text:
\section{Introduction}
\label{sec:intro}
Emergence of social media has resulted in a large-scale, heterogeneous and dynamic space for the users to get engaged in different activities.
Studying engagement patterns in such platforms has its own merit for multiple purposes: market researchers can identify their potential audience for advertising campaigns and lucrative strategies; political campaigners can develop wide-scale trend analysis of the mass on the effect of their propaganda, etc.

Engagement dynamics in social media has thus attracted wide attention over a decade. Past studies attempted to predict (i) which pair of users is more likely to get engaged with each other based on their history \cite{schantl2013utility, yuan2016will}, and (ii) which posts will engage more users \cite{rowe2014mining, noguti2016post}.
All these studies tackled the engagement prediction problem in a {\em static manner} by considering the entire discussion as a whole, thus ignoring dynamic user engagement and the micro-dynamics controlling temporal growth.
The growth rate of a discussion, i.e., how many comments are being posted per unit time, varies over time, so as the user engagement. As the discussion continues, it unfolds diverse topics and user interactions, thus attracting different types of users over time. If we imagine users located in different points on a multidimensional space and clustered based on their coherent activities, a discussion can then be intuitively thought of as a growing and moving cloud in that space, attracting different sets of users in varying rates over time. \textit{The aim of the present work is to model the \textbf{time-varying engagement dynamics} of users with ongoing discussion -- a completely novel problem without any previous work, to the best of our knowledge.}  We build a framework which jointly models two phenomena -- user engagement from different clusters of users, and the rate of growth of discussions over time.

Different discussions {\em attract} different users at different rates. Although an individual user may get repelled by a particular discussion, the idea of repulsion cannot be consistently modeled without access to his/her cognitive data, or some platform-specific  features such as {\em dislike}. This implies that the interaction between a user and a discussion is essentially {\em attraction}, which can be zero but always non-negative. This motivates us to imagine a discussion to induce a gravity-like force towards the users. In fact, if we rely on the relativistic definition of gravity (explained in Sec. \ref{subsec:EFE}), it is even possible to adapt repulsion as a positive curvature in user manifold; however, in this work, we restrict ourselves to model interaction as `attraction' only.

Newtonian model of gravitation describes gravity as a force following inverse squared distance law between particles -- proportional to the mass of the particles and inversely proportional to their distance squared. Given two point particles of mass $m_1$ and $m_2$ placed at positions $\Vec{r_1}$ and $\Vec{r_2}$ respectively, the magnitude of the force of gravity between them, denoted by $F$ is given by,
\begin{equation}
    \label{eq:newton}
    F = \frac{Gm_1m_2}{|\Vec{r_1}-\Vec{r_2}|^2}
    \vspace{-2mm}
\end{equation}
where $G$ is the gravitational constant. In our hypothesis, discussions have some mass-like property which changes over time. Users `near' to a discussion get attracted more. A  `massive' discussion tends to attract more users and therefore  would achieve more  growth rate. The degree of this `massiveness' can be a function of the topic, relevance, properties of engaged users, etc. But Newtonian model does not explain how mass and distance (or spacetime, to be precise) interact with each other. In case of online discussions, users are not mere objects, rather they have  histories, which bear complex connection with each other and the discussion itself.

In physics, cosmic phenomena such as motion of the Mercury around the Sun \cite{le1859lettre}, bending of light passing near stars \cite{soldner1804deflection},
etc. cannot be explained by Newtonian model of gravity. A more sophisticated understanding of gravitation, which explains the failure of Newtonian model, was given by Einstein with his `General Theory of Relativity' \cite{einstein1915feldgleichungen} (GR Theory). Intuitively, relativistic theory of gravitation describes spacetime as an $(n+1)$-dimensional Riemannian manifold, with $n$ dimensions for space and one dimension for time. Gravity is simply the \textit{curvature} of this manifold at any point. According to this theory, the curvature can be caused by an object with mass and/or energy.
Any object free-falling through this spacetime manifold must follow the `straightest' path or \textit{geodesic} -- a path with constant directional derivative w.r.t. the manifold. More the mass/energy content of an object, more curved the spacetime will be around it, and hence more will be the effect of gravity. {\bf This `fusion' of seemingly heterogeneous physical properties like mass/energy and  spacetime by GR theory is the primary motivation behind our proposed model \texttt{RGNet}, which learns to efficiently fuse textual features of discussion with activity history of users in a temporal fashion to predict engagement dynamics.}

\begin{figure}[!t]
\centering
\includegraphics[width=0.45\textwidth]{Figures/model.pdf}
\vspace{-2mm}
\caption{(Color online) Problem of temporal engagement dynamics explained: comments and their replies form a tree, with the post as root; each contour in the figure depicts a comment window of size 5; contours grow temporally; color intensity of the contour corresponds to the rate of growth, i.e.,  how fast the comments are added in that window; lines from the user cluster to contour indicate that users from that cluster commented on that window.}
\label{fig:model}
\vspace{-5mm}
\end{figure}

Fig. \ref{fig:model} explains how we model engagement dynamics as a time-varying process (Table \ref{tab:notation} summarizes important notations). In particular,
our major contributions are five-fold:
\begin{itemize}
    \item  We propose \texttt{GUVec}, a novel algorithm to represent users of a discussion platform as fixed dimensional vectors based on their temporal, communicative and semantic proximity.
     \item We propose \texttt{RGNet}, an engagement prediction model which represents ongoing discussions as time-varying `dust clouds' in the user manifold

     and models them using relativistic theory of gravity to predict which clusters of users from the manifold are likely to get engaged,

     and how fast the discussion cloud will grow.
     \item We propose two deep learning based models using Long Short-Term Memory (LSTM) cells, and a model similar to \texttt{RGNet}\ based on Newtonian model of gravity. We also adopt the work by Rowe and Alani \cite{rowe2014mining} in temporal setting. All these models are considered as baselines.
     \item We  also predict user engagement by adopting \texttt{RGNet}\ in a non-temporal setting (for the sake of a direct comparison with the existing baseline)   -- given a post, whether any user will comment to that post or not.
     \item We perform comprehensive evaluation on the Reddit CMV dataset \cite{tan2016winning} (for temporal engagement prediction) and Reddit \textit{r/news} community (for non-temporal engagement prediction) to show the efficiency of \texttt{GUVec}\ and \texttt{RGNet}.

\end{itemize}

     {\bf To the best of our knowledge, \texttt{RGNet}\ is the first model of its kind which is inspired by the fundamental theories of classical mechanics.}

\begin{table}
    \centering
    \small
    \caption{Important notations used throughout the paper.}
    \vspace{-2mm}
    \label{tab:notation}
    \begin{tabular}{c|c}
    \hline
         {\bf Notation} & {\bf Denotation}  \\
         \hline
         $\mathbf{A}$ & User co-occurrence matrix\\
         $g_{\mu\nu}, g^{\mu\nu}$ & Metric tensor, inverse metric tensor\\
         $T_{\mu\nu}$ & Stress-energy tensor\\
         $R_{\mu\nu}, R$ & Ricci tensor, Ricci scalar\\
         $\sigma_1(x)$ & $max(0,x)$\\
         $\sigma_2(x)$ & $(1+\exp(-x))^{-1}$\\
         $w$ & Window size of comments\\
         $n$ & No. of user clusters \\
         $N$ & No. of windows in a discussion\\
         $\mathbf{U}$ & User set\\
         $C_i, \mathbf{C}$ & $i^{th}$ user cluster, set of cluster centers\\
         \hline
    \end{tabular}
    \vspace{-5mm}
\end{table}

 \section{\texttt{GUVec}: Global User Embedding}
 \label{sec:user_embedding}

 To compute user vectors from a discussion corpus, our proposed global user embedding method \texttt{GUVec}\ first constructs a user-user co-occurrence matrix $\mathbf{A}$. We use three different notions of proximity between two users $u_i$ and $u_j$: (i) {\bf Communicative Proximity:} they communicated with each other in a discussion; this happens when $u_i$ replied to $u_j$ in a discussion or vice versa, meaning they are present in the same chain of comments; (ii) {\bf Temporal Proximity:} they are temporally close to each other; they are engaged in same discussion (have not replied to each other) nearly at the same time; (iii) {\bf Semantic Proximity:} they are engaged in similar type of discussions.

 To construct a meaningful embedding of users, we only take those who are engaged in at least two discussions.

 Given the entire set of such users denoted by $\mathbf{U}$, the co-occurrence matrix $\mathbf{A}$ is symmetric and of dimension $|\mathbf{U}|\times|\mathbf{U}|$. To compute semantic proximity, we use ConceptNet Numberbatch word-vectors

 \cite{speer2016ensemble}. We take the words present in the discussion titles (after removing stopwords) and compute the weighted average of the corresponding word vectors. This weighted average now represents the title vector $T_k$ of  discussion $D_k$.

For any pair of users $u_i, u_j \in \mathbf{U}$, their proximity $A_{ij} = A_{ji} \in \mathbf{A}$ is computed as follows:
\begin{itemize}
     \item  Communicative Proximity: If $u_i$, $u_j$ replied to each other, then increment $A_{ij}$ by 2.
     \item Temporal Proximity: If $u_i$, $u_j$ commented on the same discussion at time $t_i$ and $t_j$ respectively, but did not reply to each other, then
     \begin{equation}
     \label{Eq:temp_sim}\small
     A_{ij}=A_{ij}+(1+\exp(-\alpha))^{-1}, \text{ where }
         \alpha = \frac{t_{end}-t_{start}+1}{|t_i-t_j|+1}
     \end{equation}
     where $t_{start}$ and $t_{end}$ are the starting and ending times of the discussion, respectively.
     \item Semantic Proximity: If $u_i$, $u_j$  commented on different discussions $D_m$ and $D_n$, respectively, then
    \begin{equation}
     \label{Eq:sem_sim}\small
         A_{ij} =
  \begin{cases}
                                   A_{ij}+\cos{\theta}& \text{if } \theta\leq\theta_0 \\
                                   A_{ij} & \text{otherwise}
  \end{cases}
  \vspace{-2mm}
\end{equation}
where $\theta = \arccos(\frac{T_m^\top  T_n}{|T_m|\cdot|T_n|})$, and  $0\leq\theta_0\leq\frac{\pi}{12}$ is a threshold angle (Sec. \ref{sec:setup} for parameter selection).

\end{itemize}

  In Eq.~\ref{Eq:temp_sim}, $\alpha$ accounts for how much temporally close two comments are w.r.t. the total time span of the discussion. This normalizes temporal proximity of discussions growing in different rates. We put highest proximity value for two users if they replied to each other. Both the terms $(1+\exp(-\alpha))^{-1}$ and $\cos\theta$ have upper bound of $1$. Therefore, for any pair of users, the contribution of their temporal and semantic proximity taken together cannot exceed their communicative proximity, which is incremented by 2.

