Why is maximum likelihood estimator suspectible to outliers? I'm new to statistics and currently learning abot MLE.
Some of the papers I read: Robust Graph Embedding with Noisy Link Weights mentioned MLEs are suspectible to contamination in data, but didn't make any further mention about it.
So I would like to know the reason for this with clear and detailed explains.

Let's consider a simple example for computing MLE of mean from a small sample dataset: $X=\{1, 2, 5, 10, 4, 8\}$.
Let's assume the data is generated with a Poisson distribution with rate $\lambda$ and we aim at finding the MLE of $\lambda$.
Now, the likelihood $L(\lambda|x_1,x_2,\ldots,x_n)=\prod\limits_{i=1}^{n}P(X=x_i)=\prod\limits_{i=1}^{n}\dfrac{e^{-\lambda}\lambda^{x_i}}{x_i!}$
s.t., the log likelihood $l(\lambda) = \sum\limits_{i=1}^{n}(-\lambda)+x_iln(\lambda)-ln(x_i!)=-n\lambda+ln(\lambda)\sum\limits_{i=1}^{n}x_i-\sum\limits_{i=1}^{n}ln(x_i!)$
$\implies \dfrac{\partial l}{\partial\lambda}=-n+\dfrac{\sum\limits_{i=1}^{n}x_i}{\lambda}=0$ at maximum
$\implies \hat{\lambda}_{MLE}=\dfrac{\sum\limits_{i=1}^{n}x_i}{n}$
Here, $n=6$, $\hat{\lambda}_{MLE}=5$, i.e., the sample mean.
Now, let's consider an outlier point $1000$ gets added to the dataset. Now, the MLE changes to $147.1429$, which clearly overfits to the noise in the data.
