Inferring Tree Structure with Hidden Traps from First Passage Times

TL;DR

This paper presents a rigorous framework for inferring the depth $L$ and geometric bias $p$ of finite Cayley trees from first-passage-time data of discrete-time random walks, including the presence of waiting (traps). By developing a generating-function approach, it derives a backward-recurrence for first-passage-time factorial moments and shows that, in the absence of waiting, two factorial moments suffice to uniquely determine the tree structure. When waiting is present, the FPT generating function factorizes as a composition, leading to nonlinear relations between waiting-time moments and nonwaiting FPTFMs; the authors propose strategies—varying initial conditions and Fourier-based fitting of the FPT distribution—to resolve identifiability even for heavy-tailed waiting times. They validate the methods with analytical results and Monte Carlo simulations, including geometric and power-law waiting, and demonstrate robust structural inference with implications for biological transport and spatial networks. The work thus provides a principled, noninvasive toolkit for reconstructing tree-like networks from stochastic timing data and outlines pathways to extend to more general network topologies.

Abstract

Tracking the movement of tracer particles has long been a strategy for uncovering complex structures. Here, we study discrete-time random walks on finite Cayley trees to infer key parameters such as tree depth and geometric bias toward the root or leaves. By analyzing first passage properties, we show that the first two first-passage-time factorial moments (FPTFMs) uniquely determine the tree structure. However, if the random walker experiences waiting phases -- due to sticky branch walls or presence of traps -- this identification becomes nontrivial. We demonstrate that the generating function of the first passage time (FPT) distribution decomposes into contributions from the waiting time distribution and the random walk without waiting, leading to a nonlinear system of equations relating the factorial moments of the waiting time distribution and the FPTFMs of random walks with and without waiting. For geometrically distributed waiting times, additional moment measurements do not suffice, but unique determination of the structure is achieved by varying initial conditions or fitting the Fourier transform of the FPT distribution to measured data. The latter method remains effective also for power-law waiting time distributions, where higher-order FPTFMs are undefined. These results provide a framework for reconstructing tree-like networks from FPT data, with applications in biological transport and spatial networks.

Figures (9)