Once $\mathbf{A}$ is computed, \texttt{GUVec}\ minimizes the following objective function
to obtain user vectors:
 \begin{equation}
 \small
 \label{Eq:embedding_loss}
 \vspace{-1mm}
     J = \sum_{i,j}^{|\mathbf{U}|} \log(1+A_{ij})(\mathbf{v_i}^\top \mathbf{v_j} +b_i+b_j -\log(1+A_{ij}))^2
     \vspace{-2mm}
 \end{equation}
 where $v_i$ and $b_i$ correspond to the $i^{th}$ user vector and bias, respectively. This objective function bears some similarity to that of GloVe embedding \cite{pennington2014glove}. Eq.~\ref{Eq:embedding_loss} uses the hypothesis that, for any two users $i$ and $j$, the term $\mathbf{v_i}^\top \mathbf{v_j}$ should be proportional to the logarithm of the probability of $j$ occurring in the context of $i$. This probability can be computed as $P_{ij}=A_{ij}/A_{i}$, thus $\log P_{ij}=\log A_{ij}-\log A_i$. Since $P_{ij}=P_{ji}$, i.e., the probability of a user $i$ appearing in user $j$'s context is same as the reverse, we need to exclude the term containing $A_{i}$. Hence we introduce the bias terms $b_i$ and $b_j$ in Eq.~\ref{Eq:embedding_loss}. We also need to assure that vectors of highly co-occurring users should be computed with greater accuracy. Therefore,  we introduce the weighing term $\log(1+A_{ij})$.

 \begin{figure}[!t]
\centering
\includegraphics[width=0.48\textwidth]{Figures/rgnet.pdf}
\vspace{-3mm}
\caption{Architecture of \texttt{RGNet}: C$_j$ signifies $j^{th}$ comment (chronologically ordered by time) in the discussion. Colored arrows represent different transformations as defined in Eq.~\ref{Eq:model_1} to Eq.~\ref{Eq:model_output}.}
\label{fig:rgnet}
\vspace{-5mm}
\end{figure}

 \section{\texttt{RGNet}: Modeling User Engagement}
 After computing user vectors, we group them into $n$ clusters using standard clustering methods (see Sec. \ref{sec:setup}). Henceforth, the cluster centers $C_1,\cdots, C_n \in \mathbf{C}$ will represent $n$-regions of user manifold. We will first explain the Einstein Field Equations and their components, followed by how \texttt{RGNet}\ incorporates them in modeling user engagement. Fig. \ref{fig:rgnet} shows a schematic architecture of \texttt{RGNet}.

 \subsection{Einstein Field Equations (EFE)}
 \label{subsec:EFE}
 In general theory of relativity, spacetime is a four-dimensional manifold $\mathcal{M}$ with one dimension of time and three dimensions of space. Gravity is not an external force (like electromagnetic or nuclear forces), rather an intrinsic property of spacetime, defined as \textit{curvature} in $\mathcal{M}$. Any object without the effect of any force, will follow a {\em geodesic} (a curve for which directional co-variant derivative along the tangents of the curve remains zero) along this manifold. The geometry of the spacetime manifold is defined by sixteen Einstein Field Equations \cite{einstein1915feldgleichungen}:
 \begin{equation}
 \small
 \label{Eq.EFE}
     R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu} =\frac{8\pi G}{c^4}T_{\mu\nu}
 \end{equation}
 This is a tensor equation, with $\mu$, $\nu$ corresponding to dimension indices of the spacetime. As there are total four dimensions (one for time and three for space), the pair $\mu,\nu$ can take sixteen different values. $G$, $\Lambda$ and $c$ are three constants -- Newtonian gravitational constant, cosmological constant and velocity of light in vacuum, respectively. $g_{\mu\nu}$ is called the \textit{metric tensor} of the manifold. This is a {\em contra-variant} tensor which gives the idea of distance between two vectors on a manifold:
 \begin{equation}
 \small
 \label{Eq:metric_tensor}
     ds^2 = \sum_{\mu}\sum_{\nu}g_{\mu\nu}dx^{\mu}dx^{\nu}
     \vspace{-2mm}
 \end{equation}
 where $dx^i$ is the difference in the $i^{th}$ component of two vectors. It has its covariant counterpart $g^{\mu\nu}$, which is  called the {\em inverse metric}.

$R_{\mu\nu}$ is the \textit{Ricci curvature tensor}. $R$ is the corresponding \textit{Ricci scalar}. The change in a vector for parallel transport (i.e., following a geodesic) along two different infinitesimal flows in a smooth manifold is given by the Riemann Curvature tensor. Ricci tensor is the contraction of Riemann tensor on the second index.
Both Ricci and Riemann tensors can be computed from second order derivatives of the metric tensor. We define {\em Christoffel symbol of second kind} as $\Gamma_{kl}^{i}$,
\vspace{-1mm}
 \begin{equation}
 \small
     \label{eq:christoffel}
     \Gamma_{kl}^{i} = \frac{1}{2}g^{im}(g_{mk,l}+ g_{ml,k}-g_{kl,m})
 \end{equation}
 where $g_{mk,l}$ is the partial derivative of $g_{mk}$ with respect to the $l^{th}$ component. Then, $R_{ij}$ is defined as,
 \begin{equation}
     \label{eq:ricci_christoffel}
     R_{ij} = \Gamma_{ij,l}^{l}-\Gamma_{il,j}^{l}+ \Gamma_{ij}^{m}\Gamma_{lm}^{l} - \Gamma_{il}^{m}\Gamma_{jm}^{l}
 \end{equation}
 Eqs.~\ref{eq:christoffel} and \ref{eq:ricci_christoffel} are seemingly very complex to directly compute using metric tensor. However, the important fact is that  Ricci tensor can be computed as a function of derivatives of the metric tensor, and therefore, as differential function of the components of points in the manifold.
 Ricci scalar is simply the trace of the Ricci tensor:
 \vspace{-1mm}
 \begin{equation*}
 \small
     R = g^{\mu\nu}R_{\mu\nu}
     \vspace{-1mm}
 \end{equation*}
 Intuitively, in a 2D manifold, a zero Ricci scalar at a point indicates that the manifold is flat at that point; negative value indicates a saddle point, and positive value indicates a hill. In Eq.~\ref{Eq.EFE}, the term $R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$ describes the curvature of the spacetime manifold at any point. Its trace with respect to the inverse metric yields negative scalar curvature:
 \vspace{-1mm}
 \begin{equation}
 \small
     g^{\mu\nu}(R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}) = -R
     \vspace{-1mm}
 \end{equation}
 \begin{figure}[!t]
\centering
\includegraphics[width=0.17\textwidth]{Figures/stress_energy.pdf}
\vspace{-2mm}
\caption{Components of the stress-energy tensor. The components in blue, yellow, green and pink represent mass density, momentum density, momentum flux and  pressure, respectively.}
\label{fig:stress-energy-tensor}
\vspace{-5mm}
\end{figure}
 $T_{\mu\nu}$ is called the \textit{stress-energy tensor}. For an infinitesimal volume of spacetime manifold, its components represent the properties as shown in Fig.~\ref{fig:stress-energy-tensor}. For an isolated massive particle, all the components except $T_{00}$ are zero. For a cloud of dust, only the diagonal elements have non-zero value.

Multiplying both sides of Eq.~\ref{Eq.EFE} by inverse metric tensor $g^{\mu\nu}$ yields,
 \begin{equation}
 \small
 \label{Eq:rel_curvature}
     \frac{8\pi G}{c^4}g^{\mu\nu}T_{\mu\nu} = -R+g^{\mu\nu}\Lambda g_{\mu\nu}
 \end{equation}

 \subsection{EFE in Discussion Spacetime}
Eq.~\ref{Eq.EFE} does not have any static solution without the cosmological constant $\Lambda$, indicating the universe is expanding. Einstein introduced $\Lambda$ to make it static, which, upon the observation of expanding universe in reality by Hubble \cite{hubble1929relation}, was discarded later. In our particular case of learning engagement dynamics in discussions using general relativity, we also omit  $\Lambda$ and reduce the constants in Eq.~\ref{Eq:rel_curvature} to yield
 \begin{equation}\small
 \label{Eq.model}
     \sum_{j=0}^{d}\sum_{k=0}^{d}g^{jk}(C_i)M_{jk}(C_i, T_i) = R'(C_i)
 \end{equation}
 where $C_i$ represents the position of $i^{th}$ user cluster in the user manifold we computed in Sec.~\ref{sec:user_embedding}, $T_i$ is the set of features representing the discussion, $d$ is the dimension of the user vectors, $g^{jk}$ is the inverse metric tensor which is computed as a function of cluster positions, $M_{jk}$ is the stress-energy tensor counterpart for discussion which is computed as a function of cluster positions and features of discussion. We prepend the time value to each user vector to convert it into a $(d+1)$-dimensional spacetime manifold. \texttt{RGNet}\ learns each component of Eq.~\ref{Eq.model} as a series of non-linear transformations: $\mathbf{X'}=\sigma(\mathbf{W}^\top \cdot \mathbf{X} + \mathbf{B})$, where $\mathbf{X}$ and $\mathbf{X'}$ are input and output of the transformation respectively, $\sigma$ is a bounded non-linear function, $\mathbf{W}$ and $\mathbf{B}$ are weight and bias matrices to be learned respectively.

It is important to note that relativistic model of spacetime requires multiple constraints to be fulfilled. First of all, physical laws should be observer independent -- one can choose any frame of reference (rotated, translated, moving w.r.t. another frame of reference) and the physics must remain the same. General Relativity requires this constraint to be local. The user manifold obtained from \texttt{GUVec}\ computes the position of a user in the manifold using the vector dot product, which is invariant to rotation and translation. Moreover, it takes into account the temporal proximity of two users. We expect this to reflect invariance to temporal transformations of the manifold as well. However, engagement over online discussions is not a deterministic physical process. We claim it to be only analogous to spacetime geometry and not an exact replica. So in our case, Einstein's Field Equations are only abstract approximation learnt by \texttt{RGNet}. An exact mathematical model of engagement is far more complex, if not intractable.

We define the temporal progress of a discussion as windows of comments of fixed size $w$. This means, at the $i^{th}$ step, the size of our discussion is $1+iw$ (post + comments), and we need to predict for the next $w$ comments. Due to variable size of discussions, we define maximum size of the discussion to be $(1+Nw)$, where $N$ is the number of windows, and hence, the number of prediction steps for a single discussion. All the discussions with size less than the maximum size are zero-padded at the end.

\subsection{Feature Selection}\label{sec:feature_selection}

For the original post and every comment in the discussion, we extract  following features based on the content, user, surface structure of the text.

\noindent{\bf (i)} {\bf Content Features:} \\

\indent{$\bullet$} {\em Average of tf-idf scores of the tokens}. This represents how many unique and relevant words are used in the comment.\\
\indent{$\bullet$} {\em LIX readability score} \cite{bjornsson1983readability}, computed as:
         $r = \frac{|w|}{|s|}+100\times\frac{|cw|}{|w|}$,
where $w$ and $s$ are the sets of words and sentences respectively, and $cw$ is the set of words with more than six characters. Larger the value of $r$, harder the comment/post is to read in a short time.\\
\indent{$\bullet$} {\em Cumulative entropy of terms}, given by $
         p=\frac{1}{|T|}\sum_{t\in T}tf_t(\log|T|-\log(tf_t))$,
where $T$ is the set of all unique tokens in the corpus, and $tf_t$ is the frequency of term $t$ in the comment/post.\\
\indent{$\bullet$} {\em Polarity of the comment/post}, i.e., sum of sentiment intensity scores of the unique terms computed using SenticNet \cite{cambria2018senticnet}. We also use the total number of positive and negative sentiment words as polarity features.

\noindent{\bf (ii)} {\bf Surface Features}: We use the following surface features -- total number of sentences in the text, average number of words per sentences, count of URLs present in the comment, depth of the comment in discussion tree, time difference of the comment with the post and the count of closing punctuation markers, i.e., `.',`!' and `?' (as different types of closing punctuation markers signify different discourse).

