Age of Broadcast and Collection in Spatially Distributed Wireless Networks

Chirag Rao and Eytan ModianoChirag Rao and Eytan Modiano are with the Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute
of Technology, Cambridge, MA, 02139, USA. The final version of this paper will
appear in the proceedings of IEEE INFOCOM 2023.
E-mail: {crao<EMAIL_ADDRESS>
We consider a wireless network with a base station broadcasting and collecting time-sensitive data to and from spatially distributed nodes in the presence of wireless interference. The Age of Information (AoI) is the time that has elapsed since the most-recently delivered packet was generated, and captures the freshness of information. In the context of broadcast and collection, we define the Age of Broadcast (AoB) to be the amount of time elapsed until all nodes receive a fresh update, and the Age of Collection (AoC) as the amount of time that elapses until the base station receives an update from all nodes.
We quantify the average broadcast and collection ages in two scenarios: 1) instance-dependent, in which the locations of all nodes and interferers are known, and 2) instance-independent, in which they are not known but are located randomly, and expected age is characterized with respect to node locations. In the instance-independent case, we show that AoB and AoC scale super-exponentially with respect to the radius of the region surrounding the base station. Simulation results highlight how expected AoB and AoC are affected by network parameters such as network density, medium access probability, and the size of the coverage region.

§ INTRODUCTION

Collection and broadcast of fresh information over spatially-distributed wireless nodes is important for proper functioning of real-time systems, such as search-and-rescue drones or environmental monitoring using IoT sensors <cit.>. Dynamic environments and the lack of wired infrastructure necessitate deployment of highly-distributed, ad-hoc network of sensors to gather and send information updates wirelessly, where nodes must communicate with minimal coordination overhead using simple random access schemes.

Such networks must also operate under wireless communication constraints, including interference, fading, and path loss. Ensuring broadcast and collection of the freshest information possible in such a setting is a considerable challenge.

A popular paradigm for measuring the freshness of information observed from a process is the Age of Information (AoI) <cit.>. The literature addressing AoI and wireless networks is extensive. Average and peak AoI in wireless networks were first characterized in <cit.>. Optimal wireless link scheduling was studied in <cit.>, relying on a centralized scheduler that is able to coordinate link activations, and the authors of <cit.> considered scheduling policies that minimize AoI in wireless networks with packets randomly arriving and queueing at the base station. In addition, the authors of <cit.> studied scheduling with random packet arrivals in a random access setting. Several works have addressed AoI and broadcast. In particular, the authors of <cit.> found optimal centralized scheduling policies for broadcast from a base station to a number of nodes, minimizing functions of AoI such as Expected Weighted Sum AoI.  In <cit.> the authors investigated AoI in multicast and broadcast networks with i.i.d. exponential (continuous-time) inter-packet delivery times. Works such as <cit.> investigated network scheduling to minimize AoI under general wireless channel unreliability, while <cit.> studied scheduling policies with random arrivals, modeling the problem as a Markov Decision Process. From an information theoretic perspective, <cit.> explored the effect of coding on the AoI in two-user broadcast networks, and  <cit.> addressed AoI for Broadcast in CSMA/CS wireless networks, assuming network connectivity follows the Protocol Model <cit.>. The authors of <cit.> explored AoI in all-to-all broadcast wireless networks, deriving average and peak AoI using fundamental properties of graphs.

More recently, AoI in spatially-distributed networks has been investigated. The authors of <cit.> investigated data dissemination and gathering, modeling spatial separation as edges on a mobility graph. The authors of <cit.>, deployed stochastic geometry analysis to capture the spatiotemporal statistics of AoI in networks where nodes are distributed as a homogeneous point process. The authors of <cit.> optimized network parameters such as the medium access probability to minimize average and peak AoI, leveraging knowledge of the interference statistics of Poisson-distributed wireless networks. While AoI has been considered in spatially-distributed wireless networks, the important cases of wireless broadcast and collection in a spatially distributed network have not been addressed.

Our main contribution in this work is to introduce the notion of Age to the broadcast and collection of information.  We define two metrics – the age of broadcast (AoB) and the age of collection (AoC) – that characterize the amount of time elapsed since all receivers successfully receive an update in the broadcast case or all transmitters successfully deliver a packet to the base station in the collection case. We consider both the instance-dependent, and the instance-independent AoB and AoC. In the instance-dependent scenario, the locations of all interferers, transmitters and receivers are fixed and known. In the instance-independent scenario, the positioning of nodes and interferers is unknown but is distributed according to a Poisson point process.

