arXiv subjects

Abel Jansma

Publications and source records attributed to Abel Jansma.

9 recordsLinked to original sources

arXiv2026

Climate-model factor separation with Shapley values and efficient sampling

Climate models often feature nonlinear responses to changes in model parameters or boundary conditions. Factor separation asks how much of the resulting change should be assigned to altered factors and their interactions. Lunt et al. (2010) showed that two factor attribution methods, the \emph{linear-sum} and \emph{shared-interaction} methods, coincide for $n\le 4$ factors and conjectured this holds for arbitrary $n$. We show that the linear-sum and shared-interaction formulas are respectively the permutation and Harsanyi-dividend representations of the Shapley value. Their equality therefore follows from a classical theorem in cooperative game theory. To our knowledge, this is the first explicit identification of the Stein--Alpert and Lunt climate-model factor-separation methods with Möbius/Harsanyi coefficients and Shapley values. The result imports many established and efficient Shapley sampling methods into climate-model experimental design, and extends to vector-valued responses.

math.OC↗

arXiv2026

Möbius transforms and Shapley values for vector-valued functions on weighted directed acyclic multigraphs

Möbius inversion and Shapley values are two mathematical tools for characterizing and decomposing higher-order structure in complex systems. The former defines higher-order interactions as discrete derivatives over a partial order; the latter provides a principled way to attribute those interactions back to the `atomic' elements of the system. Both have found wide application, from combinatorics and cooperative game theory to machine learning and explainable AI. We generalize both tools simultaneously in two orthogonal directions: 1) from real-valued functions to functions valued in any abelian group (in particular, vector-valued functions), and 2) from partial orders and lattices to directed acyclic multigraphs (DAMGs) and weighted versions thereof. The classical axioms, linearity, efficiency, null player, and symmetry, which uniquely characterize Shapley values on lattices, are insufficient in this more general setting. We resolve this by introducing projection operators that recursively re-attribute higher-order synergies down to the roots of the graph, and by proposing two natural axioms: weak elements (coalitions with zero synergy can be removed without affecting any attribution) and flat hierarchy (on graphs with no intermediate hierarchy, attributions are distributed proportionally to edge counts). Together with linearity, these three axioms uniquely determine the Shapley values via a simple explicit formula, while automatically implying efficiency, null player, symmetry, and a novel projection property. The resulting framework recovers all existing lattice-based definitions as special cases, and naturally handles settings, such as games on non-lattice partial orders, which were previously out of reach. The extension to vector-valued functions and general DAMG-structured hierarchies opens new application areas in machine learning, natural language processing, and explainable AI.

cs.GT↗

arXiv2025

Engineering Emergence

A defining property of complex systems is that they have multiscale structure. How does this multiscale structure come about? We argue that within systems there emerges a hierarchy of scales that contribute to a system's causal workings. An intuitive example is how a computer can be described at the level of its hardware circuitry (its microscale) but also its machine code (a mesoscale) and all the way up at its operating system (its macroscale). Here we show that even simple systems possess this kind of emergent hierarchy, which usually forms over only a small subset of the super-exponentially many possible scales of description. To capture this formally, we extend the theory of causal emergence (version 2.0) so as to analyze how causal contributions span the full multiscale structure of a system. Our analysis reveals that systems can be classified along a taxonomy of emergence, such as being either top-heavy or bottom-heavy in their causal workings. From this new taxonomy of emergence, we derive a measure of complexity based on a literal notion of scale-freeness (here, when causation is spread equally across the scales of a system) and compare this to the standard network science definition of scale-freeness based on degree distribution, showing the two are closely related. Finally, we demonstrate the ability to engineer not just the degree of emergence in a system, but to control it with pinpoint precision.

cs.IT↗

arXiv2025

Decomposing Interventional Causality into Synergistic, Redundant, and Unique Components

We introduce a novel framework for decomposing interventional causal effects into synergistic, redundant, and unique components, building on the intuition of Partial Information Decomposition (PID) and the principle of Möbius inversion. While recent work has explored a similar decomposition of an observational measure, we argue that a proper causal decomposition must be interventional in nature. We develop a mathematical approach that systematically quantifies how causal power is distributed among variables in a system, using a recently derived closed-form expression for the Möbius function of the redundancy lattice. The formalism is then illustrated by decomposing the causal power in logic gates, cellular automata, chemical reaction networks, and a transformer language model. Our results reveal how the distribution of causal power can be context- and parameter-dependent. The decomposition provides new insights into complex systems by revealing how causal influences are shared and combined among multiple variables, with potential applications ranging from attribution of responsibility in legal or AI systems, to the analysis of biological networks or climate models.

cs.AI↗

arXiv2025

A Mereological Approach to Higher-Order Structure in Complex Systems: from Macro to Micro with Möbius

