However, the undirected setting is often insufficient: only a causal (i.e., directed) network provides

However, in many scenarios, nodes also have attributes that are correlated with the clustering structure. Thus, network information (edges) and node information (attributes) can be jointly leveraged to design high-performance clustering algorithms. Under a general model for the network and node attributes, this work establishes an information-theoretic criterion for the exact recovery of community labels and characterizes a phase transition determined by the Chernoff-Hellinger divergence of the model.

Structured representations such as keypoints are widely used in pose transfer, conditional image generation, animation, and 3D reconstruction. However, their supervised learning requires expensive annotation for each target domain. We propose a self-supervised method that learns to disentangle object structure from the appearance with a graph of 2D keypoints linked by straight edges. Both the keypoint location and their pairwise edge weights are learned, given only a collection of images depicting the same object class. The resulting graph is interpretable, for example, AutoLink recovers the human skeleton topology when applied to images showing people. Our key ingredients are i) an encoder that predicts keypoint locations in an input image, ii) a shared graph as a latent variable that links the same pairs of keypoints in every image, iii) an intermediate edge map that combines the latent graph edge weights and keypoint locations in a soft, differentiable manner, and iv) an inpainting objective on randomly masked images. Although simpler, AutoLink outperforms existing self-supervised methods on the established keypoint and pose estimation benchmarks and paves the way for structure-conditioned generative models on more diverse datasets.

Knowledge graph embedding (KGE) relies on the geometry of the embedding space to encode semantic and structural relations. Existing methods place all entities on one homogeneous manifold, Euclidean, spherical, hyperbolic, or their product/multi-curvature variants, to model linear, symmetric, or hierarchical patterns. Yet a predefined, homogeneous manifold cannot accommodate the sharply varying curvature that real-world graphs exhibit across local regions. Since this geometry is imposed a priori, any mismatch with the knowledge graph's local curvatures will distort distances between entities and hurt the expressiveness of the resulting KGE. To rectify this, we propose RicciKGE to have the KGE loss gradient coupled with local curvatures in an extended Ricci flow such that entity embeddings co-evolve dynamically with the underlying manifold geometry towards mutual adaptation. Theoretically, when the coupling coefficient is bounded and properly selected, we rigorously prove that i) all the edge-wise curvatures decay exponentially, meaning that the manifold is driven toward the Euclidean flatness; and ii) the KGE distances strictly converge to a global optimum, which indicates that geometric flattening and embedding optimization are promoting each other. Experimental improvements on link prediction and node classification benchmarks demonstrate RicciKGE's effectiveness in adapting to heterogeneous knowledge graph structures.

In shift bribery, a briber seeks to promote his preferred candidate by paying voters to raise their ranking. Classical models of shift bribery assume voters act independently, overlooking the role of social influence. However, in reality, individuals are social beings and are often represented as part of a social network, where bribed voters may influence their neighbors, thereby amplifying the effect of persuasion. We study Shift bribery over Networks, where voters are modeled as nodes in a directed weighted graph, and arcs represent social influence between them. In this setting, bribery is not confined to directly targeted voters its effects can propagate through the network, influencing neighbors and amplifying persuasion. Given a budget and individual cost functions for shifting each voter's preference toward a designated candidate, the goal is to determine whether a shift strategy exists within budget that ensures the preferred candidate wins after both direct and network-propagated influence takes effect. We show that the problem is NP-Complete even with two candidates and unit costs, and W[2]-hard when parameterized by budget or maximum degree. On the positive side, we design polynomial-time algorithms for complete graphs under plurality and majority rules and path graphs for uniform edge weights, linear-time algorithms for transitive tournaments for two candidates, linear cost functions and uniform arc weights, and pseudo-polynomial algorithms for cluster graphs. We further prove the existence of fixed-parameter tractable algorithms with treewidth as parameter for two candidates, linear cost functions and uniform arc weights and pseudo-FPT with cluster vertex deletion number for two candidates and uniform arc weights. Together, these results give a detailed complexity landscape for shift bribery in social networks.

The alternating direction of multipliers method (ADMM) is a popular method to solve distributed consensus optimization utilizing efficient communication among various nodes in the network. However, in the presence of faulty or attacked nodes, even a small perturbation (or sharing false data) during the communication can lead to divergence of the solution. To address this issue, in this work we consider ADMM under the effect of Byzantine threat, where an unknown subset of nodes is subject to Byzantine attacks or faults. We propose Dynamically Weighted ADMM (DW-ADMM), a novel variant of ADMM that uses dynamic weights on the edges of the network, thus promoting resilient distributed optimization. We establish that the proposed method (i) produces a nearly identical solution to conventional ADMM in the error-free case, and (ii) guarantees a bounded solution with respect to the global minimizer, even under Byzantine threat. Finally, we demonstrate the effectiveness of our proposed algorithm using an illustrative numerical simulation.

However, the undirected setting is often insufficient: only a causal (i.e., directed) network provides

Here, we show that marginal variance tends to increase along the causal order for generically sampled additive noise models.

The Equivariant Quantum Circuit (EQC) for the Travelling Salesman Problem (TSP) has been shown to achieve near-optimal performance in solving small TSP problems (up to 20 nodes) using only two parameters at depth 1. However, extending EQCs to larger TSP problem sizes remains challenging due to the exponential time and memory for quantum circuit simulation, as well as increasing noise and decoherence when running on actual quantum hardware. In this work, we propose the Size-Invariant Grid Search (SIGS), an efficient training optimization for Quantum Reinforcement Learning (QRL), and use it to simulate the outputs of a trained Depth-1 EQC up to 350-node TSP instances - well beyond previously tractable limits. At TSP with 100 nodes, we reduce total simulation times by 96.4%, when comparing to RL simulations with the analytical expression (151 minutes using RL to under 6 minutes using SIGS on TSP-100), while achieving a mean optimality gap within 0.005 of the RL trained model on the test set. SIGS provides a practical benchmarking tool for the QRL community, allowing us to efficiently analyze the performance of QRL algorithms on larger problem sizes. We provide a theoretical explanation for SIGS called the Size-Invariant Properties that goes beyond the concept of equivariance discussed in prior literature.