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\title{A Mutually Exciting Latent Space Hawkes Process Model\\
for Continuous-time Networks (Supplementary material)}


\author[1]{\href{mailto:<zhipeng.huang@rockets.utoledo.edu>?Subject=Your UAI 2022 paper}{Zhipeng Huang}{}}
\author[1]{Hadeel Soliman}
\author[2]{Subhadeep Paul}
\author[1]{Kevin S. Xu}

% Add affiliations after the authors
\affil[1]{%
    Department of Electrical Engineering and Computer Science\\
    University of Toledo\\
    Toledo, OH, USA
}
\affil[2]{%
    Department of Statistics\\
    The Ohio State University\\
    Columbus, OH, USA
}

  
  \begin{document}

\onecolumn
\maketitle

\begin{abstract}
Networks and temporal point processes serve as fundamental building blocks for modeling complex dynamic relational data in various domains. 
We propose the \emph{latent space Hawkes (LSH)} model, a novel generative model for continuous-time networks of relational events, using a latent space representation for nodes. 
We model relational events between nodes using mutually exciting Hawkes processes with baseline intensities dependent upon the distances between the nodes in the latent space and sender and receiver specific effects. 
We demonstrate that our proposed LSH model can replicate many features observed in real temporal networks including reciprocity and transitivity,  while also achieving superior prediction accuracy and providing more interpretable fits than existing models.
\end{abstract}



\appendix


\section{Additional Model and Estimation Details}

\subsection{Proof of Theorem \ref{thm:identifiability}}
\label{sec:supp_proof}

\begin{proof}
Suppose two sets of parameters $\theta_2, Z,\gamma,\delta$ and $\theta_2', Z',\gamma',\delta'$ lead to the same $\log (\mu)$:
\begin{equation*}
\theta_2 J_n + ZZ^T + \tilde{\delta}1_n^T + 1_n \tilde{\gamma}^T = \theta_2' J_n + Z'Z'^T + \tilde{\delta}'1_n^T + 1_n \tilde{\gamma}'^T.
\end{equation*}
Left multiplying both sides by $1_n^T$ and noting that (i) $1_n^T Z = 1_n^T H Z = 1_n^T (I_n - \frac{1}{n}1_n 1_n^T)Z = 0$ by assumption 1 and (ii) $1_n^T \tilde{\delta}=0$ by assumption 2,  we get
\begin{align}
& 1_n^T\theta_2 J_n  + 1_n^T 1_n \tilde{\gamma}^T = 1_n^T \theta_2' J_n  + 1_n^T 1_n \tilde{\gamma}'^T \nonumber\\
& \Rightarrow n (\theta_2 - \theta_2')1_n^T  + n(\tilde{\gamma}^T - \tilde{\gamma}'^T) = 0
\label{eq:identify}
\end{align}
Now, right multiplying by $1_n$ we get 
\begin{equation*}
n^2 (\theta_2 - \theta_2') =0
\end{equation*}
because $\gamma^T 1_n = 0$ and $\tilde{\gamma}^T 1_n = 0$ by our identifiability constraints.
This implies that $\theta_2 = \theta_2'$. With this, using \ref{eq:identify}, we have
\begin{equation*}
n(\tilde{\gamma}^T - \tilde{\gamma}'^T) = 0 \Rightarrow \tilde{\gamma} =\tilde{\gamma}'.
\end{equation*}
Finally, right multiplying by $1_n$ from the beginning and then using the above result $\theta_2 = \theta_2'$ we have
\begin{align*}
& \theta_2 J_n1_n  + \tilde{\delta} 1_n^T 1_n  =  \theta_2' J_n 1_n +  \tilde{\delta}' 1_n^T1_n \\
& \Rightarrow  n(\tilde{\delta} - \tilde{\delta}') = 0, \quad \Rightarrow \tilde{\delta} = \tilde{\delta}'.
\end{align*}
In light of the above results we then conclude
\[
ZZ^T = Z'Z'^T. 
\]
which means $Z=Z'O$ where $O_{d \times d}$ is an orthogonal matrix such that $OO^T =I$.
Furthermore, from the results of \citet{ozaki1979maximum}, the baseline intensity $\mu_{uv}$ and jump size parameters $\alpha_1, \alpha_2$ are
identified, i.e.~if two sets of $\mu_{uv}, \alpha_1, \alpha_2$ lead to the same probability density function (log-likelihood), then the two
sets must be identical.
\end{proof}

