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\title{Content Sharing Design for Social Welfare in Networked Disclosure Game\\(Supplementary Material)}

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\author[1]{\href{mailto:<feiran.jia@psu.edu>}{Feiran Jia}{}}
\author[2]{Chenxi Qiu}
\author[1]{Sarah Rajtmajer}
\author[1]{Anna Squicciarini}
% Add affiliations after the authors
\affil[1]{%
    Information Sciences and Technology,
    Pennsylvania State University,
    Pennsylvania, USA
}
\affil[2]{%
    Computer Science and Engineering,
    University of North Texas,
    Texas, USA
} 

  
\begin{document}
  
\onecolumn %% Turn this off if single column is desired for the supplement
\maketitle

\section{Omitted Details of Theoretical Results}
\subsection*{Proof of Theorem \ref{clm:tc}} 
\begin{proof}
$\Rightarrow$: (1) Everyone who invests satisfies the \textbf{threshold condition}, implying that if $x_i = 1$, $U_i(x_i, \mathbf{x}_{-i}| \mathcal{G}) \geq U_i(1- x_i, \mathbf{x}_{-i}| \mathcal{G}) \Rightarrow U_i(1, \mathbf{x}_{-i}| \mathcal{G}) \geq U_i(0, \mathbf{x}_{-i}| \mathcal{G})$. (2) other agents do not satisfy \textbf{threshold condition}, implying that if $x_i = 0$, $U_i(x_i, \mathbf{x}_{-i}| \mathcal{G}) < U_i(1- x_i, \mathbf{x}_{-i}| \mathcal{G}) \Rightarrow U_i(0, \mathbf{x}_{-i}| \mathcal{G}) > U_i(1, \mathbf{x}_{-i}| \mathcal{G})$. Therefore, $\mathbf{x}$ is a PSNE. 
\newline $\Leftarrow$: If $\mathbf{x}$ is a PSNE and each user $i$ breaks ties in favor of disclosing, then if $x_i = 1$, $U_i(x_i, \mathbf{x}_{-i}| \mathcal{G}) \geq U_i(1- x_i, \mathbf{x}_{-i}| \mathcal{G})$, implying user $i$ satisfies the threshold condition; if $x_i = 0$, $U_i(x_i, \mathbf{x}_{-i}| \mathcal{G}) < U_i(1- x_i, \mathbf{x}_{-i}| \mathcal{G})$, implying user $i$ doesn't satisfy the threshold condition. 
\newline The proof is completed. 
\end{proof}

\subsection*{Proof of Lemma \ref{lem:}} 
\begin{proof}
Case (2) can be achieved when no edge is promoted. 

For Case (1), we can promote the edge $e_{2i-1, 2i}$ in $\mathcal{G}^{\text{in}}_i$. In this case, if both nodes disclose themselves, then $x_{2i} = 1$, and we can derive both $U_{2i}(x_{2i}, \mathbf{x}_{-2i}| \mathcal{G})$ and $U_{2i}(1- x_{2i}, \mathbf{x}_{-2i}| \mathcal{G})$: 
\begin{eqnarray}
\nonumber  U_{2i}(x_{2i}, \mathbf{x}_{-2i}| \mathcal{G}) &=& w_{2i-1,2i} x_{2i-1}x_{2i} - c_{2i} x_{2i} \\
&=& 2a_i - 1.5a_i = 0.5a_i, \\
U_{2i}(1- x_{2i}, \mathbf{x}_{-2i}| \mathcal{G}) &=& 0, 
\end{eqnarray}
indicating that $U_{2i}(x_{2i}, \mathbf{x}_{-2i}| \mathcal{G}) > U_{2i}(1- x_{2i}, \mathbf{x}_{-2i}| \mathcal{G})$. Similarly, we can prove that $U_{2i-1}(x_{2i-1}, \mathbf{x}_{-(2i-1)}| \mathcal{G}) > U_{2i-1}(1- x_{2i-1}, \mathbf{x}_{-(2i-1)}| \mathcal{G})$. Therefore, Case (1) is a PSNE. 

