Walsh Hadamard Agreement & Disagreement for the iterated system HD

 

This interactive sketch explores how a binary diagonal matrix D interacts with the Walsh–Hadamard mixing matrix H. 

Each entry of D can be flipped between +1 and -1, dividing the nodes into two groups. 
The code visualizes two derived operators: 
●​ X = HD + DH 
●​ Y = HD - DH 

These have a surprisingly clean interpretation: 
●​ X keeps interactions between nodes with the same sign (“agreement”). 
●​ Y keeps interactions between nodes with opposite signs (“disagreement”).

Because the Walsh–Hadamard matrix contains only ±1 entries, the resulting matrix values become simple: 
●​ reinforcement: +2 
●​ cancellation: 0 
●​ opposite reinforcement: -2 

The graph views make this intuitive:

●​ The graph of X splits into same-sign clusters. 
●​ The graph of Y becomes a bipartite interaction graph between opposite-sign groups. 

As entries of D are flipped, the system transitions between: 
●​ uniform structure (all signs equal), 
●​ clustered structure, 
●​ and maximal cross-group interaction when the signs are balanced. 

View Online:
https://editor.p5js.org/seanhaddps/full/um-DEnBEo

Code:
https://editor.p5js.org/seanhaddps/sketches/um-DEnBEo

Overview:
https://archive.org/details/@seanc4s




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