\subsection{Full log-likelihood expression}
\label{sec:appendix_ll}


Let $\Lambda_{uv}(t_k^{uv}) = \int_0^{t_k} \lambda_{uv}^* (t) \,dt$. We can break the time interval $[0,t_k]$ to $[0,t_1],[t_1, t_2], \ldots , [t_{k-1}, t_k]$, resulting in
\begin{equation}
\begin{split}
\Lambda_{uv}(t_k^{uv}) & = \int_0^{t_k^{uv}} \lambda_{uv}^* (t) \,dt \\
&= \int_0^{t_1^{uv}} \lambda_{uv}^*(t)\,dt + \sum^{k-1}_{i=1} \int^{t_{i+1}^{uv}}_{t_i^{uv}}  \lambda^*_{uv}(t)\,dt  \\
                  & = \int_0^{t_1^{uv}}\mu_{uv} \,dt + \sum^{k-1}_{i=1} \int^{t_{i+1}^{uv}}_{t_i^{uv}} \left[\mu_{uv} +  \sum_{j:t^{uv}_j < t_i^{uv}}\sum_b^B C_b\alpha_1\beta_b e^{-\beta_b(t-t^{uv}_j)} + \sum_{j:t^{vu}_j < t^{uv}_i}\sum_b^B C_b\alpha_2\beta_b e^{-\beta_b(t-t^{vu}_j)}\right]dt \\
                  & = \mu_{uv}t_k^{uv} + \sum^{k-1}_{i=1} \int^{t_{i+1}^{uv}}_{t_i^{uv}}\sum_b^B \left[ \sum_{j:t^{uv}_j < t_i^{uv}} C_b\alpha_1\beta_b e^{-\beta_b(t-t^{uv}_j)} + \sum_{j:t^{vu}_j < t^{uv}_i} C_b\alpha_2\beta_b e^{-\beta_b(t-t^{vu}_j)}\right]dt \\
                  & = \mu_{uv}t_k^{uv} + \sum^{k-1}_{i=1}\sum_{j:t^{uv}_j < t_i^{uv}}\sum_b^B \int^{t_{i+1}^{uv}}_{t_i^{uv}} \left[  C_b\alpha_1\beta_b e^{-\beta_b(t-t^{uv}_j)}\right] dt +    \sum^{k-1}_{i=1}\sum_{j:t^{vu}_j < t^{uv}_i}\sum_b^B \int^{t_{i+1}^{uv}}_{t_i^{uv}}\left[ C_b\alpha_2\beta_b e^{-\beta_b (t-t^{vu}_j)}\right]dt \\
                  & = \mu_{uv}t_k^{uv} - \sum^{k-1}_{i=1}\sum_b^B\sum_{j:t^{uv}_j < t_i^{uv}}{C_b\alpha_1} \left[ e^{-\beta_b(t_{i+1}^{uv}-t^{uv}_j)} - e^{-\beta_b(t_i^{uv}-t^{uv}_j)}\right] \\
                  &\qquad\qquad\qquad - \sum^{k-1}_{i=1}\sum_b^B\sum_{j:t^{vu}_j < t^{uv}_i}{C_b\alpha_2} \left[  e^{-\beta_b(t_{i+1}^{uv}-t^{vu}_j)} - e^{-\beta_b(t_i^{uv}-t^{vu}_j)}\right]
\end{split}
\end{equation}

By extending the summation, many of terms can cancel out, and then we can simplify $\Lambda_{uv}(t_k^{uv})$ as follows:
\begin{equation} \label{eq:4}
\Lambda_{uv}(t_k^{uv}) = \mu _{uv} t_k^{uv} - \sum_b^B C_b\alpha_1 \sum_{j:t^{uv}_j<t_k^{uv}} \left[ e^{-\beta_b(t_k^{uv} - t^{uv}_j)} -1 \right] - \sum_b^B C_b\alpha_2 \sum_{j:t^{vu}_j<t_k^{uv}} \left[ e^{-\beta_b(t_k^{uv} - t^{vu}_j)} -1 \right]. 
\end{equation}
Substituting \eqref{eq:4} into \eqref{eq:3} gives the following simplified expression for the log-likelihood.
\begin{equation} \label{eq:5}
\begin{split}
l = \sum_{u\neq v} \Bigg\{ & \sum_{i=1}^k \log\left[\mu_{uv} +  \sum_{j:t^{uv}_j < t_i^{uv}}\sum_b^B  C_b\alpha_1\beta_b e^{-\beta_b(t_i^{uv}-t^{uv}_j)} + \sum_{j:t^{vu}_j < t^{uv}_i} \sum_b^B C_b\alpha_2\beta_b e^{-\beta_b(t^{uv}_i-t^{vu}_j)}\right] \\
    & -\mu _{uv} t_k^{uv} + \sum_b^B C_b\alpha_1 \sum_{j:t^{uv}_j<t_k^{uv}} \left[ e^{-\beta_b(t_k^{uv} - t^{uv}_j)} -1 \right] - \sum_b^B C_b\alpha_2 \sum_{j:t^{vu}_j<t_k^{uv}} \left[ e^{-\beta_b(t_k^{uv} - t^{vu}_j)} -1 \right]   \Bigg\}.
\end{split}
\end{equation}
We use the negative of the log-likelihood expression in \eqref{eq:5} as the objective function for the L-BFGS-B minimization.


\subsection{Alternating Minimization}
\label{sec:alternating}

An alternative approach to obtaining the maximum likelihood estimate (MLE) is to partition our parameters into two sets: the latent node positions $Z$ and all other parameters $\Theta = (\alpha_1, \alpha_2, \theta_1, \theta_2, \delta, \gamma)$. 
We propose an alternating minimization approach to obtain the MLE. 
It alternates between optimizing the NLL over the estimated latent positions $\hat{Z}$ while holding all other parameters fixed and optimizing all other parameters $\hat{\Theta}$ while holding the latent positions fixed. 

We run each minimization over a fixed number of iterations and then switch to the other minimization. 
We experiment with different values for the number of steps $(s_{\Theta}, s_Z)$ denoting the number of iterations to run the optimization over $\Theta$ and $Z$, respectively. 
We find that there is not much difference in the performance for different numbers of steps for the directions in alternate minimization.  Taking 2 steps in each directions, i.e.~$s_{\Theta} = 2, s_Z = 2$, seems to work well.
Pseudocode for our alternating minimization estimation procedure is shown in Algorithm \ref{alg:alt_min}. 



\begin{algorithm}[t]
   \caption{Alternating minimization estimation algorithm}
   \label{alg:alt_min}
\begin{algorithmic}[1]
   \STATE {\bfseries Input:} Relational event triplets $(u, v, t)$, latent dimension $d$, kernel decays $\beta$, kernel scale parameters $C$
   \STATE Initialize $ \hat{Z}$ using multidimensional scaling
   \STATE Initialize $ \hat{\Theta}$ randomly
   \REPEAT
   \STATE $\hat{\Theta}\leftarrow$ $s_\theta$ steps of L-BFGS-B over $\hat{\Theta}$ on log-likelihood \eqref{eq:5} while keeping $\hat{Z}$ fixed\;
   \STATE $\hat{Z}\leftarrow$ $s_Z$ steps of L-BFGS-B over $\hat{Z}$ on log-likelihood \eqref{eq:5} while keeping $\hat{\Theta}$ fixed\;
   \UNTIL{log-likelihood \eqref{eq:5} converges}
   \STATE {\bfseries Return:} Estimated model parameters $(\hat{Z}, \hat{\Theta})$
\end{algorithmic}
\end{algorithm}

