Interpreting network structure

Context

  • We are given a network which performs some task
     
  • We want to understand how the network’s structure relates to that task

(e.g. trained from Moritz or Mihai, or experimentally measured from Hamutal or Mina)

Local motifs

Global structure

  • Laplacian



     
  • Hodge Laplacian
  • triangles, stars, chains, …



     
  • simplices,
    cells
  • graph



     
  • simplicial complex,
    cell complex

Functional decomposition

See also (Hoppe, Grande, 2025, Don’t be afraid of Cell Complexes)

Bridging structure & function with a shift operator 

Discrete Fourier transform

Decompose into modes

Graph Laplacian

Hodge Laplacian

This is a global description of the graph

Modes

Low dim representation of functions on the graph

Possible direction: Can we go more local w/ subpops?

Linear, random connectivity within each subpopulation

\dot{\bm{r}}(t) = -\bm{r} + {\color{blue} \bm{W}}\bm{r}(t) + \bm{ξ}(t)

Transfer function in terms of functional Fourier modes

H(ω) = \frac{Y(ω)}{X(ω)} = \frac{z_α}{iωτ + 1 - μ_α^{(2)}} \frac{P_{αβ}b_β}{iωτ + 1 - λ_β^{(1)}}

What we might explore

  • Generalize to a (Hodge) Laplacian shift operator?
  • Does the basis also work with sharp activity spikes?

Daniel Moreno Soto

New tool: Model comparison under uncertainty

Consistent criterion even when no model is correct

Strength of criterion depends on dataset size

More samples

One model is correct

No model is correct

Selection criterion (lower ⇒ better model)

BIC

Bayes factor

MDL

AIC

elpd

EMD
(ours)

May be useful for testing model hypotheses

(René, Longtin, 2025; Selecting fitted models under epistemic uncertainty)