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-\documentclass[10pt]{article}
-\usepackage[utf8x]{inputenc}
-\usepackage{amsmath,amsfonts,amsthm}
-\usepackage[english]{babel}
-\usepackage[capitalize, noabbrev]{cleveref}
-\usepackage{algorithm}
-\usepackage{algpseudocode}
-
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-  {\vspace{-2ex} \begin{enumerate} %
-     \setlength{\itemsep}{-1ex} \setlength{\parsep}{0pt}}%
-  {\end{enumerate}}
- 
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-  {\end{itemize}}
- 
-\newenvironment{description*}%
-  {\vspace{-2ex} \begin{description} %
-     \setlength{\itemsep}{-1ex} \setlength{\parsep}{0pt}}%
-  {\end{description}}
-
-\DeclareMathOperator{\E}{\mathbb{E}}
-\let\P\relax
-\DeclareMathOperator{\P}{\mathbb{P}}
-\newcommand{\ex}[1]{\E\left[#1\right]}
-\newcommand{\prob}[1]{\P\left[#1\right]}
-\newcommand{\inprod}[1]{\left\langle #1 \right\rangle}
-\newcommand{\R}{\mathbf{R}}
-\newcommand{\N}{\mathbf{N}}
-\newcommand{\C}{\mathcal{C}}
-\newcommand{\eps}{\varepsilon}
-\newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}}
-\newcommand{\llbracket}{[\![}
-
-\newtheorem{theorem}{Theorem}
-\newtheorem{lemma}{Lemma}
-\algrenewcommand\algorithmicensure{\textbf{Output:}}
-\algrenewcommand\algorithmicrequire{\textbf{Input:}}
-\DeclareMathOperator*{\argmax}{arg\,max}
-
-\author{}
-\title{}
-\date{}
-
-\begin{document}
-
-\paragraph{Notations}
-\begin{itemize}
-    \item $n$ criminals, $V = \{1,\dots,n\}$
-    \item $\alpha_{i,j}$ influence of criminal $i$ on criminal $j$.
-    \item the infections form a set of pairs $C = \{(t_i, v_i),\; 1\leq i\leq m\}$
-        where $t_i$ is the time and $v_i\in\{1,\dots, n\}$ is the identifier of
-        the criminal.
-\end{itemize}
-
-\paragraph{Model}
-The infections are modeled as a multivariate Hawkes process. The instantaneous rate
-of infection of user $j$ at time $t$ is given by:
-\begin{displaymath}
-    \lambda_j(t) = \lambda(t) + \sum_{t_i<t}\alpha_{v_i, j}
-    \phi(t-t_i)
-\end{displaymath}
-
-\emph{Technical concern:} stability of the process, there is a condition on the
-$\alpha_{i,j}$ parameters so that the process is stable, i.e does not keep
-accelerating forever. We probably do not need to worry about it too much:
-empirically the process is stable.
-
-\paragraph{Time kernel}
-we choose an exponential kernel for $\phi$ (will allow the recursive trick
-later on) $ \phi(t) = \mu e^{-\mu t} $
-
-\paragraph{Base rate} $\lambda(t)$ captures the seasonal effect. TBD, worst
-case is to choose a constant approximation, $\lambda(t) = \lambda_0$. We
-define $\Lambda(T) \eqdef \int_{0}^T \lambda(t)dt$.
-
-\paragraph{Edge model} TBD
-
-\paragraph{Likelihood} The log-likelihood for observation horizon $T$ is given
-by:
-\begin{displaymath}
-    \mathcal{L}(C) = \sum_{i=1}^m \log \lambda_{v_i}(t_i)
-    - \sum_{j=1}^n\int_{0}^T \lambda_j(t)dt
-\end{displaymath}
-Note that we have:
-\begin{displaymath}
-    \int_{0}^T \lambda_j(t)dt = \int_{0}^T\lambda(t)dt + \sum_{i=1}^m
-        \alpha_{v_i, j}\big[1-e^{-\mu(T-t_i)}\big]
-\end{displaymath}
-
-Hence the log-likelihood can be rewritten (after reordering a little bit):
-\begin{displaymath}
-    \mathcal{L}(C) = \sum_{i=1}^m\log \lambda_{v_i}(t_i)
-    - \sum_{i=1}^m A_{v_i} \big[1-e^{-\mu(T-t_i)}\big]
-    - n\Lambda(T)
-\end{displaymath}
-where $A_{v_i} = \sum_{j=1}^n \alpha_{v_i, j}$ (out-weight of criminal $v_i$),
-or in complete form:
-\begin{displaymath}
-    \mathcal{L}(C) = \sum_{i=1}^m\log\left[ \lambda(t_i)
-    + \sum_{j=1}^{i-1}\alpha_{v_j, v_i}\mu e^{-\mu(t_i-t_j)}\right]
-    - \sum_{i=1}^m A_{v_i} \big[1-e^{-\mu(T-t_i)}\big]
-    - n\Lambda(T)
-\end{displaymath}
-
-\paragraph{Computational trick} As written above, the log-likelihood can be
-computed in time $O(m^2 + n^2)$. This is not linear in the input size, which is
-$O(m + n^2)$. The $O(m^2)$ term comes from the first sum in the definition of
-the log-likelihood. However, in the case of an exponential time kernel, using
-a recursive trick, the complexity of this term can be reduced to $O(mn)$ for an
-overall running time of $O(mn + n^2)$.
-
-\emph{Question:} Is it worth it in our case? Yes if the number of victims
-is significantly smaller than the number of infections. Is it the case?
-
-
-\end{document}