Community detection is an essential tool for unsupervised data exploration and revealing theorganisational structure ofnetworkedsystems. Withalong history innetwork science, community detection typically relies on objectivefunctions, optimised with custom-tailored search algorithms, but often without leveraging recentadvancesindeeplearning.

Community detection is an essential tool for unsupervised data exploration and revealing the organisational structure of networked systems. With a long history in network science, community detection typically relies on objective functions, optimised with custom-tailored search algorithms, but often without leveraging recent advances in deep learning. Recently, first works have started incorporating such objectives into loss functions for deep graph clustering and pooling. 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. Our formulation turns the map equation compatible with any neural network architecture, enables end-to-end learning, incorporates node features, and chooses the optimal number of clusters automatically, all without requiring explicit regularisation. Applied to unsupervised graph clustering tasks, we achieve competitive performance against state-of-the-art deep graph clustering baselines in synthetic and real-world datasets.

Community detection is an essential tool for unsupervised data exploration and revealing the organisational structure of networked systems. With a long history in network science, community detection typically relies on objective functions, optimised with custom-tailored search algorithms, but often without leveraging recent advances in deep learning. Recently, first works have started incorporating such objectives into loss functions for deep graph clustering and pooling. 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. Our formulation turns the map equation compatible with any neural network architecture, enables end-to-end learning, incorporates node features, and chooses the optimal number of clusters automatically, all without requiring explicit regularisation.

Graph pooling is an essential part of deep graph representation learning. We introduce MapEqPool, a principled pooling operator that takes the inherent hierarchical structure of real-world graphs into account. MapEqPool builds on the map equation, an information-theoretic objective function for community detection based on the minimum description length principle which naturally implements Occam's razor and balances between model complexity and fit. We demonstrate MapEqPool's competitive performance with an empirical comparison against various baselines across standard graph classification datasets.

Community detection and graph clustering are essential for unsupervised data exploration and understanding the high-level organisation of networked systems. Recently, graph clustering has received attention as a primary task for graph neural networks. Although hierarchical graph pooling has been shown to improve performance in graph and node classification tasks, it performs poorly in identifying meaningful clusters. Community detection has a long history in network science, but typically relies on optimising objective functions with custom-tailored search algorithms, not leveraging recent advances in deep learning, particularly from graph neural networks. In this paper, we narrow this gap between the deep learning and network science communities. We consider the map equation, an information-theoretic objective function for unsupervised community detection. Expressing it in a fully differentiable tensor form that produces soft cluster assignments, we optimise the map equation with deep learning through gradient descent. More specifically, the reformulated map equation is a loss function compatible with any graph neural network architecture, enabling flexible clustering and graph pooling that clusters both graph structure and data features in an end-to-end way, automatically finding an optimum number of clusters without explicit regularisation by following the minimum description length principle. We evaluate our approach experimentally using different neural network architectures for unsupervised clustering in synthetic and real data. Our results show that our approach achieves competitive performance against baselines, naturally detects overlapping communities, and avoids over-partitioning sparse graphs.

As the No Free Lunch theorem formally states [1], algorithms for detecting communities in networks must make tradeoffs. In this work, we present a method for using metadata to inform tradeoff decisions. We extend the content map equation, which adds metadata entropy to the traditional map equation, by introducing a tuning parameter allowing for explicit specification of the metadata's relative importance in assigning community labels. On synthetic networks, we show how tuning for node metadata relates to the detectability limit, and on empirical networks, we show how increased tuning for node metadata yields increased mutual information with the metadata at a cost in the traditional map equation. Our tuning parameter, like the focusing knob of a microscope, allows users to "zoom in" and "zoom out" on communities with varying levels of focus on the metadata.

Comprehending complex systems by simplifying and highlighting important dynamical patterns requires modeling and mapping higher-order network flows. However, complex systems come in many forms and demand a range of representations, including memory and multilayer networks, which in turn call for versatile community-detection algorithms to reveal important modular regularities in the flows. Here we show that various forms of higher-order network flows can be represented in a unified way with networks that distinguish physical nodes for representing a~complex system's objects from state nodes for describing flows between the objects. Moreover, these so-called sparse memory networks allow the information-theoretic community detection method known as the map equation to identify overlapping and nested flow modules in data from a range of~different higher-order interactions such as multistep, multi-source, and temporal data. We derive the map equation applied to sparse memory networks and describe its search algorithm Infomap, which can exploit the flexibility of sparse memory networks. Together they provide a general solution to reveal overlapping modular patterns in higher-order flows through complex systems.