\noindent{\bf (iii)} {\bf Latent Semantics}: We use the pre-trained word vectors mentioned in Sec. \ref{sec:user_embedding} to represent the latent semantics of the text. Every comment is represented as a vector: $
         V = \frac{1}{|C|}\sum_{t\in C} (tf\text{-} idf_t\cdot W_t$),
where $C$ is the set of unique terms in the comment, and $W_t$ is the word vector of term.
For the post, we also use the title vectors mentioned in Sec. \ref{sec:user_embedding} as features.

\noindent{\bf (iv) User Features}: We use user vectors computed by \texttt{GUVec}\ as user-based features, which reflect past activity and connections of a user.

For a total $f$ number of features representing each comment, the representation of a post $\mathbf{X_1}$ is then an array of size $(f+d_w)$ with $d_w$ being the size of word vectors used; all the comments taken together $\mathbf{X_2}$ is an array of size $N\times w\times f$, and user manifold regions $\mathbf{C}$ are represented as an array of size $N\times n\times (d+1)$ for a single discussion.

 \subsection{Stress-Energy Tensor of Discussion}
 First, we compute an intermediate representation of the post and the comments with dimension:
 \vspace{-1mm}
 \begin{equation}
 \small
 \label{Eq:model_1}
 \begin{split}
     \mathbf{X'_1} &= \sigma_1(\mathbf{W_1}\cdot \mathbf{X_1}^\top + \mathbf{B_1})\\
     \mathbf{X'_2}[i] &= \sigma_1(\mathbf{W_2}\cdot \mathbf{X_2}[i]^\top + \mathbf{B_1})\\
     \mathbf{X'} &= (\mathbf{X'_1}, \mathbf{X'_2})
 \end{split}
 \vspace{-1mm}
 \end{equation}
$\mathbf{X'}$ now contains representation of the post and each of the $N$ comment windows. $\mathbf{W_k}$ and $\mathbf{B_k}$ (where $k=1,2,\cdots$) mentioned throughout the paper indicate the learnable weight and bias matrices, respectively.
$\sigma_1$ is the rectified linear unit function. Now all the representations from 0 to $(i-1)^{th}$ steps should contribute at  $i^{th}$ step. Therefore, we take a weighted cumulative sum of $\mathbf{X'}$:
\begin{equation}
\small
 \label{Eq:model_2}
 \mathbf{X''}[i]= \frac{\sum_{j=0}^{i}\mathbf{W_3}[j]\mathbf{X'}[j]}{\sum_{j=0}^{i}w_j}
 \end{equation}

 Next, we concatenate $\mathbf{X''}[i]$ to each $C_l\in \mathbf{C}$ and compute the corresponding stress-energy tensor:
 \begin{equation}
 \small
 \label{Eq:model_3}
 \begin{split}
 \mathbf{M}[i][l] &= \sigma_2(\mathbf{W_4}\cdot\sigma_2(\mathbf{W_5}\cdot (\mathbf{X''}[i],C_l)^\top\\
 &+\mathbf{B_4})^\top+\mathbf{B_3})
 \end{split}
 \end{equation}
 where $\sigma_2(x)=(1+\exp(-x))^{-1}$. Each $\mathbf{M}[i][l]$ is a $(d+1)$-dimensional vector representing the diagonal elements of the stress-energy tensor $M_{jk}$ of Eq.~\ref{Eq.model}.

\subsection{Inverse  Metric Tensor}
 We compute the values of inverse metric tensor $\mathbf{g}_{inv}$ for $i^{th}$ prediction step at $l^{th}$ cluster region as a function of the cluster center as follows:
 \vspace{-1mm}
 \begin{equation}
 \small
 \mathbf{g}_{inv}[i][l] = \sigma_2(\mathbf{W_6}\cdot\sigma_2(\mathbf{W_7}\cdot C_l^\top+\mathbf{B_6})^\top +\mathbf{B_5})
 \vspace{-1mm}
 \end{equation}
 Again, this is a $(d+1)$-dimensional vector which represents the diagonal of the inverse metric tensor $g^{jk}$ of Eq.~\ref{Eq.model}.

 \subsection{Curvatures of Manifold}
Once we obtain the stress-energy tensor $\mathbf{M}[i][l]$ and the inverse matrix tensor $\mathbf{g}_{inv}[i][l]$ of the discussion at $i^{th}$ prediction step for $l^{th}$ cluster, we  compute the scalar curvature of the manifold at $l^{th}$ cluster based on Eq. \ref{Eq.model} as,
\vspace{-1mm}
 \begin{equation}
 \small
     R'[l] = \sum_{j=0}^{d}\mathbf{M}[i][l][j]\cdot\mathbf{g}_{inv}[i][l][j]
     \vspace{-1mm}
 \end{equation}
 This step actually performs the fusion of textual and user interaction features. The extent to which the discussion attracts users towards it for the entire manifold can be computed as the weighted sum of each of $R'[l]$, given by,
 \vspace{-1mm}
 \begin{equation}
 \label{eq:r_total}
 \small
     R_{total} = \sum_{l=1}^{n}\mathbf{W_8}[l]\cdot R'[l]
     \vspace{-1mm}
 \end{equation}
 Finally, we define cluster engagement probability $y_1$ and discussion growth velocity $y_2$ as two nonlinear functions of cluster-wise scalar curvature and total curvature respectively:
 \vspace{-1mm}
 \begin{equation}
 \small
 \label{Eq:model_output}
         y_1 = \sigma_2(R'),\ \ \
         y_2 = \sigma_1(R_{total})
         \vspace{-1mm}
 \end{equation}
 so that $0\leq y_1\leq 1$ and $0\leq y_2$, befitting to both user cluster engagement prediction and growth rate forecasting tasks. Altogether, we train \texttt{RGNet}\ to learn the following function:
 \vspace{-2mm}
 \begin{equation}
 \small
     (y_1,y_2)=\mathcal{F}(\mathbf{X_1},\mathbf{X_2},\mathbf{C}|\mathbf{W_{1,\cdots 8}}, \mathbf{B_{1,\cdots 6}})
     \vspace{-1mm}
 \end{equation}

For user cluster engagement prediction task (multi-label classification), we use \textit{binary cross-entropy} loss with $0.5$ as the threshold, and for the growth rate forecasting (regression),  we use \textit{mean squared error} loss to train \texttt{RGNet}.

 \section{Baselines for Temporal Engagement}
 \label{sec:baseline}
 Due to the lack of existing baseline in predicting temporal user engagement, we design four baselines:

 (i) {\bf Newtonian Model:} This model is similar  to \texttt{RGNet}\ except it uses  Newtonian model over flat space instead of spacetime manifold. We compute $\mathbf{X''}$ based on Eqs. \ref{Eq:model_1} and \ref{Eq:model_2} using same set of features, except instead of computing a stress-energy tensor for each cluster (Eq. \ref{Eq:model_3}), we compute a scalar mass $M_i$ at the $i^{th}$ prediction step:
 \begin{equation}\small
 \label{Eq:New_model_3}
 \begin{split}
 M_i &= \sigma_2(\mathbf{W'_1}\cdot\sigma_2(\mathbf{W'_2}\cdot \mathbf{X''}[i]^\top +\mathbf{B'_1})^\top+\mathbf{B'_2})
 \end{split}
 \end{equation}

 We compute the position of the discussion at $i^{th}$ prediction step on the $d$-dimensional space as a weighted average of user vectors $\mathbf{r}$ commented till $i^{th}$ prediction,
 \begin{equation}
 \small
     r_i = \Big(\sum_{j=0}^{iw-1}\mathbf{W'_3}[j]\Big)^{-1} \Big(\sum_{j=0}^{iw-1}\mathbf{W'_3}[j]\cdot \mathbf{r}[j] \Big)
 \end{equation}

 We then define $y_1$ and $y_2$ from Eq.~\ref{Eq:model_output} as,
 \vspace{-1mm}
 \begin{equation}
 \small
 \label{Eq:newmodel_output}
 y_1 = \sigma_2(\frac{M_i}{|r_i-C_l|^2});\ \ y_2 = \sigma_1(\sum_{l=1}^{n}\mathbf{W'_4}[l]\cdot \frac{M_i}{|r_i-C_l|^2})
 \vspace{-1mm}
 \end{equation}
 and train the model using same loss functions.

(ii) {\bf LSTM Models:} We implement two LSTM models; one with the features we defined in Sec. \ref{sec:feature_selection} ({\bf LSTM-f}), another using raw text data ({\bf LSTM-r}). To input raw text data, we use one-hot encoding of each word and initialize an embedding layer with pre-trained word vectors mentioned in Sec.~\ref{sec:user_embedding}. This model uses an extra layer of LSTM cells to compute the representation of comments from words. Both these models use same loss functions (binary cross-entrpoy and mean squared error). Fig. \ref{fig:rnn_baseline} shows the architecture.
 \begin{figure}[!t]
\centering
\includegraphics[width=0.48\textwidth]{Figures/rnn_baseline.pdf}
\vspace{-2mm}
\caption{Architecture of the LSTM-f and LSTM-r models. LSTM layer in dotted region present in the left is used only for LSTM-r.}
\label{fig:rnn_baseline}
\vspace{-5mm}
\end{figure}

(iii) {\bf Logistic Regression:} Lastly, we implement a logistic regression classifier adapted from the model proposed by Rowe and Alani \cite{rowe2014mining}. We consider the same set of features except {\em the duration of a user in the community} as Reddit does not provide this data. The authors broadly categorized the features used as \textit{social} and \textit{content} features. Their original work is not designed for temporal engagement modeling. Also, they performed a binary classification of whether a post will get commented or not. We adopt this model for our task with two modifications: (a) we take each user cluster and predict whether a user from this cluster will comment (user-network based features are calculated for each cluster separately, not the whole user-user interaction network) and (b) at each prediction step for a particular discussion, we take the post and comments (if any) till that instance as a single entity -- content features are calculated from the merged texts of post and comments, and the average of social features of the users who posted/commented is considered as the cumulative social feature. {\em This model is  made only for predicting engagement of user clusters}.

\section{Non-temporal Engagement Prediction}
As already stated, our defined problem of predicting temporal engagement dynamics is novel, and there is no existing work which can be directly considered to compare with \texttt{RGNet}. Rowe and Alani \cite{rowe2014mining} (henceforth, referred as R\&A) proposed a framework to predict engagement in a non-temporal manner. Given a post, their model predicts whether it will attract any user or not. We modify \texttt{RGNet}\ to suit this task and compare the performance.

We hypothesize that, if a post fails to curve the user manifold `effectively', it will not attract any users in the future. For this, we input the post feature $\mathbf{X_1}$ to \texttt{RGNet}. As this is a one-shot prediction for the post only, the comment feature $\mathbf{X_2}$ as well as the comment window are irrelevant here. Also, all the occurrences of $i$ in the governing equations of \texttt{RGNet}\ have a single value (i.e., $0$), as this is the first prediction step in the full implementation of \texttt{RGNet}. Therefore, the stress-energy tensor $\mathbf{M}$ in Eq. \ref{Eq:model_3} is computed from $\mathbf{X'_1}$ only (first part of Eq. \ref{Eq:model_1}). Total curvature $R_{total}$ (in Eq. \ref{eq:r_total}) estimates the degree of total attraction generated by the post. We compute the probability of a post to attract any user at all from the total curvature as:
\vspace{-1mm}
\begin{equation}
\small
    y_3 = \sigma_2(R_{total})
    \vspace{-1mm}
\end{equation}
Here, $y_3$ ranges in interval $(0,1)$. We take $y_3\leq 0.5$ as negative class (post fails to attract any user), and positive class, otherwise.

