arXiv subjects

Christopher Blöcker

Publications and source records attributed to Christopher Blöcker.

14 recordsLinked to original sources

arXiv2026

Dimensions of Power: A Systematic Guide to Power Indices for Explainable AI

Power indices, originating in cooperative game theory, quantify each player's influence on the outcome of a given game. Originally designed to distribute profits or costs among players and to analyse the fairness of voting systems, power indices have recently gained prominence as methods for attributing outputs of AI-based systems to inputs, thus facilitating explainability. However, selecting the appropriate power index for a given explanation task is an understudied problem. To address this, we organise power indices along three attribution dimensions: single-player, set-based, and cardinality-based. For each dimension, we review the corresponding power indices, generalise existing ones where applicable, and analyse which formal principles they satisfy. We provide proofs for properties that are missing in the literature and show that moving to the cardinality-based setting removes player-identity information while preserving some index-level distinctions. Using concrete examples, we illustrate how the choice of dimension and index affects the resulting attributions in practice, and offer guidance for practitioners seeking to select a suitable power index for a given application context.

cs.GT↗

arXiv2026

The Role of Node Features in Graph Pooling

Graph pooling is commonly applied in graph classification, yet its empirical gains over standard WL-1 expressive GNNs are often marginal or inconsistent. We study this gap by analysing the interaction between node features and graph topology and their effect on pooling objectives. Our analysis reveals that pooling operators require node features that are well-aligned with the graph's topology -- a condition often overlooked and not guaranteed in empirical networks. We formalise fundamental requirements for node features to enable effective pooling, and introduce a quantitative measure of feature quality. Our empirical evaluation shows that, when these requirements are satisfied, pooling can be beneficial and improve performance on appropriate datasets.

cs.LG↗

arXiv2026

From Link Prediction to Forecasting: Addressing Challenges in Batch-based Temporal Graph Learning

Dynamic link prediction is an important problem considered in many recent works that propose approaches for learning temporal edge patterns. To assess their efficacy, models are evaluated on continuous-time and discrete-time temporal graph datasets, typically using a traditional batch-oriented evaluation setup. However, as we show in this work, a batch-oriented evaluation is often unsuitable and can cause several issues. Grouping edges into fixed-sized batches regardless of their occurrence time leads to information loss or leakage, depending on the temporal granularity of the data. Furthermore, fixed-size batches create time windows with different durations, resulting in an inconsistent dynamic link prediction task. In this work, we empirically show how traditional batch-based evaluation leads to skewed model performance and hinders the fair comparison of methods. We mitigate this problem by reformulating dynamic link prediction as a link forecasting task that better accounts for temporal information present in the data.

cs.LG↗

arXiv2026

Community Detection with the Map Equation and Infomap: Theory and Applications

Real-world networks have a complex topology comprising many elements often structured into communities. Revealing these communities helps researchers uncover the organizational and functional structure of the system that the network represents. However, detecting community structures in complex networks requires selecting a community detection method among a multitude of alternatives with different network representations, community interpretations, and underlying mechanisms. This tutorial focuses on a popular community detection method called the map equation and its search algorithm Infomap. The map equation framework for community detection describes communities by analyzing dynamic processes on the network. Thanks to its flexibility, the map equation provides extensions that can incorporate various assumptions about network structure and dynamics. To help decide if the map equation is a suitable community detection method for a given complex system and problem at hand - and which variant to choose - we review the map equation's theoretical framework and guide users in applying the map equation to various research problems.

physics.soc-ph↗

arXiv2026

Deep Graph Learning will stall without Network Science

Deep graph learning focuses on flexible and generalizable models that learn patterns in an automated fashion. Network science focuses on models and measures revealing the organizational principles of complex systems with explicit assumptions. Both fields share the same goal: to better model and understand patterns in graph-structured data. However, deep graph learning prioritizes empirical performance but ignores fundamental insights from network science. Our position is that deep graph learning will stall without insights from network science. In this position paper, we formulate six Calls for Action to leverage untapped insights from network science to address current issues in deep graph learning, ensuring the field continues to make progress.

cs.LG↗

arXiv2025

Mapping memory-biased dynamics with compact models reveals overlapping communities in large networks

