1 files changed, 57 insertions, 0 deletions
diff --git a/supplements/main.tex b/supplements/main.tex
index 1f3e69f..17a91bb 100644
--- a/supplements/main.tex
+++ b/supplements/main.tex
@@ -357,4 +357,61 @@ We also simulated contagions on the co-offending network. Since we are most inte
 
 \subsection{Comments on Causality}
 
+The interpretation of the cascades obtained by applying the methodology of
+Section~\ref{sec:meth} requires a nuanced understanding of the notion of
+causality. In particular, the meaning of assigning a single cause to an
+infection needs to be qualified.
+
+A common definition of causation is the so-called \emph{counter-factual} one
+whose first formalization can be traced back to the philosopher David Hume
+(CITE):
+\begin{quote}
+    \emph{\ldots we may define a cause to be an object followed by another
+        [\ldots] where, if the first object had not been, the second never
+    had existed.}
+\end{quote}
+The limits of this definition have been the subject of many philosophical
+works, starting from David Lewis' paper \emph{Causation} (CITE). In particular,
+the problem of \emph{late preemption} is made apparent by the following thought
+experiment due to Ned Hall (CITE):
+\begin{quote}
+    \emph{Suzy and Billy, two friends, both throw rocks at a bottle.  Suzy is
+        quicker, and consequently it is her rock, and not Billy’s, that breaks
+        the bottle.  But Billy, though not as fast, is just as accurate: Had
+        Suzy not thrown, or had her rock somehow been interrupted mid-flight,
+    Billy’s rock would have broken the bottle moments later.}
+\end{quote}
+While Suzy's throw is obviously the cause of the bottle shattering, it fails
+the counter-factual definition of causation, since had her throw not occurred
+the bottle would have been shattered all the same by Billy's throw.
+
+Without entering the philosophical debate about defining causation, we note
+that, at least for the purpose of assigning responsibility, Billy and Suzy
+should be considered jointly responsible for the bottle breaking: they were
+both throwing rocks at it, and the fact that Suzy's rock reached the bottle
+first is coincidental. Of course, if Suzy was throwing rocks twice as
+frequently as Billy, without observing whose rock hit the bottle first, it
+would be tempting to say that it is more likely that a rock thrown by Suzy hit
+the bottle first, or that she is ``more responsible''. But it is still a fact
+that at the time the bottle was broken, it was the target of the combined
+throws of Billy and Suzy.
+
+The Hawkes contagion model introduced in \eqref{eq:hawkes} can be
+re-interpreted in light of this example: as they become infected, victim nodes
+start throwing rocks at their associates with a frequency which decreases over
+time and a node becomes infected when it is hit by a rock. Since we do not
+observe whose rock hits first, the only thing we can say for certain is that at
+the time it was infected, the individual was subject to the combined throws of
+its neighbors who were infected in the past. This combined effect is expressed
+mathematically by the sum in \eqref{eq:hawkes}. The Hawkes model also allows us
+to order the nodes by the frequency at which they are throwing rocks.
+
+It is now clear which interpretation to give to the cascades extracted in
+Section~\ref{sec:meth}: it is a simplification where we assign the
+responsibility of an infection to the candidate cause which was throwing rocks
+with the highest frequency at the time of the infection. This simplification is
+acceptable in the sense that it is more likely that the infection was the
+effect of a rock coming from this cause. But even though it makes tracing the
+paths of violence easier, a more conservative statement would instead consider
+all the throws to be jointly responsible for the infection.
 \end{document}