Rasmus<p>The "Maximum Entropy Stress"-algorithm for graph layouts, just keeps blowing my mind.<br> <br>It lays out a graph by trying to keep Euclidean distance between nodes close to their graph-theoretic distances (shortest paths between the nodes along the edges). <br>When laying out bigger graphs one can limit the computational load by introducing a cutoff to the paths walked. So for example, a cutoff of 4 reduces the neighbourhood of a node to other nodes reachable in 4 steps along edges. When using a cutoff the algorithm thus acts locally.</p><p>Now have a look at the example below where we build a graph made up of 7-cycles. We start with one 7-cycle, then grow the graph step by step adding rings of 7-cycles along the boundary. Occasionally, the graph will loop and the boundary will self-connect based on rules defined by a group with exponential growth. </p><p>The frames in the animation below show the addition of rings and the subsequent layout proposed by the layout algorithm.</p><p>While the first ring additions lead to fairly chaotic layouts with denser and denser boundaries, a clear break occurs after the addition of ring 8.<br>Suddenly a new order emerges due to the symmetries of the underlying group and the local action of the "MaxEnt Stress"-Algorithm. </p><p>The transition is so drastic that it appears not as a shift in layout, but rather as a fundamental change in the underlying structure.</p><p>A boundary materializes that seems to pull the inside apart. All of a sudden there is a marked difference in the area occupied by 7-cycles of inner rings, as opposed to the ones further towards the boundary. (1/2)<br><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>math</span></a> <a href="https://mathstodon.xyz/tags/geometry" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>geometry</span></a></p>