Lightnews — Scholar-powered news

Abel Jansma @abelaer.bsky.social · 15h

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!

Abel Jansma @abelaer.bsky.social · 15h

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.

1 1

Abel Jansma @abelaer.bsky.social · 15h

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*!

1 1

Abel Jansma @abelaer.bsky.social · 15h

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:

1

Abel Jansma @abelaer.bsky.social · 15h

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!

1

Abel Jansma @abelaer.bsky.social · 15h

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.

1 2

Abel Jansma @abelaer.bsky.social · 15h

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.

1 1

Abel Jansma @abelaer.bsky.social · 15h

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...

1

Abel Jansma @abelaer.bsky.social · 15h

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. 🧮

1

Abel Jansma @abelaer.bsky.social · 15h

🚨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: 🧵

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...

arxiv.org

1 3 7

Abel Jansma @abelaer.bsky.social · 15d

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?

Abel Jansma @abelaer.bsky.social · 15d

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.

Abel Jansma @abelaer.bsky.social · 16d

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):

1

Abel Jansma @abelaer.bsky.social · 16d

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):

3

Abel Jansma @abelaer.bsky.social · 19d

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!

Abel Jansma @abelaer.bsky.social · Jan 22

🚨New paper! The “partial causality decomposition”

When many things influence an outcome, how to distinguish individual vs collective #causality?

A framework to disentangle these complex relationships into synergistic, redundant, and unique components: arxiv.org/abs/2501.11447 🧵

arxiv.org

1

Abel Jansma @abelaer.bsky.social · Aug 12

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

Dutch Institute for Emergent Phenomena

www.youtube.com

1

Abel Jansma @abelaer.bsky.social · Aug 5

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

Abel Jansma @abelaer.bsky.social · May 14

Yes!

1

Abel Jansma @abelaer.bsky.social · May 6

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...

WORKSHOPANALOGIES | DIEP

www.d-iep.org

2 1 5

Abel Jansma @abelaer.bsky.social · Apr 8

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...

Abel Jansma @abelaer.bsky.social · Apr 8

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

Mereological approach to higher-order structure in complex systems: From macro to micro with M\"obius

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 understan...

journals.aps.org

1 1 2

Abel Jansma @abelaer.bsky.social · Feb 7

The link doesn’t seem to work for me

1

Abel Jansma @abelaer.bsky.social · Feb 6

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

1

Abel Jansma @abelaer.bsky.social · Feb 4

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...

Complex Systems and Quantitative Mereology

abeljansma.nl

1 2

Abel Jansma @abelaer.bsky.social · Jan 22

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

1 1

Abel Jansma @abelaer.bsky.social · Jan 22

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!

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...

arxiv.org