Note that there exists no PNSE in $\mathcal{G}^{\text{in}}_i$ such that one node discloses and the other does not. For the sake of contradiction, consider the case that $v_{2i}$ discloses and $v_{2i-1}$ does not. Then, $x_{2i} = 1$ and $x_{2i-1} = 0$,  
\begin{eqnarray}
\nonumber && U_{2i}(x_{2i}, \mathbf{x}_{-2i}| \mathcal{G}) = 0 - c_{2i} \leq 0 - 1.5a_i = -1.5a_i \\ 
\nonumber && U_{2i}(1- x_{2i}, \mathbf{x}_{-2i}| \mathcal{G}) = 0
\end{eqnarray}
indicating that $U_{2i}(x_{2i}, \mathbf{x}_{-2i}| \mathcal{G}) < U_{2i}(1 - x_{2i}, \mathbf{x}_{-2i}| \mathcal{G})$ and hence $x_{2i}$ should not be 1 in this PNSE, which is a contradiction. The proof of Lemma \ref{lem:} is completed. 
\end{proof}

\subsection*{Proof of Lemma~\ref{lem:maxInvestRel}}
% \begin{lemma} \label{lem:maxInvestRel}
% Suppose $\bx$ is the profile returned by the MaxInvest. For any PSNE profile $\bx'$, we have $\bx' \leq \bx$.%$I(x') \subseteq I(x)$.
% \end{lemma}

\begin{proof}
\textbf{Base case} - In the first iteration, each node $i$ deactivated by MaxInvest %, i.e., $x_i^{(0)} = 0$, 
cannot disclose in any PSNE. 
\newline \textbf{The induction step} - Assuming that the nodes deactivated by MaxInvest in the first $k$ iterations cannot disclose in any PSNE, then the nodes deactivated by MaxInvest in the $(k+1)$th iteration cannot disclose in any PSNE. 
Therefore, any node deactivated by MaxInvest cannot disclose in any other PSNE. The proof is completed. 
\end{proof}

\subsection*{Proof of Theorem~\ref{thm:optimal}}
\begin{proof}
% \fj{We do not use P, need to change the notation }
For any PSNE $\mathbf{x}'$, we have $\mathbf{x}' \leq \mathbf{x}$ (according to Lemma \ref{lem:maxInvestRel}). We let $\mathcal{A}$ and $\mathcal{A}'$ denote the set of nodes disclosed in $\mathbf{x}$ and $\mathbf{x}'$, respectively, i.e., $\mathcal{A}' \subseteq \mathcal{A}$. Then, 
\begin{eqnarray}
SW(\mathbf{x}|\mathcal{G}) - SW(\mathbf{x}'|\mathcal{G}) & = &\sum_{i\in \mathcal{A}}U_i\left(\mathbf{x}|\mathcal{G}\right) - \sum_{i\in \mathcal{A}'}U_i\left(\mathbf{x}'|\mathcal{G}\right) \\ \nonumber
&=& \sum_{i\in \mathcal{A}\backslash \mathcal{A}'}\underbrace{U_i\left(\mathbf{x}|\mathcal{G}\right)}_{\small \geq 0 ~\mbox{since each $i$ discloses}} + \sum_{i\in \mathcal{A}'}\left(U_i\left(\mathbf{x}|\mathcal{G}\right)- U_i\left(\mathbf{x}'|\mathcal{G}\right)\right) \\ 
&\geq & \sum_{i\in \mathcal{A}'}\left( x_i \sum_{j\in \mathcal{N}_i} w_{j,i}x_{j} - c_i x_i -  x'_i \sum_{j\in \mathcal{N}_i} w_{j,i}x'_{j} + c_ix'_i\right) \\
&=& \sum_{i\in \mathcal{A}'} \left(\sum_{j\in \mathcal{N}_i} w_{j,i}\underbrace{\left(x_j - x'_j\right)}_{\geq 0}\right) \geq 0. 
\end{eqnarray}
indicating that $SW(\mathbf{x}|\mathcal{G}) \geq SW(\mathbf{x}'|\mathcal{G})$ for any PSNE $\mathbf{x}'$. 
\end{proof}



\subsection*{Proof of Theorem~\ref{thm:monotonicity}}
\begin{proof}
Let $\bx$ and $\by$ denote the returned profiles of $MaxInvest(\mathcal{S})$ and $MaxInvest(\mathcal{T})$. 