The rest of the paper is organized as follows. In <Ref>, we introduce the system model and define AoB and AoC. We then detail preliminaries in <Ref>. In <Ref>, we characterize the expected AoB, then characterize AoC in <Ref>. Numerical results from simulation are presented in <Ref>, and concluding remarks and future directions are stated in <Ref>.

§ SYSTEM MODEL

We now introduce the network model, the traffic model, as well as AoI before formally defining AoB and AoC.

Notation: Common notation can be found in <Ref>. Whenever necessary for clarity, the expected value operator with respect to the distribution of some random element $X$ will be denoted by $\mathbb{E}_X[\cdot]$. The spatial point process models in this work are simple point processes, meaning node positions are distinct almost everywhere. Therefore, the convention will be that a node located at position $y\in\mathbb{R}^2$ will simply be referred to as node $y$. The $\ell$-2 norm will be denoted by $\|\cdot\|$. Random elements will generally be represented with an uppercase letter, a realization of which will be represented with a lowercase letter. For example, a realization of a point process $\Phi$ is $\phi$. For some set $\mathcal{W}$, the operator $[\cdot]_k$ produces $\left[\mathcal{W}\right]_k=\left\{A\subseteq\mathcal{W}\,\text{s.t.}\,|A|=k\right\}$, the set of subsets of $\mathcal{W}$ with cardinality $k$.

2cCommon notation
Notation     Description
$\Phi$     x]@l@Poisson Point process in $\mathbb{R}^2$ composed of
two independent processes $\Phi\stackrel{\Delta}{=}\Phi_N\cup\Phi_I$

$\lambda$     x]@l@Intensity of $\Phi_I$ and $\Phi_N$

$b_2(x,r)$     x]@l@Disk in $\mathbb{R}^2$ centered at $x$ with radius $r$

$H_{ij}$     x]@l@Channel fading coefficient between a
transmitter $i$ and receiver $j$

$\mu^{\Phi_I}_{ji}$     x]@l@Probability of successful delivery of packet
from $j$ to $i$ in the presence of interferers $\Phi_I$

$\theta$     x]@l@SIR threshold value; $\theta>1$

$p$     x]@l@medium access probability common to all
nodes and interferers, including the base
station when broadcasting

$\beta$     x]@l@Path loss exponent

$\mathcal{O}$     x]@l@Base station situated at the origin; $\mathcal{O}=(0,0)$

$X_i[t]$     x]@l@Inter-packet reception duration for the packet
reception process of receiver $i$ at time $t$

$A_{ji}(k)$     x]@l@AoI at receiver $i$ transmitted from $j$

$\ell(x)$     x]@l@Path loss function
$\ell(x)=\|x\|^{-\beta},\ x\in\mathbb{R}^2$

Example of a spatial realization $\phi_N$ of nodes (black circles), confined to a disk $b_2(0,r)$, and interferers (crosses) $\phi_I$, distributed across the Euclidean plane, with the base station (black triangle) in the center

§.§ Network Model

Consider a base station, denoted by $\mathcal{O}$, in the Euclidean plane situated at the origin, with a finite set of nodes randomly distributed in a disk $b_2(0,r)$ of finite radius $r$. The nodes are distributed within a disk as a homogeneous Poisson Point Process with intensity $\lambda$, denoted by $\Phi_N$. Interferers are also distributed according to a homogeneous Poisson Point Process $\Phi_I$ that is distributed across $\mathbb{R}^2$ with intensity $\lambda$ (see <Ref>). We denote the combined point process of nodes and interferers by

\begin{align}\Phi\stackrel{\Delta}{=}\Phi_N\cup\Phi_I\,.\end{align}

This spatial model captures a scenario in which the base station may be one of many broadcast and collection nodes in a spatially-large wireless network, and where the base station is only interested in communicating with nodes within its vicinity. Each information update consists of a single, timestamped packet. When broadcasting, the base station attempts transmission of a packet to all nodes in the disk; the packet is successfully received at a receiver if the signal-to-interference ratio (SIR) exceeds a fixed threshold $\theta>1$. Similarly, during collection, the base station successfully receives a packet from a given transmitter when the SIR exceeds $\theta$.  All transmission attempts occur at the start of discrete time-slots, the packet duration and the slot length both normalized to $1$. Therefore, time $t$ is defined to be discrete, denoting the $t$th slot. Medium access is granted to a transmitter – including the base station when transmitting– via an ALOHA-type random access scheme with a fixed common medium access probability (MAP) of $p$. That is, in any given time-slot the probability that a given transmitter attempts transmission is $p$, independent of all other time-slots and users in the network. In all subsequent sections we assume the packet delivery process is at steady state, having started at time $t=-\infty$.

