---
title: Object-Centric Directly-Follows Graphs
url: https://www.emergentmind.com/topics/object-centric-directly-follows-graphs-oc-dfgs
type: topic
---

# Object-Centric Directly-Follows Graphs

Object-Centric Directly-Follows Graphs (OC-DFGs) and their Markov-based multigraph variant, the Markov Directly-Follows Multigraph (MDFMG), are advanced modeling constructs in object-centric process mining. OC-DFGs extend the classic Directly-Follows Graph (DFG) model to handle event data with multiple, co-evolving object types. MDFMG enhances OC-DFGs by assigning transition probabilities and enabling quantitative comparison between the behavioral control flows of different object types. These structures support principled, automated clustering of object types with similar control-flow behavior, transforming the interpretability and scalability of process mining in complex real-world settings [2206.11017].

## 1. Classic DFGs and Their Limitations

The Directly-Follows Graph (DFG) is a fundamental tool in classical process mining. For an event log $L$ over activity set $T$ and a case identifier $c$, the DFG $G = (T, E, f)$ is a directed graph where:
- Vertices represent activities $t \in T$.
- $E \subseteq T \times T$ consists of pairs $(t, t')$ if $t$ is directly followed by $t'$ in a trace.
- $f: E \to \mathbb{N}$ records the frequency of each directly-follows relation.

This formalism assumes a single case notion, constraining its expressiveness in domains where multiple interacting entities (e.g., orders, items, packages) participate in events. The necessity for a more flexible representation capable of modelling "object-centric" event data motivates the development of OC-DFGs.

## 2. Object-Centric Event Logs and Directly-Follows Multigraphs

In Object-Centric Event Logs (OCEL), each event $e$ logs an activity $a(e) \in T$ and can refer to multiple objects. Every object $o$ possesses a type $\theta(o) \in OT$, where $OT$ denotes the set of object types. Rather than treating each trace as a case, OCEL enables extraction of a local event sequence $\sigma_o$—the ordered list of events relevant to each object $o$.

The Directly-Follows Multigraph (DFM), the "object-centric DFG," is formally defined as $G = (OT, T, R, f)$ with:
- $R \subseteq T \times OT \times T$ encoding directly-follows relations for each object type: $(t, \theta, t')$ represents activity $t$ directly preceding $t'$ for objects of type $\theta$.
- $f: R \to \mathbb{N}$ counts occurrences of each triple.

This model captures the possibility that each object type induces its own characteristic directly-follows behavioral pattern.

## 3. Markov Directly-Follows Multigraph: Probabilistic Extension

The Markov Directly-Follows Multigraph (MDFMG) extends the DFM to support quantitative comparison between object types by introducing "Markovization"—the normalization of transition frequencies:
- For each $(t, \theta, t') \in R$, define the local transition probability
  $$
  p((t, \theta, t')) = \frac{f((t, \theta, t'))}{\sum_{u \in T} f((t, \theta, u))}
  $$
  ensuring that, for fixed $(\theta, t)$, the outgoing probabilities sum to 1.

Each object type $\theta$ is thus associated with a row-stochastic matrix $P_\theta$ where entries $(i, j)$ give the probability of $t_i$ directly preceding $t_j$ for this type.

Pairwise similarity between object types is quantified using a cosine-like normalized inner product:

$$
sim(\theta_1, \theta_2) = \frac{\sum_{i, j} P_{\theta_1, ij} P_{\theta_2, ij}}{\frac{1}{2} \left( \sum_{i, j} P_{\theta_1, ij}^2 + \sum_{i, j} P_{\theta_2, ij}^2 \right)}
$$

This measure enables rigorous behavioral comparison between the Markov chains induced by each object type.

