Low-dimensional embeddings, from classical spectral embeddings to modern neural-net-inspired methods, are a cornerstone in the modeling and analysis of complex networks. Recent work by Seshadhri et al. (PNAS 2020) suggests that such embeddings cannot capture local structure arising in complex networks. In particular, they show that any network generated from a natural low-dimensional model cannot be both sparse and have high triangle density (high clustering coefficient), two hallmark properties of many real-world networks. In this work we show that the results of Seshadhri et al. are intimately connected to the model they use rather than the low-dimensional structure of complex networks. Specifically, we prove that a minor relaxation of their model can generate sparse graphs with high triangle density. Surprisingly, we show that this same model leads to of many real-world networks. We give a simple algorithm based on logistic principal component analysis (LPCA) that succeeds in finding such exact embeddings. Finally, we perform a large number of experiments that verify the ability of very low-dimensional embeddings to capture local structure in real-world networks.

Recent advances in deep reinforcement learning for autonomous cyber defence have resulted in agents that can successfully defend simulated computer networks against cyber-attacks. However, many of these agents would need retraining to defend networks with differing topology or size, making them poorly suited to real-world networks where topology and size can vary over time. In this research we introduce a novel set of Topological Extensions for Reinforcement Learning Agents (TERLA) that provide generalisability for the defence of networks with differing topology and size, without the need for retraining. Our approach involves the use of heterogeneous graph neural network layers to produce a fixed-size latent embedding representing the observed network state. This representation learning stage is coupled with a reduced, fixed-size, semantically meaningful and interpretable action space. We apply TERLA to a standard deep reinforcement learning Proximal Policy Optimisation (PPO) agent model, and to reduce the sim-to-real gap, conduct our research using Cyber Autonomy Gym for Experimentation (CAGE) Challenge 4. This Cyber Operations Research Gym environment has many of the features of a real-world network, such as realistic Intrusion Detection System (IDS) events and multiple agents defending network segments of differing topology and size. TERLA agents retain the defensive performance of vanilla PPO agents whilst showing improved action efficiency. Generalisability has been demonstrated by showing that all TERLA agents have the same network-agnostic neural network architecture, and by deploying a single TERLA agent multiple times to defend network segments with differing topology and size, showing improved defensive performance and efficiency.

Abstract--Understanding and quantifying node importance is a fundamental problem in network science and engineering, underpinning a wide range of applications such as influence maximization, social recommendation, and network dismantling. Prior research often relies on centrality measures or advanced graph embedding techniques using structural information, followed by downstream classification or regression tasks to identify critical nodes. However, these methods typically decouple node representation learning from the ranking objective and rely on the topological structure of target networks, leading to feature-task inconsistency and limited generalization across networks. This paper proposes a novel framework that leverages causal representation learning to get robust, invariant node embeddings for cross-network ranking tasks. Firstly, we introduce an influence-aware causal node embedding module within an autoencoder architecture to extract node embeddings that are causally related to node importance. Moreover, we introduce a causal ranking loss and design a unified optimization framework that jointly optimizes the reconstruction and ranking objectives, enabling mutual reinforcement between node representation learning and ranking optimization. This design allows the proposed model to be trained on synthetic networks and to generalize effectively across diverse real-world networks. Extensive experiments on multiple benchmark datasets demonstrate that the proposed model consistently outperforms state-of-the-art baselines in terms of both ranking accuracy and cross-network transferability, offering new insights for network analysis and engineering applications--particularly in scenarios where the target network's structure is inaccessible in advance due to privacy or security constraints. Complex networks provide a powerful framework for modeling and analyzing a wide range of systems across diverse domains, including social networks, transportation systems, and biological networks [1]. In these networks, nodes represent entities within a real system such as individuals, infrastructure components, or functional units, while edges capture interactions or relationships between them. A key challenge in network science and engineering is identifying important nodes, as they play pivotal roles in maintaining network functionality, performance, stability, and robustness [2].

One of the most persistent challenges in network science is the development of various synthetic graph models to support subsequent analyses. Among the most notable frameworks addressing this issue is the Artificial Benchmark for Community Detection (ABCD) model, a random graph model with community structure and power-law distribution for both degrees and community sizes. The model generates graphs similar to the well-known LFR model but it is faster, more interpretable, and can be investigated analytically. In this paper, we use the underlying ingredients of ABCD and introduce its variant, mABCD, thereby addressing the gap in models capable of generating multilayer networks. The uniqueness of the proposed approach lies in its flexibility at both levels of modelling: the internal structure of individual layers and the inter-layer dependencies, which together make the network a coherent structure rather than a collection of loosely coupled graphs. In addition to the conceptual description of the framework, we provide a comprehensive analysis of its efficient Julia implementation. Finally, we illustrate the applicability of mABCD to one of the most prominent problems in the area of complex systems: spreading phenomena analysis.

Thank you all for the helpful reviews. "and the goal is to figure out what type of object each pixel corresponds to" [Forouzan, 2015]. As suggested, we will run and report experiments on more networks, for a more comprehensive picture of our algorithm. We will focus more on the real-world networks. Segmentation-11 details... See above section on experiments .

We prove that our model can exactly represent any graph with low arboric-ity, a property that many real-world networks satisfy; our proof also applies to related models but has much greater scope than the closest prior bound, which is based on low max degree .

We consider the map equation, a popular information-theoretic objective function for unsupervised community detection, and express it in differentiable tensor form for optimisation through gradient descent.