Emergence of generic first-passage time distributions for large Markovian networks

TL;DR

This work builds on the connection between first-passage times and graph theory to show that a deterministic peak emerges when infinitely many eigenvalues contribute, while the exponential limit arises from a single dominant eigenvalue.

Abstract

First-passage times are often the most relevant aspect of a complex Markovian network, because they signify when information processing has resulted in a definite decision. Previous studies have shown that for kinetic proofreading networks in the limit of large network size the first-passage time distribution converges either to a delta or to an exponential distribution. Remarkably, these two forms correspond to the two extreme distributions of minimal and maximal entropy for a fixed mean, respectively. Here we build on the connection between first-passage times and graph theory to show that these two limits are not model-specific, but arise generically in Markovian networks from the distribution of the eigenvalues of the generator matrix. A deterministic peak emerges when infinitely many eigenvalues contribute, while the exponential limit arises from a single dominant eigenvalue. We also show that the exponential limit emerges robustly for reversible networks when a backward bias exists. In contrast, the deterministic limit is not obtained from a simple reversal of this condition, but under structurally tighter conditions, revealing a fundamental asymmetry between both regimes. Our theoretical analysis is illustrated and validated by computer simulations of one-step master equations and random networks.

Figures (7)

• Figure 1: Generic first-passage time (FPT) distributions for large Markovian networks. (a) FPT in a large network: a trajectory starting at an initial state (square) reaches the target state $N$ (diamond) for the first time. (b) Two generic distributions have been observed before to emerge for kinetic proofreading networks in the limit of very large systems: the delta and the exponential distributions. (c) Here we show that in the limit $N\to\infty$, the FPT-distribution is determined by the eigenvalue structure of the generator matrix. If infinitely many eigenvalues contribute, the distribution collapses to a delta peak. If $\lambda_1^{-1}$ dominates the sum of inverse eigenvalues, the FPT rescaled by its mean converges to an exponential law. (d) Examples of graph theoretical concepts used here: a graph, a spanning tree and a two-tree spanning forest. A graph-theoretical decomposition provides an exact representation of the FPT-moments and the Laplace transform of the FPT-density.
• Figure 2: The first-passage time (FPT) of a one-step master equation compared to its mean as system size $N$ increases. (a) The network of the one-step master equation with site-dependent forward rates $k_i$ and backward rates $r_i$. (b) FPT-statistics for $10^5$ runs with randomly drawn rates. Forward bias: rates drawn from uniform distributions, $k_i\sim U[1,2]$, $r_i\sim U[0,1]$, showing convergence to a delta-distribution (deterministic limit). Backward bias: rates drawn from uniform distributions, $k_i\sim U[0.5,1]$, $r_i\sim U[0.5,1.5]$, showing convergence to an exponential distribution (exponential limit). (c) FPT-statistics for $10^5$ runs with constant rates. Forward bias: rates chosen as $k_i=\frac{3}{2}$, $r_i=\frac{1}{2}$, showing convergence to a delta-distribution (deterministic limit). Backward bias: rates chosen as $k_i=\frac{3}{4}$, $r=1$, showing convergence to an exponential distribution (exponential limit).
• Figure 3: Illustration of graph theoretical concepts. (a) Example of a Markovian network. (b) A spanning tree of the graph with root in $5$. (c) A two-tree spanning forest with roots in $3$ and $5$. (d) Network with an outgoing edge to $5$ with weight $s$ added to all vertices, which can be used to calculate the Laplace transform of the first-passage time.
• Figure 4: A minimal example to illustrate how irreversible transitions can result in two delta peaks. (a) A network branching into two Poisson processes with rates $k_1$ and $k_2$, both starting at vertex $1$ and ending in vertex $N$. (b) The first-passage time density results in two distinct delta peaks as $N$ increases, corresponding to the two chains. Here shown for $k_1=1$ and $k_2=2$.
• Figure 5: FPT-statistics for a one-step master equation with $N=50$ for $10^5$ trials with rates as specified in Eq. (\ref{['eq:apparent_forward_bias']}). Despite an apparent forward bias, the limiting behavior of the distribution is exponential, showing that the expectation that a forward bias would always lead to a quasi-deterministic behavior is not correct.