(1) We prove $\bx \leq \by$ by induction. % We let $\mathcal{V}$
\textbf{Base Case:} In the first iteration of MaxInvest($\mathcal{T}$), for each node $i$ satisfying $\sum_{e_{i,j} \in \mathcal{T}} w_{j,i} < c_i$, we set $y^{(0)}_i = 0$. Correspondingly, each $x_i = 0$ since $\mathcal{S}\subseteq \mathcal{T}$ and $\sum_{e_{i,j} \in \mathcal{S}} w_{j,i} \leq \sum_{e_{i,j} \in \mathcal{T}} w_{j,i} < c_i$. Therefore $\bx \leq \by^{(0)}$.
\newline \textbf{The induction step:} Assuming that in the $k$th iteration of MaxInvest($\mathcal{T}$), $\bx \leq \by^{(k)}$. We will then prove in the $k+1$th iteration, $\bx \leq \by^{(k+1)}$. In the $(k+1)$th iteration, suppose that node $i$ is popped off $Q$, we have $\sum_{e_{i,j} \in \mathcal{T}} w_{j,i}y^{(k)}_j < c_i$ and $y^{(k+1)}_i = 0$. Given $\bx \leq \by^{(k)}$ and $\mathcal{S}\subseteq \mathcal{T}$, we have $\sum_{e_{i,j} \in \mathcal{S}} w_{j,i}x_j \leq \sum_{e_{i,j} \in \mathcal{T}} w_{j,i}y^{(k)}_j < c_i$. Since $\bx$ is a PSNE, we have $y^{(k+1)}_i = x_i = 0$. Then in the iteration $k+1$, $\bx \leq \by^{(k+1)}$.

We conclude that $\forall k:\ \bx \leq \by^{(k)}$, and thus $\bx \leq \by$. 

(2) As $\sum_{i\in \mathcal{V}} x_i \leq \sum_{i\in \mathcal{V}} y_i \Rightarrow I(\mathcal{S}) \leq I(\mathcal{T})$.  

(3) Then we show that $\sigma(\mathcal{T}) \geq \sigma(\mathcal{S})$. Let $\mathcal{A}(\mathcal{T})$ and $\mathcal{A}(\mathcal{S})$ denote the set of disclosed nodes in MaxInvest($\mathcal{T}$) and MaxInvest($\mathcal{S}$). % \fj{Should we explicitly define $A$ for convinence?} 
%\begin{equation}
\begin{eqnarray}
\sigma(\mathcal{T}) - \sigma(\mathcal{S}) &=& SW(\by| (\mathcal{V}, \mathcal{T}))  - SW(\bx| (\mathcal{V}, \mathcal{S})) \\
& = & \sum_{i\in \mathcal{A}(\mathcal{S})} \Big( U_i(\by|(\mathcal{V}, \mathcal{T})) -  U_i(\bx|(\mathcal{V}, \mathcal{S})) \Big) + \sum_{i\in \mathcal{A}(\mathcal{T})\backslash \mathcal{A}(\mathcal{S})} U_i(\by|(\mathcal{V}, \mathcal{T}))\\
&\geq& \sum_{i\in \mathcal{A}(\mathcal{S})} \Big( y_i \sum_{e_{i,j}\in \mathcal{T}}w_{j,i}y_j-c_iy_i - x_i \sum_{e_{i,j}\in \mathcal{S}}w_{j,i}x_j+c_ix_i \Big) \\
% & = \sum_{i\in \mathcal{A}(\mathcal{S})} \Big(\sum_{e_{i,j}\in \mathcal{T}}w_{j,i}y_j -\sum_{e_{i,j}\in \mathcal{S}}w_{j,i}x_j\Big)\\
& =& \sum_{i\in \mathcal{A}(\mathcal{S})} \Big(\sum_{e_{i,j}\in \mathcal{T} \backslash \mathcal{S}}w_{j,i}y_j +\sum_{e_{i,j}\in \mathcal{S}}w_{j,i}(y_j - x_j)\Big) \\
&\geq& 0
\end{eqnarray}
%\end{equation}
\end{proof}


\subsection*{Proof of Theorem \ref{thm:sup}}


\begin{proof}
To demonstrate the super-modularity, we would like to prove that for any edge $e^* \in \mathcal{E}^{(in)}$, and all pairs of the set $\mathcal{S}\subseteq \mathcal{T} \subseteq \mathcal{E}^{(in)}$, $\sigma(\cdot)$ satisfies 
\begin{equation}
    \sigma(\mathcal{S}\cup \{e^*\}) - \sigma(\mathcal{S}) \leq \sigma(\mathcal{T}\cup \{e^*\}) - \sigma(\mathcal{T})
\end{equation}

% Since $\frac{c_i}{w_{j,i}} \leq 1, \forall (i, j) \in \mathcal{E}^{in}$, for any existing edge $e = (u,v)$, we have $x_v = x_w = 1$. 

