Fig. 2: Wavefront propagation for κ-dimensional simplicial threshold model cascades on noisy ring complexes.

a Each k-simplex with k ≤ κ is given a binary state \({x}_{i}^{k}(t)\in \{0,1\}\) indicating whether it is inactive or active, respectively, at time t = 0, 1, 2, …. Cascade propagation occurs when an inactive boundary vertex is adjacent to sufficiently many active k-simplices, in which case it (and possibly some of its adjacent k-simplex neighbors) will become active upon the next time step. There are different types of inactive k-simplices, depending on how many of their vertices are active. b Noisy ring complexes (see Methods section “Generative model for noisy ring complexes”) generalize noisy ring lattices²² and contain vertices that lie on a 1D ring manifold that is embedded in a 2D “ambient” space. Each vertex has d^(NG) = 1 non-geometric edge (red lines) to a distant vertex and d^(G) geometric edges (blue lines) to nearby vertices with d^(G) ∈ {2, 4, 6} (left, middle, and right columns, respectively). Higher-dimensional simplices arise in the associated clique complexes and are similarly classified as geometric/non-geometric. To simplify our illustrations, we place vertices alongside the manifold when d^(G) > 2, and we do not visualize 3-simplices. c Geometric k-simplices with k≤K compose a K-dimensional geometrical substrate. For noisy ring complexes, K = d^(G)/2 and the substrate is a K-dimensional channel—which is a non-intersecting sequence of lower-adjacent K-simplices. Channels generalize the graph-theoretic notion of a “path”. d Simplicial threshold model cascades with different dimension κ ≤ K can propagate by wavefront propagation along a K-dimensional channel. Note that a simplicial threshold model cascade does not utilize all available k-simplices when κ < K.