\section{Experimental Setup}\label{sec:setup}
We describe the datasets and parameter selection for \texttt{RGNet}: both for temporal and non-temporal engagement prediction.
\subsection{Datasets}
The Reddit CMV dataset that we used contains $18,363$ discussions from Jan 1, 2013 - May 7, 2015 for training, and $2,263$ discussions from May 8, 2015 - Sep 1, 2015 for testing. We excluded comments posted by deleted users and delta-bots (carrying author-tags ``deleted'' and ``DeltaBot'' respectively) and users who commented only once.

This leaves our training (test) set with $46,121$ ($6,044$) users and $1,011,890$ ($112,432$) comments in total.

However, this CMV dataset was originally filtered, such that there is no post which failed to attract any user comment. Therefore, we cannot use this dataset for non-temporal engagement prediction. For this task, we crawled posts from Reddit \textit{news} community. We collected a total of $43,343$ posts from Sep 1, 2016 to Jan 16, 2019, out of which $5,449$ posts do not have any comments. To avoid classification bias, we take equal number of posts containing comments. Here again, we excluded users who have commented/posted only once or carry the author tag ``deleted'' (delta-bots does not appear in this community) to compute the \texttt{GUVec}\ embeddings. This results in a total of $29,431$ users.

\subsection{Parameter Selection}
While constructing the  co-occurrence matrix $\mathbf{A}$, computing the semantic proximity is computationally the most expensive part as we need to count for all possible pairs of users between every pair of discussions. The choice of  $\theta_0$ in Eq.~\ref{Eq:sem_sim} can significantly reduce this cost if we pre-compute  $\cos\theta$ between pairs of  discussion titles and take into account only those discussions having $\theta \leq \theta_0$. In Fig.~\ref{fig:parameter}(a), we plot the number of discussion title pairs with $\theta$ between them. Discussion pairs with $\theta\leq \frac{\pi}{12}$ amounts to $1.5\%$ of the total pairs. We find that the number of user-pairs for this subset is $\mathcal{O}(|\mathbf{U}|^{\frac{3}{2}})$. Therefore we choose $\theta_0$ to be $\frac{\pi}{12}$.

We vary the embedding dimension $d$ from 16 to 256.
Fig.~\ref{fig:parameter}(b) shows that the performance of \texttt{RGNet}\ does not change much after $d=128$.
We cluster user embedding space using K-means by varying K from $8$, $16$, $24$ to $32$.\footnote{We also tried with agglomerative and DBSCAN methods for user clustering and observed similar results.}
In case of engagement modeling, we vary window size from $5$ to $20$.  All the models except the logistic regression were optimized using Adam \cite{kingma2014adam} optimization algorithm. Unless otherwise stated, we use the following parameter values as default: $\theta_0 = \frac{\pi}{12}$, $d=128$, $w=15$, and $K=n=8$.

\begin{figure*}[!t]
\centering
\includegraphics[width=0.8\textwidth]{Figures/parameter.pdf}
\vspace{-2mm}
\caption{(a) Selection of $\theta_0$, (b) variation of \texttt{RGNet}'s performance w.r.t. $d$, (c) performance of user embedding methods for both the temporal tasks with \texttt{RGNet}.}
\label{fig:parameter}
\vspace{-5mm}
\end{figure*}

\begin{table*}[!t]
     \centering
     \caption{Evaluation metrics used for multi-label classification. There are $n$ multi-label instances $(x_i, y_i)$ with $y_i$ being a binary vector of size $L$, and $h(x_i)$ being the predicted set of labels for $x_i$. $\lVert a\rVert_1$ denotes L1 norm of vector $a$, $\bigoplus$ signifies element-wise XOR, $y_k$ and $h^k(x_i)$ are $k^{th}$ entries of true label set and predicted label set respectively, and $I(s)=1$ if $s$ is true, and $0$ otherwise.}
     \small
     \begin{tabular}{|l|l|l|}
     \hline
          \multicolumn{1}{|c|}{\bf Metric} & \multicolumn{1}{c|}{\bf Description} & \multicolumn{1}{c|}{\bf Formula}  \\
     \hline
          Micro F1 & Micro average of precision and recall on all binary labels & $\frac{2\times \sum_{i=1}^{n}\lVert h(x_i)\cap y_i \rVert_1}{\sum_{i=1}^{n} \lVert h(x_i)\rVert_1+\sum_{i=1}^{n}\lVert y_i\rVert_1}$\\
    \hline
          Macro F1 & Macro average of precision and recall
          & $\frac{1}{L}\sum_{k=1}^{L}\frac{2\times \sum_{i=1}^{n}h^k(x_i)y_i^k}{\sum_{i=1}^{n}h^k(x_i)+\sum_{i=1}^{n}y_i^k}$\\
     \hline
          Hamming Loss & Average error rate over all the binary labels
          & $\frac{1}{n}\sum_{i=1}^{n}\frac{1}{L}\lVert h(x_i)\bigoplus y_i\rVert_1$\\
     \hline
          Subset $0/1$ Loss & Average \% when predicted label set is exactly correct
          & $\frac{1}{n}\sum_{i=1}^{n}I(h(x_i)\neq y_i)$\\
     \hline
     \end{tabular}
     \vspace{-5mm}
     \label{tab:metrics}
 \end{table*}

\section{Experimental Results}

We perform comparative evaluation for three tasks separately. For temporal engagement dynamics, we compare \texttt{RGNet}\ with other baselines to predict user cluster engagement and growth rate forecasting. For non-temporal engagement, we present the performance of \texttt{RGNet}\ for different number of clusters and compare it with R\&A  \cite{rowe2014mining}. We study the importance of different features for these tasks. We also show the efficiency of \texttt{GUVec}\ compared to other embedding methods for temporal engagement tasks mentioned above, and empirically show the complexity of \texttt{GUVec}. In the end, we present a case study of user cluster engagement prediction obtained from \texttt{RGNet}.

\begin{table}[]
    \centering
    \caption{Multi-label classification performance of the competing methods for temporal user engagement prediction. $\downarrow$ ($\uparrow$) indicates the smaller (larger) the value, the better the performance.}
    \scalebox{0.9}{
    \begin{tabular}{|l|c|c|c|c|}
    \hline
         {\bf Method} & {\bf HL} $\downarrow$ & {\bf MiF} $\uparrow$ & {\bf MaF} $\uparrow$ & {\bf 0/1} $\downarrow$   \\\hline
         \texttt{RGNet} & {\bf 0.27} & {\bf 0.72} & {\bf 0.65} & {\bf 0.78} \\\hline
         LSTM-f & 0.36 & 0.64 & 0.57 & 0.81 \\
         LSTM-r & 0.37 & 0.61 & 0.56 & 0.81  \\
         Newton. & 0.40 & 0.56 & 0.51 & 0.86   \\
         Logistic Regression & 0.36 & 0.62 & 0.56 & 0.85 \\\hline
    \end{tabular}}
    \vspace{-5mm}
    \label{tab:accuracy_cluster}
\end{table}

\subsection{Predicting Temporal Engagement of User Clusters}
We pose user cluster engagement prediction problem as a multi-label classification problem.
At $i^{th}$ prediction step, let there be $m$ comments in the $(i+1)^{th}$ window, with $m\leq w$. Let there be $n$ clusters of the user manifold. Each instance in our dataset corresponds to a window. For $(i+1)^{th}$ window, we  create the ground-truth binary vector $\mathbf{Y_{i+1}}$ of size $n$ such that, $Y_{i+1}[j]=1$ if there is at least one comment in the $(i+1)^{th}$ window from a user belonging to $j^{th}$ cluster, $0$ otherwise.

Table \ref{tab:accuracy_cluster} reports the performance of the competing methods based on four standard metrics used for multi-label classification \cite{Shi:2014:MCB:2611448.2505272} (see Table \ref{tab:metrics} for the description) -- Hamming Loss (HL), Micro F1 (MiF), Macro F1 (MaF), and Subset 0/1 (0/1). \texttt{RGNet}\ outperforms others across all the metrics -- it beats the best baseline (LSTM-f) by 12.5\% (14.03\%) higher  Micro (Macro) F1 .

Table \ref{tab:varyingwindowcluster}(a) shows that as the number of clusters $n$ grows, the average degradation of performance for \texttt{RGNet}\ and Newtonian model is minimum (15.22\% and 10.15\% respectively averaged across consecutive values of $n$) compared to others (15.14\% for LSTM-f).  These two models  benefit from the fact that, with smaller cluster size, cluster centers exhibit accurate locality of the cluster, which helps them compute more accurate curvature and the distance vector.  Table \ref{tab:varyingwindowcluster}(b) shows that for most of the models, the performance increases as the window size $w$ grows.

To check how homogeneity of users (in terms of their clusters) already engaged till $i^{th}$ window affects the performance for $(i+1)^{th}$ window, we compute the
entropy of the cluster membership of users $\mathbf{U}_{i}$ engaged till $i^{th}$ window, as $H_i=-\sum_{c\in \mathbf{C}} p_c \log p_c$, where $p_c$ is the fraction of users in $\mathbf{U}_{i}$ belonging to cluster $c$. Fig. \ref{fig:homogeneity}(a) indicates that as $H_i$ increases (users already engaged tend to be members of same cluster), the performance decreases (Pearson $\rho=-0.632$) since the model tends to predict more to the cluster whose members are engaged more in discussion. This further results more mistakes for those clusters which have not been engaged so far in the discussion. However, the decrease in performance  is less for \texttt{RGNet}\ compared to the best baseline.

\begin{table}[]
\centering
\caption{Performance with different number of clusters and window size for both the temporal tasks.}

\begin{tabular}{|l|c|c|c|c|}
\hline
\multirow{2}{*}{\textbf{Model}} & \multicolumn{4}{c|}{{\bf (a) \# of clusters, $n$ (Micro F1, $\mathcal{E}$)}}\\\cline{2-5}
&
\textbf{8} & \textbf{16} & \textbf{24} & \textbf{32}\\\hline
\texttt{RGNet} & 0.72,11.02 & 0.62,7.93 & 0.53,6.54 & 0.44, 6.01 \\
LSTM-f & 0.64,11.23 & 0.57,10.87 & 0.46,9.92 & 0.39,10.56 \\
LSTM-r & 0.61,21.55 & 0.53,19.12 & 0.46,20.04 & 0.38,21.87 \\
Newt. & 0.56,24.66 & 0.48,28.54 & 0.42,25.01 & 0.40,23.12  \\\hline
\end{tabular}

\begin{tabular}{|l|c|c|c|c|}
\hline
\multirow{2}{*}{\textbf{Model}} & \multicolumn{4}{c|}{{\bf (b) Window size, $w$ (Micro F1, $\mathcal{E}$)}}\\\cline{2-5}
&
\textbf{5} & \textbf{10} & \textbf{15} & \textbf{20}\\\hline
\texttt{RGNet} & 0.54,8.04 & 0.66,6.07 & 0.72,6.01 & 0.69,6.12 \\
LSTM-f & 0.49,12.56 & 0.55,11.20 & 0.64,11.23 & 0.65,10.98 \\
LSTM-r & 0.49,24.32 & 0.54,22.87 & 0.61,21.55 & 0.61,21.45 \\
Newt. & 0.32,29.57 & 0.46,27.01 & 0.56,24.66 & 0.57,25.03
\\\hline
\end{tabular}
\label{tab:varyingwindowcluster}
\vspace{-5mm}

\end{table}

\subsection{Growth Rate Forecasting for Temporal Engagement}

We define the {\em growth rate} of engagement for a discussion  at $(i+1)^{th}$ window as $v^{i+1} = \log(\frac{m}{\Delta t})$, where $\Delta t$ is the time difference of first and last comments in $(i+1)^{th}$ window, and $m$ is the window size.
To test how effectively each competing model predicts the growth rate for $(i+1)^{th}$ window, we use relative \%-error in prediction given by $\mathcal{E}^{i+1}=\frac{|v^{i+1}_{true}-v^{i+1}_{predict}|}{|v^{i+1}_{true}|}\times100\%$, where $v^{i+1}_{true}$  ($v^{i+1}_{predict}$) is the actual (predicted) value. Table \ref{tab:varyingwindowcluster}(a) shows the average error $\mathcal{E}$ across all the windows incurred by the models trained with different number of clusters $n$. We observe that as $n$ increases, the average error for \texttt{RGNet}\ decreases. The reason is that more the number of clusters obtained from user embedding, more precisely \texttt{RGNet}\ can compute the curvature throughout the manifold.

