test about ratios of variance components in the unbalanced three-stage nested design is given.The description of the least squares method as a unified procedure for the estimation of variance components and for the derivation of permissible (non-negative) estimators of variance components is given in chapter 8. The use of the main estimation procedures such as iterated Least Squares (or I-MINQUE), Maximum Likelihood (ML) and Restricted Maximum Likelihood (REML) estimators, is also discussed there.Finally in chapter 9 the use of variance components in predicting the random effects is discussed. Best Linear Unbiased Prediction (BLUP), which is extensively used in animal breeding, has not so far been applied on the same scale in plant breeding. How to use it is the subject of the last section 9.2.

Bayesian estimation for jump Markov linear systems with non-homogeneous transition probabilities

There have been recent calls for wider application of generative modelling approaches in applied social network analysis. At present, however, it remains difficult for typical end users-for example, field researchers-to implement generative network models, as there is a dearth of openly available software packages that make application of such models as simple as other, permutation-based approaches. Here, we outline the STRAND R package, which provides a suite of generative models for Bayesian analysis of animal social network data that can be implemented using simple, base R syntax. To facilitate ease of use, we provide a tutorial demonstrating how STRAND can be used to model proportion, count or binary network data using stochastic block models, social relation models or a combination of the two modelling frameworks. STRAND facilitates the application of generative network models to a broad range of data found in the animal social networks literature.

Motivated by growing concerns over ensuring privacy on social networks, we develop new algorithms and impossibility results for fitting complex statistical models to network data subject to rigorous privacy guarantees. We consider the so-called node-differentially private algorithms, which compute information about a graph or network while provably revealing almost no information about the presence or absence of a particular node in the graph. We provide new algorithms for node-differentially private estimation for a popular and expressive family of network models: stochastic block models and their generalization, graphons. Our algorithms improve on prior work [15], reducing their error quadratically and matching, in many regimes, the optimal nonprivate algorithm [37]. We also show that for the simplest random graph models (G(n, p) and G(n, m)), node-private algorithms can be qualitatively more accurate than for more complex models—converging at a rate of 1/ e^2 n^3 instead of 1/e^2 n^2. This result uses a new extension lemma for differentially private algorithms that we hope will be broadly useful.

Social influence cannot be identified from purely observational data on social networks, because such influence is generically confounded with latent homophily, that is, with a node’s network partners being informative about the node’s attributes and therefore its behavior. If the network grows according to either a latent community (stochastic block) model, or a continuous latent space model, then latent homophilous attributes can be consistently estimated from the global pattern of social ties. We show that, for common versions of those two network models, these estimates are so informative that controlling for estimated attributes allows for asymptotically unbiased and consistent estimation of social-influence effects in linear models. In particular, the bias shrinks at a rate that directly reflects how much information the network provides about the latent attributes. These are the first results on the consistent nonexperimental estimation of social-influence effects in the presence of latent homophily, and we discuss the prospects for generalizing them.

Estimation of the cell probabilities in a two-way contingency table is considered when it is plausible that the table might have an independence structure relating to the two traits. In a classical nonparametric setup, the unrestricted maximum likelihood estimators of the cell probabilities are the corresponding sample proportions; under the assumption of independence, the restricted estimators are the product of the respective row and column sample proportions. The latter estimators behave better than the former when independence actually holds, but a different picture may emerge for possible departure from the assumed independence structure; the restricted estimators may be heavily biased, inefficient, and even inconsistent. For this reason, a preliminary test on independence based on the classical contingency chi-squared statistic may be conveniently incorporated in the formulation of a preliminary test estimator of the matrix of cell probabilities. Since, typically, we have a multiparameter estimation problem, a pretest or shrinkage estimator based on the classical Stein rule may also be formulated in the same vein. The primary objective of this article is to focus on the asymptotic distribution theory of all four estimators under the null hypothesis of independence as well as local Pitman-type alternatives. The bias and risk based on suitable quadratic loss functions of these estimators are considered in the same asymptotic setup, and these are then incorporated in the study of the asymptotic relative performance of these estimators. In the light of their asymptotic risks, neither the preliminary test nor the shrinkage estimators dominates the other, though each fares well relative to the unrestricted or the restricted maximum likelihood estimators.

Stratigraphic uncertainty widely exists in geotechnical structures. An efficient generalized coupled Markov chain (GCMC) model was proposed to simulate the stratigraphic uncertainty in the past. This model, however, cannot be directly applied to geotechnical problems because of the difficulty in directly estimating the horizontal transition probability matrices (HTPMs) of the GCMC from boreholes with large hole-to-hole distance. This paper, therefore, aims at addressing the above problem by using a maximum likelihood estimation (MLE) method and utilizing the MLE modified GCMC model to characterize stratigraphic uncertainty. In the framework of the GCMC, the validity of the proposed method is verified, and information entropy method is adopted to quantify the stratigraphic uncertainty. As an illustration, the proposed model is applied to a construction site in Hong Kong to characterize the stratigraphic uncertainty. A systematic comparison of coupled Markov chain (CMC) and GCMC model for stratigraphic uncertainty simulation was conducted. The influences of borehole layout scheme on transition probability matrices (TPMs) estimation and stratigraphic uncertainty simulation are explored. Results showed that the result of stratigraphic uncertainty simulation of the GCMC model is more reasonable than that of the CMC model. The borehole layout scheme is very important for estimating TPMs and significantly influences the stratigraphic uncertainty simulation. The information entropy map has the potential to become a useful guideline for the design of the borehole layout scheme in geotechnical engineering practice.