Relating macroscopic observables to microscopic interactions is a central challenge in the study of complex systems. While current approaches often focus on pairwise interactions, a complete understanding requires going beyond these to capture the full range of possible interactions. We present a unified mathematical formalism, based on the Möbius inversion theorem, that reveals how different decompositions of a system into parts lead to different, but equally valid, microscopic theories. By providing an exact bridge between microscopic and macroscopic descriptions, this framework demonstrates that many existing notions of interaction, from epistasis in genetics and many-body couplings in physics, to synergy in game theory and artificial intelligence, naturally and uniquely arise from particular choices of system decomposition, or mereology. By revealing the common mathematical structure underlying seemingly disparate phenomena, our work highlights how the choice of decomposition fundamentally determines the nature of the resulting interactions. We discuss how this unifying perspective can facilitate the transfer of insights across domains, guide the selection of appropriate system decompositions, and enable the search for new notions of interaction. To illustrate the latter in practice, we decompose the Kullback-Leibler divergence, and show that our method correctly identifies which variables are responsible for the divergence. In addition, we use Rota's Galois connection theorem to describe coarse-grainings of mereologies, and efficiently derive the renormalised couplings of a 1D Ising model. Our results suggest that the Möbius inversion theorem provides a powerful and practical lens for understanding the emergence of complex behaviour from the interplay of microscopic parts, with applications across a wide range of disciplines.

physics.data-an↗

arXiv2024

The Fast Möbius Transform: An algebraic approach to information decomposition

The partial information decomposition (PID) and its extension integrated information decomposition ($Φ$ID) are promising frameworks to investigate information phenomena involving multiple variables. An important limitation of these approaches is the high computational cost involved in their calculation. Here we leverage fundamental algebraic properties of these decompositions to enable a computationally-efficient method to estimate them, which we call the fast Möbius transform. Our approach is based on a novel formula for estimating the Möbius function that circumvents important computational bottlenecks. We showcase the capabilities of this approach by presenting two analyses that would be unfeasible without this method: decomposing the information that neural activity at different frequency bands yield about the brain's macroscopic functional organisation, and identifying distinctive dynamical properties of the interactions between multiple voices in baroque music. Overall, our proposed approach illuminates the value of algebraic facets of information decomposition and opens the way to a wide range of future analyses.

cs.IT↗

arXiv2024

Superdense Coding and Stabiliser Codes with Ising-coupled Entanglement

A new class of quantum states is introduced by demanding that the computational measurement statistics approach the Boltzmann distribution of higher-order strongly coupled Ising models. The states, referred to as $n$-coupled states, are superpositions of even or odd parity $n$-qubit states, generalize Bell states, and form an orthonormal basis for the $n$-qubit Hilbert space. For any $n$, the states are maximally connected and locally maximally entangled. It is proven that the $n$-qubit W and GHZ multipartite entanglement classes have vanishing hyperdeterminant for all $n\geq 3$ and $n\geq 4$, respectively, and that the $n$-coupled states fall in the latter. Still, multiple novel protocols for multi-party secure dense coding and stabiliser code construction are presented, which rely on the structure of $n$-coupled states as well as symmetry-breaking phase perturbations.

quant-ph↗

arXiv2023

A compositional game to fairly divide homogeneous cake

The central question in the game theory of cake-cutting is how to fairly distribute a finite resource among multiple players. Most research has focused on how to do this for a heterogeneous cake in a situation where the players do not have access to each other's valuation function, but I argue that even sharing homogeneous cake can have interesting mechanism design. Here, I introduce a new game, based on the compositional structure of iterated cake-cutting, that in the case of a homogeneous cake has a Nash equilibrium where each of $n$ players gets $1/n$ of the cake. Furthermore, the equilibrium distribution is the result of just $n-1$ cuts, so each player gets a contiguous piece of cake. Naive composition of the `I cut you choose' rule leads to an exponentially unfair cake distribution with a Gini-coefficient that approaches 1, and suffers from a high Price of Anarchy. This cost is completely eliminated by the proposed \textit{Biggest Player} rule for composition which achieves decentralised and asynchronous fairness at linear Robertson-Webb complexity. After introducing the game, proving the fairness of the equilibrium, and analysing the incentive structure, the game is implemented in Haskell and the Open Game engine to make the compositional structure explicit.

cs.GT↗

arXiv2023

Higher-Order Interactions and Their Duals Reveal Synergy and Logical Dependence beyond Shannon-Information

Information-theoretic quantities reveal dependencies among variables in the structure of joint, marginal, and conditional entropies, but leave some fundamentally different systems indistinguishable. Furthermore, there is no consensus on how to construct and interpret a higher-order generalisation of mutual information (MI). In this manuscript, we show that a recently proposed model-free definition of higher-order interactions amongst binary variables (MFIs), like mutual information, is a Möbius inversion on a Boolean algebra, but of surprisal instead of entropy. This gives an information-theoretic interpretation to the MFIs, and by extension to Ising interactions. We study the dual objects to MI and MFIs on the order-reversed lattice, and find that dual MI is related to the previously studied differential mutual information, while dual interactions (outeractions) are interactions with respect to a different background state. Unlike mutual information, in- and outeractions uniquely identify all six 2-input logic gates, the dy- and triadic distributions, and different causal dynamics that are identical in terms of their Shannon-information content.

cs.IT↗