Weakly Reversible Matrices

• Weakly Reversible Matrices are stochastic matrices that satisfy a relaxed detailed-balance condition, allowing zeros and reducible chains.
• Key insights include spectral criteria such as positive spectra and principal logarithm, along with algebraic characterization using Vandermonde systems.
• Computational frameworks like LP and MILP enable dynamic network realizations that enforce strong connectivity and ensure unique, stable steady states.

A weakly reversible matrix is a stochastic matrix (with non-negative entries and unit row sums) that generalizes the classical notion of reversibility for Markov chains and mass-action systems. The weakly reversible condition relaxes the requirement for strictly positive weights in the detailed-balance condition, permitting zeros and thereby accommodating reducible chains and more general network topologies. Weakly reversible matrices play a central role in both stochastic processes and deterministic dynamical systems, underpinning structural and dynamical properties such as complex balancing, globally attracting steady states, and embeddability into continuous-time Markov semigroups. Modern investigations establish sharp algebraic, spectral, and combinatorial characterizations, connect weak reversibility to dynamical equivalence, and provide computational frameworks for realizing maximal reaction networks.

1. Definition and Algebraic Characterization

A Markov matrix $M\in M_d$ is weakly reversible if there exists a probability vector $\mathbf{p}\geq0$ (with $\sum_i p_i = 1$ and $p_i \geq 0$) such that the detailed-balance condition

$p_i M_{ij} = p_j M_{ji}, \qquad \forall\,i,j$

holds (Baake et al., 27 Nov 2025). This reduces to the classical reversible case when all $p_i>0$. In matrix form, denoting $D=\operatorname{diag}(p_1,\dots,p_d)$, the condition becomes

$D M = M^{\mathsf T} D,$

or, when $D$ is invertible, $M = D^{-1} M^{\mathsf T} D$. In network terms, weak reversibility corresponds to the requirement that each connected component (linkage class) of the reaction graph is strongly connected; for every directed edge $y \to y'$, there exists a path from $y'$ back to $y$ (Craciun et al., 2018, Kothari et al., 7 Sep 2024, Szederkenyi et al., 2011). Algebraically, a realization is weakly reversible if there exists a positive complex weighting $c$ in the kernel of the incidence matrix $I$ restricted to each linkage class, i.e. $I_{\ell}\,c_{\ell} = 0$ where $I_{\ell}$ is irreducible (Szederkenyi et al., 2011).

2. Spectral and Matrix-Theoretic Criteria

The embeddability of a weakly reversible $M$ into a continuous-time Markov semigroup requires the existence of a generator $Q$ such that $M=e^Q$, $Q_{ij}\ge0$ for $i \ne j$, $Q_{ii}=-\sum_{j\ne i} Q_{ij}$, and $Q$ satisfies the same weak detailed-balance (Baake et al., 27 Nov 2025).

Key spectral criteria are:

• Non-singularity: $\det M > 0$ is necessary (embeddable $M$ cannot have eigenvalues in $(-\infty,0]$).
• Positive spectrum: For reversible embeddings, all eigenvalues must be in $\mathbb{R}_+$; i.e., $\sigma(M)\subset\mathbb{R}_+$.
• Negative eigenvalues: These may appear (in even multiplicity, as per Culver’s criterion) for embeddable $M$ only in non-reversible balanced generators; reversible embeddings are forbidden if $\lambda<0$ occurs.
• Principal logarithm: If $\sigma(M)\subset\mathbb{R}_+$, the matrix logarithm

$$\log(M) = \sum_{n=1}^\infty \frac{(-1)^{n-1} n (M-I)^n}{n}$$

is real, has zero row sums, and yields the unique reversible real logarithm.

• Vandermonde test: The explicit coefficients of the log expansion are determined via a linear system using the shifted Vandermonde matrix of the eigenvalues.

Embeddability for weakly reversible $M$ is certified if $\sigma(M)\subset\mathbb{R}_+$ and the principal logarithm is a rate matrix (zero row sums and non-negative off-diagonal entries).

3. Network Realizations and Dynamical Equivalence

In the context of mass-action kinetics, weak reversibility is defined graph-theoretically: an E-graph $G=(V,E)$ is weakly reversible if each linkage class is strongly connected (Kothari et al., 7 Sep 2024, Craciun et al., 2018, Szederkenyi et al., 2011). For a given polynomial or power-law system $dx/dt = \sum w_i x^{y_i}$, the question of weakly reversible realizability reduces to solving the bilinear system:

1. $\sum_{j\ne i} k'_{ij}(y_j - y_i)=w_i$ for $i=1,\ldots,N$
2. $$\sum_{j\ne i} \alpha_i k'_{ij} = \sum_{j\ne i} \alpha_j k'_{ji}$$ for $i=1,\ldots,N$ with $k'_{ij}\ge0$, $\alpha_i > 0$ (Craciun et al., 2018). No new complexes are required beyond those present in the original system.