The transmission power from every transmitter, including interferers, is fixed and normalized to 1. The wireless channel experiences Rayleigh fading and path loss attenuation. The fading loss random variable $H$ is i.i.d. exponentially distributed with mean 1. For a transmission from a transmitter $x$ to a receiver $y$, the path loss is defined to be

The path loss exponent $\beta$ is generally chosen to be in the interval $(2,4)$. At time-slot $t$ the medium access indicator random variable $Z_x[t]$ is 1 if a transmitter $x$ attempts transmission and 0 otherwise. Given the realization of node and interferer locations $\phi$ and including medium access probability, transmission power, fading, and path loss, we may represent the signal power observed at receiver $y$ for a broadcast from the base station to be
$$S^{\phi_I}_{\mathcal{O}y}[t]=Z_\mathcal{O}[t] H_{\mathcal{O}y}[t]\ell(y).$$
Similarly, the interference observed at $y$ is given by
$$I^{\phi_I}_{\mathcal{O}y}[t]=\sum_{x\in\phi_I}Z_x[t] H_{xy}[t]\ell(x-y)\,.$$
Therefore, the SIR is given by the ratio of $S^\phi_{\mathcal{O}y}[t]$ and $I^\phi_{\mathcal{O}y}[t]$,

\begin{align}\label{eq:sir_b}
SIR^{\phi_I}_{\mathcal{O}y}[t]=\frac{S^{\phi_I}_{\mathcal{O}y}[t]}{I^{\phi_I}_{\mathcal{O}y}[t]}=\frac{Z_\mathcal{O}[t]H_{\mathcal{O}y}[t]\ell(y)}{\sum_{x\in\phi_I}Z_x[t] H_{xy}[t]\ell(x-y)}\,.
\end{align}

For collection, the transmission signal from a transmitter $x$ in $\phi_N$ is subject to interference from both the field of interferers as well as other transmitters in $\phi_N$. Therefore, the SIR is

\begin{align*}
SIR^{\phi}_{x\mathcal{O}}[t]=\frac{Z_x[t]H_{x\mathcal{O}}[t]\ell(x)}{\sum_{y\in\phi\setminus x}Z_y[t] H_{y\mathcal{O}}[t]\ell(y)}\,.
\end{align*}

In the following subsection, we formally define the AoI metric, which will then be used to define AoB and AoC.

§.§ Age of Information

AoI is denoted by $A[t]$. Let $G[t]$ be the time stamp of the most recent packet successfully received as of time $t$. The time evolution of AoI is then defined in <Ref>:

A[t+1]=A[t]+1,    if no reception
min{t-G[t], A[t]}+1,    if reception .

The AoI at a receiver $x$ and at time $t$ corresponding to information updates from some node $y$ is denoted by $A_{yx}[t]$.

Age evolution over discrete time-slots

We assume any information source node can generate an update at-will, i.e. at each time $t$, an information update packet is instantaneously generated and transmitted with probability $p$. Therefore, the time stamp associated with a packet transmitted at time $t$ will always be $t$, and <Ref> becomes

A[t+1]=A[t]+1,   if no reception
1,   if reception .

An example of the age evolution over time is provided in <Ref>.

§.§ AoB and AoC

Having defined AoI and the network model, we now define Age of Broadcast and Age of Collection.

