RCDS Matrices: Structure and Switching

• RCDS matrices are binary matrices defined by prescribed row and column sums that satisfy the Gale–Ryser theorem and are key in combinatorial matrix theory.
• The checkerboard switching operation transforms 2×2 submatrices while preserving marginals, enabling navigation through a directed acyclic graph of feasible configurations.
• These matrices are applied in network sampling, graph optimization, and even in modeling correlated Wishart matrices for computer vision problems.

RCDS Matrices (Row-Column Degree Specified Matrices) refer to the set of binary (zero–one) matrices with prescribed row and column sum vectors. Such matrices, broadly denoted as $\mathbf{M}(R,C)$, are central to combinatorial matrix theory, graph theory, and applied fields such as network science and statistical modeling. Their structure and transformations—most notably via checkerboard switchings—play a decisive role in characterizing the configuration spaces of graphs and matrices with fixed marginals, as well as in the analysis and optimization of network properties and applications such as sampling, algorithmic enumeration, and the study of extremal spectral quantities (Ellison et al., 2022).

1. Formal Definition and Existence Criteria

For vectors $R = (r_1, \ldots, r_p) \in \mathbb{N}^p$ and $C = (c_1, \ldots, c_q) \in \mathbb{N}^q$ such that $\sum_i r_i = \sum_j c_j$, define the class

$$\mathbf{M}(R,C) = \left\{ A \in \{0,1\}^{p \times q} : \sum_j a_{ij} = r_i \ \forall i, \quad \sum_i a_{ij} = c_j \ \forall j \right\}.$$

A necessary and sufficient condition for $\mathbf{M}(R,C) \neq \varnothing$ is given by the Gale–Ryser theorem, which requires that $R$ be majorized by the conjugate of $C$. However, many analyses assume a non-empty class given $(R,C)$ fixed (Ellison et al., 2022).

2. Checkerboard Switching and the Directed Graph $G(R,C)$

A fundamental local operation is the checkerboard switch, acting on $2 \times 2$ submatrices:

• A positive checkerboard has the form $$\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$$.
• A negative checkerboard has the form $$\begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}$$.

Switching transforms one pattern into the other, preserving the row and column sums. The primitive switching matrix is $_{i,j,k,\ell}$, with $+1$ at (i,k) and (j,ℓ), $-1$ at (i,ℓ) and (j,k); the switch is feasible iff these entries of $A$ form a negative checkerboard.

Define the oriented graph $G(R,C)$ on vertex set $\mathbf{M}(R,C)$, where an arc $A \to A'$ exists iff $A'$ is obtained from $A$ by a positive switch. This structure forms the basis for systematic navigation and analysis of the space of RCDS matrices (Ellison et al., 2022).

3. Structure and Dynamics of $G(R,C)$: Acyclicity, Sources, and Sinks

The oriented switching graph $G(R,C)$ is a directed acyclic graph (DAG). Define $I(A) = \sum_{i,j} i j a_{ij}$. Any positive switch $A \to A'$ increases $I$ by $(i-j)(k-\ell) > 0$, precluding cycles. The undirected version is connected by classical results (e.g., Ryser 1957).

• A source (no incoming arcs) corresponds to a matrix with no positive checkerboards (anti-nested pattern).
• A sink (no outgoing arcs) is a matrix with no negative checkerboards (nested or zebra pattern, under potential geometric splitting).

In certain configurations, notably when $\mathbf{M}(R,C)$ allows a split zebra or split anti-zebra pattern, such a matrix is the unique sink or source, respectively (Ellison et al., 2022).

4. Path-Existence and Characterization via Switching Decomposition

Given matrices $A, A' \in \mathbf{M}(R,C)$, one seeks necessary and sufficient conditions for a path $A \to \cdots \to A'$ in $G(R,C)$ via positive switches. The difference $M = A' - A$ admits a unique decomposition as a non-negative linear combination of unit switching matrices: $$M = \sum_{i=1}^{p-1} \sum_{k=1}^{q-1} t_{i,k} \cdot {}_{i,i+1,k,k+1}$$ for $t_{i,k} \in \mathbb{N}$.

A necessary condition is the existence of such $T = (t_{i,k})$. Sufficient conditions (Theorem 3.9) require that for each level $\ell$ up to $\max t_{i,k}$, the super-level set $S_\ell = \{ (i, k) : t_{i,k} \geq \ell \}$ is a union of simply connected polyominoes, and that adjacent $t$-values differ by at most one. These ensure the existence of a sequence of valid positive switches from $A$ to $A'$. This decomposition is structurally analogous to Young diagram (cumulative sums), dual to the majorization (Gale–Ryser) approach (Ellison et al., 2022).

5. Specialization: Symmetric Adjacency Matrices, Zagreb Index, and Spectral Radius

For degree sequence $D = (d_1 \ge \cdots \ge d_n)$, the set $\mathbf{M}(D)$ comprises all symmetric zero-diagonal binary matrices with given row-sums $D$, i.e., adjacency matrices of simple graphs with degree sequence $D$. Checkerboard switches must be performed simultaneously with their transposes to maintain symmetry and zero-diagonal structure.

A key result is that the second Zagreb index $M_2(A) = \sum_{ij \in E(A)} d_i d_j$ and its derived quantity $Z_2(A) = \sqrt{M_2/m}$ with $m = |E(A)|$ are non-decreasing along arcs of $G(D)$. All sinks of $G(D)$ are exactly the global maxima of $Z_2$ on $\mathbf{M}(D)$. The spectral radius $\lambda_1(A)$ also tends to increase along arcs, and any global maximizer of $\lambda_1$ must be a sink of $G(D)$. This suggests a natural algorithmic approach for maximizing spectral or combinatorial quantities over the space of RCDS matrices (Ellison et al., 2022).

6. Algorithmic and Combinatorial Applications

The acyclic structure of $G(R,C)$ enables:

• Uniform sampling within $\mathbf{M}(R,C)$ by random mixes of positive and negative switches.
• Optimization heuristics (e.g., “climbing” to the sink) for maximizing $Z_2$ or approximating extremal spectral radius.
• Efficient enumeration of all matrices with fixed $(R,C)$ via dynamic programming on the DAG.

Such methods underlie approaches for sampling bipartite graphs with fixed degree sequences, counting magic squares (via rook-theory), and re-wiring heuristics in network analysis. The necessary and sufficient conditions on $M = A' - A$ provide exact certificates for reachability via positive switches, offering both theoretical and computational utility (Ellison et al., 2022).

7. Extensions and Related Modeling: Covariance Matrices and Wishart Representations

In statistical and applied settings, particularly in computer vision, the term RCD is also used for Region Covariance Descriptors, which are $D \times D$ positive-definite matrices computed from per-pixel features of an image. Such matrices are often modeled as Wishart-distributed random matrices, $W \sim \mathcal{W}_p(\Sigma, M)$ (lan, 2020).

When auxiliary covariates (e.g., subject demographics) introduce inter-instance correlation, independent Wishart models are insufficient. Composite likelihood-based EM algorithms with explicit pairwise covariance structure have been proposed to accommodate such correlated Wishart matrices, improving classification performance in high-dimensional covariate-rich settings such as face recognition tasks.

A plausible implication is that ideas from RCDS matrix combinatorics (switching operations, majorization, reachability certificates) may find applications in the efficient sampling and testing of hypotheses involving structured covariance matrices (lan, 2020).

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References:

• "Switching Checkerboards" (Ellison et al., 2022)
• "Correlated Wishart Matrices Classification via an Expectation-Maximization Composite Likelihood-Based Algorithm" (lan, 2020)