# Temporal Label Walk for Community Detection and Tracking in Temporal Network
## 4 The Effects of Parameters  
Three parameters, \(c\), \(\beta\) and \(\alpha\) should be set manually in TLW, where \(\alpha\) uses to control information intensity. A larger \(\alpha\) means that information received recently is more important. Using artificial data set generated in Section 3.1, label value corresponding to different \(\alpha\) is shown in Figure 11.  
Figure 11: The effect of \(\alpha\) on the label value of node 3. The larger \(\alpha\) makes node 2 have more influence on node 3 than the others.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Method** & **K-Clique** & **Asynchronous** & **Label Propagation** & **TLW** \\ \hline
1 day (s) & 0.021 \(\pm\) 0.001 & 0.030 \(\pm\) 0.021 & 0.024 \(\pm\) 0.002 & 0.049 \(\pm\) 0.005 \\
1 week (s) & 0.036 \(\pm\) 0.008 & 0.077 \(\pm\) 0.017 & 0.041 \(\pm\) 0.012 & 0.106 \(\pm\) 0.008 \\
1 month (s) & 0.098 \(\pm\) 0.006 & 0.108 \(\pm\) 0.009 & 0.097 \(\pm\) 0.007 & 0.121 \(\pm\) 0.007 \\
2 months (s) & 0.127 \(\pm\) 0.312 & 0.156 \(\pm\) 0.013 & 0.134 \(\pm\) 0.011 & 0.193 \(\pm\) 0.011 \\
3 months(s) & 0.264 \(\pm\) 0.008 & 0.284 \(\pm\) 0.075 & 0.340 \(\pm\) 0.132 & 0.421 \(\pm\) 0.013 \\
4 months (s) & 0.455 \(\pm\) 0.059 & 0.665 \(\pm\) 0.072 & 0.551 \(\pm\) 0.072 & 0.879 \(\pm\) 0.022 \\ \hline \end{tabular}
\end{table}
Table 3: The running time with the increase of \(\Delta t\) about different methods.  
Figure 10: Community structure of _Students_ and _Facebook_ in one day, one week, and one month, respectively. Nodes in black denote that they do not belong to any community obviously yet.  
\(\beta\) and \(c\) are time decay parameters which control the speed of exponential decay of information. \(c\) depends on time interval \(\tau\) of data streams. We generate a new data streams to illustrate the effect of different \(\beta\) and \(c\). Similar to the artificial data streams generated in Section 3.1, at each \(t_{i}\), now only one message passes in turn, we show this data streams in Figure 12.  
Let \(\alpha=0.1\). Fix \(\beta\) and \(c\) respectively, the first component of \(X_{3}(t)\) for different \(c\) and \(\beta\) is given in Figure 13a,b. Variation of label value from peak to trough is not violently with \(\beta=1,c=0.25\) or \(\beta=1,c=0.1\), here \(\tau=t_{i+1}-t_{i}\) is 1 s. \(c\) is more important than \(\beta\) because \(\tau\) range from 1 s to 1000 s or more in real data sets, it is suitable to consider \(c\) and \(\tau\) together so that \(e^{c\tau}\) can be a number between 0 and 1.  
To make a further analysis of influence of different parameters on community detection, we choose \(Students\) data set of the first day and fixing \(\beta=1\). Using Mini Batch K-means methods and setting the number of clusters \(n=14\), we show the effect of different \(c\) and \(\alpha\) on modularity with Mini Bath  
Figure 12: Three nodes send messages in turn. The order is \(2\xrightarrow{t_{3}}1,1\xrightarrow{t_{3}+1}3,3\xrightarrow{t_{3}+2}2,k=0,1, \cdots,n\).  
Figure 13: Fix \(\beta\) and \(c\) respectively, and calculate the first component of \(X_{3}(t)\). Two parameters are set to 0.25, 0.5, 0.75 and 1. Smaller \(c\) and larger \(\beta\) both have smaller peak value and larger valley value.  
K-means in Table 4. Please note that modularity does not change linearly with \(c\) and \(\alpha\), but depend on the data sets.