We study a broad class of graphical models whose independencies correspond to vertex separation in mixed graphs with directed, undirected, and bidirected edges, that are capable of encoding independence structures arising from feedback, latent and selection mechanisms. In particular, we introduce separable graphs, in which each missing edge implies the existence of a separating set for its endpoints, and essentially separable graphs, those graphs separation equivalent to a separable graph. We show that these models include many existing graph families used to define graphical models an provide several characterizations of separable graphs and essentially separable graphs. We also provide multiple characterizations of separation equivalence for separable graphs. One is a graphical characterization in terms of ordinary graph properties, extending earlier results for specific subfamilies Another is a separational characterization depending only on graph separation properties. Finally, we provide a canonical representation for the equivalence classes of essentially separable graphs and develop an algorithm that, under suitable assumptions, identifies the equivalence class of any essentially separable graph.

We study the problem of estimating the arrival times of vertices in a uniform random recursive tree from its unlabeled structure. We adopt a pointwise perspective and analyze the distribution of the relative estimation error, and derive tail bounds that are uniform in both the vertex and the tree size. For the ranking induced by Jordan centrality, the probability that the estimate exceeds the true arrival time by a factor $S$ decays on the order of $1/S$, while the probability of underestimating the arrival time by a factor $1/S$ decays exponentially in $S$. We introduce a refined centrality measure whose overestimation tail decays on the order of $(\log S)/S^{2}$, at the cost of a heavier lower tail of order $1/S^{2}$. These results reveal a tradeoff between upper- and lower-tail performance in arrival-time estimation that is invisible to the previously studied risk functional. Nevertheless, the refined centrality measure attains the optimal order of the risk for all its parameter values.

Self-Supervised Learning (SSL) for Combinatorial Optimization (CO) is an emerging paradigm for solving combinatorial problems using neural networks. In this paper, we address a central challenge of SSL for CO: solving problems with discrete constraints. We design an end-to-end differentiable framework that enables us to solve discrete constrained optimization problems with neural networks. Concretely, we leverage algorithmic techniques from the literature on convex geometry and Carathéodory's theorem to decompose neural network outputs into convex combinations of polytope corners that correspond to feasible sets. This decomposition-based approach enables self-supervised training but also ensures efficient quality-preserving rounding of the neural net output into feasible solutions. Extensive experiments in cardinality-constrained optimization show that our approach can consistently outperform neural baselines. We further provide workedout examples of how our method can be applied beyond cardinality-constrained problems to a diverse set of combinatorial optimization tasks, including finding independent sets in graphs, and solving matroid-constrained problems.

A major goal of neuroscience is to understand brain computations during visual processing in naturalistic settings. A dominant approach is to use image-computable deep neural networks trained with different task objectives as a basis for linear encoding models. However, in addition to requiring estimation of a large number of linear encoding parameters, this approach ignores the structure of the feature maps both in the brain and the models. Recently proposed alternatives factor the linear mapping into separate sets of spatial and feature weights, thus finding static receptive fields for units, which is appropriate only for early visual areas. In this work, we employ the attention mechanism used in the transformer architecture to study how retinotopic visual features can be dynamically routed to category-selective areas in high-level visual processing. We show that this computational motif is significantly more powerful than alternative methods in predicting brain activity during natural scene viewing, across different feature basis models and modalities. We also show that this approach is inherently more interpretable as the attentionrouting signals for different high-level categorical areas can be easily visualized for any input image. Given its high performance at predicting brain responses to novel images, the model deserves consideration as a candidate mechanistic model of how visual information from retinotopic maps is routed in the human brain based on the relevance of the input content to different category-selective regions.

We study the problem of approximating all-pair distances in a weighted undirected graph with differential privacy, introduced by Sealfon [Sea16]. Given a publicly known undirected graph, we treat the weights of edges as sensitive information, and two graphs are neighbors if their edge weights differ in one edge by at most one. We obtain efficient algorithms with significantly improved bounds on a broad class of graphs which we refer to as recursively separable. In particular, for any n-vertex Kh-minor-free graph, our algorithm achieve an additive error of eO(h(nW)1/3), where W represents the maximum edge weight; For grid graphs, the same algorithmic scheme achieve additive error of eO(n1/4 W). Our approach can be seen as a generalization of the celebrated binary tree mechanism for range queries, as releasing range queries is equivalent to computing all-pair distances on a path graph. In essence, our approach is based on generalizing the binary tree mechanism to graphs that are recursively separable. JL and ZZ have been supported by National Science Foundation of China under Grant No. 62472212 and the New Cornerstone Science Foundation. Supported in part by NSF award 2228995 JU's research was funded by the NSFCNS 2433628, Google Seed Fund grant, Google Research Scholar Award, Dean Research Seed Fund, and Rutgers Decanal Grant no.