Let $\mathcal{A}(\mathcal{T})$ and $\mathcal{A}(\mathcal{S})$ denote the set of disclosed nodes in MaxInvest($\mathcal{T}$) and MaxInvest($\mathcal{S}$).
Suppose $e^* = (u,v)$, we discuss the following cases. According to the monontoncity, we have $\mathcal{A}(\mathcal{S}) \subseteq \mathcal{A}(\mathcal{T})$. %There are following cases.
\newline (1) If $e^*$ connects the disclosed nodes in $\mathcal{A}(\mathcal{S})$, $\sigma(\mathcal{S}\cup \{e^*\}) - \sigma(\mathcal{S}) = \sigma(\mathcal{T}\cup \{e^*\}) - \sigma(\mathcal{T}) = w_{u,v} + w_{v,u}$. 
\newline (2) If $e^*$ connects two nodes outside of $\mathcal{A}(\mathcal{T})$, then $\sigma(\mathcal{S}\cup \{e^*\}) - \sigma(\mathcal{S}) = \sigma(\mathcal{T}\cup \{e^*\}) - \sigma(\mathcal{T}) = w_{u,v}-c_v + w_{v,u}-c_w$. 
\newline (3) If $e^*$ connects one node in $\mathcal{A}(\mathcal{S})$ and another node outside the set $\mathcal{A}(\mathcal{T})$ (i.e. $\mathcal{V}\backslash\mathcal{A}(\mathcal{T})$), $\sigma(\mathcal{S}\cup \{e^*\}) - \sigma(\mathcal{S}) = \sigma(\mathcal{T}\cup \{e^*\}) - \sigma(\mathcal{T}) = w_{u,v}-c_v + w_{v,u} \text{~or~} w_{v,u}-c_w + w_{u,v}$.
\newline (4) If $e^*$ connects one node in $\mathcal{A}(\mathcal{T})\backslash \mathcal{A}(\mathcal{S})$ and another node outside the set $\mathcal{A}(\mathcal{T})$, we have $\sigma(\mathcal{S}\cup \{e^*\}) - \sigma(\mathcal{S}) = w_{u,v}-c_v + w_{v,u}-c_w  \leq \sigma(\mathcal{T}\cup \{e^*\}) - \sigma(\mathcal{T}) = w_{u,v} + w_{v,u}-c_w \text{~or~} w_{u,v}-c_v + w_{v,u} $ .
\newline (5) If $e^*$ connects two nodes in $\mathcal{A}(\mathcal{T})\backslash \mathcal{A}(\mathcal{S})$, we have $\sigma(\mathcal{S}\cup \{e^*\}) - \sigma(\mathcal{S}) = w_{u,v}-c_v + w_{v,u}-c_w  \leq \sigma(\mathcal{T}\cup \{e^*\}) - \sigma(\mathcal{T}) =  w_{u,v}+ w_{v,u}$. 
\end{proof}
\subsection*{Proof of Theorem \ref{thm:sub}}

\begin{theorem}
\label{thm:sub}

The optimal investment function $I(\mathcal{E})$ is sub-modular when $\frac{c_i}{w_{j,i}} \leq 1, \forall (i, j) \in \mathcal{E}^{in}$.  
\end{theorem}


\begin{figure}[h]
    \centering
    \begin{minipage}{.5\textwidth}
        \centering
       \includegraphics[width = 0.80\linewidth]{./images/counter_example.pdf}
        \caption{Instance of Remark~\ref{thm:non_sup} and Remark~\ref{thm:non_sub}.}
        \label{fig:non}
    \end{minipage}
\end{figure}
% The idea is similar to the proof of Theorem~\ref{thm:sup}. \fj{put in appendix}
\begin{proof}
To demonstrate the sub-modularity, we would like to prove that for any edge $e^* \in \mathcal{E}^{(in)}$, and all pairs of the set $\mathcal{S}\subseteq \mathcal{T} \subseteq \mathcal{E}^{(in)}$, $\sigma(\cdot)$ satisfies 
\begin{equation}
    I(\mathcal{S}\cup \{e^*\}) - \sigma(\mathcal{S}) \geq I(\mathcal{T}\cup \{e^*\}) - \sigma(\mathcal{T})
\end{equation}

