Abel Jansma

I know you like showing pictures of lenses but this seems a little excessive

Many sea stars begin life as young fairy-like creature (called a brachiolaria) that float through the open ocean. Eventually, a small star forms within them (here in yellow). The fairy-like brachiolaria sinks under the star’s weight, and the star pops out!
🎥@the_story_of_a_biologist (on Insta)

There's so much more in the paper, largely thanks to Patrick who really pushed this to the next level. We're already working on applications: SVs are often used for XAI, but now we can do this for vector-valued functions--the kind implemented by transformers... Stay tuned!

To summarise:
⬆️Möbius inversions construct higher-order structure.
⬇️Shapley values project this down again, in the 'right' way.

We derive generalisations of both, to directed acyclic multigraphs, and group-valued functions.

This shows how intimately related Shapley values and Möbius inversions are: we derive an expression that expresses Shapley values *purely in terms of the incidence algebra*!

Doing so required also generalising the Möbius inversion theorem to this setting (prev. only defined for ring-valued functions). We show that it's a natural theorem in the *path algebra* of the graph:

But we go further.
Classical Shapley values only work for real-valued functions on power sets of players (or lattices).

We generalise them even beyond posets to
✅vector/group-valued fns
✅weighted directed acyclic multigraphs
, and prove uniqueness!

That’s exactly what we do.
We reinterpret Shapley values as projection operators: a recursive re-attribution of higher-order synergy to lower-order parts.

This turns Shapley values into a general projection framework for hierarchical structure, valid far beyond game theory.

Möbius inversions are a way to derive higher-order interactions ion a system's mereology. I wrote a blog post about this here 👉https://abeljansma.nl/2025/01/28/mereoPhysics.html

If Shapley values are truly general, we should be able to express them for any Möbius inversion/higher-order structure.

But Shapley values (SVs) aren’t just about fairness.
They're really a projection operator: the right way to push higher-order structure back down to lower levels.
So… can we do this more generally? 🤔

Enter Möbius inversions...

If a group of people earn a payoff together, how should it be fairly distributed?
Shapley values are weighted sums of sub-coalition "synergies", and provably the fairest possible distribution.
It earned Shapley the Nobel Prize. 🧮

Möbius transforms and Shapley values for vector-valued functions on weighted directed acyclic multigraphs We generalize the concept of Möbius inversion and Shapley values to directed acyclic multigraphs and weighted versions thereof. We further allow value functions (games) and thus their Möbius transform...

🚨New paper: arxiv.org/abs/2510.05786

*Shapley values beyond game theory*
We show that Shapley values aren’t just about dividing payoffs--they are the right way to project down any higher-order structure.

We generalise them, and Möbius inversions, in important ways: 🧵

This is how Rota originally introduced the incidence algebra. Everyone since has (correctly) required the ring to be commutative. Did people in the '60s just refer to commutative rings as associative rings?

Just to balance out the discourse: the #NeurIPS2025 review process for this paper went great. Fair reviews, mostly responsive reviewers, and a thoughtful AC that caught a possible conflict of interest. Definitely improved the paper.

The #NeurIPS2025 version is now online: arxiv.org/pdf/2501.11447

It includes a new analysis to show that LLM semantics can be decomposed: the negativity of "horribly bad" is redundantly encoded in the two words, whereas "not bad" has synergistic semantics (i.e. negation):

The "partial causality decomposition" was just accepted for a spotlight at #NeurIPS2025!

The final version includes a decomposition of LLM semantics---the Arxiv version should be updated soon. Stay tuned!

Dutch Institute for Emergent Phenomena

Hi! sorry I'm not on often on this site. They're now online: www.youtube.com/@dutchinstit...

While I'm flattered, it's a bit weird that google's AI defers to me when you search for this:

Yes!

Next week we're organising a workshop on the role of analogies in (artificial) intelligence, with:

Melanie Mitchell (@melaniemitchell.bsky.social), Martha Lewis, Jules Hedges (‪@julesh.mathstodon.xyz.ap.brid.gy‬), and Han van der Maas.

Register here: www.d-iep.org/workshopanal...

Don't take my word for it--take Reviewer 2's: "I found the paper extremely interesting and deep"

A gentle introduction is available at abeljansma.nl/2025/01/28/m...

My new approach to higher-order interactions in complex systems is now published in Physical Review Research:
journals.aps.org/prresearch/a...

The link doesn’t seem to work for me

just one more index bro I swear just one more subscript and it's gonna be so clear just one more index please bro

New blog post 🚨

A gentle dive into the mereology of complex systems, Möbius inversion, and a new way to think about higher-order interactions—no prior knowledge needed!
abeljansma.nl/2025/01/28/m...

New year, new Möbius inversion!

This new paper is an example of my more general proposal that you should study complex systems with 'quantitative mereology' by applying the Möbius inversion theorem: arxiv.org/abs/2404.14423

Decomposing causality into its synergistic, unique, and redundant components Causality lies at the heart of scientific inquiry, serving as the fundamental basis for understanding interactions among variables in physical systems. Despite its central role, current methods for ca...

This project was inspired by a recent method called SURD (arxiv.org/abs/2405.12411) that also aims to decompose causal synergy and redundancy, but doesn't manage to. More on this in the discussion of my paper. Let me know what you think, and if you have a system to analyse!

In chemical networks, reaction rates determine whether control is redundant (independent but similar pathways, in red) or synergistic (cooperative synthesis, in green).

In cellular automata, causal power is strongly context-dependent. The same rule can show different causal decompositions on different initial conditions!

In logic gates, we found that causal power shifts between redundant and synergistic depending on input probabilities. XOR gates only show pure synergy at p=0.5! (synergy in green, redundancy in red)