Query sensitivity of time series under differential privacy I stumbled upon a paper that proposes local DP around this argument:

A user $u_i$ generates a sequence $s_{i}$ of observations at certain timestamps:

$$
s = ((t_1, x_1), (t_2, x_2), \dots, (t_n, x_n))
$$

The authors apply $(\varepsilon/n, 0)$-DP to each sequence by adding Laplacian noise
As widely known, Laplacian must be of scale $b = \frac{\Delta f}{ \text{budget}}$

The authors propose budget of $\varepsilon / n$, which is IMO correct. But they also define $\Delta f$, aka the sensitivity of the query, as simply the range of any value at any timestamp, $\text{max}(x) - \text{min}(x)$.
I'm not convinced that this is the true sensitivity. To my understanding, the query output is (ignoring the timestamps) not a single value $\mathbb{R}$ but rather the vector of outputs $\mathbb{R}^n$, so per definition of sensitivity $\ell_1$-sensitivity of a function $f : \mathbb{N}^{|\mathcal{X}|} \rightarrow \mathbb{R}^k$:
$$
\Delta f = \max_{x, y \in \mathbb{N}; \| x - y\|_1 = 1} \| f(x) - f(y) \|_1
$$
and properly computing the $\ell_1$ norm as $\| x - y\|_1 = \sum_{i = 1}^{k} | x_i - y_i |$, the sensitivity should be
$$(\text{max}(x) - \text{min}(x))^n$$
Is my reasoning correct (and the paper's DP potentially wrong), or am I missing something? (I don't reveal the paper on purpose.)

Update: Claryfing context of the time series.

The health data stream each user is represented as a sequence $s = ((t_1, x_1), (t_2, x_2), \dots, (t_n, x_n))$


Here, $(t_d, x_d)$ represents the $d$-th point in the stream where $x_d$ denotes the value measured by the wearable health device at timestamp $t_d$.


We further assume that $x_d$, which is measured by the specific sensor in a wearable health device, is within the predefined range $[x_{min}, x_{max}]$.

Their particular use-case is collecting heart rate ($x$) over time ($t$).
I think you should explain what is $x_i$ in this time series. I would recommend linking the paper. I think without a strong context we could answer wrongly because maybe you could have understood something wrong as it has happened to me sometimes. Some points:

Query output could be $\mathbb{R}$ or $\mathbb{R}^n$, depends what kind of query you are defining and using.

You are right, I suspect they are defining a global sensitivity because maybe is easier to compute. But this depends on the kind of query they are defining on the paper, is a linear query, an only-one query  or a non-linear query