\section{Additional Experiment Results}


\subsection{Simulated Networks}
\label{sec:sim_appendix}


The generative process for the simulated networks is shown in Algorithm \ref{alg:sim_exp}.
As discussed in Theorem \ref{thm:identifiability}, the latent positions $Z$ are only identifiable up to a rotation. 
Furthermore, the slope parameter $\theta_1$ is not identifiable, so we absorb it into the latent positions by setting
\begin{equation*}
\hat{Z} \leftarrow \hat{Z} \sqrt{|\hat{\theta}_1}| \qquad \text{and} \qquad \hat{\theta}_1 \leftarrow \frac{\hat{\theta}_1}  {\sqrt{|\hat{\theta}_1|}}. 
\end{equation*}
Then, to compare the estimated and actual latent positions, we apply a Procrustes transform \citep{gower1975generalized} to the estimated latent positions $\hat{Z}$. 

\begin{algorithm}[t]
   \caption{Generative procedure for Latent space Hawkes network}
   \label{alg:sim_exp}
\begin{algorithmic}[1]
   \STATE {\bfseries Input:} Number of nodes $n$, time duration $T$, latent dimension $d$, kernel decays $\beta$, kernel scaling parameters $C$, model parameters $(\theta_1,\theta_2, \alpha_1, \alpha_2)$
   \STATE Sample latent positions $Z$ and sender and receiver effects $\delta$, $\gamma$ from Normal distributions: $Z\leftarrow \mathcal{N}(0,\sigma_z^2I_{nd}), \delta \leftarrow \mathcal{N}(0,\sigma_\delta^2I_n), \gamma \leftarrow \mathcal{N}(0,\sigma_\gamma^2I_n)$
   \STATE Set $Z \leftarrow Z \sqrt{|\theta_1}|, \theta_1 \leftarrow \theta_1 / \sqrt{|\theta_1|}$ to remove identifiability issues
   \FORALL{node pairs $u \neq v$}
   \STATE $ \mu_{uv} \leftarrow e^{-\theta_1||z_u - z_v||^2_2 + \theta_2 + \delta_u
    +\gamma_v}$
   \STATE $\mathcal{H}(u,v) \leftarrow$ Ogata's thinning algorithm ($\mu_{uv}, \alpha_1, \alpha_2, T, \beta, C$)\;
   \ENDFOR

   \STATE {\bfseries Return:} Simulated network $\mathcal{H}$ containing events $\mathcal{H}(u,v)$ for all $u \neq v$
\end{algorithmic}
\end{algorithm}

The root mean squared error (RMSE) for the latent positions, each of the model parameters, and baseline intensities $\mu_{uv}$ for all node pairs is shown in Figure \ref{fig:sim1_supp}. 
As the time duration $T$ increases, more events are generated. 
Notice that the errors are all decreasing with increasing $T$ as one would expect.


\begin{figure*}[p]
\centering %centers figure
    \begin{subfigure}[c]{2.5in}
        \newcommand{\figwidth}{2.5in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/RMSE_simulation/RMSE_z.pdf}
        \caption{latent positions $z$}
    \end{subfigure}
    \begin{subfigure}[c]{2.5in}
        \newcommand{\figwidth}{2.5in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/RMSE_simulation/RMSE_delta_1.pdf}
        \caption{sender effects $\delta$}
    \end{subfigure}
        \begin{subfigure}[c]{2.5in}
        \newcommand{\figwidth}{2.5in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/RMSE_simulation/RMSE_delta_2.pdf}
        \caption{receiver effects $\gamma$}
    \end{subfigure}
    \begin{subfigure}[c]{2.5in}
        \newcommand{\figwidth}{2.5in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/RMSE_simulation/RMSE_theta_1.pdf}
        \caption{intercept parameter $\theta_2$}
    \end{subfigure}
    \begin{subfigure}[c]{2.5in}
        \newcommand{\figwidth}{2.5in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/RMSE_simulation/RMSE_alpha_1.pdf}
        \caption{jump size for self-exciting term $\alpha_1$}
    \end{subfigure}
    \begin{subfigure}[c]{2.5in}
        \newcommand{\figwidth}{2.5in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/RMSE_simulation/RMSE_alpha_2.pdf}
        \caption{jump size for reciprocal term $\alpha_2$}
    \end{subfigure}
    \begin{subfigure}[c]{2.5in}
        \newcommand{\figwidth}{2.5in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/RMSE_simulation/RMSE_mu.pdf}
        \caption{baseline intensity $\mu$}
    \end{subfigure}
\caption{Root mean square error over 30 simulated networks ($\pm$ 2 standard errors) for different parameters}
\label{fig:sim1_supp}
\end{figure*}


\subsection{Real Network Dataset Descriptions}
\label{sec:supp_datasets}
We perform experiments on several real network datasets: 
\begin{itemize}
  \item Reality mining dataset \citep{eagle2006reality} is derived from the Reality Commons project from 14 September 2004 to 5 May 2005. The dataset we use contains 65 students and 2150 communications (Denote as Reality).
  \item Enron email dataset \citep{klimt2004enron} consists communications among 155 Enron individuals. It contains 9,646 email message spanning a period of 453 days. This dataset is tested in Yang's paper for their DLS model (Denote as Enron)
  \item The Militarized Interstate Disputes (MID) dataset \citep{palmer2021mid5} consists of 145 countries with 5088 Disputes among them with a period of 8380 days (Denote as MID). 
  \item Facebook-forum dataset \citep{nr} consists of 899 students posted 33,720 broadcast messages in the forum over 165 days (Denote as FB-forum).
\end{itemize}

For the Reality, FB-forum, and MID datasets, we re-scale the timestamps so that they are in all $[0, 1000]$ in the same manner as \citet{arastuie2020chip}. For Enron, we keep the same scale as \citet{yang2017decoupling} to make a fair comparison against the DLS model. 

