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In this section, I propose some applications of Hawkes process to twitter/hashtag data.   \subsection{Hawkes process decomposition} decomposition of a debate}  For some hashtags, especially those related to breaking events, the series is driven by the \textbf{debate} of two or more groups of users. The idea is to decompose a hashtag series into the sum of a multivariate Hawkes process. For simplicity, we restrict ourselves to the debate between two parties of users.   Firstly, we show that the sum of two interacting Hawkes process is \textbf{not} necessarily Hawkes process, which raises question the validity of modeling the total as a self-exciting process. Consider a bivariate Hawkes process $(N_1(t), N_2(t))$, which are related by   \[ \pmatrix{\mathbb{E} \frac{dN_1(t)}{dt} \\ \mathbb{E} \frac{dN_2(t)}{dt}} = \pmatrix{\mu_1 \\ \mu_2} + \int_{-\infty}^{t} \mathbf{\Phi(t-s)} \pmatrix{dN_1(s) \\ dN_2(s)}, \]  where $\mathbf{\Phi} = \pmatrix{\phi_{11} & \phi_{21} \\ \phi_{12} & \phi_{22}}$ is the kernel capturing self and mutual excitations. Now consider summing the two into $N(t) = N_1(t) + N_2(t)$ by left multiplying a vector $\mathbf{u}^T = (1,1)$, we have   \[ \mathbb{E} \frac{dN(t)}{dt} = (\mu_1 + \mu_2) + \int_{-\infty}^{t} \mathbf{u}^T \mathbf{\Phi(t-s)} \pmatrix{dN_1(s) \\ dN_2(s)}. \]  From RHS, this reduces to a self-excitation form only if $\mathbf{u}^T$ is a left eigenvector of $\mathbf{\Phi}$ ($\phi_{11} + \phi_{21} = \phi_{12} + \phi_{22}$ in this case).