This paper develops a unified framework for measuring concentration in weighted systems embedded in networks of interactions. While traditional indices such as the Herfindahl-Hirschman Index capture dispersion in weights, they neglect the topology of relationships among the elements receiving those weights. To address this limitation, we introduce a family of topology-aware concentration indices that jointly account for weight distributions and network structure. At the core of the framework lies a baseline Network Concentration Index (NCI), defined as a normalized quadratic form that measures the fraction of potential weighted interconnection realized along observed network links. Building on this foundation, we construct a flexible class of extensions that modify either the interaction structure or the normalization benchmark, including weighted, density-adjusted, null-model, degree-constrained, transformed-data, and multi-layer variants. This family of indices preserves key properties such as normalization, invariance, and interpretability, while allowing concentration to be evaluated across different dimensions of dependence, including intensity, higher-order interactions, and extreme events. Theoretical results characterize the indices and establish their relationship with classical concentration and network measures. Empirical and simulation evidence demonstrate that systems with identical weight distributions may exhibit markedly different levels of structural concentration depending on network topology, highlighting the additional information captured by the proposed framework. The approach is broadly applicable to economic, financial, and complex systems in which weighted elements interact through networks.

In distributed machine learning (DML), the network performance between machines significantly impacts the speed of iterative training. In this paper we propose BML, a new gradient synchronization algorithm with higher network performance and lower network cost than the current practice. BML runs on BCube network, instead of using the traditional Fat-Tree topology.

Source code for the training pipeline, tasks, and models used in this work, is available as part of the supplementary material. We used the same Adam [48] optimizer for all our experiments and a learning rate of 0.001, and a batch size of 128. For solving the differential equations both during ground truth data generation as well as with the neural ODEs, we use the Tsitouras 5/4 Runge-Kutta (Tsit5) method from DifferentialEquations.jl [36]. A.1 Coupled Pendulum The coupled pendulum dynamics are defined as We train the MP-NODE on a dataset of 500 trajectories, each randomly initialized with state values between [ π/2, π/2] for the θ and [ 1, 1] for θ, with a time step of 0.1s and each trajectory 10s long. The dataset is normalized through Z-score normalization.

Quantum networks are becoming increasingly important because of advancements in quantum computing and quantum sensing, such as recent developments in distributed quantum computing and federated quantum machine learning. Routing entanglement in quantum networks poses several fundamental as well as technical challenges, including the high dynamicity of quantum network links and the probabilistic nature of quantum operations. Consequently, designing hand-crafted heuristics is difficult and often leads to suboptimal performance, especially if global network topology information is unavailable. In this paper, we propose RELiQ, a reinforcement learning-based approach to entanglement routing that only relies on local information and iterative message exchange. Utilizing a graph neural network, RELiQ learns graph representations and avoids overfitting to specific network topologies - a prevalent issue for learning-based approaches. Our approach, trained on random graphs, consistently outperforms existing local information heuristics and learning-based approaches when applied to random and real-world topologies. When compared to global information heuristics, our method achieves similar or superior performance because of its rapid response to topology changes.

This paper proposes an optimization of Quantum Key Distribution (QKD) Networks using Graph Neural Networks (GNN) framework. Today, the development of quantum computers threatens the security systems of classical cryptography. Moreover, as QKD networks are designed for protecting secret communication, they suffer from multiple operational difficulties: adaptive to dynamic conditions, optimization for multiple parameters and effective resource utilization. In order to overcome these obstacles, we propose a GNN-based framework which can model QKD networks as dynamic graphs and extracts exploitable characteristics from these networks' structure. The graph contains not only topological information but also specific characteristics associated with quantum communication (the number of edges between nodes, etc). Experimental results demonstrate that the GNN-optimized QKD network achieves a substantial increase in total key rate (from 27.1 Kbits/s to 470 Kbits/s), a reduced average QBER (from 6.6% to 6.0%), and maintains path integrity with a slight reduction in average transmission distance (from 7.13 km to 6.42 km). Furthermore, we analyze network performance across varying scales (10 to 250 nodes), showing improved link prediction accuracy and enhanced key generation rate in medium-sized networks. This work introduces a novel operation mode for QKD networks, shifting the paradigm of network optimization through adaptive and scalable quantum communication systems that enhance security and performance.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/