\subsection{Descriptions of Other Models}
\label{sec:supp_models}

\paragraph{Dual Latent Space (DLS) Model}
We use the implementation at \url{https://github.com/jiaseny/lspp} for the DLS model. 
We provide a detailed comparison of the DLS model with our proposed LSH model in Section \ref{sec:relation_DLS}. 
It adopts a bivariate Hawkes process and latent space-based approach to capture homophily and reciprocity of continuous-time dynamic networks. Unlike our proposed LSH model, the DLS model does not have the self-excitation term. Their reciprocal terms are parameterized by multiple latent spaces associated with different decay values.

\paragraph{Hawkes Process-based Block Models}
We use the implementation at \url{https://github.com/IdeasLabUT/CHIP-Network-Model} for the Community Hawkes Independent Pairs (CHIP) model  \citep{arastuie2020chip} and the Block Hawkes Model (BHM) \citep{junuthula2019block}. 
CHIP is a univariate Hawkes process network model with block structure where each node pair is independent of all others. 
The BHM is also a univariate Hawkes process network model with block structure; however, an event between a node pair equally excites all node pairs in the same block pair.

\paragraph{Continuous-time Dynamic Network Embeddings (CTDNE)}
We used the same hyperparameters ($d=128, R=10, L=80, \omega=10$) as mentioned in \citet{nguyen2018continuous}. We used the implementation from the StellarGraph package \citep{StellarGraph}. We used timestamps up to $t_i$ (beginning of test window) to create both temporal walks and the classifier's positive and negative examples. Edges feature vector is computed using weighted-L2 operation. To create walks, we test using directed/undirected graphs, also varied the neighbor selection distribution between biased(exponential)/unbiased. Best results are reported for each dataset.





\subsection{Dynamic Link Prediction}
\label{sec:supp_dynamic_lp}

\begin{figure}[t]
\centering
\includegraphics[width=4in]{Figures/Dynamic link prediction/boxcompare.pdf}
\caption{AUC values for dynamic link prediction over 100 random time intervals}
\label{fig:dlp_boxplots}
\end{figure}


In section \ref{sec:real_experiment}, we perform dynamic link prediction experiments. In this section, we demonstrate more results of the AUC values and the ROC curves over 100 time intervals for different models and different real world datasets. 

Figure \ref{fig:dlp_boxplots} shows the boxplot of the AUC values for dynamic link prediction over 100 random time intervals. 
The DLS model does not scale to the fb-forum dataset. 
The box plot indicates that our proposed model achieves the best in MID and fb-forum dataset and is competitive in Reality and Enron datasets. Moreover, in the fb-forum, MID, and Enron datasets, the dispersion in AUC values from our model is less as evidenced by the low width of the boxes. In all datasets, the bulk of the distribution of AUC values for our LSH model is above 90\% indicating a superior performance in the dynamic link prediction task.

Figure \ref{fig:dlp_reality}-\ref{fig:dlp_fb-forum} demonstrate the corresponding ROC curves for different models and real world datasets. The curves matches what we observed in Table \ref{tab:predictive_accuracy} that our proposed model outperforms other models on MID and fb-forum and is competitive on Reality and Enron.



\begin{figure}[p]
\newcommand{\figwidth}{3.0in}
\centering %centers figure
    \begin{subfigure}[c]{\figwidth}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/Dynamic link prediction/LSH_ROC_reality.pdf}
        \caption{ROC curve of LSH model for Reality Mining}
    \end{subfigure}
    \begin{subfigure}[c]{\figwidth}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/Dynamic link prediction/DLS_reality_auc.pdf}
        \caption{ROC curve of DLS model for Reality Mining}
    \end{subfigure}
    
    \begin{subfigure}[c]{\figwidth}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/Dynamic link prediction/CHIP_AUC_reality.pdf}
        \caption{ROC curve of CHIP model for Reality Mining}
    \end{subfigure}
    \begin{subfigure}[c]{\figwidth}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/Dynamic link prediction/RealityMining_BHM_ROC.pdf}
        \caption{ROC curve of BHM model for Reality Mining}
    \end{subfigure}
    
\caption{Dynamic link prediction on 100 random time intervals on Reality Mining}
\label{fig:dlp_reality}
\end{figure}


\begin{figure}[p]
\centering %centers figure
    \begin{subfigure}[c]{3.0in}
        \newcommand{\figwidth}{3.0in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/Dynamic link prediction/LSH_ROC_Enron-Yang.pdf}
        \caption{ROC curve of LSH model for Enron}
    \end{subfigure}
    \begin{subfigure}[c]{3.0in}
        \newcommand{\figwidth}{3.0in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/Dynamic link prediction/DLS_Enron-Yang_auc.pdf}
        \caption{ROC curve of DLS model for Enron}
    \end{subfigure}
    
    \begin{subfigure}[c]{3.0in}
        \newcommand{\figwidth}{3.0in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/Dynamic link prediction/CHIP_AUC_Enron-Yang.pdf}
        \caption{ROC curve of CHIP model for Enron}
    \end{subfigure}
    \begin{subfigure}[c]{3.0in}
        \newcommand{\figwidth}{3in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/Dynamic link prediction/Enron-15_BHM_ROC.pdf}
        \caption{ROC curve of BHM model for Enron}
    \end{subfigure}
\caption{Dynamic link prediction on 100 random time intervals on Enron}
\label{fig:dlp_enron}
\end{figure}

\begin{figure}[p]
\centering %centers figure
    \begin{subfigure}[c]{3.0in}
        \newcommand{\figwidth}{3.0in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/Dynamic link prediction/LSH_ROC_MID.pdf}
        \caption{ROC curve of LSH model for MID}
    \end{subfigure}
    \begin{subfigure}[c]{3.0in}
        \newcommand{\figwidth}{3.0in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/Dynamic link prediction/DLS_MID_auc.pdf}
        \caption{ROC curve of DLS model for MID}
    \end{subfigure}
    