Fig.~\ref{fig:homogeneity}(b) shows the correlation of per-window error $\mathcal{E}^{i+1}$ and true growth rate $v^{i+1}_{true}$ (for better visualization, we normalize $v^{i+1}_{true}$ by its maximum value obtained). We observe that for higher values of growth rate, prediction is more erroneous (Pearson $\rho=0.178$). We empirically observe that such a large growth rate occurs when more that $50$ comments appear per second. Such instances (discussions) seldom appear in our dataset (1.74\% of total discussions).

\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{Figures/entropy.pdf}
\caption{(a) Accuracy of \texttt{RGNet}\ with the change in homogeneity (measured by entropy), and (b) per-window error with increasing growth rate.}
\label{fig:homogeneity}
\vspace{-5mm}
\end{figure}

\subsection{Non-temporal Engagement Prediction}
\begin{table}[!t]
    \centering
    \caption{Evaluation results for predicting non-temporal engagement. \texttt{RGNet}-$n$ signifies \texttt{RGNet}\ with $n$ number of clusters.}
    \vspace{-1mm}
    \begin{tabular}{|c|c|c|c|}
    \hline
    {\bf Model} & {\bf F1-score} & {\bf AUC} & {\bf Accuracy}\\
    \hline
         \texttt{RGNet}-8 & 0.51 & 0.52 & 0.52 \\
         \texttt{RGNet}-16 & 0.54 & 0.55 & 0.57 \\
         \texttt{RGNet}-24 & {\bf 0.61} & {\bf 0.62} & {\bf 0.63} \\
         \texttt{RGNet}-32 & 0.59 & 0.59 & 0.60 \\
         R\&A & 0.56 & 0.57 & 0.58\\\hline
    \end{tabular}
    \vspace{-5mm}
    \label{tab:non-temporal-engagement}
\end{table}
In Table \ref{tab:non-temporal-engagement}, we present the performances of different models for predicting whether a given post will attract users or not. We observe that \texttt{RGNet}\ with 24 clustering of the manifold
performs the best. Moreover, both settings with 24 and 32 clusters outperform R\&A by a significant margin.

 In Table \ref{tab:non-temporal-engagement}, we can observe an increase in performance of \texttt{RGNet}\ for this task as the number of clusters grows. A possible reason  might be the heterogeneous distribution of users over the manifold and how accurately \texttt{RGNet}\ is being informed about this heterogeneity. It is already explained that more closely located users (i.e., users in a dense cluster) are more likely to interact with each other in near future. Therefore, a post coming from an outlier user is less likely to be replied by other users. With less number of clusters, sparsely separated users are identified to be members of a cluster. This results in \texttt{RGNet}\ assigning wrong curvature value for those users. With more clusters, this error is minimized. However, with increasing number of clusters, errors in curvature computation for each cluster get accumulated  and affect the total curvature $R_{total}$. This possibly explains the performance drop for \texttt{RGNet}\ with $32$ clusters in Table \ref{tab:non-temporal-engagement}.

\subsection{Feature Importance}
We perform feature ablation study for both temporal and non-temporal engagement prediction tasks. For the former case, we study feature importance only for \texttt{RGNet}, whereas for the latter case, the analysis is done for both \texttt{RGNet}\ and R\&A.

\subsubsection{Feature Ablation for Temporal Engagement}
We study the importance of different features for \texttt{RGNet}\ in two settings. In the first setting,
we drop each group of features (mentioned in Sec. \ref{sec:feature_selection}) in isolation and report the accuracy. In the second setting, we add random noise to each feature in isolation --  we draw random samples from Gaussian distributions with same mean and standard deviation as that of the original distribution of the feature (this experiment was repeated 10 times, and the average result was reported). Table~\ref{tab:feature_study} indicates that user features bear utmost importance for both the tasks, though its effect is more visible in user cluster engagement prediction than the growth rate prediction. This is quite consistent with our intuition that {\em similar types of users tend to flock together}.

\begin{table}[]
\centering
\small
\caption{Performance of \texttt{RGNet} with each feature set (i) removed in isolation, and (ii) replaced by noisy feature set in isolation for both user cluster prediction (Micro F1) and growth rate forecasting ($\mathcal{E}$). }
\vspace{-2mm}

\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{2}{*}{{\bf Feature set}} & \multicolumn{2}{c|}{\specialcell{{\bf Feature removed}}} & %
    \multicolumn{2}{c|}{\specialcell{{\bf Noise added}}}\\
\cline{2-5}
& {\bf Micro F1} & {\bf $\mathcal{E}$} & {\bf Micro F1} & {\bf $\mathcal{E}$}\\
\hline
Latent  & 0.62 & 8.33 & 0.57 & 16.56 \\
User & {\bf 0.54} & {\bf 8.34} & {\bf 0.52} & {\bf 17.15} \\
Content & 0.59 & 7.12 & 0.56 & 15.42  \\
Surface & 0.63 & 8.43 & 0.57 & 14.89 \\
\hline

\end{tabular}
\vspace{-5mm}
\label{tab:feature_study}
\end{table}
\begin{table}[!t]
\centering
\caption{Feature ablation of  \texttt{RGNet}\ and R\&A for the task of non-temporal user engagement prediction.}
\vspace{-2mm}
\begin{tabular}{|c|c|c|}
\hline
\multicolumn{1}{|c|}{\bf Model} & \multicolumn{1}{c|}{\bf Features used} & {\bf F1-score} \\ \hline
\multirow{2}{*}{\texttt{RGNet}-24} & Content & 0.55 \\
 & Social & 0.58 \\ \hline
\multirow{2}{*}{R\&A} & Content & 0.52 \\
 & Social & 0.55 \\ \hline
\end{tabular}
\label{tab:non-temporal-feature-study}
\vspace{-5mm}
\end{table}
\subsubsection{Feature Ablation for Non-temporal Engagement}
For the non-temporal engagement prediction task, we perform feature ablation study for both \texttt{RGNet}\ and R\&A.  Rowe and Alani \cite{rowe2014mining} grouped the features into two categories -- content features and social features. Content features of R\&A are closely similar to content, surface and latent features of \texttt{RGNet}\ (see Sec. \ref{sec:feature_selection}), with many features common in both the models. We group these features as content features as a whole in this study, and name the user features as social features, for a better comparison between these two models. We use the best performing version of our model (\texttt{RGNet}-24) for the feature importance study. Table \ref{tab:non-temporal-feature-study} shows that both the models treat social features with higher importance compared to content features for this task.

\subsection{Performance of \texttt{GUVec}}
We compare \texttt{GUVec}\ with three baselines: Node2Vec \cite{Grover:2016} is run on (i) our co-occurrence matrix $\mathbf{A}$ ({\bf Node2Vec+$A$}) and (ii) a user-user matrix $\mathbf{A'}$, where $A'_{i,j}$ indicates the number of discussions where users $i$ and $j$ participated together ({\bf Node2Vec+$A'$}); third baseline is designed by aggregating all the comments/posts by user $i$ in the training set and running Doc2Vec \cite{le2014distributed} on the aggregated text to obtain user embedding ({\bf Doc2Vec})\footnote{The results of Node2Vec and Doc2Vec were reported after appropriate parameter tuning.}. Fig. \ref{fig:parameter}(c) shows that \texttt{GUVec}\ performs the best in both the tasks.

We also present an empirical study on the complexity of \texttt{GUVec}. Intuitively, building the user-user co-occurrence matrix $\mathbf{A}$ is computationally most expensive, as it needs pair-wise comparison between users. Any pair-wise computation from an input of size $n$ results in an worst case time complexity of $\mathcal{O}(n^2)$. As we compute the full matrix, this bound should be same for space complexity, too. However, \texttt{GUVec}\ does not take pairs from the full user set $\mathbf{U}$ but only a finite subset $\mathbf{U}'\subset\mathbf{U}$. Complexity of building $\mathbf{A}$ is readily reflected by the number of non-zero elements in $\mathbf{A}$, because only these elements correspond to a pair-wise comparison between users.
\begin{figure}
    \centering
    \includegraphics[width=0.75\columnwidth]{Figures/complexity_guvec.pdf}
    \vspace{-2mm}
    \caption{(Color online) Plot showing the number of non-zero elements in $\mathbf{A}$ versus total number of users (red dots); blue, black and green dashed lines represent the curves of $\lvert\mathbf{U}\rvert^2$, $\lvert\mathbf{U}\rvert^{\frac{5}{3}}$ and $\lvert\mathbf{U}\rvert^{\frac{3}{2}}$ respectively.}
    \vspace{-5mm}
    \label{fig:guvec_complexity}
\end{figure}
In Fig. \ref{fig:guvec_complexity}, we plot the number of non-zero elements in the co-occurrence matrix with varying sizes of user sets. For a comparative understanding, we also plot the curves of $\lvert\mathbf{U}\rvert^2$, $\lvert\mathbf{U}\rvert^{\frac{5}{3}}$ and $\lvert\mathbf{U}\rvert^{\frac{3}{2}}$. As we can see, the complexity of \texttt{GUVec}\ falls in between $\mathcal{O}(\lvert\mathbf{U}\rvert^{\frac{5}{3}})$ and $\mathcal{O}(\lvert\mathbf{U}\rvert^{\frac{3}{2}})$. This is due to high clustering of users in the user-user interaction network; users tend to form groups and their interactions remain mostly within the group. For this, \texttt{GUVec}\ needs to compute pair-wise proximity values for pairs from very small subsets of $\mathbf{U}$. Let us assume that the user set $\mathbf{U}$ is fragmented into $k$ equal-size partitions. Then, the total number of pair-wise computations \texttt{GUVec}\ needs will be $\frac{\lvert\mathbf{U}\rvert(\lvert\mathbf{U}\rvert-1)}{2k}$. From Fig. \ref{fig:guvec_complexity}, we know that,
$\lvert\mathbf{U}\rvert^{\frac{3}{2}} < \frac{\lvert\mathbf{U}\rvert(\lvert\mathbf{U}\rvert-1)}{2k} < \lvert\mathbf{U}\rvert^{\frac{5}{3}}$,
which further simplifies to bounds of $k$ itself, given by,
$\mathcal{O}(\lvert\mathbf{U}\rvert^{\frac{1}{3}}) < k < \mathcal{O}(\lvert\mathbf{U}\rvert^{\frac{1}{2}})$.
\subsection{Diagnostics with a Case Study}
Fig.~\ref{fig:example_engagement} presents an example of  the user cluster engagement prediction results by \texttt{RGNet}\ for first three consecutive windows.  We observe that \texttt{RGNet}\ always computes high curvature for the cluster containing the users who started the discussion. It thus leads to an erroneous prediction for $1^{st}$ window, where \texttt{RGNet}\ predicts that users from cluster-$1$ will be engaged. Even in the $2^{nd}$ window, a high curvature value is assigned to this cluster (darkest shade compared to rest of the clusters). Moreover, clusters from which users have been engaged in a window, tend to hold a high curvature value in the successive steps (cluster-$3$, for example). It is important to note that these are absolute values of curvature; originally, more attraction means more negative curvature.