% Since $\frac{c_i}{w_{j,i}} \leq 1, \forall (i, j) \in \mathcal{E}^{in}$, for any existing edge $e = (u,v)$, we have $x_v = x_w = 1$. 
%Let $\mathcal{A}(\mathcal{T})$ and $\mathcal{A}(S)$ denote the set of disclosed nodes in MaxInvest($\mathcal{T}$) and MaxInvest($S$).
 According to the monontoncity, we have $\mathcal{A}(S) \subseteq \mathcal{A}(\mathcal{T})$. Suppose $e^* = (u,v)$, we discuss the following cases.
%There are following cases.
(1) If $e^*$ connects the disclosed nodes in $\mathcal{A}(S)$, then there are no new disclosing nodes. $I(S\cup \{e^*\}) - I(S) = I(\mathcal{T}\cup \{e^*\}) - I(\mathcal{T}) = 0$. 
(2) If $e^*$ connects two nodes outside of $\mathcal{A}(\mathcal{T})$, then $I(S\cup \{e^*\}) - I(S) = I(\mathcal{T}\cup \{e^*\}) - I(\mathcal{T}) = 2$. 
(3) If $e^*$ connects one node in $\mathcal{A}(S)$ and another node outside the set $\mathcal{A}(\mathcal{T})$ (i.e. $\mathcal{V}\backslash\mathcal{A}(\mathcal{T})$), $I(S\cup \{e^*\}) - I(S) = I(\mathcal{T}\cup \{e^*\}) - I(\mathcal{T}) = 1$.
(4) If $e^*$ connects one node in $\mathcal{A}(\mathcal{T})\backslash \mathcal{A}(S)$ and another node outside the set $\mathcal{A}(\mathcal{T})$, we have $I(S\cup \{e^*\}) - I(S) = 2 \geq I(\mathcal{T}\cup \{e^*\}) - I(\mathcal{T}) = 1 $ .
(5) If $e^*$ connects two nodes in $\mathcal{A}(\mathcal{T})\backslash \mathcal{A}(S)$, we have $I(S\cup \{e^*\}) - I(S) = 2  \geq I(\mathcal{T}\cup \{e^*\}) - I(\mathcal{T}) =  0$. 
\end{proof}



\subsection*{Examples of Remark \ref{thm:non_sup} and Remark \ref{thm:non_sub}}

\begin{remark}
\label{thm:non_sup}
$\sigma(\cdot)$ is in general non-supermodular.  
\end{remark}
\begin{remark}
\label{thm:non_sub}
$I(\cdot)$ is in general non-submodular.  
\end{remark}


Figure~\ref{fig:non} gives an example of Remark \ref{thm:non_sup}: We assume that $\mathcal{T} = \{ e_1. e_2, e_3, e_4, e_5, e_6\}$, $S = \{ e_1. e_2, e_3, e_4\}$. All the nodes have the same cost $c$, and all the weights are $w \geq 0$. We assume that $c = 1.2w$. The node will disclose only when there are more than two neighbours disclose. We have $\sigma(S\cup \{e^*\}) - \sigma(S) = 5 (2w - c) = 4w > \sigma(\mathcal{T}\cup \{e^*\}) - \sigma(\mathcal{T}) = 2w$. 


Figure~\ref{fig:non} also gives an example of Remark \ref{thm:non_sup}: Assuming $\mathcal{T} = \{ e_1. e_2, e_3, e_4\}$, $S = \{e_1\}$, $\frac{c_i}{w_{j,i}} = 2, \forall (i, j) \in \mathcal{E}^{in}$, we have $I(S\cup \{e^*\}) - I(S) = 0 < I(\mathcal{T}\cup \{e^*\}) - I(\mathcal{T}) = 5$. 