§.§.§ Age of Broadcast

The AoB $B_\mathcal{O}^{\phi_N}[t]$ with respect to some realization of the receiver locations $\phi_N$ and base station $\mathcal{O}$ at time-slot $t$ is defined as

\begin{align}
B_\mathcal{O}^{\phi}[t]\stackrel{\Delta}{=}\max_{i\in\phi_N}A^{\phi_I}_{\mathcal{O} i}[t]\,.
\end{align}

Given the base station begins broadcasting at $t-B_\mathcal{O}^{\phi}[t]$, the time until all receivers get an update cannot be less than $B_\mathcal{O}^{\phi}[t]$. Moreover, at least one base station has an update that is no greater than $t-B^{\phi}_{\mathcal{O}}[t]$

§.§.§ Age of Collection

The AoC $C_{\mathcal{O}}^{\phi}[t]$ with respect to the base station $\mathcal{O}$ and the realization of node and interferer locations $\phi$ at time $t$ is defined as

\begin{align}
C_{\mathcal{O}}^{\phi}[t]\stackrel{\Delta}{=}\max_{j\in\phi_N}A^{\phi}_{j \mathcal{O}}[t]\,.
\end{align}

For a given time $t$, the base station will have received at least one update from all but one transmitter since $t-C^{\phi}_{\mathcal{O}}[t]$.

In both broadcast and collection settings, we adopt the convention that if $\phi_N=\emptyset$, then AoB and AoC is 0 for all time. Having defined AoB and AoC, we establish preliminary results that are used in subsequent sections to analyze AoB and AoC.

§ PRELIMINARIES

When broadcasting, the probability the base station successfully delivers an update to an arbitrary receiver $y$  given the locations of the interferer positions $\phi_I$ is determined by the medium access probability and the channel characteristics. Since transmission attempts from transmitters and interferers alike are i.i.d. and Bernoulli in each time slot, the probability of successful delivery to $y$ is time-invariant and the time index $t$ can be dropped. Given the spatial realization of interferers $\phi_I$, the success probability is given by,

\begin{align}
\mu^{\phi_I}_{\mathcal{O}y}=\mathbb{P}\left(SIR_{\mathcal{O}y}>\theta\right)\,.
\end{align}

Averaging over the channel fading and the random access, and given the spatial realization $\phi_I$, the conditional reception success probability at a receiver $y$ is given by

\begin{align}
\mu_{\mathcal{O}y}^{\phi_I}&=\mathbb{P}\left(SIR^{\phi_I}_{\mathcal{O}y}>\theta\,\big|\,\Phi_I=\phi_I\right)\\&=p\prod_{x\in\phi_I}\left(1-\frac{p}{1+\theta\frac{\ell(y)}{\ell(x-y)}}\right)\,.\label{eq:lemma_succ}
\end{align}

The derivation of <Ref> is omitted due to space constraints but is similar to the analysis in <cit.> Lemma 1.
Through an identical line of reasoning, the conditional success probability during collection with respect to transmitter $x\in\phi_N$ is given by

\begin{align}\label{eq:coll_cond_succ}
\mu^\phi_{x\mathcal{O}}=p\prod_{y\in\phi\setminus x}\left(1-\frac{p}{1+\theta\frac{\ell(x)}{\ell(y)}}\right)\,.
\end{align}

where the sources of interference are both $\phi_I$ and $\phi_N\setminus x$; thus success probability is conditioned on $\phi$ instead of $\phi_I$.

Note that $\mu^{\phi_I}_{\mathcal{O}y}$ and $\mu^{\phi}_{x\mathcal{O}}$ are dependent on the realization $\phi$. Thus, when not given $\phi$, the reception success probability is a random variable.

Next, we de-condition <Ref> and <Ref> on $\Phi_I$ by taking the average over all realizations of the interferer locations. The packet reception success probability from the base station $\mathcal{O}$ to a receiver $y$ is then given by

\begin{align}\label{eq:av_succ}
\mu(\|y\|)= p\exp\left(-p\lambda\pi C \|y\|^2\right)\,,
\end{align}

\begin{align}
\end{align}

and the gamma function  $\Gamma(\cdot)$ is defined as ${\Gamma(x)\stackrel{\Delta}{=}\int_0^\infty t^{x-1}e^{-t}\,dt}$, and $\delta=\frac{2}{\beta}$.

The proof is omitted for brevity, but is similar to the analysis in <cit.> (see Section 3.2.3).

Note that $\mu$ no longer depends explicitly on node or interferer geometry and is instead only a function of the distance between $\mathcal{O}$ and $y$. Thus, in the instance-independent analysis we will express the success probability purely as a function of distance between the base station and the node.

