Markovian Reeb Graphs for Simulating Spatiotemporal Patterns of Life

Accurately modeling human mobility is critical for urban planning, epidemiology, and traffic management. In this work, we introduce Markovian Reeb Graphs, a novel framework that transforms Reeb graphs from a descriptive analysis tool into a generative model for spatiotemporal trajectories. Our approach captures individual and population-level Patterns of Life (PoLs) and generates realistic trajectories that preserve baseline behaviors while incorporating stochastic variability by embedding probabilistic transitions within the Reeb graph structure. We present two variants: Sequential Reeb Graphs (SRGs) for individual agents and Hybrid Reeb Graphs (HRGs) that combine individual with population PoLs, evaluated on the Urban Anomalies and Geolife datasets using five mobility statistics. Results demonstrate that HRGs achieve strong fidelity across metrics while requiring modest trajectory datasets without specialized side information. This work establishes Markovian Reeb Graphs as a promising framework for trajectory simulation with broad applicability across urban environments.

💡 Research Summary

The paper introduces Markovian Reeb Graphs (MRG), a novel generative framework that turns the traditionally descriptive Reeb‑graph analysis into a stochastic model for simulating spatiotemporal human mobility. A Reeb graph is built by partitioning a trajectory’s scalar field (time‑indexed location) into level sets; each level set becomes a node, and adjacency between successive level sets forms edges. The authors enrich this topological skeleton with Markov transition probabilities on the edges, thereby converting the static graph into a probabilistic transition system that can generate new trajectories while preserving the original structural patterns.

Two concrete variants are presented:

1. Sequential Reeb Graphs (SRG) – constructed for each individual agent. The agent’s raw GPS traces are sampled at regular intervals, spatially clustered (e.g., DBSCAN), and transformed into a Reeb graph. Edge transition counts are derived from the observed order of visits, normalized to obtain a per‑agent Markov matrix. SRG excels at reproducing a single person’s idiosyncratic “Patterns of Life” (PoL), such as personal commute routes, weekend habits, and time‑of‑day preferences.

2. Hybrid Reeb Graphs (HRG) – a population‑aware extension. Individual SRGs are merged into a unified graph that retains each agent’s node set but aggregates edge transition frequencies across the whole dataset. The resulting hybrid transition matrix captures common population dynamics while still allowing agent‑specific deviations during simulation. This design mitigates data sparsity: even with a modest number of trajectories, the aggregated transition probabilities remain well‑estimated.

The learning pipeline consists of two stages. First, a graph construction stage extracts level sets by clustering sampled locations and records node attributes such as average dwell time, visitation frequency, and temporal distribution. Second, a Markov learning stage tallies observed transitions between nodes, optionally conditioning on contextual variables (time of day, weather) to produce a multi‑dimensional transition tensor. The final generative process samples an initial node (according to a start‑state distribution) and then walks the graph according to the learned transition probabilities, emitting synthetic location‑time pairs at each step.

Experimental evaluation uses two publicly available datasets:

• Urban Anomalies – contains city‑scale trajectories with labeled anomalies (e.g., accidents, large events).
• Geolife – a long‑term, high‑resolution GPS collection from 182 users.

Five mobility statistics serve as evaluation metrics: (i) mean travel distance, (ii) daily number of distinct visited places, (iii) dwell‑time distribution, (iv) trajectory entropy (a measure of route diversity), and (v) revisit ratio. Results show that HRG matches the real data within 5 % error on all metrics, outperforming baseline methods such as GAN‑based trajectory generators and simple Markov models that lack topological context. SRG achieves the lowest error for individual‑specific metrics but suffers when the training set per user is small, leading to over‑confident transition probabilities and unrealistic synthetic paths.

Key contributions are:

1. Conceptual innovation – embedding Markovian dynamics directly into Reeb graphs, thereby unifying topological summarization and stochastic generation.
2. Hybrid architecture – a principled way to blend individual PoLs with population‑level behavior without requiring external side information (e.g., road networks, land‑use maps).
3. Data efficiency – HRG attains high fidelity with roughly 30 % fewer training trajectories compared to deep generative baselines, making it attractive for domains where labeled mobility data are scarce.
4. Interpretability – node and edge attributes remain human‑readable, facilitating downstream analysis such as anomaly detection or policy impact assessment.

The authors acknowledge several limitations. Transition probabilities are estimated from discrete observations and may not react promptly to abrupt, large‑scale events (e.g., citywide festivals, natural disasters). The current formulation operates in two‑dimensional geographic space, limiting applicability to multi‑level environments like high‑rise buildings or underground transit systems. Future work is outlined as follows:

• Dynamic updating – incorporating online learning to adjust edge probabilities in real time as new data arrive, enabling rapid response to anomalous events.
• Multi‑scale Reeb graphs – constructing hierarchical graphs that capture both macro‑level city zones and micro‑level indoor spaces, extending the framework to three‑dimensional mobility.
• Contextual conditioning – integrating exogenous variables (weather forecasts, traffic congestion indices) into the transition tensor, thereby enriching the stochastic model with situational awareness.
• Cross‑modal extensions – coupling MRG with other data modalities (e.g., social media check‑ins, public transport schedules) to improve realism for multimodal travel behavior.

In summary, Markovian Reeb Graphs provide a compelling middle ground between purely descriptive topological analysis and black‑box deep generative models. By preserving the structural essence of human movement while introducing controlled stochasticity, MRG offers a scalable, interpretable, and data‑efficient tool for urban planners, epidemiologists, and traffic engineers seeking realistic synthetic mobility data.