    \begin{subfigure}[c]{3.0in}
        \newcommand{\figwidth}{3.0in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/Dynamic link prediction/CHIP_AUC_MID.pdf}
        \caption{ROC curve of CHIP model for MID}
    \end{subfigure}
    \begin{subfigure}[c]{3.0in}
        \newcommand{\figwidth}{3.0in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/Dynamic link prediction/MID_BHM_ROC.pdf}
        \caption{ROC curve of BHM model for MID}
    \end{subfigure}
\caption{Dynamic link prediction on 100 random time intervals on MID}
\label{fig:dlp_mid}
\end{figure}

\begin{figure}[p]
\centering %centers figure
    \begin{subfigure}[c]{3.0in}
        \newcommand{\figwidth}{3.0in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/Dynamic link prediction/LSH_ROC_fb-forum.pdf}
        \caption{ROC curve of LSH model for fb-forum}
    \end{subfigure}

    \begin{subfigure}[c]{3.0in}
        \newcommand{\figwidth}{3.0in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/Dynamic link prediction/CHIP_AUC_fb-forum.pdf}
        \caption{ROC curve of CHIP model for fb-forum}
    \end{subfigure}
    \begin{subfigure}[c]{3.0in}
        \newcommand{\figwidth}{3.0in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/Dynamic link prediction/fb-forum_BHM_ROC.pdf}
        \caption{ROC curve of BHM model for fb-forum}
    \end{subfigure}
\caption{Dynamic link prediction on 100 random time intervals on fb-forum}
\label{fig:dlp_fb-forum}
\end{figure}



\subsection{Posterior Predictive Checks}
\label{sec:supp_ppc}
As we discussed in Section \ref{sec:generative_accuracy}, we simulate 15 networks from the fitted model and demonstrate the PPCs on the statistics in Table \ref{tab:generative_accuracy}. We show the corresponding histogram of the PPCs for Reality Mining in Figures \ref{fig:ppc_lsh} and \ref{fig:ppc_dls} for LSH and DLS, respectively. The actual value of the statistics is plotted as the blue vertical line. The red vertical line indicates the mean value of the statistics observed over the 15 simulated networks. Each figure consists of five subplots. The first, third, fourth, and fifth subplots give the histograms of average clustering coefficient, counts of events, reciprocity, and transitivity observed over the 15 simulated networks. The second subplot in each figure shows the histogram of the degree distribution. The left is the actual degree distribution, and the right is the degree distribution of the mean degree for each node of 15 simulated networks. 

In general, the LSH performs quite well on generating average clustering coefficients, number of events, reciprocity, transitivity, and degree distribution that are similar to what is observed in the corresponding real dataset. The DLS performs well on the average clustering coefficients, the transitivity, and the degree distribution but simulates orders of magnitude more events. The DLS model also fails to simulate the high reciprocity observed in the data. 



\begin{figure*}[p]
\centering %centers figure
    \begin{subfigure}[c]{3.0in}
        \newcommand{\figwidth}{3.0in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/PPC/LSH_coef_reality.pdf}
        \caption{Histogram of average clustering coefficients}
    \end{subfigure}
    \begin{subfigure}[c]{3.0in}
        \newcommand{\figwidth}{3.0in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/PPC/LSH_degree_distribution_reality.pdf}
        \caption{Histogram of degree distribution for real network(left) and mean of 15 simulated networks (right)}
    \end{subfigure}
    \begin{subfigure}[c]{3in}
        \newcommand{\figwidth}{3in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/PPC/LSH_events_reality.pdf}
        \caption{Histogram of number of counts}
    \end{subfigure}
        \begin{subfigure}[c]{3in}
        \newcommand{\figwidth}{3in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/PPC/LSH_reciprocity_reality.pdf}
        \caption{Histogram of reciprocity}
    \end{subfigure}
        \begin{subfigure}[c]{3in}
        \newcommand{\figwidth}{3in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/PPC/LSH_trans_reality.pdf}
        \caption{Histogram of transitivity}
    \end{subfigure}
\caption{Histogram of 15 simulations from the LSH model fitted to Reality mining (blue line: actual value; red line: mean of 15 simulated values).}
\label{fig:ppc_lsh}
\end{figure*}

\begin{figure*}[p]
\centering %centers figure
    \begin{subfigure}[c]{3.0in}
        \newcommand{\figwidth}{3.0in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/PPC/DLS_coef_reality.pdf}
        \caption{Histogram of average clustering coefficients}
    \end{subfigure}
    \begin{subfigure}[c]{3.0in}
        \newcommand{\figwidth}{3.0in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/PPC/DLS_degree_distribution_reality.pdf}
        \caption{Histogram of degree distribution for real network(left) and mean of 15 simulated networks (right)}
    \end{subfigure}
    \begin{subfigure}[c]{3in}
        \newcommand{\figwidth}{3in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/PPC/DLS_events_reality.pdf}
        \caption{Histogram of number of counts}
    \end{subfigure}
        \begin{subfigure}[c]{3in}
        \newcommand{\figwidth}{3in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/PPC/DLS_reciprocity_reality.pdf}
        \caption{Histogram of reciprocity}
    \end{subfigure}
        \begin{subfigure}[c]{3in}
        \newcommand{\figwidth}{3in}
        \centering
        \hfill
        \includegraphics[width=\figwidth]{Figures/PPC/DLS_trans_reality.pdf}
        \caption{Histogram of transitivity}
    \end{subfigure}
\caption{Histogram of 15 simulations from the DLS model fitted to Reality mining (blue line: actual value; red line: mean of 15 simulated values).}
\label{fig:ppc_dls}
\end{figure*}



\section{Additional Case Study Results}
\label{sec:supp_case_study}



\begin{figure}[tp]
\centering %centers figure
\newcommand{\figwidth}{6.7in}
     \includegraphics[width=\figwidth]{Figures/LSP/LSP_MID_zoomed.pdf}
\caption{2-D latent space plot for MID data with positive slope (zoomed in version of Figure \ref{fig:mid_latent_pos}).}
\label{fig:mid_latent_zoomed}
\end{figure}

\begin{figure}[tp]
\centering %centers figure
\newcommand{\figwidth}{6.7in}
     \includegraphics[width=\figwidth]{Figures/LSP/LSP_MID_cont_zoomed.pdf}
\caption{2-D latent space plot for MID data with positive slope and countries colored by continent (zoomed in version of Figure \ref{fig:mid_latent_continents}).}
\label{fig:mid_latent_cont_zoomed}
\end{figure}

