Circular Electrical Networks

• Circular electrical networks are finite, weighted, connected graphs embedded in a disk with cyclically ordered boundary nodes and positive conductance assignments.
• They facilitate network analysis through Dirichlet-to-Neumann maps, Kron reduction, and the use of circular minors to certify total nonnegativity.
• These networks underpin advanced studies in total positivity, Grassmannians, and phylogenetics, offering insights into inverse problems and network stratification.

A circular electrical network is a finite, weighted, connected graph embedded in the closed disk with a distinguished cyclically ordered subset of boundary vertices on the boundary circle. Each edge is assigned a positive conductance. Applying potentials to the boundary nodes and measuring the resulting currents defines the Dirichlet–to–Neumann (response) map. The class of circular planar (CP) networks—those that admit an embedding with non-crossing edges and boundary nodes in fixed cyclic order—serves as a foundational model in combinatorial, algebraic, and inverse problems for planar network theory, with deep links to total positivity, Grassmannians, and applications in phylogenetics.

1. Definitions and Network Data

A circular planar electrical network is defined by a triple $(G, N, c)$ with:

• $G = (V,E)$ a finite, connected planar graph embedded in the closed disk,
• $N \subset V$, $|N| = n$, the set of boundary (node) vertices, ordered cyclically on the boundary,
• edge weights $c: E \to \mathbb{R}_{>0}$ (conductances).

The harmonic extension of a boundary voltage vector $u:N \to \mathbb{R}$ to all vertices minimizes network energy, leading to the response matrix $L$ (or $\Lambda_G$), defined so that the net boundary current at node $i$ is $L(u)_i = -\Delta f(i)$, with the network Laplacian operator $\Delta$. For CP networks, $L$ is symmetric, each row and column sum to zero, and all off-diagonal entries are non-positive (Kenyon et al., 2014).

The Kron (Schur) reduction eliminates interior nodes, yielding a reduced Laplacian on the boundary. The effective resistance between boundary nodes $i, j$ is $R_{ij} = (e_i - e_j)^T \Lambda^\dagger (e_i - e_j)$, where $e_i$ is the standard basis and $\Lambda^\dagger$ the Moore–Penrose pseudoinverse (Forcey, 2021).

2. Positivity, Circular Minors, and Invariants

Response matrices of CP networks are characterized by Curtis–Ingerman–Morrow: an $n \times n$ symmetric matrix with zero row/column sums arises as a response matrix if and only if all circular minors are nonnegative. For a pair of ordered, disjoint boundary subsets $I, J$, the circular minor $\Delta_{I,J}(L)$ is the determinant of the submatrix indexed by $I$ and $J$, and is nonnegative only if a corresponding family of vertex-disjoint paths exists. This total nonnegativity is reflected algebraically and combinatorially (Alman et al., 2013, Kenyon et al., 2014).

Electrical positroids encode which sets of circular minors can be simultaneously positive, forming an analogue of positroid theory for the Grassmannian. Minimal test sets of minors, such as the diametric circular pairs or the Kenyon–Wilson small central minors (CM_{x,y}), allow efficient positivity certification and stratification of the space of response matrices into semialgebraic cells (Kenyon et al., 2014, Alman et al., 2013).

3. Cell Decompositions, Standard Networks, and the Electrical Poset

The set of all $n$-node circular planar networks, up to electrical equivalence (generated by local series-parallel reductions, Y–Δ moves, removal of loops/spikes), carries a stratification indexed by equivalence classes of minimal (critical) representatives. Each critical graph defines a cell $\Omega(\Gamma)$ diffeomorphic to $(\mathbb{R}_{>0})^{m}$, where $m$ is the number of edges. The closure order on these cells defines the electrical poset $EP_n$, graded by the number of edges in the critical representative (Alman et al., 2013).

The standard network in each class is constructed canonically from the medial strand matching, and every equivalence class corresponds uniquely to a standard network and its cell. Cell boundaries are described by deletion/contraction, corresponding to edge weights tending to 0 or ∞ (Kenyon et al., 2014, Alman et al., 2013).

Enumeration of critical networks is governed by recurrences analogous to those for non-crossing matchings and SIF permutations. For example, for $n = 3,4$ the poset $EP_n$ matches Boolean lattices in structure for small $n$ (Alman et al., 2013).

4. Canonical Coordinates: Pfaffians, Minimal Groves, and the Chamber Ansatz

Canonical coordinates for a standard CP network can be computed as biratios of Pfaffians built from the response matrix $L$. Each crossing in the medial-strand diagram determines a "tripod" partition and a unique spanning grove (minimal forest with all components meeting the boundary); its weight is the product of conductances along its edges. The conductance at a crossing $e$ is computed as

$$c_e = \frac{Z_{\tau_e} \cdot Z_{\tau_f}}{Z_{\tau_a} \cdot Z_{\tau_b}}$$

where $Z_\tau$ are grove partition functions indexed by partition types, each expressible as a Pfaffian of a specialized skew-symmetric matrix defined from $L$. Thus, all edge weights in a standard network can be recovered from $L$ via biratios of Pfaffians. For the top cell, positivity of the $\binom{n}{2}$ central minors suffices for certification of well-connectedness; for a given standard network with $m$ edges, $m+1$ Pfaffians suffice to guarantee positivity of conductances (Kenyon et al., 2014).

