arxiv: 2605.00135 · v1 · submitted 2026-04-30 · ❄️ cond-mat.stat-mech · math.PR

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Graph theoretic derivation of mutual linearity for transient probabilities and hitting time distributions in Markov networks

Julian B. Voits , Ulrich S. Schwarz (Heidelberg University)

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Pith reviewed 2026-05-09 20:36 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math.PR

keywords Markov networksmutual linearitymatrix-tree theoremhitting timestransient probabilitiesresponse ratiosgraph theory

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The pith

Mutual linearity of response ratios holds for transient probabilities and hitting times in Markov networks via graph theory.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that mutual linearity, a property relating how occupation probabilities and currents respond to changes in individual transition rates, extends from the stationary case to the full transient regime and to hitting time distributions. It reaches this by applying the all-minors matrix-tree theorem to the network graph, which supplies explicit combinatorial formulas for the non-stationary response ratios in Laplace space. The stationary results emerge as the long-time limit, while short-time behavior is fixed by the shortest paths in the graph. A reader cares because the same combinatorial structure governs both equilibrium and non-equilibrium dynamics without needing trajectory-level decompositions.

Core claim

For irreducible, time-homogeneous Markov networks, mutual linearity holds for occupation probabilities and network currents in the stationary regime and in the non-stationary regime in Laplace space. The all-minors matrix-tree theorem yields explicit combinatorial expressions for the non-stationary response ratios under variation of a single transition rate. The stationary result follows as the long-time limit, small-time asymptotics are set by minimal path distances, and mutual linearity extends to hitting time densities.

What carries the argument

The all-minors matrix-tree theorem applied to the transition graph, which expresses the relevant determinants and cofactors as sums over spanning trees and thereby produces the combinatorial response ratios.

If this is right

Response ratios under single-rate perturbations admit explicit sums over spanning trees at any finite time.
The long-time limit of these expressions recovers the known stationary mutual linearity.
Short-time expansions of the ratios are completely determined by the graph distance between states.
Hitting-time densities satisfy the same linear response relations as the occupation probabilities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The combinatorial formulas could be used to design efficient numerical schemes that avoid integrating the full master equation for sensitivity calculations.
Similar graph-theoretic arguments may apply to other linear response properties in continuous-time Markov chains on networks.
Relaxing time-homogeneity would require a different combinatorial tool, since the matrix-tree theorem relies on constant rates.

Load-bearing premise

The Markov network must be irreducible and time-homogeneous so that the all-minors matrix-tree theorem applies directly to the transient quantities.

What would settle it

Construct a small reducible Markov network, compute its transient probabilities after a rate perturbation, and check whether the response ratios remain mutually linear.

Figures

Figures reproduced from arXiv: 2605.00135 by Julian B. Voits, Ulrich S. Schwarz (Heidelberg University).

**Figure 1.** Figure 1: Illustration of relevant graph theoretic concepts. (a) An example of a directed graph G with N = 7 vertices. (b) A path in G from vertex 1 to vertex 7. (c) A spanning forest of G with roots 1 and 6. (d) A spanning tree of G with root 5. The all-minors matrix-tree theorem [12] gives a combinatorial expression for the determinants of the minors of W in terms of sums over spanning forests. In particular, the … view at source ↗

**Figure 2.** Figure 2: Illustration of the construction that allows to express the Laplace transform of the occupation probabilities in graph theoretic terms. (a) An example for a Markov network on four states represented as a weighted and directed graph G with the states as vertices, the transitions as directed edges with the transition rates as edge weights. (b) The augmented graph G ∗ is constructed by including an additional… view at source ↗

**Figure 3.** Figure 3: Illustration of the mutual linearity for an example network. (a) Network with four vertices, where the rate k13 is varied. (b) Affine relation (mutual linearity) between the Laplace transforms of the occupation probabilities L{p}(s)1 and L{p}(s)2 for different values of s. The parameters are k12 = 1, k21 = 0.4, k23 = 1.2, k32 = 0.5, k34 = 0.9, k43 = 0.6, k41 = 1.1, k14 = 0.3, k31 = 0.2 and k13 ∈ [0, 5]. Th… view at source ↗

**Figure 4.** Figure 4: Illustration of the auxiliary map H ij nl which exchanges the outgoing edges of the vertices on the i → l-path in Fi→l [j,l] until it reaches the n-tree F j→n [i,n] (blue path) with the corresponding edges in F j→n [i,n] , and also exchanges the outgoing edges of the vertices on the j → n-path in F j→n [i,n] until it reaches the l-tree Fi→l [j,l] (green path) with the corresponding edges in Fi→l [j,l] . Th… view at source ↗

**Figure 5.** Figure 5: Illustration of the auxiliary map A ij nl which exchanges the edges on the i → l-path in Fi→l [j,l] with the corresponding edges in F j→n [i,n] or on the j → n-path in F j→n [i,n] with the corresponding edges in Fi→l [j,l] , yielding a tuple (F[i,j] , F i→l,j→n [n,l] ). Short-time limit The short-time scaling follows from the large-s behavior of Eq. (20). In this limit, the combinatorial terms are dominate… view at source ↗