For every window, \texttt{RGNet}\ computes $g^{\mu\nu}$ for each cluster center. Using Eq.~\ref{Eq:metric_tensor}, we then compute intra-cluster distance for each cluster at every window. Table~\ref{tab:cluster_distance} shows that {\em metric distance} (distance between two vectors computed using Eq.~\ref{Eq:metric_tensor}) is always greater than flat Euclidean distance; more the curvature for a cluster (hence more probable the users from that cluster are to get engaged), more is the stretching of the intra-cluster distance.

 \section{Related Work}
Various social media platforms enable users with different types of activities.  In case of Twitter, a large body of literature address the problem of retweet prediction and user influence detection \cite{zhang2016retweet,cheng2014can}. Liu et al. \cite{liu2018c} proposed a user behavior model for retweet prediction. Recently, studies on the role of multimodality in retweet prediction have gained much focus \cite{wang2018retweet, zhao2018attentional}.
Another problem, which is much similar to ours, is the reply prediction \cite{nishi2016reply,yuan2016will,schantl2013utility,rowe2014mining}.
For both the tasks, various sets of features were employed, which  can be broadly categorized into two groups -- {\em content features}
and {\em social features}.
As Cha et al. \cite{cha2010measuring} and Macskassy and Michelson  \cite{macskassy2011people} suggested, content features play a vital role in retweet prediction; whereas replies are more dominated by social features \cite{sousa2010characterization, schantl2013utility}. Reply networks were studied in various other platforms: \url{Boards.ie}, SAP community network\footnote{https://www.sap.com/index.html}, Facebook and many others \cite{rowe2014mining}. Most of these studies  predict which post is going to get more replies, typically ignoring the temporal dynamics of discussions. Other studies explored the evolving structural properties of reply network  \cite{bakshy2011everyone,banos2013role,cogan2012reconstruction}. Purohit et al.
\cite{purohit2011understanding} proposed a framework to predict user engagement in clusters formed from topic-based discussions over Twitter.

\begin{figure}[!t]
\centering
\includegraphics[width=0.5\textwidth]{Figures/example_engagement.pdf}
\caption{(Color online) An example prediction of user clusters by \texttt{RGNet}\ for first three consecutive windows. Eight clusters are marked by the indices in $0^{th}$ window.  Darker shades represent clusters with higher curvature value.}
\label{fig:example_engagement}
\vspace{-5mm}
\end{figure}
\begin{table}[!t]
\centering
\caption{Average intra-cluster distances (FD represents Euclidean distances, MD-$i$ represents metric distance at $i^{th}$ window).}
\vspace{-3mm}
\scalebox{0.8}{
\begin{tabular}{|c|c|c c c|}
\hline
{\bf Cluster ID} & {\bf ED} & {\bf MD-0} & {\bf MD-1} & {\bf MD-2}\\
\hline
1 &0.259 &0.318 &0.323 &0.298\\
2 &0.179 &0.222 &0.226 &0.239\\
3 &0.210 &0.254 &0.247 &0.245\\
4 &0.286 &0.298 &0.301 &0.311\\
5 &0.232 &0.261 &0.275 &0.279\\
6 &0.175 &0.185 &0.181 &0.192\\
7 &0.198 &0.220 &0.223 &0.267\\
8 &0.205 &0.237 &0.229 &0.229\\
\hline
\end{tabular}}
\vspace{-5mm}

\label{tab:cluster_distance}

\end{table}
User-user engagement dynamics is a much studied problem, where the target is to predict the probability of future interaction between a pair of users based on their friendship history \cite{schantl2013utility, yuan2016will}. This is closely similar to link prediction in dynamic social networks \cite{ zhu2016scalable}.

Another related problem is comment popularity prediction in discussion forums. He et al. \cite{he2016deep} proposed a deep reinforcement learning model for predicting popular comments in Reddit. Horne et al. \cite{horne2017identifying} reported sentiment features to be most effective for comment popularity ranking in Reddit.

\section{Conclusion}
In this work, we adopted General Theory of Relativity to devise efficient fusion of heterogeneous features for modeling temporal and non-temporal dynamics of user engagement in online discussions. Our contributions in this work are: (i) \texttt{GUVec}, a novel user embedding method to represent users in a discussion platform as distributed vectors based on three different notions of proximity, (ii) \texttt{RGNet}, a novel user engagement model inspired by Einstein Field Equations, (iii) a comprehensive set of features characterizing a discussion (post, comments, and users), and (iv) an exhaustive comparative analysis to show the superior performance of \texttt{RGNet}\ compared to other baselines for temporal and non-temporal user engagement prediction and growth rate forecasting.

redpajama_set_name: RedPajamaArXiv


Fluid Mechanics, as a scientific discipline in a modern sense, was established between the last third of the 17th century and the first half of the 18th century. This book analyses its genesis from two lines: resistance and discharge. This approach highlights the existence of a remarkable experimental aspect in the aforementioned research lines, together with their link with problems of a practical nature, such as ballistics, hydraulics, fluid-using machines or naval theory.


text:
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The Shiretoko-hanto Peninsula in northeastern Hokkaido juts out into the Sea of Okhotsk. It is said to be the last unexplored region of Japan, and consists of steep mountain peaks covered with virgin forests.
Cormorants and white-tailed sea eagles live there, and the whole area has been designated as a national park. The towns of Utoro and Rausu on the eastern side of the peninsula are the starting points for tourists.
Visitors can see the whole of the peninsula and its cliffs by boarding a sightseeing boat that departs from Utoro Port.Shiretoko is known as a "waterfall kingdom," and many waterfalls can be seen from the sightseeing boat: the warm-water Kamuiwakkayu-no-taki Falls, which gives off steam; Furepe-no-taki Falls, which is so beautiful it is called "maiden's tears;" and Oshinkoshin-no-taki Falls, which flows alongside a road and drops magnificently towards the sea.
There are also the Shiretoko-Goko (five lakes), which are small, quiet lakes surrounded by a virgin forest, and known as "the five jewels." You can walk round these five lakes in about an hour. In winter, the coasts of Shiretoko are sometimes surrounded by ice floes.
Shiretoko

redpajama_set_name: RedPajamaCommonCrawl


This summer, the self proclaimed queen of rap, Nicki Minaj, brings her Barbies, her pink and some friends to the Molson Canadian Amphitheater for her PinkPrint Tour. The show features an array of hip hop most eccentric artists including; Meek Mill, Dej Loaf, Tinashe and duo Rae Sremmud. The lineup alone showcases exactly what the night will present which is non stop dancing and good vibes. What better way to sign off Summer 2015 than attending the show? For ticket information and artist videos take a look after the jump!
Def Jam recording artist and DMV native Logic has made a lot of noise this past year with the release of his first studio album "Under Pressure." Debuting at peak spots on the chart it was only right that he took his music onto the road with a fourty-nine city world tour that makes it's way to Toronto for two days beginning on Tuesday, March 10th at the Danforth Music Hall and Wednesday, March 11th at Tattoo Queen West. The show will be supported by Monster Energy tours and will feature Michael Christmas as well as Logic's tour DJ, DJ Rhetorik who recently made waves when Childish Gambino shouted him out on his mixtape "STN MTN" for doing all the cuts on "Dream/Southern Hospitality/Partna Dem," as well as "Move That Dope/Nextel Chirp/Let Your Hair Blow." The concert is sure to have all fans in the building at full attention and new fans in adoration with the rap artist making this one show you'll definitely be leaving satisfied with! For full event details and ticket information, keep reading after the jump.
On Tuesday, March 10, 2015, GOOD beats producer, Kanye's protege, and Grand Hustle artist Travis Scott will be making his way alongside Young Thug for a performance inside The Danforth Music Hall in Toronto. The concert is apart of the artists' 2015 Rodeo Tour that will be travelling to 13 different cities including the six. With the dynamic energy that Travis and Young Thug are bound to bring this is one concert you don't want to miss! For concert information and ticket details be sure to take a look after the jump.


It took Virginia Governor, Terry McAuliffe, just few hours to tells white nationalists to go home (also they are labelled supremacists, Nazis, Ultra-Right, Far Right, Racists, Fascists, or Alt-Right).
My grandmother can tell that this governor is either unaware of what he is doing, or being paid to do so.
Charlottesville is not a race issue, it is about globalist control over USA and the world.
It is an opportunity to unite USA. If African American and Hispanic are smart enough they must support White nationalists because if they don't and let the Whites loose then definitely they will be next.
To explain Virginia Violence in few simple words here it is: "It is a clash between Globalism led by Jews against Nationalism led by Whites". It is a political confrontation but unfortunately some African Americans got it totally wrong.
Jewishness or Judaism (earlier form of Zionism) is, like Zionism, not a religion, ethnicity, or nationality, it is squarely a political association nuder veils of religion and ethnicity.
Those Democrats, Liberals, Soros' people were and are the masters of slavery, human trafficking, exploitation not only in USA, but also in Ireland, Spain, Portugal, Italy, France, all Eastern Europe, Africa, and Asia.
Now they pretend not to be racists and defend human rights and liberties. They have very dirty history and they are anti-Whites and anti-Europeans out of greed and evil.
It could be expected to see arson attacks on Blacks to blame Whites, or the other way around; or even to fabricate attacks against Jews to accuse Whites or President Trump.
Let us assume we are totally wrong and the Whites are terrible. Then how that governor is going to manage his state after he sided and decided that X group has no place in Virginia? Terry McAuliffe is surely unfit to govern.
The White nationalists were protesting against the removal of a statue in a park and announced that for some days. It was legal and peaceful and has no intention to attack anybody.
The groups of Soro' s people including liberals, democrats, and gays organized and intentionally attacked the White nationalists. Any group is allowed to protest, but no group is allowed to confront and attack protestors in the same area.
The White nationalist were right and Trump should had condemned attacking protestors by Soro's thugs.
White nationalists are not racists or fascists or whatever they label them, they are just patriots. The real racists are the slave dealers who are hiding behind fake religion, ethnicity, and nationality. White nationalists are hard working people they don't live on usury and plotting.
The smear campaign against Gaddafi was designed and carried out by the Jews who wanted to attack Europe by refugee flood to change its demography and politics. For them Libya was an obstacle that had to be removed. Now they are doing it via Libya, Morocco, all North Africa, and Turkey.


