Markovian Reeb Graphs: Probabilistic Topology

• Markovian Reeb Graphs are probabilistic constructs that couple topological summaries of level-set connectivity with stochastic Markov transitions to model spatiotemporal data.
• They are applied in mobility modeling and surface diffusion, enabling simulation of individual and population dynamics via empirical conditional probabilities.
• Their construction employs efficient clustering and graph-generation algorithms, with evaluations using metrics like Jensen-Shannon Divergence and GPU-friendly sparse linear algebra.

Searching arXiv for papers on Markovian Reeb Graphs and related diffusion-based Reeb graph construction. Markovian Reeb Graphs are probabilistic Reeb-graph constructions in which topological state spaces are endowed with Markov dynamics. In the mobility-modeling formulation introduced by "ReeMark: Reeb Graphs for Simulating Patterns of Life in Spatiotemporal Trajectories" (Subrahmanya et al., 3 Oct 2025), the states are trajectory bundles derived from temporal Reeb-graph structure, and the edge weights are empirical conditional probabilities, yielding a discrete-time Markov chain over topologically motivated bundles. A distinct but related use of a Markovian perspective appears in "A Variational Loop Shrinking Analogy for Handle and Tunnel Detection and Reeb Graph Construction on Surfaces" (Weinrauch et al., 2021), where diffusion-front evolution on a surface is rewritten in Markov-chain form and used to recover Reeb-graph structure from split and merge events. Taken together, these works show that the term denotes not a single fixed formalism, but a family of Reeb-graph methods in which topological summarization is coupled to stochastic or diffusion-driven state evolution.

1. Conceptual scope

The classical Reeb graph collapses connected components of level sets into graph nodes. In the trajectory setting of (Subrahmanya et al., 3 Oct 2025), this idea is specialized to a Temporal Reeb Graph whose nodes correspond precisely to bundles $b$ where the set of connected trajectories changes from the previous time index. The Markovian extension is obtained by assigning directed transitions between such bundles whenever at least one trajectory visits the predecessor bundle at index $i-1$ and the successor bundle at index $i$, with each edge weighted by an empirical conditional probability.

In that formulation, the graph is not merely a topological descriptor but a probabilistic generative model. The resulting triple $(V,E,P)$ is a discrete-time Markov chain whose states are “critical” bundles and whose transitions encode both frequent and infrequent mobility changes (Subrahmanya et al., 3 Oct 2025). This makes the Reeb graph operational as a simulator rather than solely as a summary of connectivity changes.

A different Markovian viewpoint appears in the surface-based construction of (Weinrauch et al., 2021). There, a diffusion process over a surface mesh is written as $u^{t+1}=Pu^t$ for a row-stochastic update matrix $P$, and in the multi-field setting as $\Phi^{t+1}=P\Phi^t$. Split and merge events in the evolving fronts induce branch and merging nodes in a Reeb graph. This suggests that “Markovian Reeb Graph” can refer either to a Reeb graph equipped with a transition kernel, or to a Reeb graph extracted from dynamics cast in Markov form.

2. Trajectory bundles and temporal Reeb structure

In (Subrahmanya et al., 3 Oct 2025), each raw trajectory is written as $T=(p_0,p_1,\dots,p_{L-1})$, where $p_i=(\text{latitude},\text{longitude})$ at discrete time index $i$. Across $i-1$0 trajectories, bundles are formed at each time index $i-1$1, and two points at the same index fall in the same equivalence class whenever $i-1$2, using Euclidean distance and a threshold $i-1$3. Each bundle $i-1$4 has an index $i-1$5, a centroid $i-1$6, and a connected-component set $i-1$7.

The node set for the Markovian construction is defined from this temporal topology. Specifically, $i-1$8 consists of bundles at index $i-1$9 whose connected set differs from all bundles at $i$0. Directed edges $i$1 are created whenever there exists at least one trajectory $i$2 that visits bundle $i$3 at index $i$4 and bundle $i$5 at index $i$6. The edge weight is

$i$7

which is exactly the empirical conditional probability $i$8 (Subrahmanya et al., 3 Oct 2025).

This construction preserves the Reeb-graph intuition that nodes arise at changes in connectedness, while replacing a purely combinatorial adjacency relation with a stochastic one. A plausible implication is that the model occupies an intermediate position between topological data analysis and sequential probabilistic modeling: topology determines the state abstraction, and empirical transition frequencies determine the dynamics.

3. Markov-chain formulation

The central algebraic object in (Subrahmanya et al., 3 Oct 2025) is the transition-probability matrix $i$9, where $(V,E,P)$0 and $(V,E,P)$1 for all $(V,E,P)$2. The resulting structure $(V,E,P)$3 is explicitly a discrete-time Markov chain. Its states are not arbitrary spatial cells, but bundles selected by changes in connectivity, which gives the chain a topological semantics absent from standard mobility-transition graphs.

