Compactifications of phylogenetic systems and species of electrical networks

TL;DR

This work builds a precise bridge between circular electrical networks and circular phylogenetic split systems by introducing maps $oldsymbol{ξ}$, $oldsymbol{ρ}$, and $oldsymbol{σ}$ and extending them to cactus (compactified) and global spaces. It shows that for connected circular planar networks the graphical, Kalmanson, and induced constructions coincide up to duals, and it extends these correspondences component-wise to cactus networks, preserving cactus structure. The paper develops a rich enumerative framework using combinatorial species and Lagrange inversion to count cactus splits and their global counterparts, revealing Bell-number counts for 0-cells and establishing global compositions $oldsymbol{Ψ}^{global} = E_+ oldsymbol{Ψ}^{global}$ and $oldsymbol{Ω}^{global} = oldsymbol{Ω}^{global} E_+$. These results illuminate planarity obstructions, enable modular network reconstruction from boundary data, and connect to the Symmetric Traveling Salesman Polytope, offering practical pathways for circuit design and phylogenetic reconstruction in a unified mathematical setting.

Abstract

We describe new spaces and maps. Our graphical map is a visual and numerical correspondence between spaces of circular electrical networks and circular planar split systems. When restricted to the planar circular electrical case, this graphical map finds the split system uniquely associated with the Kalmanson resistance distance of the dual network, matching the induced split system familiar from phylogenetics. This correspondence is extended to compactifications of the respective spaces, taking cactus networks to the cactus split systems defined herein. The graphical map preserves both network components and cactus structure, allowing an elegant enumeration of induced phylogenetic split systems via combinatorial species. We introduce the global spaces of circular planar electrical networks and circular split systems. These new spaces are also CW complexes, but the 0-cells of each are counted by the Bell numbers as opposed to the Catalan numbers. As species, the two sorts of global cacti are seen to be compositions in complementary ways.

Figures (21)

• Figure 1: Example labels for the cells in the spaces we consider, for $n=4$: (a) circular planar electrical networks, (b) cactus networks (c) circular split systems and (d) circular cactus split systems . The total numbers of these cells are listed in Table \ref{['topper']}, row 4, and the entire sets for (a) and (c) are shown in Figure \ref{['fig:fours']}.
• Figure 2: Domains and ranges for the maps in this paper.
• Figure 3: A planar network and the related structures that we will calculate. Top row shows (a) a clockwise planar electrical network $N$ (with conductance 1 for non-labeled edges), (b) its splits $\mathbf{\sigma}(N)$, (c) the polygon diagram of $\mathbf{\xi}'(N^*) = \rho(N) = \mathbf{\sigma}(N)$, (d) the dual $N^*$, and (e) the Kron reduction $K(N^*)$.
• Figure 4: (a) Non-planar (with respect to the clockwise order) circular electrical network $N$.
• Figure 5: Cells of the space $\mathbf{\Omega}_4$ on the left, $\mathbf{\Psi}_4$ on right. The image $\mathbf{\xi}(\mathbf{\Omega}_4) \subset \mathbf{\Psi}_4$ is inside the dashed line. The highlighted system on lower right is the image of the 4 networks highlighted at the top on the left.

Theorems & Definitions (56)

• Theorem 1
• Corollary 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Remark
• Corollary 7
• proof
• Theorem 8