Figures \ref{fig:mid_latent_zoomed} and \ref{fig:mid_latent_cont_zoomed} show zoomed in versions of the latent space for the positive slope model shown in Figures \ref{fig:mid_latent_pos} and \ref{fig:mid_latent_continents}, respectively. 
In Table \ref{tab:mid_pair_incidence}, we show pairs of countries that have the top 50 number of incidents by one country towards another. 
In Figures \ref{fig:mid_latent} and \ref{fig:mid_latent_zoomed}, we draw edges to represent the top 10 frequently occurring incidence between pairs of countries in Table \ref{tab:mid_pair_incidence}. The edges with two-way arrows indicate that they serve both as initiator and receiver to the other (e.g.~PRK and ROK both have disputes with the other with the number of incidences 109 and 95, respectively).

Tables \ref{tab:mid_nodal_pos} and \ref{tab:mid_nodal_neg} show the estimated sender and receiver effects parameters from the LSH model with positive and negative slope, respectively. To compare with the highlighted countries in Figure \ref{fig:mid_latent} and \ref{fig:mid_latent_zoomed}, countries that initiate more incidents tend to have higher sender effect and countries that receive more incidents tend to have higher receiver effect, e.g.~USA has sender effect 4.18, and Yugoslavia (YUG) has receiver effect 3.03.


\begin{table}[tp]
    \caption{Pairs of countries that have the top 50 highest incidents}
    \centering
    \begin{tabular}{lll|lll}
    \hline
    Initiator  & Receiver & \# of incidents & Initiator & Receiver & \# of incidents \\ 
    \hline
    ISR  & LEB     & 588             & CHN  & TAW     & 38               \\
    PAK  & IND     & 174             & YUG  & ALB     & 37               \\
    USA  & IRQ     & 174             & SYR  & TUR     & 34               \\
    USA  & PAK     & 140             & ALB  & YUG     & 31               \\
    ARM  & AZE     & 128             & JPN  & CHN     & 31               \\
    IND  & PAK     & 116             & USA  & PRK     & 29               \\
    CHN  & JPN     & 110             & IRQ  & USA     & 29               \\
    PRK  & ROK     & 109             & UKG  & YUG     & 29               \\
    ROK  & PRK     & 95              & KUW  & IRQ     & 28               \\
    RUS  & GRG     & 88              & LEB  & ISR     & 28               \\
    TUR  & GRC     & 85              & FRN  & YUG     & 27               \\
    UKG  & IRQ     & 83              & AZE  & ARM     & 26               \\
    TUR  & IRQ     & 80              & RUS  & JPN     & 26               \\
    GRC  & TUR     & 73              & PHI  & CHN     & 26               \\
    SYR  & LEB     & 61              & NTH  & YUG     & 26               \\
    PAK  & AFG     & 61              & ITA  & YUG     & 26               \\
    TUR  & SYR     & 54              & CAM  & THI     & 25               \\
    TUR  & CYP     & 53              & TAW  & CHN     & 25               \\
    IRN  & IRQ     & 53              & SUD  & UGA     & 25               \\
    THI  & CAM     & 52              & GMY  & YUG     & 25               \\
    AFG  & PAK     & 51              & TUR  & YUG     & 25               \\
    ISR  & SYR     & 50              & BEL  & YUG     & 24               \\
    RUS  & AFG     & 44              & UKR  & RUS     & 24               \\
    IND  & BNG     & 44              & GRG  & RUS     & 24               \\
    USA  & YUG     & 39              & GRC  & YUG     & 24              
    \end{tabular}
    \label{tab:mid_pair_incidence}
\end{table}


