Bootcamp

Networks are now a common language for social systems, markets, biological interaction maps, online platforms, and communication infrastructure. Yet a graph drawing or simulation can conceal precisely the structure we want to understand. When does a local MCMC sampler reach its intended distribution quickly enough to be useful? When do simple local statistics force global random-like behavior? Which traces of a network’s early history survive a change in its dynamics? How do latent attributes alter what degree, PageRank, and network samples record? I will show how probability, asymptotics, and stochastic-process representations turn these questions into sharp qualitative predictions.

Through four vignettes: exponential random graphs, pseudorandomness and motif counts in high dimensions, change points in preferential attachment, and attributed network models, I will illustrate a recurring principle: the right reduced mathematical object can reveal phenomena that are difficult to infer from data or simulation alone. Fixed-point structure governs whether local MCMC mixes rapidly; local regularity forces random-like motif counts; branching-process time explains the persistence of early history; and local weak limits expose the different biases of ranking and sampling procedures. The goal is not a survey, but a sequence of mathematical explanations for why mathematics can be unexpectedly effective in network science.

Networks are often treated as observed objects, but in scientific applications they are typically incomplete, biased, and noisy measurements of an underlying relational system. This talk highlights Bayesian approaches to learning network structure while representing that uncertainty explicitly. I will use two motivating examples: predicting unobserved animal–plant interactions from highly taxonomically biased ecological studies, and characterizing how weighted brain-connectome structure varies across groups and covariates. In both settings, latent structure permits information to be shared across a large number of potential edges, while probabilistic models distinguish absent links from missed observations and propagate uncertainty into predictions and scientific contrasts. I will close with open challenges in scalable computation for large sparse networks, flexible modeling of weighted and multiplex structure, and principled inference on group differences.

Networks often encode local relationships, while global structure can emerge through repeated propagation over those local connections. In this talk, I will discuss this perspective in a geometric setting, where a graph is constructed from samples on an unknown low-dimensional manifold. We show that the iterated graph diffusion operator approximates the manifold heat semigroup at finite diffusion times, under minimal regularity assumptions on the test function. I will also discuss how this can be used to estimate the manifold heat kernel. In particular, the construction uses only ambient Euclidean kernels without estimating the manifold geometry or graph Laplacian, while its statistical complexity is governed by the intrinsic low dimensionality of the manifold.

This talk will examine some approaches and existing problems in theorizing and modeling social influence processes when attitudes are interconnected.

Networks often come with a some amount of side information. For example, the community labels of a subset of their vertices. How can this partial information be used to improve the recovery of the remaining labels? In this talk, we present a random-walk approach using the partial labels to improve community detection. We treat vertices with revealed labels as absorbing sets and use the associated quasi-stationary distributions to infer the unknown labels. This perspective leads to an intuitive algorithm as well as a tractable framework for its analysis. For the partially labeled stochastic block model, we obtain explicit recovery error rates that quantify the benefit of the revealed information. This is joint work with Michael Nisenzon.

I will discuss some recent works on learning dynamics with memory, in the framework of generalized Langevin equations, based on joint works with a former postdoc Quanjun Lang (currently Penn State).

This paper systematically studies the behavior of the leading eigenvectors for independent edge undirected random graphs generated from a general latent position model whose link function is possibly infinite rank and also possibly indefinite. We first derive uniform error bounds in the two-to-infinity norm as well as row-wise normal approximations for the leading sample eigenvectors. We then build on these results to tackle two graph inference problems, namely (i) entrywise bounds for graphon estimation and (ii) testing for the equality of latent positions, the latter of which is achieved by proposing a rank-adaptive test statistic that converges in distribution to a weighted sum of independent chi-square random variables under the null hypothesis. Our fine-grained theoretical guarantees and applications differ from the existing literature which primarily considers first order upper bounds and more restrictive low rank or positive semidefinite model assumptions. Further, our results collectively quantify the statistical properties of eigenvector-based spectral embeddings with growing dimensionality for large graphs.

