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Enrico @enricocaprioglio.bsky.social · 18/09/2025

To get some analytical results, we studied Ornstein–Uhlenbeck (OU) processes for which we have a simple expression for the stationary Σ (symmetric interactions). Using this, we get to two more theorems, one for systems of size 3, and the other for any size in the limit of weak interactions:

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Enrico @enricocaprioglio.bsky.social · 18/09/2025

Theo. 2: if a OU process with N=3 is synergistic, then interactions must be antibalanced Theo. 3: antibalanced interaction motifs are necessary for OU processes to be synergy-dominated For any strength of interactions, here are some numerical results when N = 8 (using every possible sign pattern):

Antibalanced interaction motifs are necessary for synergy-dominance in dynamical Gaussian systems. Mean Ω(X) as a function of the number of antibalanced triangles in the interaction matrix. Lower bounds indicate the lowest Ω(X) encountered in our numerical exploration across all configurations with a specific no. of antibalanced triangles. Each mean Ω(X) is coloured according to its mean balance-energy value U.

Here, Ω(X) is anticorrelated with the number of antibalanced triangle motifs in the interaction matrix of the OU process. When all triangles are balanced, the system is very redundant. When all triangles are antibalanced, the system displays the highest synergy (i.e., lowest Ω(X))

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Enrico @enricocaprioglio.bsky.social · 18/09/2025

Is there any reason to believe that this intuition works for other complex systems? And does it work using real data (our proofs do require the correlation matrices to have a spectral radius < 2)? Surprisingly, yeah it works!

(LEFT) Antibalanced interaction motifs characterize synergy-dominated complex systems with pairwise coupling. Plots show Ω versus the number of antibalanced triangles in the pairwise coupling matrix. (a) For the Ising model we compute Ω exactly for each possible coupling configuration (β = 0.25). (b) For the SL network of oscillators we report the mean Ω (circle) over 100 initial conditions and the minimum Ω (star) found (lower bound) across configurations with a specific no. of antibalanced triangles.

The curves in both (a) and (b) are similar to the one for the OU process. With no antibalanced triangles, the system is displaying the highest redundancy. While for the largest number of antibalanced triangles, the system show the lowest Ω(X) (most synergistic)

(RIGHT) Structural balance-energy predicts synergy: the most synergistic N-plets maximise U and positive U guarantees synergy-dominance. (a) N-plets from a multivariate time-series of 9 foreign-exchange (FX) rate logarithmic returns. (b) N-plets from resting-state fMRI functional-connectivity matrices (229 brain regions).
Here we see that Ω(X) and a measure of balanced U behave very similarly (strongly anticorrelated). Near zero however, interesting stuff happens.

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