• Figure 1: (color online) Schematic illustration of the problem and modeling framework. (a) A Cayley tree of unknown structure (represented by the gray shading), characterized by its depth $L$, upward hopping probability $p$ (in conjunction with the coordination number $\mathfrak z$), and the distribution $w(t)$ of waiting times induced by stickiness or hidden traps along the branch walls. Random walkers are released from a specific level of the tree---here, the leaf nodes (dark purple)---and eventually reach the root (yellow). The distribution of their first passage times (inset graph) is used to infer the structural parameters $L$, $p$ and $w(t)$. (b) A single subtree of a Cayley tree of depth $L=4$ and coordination number $\mathfrak z=4$. The leaf nodes ($i=L=4$) are displayed in dark purple, the root node ($i=0$) in yellow. Transition probabilities are indicated on the left. On the right, a sample trajectory of a walker moving from a leaf to the root is depicted with footsteps and pentagons; the size of each pentagon reflects the random waiting time at that node, while the color of the pentagons and footsteps transitions from purple to yellow to indicate the progression of time. A cycle-free sample path is chosen to clearly illustrate the waiting behavior.
• Figure 2: (color online) First passage time distributions and their Fourier transforms for random walks on regular trees without waiting. (a-d) Examples of the FPT distribution $\mathfrak f_L(t;L)$ from the leaves to the root of a regular Cayley tree, for a random walker without waiting times. $\mathfrak f_L(t;L)$ is nonzero for $t\,{\geq}\,L$ and only for even or odd $t$ depending on $L$. (e-h) Fourier transforms $\tilde{\mathfrak f}_L(\omega; L)$ of the corresponding FPT distributions (from panels a-d), shown in the complex plane. Each contour encloses the origin and loops exactly $L$ times. For even values of $L$, the contour is traced twice. The parameters $L$ and $p$ used for each column are indicated at the top.
• Figure 3: (color online) First-passage-time factorial moments as functions of tree depth $L$ and upward hopping bias $p$. (a,b) Logarithm of the first factorial moment for walkers starting from the leaf nodes ($\log_{10} \left\langle{} \mathfrak{t}_L^{\underline{1}} \right\rangle{}$, panel a) or starting one level below the root ($\log_{10} \left\langle{} \mathfrak{t}_1^{\underline{1}} \right\rangle{}$, panel b). Lines are contours of constant $\log_{10} \left\langle{} \mathfrak{t}_L^{\underline{1}} \right\rangle{}$ or $\log_{10} \left\langle{} \mathfrak{t}_1^{\underline{1}} \right\rangle{}$ respectively. (c) Normalized second factorial moment $\frac{ \left\langle{} \mathfrak{t}_L^{\underline{2}} \right\rangle{}}{ \left\langle{} \mathfrak{t}_L^{\underline{1}} \right\rangle{}^2}$ for walkers starting from the leaves. This quantity saturates at large $L$ and small $p$, limiting its utility for parameter inference in that regime. (d) Ratio $\frac{ \left\langle{} \mathfrak{t}_1^{\underline{L}} \right\rangle{}}{ \left\langle{} \mathfrak{t}_L^{\underline{1}} \right\rangle{}}$, comparing first factorial moments for walkers starting just below the root versus from the leaves. This ratio becomes independent of $L$ for small $p$ and sufficiently deep trees, making it a robust alternative for inferring tree parameters when the normalized second moment is saturated. (e,f) Contour plots showing combinations of moment-based observables. In (e), contours of $\log_{10} \left\langle{} \mathfrak{t}_L^{\underline{1}} \right\rangle{}$ and $\frac{ \left\langle{} \mathfrak{t}_L^{\underline{2}} \right\rangle{}}{ \left\langle{} \mathfrak{t}_L^{\underline{1}} \right\rangle{}^2}$ intersect at a unique point corresponding to the true values of $L$ and $p$. Similarly, (f) shows that intersections of $\log_{10} \left\langle{} \mathfrak{t}_1^{\underline{1}} \right\rangle{}$ and $\frac{ \left\langle{} \mathfrak{t}_1^{\underline{L}} \right\rangle{}}{ \left\langle{} \mathfrak{t}_L^{\underline{1}} \right\rangle{}}$ contours provide an alternative route to identify the tree parameters.
• Figure 4: (color online) Isosurfaces of constant first passage quantities in the $(L,p,q)$ phase space. Each surface corresponds to a fixed value of a given FPT observable, with color indicating the specific isovalue. (a) Logarithm of the first factorial moment, $\log_{10}\left( \left\langle{} t_L^{\underline{1}} \right\rangle{}\right)$. As predicted by Eq. (\ref{['Eq:FM_Waiting_first3']}), all isosurfaces are parallel, reflecting the linear dependence on the first factorial moment of the waiting time $\left\langle{} \tau_w^{\underline{1}} \right\rangle{} \,{=}\, \frac{1}{q}$. (b) Ratio $\frac{ \left\langle{} t_1^{\underline{1}} \right\rangle{}}{ \left\langle{} t_L^{\underline{1}} \right\rangle{}}$, comparing first factorial moments for walkers starting near the root versus from the leaves. (c) Normalized second factorial moment, $\frac{ \left\langle{} t_L^{\underline{2}} \right\rangle{}}{ \left\langle{} t_L^{\underline{1}} \right\rangle{}^2}$. (d) Normalized third factorial moment, $\frac{ \left\langle{} t_L^{\underline{3}} \right\rangle{}}{ \left\langle{} t_L^{\underline{1}} \right\rangle{}^3}$. Panels (c) and (d) show that the surfaces quickly lose their dependence on the waiting probability $q$ at low values of $q$, making them less informative in that regime. Additionally, the similarity in behavior suggests that factorial moments beyond the second contribute little new information for structural inference.
• Figure 5: (color online) Isosurfaces of constant first passage quantities in the $(L,p,q)$ phase space. The intersection point of the three surfaces identifies the parameters $L$, $p$, and $q$. This provides a 3D analog to the contour plots shown in Figs. \ref{['Fig:3']}(e) and (f). (a) Isosurfaces for the logarithm of the first factorial moment $\log_{10}( \left\langle{} t_L^{\underline{1}} \right\rangle{})$ (blue), the second normalized factorial moment $\frac{ \left\langle{} t_L^{\underline{2}} \right\rangle{}}{ \left\langle{} t_L^{\underline{1}} \right\rangle{}^2}$ (red), and the third normalized factorial moment $\frac{ \left\langle{} t_L^{\underline{3}} \right\rangle{}}{ \left\langle{} t_L^{\underline{1}} \right\rangle{}^3}$ (yellow). The values these quantities attain belong to a system with $p\,{=}\,0.6$, $L\,{=}\,10$ and $q\,{=}\,0.01$. As the second and third normalized moments produce nearly overlapping surfaces, measuring these three quantities does not allow for a unique reconstruction of the tree parameters in this regime. (b-d) Isosurfaces for $\log_{10}( \left\langle{} t_L^{\underline{1}} \right\rangle{})$ (blue), $\frac{ \left\langle{} t_L^{\underline{2}} \right\rangle{}}{ \left\langle{} t_L^{\underline{1}} \right\rangle{}^2}$ (red), and the ratio $\frac{ \left\langle{} t_1^{\underline{1}} \right\rangle{}}{ \left\langle{} t_L^{\underline{1}} \right\rangle{}}$ (yellow). The values these quantities attain belong to a system with $L\,{=}\,10$, $q\,{=}\,0.01$, and (b) $p\,{=} \,0.4$, (c) $p\,{=}\,0.5$, and (d) $p\,{=}\,0.6$. In panels (b) to (d), the three surfaces intersect at a single point, confirming that the combination of these observables enables unambiguous determination of $L$, $p$ and $q$.