The chamber ansatz (Kazakov, Müller–Speyer–Talaska), via embedding into the non-negative Grassmannian, reconstructs each conductance as the reciprocal of certain twisted Plücker coordinates evaluated on face labels in the network's median graph (Kazakov, 23 Feb 2025).

5. Metric Geometry: Kalmanson Inequalities, Split Systems, and Phylogenetic Analogues

The effective resistance matrix $R$ on the boundary nodes of any CP network is a Kalmanson metric with respect to the circular ordering. The Kalmanson four-point inequalities state that for any cyclically ordered $i<j<k<\ell$: $$R_{ik} + R_{j\ell} \geq R_{ij} + R_{k\ell}, \quad R_{ik} + R_{j\ell} \geq R_{jk} + R_{i\ell}$$ These metric constraints correspond bijectively, via the Farris transform (Gromov product), to weighted circular split systems. Each split corresponds to a cut in the network; the unique weighted circular split system (combinatorial invariant) is fully determined by the response or resistance matrix (Forcey, 2021, Gorbounov et al., 2024, Devadoss et al., 2024).

A network is $k$-nested if each edge lies on at most $k$ independent simple cycles; series–parallel networks are precisely the 1-nested case. The inclusion chain $$\{\text{trees}\} \subset \{\text{series–parallel}\} \subset \cdots \subset \{\text{all circular planar}\}$$ reflects progressive circuit complexity with implications for reconstructibility from partial boundary data (Forcey, 2021). Inverse and splitting technology directly port to settings in phylogenetic network analysis, where Kalmanson metrics characterize split-decomposable genetic distance matrices (Gorbounov et al., 2024, Devadoss et al., 2024).

6. Inverse Problems, Scaffolds, Recoverability, and Algorithmics

Circular planar networks can be explicitly and algorithmically reconstructed from their response matrices. Key foundational theorems (Curtis–Ingerman–Morrow, de Verdière–Gitler–Vertigan) establish that if the medial graph of the embedding is lensless (critical), the network is recoverable via layer-stripping—a sequence of edge deletions/contractions guided by carefully constructed scaffolds (oriented sequences of edges). Each scaffold determines an order for harmonic continuation, systematically exposing and allowing recovery of edge weights via specialized boundary value problems.

Every Lagrangian subspace in the electrical Grassmannian—encapsulating boundary behaviors of networks—can, over fields other than $\mathbb{F}_2$, be realized by a circular planar network, albeit possibly with signed conductances. This functorial framework extends to the composition of IO-morphisms, symplectic characterizations, and the electrical linear group EL_n (Jekel, 2016, Johnson, 2012).

Algorithmic implementations include Gröbner basis approaches for reconstructing network topology from partial Thevenin boundary measurements under triangle and Kalmanson inequality constraints, even in RL and RC circuit classes (Biradar et al., 2024).

7. Embedding in the Non-negative Grassmannian and Compactifications

CP network data admit embeddings into the totally non-negative Grassmannian Gr_{$\geq 0$}(n−1,2n) via the boundary measurement (Temperley) matrices, with Plücker coordinates encoding weighted sums over almost-perfect matchings or minimal groves. The image of all (cactus) networks under this map is the electroid variety, a linear section of the Grassmannian stratified by positroids and indexed by medial matchings. Compactification is achieved by allowing some edge weights to vanish or diverge, producing "cactus" networks—networks on trees of disks with boundary node identifications determined by noncrossing partitions (Lam, 2014, Devadoss et al., 2024).

Cell closures in the compactified space and the duality between grove measurements and boundary measurements identify closure orders on strata with the uncrossing order on matchings and the affine Bruhat order. Species-theoretic and combinatorial frameworks count configurations via Bell and Catalan numbers, capturing the full global structure of cactus split systems and their network realizations (Devadoss et al., 2024, Lam, 2014).

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Key References:

• (Kenyon et al., 2014) Kenyon–Wilson, “The space of circular planar electrical networks”
• (Alman et al., 2013) Alman–Lian–Tran, “Circular Planar Electrical Networks II: Positivity Phenomena”
• (Alman et al., 2013) Alman–Lian–Tran, “Circular Planar Electrical Networks I: The Electrical Poset EP_n”
• (Forcey, 2021) Forcey, “Circular planar electrical networks, Split systems, and Phylogenetic networks”
• (Devadoss et al., 2024) Forcey–Scalzo–Devadoss, “Compactifications of phylogenetic systems and species of electrical networks”
• (Gorbounov et al., 2024) Gorbounov–Kazakov–Talalaev, “Electrical networks and data analysis in phylogenetics”
• (Gorbounov et al., 13 Jan 2026) Gorbounov–Kazakov, “Metric properties of electrical networks and the graph reconstruction problems”
• (Kazakov, 23 Feb 2025) Kazakov, “Inverse problems related to electrical networks and the geometry of non-negative Grassmannians”
• (Jekel, 2016) Baker–Wethington–Irving, “Layering ∂-Graphs and Networks”
• (Johnson, 2012) Pachter, “Circular planar resistor networks with nonlinear and signed conductors”
• (Lam, 2014) Lam, “Electroid varieties and a compactification of the space of electrical networks”