Conditioned on $\Phi_N$, we find the success probability in the collection case de-conditioning on $\Phi_I$ to be

\begin{align}
\mu^{\phi_N}_{y\mathcal{O}}&=\mathbb{P}\left(H_{y\mathcal{O}}\geq \frac{\theta\left(I^{\phi_I}_{y\mathcal{O}}+I^{\phi_N\setminus\{y\}}_{y\mathcal{O}}\right)}{\ell(y)}\right)\\
&=\mathbb{E}\left[\exp\left(-\frac{\theta I^{\phi_I}_{y\mathcal{O}}}{\ell(y)}\right)\right]\cdot \mathbb{E}\left[\exp\left(-\frac{\theta I^{\phi_N\setminus\{y\}}_{y\mathcal{O}}}{\ell(y)}\right)\right]\\
&=p\exp\left(-p\lambda\pi C \|y\|^2\right)\cdot\prod_{j\in\phi_N\setminus\{y\}}\left(1-\frac{p}{1+\theta\frac{\ell(y)}{\ell(j)}}\right)\\
\end{align}

where $I^{\phi_N\setminus\{y\}}_{y\mathcal{O}}$ denotes the interference induced by the transmission of the nodes in $\phi_N\setminus\{y\}$.
Having established packet reception probabilities results in both instance-dependent and instance-independent cases, we leverage this insight in deriving AoB and AoC in the following sections, starting with broadcast.

§ BROADCAST

In this section we characterize the Expected AoB (EAoB), the expectation taken with respect to the ALOHA network traffic. In the instance-dependent case, we analyze EAoB given perfect knowledge of node and interferer locations. In the instance-independent case, node and interferer locations are unknown, so we find EAoB in expectation over the node and interferer point processes.

§.§ Instance-dependent (BD)

We assume the locations of all interferers and receivers are known. Interferers are located according to a realization of $\Phi_I$, denoted $\phi_I$. Receivers are also distributed according to a realization of $\Phi_N$ and is denoted $\phi_N$. Recall that the network is at steady state, having started at $t=-\infty$. Therefore, the AoB process is stationary and EAoB, defined as $\mathbb{E}\left[B^{\phi}_{\mathcal{O}}[t]\right]$, is the same for all finite $t$ and the dependence on time can be dropped to give $\mathbb{E}\left[B^{\phi}_\mathcal{O}\right]$.

An illustrative example supporting <Ref>, where (a) is the age evolution of three receivers that form the set $\phi_N$, (b) describes the packet arrival process up to time $t$, and (c) is the reverse of the process in (b).

To determine the EAoB, it is helpful to connect average broadcast age to the average broadcast delay, a related-yet-distinct metric <cit.>. For each receiver  $y\in\phi_N$, define $X^{\phi_I}_{\mathcal{O}y}[t]$ to be the time elapsed until successful reception of the next packet at receiver $i$ since time $t$. Broadcast delay $D^{\phi
}_\mathcal{O}[t]$ is then the time elapsed from time $t$ until the time at which all receivers in $\phi_N$ have received the next packet. That is,

\begin{align}\label{eq:bdelay}
\end{align}

Since the packet reception process is stationary – by virtue of the ALOHA random access and i.i.d. fading –  the average broadcast delay $\mathbb{E}\left[D^{\phi}_\mathcal{O}[t]\right]$ is the same for all finite $t$ and the dependence on $t$ can be dropped to give $\mathbb{E}\left[D^{\phi}_{\mathcal{O}}\right]$. We now reason that the average broadcast delay is equivalent to EAoB. This is evident by observation of the reverse packet reception process (see <Ref>).
Consider the instantaneous AoB at time $t$, which is the maximum AoI in $\phi_N$, and is exactly the time that elapsed between the current time point and the first time point in the past since which all receivers in the receiver set have successfully received at least one information update (see <Ref>). Now consider the broadcast delay, given by <Ref> and shown in <Ref>. Since the packet reception process's evolution in the forward direction is identical in distribution to that of its reverse process looking back in time, we may conclude the average broadcast delay must be equal to the EAoB. The formal claim is outlined below.

\begin{align}\mathbb{E}\left[B^{\phi}_\mathcal{O}\right] =\mathbb{E}\left[D^{\phi}_\mathcal{O}\right]\,.\end{align}

See Appendix <Ref>. Refer to <Ref> for an intuitive illustration of the proof.