\begin{table}[p]
    \caption{Nodal effect parameters estimated by the model with positive slope}
    \label{tab:mid_nodal_pos}
    \centering
    \begin{tabular}{lll|lll|lll|lll}
    \hline
       Country & send & receive & Country & send & receive & Country & send & receive & Country & send & receive \\ \hline
        AFG & -3.64 & 1.27 & DRV & -1.30 & -0.67 & LUX & -0.40 & -4.26 & SPN & 1.31 & -0.85 \\ 
        ALB & 0.34 & -0.11 & ECU & 2.41 & -1.91 & MAA & -3.89 & 2.14 & SRI & -0.08 & -1.67 \\ 
        ALG & 0.63 & -1.72 & EGY & 0.78 & 0.14 & MAC & -2.13 & 0.93 & SUD & 1.56 & 2.03 \\ 
        ANG & 0.81 & 0.69 & EQG & 2.07 & -2.21 & MAL & 0.89 & -4.16 & SUR & 1.75 & -0.94 \\ 
        ARG & 0.50 & -5.22 & ERI & 1.95 & 1.63 & MLD & -0.49 & 1.51 & SWD & -1.10 & -0.62 \\ 
        ARM & -0.24 & -1.25 & EST & -1.10 & -0.59 & MLI & 1.62 & 4.58 & SWZ & -4.77 & -0.10 \\ 
        AUL & 1.59 & -2.00 & ETH & 1.77 & 0.72 & MNG & -1.01 & -3.76 & SYR & 0.53 & 0.45 \\ 
        AUS & -4.54 & -0.48 & FIN & -4.57 & 0.24 & MOR & -0.97 & -1.71 & TAJ & 2.03 & 0.20 \\ 
        AZE & 0.10 & -0.44 & FRN & 1.44 & -0.51 & MYA & -3.80 & 5.48 & TAW & 1.62 & 1.35 \\ 
        BAH & -0.20 & -5.32 & GHA & 3.34 & 2.57 & NAM & 0.95 & 1.69 & TAZ & 0.49 & 2.61 \\ 
        BEL & 0.22 & 0.13 & GMY & 0.93 & -1.88 & NEP & -0.59 & -1.87 & THI & -0.71 & 5.22 \\ 
        BEN & -2.42 & -0.74 & GRC & 1.04 & -1.54 & NIC & 3.32 & 4.88 & TKM & -0.29 & -5.54 \\ 
        BHU & -4.13 & -1.45 & GRG & -4.07 & 1.20 & NIG & 5.07 & 0.91 & TOG & 2.51 & 1.24 \\ 
        BLR & 1.53 & -2.74 & GUI & 1.96 & -0.64 & NIR & 0.28 & 1.65 & TRI & -3.71 & 0.22 \\ 
        BNG & 0.24 & 2.57 & GUY & -2.08 & 1.76 & NOR & 1.17 & -1.04 & TUN & -1.01 & -3.70 \\ 
        BOS & -5.06 & -0.15 & HAI & -4.64 & 0.10 & NTH & 1.28 & -1.90 & TUR & 2.78 & -0.15 \\ 
        BOT & 0.36 & -2.67 & HON & 1.45 & 5.06 & OMA & -0.12 & -5.26 & UAE & 0.69 & -1.67 \\ 
        BRA & 1.48 & -0.80 & HUN & 0.35 & -4.19 & PAK & 2.60 & 2.28 & UGA & 1.50 & 3.01 \\ 
        BUI & 0.89 & 3.81 & ICE & -0.41 & -4.25 & PAL & -0.81 & -2.12 & UKG & 1.96 & -1.84 \\ 
        BUL & -0.26 & -0.92 & IND & 3.45 & 5.07 & PER & -1.27 & 2.49 & UKR & 0.88 & 2.62 \\ 
        CAM & -3.86 & 6.68 & INS & 1.78 & 0.67 & PHI & -0.57 & 2.52 & USA & 4.18 & 1.10 \\ 
        CAN & 1.81 & -1.28 & IRE & -4.26 & -0.67 & PNG & -1.65 & 1.71 & UZB & 2.75 & 0.95 \\ 
        CAO & 6.17 & -0.29 & IRN & 2.67 & 0.91 & POL & 0.33 & -1.41 & VEN & 2.45 & 1.92 \\ 
        CDI & 1.28 & 0.85 & IRQ & 0.82 & 2.53 & POR & 0.60 & -1.12 & VTM & 0.04 & 2.75 \\ 
        CEN & 3.80 & -4.27 & ISR & 2.19 & -0.13 & PRK & 2.00 & 0.68 & YEM & -3.84 & 0.63 \\ 
        CHA & 2.67 & 2.92 & ITA & 0.63 & -1.73 & QAT & 0.68 & -4.92 & YUG & 2.78 & 3.03 \\ 
        CHL & -4.97 & -1.65 & JAM & 1.47 & -1.14 & ROK & 1.30 & -0.10 & ZAM & -1.12 & 1.43 \\ 
        CHN & 3.76 & 1.53 & JOR & -0.31 & -1.25 & ROM & -0.98 & -3.77 & ZIM & -0.33 & 0.32 \\ 
        COL & 2.77 & 4.73 & JPN & 2.25 & 1.68 & RUM & -3.02 & 1.13 & ~ & ~ & ~ \\ 
        CON & -4.47 & 1.89 & KEN & 0.55 & 1.62 & RUS & 4.01 & 1.82 & ~ & ~ & ~ \\ 
        COS & -1.90 & 0.12 & KOS & -4.18 & -0.56 & RWA & 0.62 & 3.04 & ~ & ~ & ~ \\
        CRO & 0.61 & -0.45 & KUW & 0.17 & -5.34 & SAL & -4.19 & 2.05 & ~ & ~ & ~ \\ 
        CUB & 0.02 & -5.59 & KYR & 2.01 & 0.08 & SAU & 0.56 & -0.10 & ~ & ~ & ~ \\ 
        CYP & -4.62 & -1.33 & KZK & 0.20 & -5.64 & SIE & -1.30 & 2.40 & ~ & ~ & ~ \\ 
        CZR & -0.44 & -3.13 & LAT & -0.83 & -1.44 & SIN & -4.26 & 1.08 & ~ & ~ & ~ \\
        DEN & 1.09 & -1.59 & LBR & 1.94 & 1.34 & SLO & -1.11 & -0.60 & ~ & ~ & ~ \\ 
        DJI & -1.61 & 0.25 & LEB & -3.94 & -0.30 & SLV & -1.14 & -0.11 & ~ & ~ & ~ \\ 
        DOM & -0.35 & -5.66 & LIB & 0.30 & 2.93 & SOL & -4.57 & 3.22 & ~ & ~ & ~ \\ 
        DRC & 1.50 & 3.37 & LIT & 0.56 & 1.77 & SOM & -4.16 & 1.04 & ~ & ~ & ~ \\ 

