Object-Centric Directly-Follows Graphs

• OC-DFGs are advanced modeling constructs that capture multi-object event sequences by extending traditional DFGs to handle object-centric data.
• The Markov Directly-Follows Multigraph (MDFMG) assigns local transition probabilities, enabling quantitative comparisons of control-flow behavior across object types.
• Automated threshold-based clustering using cosine-like similarity measures groups object types with similar patterns, enhancing scalability and interpretability.

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 (Jalali, 2022).

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 $T$0 logs an activity $T$1 and can refer to multiple objects. Every object $T$2 possesses a type $T$3, where $T$4 denotes the set of object types. Rather than treating each trace as a case, OCEL enables extraction of a local event sequence $T$5—the ordered list of events relevant to each object $T$6.

The Directly-Follows Multigraph (DFM), the "object-centric DFG," is formally defined as $T$7 with:

• $T$8 encoding directly-follows relations for each object type: $T$9 represents activity $c$0 directly preceding $c$1 for objects of type $c$2.
• $c$3 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 $c$4, define the local transition probability

$c$5

ensuring that, for fixed $c$6, the outgoing probabilities sum to 1.

Each object type $c$7 is thus associated with a row-stochastic matrix $c$8 where entries $c$9 give the probability of $G = (T, E, f)$0 directly preceding $G = (T, E, f)$1 for this type.

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

$G = (T, E, f)$2

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 $G = (T, E, f)$3 of type $G = (T, E, f)$4, extract the sequence of events $G = (T, E, f)$5 sorted by timestamp, then for every consecutive pair of events $G = (T, E, f)$6, increment $G = (T, E, f)$7.
2. Aggregating relations: Form $G = (T, E, f)$8.
3. Computing transition probabilities: For every $G = (T, E, f)$9 in $t \in T$0, compute $t \in T$1 as above.
4. Building probability matrices: Assemble $t \in T$2 for each object type.
5. Calculating pairwise similarities: Compute $t \in T$3 for all $t \in T$4.

Efficient sparse-array or map-based data structures are recommended for $t \in T$5 and $t \in T$6. The $t \in T$7 similarity matrix enables subsequent clustering.

5. Threshold-Based Clustering of Object Types

The similarity matrix $t \in T$8 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 $t \in T$9 with $E \subseteq T \times T$0, their clusters are merged.

Selection of the threshold $E \subseteq T \times T$1 has critical impact on the number and composition of clusters. A half-interval search algorithm systematically identifies all breakpoints—values of $E \subseteq T \times T$2 where the number of clusters changes:

1. For given $E \subseteq T \times T$3, compute clusters at both thresholds.
2. If cardinalities agree, no further subdivision is required.
3. If not, test the midpoint $E \subseteq T \times T$4, and recurse on $E \subseteq T \times T$5 and $E \subseteq T \times T$6.
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 $E \subseteq T \times T$7 and $E \subseteq T \times T$8, where $E \subseteq T \times T$9 yields different directly-follows frequencies by type (see (Jalali, 2022)). Transition-probability matrices $(t, t')$0 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 $(t, t')$1, all are merged: $(t, t')$2.
• At $(t, t')$3, $(t, t')$4 and $(t, t')$5 cluster ($(t, t')$6), $(t, t')$7 is separate.
• At $(t, t')$8, all are singletons.

Flattening the log on $(t, t')$9 yields a simpler DFG than on $t$0; isolating $t$1 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 $t$2, 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 (Jalali, 2022).