Justice Sherifat Solebo of the Special Offences Court sitting in Ikeja, last week, ordered a suspended judge of the Federal High Court in Lagos, Justice Mohammed Yunusa, to face trial for the alleged corruption charges filed against him by the Economic and Financial Crimes Commission (EFCC). She gave the order while ruling on the preliminary objection filed by the embattled judge and his co-accused person.
The EFCC had arraigned Yunusa alongside Esther Agbo, a staff of the law chambers of Mr. Rickey Tarfa (SAN), on January 17, 2018 on a four-count charge bordering on attempted perversion of the course of justice and corruption by a public official, while Agbo was charged with offering gratification to a public official. They, however, denied the charges.
According to the EFCC, Yunusa had constant and confidential communications with Tarfa, who was handling three lawsuits marked FHC/L/CS/714/2015, FHC/L/CS/715/2015 and FHC/L/CS/716/2015 before him. It also alleged that Yunusa collected N1.5 million as bribe from Tarfa for the purpose of giving favourable rulings and judgments in the cases. The judge was also accused of receiving N750, 000 from another senior advocate between March 2015 and September, 2015, to get favourable judgment in some cases.
But the judge, through his counsel, Chief Robert Clarke (SAN), filed a preliminary objection on March 9, arguing that the embattled judge had been absolved of the corruption charges by the National Judicial Council (NJC). Citing the case of Federal Republic of Nigeria vs Justice Hyeladzira Nganjiwa, the counsel said for Yunusa's dismissal to be valid, President Muhammadu Buhari had to give a recommendation for his suspension.
The counsel to Agbo, Mr. John Odubela (SAN), also asked the court to dismiss the charges against her on three grounds. The first is that the amended information was incompetent, because it was signed by Mr. E. E. Iheanacho, an EFCC official, instead of the Attorney-General (A-G) or an official from the A-G's office. The other two grounds were that the National Industrial Court is the court with jurisdiction, because the alleged crime was made in the course of Agbo's employment and that the case against her is unknown to law.
But Solebo, in a two-hour ruling, held a contrary view. Referring to a November 8, 2016 letter from the NJC to the EFCC suspending the judge, she noted that the NJC had already exercised disciplinary action against the embattled judge.
"I do not think that the defendant is a sitting judge having been suspended. I thereby agree that prosecution cannot be brought until the NJC exercises its disciplinary powers against a judicial officer. Going through Nganjiwa's case, there is no evidence that the NJC investigated Nganjiwa but in the case against the first defendant (Yunusa) the NJC in its 76th meeting investigated him.
"What is left to be done? The NJC has done what it is empowered to do; it is my decision that I am not bound by Nganjiwa Vs Federal Republic of Nigeria. It is my conclusion that the first defendant has to defend himself in the information filed by the prosecution. It is my decision that this court has jurisdiction on the charges filed against the first defendant in this case and the notice of preliminary objection is dismissed," Solebo said.
On Agbo, the judge dismissed the three grounds of the preliminary objection, citing Section 174 of the Constitution and maintained that. "The EFCC does not require a fiat or the consent of the A-G to prosecute as contained in Section 43 of the EFCC Act, noting that the offence for which Agbo was charged under Section 64(1)(a) of the Criminal Law of Lagos is known to law.
"General prosecutorial power has been delegated by the A-G to the EFCC. It is my modest view as presented, that the amended charge filed by the applicant is competent and validly signed by Iheanacho and this court is competent to adjudicate same," Solebo said.
On the issue of jurisdiction, the judge said, "It is my modest opinion that for criminal matters, exclusive jurisdiction is not conferred on the National Industrial Court according to Section 254(C)(5) of the Constitution.
Reacting to the ruling, while the prosecution praised the judge for the courage to ignore the precedence set by the Court of Appeal in the case of Justice Nganjiwa, the counsel to Justice Yunusa sees it as judicial rascality, promising to appeal the decision at the Appeal Court.
Many analysts have also expressed reservations over the ruling. While some have argued that since Yunusa is still in service, he deserves to benefit from the judgment of Nganjiwa. According to them, the fact that the judge has not retired means that he should benefit from the judgment of the Court of Appeal.
Analysts further argued that the suspension placed on the judge by the NJC does not preclude him from benefitting from the judgment of the Court of Appeal.
But there are those who felt otherwise. They argued that the fact that senior lawyers, who were alleged to have bribed the judge are standing trial, makes dismissing his case impossible. They stated that in the case of Nganjiwa, those alleged to have bribed him were not put on trial.
"If you set Yunusa free, what happens to the lawyers, who were alleged to have given him bribe?" asked a source, who spoke to THISDAY off the record.
Nganjiwa's case has been a major reference point on which corruption cases are now determined in the judiciary. The EFCC had arraigned the judge also of the Federal High Court in Lagos on a 14-count charge before Justice Adedayo Akintoye of the Lagos State High Court in Igbosere on June 23, 2017.
In the charges, the EFCC accused him of unlawfully enriching himself as a public official by allegedly receiving $260,000 and N8.65million through his bank account between 2013 and 2015. The judge pleaded not guilty and trial commenced.
But the judge's counsel, Clarke, argued a preliminary objection challenging Justice Akintoye's jurisdiction to hear the case. The SAN contended that by virtue of Section 158 of the 1999 Constitution, only the NJC has the powers to deal with the kind of allegations brought by the EFCC against his client, a serving judge.
But the prosecuting counsel for the EFCC, Mr. Rotimi Oyedepo, disagreed and maintained that despite being a serving judge, Nganjiwa did not have immunity against criminal prosecution.
"We must be bold enough to concede that the NJC has administrative power in the appointment and administrative discipline of a judicial officer; but Section 158 of the Constitution did not clothe the applicant with immunity from criminal trial," Oyedepo had argued.
In her ruling, Akintoye upheld Oyedepo's argument and dismissed Nganjiwa's objection.
Displeased, the judge, through his counsel, appealed the decision. In its judgment, the Court of Appeal struck out the corruption charges. A panel presided over by Justice Adejumo Obaseki quashed the 14-count charge pressed against Nganjiwa on the grounds that a serving judge could not be investigated or prosecuted without being first disciplined by the NJC.
The appellate court agreed with Clarke that a judge could not be prosecuted until he or she had either been dismissed or compulsorily retired by the NJC. The court upheld Clarke's argument that the case should have been referred to the NJC.
It was based on the above decision by the Court of Appeal that Justice John Tsoho of the Federal High Court in Abuja recently dismissed the 13-count charge of money laundering and passport fraud instituted by the federal government against Justice Sylvester Ngwuta, a serving justice of the Supreme Court, The jurist was among the judicial officers, whose houses were raided by the operatives of the Department of State Services (DSS) in October 2016. He was subsequently charged with various corruption offences by the Office of the Attorney-General of the Federation.
His counsel, Chief Kanu Agabi (SAN), had while arguing his client's motion, submitted that by virtue of the said Court of Appeal's judgment in Nganjiwa's case and the provisions of section 158(1), Paragraph 21(8) of the Third Schedule to the Constitution, the charges instituted against his client were incompetent.
He maintained that going by the judgment of the Court of Appeal, the complaints forming the basis of the charges instituted against him ought to have been referred to the NJC and allowed the NJC to discipline him before instituting the charges.
In his ruling, Tsoho dismissed the charges, anchoring his decision on the Court of Appeal decision. He held that the pre-conditions for instituting charges against a judicial officer were not met before charging Ngwuta.
The way the EFCC asked the Supreme Court to upturn the judgment of the Court of Appeal and Court of Appeal to review the judgment in favour of Ngwuta is the same way Yunusa has promised to appeal the decision of the Special Offences Court sitting in Ikeja.
But from the way the cases are going, it is becoming obvious that it is only the Supreme Court that has the ultimate power to resolve the contentions.


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THE YOUNG & THE RESTLESS Spoilers — Traci Returns!
When people are confronted with the truth in these spoilers from THE YOUNG & THE RESTLESS, will they listen?
Jack tries to convince Traci that his relationship with Phyllis now is strictly business. "That is the only reason she's here," he insists. "There is nothing between us." Traci smiles and tells her brother, "You are a terrible liar." Can she talk some sense into Jack?
MUST SEE: The Stars of Y&R Did the Best Mannequin Challenge Video!
Ashley warns Michelle to do the right thing and walk away from Travis, but Michelle admits, "I can't." Meanwhile, Billy warns Travis, "You should know there's no secrets in this town." Travis holds his ground and reminds Billy that it's really none of his business, but Billy shoots back, "The hell it isn't." Will he tell Victoria what Travis has been up to?


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Dr. Dominic Palazzolo: Metallic content of e-cig vapor is 'negligible'
by Matt Rowland May 12, 2017
Public Health agencies like the U.S. Food and Drug Administration (FDA) and the Centers for Disease Control and Prevention (CDC) have often claimed that e-cig vapor is just as toxic as conventional tobacco smoke. However, when they are asked for supporting documentation to prove these claims, they often state that further research is still needed. So, a team of scientists led by Dr. Dominic L. Palazzolo from Lincoln Memorial University in Harrogate, Tennessee, decided to conduct their own research.
Outline of the Palazzolo study
Palazzolo and his team decided to measure the levels of at least nine different trace metals in both e-cig vapor and the smoke from conventional tobacco cigarettes. The metals included Aluminum, Arsenic, Cadmium, Copper, Iron, Manganese, Nickel, Lead, and Zinc. After conducting an extensive series of laboratory tests, the scientists determined that the levels of each of these trace metals from vaping is essentially "negligible" and pose no threat to public health.
"The contents of Al, As, Cu, Fe, and Mn on smoke-exposed MCE membranes were also significantly higher (p < 0.05) than their content on aerosol-exposed membranes. The contents per cigarette equivalent of metals in E-liquid before aerosolization were negligible compared to amounts of aerosolized E-liquid, except for Fe (0.002 μg before and 0.001 μg after). Elemental analysis of the core assembly reveals the presence of several of these trace metals, especially Al, Fe, Ni, and Zn."
The scientists also note that the levels of arsenic fell "below the detection limit" even for the smokers. The e-cig study entitled Trace Metals Derived from Electronic Cigarette (ECIG) Generated Aerosol: Potential Problem of ECIG Devices That Contain Nickel is published on the Frontiers in Physiology website.
To obtain their research, Palazzolo and his team solicited the help of volunteers from both the vaping and smoking communities. The vapers were asked to vape the same e-liquid of 7ml nicotine concentrations heated to temperatures of 350 degrees Celsius. The e-liquid also consisted of 20% Vegetable Glycerin and 80% Propylene Glycol, which is fairly standard.
The smokers were each given Marlboro cigarettes purchased from a local convenience store. Marlboros, by the way, offer approximately 1mg of nicotine per cigarette. Meanwhile, each different group was given a completely separate room to engage in their related activities so that there would be no cross-contamination. The smoke and e-cig vapor were measured using a piece of scientific equipment called a Thermo Scientific Hamilton SafeAire II.
Related Dr. Michael Siegel: CA vaping study shows second-hand vapor is harmless to public health