Many real-world systems, from social networks to protein-protein interactions and species distributions, exhibit overlapping flow-based communities that reflect their functional organisation. However, reliably identifying such overlapping flow-based communities requires higher-order relational data, which are often unavailable. To address this challenge, we capitalise on the flow model underpinning the representation-learning algorithm node2vec and model higher-order flows through memory-biased random walks on first-order networks. Instead of simulating these walks, we model their higher-order dynamic constraints with compact models and control model complexity with an information-theoretic approach. Using the map equation framework, we identify overlapping modules in the resulting higher-order networks. Our compact-model approach proves robust across synthetic benchmark networks, reveals interpretable overlapping communities in empirical networks, and scales to large networks.

physics.soc-ph↗

arXiv2025

MDL-Pool: Adaptive Multilevel Graph Pooling Based on Minimum Description Length

Graph pooling compresses graphs and summarises their topological properties and features in a vectorial representation. It is an essential part of deep graph representation learning and is indispensable in graph-level tasks like classification or regression. Current approaches pool hierarchical structures in graphs by iteratively applying shallow pooling operators up to a fixed depth. However, they disregard the interdependencies between structures at different hierarchical levels and do not adapt to datasets that contain graphs with different sizes that may require pooling with various depths. To address these issues, we propose MDL-Pool, a pooling operator based on the minimum description length (MDL) principle, whose loss formulation explicitly models the interdependencies between different hierarchical levels and facilitates a direct comparison between multiple pooling alternatives with different depths. MDP-Pool builds on the map equation, an information-theoretic objective function for community detection, which naturally implements Occam's razor and balances between model complexity and goodness-of-fit via the MDL. We demonstrate MDL-Pool's competitive performance in an empirical evaluation against various baselines across standard graph classification datasets.

cs.LG↗

arXiv2025

Compressing regularized dynamics improves link prediction with the map equation in sparse networks

Predicting future interactions or novel links in networks is an indispensable tool across diverse domains, including genetic research, online social networks, and recommendation systems. Among the numerous techniques developed for link prediction, those leveraging the networks' community structure have proven highly effective. For example, the recently proposed MapSim predicts links based on a similarity measure derived from the code structure of the map equation, a community-detection objective function that operates on network flows. However, the standard map equation assumes complete observations and typically identifies many small modules in networks where the nodes connect through only a few links. This aspect can degrade MapSim's performance on sparse networks. To overcome this limitation, we propose to incorporate a global regularization method based on a Bayesian estimate of the transition rates along with three local regularization methods. The regularized versions of the map equation compensate for incomplete observations and mitigate spurious community fragmentation in sparse networks. The regularized methods outperform standard MapSim and several state-of-the-art embedding methods in highly sparse networks. This performance holds across multiple real-world networks with randomly removed links, simulating incomplete observations. Among the proposed regularization methods, the global approach provides the most reliable community detection and the highest link prediction performance across different network densities. The principled method requires no hyperparameter tuning and runs at least an order of magnitude faster than the embedding methods.

cs.SI↗

arXiv2024

The Map Equation Goes Neural: Mapping Network Flows with Graph Neural Networks

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.

cs.LG↗

arXiv2024

Flow Divergence: Comparing Maps of Flows with Relative Entropy

Networks represent how the entities of a system are connected and can be partitioned differently, prompting ways to compare partitions. Common approaches for comparing network partitions include information-theoretic measures based on mutual information and set-theoretic measures such as the Jaccard index. These measures are often based on computing the agreement in terms of overlap between different partitions of the same set. However, they ignore link patterns which are essential for the organisation of networks. We propose flow divergence, an information-theoretic divergence measure for comparing network partitions, inspired by the ideas behind the Kullback-Leibler divergence and the map equation for community detection. Similar to the Kullback-Leibler divergence, flow divergence adopts a coding perspective and compares two network partitions $\mathsf{M}_a$ and $\mathsf{M}_b$ by considering the expected extra number of bits required to describe a random walk on a network using $\mathsf{M}_b$ relative to reference partition $\mathsf{M}_a$. Because flow divergence is based on random walks, it can be used to compare partitions with arbitrary and different depths. We show that flow divergence distinguishes between partitions that traditional measures consider to be equally good when compared to a reference partition. Applied to real networks, we use flow divergence to estimate the cost of overfitting in incomplete networks and to visualise the solution landscape of network partitions.