% \subsection*{Time Complexity Analysis} 
% % \textcolor{red}{@chenxi: Need Double-check}

% \noindent \textbf{Time complexity of MaxInvest.} % Checking the threshold condition takes $O(|\mathcal{V}| +|\mathcal{E}|)$. 
% Each node can be added to $Q$ for at most once. For each node $i$ added to $Q$, we need to update its neighbors' threshold conditions and check whether the conditions are satisfied, which takes total $O(|\mathcal{E}|)$. Therefore, the time complexity of MaxInvest is $O(|\mathcal{V}| +|\mathcal{E}|)$.
% % The key idea of the algorithm is to keep the upper bound of possible benefit can be obtained in $H$...

% \noindent \textbf{Time complexity of Greedy}. To add an edge to $\mathcal{E}$, we need to pick up the edge with the highest SW marginal gain in $\mathcal{E}^{\mathrm{in}}$ and outside $\mathcal{E}$, which needs to call the MaxInvest algorithm (Algorithm \ref{alg:MaxInvest}) $O(|\mathcal{E}^{\mathrm{in}}|)$ times. The algorithm adds totally $B$ edges to $\mathcal{E}$, therefore the complexity is $O(|\mathcal{E}^{\mathrm{in}}|)\times O(B) \times O(|\mathcal{V}|+|\mathcal{E}|) = O(|\mathcal{E}^{\mathrm{in}}|(|\mathcal{V}|+|\mathcal{E}|)B)$. 

% \noindent \textbf{Time complexity of RankHeuristic}. In each iteration, the algorithm needs to call the MaxInvest function (Algorithm \ref{alg:MaxInvest}, which has $O(|\mathcal{V}| + |\mathcal{E}^{in}|)$ operations) to obtain the optimal action profile, compute the utility of each user ($O(|\mathcal{V}|)$ operations), and sort the users based on their utility ($O(|\mathcal{V}|\log |\mathcal{V}|)$  operations). The number of iterations is up to $O(|\mathcal{V}|)$. Therefore, the time complexity is $(O(|\mathcal{V}| + |\mathcal{E}^{in}|)+O(|\mathcal{V}|)+O(|\mathcal{V}|\log |\mathcal{V}|))\times O(|\mathcal{V}|) = O((|\mathcal{E}^{in}|+|\mathcal{V}|\log |\mathcal{V}|) |\mathcal{V}|)$. 

\section{Omitted Details of Experiments}



\subsection*{Social Interaction Graphs}
Figures \ref{fig:April}, \ref{fig:August}, \ref{fig:September} illustrate the respective social interaction graphs. Node size is proportional to node degree. Red nodes represent users with a high self-disclosure rate (between 0.95 and 1).
 \begin{figure}[h!]
     \centering
     \begin{minipage}{.30\textwidth}
         \centering
         \includegraphics[width = .90\linewidth]{./images/april_colored.pdf}
         \caption{April 2021.}
         \label{fig:April}
     \end{minipage}
     \hfill
     \begin{minipage}{.32\textwidth}
         \centering
      \includegraphics[width = .81\linewidth]{./images/august_colored.pdf}
         \caption{August 2020.}
         \label{fig:August}
     \end{minipage}
     \hfill
     \begin{minipage}{.32\textwidth}
         \centering
      \includegraphics[width = .81\linewidth]{./images/sep_colored.pdf}
         \caption{September 2020.}
         \label{fig:September}
     \end{minipage}
 \end{figure}

\subsection*{Omitted Details of Section~\ref{subsec:realexp} (Label Generation)%\ref{sec:label_generation}
}

 
In the original dataset, each sentence is associated with 6 labels: informational disclosure, emotional disclosure, support, general support, informational support, and emotional support.
We use three of them in our task: informational disclosure, emotional disclosure, and emotional support.
For example, the following training sentence is labeled for emotional disclosure and emotional support: 
%\begin{displayquote}
\begin{itemize} 
\item \textit{I hope this chapter results in a better, healthier, more fulfilled you!!} 
\end{itemize} 
While our initial study only focused on the disclosure label, we believe that the labeled dataset created for this study can be of great value for future research in this area. 
\subsection*{Computation time}
In table~\ref{tab:run-time}, we present the computation time (in seconds) of our algorithm for the set of experiments in Section~\ref{subsec:realexp}, which demonstrates that our heuristics work efficiently in large-scale networks. We conducted 20 trials for each experiment. Notably, our heuristics can solve the problem extremely quickly, particularly when the budget is large.