Under independent treatment assignment without interference, cross-fitting enables flexible covariate adjustment while preserving finite-sample unbiasedness. Under network interference, however, out-of-sample prediction alone is no longer sufficient: treatment assignments in the evaluation sample can affect outcomes in the training sample, inducing dependence between the fitted predictions and the Horvitz–Thompson weights. To restore finite-sample unbiasedness, we propose neighborhood-excluded cross-fitting, which removes from each training set units whose outcomes may depend on evaluation-fold assignments without requiring a correctly specified outcome model. We establish asymptotically valid design-based Wald inference for direct and indirect effects under Bernoulli randomization and the global average treatment effect under Bernoulli cluster randomization. Our theory characterizes how the number of folds should scale with the exclusion neighborhoods: unit-level splitting may require a growing number of folds, whereas cluster-level splitting substantially relaxes this requirement and permits a fixed number of folds under partial interference. For linear adjustment, we derive variance-optimal and confidence-interval-length-optimal procedures and establish explicit rate conditions under which they remain valid with diverging covariate dimension.
The variance-optimal procedure is also asymptotically no-harm. Simulations confirm that neighborhood exclusion removes the empirical bias of standard cross-fitting while substantially improving precision, and an application to a social network experiment yields substantially shorter confidence intervals than the unadjusted estimator.

In a random geometric graph model, vertices are associated to random points in Euclidean space, and edges are drawn between pairs of vertices whose points are sufficiently close together. Classically, these models were studied for Euclidean spaces of fixed dimension, but more recently, problems in statistics and learning theory have motivated consideration of high-dimensional spaces. I’ll give an overview of these models, their connection with other latent-space random graph models, the main problems of interest, and what’s currently known. Time permitting, I’ll describe recent work with Sofia Poinelli and Tatiana Brailovskaya on a recovery problem for random geometric graphs with latent manifold structure.

Counterfactual reasoning asks a fundamental question: what would need to be different for an outcome to change? Most existing approaches answer this question by perturbing an input until a model changes its prediction. For graph-structured data, however, such perturbations can be model-dependent, structurally implausible, and disconnected from meaningful relational patterns. In this talk, I present a data-grounded perspective on counterfactual reasoning that shifts the paradigm from perturbation to retrieval: rather than synthesizing hypothetical alternatives, we search observed graph data for semantically similar structures associated with different outcomes.

I will discuss the key technical challenges underlying this shift. We develop semantic counterfactual dimensions and multi-scale graph concepts that provide meaningful coordinates for graph comparison, and use hypergraph-based optimal transport to characterize interpretable structural transformations. To support large-scale retrieval, we introduce certified lower and upper bounds for efficient pruning and ranking, as well as concept-based indexing for searching subgraphs within a single large network. We further incorporate robustness and diversity to identify stable and non-redundant counterfactual alternatives. Together, these ideas suggest a broader view of counterfactual analysis—not merely as a technique for explaining individual predictions, but as a data querying and reasoning paradigm for discovering alternative outcomes and the structural differences that distinguish them.

Researchers routinely use parameter estimates from statistical models for social network data to initiate empirically calibrated social simulations. This talk will describe recent work considering under what conditions such simulations support a causal interpretation. It positions such simulations in the potential outcomes framework, formalizes implicit target quantities, introduces identification results, discusses sufficient conditions, and highlights philosophical considerations when drawing a causal interpretation. The talk demonstrates that causal inference is theoretically possible in these types of network simulations, but the empirical requirements to draw causal inference will rarely be met in practice. It concludes by highlighting practical approaches to empirical network simulation concerned with causality that focus on robustness testing with sensitivity analyses.

I this talk, I will present a model to study how a platform should incentivize strategic agents to truthfully contribute high-quality data in collaborative learning environments, where each agent benefits from improved estimation based on others’ data. Data quality is privately observed and can be misreported, creating a joint problem of statistical aggregation and incentive design. We formulate this setting as a Bayesian mechanism design problem in which the platform jointly determines allocation and payment rules to maximize revenue, while ensuring truthful collaboration. We show that the optimal data-sharing mechanism takes an implementable form: a personalized threshold-and-pricing rule that allocates the learned estimator to an agent only if her reported quality exceeds a personalized cutoff and charges a price reflecting the relevance of her data. In a canonical Gaussian mean estimation setting, we derive a closed-form characterization and quantify how correlation across datasets shapes allocation and payments. We then allow agents to exert costly effort to improve data quality and show that the optimal mechanism fundamentally alters incentives: while free-riding arises without the optimal design, the optimal design induces a supermodular effort game, mitigating free-riding: each agent is incentivized to exert more effort when others exert more. Finally, we show that equilibrium efforts form a complete lattice, and in the highest-effort equilibrium, each agent increases effort as others’ data becomes more relevant. Based on joint work with Saeed Alaei, Ali Daei Naby, and Azarakhsh Malekian (EC ’26).

