Directed walks shape a universal square-root law of entropy production rate in nonreciprocal systems

Thiparat Chotibut; Ewa Gudowska-Nowak; Maciej A. Nowak

cond-mat.stat-mechAug 25, 2026

Directed walks shape a universal square-root law of entropy production rate in nonreciprocal systems

Authors: Thiparat Chotibut, Ewa Gudowska-Nowak, Maciej A. Nowak

Organizations: Chula Intelligent and Complex Systems Center of Excellence, Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand · Institute of Theoretical Physics and Mark Kac Center for Complex Systems Research, Jagiellonian University, 30-348 Krak´ow, Poland

Abstract

The entropy production rate (EPR) quantifies irreversibility of a nonequilibrium steady state, yet standard formulas obscure how a complex interaction network generates it. For multivariate Ornstein-Uhlenbeck dynamics on such networks, we express the EPR as a quadratic form in antisymmetric matrices measuring the nonreciprocity of aggregate directed walks at every length, and, equivalently, as two weighted-walk quantities: pairs of directed walks sharing both endpoints, and directed closed walks. For diagonalizable interactions, an exact correspondence translates these walk quantities into eigenvalues and biorthogonal eigenvector overlaps. Across dense, sparse, and deep acyclic random interactions satisfying matched-walk conditions, the mean EPR per node universally follows the square-root law $φ_*(g)=1-\sqrt{1-g^2}$ , where $g \in [0,1)$ parametrizes the interaction strength. Deep acyclic interaction matrices are nilpotent, with all eigenvalues fixed at zero for every $g$ , yet, as their depth increases, their mean EPR per node approaches $φ_*(g)$ . Thus, the square-root law arises from directed walk properties, rather than from a shared spectral density or specific network topology.

Figures & tables

Figure 1: The matched-walk prediction and the square-root law. Numerical ensemble mean EPR per node (symbols) for acyclic networks and large- $N$ contractive recurrent ensembles agree with the analytic predictions shown in curves. The finite-depth feedforward curves follow Eq. ( 21 ) and converge to the square-root law $\phi_{*}(g)$ as $L\to\infty$ . The vertical dashed line at $g^{2}=1$ marks the radius-of-convergence boundary of the Catalan series resummation over all walk lengths, indicating the divergence of the susceptibility $\frac{d\phi_{*}}{d(g^{2})}=\frac{1}{2\sqrt{1-g^{2}}}$ as $g\rightarrow 1^{-}$ . Table 2 summarizes the ensemble construction. Numerical details are in SM M.

Figure 2: Dissipation is non-monotonic in edge probability. For centered dense Erdős-Rényi networks the effective interaction is

g^{2}=g_{0}^{2}p(1-p)

, so the EPR rises then falls with edge probability, peaking at

p=1/2

: beyond half-filling, adding edges dilutes the effective interaction strength and lowers dissipation. The square-root law

\phi_{*}(g)

predicts the full EPR (solid lines) at fixed

g_{0}^{2}\in\{1.50^{2},1.75^{2},1.90^{2},1.94^{2}\}

. Symbols are large-

N

ensemble means from direct Lyapunov solution, conditioned on

\rho(W)<1

, with error bars showing SD across retained realizations. The dashed

g_{0}^{2}=4

envelope touches the Catalan-series convergence boundary

g^{2}=1

p=1/2

; see SM M. This is an analytic benchmark for the edge-probability dependence reported numerically in [ 36 ] .

Ensemble	Construction	Details
Real Gaussian	$W_{ij}=gZ_{ij}/\sqrt{N}$ for $i\neq j$ , $Z_{ij}\stackrel{{\scriptstyle\rm iid}}{{\sim}}\mathcal{N}(0,1)$ , $W_{ii}=0$	SM I
Rademacher	$W_{ij}=g\varepsilon_{ij}/\sqrt{N}$ , $\mathbb{P}(\varepsilon_{ij}=\pm 1)=1/2$ , $W_{ii}=0$	SM I 4
Centered dense ER	$W_{ij}=g(A_{ij}-p_{})/\sqrt{Np_{}(1-p_{})}$ for $i\neq j$ , $A_{ij}\sim{\rm Bernoulli}(p_{})$ , $W_{ii}=0$ ; $p_{*}=1/4$ in Fig. 1	SM I 3
Sparse directed ER	$W_{ij}=A_{ij}\xi_{ij}$ for $i\neq j$ , with $A_{ij}\stackrel{{\scriptstyle\rm iid}}{{\sim}}{\rm Bernoulli}(c/N)$ ; the $\xi_{ij}$ are iid, independent of the $A_{ij}$ , and satisfy $\mathbb{E}\xi_{ij}=0$ and $\mathbb{E}\xi_{ij}^{2}=g^{2}/c$ . Fig. 1 uses the mean degree $c=8$ and $\xi_{ij}\sim\mathcal{N}(0,g^{2}/c)$ ; $W_{ii}=0$	SM J
Layered feedforward	$N=Ln$ : $L$ ordered layers $V_{1},\ldots,V_{L}$ of width $n$ ; the only nonzero blocks are the forward blocks $V_{\ell}\to V_{\ell+1}$ , $\widetilde{G}_{\ell}:=\widetilde{W}_{\ell+1,\ell}=gX_{\ell}/\sqrt{n}$ for $\ell=1,\ldots,L-1$ , where $X_{\ell}\in\mathbb{R}^{n\times n}$ and each element in $\{(X_{\ell})_{ij}\}$ is iid $\mathcal{N}(0,1)$	SM K 1
Oriented out-tree	Rooted tree with edges directed parent-to-child and $W_{e}=g\xi_{e}$ ; the $\xi_{e}$ are iid, and $\mathbb{E}\xi_{e}^{2}=1$ . Fig. 1 uses uniform Cayley trees with $\|\xi_{e}\|=1$	SM K 2

Table 2: Ensembles used in Fig. 1 . Each realizes the common asymptotic mean-squared interaction strength per node

g^{2}

in Eq. ( 24 ). The last column lists the corresponding section number in the SM.

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Dynamical systems studies how states evolve over time, including stability, feedback, chaos, and transitions in physical, biological, and engineered systems.

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The node2vec random walk is a non-Markovian random walk on the vertex set of a graph, widely used for network embedding and exploration. This random walk model is defined in terms of three parameters which control the probability of, respectively, backtracking moves, moves within triangles, and moves to the remaining neighboring nodes. From a mathematical standpoint, the node2vec random walk is a nontrivial generalization of the non-backtracking random walk and thus belongs to the class of second-order Markov chains. Despite its widespread use in applications, little is known about its long-run behavior. The goal of this paper is to begin exploring its fundamental properties on arbitrary graphs. To this aim, we show how lifting the node2vec random walk to the state spaces of directed edges and directed wedges yields two distinct Markovian representations which are key for its asymptotic analysis. Using these representations, we find mild sufficient conditions on the underlying finite or infinite graph to guarantee ergodicity, reversibility, recurrence and characterization of the invariant measure. As we discuss, the behavior of the node2vec random walk is drastically different compared to the non-backtracking random walk. While the latter simplifies on arbitrary graphs when using its natural edge Markovian representation thanks to bistochasticity, the former simplifies on regular graphs when using its natural wedge Markovian representation. Remarkably, this representation reveals that a graph is regular if and only if a certain weighted Eulerianity condition holds.

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