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Social Diffusers, Bridges, and Cohesion

This tutorial explains SHE's social-analysis layer: what the scores mean, how they differ from graph centrality, and how temporal features fit in.

What is a diffuser in SHE?

A diffuser is a simplex (node, edge, triangle, ...) ranked by its structural importance for diffusion on the Hodge Laplacian.

SHE solves (I + L) x = w where L is the Hodge Laplacian at a given dimension and w is the weight vector. The solution x is then rank-percentile normalised to produce scores in [0, 1].

  • Dimension 0: ranks individual entities (nodes).
  • Dimension 1: ranks pairwise relations (edges).
  • Dimension 2: ranks triadic relations (triangles).

This is not the same as graph centrality. Eigenvector centrality on the 1-skeleton ranks nodes by pairwise connectivity. Hodge diffusion centrality ranks simplices at any dimension by their role in the simplicial diffusion process. They can disagree substantially.

What is a bridge simplex?

A bridge simplex is a relation whose members span multiple communities.

SHE's bridge score is a heuristic:

bridge_score = (number of communities spanned / number of members) * weight

This rewards simplices that span many communities with few members and high weight. It is intentionally simple — a legible first-pass signal, not a topological invariant.

Community membership comes from entity attributes (the community field). You assign these when building the hyperstructure.

What is group cohesion?

group_cohesion(hs, group) scores how tightly a candidate group is bound in the hyperstructure. It combines three signals:

  1. Relation weight — total weight of relations fully contained in the group, normalised by group size.
  2. Sub-relation density — fraction of possible pairs within the group that actually exist as relations.
  3. Higher-order support — number of relations of dimension >= 2 fully contained in the group, normalised by group size.

The final score is the geometric mean of these three components. This is a deliberately simple composite, not a formal topological measure.

Temporal features

Hard windows

window(hs, start, end) returns a new hyperstructure containing only relations within the time range [start, end). Entity attributes are preserved for all entities that appear in the window.

rolling_windows(hs, window_size, step) produces a sequence of these snapshots.

Decay-weighted windows

decay_window(hs, reference_time, half_life) builds a snapshot where relation weights decay exponentially with age:

decayed_weight = original_weight * 2^(-age / half_life)

Relations decayed below 1% of their original weight are dropped. This is more realistic than a hard cutoff — recent interactions dominate, but older ones still contribute.

Which example demonstrates this?

  • Script: examples/social_media_diffusers.py — synthetic scenario, graph vs simplex comparison
  • Notebook: notebooks/eu_email_analysis.ipynb — real email data, temporal bridge/cohesion plots
  • Notebook: notebooks/temporal_diffusion_analysis.ipynb — synthetic temporal scenario showing bridge formation over three periods

Caveats

  • Bridge detection and cohesion are heuristic scores, not topological invariants. They are useful first-pass signals.
  • Diffusion centrality depends on the Hodge Laplacian structure and weight distribution. On very small complexes, rankings can be sensitive to weight tuning.
  • These tools are research-grade. The API may change.