## 4. Algorithmic Construction from OCEL

The construction of MDFMG from an OCEL involves several steps:

1. **Counting directly-follows triples:** For each object $o$ of type $\theta(o)$, extract the sequence of events $\sigma_o$ sorted by timestamp, then for every consecutive pair of events $(e_i, e_{i+1})$, increment $f((a(e_i), \theta(o), a(e_{i+1})))$.
2. **Aggregating relations:** Form $R = \{(t, \theta, t') \mid f((t, \theta, t')) > 0\}$.
3. **Computing transition probabilities:** For every $(t, \theta, t')$ in $R$, compute $p((t, \theta, t'))$ as above.
4. **Building probability matrices:** Assemble $P_\theta$ for each object type.
5. **Calculating pairwise similarities:** Compute $sim(\theta_1, \theta_2)$ for all $\theta_1, \theta_2 \in OT$.

Efficient sparse-array or map-based data structures are recommended for $f$ and $P_\theta$. The $|OT| \times |OT|$ similarity matrix enables subsequent clustering.

## 5. Threshold-Based Clustering of Object Types

The similarity matrix $sim$ supports partitioning of object types into clusters that share control-flow behavior. The clustering procedure is single-linkage based:
- Each object type initially forms its own cluster.
- For every pair $(\theta_i, \theta_j)$ with $sim(\theta_i, \theta_j) \geq \tau$, their clusters are merged.

Selection of the threshold $\tau$ has critical impact on the number and composition of clusters. A half-interval search algorithm systematically identifies all breakpoints—values of $\tau$ where the number of clusters changes:
1. For given $\tau_{low}, \tau_{high}$, compute clusters at both thresholds.
2. If cardinalities agree, no further subdivision is required.
3. If not, test the midpoint $\tau_{mid}$, and recurse on $[\tau_{low}, \tau_{mid}]$ and $[\tau_{mid}, \tau_{high}]$.
4. Continue until all breakpoints are discovered.

This automated threshold-tuning framework removes reliance on ad-hoc analyst choices and exposes "meaningful" behavioral groupings inherent in the data.

## 6. Empirical Example and Interpretive Impact

Consider $OT = \{o, i, p\}$ and $T = \{\text{po}, \text{ca}, \text{pi}, \text{sp}, \text{sr}\}$, where $f$ yields different directly-follows frequencies by type (see [2206.11017]). Transition-probability matrices $P_o, P_i, P_p$ are computed and the similarity matrix constructed:

|        | o    | i    | p    |
|--------|------|------|------|
| **o**  | 1.00 | 0.76 | 0.00 |
| **i**  | 0.76 | 1.00 | 0.00 |
| **p**  | 0.00 | 0.00 | 1.00 |

- At $\tau=0$, all are merged: $\{o, i, p\}$.
- At $\tau=0.01$, $o$ and $i$ cluster ($sim=0.76 > 0.01$), $p$ is separate.
- At $\tau=0.77$, all are singletons.

Flattening the log on $\{o, i\}$ yields a simpler DFG than on $\{o, i, p\}$; isolating $\{p\}$ focuses on "shipping" behavior. This illustrates MDFMG’s utility in decomposing process complexity.

## 7. Evaluation and Significance

Application of MDFMG to a real Purchase-to-Pay (P2P) OCEL with nine object types and ~24,000 events revealed multiple behavioral groupings as the threshold varied—e.g., at $\tau=0.31$, object types EBELN and EBELN_EBELP merged, reflecting nearly identical control-flow; other types remained distinct. Footprint-based conformance checking via inductive miner validated these groupings, with low model-difference for merged clusters and pronounced divergence otherwise.

In summary, the Markov Directly-Follows Multigraph extends the DFG framework by:
1. Recording directly-follows behavior per object type, supporting the object-centric paradigm.
2. Assigning local transition probabilities and facilitating Markovian behavioral similarity analysis.
3. Enabling automated, data-driven clustering, reducing interpretational and technical complexity inherent in object-centric process mining contexts.

This approach substantially streamlines model discovery and analysis for data sets with multiple, intertwined case notions [2206.11017].

Source: https://www.emergentmind.com/topics/object-centric-directly-follows-graphs-oc-dfgs