Random Walks Across Dimensions: Exploring Simplicial Complexes

Abstract

We introduce a novel operator to describe a random walk process on a simplicial complex. Walkers are allowed to wonder across simplices of various dimensions, bridging nodes to edges, and edges to triangles, via a nested organization that hierarchically extends to higher structures of arbitrary large, but finite, dimension. The asymptotic distribution of the walkers provides a natural ranking to gauge the relative importance of higher order simplices. Optimal search strategies in presence of stochastic teleportation are addressed and the peculiar interplay of noise with higher order structures unraveled.

Figures (5)

• Figure 1: Example of a random walk process across the dimensions of a simplicial complex. Here, the walker is initialized on the node $0$ (panel (a)). Then, it reaches the connected link $[0,1]$ (panel (b)), heads towards the triangle $[0,1,2]$ (panel (c)) and finally lands on link $[1,2]$ (panel (d)).
• Figure 2: Correspondence between the asymptotic frequency of occupation on the simplices (symbols) and the corresponding generalized degree (solid line). The degree centrality, corresponding to the occupation probability of the simplices, naturally mixes the various dimensions. Here we set, $p_1 = p_2 = p_3 = 1/3$ and $N_0 = 50$.
• Figure 3: Each point of these triangles corresponds to a choice of the parameter probabilities defining the topology of the generated simplex $\mathcal{X}$. Panel (a): Mean node FPT for the generated simplicial complex divided by the total number of simplices. Panel (b): Mean FPT for $\mathcal{G}$, divided by the number of nodes (the sub-structures on which the walker is confined, in this case), fixed at $N_0=20$.
• Figure : 4. Panel (a): Size-rescaled explorability for a simplicial complex and its corresponding graph, as a function of $\alpha$. The red dot identifies the minimum of the reported curves, namely the optimal search strategies (OSS). Here, we set $p_3=1$ and $\delta=0.05$. Panel (b): Structural gain $g_s = \frac{g^\mathcal{X}}{g^\mathcal{G}}$, computed as the ratio between the gain of the simplicial complex topology and that of the corresponding graph topology, for $\delta=0.5$. Panel (c) $g_s$ as a function of $\delta$. The shaded region traces the variability across different realizations of the generated simplices.
• Figure : 4. Panel (a): Size-rescaled explorability for a simplicial complex and its corresponding graph, as a function of $\alpha$. The red dot identifies the minimum of the reported curves, namely the optimal search strategies (OSS). Here, we set $p_3=1$ and $\delta=0.05$. Panel (b): Structural gain $g_s = \frac{g^\mathcal{X}}{g^\mathcal{G}}$, computed as the ratio between the gain of the simplicial complex topology and that of the corresponding graph topology, for $\delta=0.5$. Panel (c) $g_s$ as a function of $\delta$. The shaded region traces the variability across different realizations of the generated simplices.