cs.SI↗

arXiv2023

Similarity-based Link Prediction from Modular Compression of Network Flows

Node similarity scores are a foundation for machine learning in graphs for clustering, node classification, anomaly detection, and link prediction with applications in biological systems, information networks, and recommender systems. Recent works on link prediction use vector space embeddings to calculate node similarities in undirected networks with good performance. Still, they have several disadvantages: limited interpretability, need for hyperparameter tuning, manual model fitting through dimensionality reduction, and poor performance from symmetric similarities in directed link prediction. We propose MapSim, an information-theoretic measure to assess node similarities based on modular compression of network flows. Unlike vector space embeddings, MapSim represents nodes in a discrete, non-metric space of communities and yields asymmetric similarities in an unsupervised fashion. We compare MapSim on a link prediction task to popular embedding-based algorithms across 47 networks and find that MapSim's average performance across all networks is more than 7% higher than its closest competitor, outperforming all embedding methods in 11 of the 47 networks. Our method demonstrates the potential of compression-based approaches in graph representation learning, with promising applications in other graph learning tasks.

cs.SI↗

arXiv2022

Map Equation Centrality: Community-aware Centrality based on the Map Equation

To measure node importance, network scientists employ centrality scores that typically take a microscopic or macroscopic perspective, relying on node features or global network structure. However, traditional centrality measures such as degree centrality, betweenness centrality, or PageRank neglect the community structure found in real-world networks. To study node importance based on network flows from a mesoscopic perspective, we analytically derive a community-aware information-theoretic centrality score based on network flow and the coding principles behind the map equation: map equation centrality. Map equation centrality measures how much further we can compress the network's modular description by not coding for random walker transitions to the respective node, using an adapted coding scheme and determining node importance from a network flow-based point of view. The information-theoretic centrality measure can be determined from a node's local network context alone because changes to the coding scheme only affect other nodes in the same module. Map equation centrality is agnostic to the chosen network flow model and allows researchers to select the model that best reflects the dynamics of the process under study. Applied to synthetic networks, we highlight how our approach enables a more fine-grained differentiation between nodes than node-local or network-global measures. Predicting influential nodes for two different dynamical processes on real-world networks with traditional and other community-aware centrality measures, we find that activating nodes based on map equation centrality scores tends to create the largest cascades in a linear threshold model.

cs.SI↗

arXiv2021

Mapping flows on weighted and directed networks with incomplete observations

Detecting significant community structure in networks with incomplete observations is challenging because the evidence for specific solutions fades away with missing data. For example, recent research shows that flow-based community detection methods can highlight spurious communities in sparse undirected and unweighted networks with missing links. Current Bayesian approaches developed to overcome this problem do not work for incomplete observations in weighted and directed networks that describe network flows. To address this gap, we extend the idea behind the Bayesian estimate of the map equation for unweighted and undirected networks to enable more robust community detection in weighted and directed networks. We derive a weighted and directed prior network that can incorporate metadata information and show how an efficient implementation in the community-detection method Infomap provides more reliable communities even with a significant fraction of data missing.

cs.SI↗

arXiv2020

Mapping Flows on Bipartite Networks

Mapping network flows provides insight into the organization of networks, but even though many real-networks are bipartite, no method for mapping flows takes advantage of the bipartite structure. What do we miss by discarding this information and how can we use it to understand the structure of bipartite networks better? The map equation models network flows with a random walk and exploits the information-theoretic duality between compression and finding regularities to detect communities in networks. However, it does not use the fact that random walks in bipartite networks alternate between node types, information worth 1 bit. To make some or all of this information available to the map equation, we developed a coding scheme that remembers node types at different rates. We explored the community landscape of bipartite real-world networks from no node-type information to full node-type information and found that using node types at a higher rate generally leads to deeper community hierarchies and a higher resolution. The corresponding compression of network flows exceeds the amount of extra information provided. Consequently, taking advantage of the bipartite structure increases the resolution and reveals more network regularities.

cs.SI↗