Evanescent random walker on networks: Hitting times, budget renewal, and survival dynamics

TL;DR

This work formulates a mortal random-walker model on ergodic networks where a positive budget must be maintained for survival, with budget renewals triggered by visits to designated target nodes. It develops a unified THCP framework for immortal walkers and extends it to evanescent walkers (MRW), deriving analytical expressions for the evanescent propagator, survival probability, mean residence time, and lifetime, as well as the distribution of target hits and related quantities. The analysis leverages defective transition matrices and generating-function techniques to handle arbitrary target configurations and connects budget renewal to stochastic resetting, yielding rich dynamics including forager-like, neutral, and detrimental regimes depending on the renewal distribution. The results are corroborated by numerical simulations on Barabási–Albert graphs, and the framework opens pathways to continuous-time extensions, resetting strategies, and broader biological and socio-economic applications where agent lifetimes interact with resource renewal mechanisms.

Abstract

We consider a mortal random walker evolving with discrete time on a network, where transitions follow a degree-biased Markovian navigation strategy. The walker starts with a random initial budget $T_1 \in \mathbb{N}$ and must maintain a strictly positive budget to remain alive. Each step incurs a unit cost, decrementing the budget by one; the walker perishes (is ruined) upon depletion of the budget. However, when the walker reaches designated target nodes, the budget is renewed by an independent and identically distributed (IID) copy of its initial value. The degree bias is tuned to either favor or disfavor visits to these target nodes. Our model exhibits connections with stochastic resetting. The evolution of the budget can be interpreted as a deterministic drift on the integer line toward negative values, where the walker is intermittently reset to positive IID random positions and dies at the first hit of the origin. The first part of the paper focuses on the target-hitting statistics of an immortal Markovian walker. We analyze the \textit{target hitting counting process} (THCP) for an arbitrary set of target nodes. Within this framework, the second part of the paper addresses the dynamics of the evanescent walker. We derive analytical results for arbitrary configurations of target nodes, including the evanescent propagator matrix, the survival probability, the mean residence time on a set of nodes during the walker's lifetime, and the expected lifetime itself. Additionally, we compute the expected number of target hits (i.e., budget renewals) in a lifetime of the walker and related distributions. We explore both analytically and numerically various scenarios affecting the life expectancy of the walker.

Figures (10)

• Figure 1: Numerical results for the target hitting counting process. We consider the same BA network and setting as in Fig. \ref{['fig2']}. (Left frame) Numbers of distinct nodes visited $N_d(i=486;t=1000)$ and number of target hits ${\cal N}_{i=486}(t=1000;{\cal B})$, respectively, in a power-law degree-biased walk [Eq. (\ref{['preferential_steps']})], plotted as functions of the tuning parameter $\alpha$, as obtained from random walk simulations. (Right frame) MFPT to the target set ${\cal B}$ as a function of $\alpha$, from the analytical expression in Eq. (\ref{['MFPT_to_B']}) (blue curve), compared with first hitting times (FHTs) obtained using Monte Carlo simulations (orange curve). For $\alpha <-2$, the target is not reached within the simulation time (FHTs $\gg$ runtime).
• Figure 2: THCP of $10$ t-nodes for the biased walk (\ref{['preferential_steps']}) in a BA network (Python NetworkX library $G=nx.barabasi\_albert\_graph(1000, m=7, seed=0)$] of $1000$ nodes. Left frames: Highly connected target ${\cal B}$ in cyan color. Departure node ($i=486$) in blue, distinct nodes visited during the runtime $1000$ are drawn in red. Right frames: Orange curves: ${\cal N}_{i=486}(t;{\cal B})$ (\ref{['arrivals_on_B']}) recorded in random walk simulations; Green curves: expected hitting number $\langle {\cal N}_{i=486}(t;{\cal B})\rangle$ of Eq. (\ref{['total_hits']}).
• Figure 3: We depict the average number of distinct nodes visited $\langle N_d(486;1000) \rangle$ from Eq. (\ref{['dis_nodes_vis']}), and the recorded values of $N_d(486;1000)$ from Fig. \ref{['fig3']} versus $\alpha$. For this setting and observation time, the exploration of the network is maximal at values slightly shifted to negative values from $\alpha=0$ (unbiased walk).
• Figure 4: Left frame: Schematic: Sample budget evolution of (\ref{['budget']}) until walker's death at $t=t_{*}$ when ${\cal C}(t_{*})=0$. Right frame: Budget evolution (\ref{['budget']}) until walker's death ($t_* = 559$) recorded in a random walk simulation for power-law biased steps (bias parameter $\alpha=1.6$) for the same BA network, target and departure node as in Figs. \ref{['fig3']} and \ref{['fig2']}. In this simulation, the budget is renewed with uniformly distributed random integers ($T \in [14, 24]$).
• Figure 5: Time dependence of $\Phi_{\cal S}(t; Q_1,Q_2)$ (Eq. (\ref{['adva']})) for some values of $Q_2$ with fixed $Q_1=0.9$ and $t_0=4$. The blue shaded region corresponds to forager's scenarios ($\Phi_{\cal S}(t) < (\Phi_{\cal S}(1))^t$), the red shaded region corresponds to detrimental scenarios. Both scenarios are separated by the neutral scenario (black dashed line).