The same paper distinguishes three graph variants. A Sequential Reeb Graph (SRG) $(V,E,P)$4 is built on the trajectories of one agent and captures that agent’s idiosyncratic patterns. A Multi-Agent Reeb Graph (MARG) $(V,E,P)$5 is built on the concatenation of all agents’ trajectories and captures population-level regularities. A Hybrid Reeb Graph (HRG) $(V,E,P)$6 merges the two by mapping each SRG node to the closest MARG node by centroid distance, retaining reachable population edges, boosting the SRG-inherited edges by a factor $(V,E,P)$7, and renormalizing each row so that outgoing weights sum to one (Subrahmanya et al., 3 Oct 2025).

The HRG is therefore not just a union of two graphs. It is a renormalized interpolation between individual-level and population-level dynamics. The paper states that $(V,E,P)$8 interpolates between pure individual fidelity and pure population generality. This suggests that the Markovian aspect is essential not merely for simulation, but also for expressing trade-offs between personalization and aggregate regularization.

4. Construction algorithms

The SRG construction in (Subrahmanya et al., 3 Oct 2025) has two phases. In the bundling phase, for each time $(V,E,P)$9, the set $u^{t+1}=Pu^t$0 is clustered via a spatial index such as a kd-tree in $u^{t+1}=Pu^t$1, producing bundles $u^{t+1}=Pu^t$2. In the graph-generation phase, for each agent $u^{t+1}=Pu^t$3 and each time $u^{t+1}=Pu^t$4, one identifies the predecessor bundle $u^{t+1}=Pu^t$5 and successor bundle $u^{t+1}=Pu^t$6; if $u^{t+1}=Pu^t$7 or $u^{t+1}=Pu^t$8, then $u^{t+1}=Pu^t$9 and $P$0 are inserted into $P$1, and edge $P$2 is inserted or updated with $P$3. Finally, each out-row of the weight matrix is normalized. The paper states that this graph-generation phase runs in $P$4 (Subrahmanya et al., 3 Oct 2025).

The MARG is formed identically, but over the pooled set of $P$5 trajectories, replacing $P$6 with $P$7. The bundling phase then has complexity $P$8. The HRG construction maps SRG nodes to nearest MARG nodes, copies the induced subgraph of reachable MARG edges, and boosts and renormalizes edge weights. Its construction runs in $P$9 time and space, dominated by a single graph traversal (Subrahmanya et al., 3 Oct 2025).

A compact summary of the three variants is given below.

Variant	Input scope	Stated role
SRG	One agent’s $\Phi^{t+1}=P\Phi^t$0 trajectories	Captures that agent’s idiosyncratic patterns
MARG	Concatenation of all $\Phi^{t+1}=P\Phi^t$1 agents’ trajectories	Captures population-level regularities
HRG	Merge of SRG and MARG	Interpolates between individual fidelity and population generality

These procedures are specific to spatiotemporal trajectory data. They should not be conflated with the surface-based diffusion construction of (Weinrauch et al., 2021), whose Reeb graph is recovered from events in a layered field rather than from bundles of trajectories.

5. Sampling and generative semantics

Given any Markov Reeb $\Phi^{t+1}=P\Phi^t$2, (Subrahmanya et al., 3 Oct 2025) generates new trajectories by random walks in two coupled steps. First, a source node $\Phi^{t+1}=P\Phi^t$3 is chosen; for continuity, it is chosen to be the closest source to the endpoint of the previous day. Then a state sequence is sampled by repeatedly drawing

$\Phi^{t+1}=P\Phi^t$4

until a sink node is reached. Second, each edge $\Phi^{t+1}=P\Phi^t$5 is labeled by the set of original subtrajectories $\Phi^{t+1}=P\Phi^t$6. Whenever the walk traverses $\Phi^{t+1}=P\Phi^t$7, one of those trajectories is chosen uniformly at random and the segment $\Phi^{t+1}=P\Phi^t$8 is appended to the generated trace (Subrahmanya et al., 3 Oct 2025).

The paper states that, by construction, this process preserves the consistency of frequent transitions via $\Phi^{t+1}=P\Phi^t$9 and the day-to-day variability through sampling randomness. The generative interpretation is therefore explicit: the Markov chain controls macro-level state evolution, while archived trajectory fragments supply geometric realization in physical space.

A common misconception would be to regard the model as a purely spatial Markov chain over grid cells. That description would be inaccurate. The state space is instead induced by topological bundle changes, and the emitted trajectory segments are original subtrajectories associated with graph edges. Another potential misconception is that the method learns arbitrary latent dynamics; the paper instead describes an empirical transition model whose weights are conditional probabilities computed directly from bundle overlaps.