Finding dense subgraphs is a fundamental problem with applications to community detection, clustering, and data mining. Our work focuses on finding approximate densest subgraphs in directed graphs in computational models for processing massive data. We consider two such models: Massively Parallel Computation (MPC) and semi-streaming. We show how to find a (2+ε)-approximation in O( logn) MPC rounds with sublinear memory per machine.

Online social media provide a platform for people to discuss current events and exchange opinions with their peers. While interactions are predominantly positive, in recent years, there has been a lot of research to understand the conflicts in social networks and how they are based on different views and opinions. In this paper, we ask whether the conflicts in a network reveal a small and interpretable set of prevalent opinion ranges that explain the users' interactions. More precisely, we consider signed graphs, where the edge signs indicate positive and negative interactions of node pairs, and our goal is to infer opinion intervals that are consistent with the edge signs.

This paper establishes the theoretical limits of graph clustering under the PopularityAdjusted Block Model (PABM), addressing limitations of existing models. In contrast to the Stochastic Block Model (SBM), which assumes uniform vertex degrees, and to the Degree-Corrected Block Model (DCBM), which applies uniform degree corrections across clusters, PABM introduces separate popularity parameters for intra-and inter-cluster connections. Our main contribution is the characterization of the optimal error rate for clustering under PABM, which provides novel insights on clustering hardness: we demonstrate that unlike SBM and DCBM, cluster recovery remains possible in PABM even when traditional edge-density signals vanish, provided intra-and inter-cluster popularity coefficients differ. This highlights a dimension of degree heterogeneity captured by PABM but overlooked by DCBM: local differences in connectivity patterns can enhance cluster separability independently of global edge densities. Finally, because PABM exhibits a richer structure, its expected adjacency matrix has rank between k and k2, where k is the number of clusters. As a result, spectral embeddings based on the top k eigenvectors may fail to capture important structural information. Our numerical experiments on both synthetic and real datasets confirm that spectral clustering algorithms incorporating k2 eigenvectors outperform traditional spectral approaches.

Given an undirected, weighted n-vertex graph G = (V,E,w), a Gomory-Hu tree T is a weighted tree on V that preserves the Min-s-t-Cut between any pair of vertices s,t V. Finding cuts in graphs is a key primitive in problems such as bipartite matching, spectral and correlation clustering, and community detection. We design a differentially private (DP) algorithm that computes an approximate Gomory-Hu tree. Our algorithm is ε-DP, runs in polynomial time, and can be used to compute s-tcuts that are O(n/ε)-additive approximations of the Min-s-t-Cuts in Gfor all distinct s,t V with high probability. Our error bound is essentially optimal, since [29] showed that privately outputting a single Min-s-t-Cut requires Ω(n) additive error even with (ε,δ)-DP and allowing for multiplicative error. Prior to our work, the best additive error bounds for approximate all-pairs Min-s-t-Cuts were O(n3/2/ε)for ε-DP [47] and O( mn/ε)for (ε,δ)-DP [66], both achieved by DP algorithms that preserve all cuts in the graph. To achieve our result, we develop an ε-DP algorithm for the Minimum Isolating Cuts problem with near-linear error, and introduce a novel privacy composition technique combining elements of both parallel and basic composition to handle'bounded overlap' computational branches in recursive algorithms, which maybe of independent interest.

Rectilinear Steiner Minimum Tree (RSMT) is widely used in Very Large Scale Integration (VLSI) and aims at connecting a set of pins using rectilinear edges while minimizing wirelength. Recently, learning-based methods have been explored to tackle this problem effectively. However, existing methods either suffer from excessive exploration of the search space or rely on heuristic combinations that compromise effectiveness and efficiency, and this limitation becomes notably exacerbated when extended to the obstacle-avoiding RSMT (OARSMT). To address this, we propose OAREST, a reinforcement learning-based framework for constructing an Obstacle-Avoiding Rectilinear Edge Sequence (RES) Tree. We theoretically establish the optimality of RES in obstacle-avoiding scenarios, which forms the foundation of our approach. Leveraging this theoretical insight, we introduce a dynamic masking strategy that supports parallel training across varying numbers of pins and extends to obstacles during inference. Empirical evaluations on both synthetic and real-world benchmarks show superior effectiveness and efficiency for RSMT and OARSMT problems, particularly in handling obstacles without training on them.