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Apology of the spherical cow:

simple models for complex systems

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Mario Castro

Comillas Pontifical University

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Two uncomfortable problems in complex systems

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http://biochemical-pathways.com/#/map/1

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Question 1

How much information is in the data?

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The “Rorschach” test

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The “Rorschach” test

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https://rb.gy/4bcu6

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The “Rorschach” test

Steady state

peak-to-steady state

Time to peak

Duration of the peak

Don’t expect models with more than four parameters (combinations) to learn more than this.

If you are lucky, also the

“curvature”

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Another “archetype”

Saturation

Slope at

50%

Concentration

at 50%

Hill function

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Take-home message #1

Simple

DATA REQUIRE

Simple

MODELS

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Question 2

What is as a spherical cow and

how do they emerge?

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  • Mechanism #1: Epistemic simplicity (adjust to information contained in data)
  • Mechanism #2: Aggregation creates information loss

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The theory of “sloppy” models

An the unexpected usefulness of Information Geometry

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Fisher Information Matrix

(FIM)

Parameter Space

Data Space

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The Manifold Boundary Approximation Method

effective theories as manifold boundaries

These limits are

data-driven!!!

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The Manifold Boundary Approximation Method

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The Manifold Boundary Approximation Method

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Spherical “bacteria”: the microbiome

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The generalized Lotka-Volterra

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Numerical experiment

  1. Four populations
  2. Deterministic dynamics
  3. Free of noise and uncertainty
  4. What does MBAM say?

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Numerical experiment

  • Four populations
  • Deterministic dynamics
  • Free of noise and uncertainty
  • What does MBAM say?

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Numerical experiment

  • Four populations
  • Deterministic dynamics
  • Free of noise and uncertainty
  • What does MBAM say?

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Numerical experiment

  • Four populations
  • Deterministic dynamics
  • Free of noise and uncertainty
  • What does MBAM say?

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Numerical experiment

  • Four populations
  • Deterministic dynamics
  • Free of noise and uncertainty
  • What does MBAM say?

20 parameters

10 parameters

Points to “slaving principle”

(so 7 effective parameters)

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Numerical experiment

  • Four populations
  • Deterministic dynamics
  • Free of noise and uncertainty
  • What does MBAM say?

20 parameters

10 parameters

Points to “slaving principle”

(so 7 effective parameters)

This gives un upper bound

(7 < 4*3)

...but not one less

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Another example: cyclic behavior

3 populations

12 parameters

2 populations

5 parameters

...but not one less

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Take-home message #2

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Question 2

What is as a spherical cow and

how do they emerge?

  • Mechanism #1: Epistemic simplicity (adjust to information contained in data)
  • Mechanism #2: Aggregation creates information loss

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What can we expect from empirical distributions?

A digression

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What can we expect from empirical distributions?

A digression

Can we go from the final distribution to the original one?

No!!!

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An archetypical “spherical cow”

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SIR is unreasonably good

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Epidemics in Free-land

  1. Take 21 districts in Madrid
  2. Simulate exactly the same SEIR pandemic (same parameters)
  3. Do it stochastically (life is stochastic)
  4. Aggregate

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Epidemics in Free-land

  • Take 21 districts in Madrid
  • Simulate exactly the same SEIR pandemic (same parameters)
  • Do it stochastically (life is stochastic)
  • Aggregate

Frequency

1 month!!

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Epidemics in the multiverse

  • Take 21 districts in Madrid: 21 replicas!

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  1. The “effective” pandemic underestimates the peak
  2. It fits well to a SIR model
  3. BUT, it does not share the same parameters as the replicas

So, by “ aggregation” 21*3 parameters (SEIR) turn into 2!!!

  • Time shift to location of the peak of the epidemic
  • Normalized to the maximum number of infected

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This sounds familiar...

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Take-home message #3

In fact, all epistemological value of the theory of probability is based on this: that large-scale random phenomena in their collective action create strict,

nonrandom regularity.

Kolmogorov, 1968

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Is too much “simplicity” killing us ?

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Conclusions

#2 Spherical cows push away all the information lost by aggregation

#1: Spherical cows adjust complexity to available information (simple but not simpler)

So... spherical cows are not only good models: they are often the best models

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Thanks!

Epidemics

Saúl Ares

José Cuesta

Susanna Manrubia

Ecology

José Cuesta

Javier Galeano

Rafael Vida

This work has been partially supported by Grant PID2022-140217NB-I00 funded by MCIN/AEI/ 10.13039/501100011033.

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