    \end{tabular}
\end{table}

\begin{table}[p]
    \caption{Nodal effect parameters estimated by the model with negative slope}
    \label{tab:mid_nodal_neg}
    \centering
    \begin{tabular}{lll|lll|lll|lll}
    \hline
        Country & send & receive & Country & send & receive & Country & send & receive & Country & send & receive \\ \hline
        AFG & -1.49 & 3.09 & DRV & -1.50 & -0.16 & LUX & 0.43 & -1.59 & SPN & 1.58 & 1.20 \\ 
        ALB & 1.12 & -0.16 & ECU & 0.27 & -2.38 & MAA & -1.55 & -0.37 & SRI & -0.51 & -1.48 \\ 
        ALG & -0.49 & -2.00 & EGY & 0.43 & 0.44 & MAC & -1.34 & 0.65 & SUD & 1.37 & 1.89 \\ 
        ANG & 1.42 & -0.42 & EQG & -0.41 & -1.40 & MAL & 0.80 & -1.39 & SUR & -0.44 & -1.67 \\ 
        ARG & 1.07 & -1.38 & ERI & 1.11 & -0.54 & MLD & -0.79 & -0.25 & SWD & -0.52 & -0.38 \\ 
        ARM & -0.28 & 0.59 & EST & -0.45 & -0.53 & MLI & -0.41 & 0.46 & SWZ & -1.55 & 0.51 \\ 
        AUL & 1.39 & -0.28 & ETH & 0.88 & -0.25 & MNG & -0.66 & -1.42 & SYR & 0.34 & 0.95 \\ 
        AUS & -1.74 & -0.30 & FIN & -1.69 & 0.43 & MOR & -0.36 & -0.54 & TAJ & 0.55 & -0.41 \\ 
        AZE & 0.02 & 1.38 & FRN & 1.94 & 0.98 & MYA & -1.53 & 1.08 & TAW & 0.59 & 0.67 \\ 
        BAH & -0.37 & -1.60 & GHA & 0.43 & -1.40 & NAM & 0.46 & -0.27 & TAZ & -0.38 & -1.69 \\ 
        BEL & 0.76 & 0.61 & GMY & 1.38 & -0.26 & NEP & -0.42 & -1.39 & THI & 0.14 & 0.91 \\ 
        BEN & -1.63 & -0.32 & GRC & 1.59 & -0.21 & NIC & 0.46 & -0.22 & TKM & -0.36 & -1.41 \\ 
        BHU & -1.49 & -0.37 & GRG & -1.62 & 1.36 & NIG & 1.34 & -0.86 & TOG & -1.56 & -0.41 \\ 
        BLR & -0.52 & -1.40 & GUI & 0.82 & -1.57 & NIR & -0.55 & -0.42 & TRI & -2.01 & -0.30 \\ 
        BNG & 0.49 & -1.80 & GUY & -1.50 & 0.54 & NOR & 0.98 & 0.17 & TUN & -0.41 & -1.53 \\ 
        BOS & -1.48 & 0.48 & HAI & -1.55 & 1.85 & NTH & 1.64 & -0.24 & TUR & 2.60 & 1.34 \\ 
        BOT & -0.51 & -1.45 & HON & -0.51 & 0.37 & OMA & -0.21 & -1.41 & UAE & 1.15 & -0.45 \\ 
        BRA & -0.49 & -1.66 & HUN & 1.29 & -1.28 & PAK & 0.47 & 0.71 & UGA & 0.48 & 1.64 \\ 
        BUI & -0.37 & 0.54 & ICE & 0.50 & -1.37 & PAL & -0.37 & -1.62 & UKG & 2.18 & -0.29 \\ 
        BUL & 0.46 & 0.36 & IND & -0.26 & 1.43 & PER & -1.56 & 0.17 & UKR & -0.37 & 1.27 \\ 
        CAM & -1.75 & 0.54 & INS & 1.26 & 1.25 & PHI & -0.36 & 1.51 & USA & 3.04 & 1.74 \\ 
        CAN & 2.13 & 0.60 & IRE & -1.66 & -0.29 & PNG & -0.36 & -0.32 & UZB & 0.95 & -1.35 \\ 
        CAO & -0.45 & -0.75 & IRN & 1.78 & 2.12 & POL & 0.92 & -0.23 & VEN & 0.93 & -0.19 \\ 
        CDI & 0.50 & 1.25 & IRQ & -0.17 & 3.36 & POR & 1.06 & 0.62 & VTM & 0.30 & 0.99 \\ 
        CEN & 0.30 & -1.38 & ISR & 2.00 & -0.26 & PRK & 0.59 & 1.69 & YEM & -1.48 & 0.57 \\ 
        CHA & 1.13 & 1.30 & ITA & 1.19 & -0.41 & QAT & 1.23 & -1.36 & YUG & 3.05 & 3.66 \\ 
        CHL & -1.49 & -0.26 & JAM & -0.53 & -1.68 & ROK & 0.47 & 0.75 & ZAM & -1.48 & 0.45 \\ 
        CHN & 2.81 & 1.04 & JOR & 0.35 & -0.26 & ROM & -0.36 & -1.85 & ZIM & 0.56 & -0.45 \\ 
        COL & -1.44 & 0.86 & JPN & -0.38 & 1.60 & RUM & -1.29 & 0.50 & ~ & ~ & ~ \\ 
        CON & -1.94 & 0.61 & KEN & 0.35 & -0.26 & RUS & 3.40 & 2.59 & ~ & ~ & ~ \\ 
        COS & -0.39 & -1.42 & KOS & -1.49 & -0.45 & RWA & -0.38 & 1.63 & ~ & ~ & ~ \\ 
        CRO & 1.27 & 0.96 & KUW & -0.29 & -1.38 & SAL & -1.72 & -0.53 & ~ & ~ & ~ \\ 
        CUB & -0.50 & -1.44 & KYR & -1.27 & 0.56 & SAU & 0.49 & 0.56 & ~ & ~ & ~ \\ 
        CYP & -1.55 & -0.22 & KZK & 0.36 & -1.71 & SIE & -1.64 & 1.18 & ~ & ~ & ~ \\ 
        CZR & 0.32 & -1.29 & LAT & -0.54 & -0.22 & SIN & -1.55 & -0.29 & ~ & ~ & ~ \\ 
        DEN & 1.39 & -0.11 & LBR & 0.30 & 1.02 & SLO & -0.51 & -0.38 & ~ & ~ & ~ \\ 
        DJI & -1.62 & -0.38 & LEB & -1.50 & -0.59 & SLV & -0.43 & 0.49 & ~ & ~ & ~ \\ 
        DOM & -0.28 & -1.72 & LIB & 0.55 & 3.34 & SOL & -1.51 & -0.34 & ~ & ~ & ~ \\ 
        DRC & 1.58 & 0.75 & LIT & 0.49 & 1.08 & SOM & -1.49 & 1.05 & ~ & ~ & ~ \\ 
    \end{tabular}
\end{table}

\end{document}