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EXCLUSIVE: Kinchant Dam operators to face inquiry
Michael Cousen said he was at his Sarina-Homebush Rd property when he saw "a wall of water" coming towards him. Emily Smith
by esmith2014
Emily Smith Journalist
EXCLUSIVE: THE role Kinchant Dam's operators played in flooding that devastated the Eton region will be scrutinised by the state's chief scientist.
The Daily Mercury can reveal that State Water Minister Mark Bailey has called on Queensland Chief Scientist Professor Suzanne Miller to launch an independent assessment into the operation of Kinchant Dam and warning systems for communities downstream, following a visit to the decimated community this week.
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Last Wednesday, residents had to be rescued by helicopter and homes were wrecked in the biggest flooding many had seen in the Eton region.
When Tropical Cyclone Debbie hit Tuesday, the upstream dam sat at 103% of its maximum operating level, prompting many to place part of the blame for the extent of the flooding on dam operators SunWater.
Although it started releasing water Monday, on Wednesday more than 1126 Olympic swimming pools worth of water started spilling.
That water only added to the massive deluge already heading down Sandy Creek following record rainfall.
The roads of Eton were totally submerged. Mitchell Cole
And as phone lines and internet were down, no warning got through to many of the downstream residents.
"It is crucial for communities living and working downstream of the dams to receive timely and clear notifications of any spills," Mr Bailey said.
"Queenslanders have shown great resilience in dealing with extreme weather, but it's important for communities to be informed and prepared and to have confidence and understanding of the operations of the dams."
The assessment would be finalised by June 30, 2017.
Yet SunWater executive general manager operations Colin Bendall thought the company had done all it needed to, to warn downstream residents.
He believed it should have been the role of emergency services and the local disaster management groups to ensure the warnings were heard, once they were told SunWater's text messages were not getting through.
"It's not SunWater's role to go out into a cyclone and knock on doors. That's the local disaster management group and emergency services would perform that role. And we wouldn't send our staff out into the cyclone until it is safe to do so," he said.
"Flood warnings statewide sit with BOM and the local disaster management groups. And SunWater feeds information into these groups as part of the bigger picture, not just the isolated section that relates to the dams."
Cyclone coverage Alan and Beverley Creber at their home thtas Flooding in North Eton near Mackay. Pic Annette Dew Annette Dew
He also wanted to point out there are three branches of Sandy Creek that flow to the Eton area and Kinchant Dam is only on one of these, and that it only made up a small portion of the creek's total catchment area.
He highlighted that when the dam is at 100% of its 'operating capacity', it is still 1m below the spillway, so can still take on a large amount of water before spilling.
Mr Bendall is in the region to hear from Eton residents about where improvements could be made and to help them understand how the dam is operated.
Eton region residents Albert and Rhonda Schmidtke. Emily Smith
Everything lost
Eton region residents Albert and Rhonda Schmidtke, who lost a couple of hundred thousand dollars worth of machinery that day, believed SunWater's decisions "definitely" made things a lot worse.
"Probably Kinchant just tipped everyone over the edge, it just made it that much worse," Mrs Schmidtke said.
"That water should have been dumped a lot earlier."
They also believed the company should have had a backup warning system in place for when technology failed.
"If I had warning, I only had to take (my machinery) half a k up the road and I'd be high and dry. (I'd have saved) a couple of hundred thousand worth of machinery, a lot of heartache and a lot of hard work," Mr Schmidke said.
"They should have gone the old fashioned way and got a car with a horn."
When the circus came to town about 10 years ago, he said, it had no problem spreading the word by driving around the region in a car with a loud speaker.
State government rescue helicopter Rescue 500 from Brisbane lands at Walkerston, after bringing residents from Eton. Emily Smith
Michael Cousen was at his Sarina-Homebush Rd property that Wednesday, when he saw "a wall of water" coming towards him.
"There was water coming from two different directions which was really strange. We saw the creek was rising definitely from rainwater. But all the Kinchant water came from a different direction," he said.
"We were told we'd have warnings and I'm a customer of SunWater and I get them by text, and there's not one text."
Haydn McNichol spent the night in a boat, with his wife and three dogs after 2m of water flooded his Eton home. Emily Smith
While he didn't blame the dam operators for the flood, he was frustrated by the poor communication. He also would have given anything to have that crucial window of time to prepare.
Water went all through his house, and he spent Wednesday night sitting at the kitchen table with flood water swirling around his knees.
His parents, Pat and Ron Cousen, spent the night sitting in the back of the ute in the garage, with the dog.
"We talked about it and said 'we hope this isn't it'," Mrs Cousen said.
"When we came back in the house we just couldn't believe what we found. And what we didn't find, things floated out the back door."
On Tuesday, Mrs Cousen was spending her 64th wedding anniversary watching as the tiles and walls were pulled out of her Homebush home. But she noted at least the upside was "everything would be cleaned out when it's all done".
Homebush residents face starting from scratch in the wake of the unprecedented flood. Emily Smith
Across the road, Heather Brady was teary as ruined furniture was thrown out of the home she'd lived in for 38 years, a former Catholic church.
As her friend Elizabeth Camilleri explained, they were throwing away "a lifetime of memories".
She said 000 hadn't worked and the couple hadn't been able to call for help, forcing them to climb into their loft and hope the water would recede. They were eventually picked up by the swift water rescue team, who dropped them at the General Gordon Hotel.
"I was terrified, I'd been through floods as a kid, so I knew what was coming," Mrs Brady said.
Eton Rural Fire Fighters Andrew Vassallo, John Muscat, Paul Vassallo and Adrian Vassallo. Emily Smith
But she was touched by support shown to her by the community; a stranger came and fixed her generator and water pump, a friend bought around a caravan for family to stay in and community members flocked to help clean up.
Around the corner, Candice Curran even practised climbing onto the roof with her husband, six-year-old and three-year-old, as the flood water raced around her home.
Cyclone coverage Felicity Allen , Ashley and Kristy Bourke evacuating their home , Flooding in North Eton near Mackay. Pic Annette Dew Annette Dew
A friend had managed to call who'd received the text from SunWater, which had recommend they "self evacuate".
But water was at the back step and they couldn't.
Ms Curran called 000, and the operator advised she prepare herself to get on the roof. "I was freaking out I was not in a good state," she said.
A home in Eton that was luckily raised, saving it from flood waters. Mitchell Cole
While her husband tried to get to the shed for a ladder, the water was waist deep and current was too strong.
"He said "I'm going to have to work out how to hoist you onto the roof," she said. "I stood on his knee and then his shoulder and he lifted me onto the roof. Then he would give me the kids and the backpack and then he was going to come up himself."
Luckily, the swift water rescue team arrived at about 9.30 that night, and took them to safety in their boat.
"I'm happy to get out with our lives really," she said.
kinchant dam
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Adam McEwen's investigation is responsible for the 1st California Kill Buyer being arrested and charged for shipping horses to slaughter! Adam did what no other law enforcement detective has ever done. A law was passed in 1998 to protect California horses from slaughter. This is the first time it has ever been enforced. Adam McEwen worked hard and dedicated a month and a half, 24/7 to this investigation to make sure justice was served.
Auctioneer 1st man charged under California Law to protect horses from slaughter.
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We certainly live in confusing times, and doubly so when it comes to financial markets.
I read Tuesday that U.S. and emerging-market equities rallied along with Treasuries on expectations the Federal Reserve would maintain its bond purchases. Then, the very same news service said the Canadian dollar fell against the greenback because of thinking the U.S. central bank would begin to curtail those very same bonds purchases. Perhaps the problem its reporters were getting mixed signals from their gleanings from subscribers' use of the company's data terminals. Whatever the case, no wonder we're befuddled.
Some things only make sense after the fact. Take Merck (ticker: MRK), which mysteriously levitated 4.7% during official trading hours Tuesday on nearly three times the average volume of the past three months, accompanied by a ramp up in buying of call options. The Big Pharma stalwart has been a distinct laggard in recent years and rarely makes such big moves; Tuesday's $2.12 surge, to $47.33, at the 4 PM close, still left the stock at just over half its peak in 2000 and more than 20% below what it fetched in 2007.
After the close, Merck said it would accelerate its share repurchase agreement with Goldman Sachs (GS), buying back $5 billion of stock as part of its previously announced $15 billion repurchase plan, and the shares tacked on an additional 47 cents, or 1%, in after-hours trading. Whatever the motivation for the fast and furious buying of Merck's shares and call options, those who got in ahead of the news participated in the $6-plus billion increase in the drug maker's stock-market value Tuesday.
Options action in the overall equity market also suggested a more bullish undertone than the fractional gains in the major indices, which still kept alive the streak of 19 straight winning Tuesday sessions, for whatever that's worth. But that belies what was going on under the surface.
As Barrons.com's options maven Steve Sears pointed out in his Tuesday Striking Price column ("Big Hedge Funds Suddenly Bullish," May 21), there was heavy buying of calls on the Standard & Poor's 500 by the market's biggest players, resulting in a rise in implied volatilities -- the crucial variable in options pricing -- suggesting a preponderance of wagers on the upside.
That's contrary to the usual tendency for options volatilities to fall in a rising market. "Vols" typically rise when traders pay up for protection in the form of puts as insurance against declines. The anomalous action suggests lots of big players' trying to catch up with the advance in which they haven't participated.
The call buying followed a widely followed call by Goldman Sachs's strategists for significantly higher stock prices by the end of 2013 and subsequent years. What was extraordinary is that they did not lift their earnings forecasts. Instead, they suggested that price-earnings multiples and dividend payouts would increase to support rising stock prices. In other words, the appreciation in equities would be based on financial factors, not fundamental ones.
Specifically, Goldman's strategists forecast the Standard & Poor's 500 will end this year at 1750, up a relatively modest 4.8% rise from here. That would still make for a vintage year for the large-capitalization-stock benchmark, which already is up 17% since the end of 2012.
Goldman further forecasts the S&P will end 2014 at 1900, an 8.5% rise from its 2013 target. And it sees the S&P 500 reaching 2100 by the end of 2015, which would figure to a 10.5%, based on its forecast numbers. That does not take into account dividends, which the firm expects to rise 30% a year, maintain equities' yield around 2% even as stocks rise.
In other words, Goldman's influential strategists predict solid, double-digit total returns for the next two-and-a-half years. To doubting laggards, this is a clarion call to board the train that's leaving without them.
That message was underlined by more Fed-speak, which gave strong indications that, notwithstanding some news outlets' confusion, no cutback in the central bank's $1 trillion-a-year liquidity provision is in immediate prospect.
Indeed, William Dudley, the president of the New York Fed (the first among equals of the 12 district banks) emphasized the Federal Open Market Committee's next move could either be to increase or decrease its bond-buying program. Recent comments from other Fed officials have suggested the latter might be possible.
But Dudley, formerly Goldman's chief U.S. economist and acutely attuned to the markets' reaction to Fed moves, emphasized in a speech Tuesday the central bank will give ample warning of any move. And, he added, they shouldn't get freaked out when the Fed starts to normalize policy.
Goldman's thesis is that investors will pay an increasing price for a stream of future corporate earnings, just as they have for a set stream of interest payments on bonds. But corporations will increase their dividend payouts to maintain a relatively constant current yield, thus supporting higher stock prices.
Underpinning all of this is the assumption that the Fed maintains its repression of interest rates, holding short-term money rates near zero and continuing to cap longer-term bond yields at historically low levels, and below likely inflation.
Given the recent experience, it's hard to argue against this scenario, however contrived it is. But, as the strategist quoted in the previous column here ("Dividend Stocks, the New Nifty 50?" May 21) suggested, investors may be scared to buy into stocks at these prices, but they are more scared of being left behind. No confusion there.


Boil sweet potatoes. Remove skin and let it cool. Add milk, sugar, ice-cubes and blend it with a blender.
Take 3 glasses. Pour 2 tablespoon of chocolate sauce in each glass. On top of it pour about half glass of sweet potato smoothie.
Finally put chopped fruits on the top and serve it immediately.
If you like it you can pour some chocolate sauce on the top also.


This jelly cake is for the Tsum Tsum lovers out there! While it may feel a bit evil to munch on these cute mini Tsum Tsum jelly slices, we're not going to lie, they taste pretty good 😆... Made using coconut milk, this Tsum Tsum Jelly Cake not only looks cute but tastes amazing as well! This colourful jelly cake is the perfect dessert for a kids party or any Tsum Tsum lovers out there! The jelly cake is best served chilled and suitable for vegetarians too. Best served chilled.