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\begin{tabular}{@{}lllllllll@{}}
\toprule
 & \textbf{b=0.2} & \textbf{b=0.4} & \textbf{b=0.6} & \textbf{b=0.8} & \textbf{b=1.0} & \textbf{b=1.5} & {\color[HTML]{333333} \textbf{b=2.0}} & \textbf{b=3.0} \\ \midrule
Aug &
  {\color[HTML]{333333} 464.27 ± 65.74} &
  282.78 ± 68.12 &
  126.40 ± 59.00 &
  28.78 ± 28.96 &
  1.42 ± 0.38 &
  1.16 ± 0.38 &
  1.06 ± 0.28 &
  1.03 ± 0.28 \\
Sep &
  77.63 ± 15.03 &
  36.91 ± 11.78 &
  9.90 ± 6.92 &
  0.52 ± 0.15 &
  0.52 ± 0.14 &
  0.38 ± 0.11 &
  0.38 ± 0.11 &
  0.38 ± 0.11 \\
Apr &
  198.19 ± 32.51 &
  132.17 ± 23.53 &
  58.84 ± 10.57 &
  18.45 ± 14.19 &
  {\color[HTML]{333333} 1.44 ± 2.05} &
  0.65 ± 0.17 &
  0.65 ± 0.17 &
  0.63 ± 0.17 \\ \bottomrule
\end{tabular}%
}
\caption{The run-time (in secs).}
\label{tab:run-time}
\end{table}
\subsection*{Model Validation}\label{sec:val}

We provided a preliminary attempt to validate our assumption that users' actions (disclose or not) depend on the threshold function (Equ. (\ref{eq:threshold})). The modeling method is the same as Section~\ref{subsec:realexp} (Algorithmic Results). However, we only consider the self-disclosure information of the posts weekly due to the lack of time data regarding comments. Users are more conservative about posting than commenting.
%We validate our assumption that users' actions (disclose or not) depend on the threshold function (Equ. (\ref{eq:threshold})). 
In each month, we pick the users who have at least one response record each week (41 users in August 2020, 52 users in September 2020, and 22 users in April 2021). 
We first estimate users' cost coefficients in the first week and use the estimated cost coefficients to predict users' responses in the remaining weeks of each month (4 weeks in April 2021, August 2020, and 2 weeks in September 2020). After, we compare whether the predicted responses are consistent with the users' actual responses, of which the results are listed in Table \ref{tab:modelvalidate} (FPR and FNR stand for \emph{false positive rate} and \emph{false negative rate}). 
\begin{table}[h!]
    \centering
    \begin{tabular}{|c|c|c|c|c|c|}
    \hline
    \hline 
  \textbf{Month}    &  FPR & FNR & F1 &  precision & recall \\ \hline
       August 2020 & 0.1388 & 0.4607 & 0.6427 & 0.7953 & 0.5393 \\ 
       September 2020  & 0.1675 & 0.6595 & 0.4516 & 0.6703 & 0.3405 \\
       April 2021  & 0.0921 & 0.8127 & 0.2928 & 0.6723 & 0.1873 \\ 
       \hline
    \end{tabular}
    \caption{Model validation results.}
    \label{tab:modelvalidate}
\end{table}
The table demonstrates that the threshold condition (Equ. (\ref{eq:threshold})) can accurately predict users' responses, especially in indicating the disclosure of users (as FPR is low). This provides empirical evidence to incentivize the disclosure of desired users by satisfying their threshold condition through edge promotion.  


This simple modeling method is implicitly based on the assumptions that (1) the user behaves strategically, and (2) we assume that the has converged to a PSNE. Filling the gap between the theoretical model and real data (real-world user behavior) is non-trivial. 



\section{Additional Related Works}
\noindent\textbf{Binary Networked Public Good Games.}
One line of relevant game theoretic research is the literature on binary networked public goods games (BNPG).%, which is a variant of public good games. \textcolor{red}{Could anyone cite some papers here?}
%In a graphical game, the player's utility just depends on the strategies of the neighbors. 
%It is NP-hard to find a pure strategy Nash equilibrium (PSNE). 
The binary networked public goods game (BNPG) is a variant of a graphical game, where players' utilities depend on the strategies of their neighbors in the social graph. Benefits are a function of accumulated efforts \cite{bramoulle2007public} and investment strategies are binary \cite{yu2020computing}. 
In \cite{kempe2020inducing}, authors study network design to induce equilibria in BNPGs.
Altruism modeling has also been considered to achieve desired investment profiles \cite{yu2021altruism}.
Our model considers a setting similar to the BNPG but with different benefit functions designed for the self-disclosure application. 