6. Relation to diffusion-based Reeb graph construction on surfaces

The surface method of (Weinrauch et al., 2021) does not define “Markovian Reeb Graphs” in the mobility sense, but it supplies a second, technically relevant interpretation of Markovianity in Reeb-graph research. There, isotropic diffusion on a surface $T=(p_0,p_1,\dots,p_{L-1})$0 is discretized as

$T=(p_0,p_1,\dots,p_{L-1})$1

and with $T=(p_0,p_1,\dots,p_{L-1})$2, this becomes $T=(p_0,p_1,\dots,p_{L-1})$3, provided $T=(p_0,p_1,\dots,p_{L-1})$4 is small enough that $T=(p_0,p_1,\dots,p_{L-1})$5 is non-negative and row-stochastic. In the multi-field loop-shrinking model, the layered field matrix $T=(p_0,p_1,\dots,p_{L-1})$6 is updated in first-order form and can likewise be rewritten as $T=(p_0,p_1,\dots,p_{L-1})$7, with non-diffusive terms handled by zeroing out or rescaling selected rows and columns of $T=(p_0,p_1,\dots,p_{L-1})$8 corresponding to base layers or colliding fronts (Weinrauch et al., 2021).

Reeb-graph extraction proceeds by event detection. If a single front develops multiple disconnected boundary loops, new layers are spawned and the event corresponds to a branching node. If two or more fronts overlap, their growth is frozen, a handle is detected, the corresponding loop is recorded, and the layers are merged; topologically, each such merging yields a merging node in the Reeb graph. The paper then defines a “height”-like function $T=(p_0,p_1,\dots,p_{L-1})$9 time when $p_i=(\text{latitude},\text{longitude})$0 was first reached by any front, and states that the graph of critical events is precisely its Reeb graph (Weinrauch et al., 2021).

This relation is important because it shows that a Markovian update law can serve either as the object represented by the graph, as in (Subrahmanya et al., 3 Oct 2025), or as the mechanism by which the graph is computed, as in (Weinrauch et al., 2021). The two works address different domains—spatiotemporal trajectories versus surface meshes—but both place stochastic or diffusion dynamics at the center of Reeb-graph methodology.

7. Evaluation, empirical findings, and interpretation

The empirical evaluation reported in (Subrahmanya et al., 3 Oct 2025) uses the Jensen-Shannon Divergence (JSD) to compare generated versus baseline trajectories under six statistics: Average Grid Activity, Temporal Accuracy of Grid Activity, Rate of Movement, Trip Duration, Distance Traveled by Agent, and Net Travel Time by Agent. Results are averaged over the “combined” month $p_i=(\text{latitude},\text{longitude})$1, described as 1,000 agents, 28 days, and 5-min GPS sampling, using the Urban Anomalies dataset for Atlanta and Berlin.

For Atlanta, the representative JSD values reported for the Hybrid Reeb Graph are 0.067 for Average Grid Activity, 0.113 for Temporal Accuracy, 0.408 for Rate of Movement, 0.325 for Trip Duration, 0.683 for Distance Traveled by Agent, and 0.685 for Net Travel Time by Agent. The corresponding average JSD across all six metrics is given as SRG 0.502, MARG 0.435, and HRG 0.379; analogous trends are stated to hold for Berlin (Subrahmanya et al., 3 Oct 2025). The paper interprets these results as showing that the Hybrid Reeb Graph achieves the best overall fidelity to both individual- and population-level patterns among the three Markov Reeb variants, while remaining data- and compute-efficient.

In (Weinrauch et al., 2021), the evaluation focus is different. The paper states that the substantiated results suggest the diffusion-based loop-shrinking approach outperforms state of the art and robustly copes with highly detailed geometric models. It further states that the diffusion-derived function is “distance-like,” avoids explicit critical-point isolation, is robust on noisy, high-resolution meshes, and maps naturally to GPU-friendly sparse linear algebra, with reported speedups on large models of 2–10 M triangles (Weinrauch et al., 2021). These claims concern Reeb-graph construction rather than probabilistic trajectory simulation.

Taken together, the available literature indicates two converging research directions. One direction uses Markovian dynamics to generate trajectories over Reeb-graph states (Subrahmanya et al., 3 Oct 2025). The other uses Markov-chain-style diffusion updates to induce the split and merge events from which a Reeb graph is assembled (Weinrauch et al., 2021). A plausible implication is that future work may further unify these perspectives, using topologically structured state spaces together with stochastic evolution laws both for inference and for synthesis.