A dynamically equivalent weakly reversible realization can be characterized and computed via mixed-integer linear programming (MILP), maximizing the number of reactions while satisfying dynamic equivalence and ensuring strong connectivity within linkage classes (Szederkenyi et al., 2011).

4. Embedding Theorems and Proof Strategies

The embedding problem for weakly reversible Markov matrices and mass-action networks centers on the existence and uniqueness of a generator or reaction realization yielding the same dynamics. For reversible $M$, the principal logarithm is unique; for weakly reversible but non-reversible cases, uniqueness may fail, especially in the presence of negative spectrum (Baake et al., 27 Nov 2025).

Proofs exploit:

• The involution $A \mapsto \widetilde{A} := D^{-1}A^{\mathsf T}D$ defines a real Jordan algebra structure for weakly reversible matrices.
• Deformation arguments and density of simple-spectrum embeddable matrices show the closedness of the rate-matrix cone.
• The Vandermonde system encapsulates the exact polynomial dependence of logarithm coefficients on the eigenvalues.
• MILP-based branch-and-cut algorithms enforce strong connectivity for reaction networks, iteratively adding cut constraints until the graph is weakly reversible (Szederkenyi et al., 2011).

5. Concrete Examples and Computational Methods

Example Table: Illustrative Weakly Reversible Matrices and Networks (from (Baake et al., 27 Nov 2025, Kothari et al., 7 Sep 2024, Szederkenyi et al., 2011))

Class & Dimension	Matrix/network form	Key spectral property
Binary case ($d=2$)	$$M = \begin{pmatrix}1-a & a \ b & 1-b \end{pmatrix}$$	$a+b<1$ for embeddability
Equal-input chain ($d$)	$M = (1-x)I + x \frac{1}{d-1}J$	1-D inequality in $x$
3×3 cycle	Cyclic $M$ with $\sigma(M)=\{1, -\epsilon, -\epsilon\}$	No reversible embedding
Reaction network	MILP maximization over $\{\delta_{i\to j}\}$, $k_{i\to j}$	Enforces weak reversibility

These instances highlight spectral obstructions and the role of network completion or matrix expansion to enforce weak reversibility.

Computational methods include:

• Bilinear and linear programming (LP, MILP) for feasibility of realizations.
• Explicit constructions using graph completion and cone decomposition for strongly endotactic networks (Kothari et al., 7 Sep 2024).
• Verification by kernel and cut-based algorithms to enforce strong connectivity (Szederkenyi et al., 2011).

6. Dynamical and Stability Implications

Weakly reversible systems are closely associated with key stability properties:

• Existence of positive steady states and complex balancing for suitable kinetic parameters (Craciun et al., 2018).
• Globally attracting steady states for strongly endotactic and certain endotactic networks, with the globally attracting locus $K(G)$ computable as the feasible region of a linear program (Kothari et al., 7 Sep 2024).
• For Markov matrices, embeddability into a reversible continuous-time process is guaranteed by the positive-spectrum plus principal-log criterion (Baake et al., 27 Nov 2025).

Complex-balanced and detailed-balanced realizations, endowed by weak reversibility, ensure uniqueness and stability of positive steady states and persistence in mass-action systems (Craciun et al., 2018).

7. Generalization of Classical Theory and Computational Complexity

Traditional treatments imposed irreducibility and positive spectrum (Baake et al., 27 Nov 2025), restricting to strictly reversible chains. The Baake–Baake–Sumner framework generalizes this to allow reducible (direct sums of reversible blocks) and weakly reversible matrices by incorporating the full spectral characterization and algebraic closure properties of the reversible cone under multiplication, powers, and limits. Negative eigenvalues are newly accommodated but may only appear in non-reversible embeddings. Complexity of the weakly reversible realization problem is generally NP-hard due to exponential cut-constraint growth in MILP formulations, but practical computation is feasible for networks of moderate size (Szederkenyi et al., 2011).

A plausible implication is the increased tractability of stability and embeddability analysis, by reducing the search space to the original set of complexes and leveraging LP/MILP solvers for weakly reversible network construction (Craciun et al., 2018, Szederkenyi et al., 2011). The intersection of spectral theory, algebraic network models, and optimization-based realization frameworks continues to deepen the understanding and applicability of weakly reversible matrices in Markov chains, mass-action systems, and beyond.