**Figure 6.** Figure 6: Illustration of the short time s → ∞ scaling of the response ratios for the example network in [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

read the original abstract

For irreducible, time-homogeneous Markov networks, mutual linearity has recently been established for both occupation probabilities and network currents in the stationary regime as well as in the non-stationary regime in Laplace space. The derivation of this property for the stationary distribution utilized the Markov-chain tree theorem, which also allows for an explicit combinatorial expression of the response ratios under variation of a single transition rate. The extension of this result was proven at the trajectory level by employing the Doob-Meyer decomposition. By employing the all-minors matrix-tree theorem, we show that this property also follows from a graph theoretic formulation and derive explicit combinatorial expressions for the non-stationary response ratios. The stationary result follows as the long-time limit and we also show that the small-time asymptotics are entirely determined by minimal path distances in the underlying graph. Finally we use the graph theoretic approach to prove that mutual linearity also extends to hitting time densities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a graph-theoretic proof via the all-minors matrix-tree theorem that mutual linearity holds for transient occupation probabilities, currents, and hitting-time densities in irreducible Markov networks, with explicit combinatorial expressions in Laplace space.

read the letter

The core result is that mutual linearity for non-stationary response ratios follows from the all-minors matrix-tree theorem applied to the resolvent of the rate matrix. This yields explicit combinatorial formulas for the ratios under single-rate perturbations, recovers the stationary case as the long-time limit, and ties the short-time behavior to minimal path distances in the graph. The same framework then proves the property for hitting-time densities. These steps are new relative to the stationary tree-theorem work and the trajectory-level Doob-Meyer proof cited in the abstract. The combinatorial expressions are the clearest addition; they make the response ratios directly readable from the weighted spanning trees of the graph. The argument is internally consistent under the stated assumptions of irreducibility and time-homogeneity, which are both necessary for the theorem to apply without modification. No hidden fitting or circular definitions appear. The only real limitation is the restriction to irreducible networks; reducible cases or absorbing states would require separate handling, but that is a standard boundary rather than a flaw in the present derivation. The paper is aimed at researchers who model physical or biological systems with Markov networks and need transient rather than steady-state tools. Anyone already working with the matrix-tree theorem or mutual-linearity results will find the explicit formulas and the hitting-time extension useful. It is worth sending to peer review; the combinatorial steps are reproducible in principle and the claims are narrowly scoped enough that referees can check them directly.

Referee Report

0 major / 0 minor

Summary. The manuscript derives mutual linearity for transient occupation probabilities, network currents, and hitting-time densities in irreducible time-homogeneous Markov networks by applying the all-minors matrix-tree theorem to the resolvent of the rate matrix. It supplies explicit combinatorial expressions for the Laplace-space response ratios under single-rate perturbations, recovers the known stationary result as the long-time limit, shows that small-time asymptotics are governed by shortest-path distances in the graph, and extends the linearity property to hitting-time densities.

Significance. If the derivations are correct, the work supplies a direct combinatorial foundation for mutual linearity outside equilibrium, with explicit expressions that could aid exact calculations or asymptotic analysis in stochastic networks. The graph-theoretic approach unifies the stationary and transient regimes and the hitting-time extension without introducing new parameters or fitting procedures.

Simulated Author's Rebuttal

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We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The referee's summary correctly identifies the key contributions: the application of the all-minors matrix-tree theorem to obtain explicit combinatorial expressions for mutual linearity of transient occupation probabilities, currents, and hitting-time densities, the recovery of the stationary case as a long-time limit, the short-time asymptotics governed by graph distances, and the unification of regimes without additional parameters.

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard theorem to extend prior result

full rationale

The paper derives mutual linearity for transient occupation probabilities, currents, and hitting-time densities by applying the all-minors matrix-tree theorem to the resolvent of the rate matrix, yielding explicit combinatorial expressions for Laplace-space response ratios. The stationary result is recovered as the long-time limit, and small-time asymptotics are tied to graph distances. These steps rest on the standard combinatorial matrix-tree theorem (not a self-citation or ansatz) under the necessary assumptions of irreducibility and time-homogeneity. No prediction reduces to a fitted input, no self-definitional loop appears, and the central claim has independent content beyond any prior mutual-linearity citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the applicability of the all-minors matrix-tree theorem to the Laplace-transformed or transient probabilities in the Markov chain graph, as well as the prior establishment of mutual linearity in other regimes.

axioms (1)

domain assumption The Markov network is irreducible and time-homogeneous.
Required for the mutual linearity property to hold as stated in the abstract.

pith-pipeline@v0.9.0 · 5461 in / 1174 out tokens · 43747 ms · 2026-05-09T20:36:13.454523+00:00 · methodology

discussion (0)

Reference graph

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