# Investigating Social Contextual Factors
## 2 Materials and Methods
### Definitions
#### 2.1.2 Survival Functions and Social Networks  
**Definition 9**. : _Survival Function. Let \(L\) represent an event log with a set of trace cycle times \(\{\tau_{cyc.1}\ldots\ \tau_{cyc.n}\}\), a trace \(\sigma_{i}\in L\) with cycle time \(\tau i.cyc\) and a random time, \(t_{r}\), the survival function \(S(t)=P\ (\tau i.cyc>t_{r})\). It gives the probability that the random time, \(t_{r}\) exceeds the trace cycle time._  
Weibull model: Let \(T=t\) denote the time-to-completion of a trace \(\sigma_{i},f(t)\) the probability density function of \(T\), the probability density function of the Weibull model is given by  
\[f(t)=\alpha\lambda t^{\alpha-1}\ e^{-\lambda t^{\alpha}}\]  
where \(\lambda>0\) represents the trace completion rate parameter, and \(\alpha>0\) represents the scale or shape parameter. **Definition 10**. : _Handover of work. Let \(P\) represent the set of performers, \(E\) represents a set of directed edges and \(\phi:E\to V^{2}\) represent an incidence function mapping edges to vertices._  
A handover-of-work graph is a directed multigraph permitting loops \(\mathrm{G}=(P,E,\phi)\). For our study, the incidence function maps the handover of work from one performer to another. A handover of work from performer \(a\) to performer \(b\) occurs if there are subsequent events (\(e_{i}\) and \(e_{i+1}\)) and \(a\) completes #activity(\(e_{i}\)), while \(b\) completes #activity(\(e_{i+1}\)). Note that the incidence function permits a performer to hand over work to themself, i.e., complete #activity (\(e_{i}\)) and (\(e_{i+1}\)). **Definition 11**. : _Group Centrality Measures. Let \(L\) represent an event log, \(P\) represents the set of performers and \(\mathrm{G}\) represent a handover of work graph derived from the log. For a trace \(o_{i}\in L\), \(\mathrm{X}=\{\#\) performer (\(\mathrm{e_{1}}\))...... \(\#\) performer (\(e_{n}\)). This denotes the subset of performers who completed the activities in a trace._  
Group degree centrality: The group degree centrality of \(\mathrm{X}\) is defined as follows:  
\[\mathrm{GD}(\mathrm{X})=\big{|}v:(u,v)\in E\wedge u\in X\wedge v\
otin P\big{|},\ X\subseteq P \tag{2}\]  
If the group is defined as the subset of performers who worked on the trace, the group degree centrality specifies the number of non-group members that are connected to group members. Group between centrality: Let \(g_{u,v}\) represent the number of geodesics connecting vertices \(u\) to \(v\) and \(g_{u,v}(X)\) represent the number of geodesics between \(u\) and \(v\) passing through some vertex of \(\mathrm{X}\). The group betweenness centrality of \(X\) is defined as follows:  
\[\mathrm{GB}(\mathrm{X})=\sum_{\{u<v\}}\frac{g_{u,v}(X)}{g_{u,v}},\ \mathrm{u},v \
otin X \tag{3}\]  
This measure shows \"the proportion of geodesics connecting pairs of non-group members that pass through the group\" [21]. Group closeness centrality: The group closeness centrality is defined as follows:  
\[\mathrm{GC}(\mathrm{X})=\frac{|P|\mathrm{X}|}{\sum_{v\in P|\mathrm{X}}dx_{v}},\ X\subseteq P \tag{4}\]  
where \(dx_{v}\) denotes the distance between \(X\) and a vertex \(v\) defined as \(dx_{v}=min_{u\in X}\ dist_{u,v}\), where \(dist_{u,v}\) is the shortest path between \(u\) and \(v\). This measures how close group members are to other non-group members. Group eigenvector centrality: Let \(\mathrm{X}^{\bullet}\) represent a super vertex such \(\mathrm{N}(\mathrm{X}^{\bullet})=\mathrm{N}(x_{1})\mathrm{UN}(x_{2})\)... \(\mathrm{UN}(x_{n})\) where \(\mathrm{N}()\) denotes the neighbourhood of the vertex and \(\mathrm{x}_{i}\) denotes the members of the set \(\mathrm{X}\). The group eigenvector centrality is defined as follows:  
\[\mathrm{GE}(\mathrm{X}^{\bullet})=\frac{1}{\lambda}\sum_{j\in m_{(i)}}x_{j} \tag{5}\]  
where \(m_{(i)}\) is a set of neighbours of \(\mathrm{X}^{\bullet}\) and \(\lambda\) is a constant  
This is a measure of how connected the members of the group are to influential vertices outside the group.