In networked valuation models, a participant’s value depends not only on her own allocation but on the allocation across the entire network. Such valuations profoundly impact the structure of optimal mechanisms and the value of private information, which comes to depend on an agent’s position in the network. While positive externalities preserve the desirable properties of classical optimal mechanisms, the negative externalities that commonly arise in modeling competitive settings create structural challenges and opportunities: (i) optimal revenue is non-monotone in connectivity, and need not be monotone in agents’ valuations; (ii) private information acts as a complexity switch: network allocation problems that are computationally tractable under complete information can become NP-hard under asymmetric private information (an informational analogue of statistical/computational tradeoffs); and (iii) connectivity can be exploited: linking previously unrelated valuations, for instance through bundling, manufactures competition and thereby improves expected revenue.

Many modern data objects, including functions and networks, have intrinsic structure that is not naturally captured by standard finite-dimensional vector representations. This raises a broad question for generative modeling: how can we synthesize complex data while preserving the structure needed for meaningful statistical analysis?

In this talk, I explore this question through the lens of functional data. I will introduce Smooth Flow Matching, a generative framework for synthesizing smooth, irregularly observed functional data without Gaussian or low-rank assumptions. The method generates infinite-dimensional functional data while preserving smoothness and can be used to create surrogate data for downstream statistical tasks. I will illustrate the framework through simulations and an application to longitudinal electronic health records. I will conclude with a brief discussion of how the challenges arising in functional data synthesis may inform broader questions in generative modeling for complex objects, such as networks.

In first passage percolation, edges are assigned i.i.d. nonnegative passage times, and one studies the minimum passage time between vertices. We consider the critical setting in which the probability that the passage time of an edge is zero equals the percolation threshold of the underlying graph. On the two-dimensional lattice, Zhang (1999) showed through examples that, at criticality, whether passage times remain bounded depends delicately on the passage time distribution near zero, and Damron, Lam, and Wang (2017) later gave a general necessary and sufficient condition for boundedness. In this talk, I will discuss an analogous problem for random graphs. We ask when the passage time between two uniformly chosen vertices in a random graph remains bounded and converges in distribution as the network size grows. There are interesting similarities between the results on the two-dimensional lattice and on random graphs, but passage times on random graphs can remain bounded under much weaker conditions.

Bayesian Nonparametrics (BNP) offers a highly flexible framework for analyzing complex network data. In this talk, I will present three recent projects that leverage BNP to tackle core network challenges: community detection, network clustering, and graph matching. Throughout the presentation, I will emphasize the methodological hurdles inherent in these approaches, focusing specifically on prior construction, inference strategies, and summarizing the posterior distribution.

When users share data with online platforms, they may reveal information not only about themselves but also about others. These privacy externalities can depress the price of data, weaken incentives to protect it, and lead to excessive data sharing. We study how market structure and platform competition shape these forces and examine their implications for users, data buyers, and overall welfare. We also explore when restricting data markets may improve welfare and consider alternative regulatory and market-design interventions that can better balance the benefits of data use against its privacy costs.

Respondent-driven sampling (RDS) is widely used to study hidden or hard-to-reach populations by incentivizing study participants to recruit their social connections. The success and efficiency of RDS can depend critically on the nature of the incentives, including their number, value, call to action, etc. Standard RDS uses an incentive structure that is set a priori and held fixed throughout the study. Thus, it does not make use of accumulating information on which incentives are effective and for whom. We propose a reinforcement learning (RL) based adaptive RDS study design in which the incentives are tailored over time to maximize cumulative utility during the study. We show that these designs are more efficient, cost-effective, and can generate new insights into the social structure of hidden populations. In addition, we develop methods for valid post-study inference which are non-trivial due to the adaptive sampling induced by RL as well as the complex dependencies among subjects due to latent (unobserved) social network structure. We provide asymptotic regret bounds and illustrate its finite sample behavior through a suite of simulation experiments.