\noindent\textbf{Network Design.}
The outer OSDSP (i.e. content-sharing network design) is intrinsically a network structure design problem.
Despite the most similar work \cite{coro2019recommending, coro2021link, yu2021altruism, kempe2020inducing} mentioned before, types of network designs include removing edges~\cite{kimura2008solving,kimura2009blocking} or nodes~\cite{jia2020blocking}, adding nodes or edges~\cite{sheldon2012maximizing,amelkin2019fighting}, and edge manipulation~\cite{chen2016eigen,castiglioni2020election}, etc.


\section{Additional Analysis of NDG}
In this section, we discover several PSNEs in NDG that can be obtained with low time complexity.

\begin{theorem}%{Trivial PSNE}
    If $c_i > 0, \forall i$, $x_i = 0, \forall i$ is a PSNE. 
\end{theorem}

\begin{proof}
    When $x_j = 0, \forall j \neq i$, we have $g_i = 0$ and $U_i = - c_i x_i$, from which we can obtain that $U_i(1, \mathbf{0})< U_i(0, \mathbf{0})$. Due to symmetric, $x_i = 0, \forall i$ is a PSNE.

To show that $x_i = 0, \forall i$ is a PSNE, we need to demonstrate that no player can gain by deviating from this strategy, given that all other players are also using this strategy.

For any player $i$, if $x_i = 0$, then $U_i = 0$. If $x_i = 1$, then player $i$ incurs a cost of $c_i$. Therefore, player $i$ has no incentive to deviate from $x_i = 0$, and this applies to all players.

Therefore, $x_i = 0, \forall i$ is a PSNE.
\end{proof}


\begin{remark}
    In cases where $c_i > 0$ for all $i$, we can generate PSNEs by considering each combination of connected components (CCs) in the optimal PSNE graph. Specifically, setting $x_i$ to 1 for nodes within these CCs and to 0 for nodes outside of them results in a PSNE.
\end{remark}

To begin, we define the optimal PSNE graph $G^*$ as a subgraph of the promotion network $G$ that includes only the nodes who invest in the optimal solution $\mathbf{x}^*$ and the corresponding edges among them. The algorithm MaxInvest is used to determine $\mathbf{x}^*$. As $\mathbf{x}^*$ is the optimal solution, nodes with $x_i^* = 0$ will not disclose in any case (From Lemma 4.2).

As per the claim of the trivial PSNE, it is evident that $x_i = 0$ for all $i$ in a connected component $C$ constitutes a PSNE among the agents in that component (trivial PSNE of this connected component). This strategy does not impact the strategies of other connected components. This is because the utility functions of nodes in different connected components are independent of one another.

Therefore, to generate PSNEs, we can select a set of connected components in the optimal PSNE graph $G^*$ and apply the strategy of setting $x_i$ to 1 for nodes within the selected connected components and 0 for other nodes in $G
$. This strategy results in a PSNE since no player can unilaterally change their strategy and increase their payoff.

Noted that we can use this approach to find PSNEs given a PSNE (not necessarily the optimal one).

\begin{remark}
     When $\exists c_i = 0$, we can use a spreading process to find a PSNE.
\end{remark}
(Spread Process) We consider the set of agents with $c_i = 0$ as a “seed set,” which are always disclosed as $U_i(1, \mathbf{x}_{-i}) \geq U_i(1, \mathbf{x}_{-i}) = 0$. We can start from them and do a process similar to the diffusion of the Linear Threshold model~\cite{Kempe:2003:MSI:956750.956769} to find a PSNE. The process can be described as follows: At each step, all nodes that have already been disclosed in the previous step remain disclosed. Any agent who satisfies the threshold condition in the current step will also disclose. We will continue iterating this process until no more agents can disclose. At this point, we have found a PSNE for the game.
\bibliography{sup}

\end{document}
