Visually Representing History Dependencies in Event Logs

Abstract

Many process mining tools produce directly-follows graphs (DFG) as visual representations of event logs. While the “directly follows” relation is a good starting point for visualizations, there are simple phenomena it does not capture, for instance, when whether or not an event directly follows another event depends on the event directly preceding it. We call this a history dependency. This paper presents an empirical study of preferences for visualizing history dependencies: plain DFGs and two enhanced variants of DFGs (with additional arcs or rectangles) are evaluated. Our empirical study provides strong support for making an effort (to discover and) to explicitly visualize history dependencies. A ProM plug-in generating such explicit visualization is described in this paper.

A Rigorous Definitions

Let $\mathcal A$ be a finite set of activities without $\rhd$ and $\square$. A trace is a finite sequence $\rhd A_0 \dots A_n \square$ which satisfies $A_i \in \mathcal A$ for every $i \le n$. An event log is a finite non-empty set of traces.

A directly-follows graph with multiplicities (DFG+) consists of a finite set V of vertices, a finite set $E \subseteq V \times V$ of edges, and a labeling function $\lambda :V \rightarrow \mathcal A \cup \{\rhd , \square \}$ such that V together with E is a directed acyclic graph (DAG), there is exactly one vertex labeled $\rhd$, and this vertex is the only vertex with no incoming edge, there is exactly one vertex labeled $\square$, and this vertex is the only vertex with no outgoing edge. A DFG+ represents the event log which consists of the labelings of all paths through it that are traces.

A DFG+ is a directly-follows graph (DFG) when every event is the label of at most one vertex. A DFG+ is reduced if every vertex is on some path from the vertex labeled $\rhd$ to the vertex labeled $\square$ and there are no two distinct vertices with the same set of successor vertices.

Fact. For every event log there is, up to isomorphism, exactly one reduced DFG+ that represents it. Such a DFG+ is called canonical for the event log.

Definition. Let $\mathcal D$ be a canonical DFG+ for an event log. 1. The event log is history-free if $\mathcal D$ is a DFG.

2. Let k be a non-negative integer and $A \in \mathcal A$. There is a k-conflict for A if there are paths $v_0 v_1 \dots v_{k}$ and $w_0 w_1 \dots w_{k}$ in $\mathcal D$ such that $v_k \ne w_k$, $\lambda (v_i) = \lambda (w_i)$ for all $i < k$, and $A = \lambda (v_k) = \lambda (w_k)$. There is a k-history dependency ($k>0$) for A if there are no k-conflicts for it, but a $(k-1)$-conflict.