This equivalence between the average broadcast delay and average AoB facilitates an analytical derivation of expected AoB. We begin by first defining the joint distribution of packet reception at each time $t$. Assuming knowledge of the locations of all nodes in $\phi_I$ and ${\phi_N}$, at time slot $t$, the joint distribution of packet reception for all the receivers, i.e. the joint distribution of $\left\{\mathbbm{1}^{\phi_I}_{\mathcal{O}i}[t]\right\}_{i\in\phi_N}$, the packet reception indicator random variables, can be obtained. Using this joint distribution, it is possible to determine the average broadcast age explicitly.

We partition the receivers into $\Xi[t]=\{i\in\phi_N\,|\,\mathbbm{1}_{\mathcal{O}i}[t]=1\}$ and $\Psi[t]=\{j\in\phi_N\,|\,\mathbbm{1}_{\mathcal{O}j}[t]=0\}$, the set of receivers that successfully received a packet at time $t$ and the set that did not receive a packet, respectively. Defining the probability of the set of receivers $\mathcal{R}$ all successfully receiving a packet at time slot $t$ as the following,

\begin{align}
\mu_{\mathcal{O}\mathcal{R}}^{\phi_I}[t]&=\mathbbm{P}\left(\cap_{i\in\mathcal{R}}\left\{SIR^{\phi_I}_{\mathcal{O}i}[t]>\theta\right\}\right)\\
&=\mathbbm{P}\left(\cap_{i\in\mathcal{R}}\left\{\mathbbm{1}^{\phi_I}_{\mathcal{O}i}[t]=1\right\}\right)\, ,
\end{align}

and conversely $w^{\phi_I}_{\mathcal{O}\mathcal{R}}[t]$ as the probability of the set of receivers $\mathcal{R}$ NOT receiving a packet at time slot $t$ as

\begin{align}
&=\mathbbm{P}\left(\cap_{i\in\mathcal{R}}\left\{\mathbbm{1}^{\phi_I}_{\mathcal{O}i}[t]=0\right\}\right)\, .
\end{align}

We next determine the probability rule of $\left\{\mathbbm{1}^{\phi_I}_{\mathcal{O}i}[t]\right\}_{i\in\phi_N}$, which is equivalent to finding the probability rule for $\Xi[t]$ without loss of generality. By the inclusion-exclusion property, the probability rule for $\Xi[t]$ can be established.

The inclusion-exclusion principle formula represents the probability of the union of a set of events ${A=\left\{A_1,\,A_2,\,\hdots,\,A_n\right\}}$ by the probabilities of intersections of its subsets:

\begin{align}
\mathbb{P}\left(\cup_{i= 1}^n A_i\right)=\sum_{j=1}^n (-1)^{n-1}\sum_{L\in[A]_k}\mathbb{P}\left(\cap_{i\in L}A_i\right)\,,
\end{align}

where $[A]_k$ denotes the set of subsets of $A$ with cardinality $k$.
Therefore, the probability rule for $\Xi[t]$ is

\begin{align}
p\left(\Xi[t]\right)&=\mathbb{P}\left(\left\{\cap_{i\in \Xi}\mathbbm{1}^{\phi_I}_{\mathcal{O}i}[t]=1\right\}\bigcap \left\{\cap_{j\in \Psi}\mathbbm{1}^{\phi_I}_{\mathcal{O}j}[t]=0\right\}\right)\\
&=\mu^{\phi_I}_{\mathcal{O}\Xi}[t]+\sum_{k=1}^{|\Psi|}(-1)^k\sum_{L\in [\Psi]_k}\mu_{\mathcal{O}\{\Xi\cup L\}}^{\phi_I}[t]\,.
\end{align}

Without loss of generality, we may define the probability rule starting at time $t=0$ for broadcast delay $D^{\phi}_{\mathcal{O}}[0]$. For the broadcast delay to be $\tau$, at least one receiver must receive a packet at time $t=\tau-1$. Moreover, of all the receivers that received a packet at time $\tau-1$, at least one must have received no packets for times $t\in\{0,\,\hdots,\,\tau-2\}$, i.e. given non-empty $\Xi[\tau-1]$, there exists some non-empty subset $J\subseteq{ \Xi[\tau-1]}$ such that $J\subseteq\cap_{i=0}^{\tau-2}\Psi[i]$. Therefore, $p^\phi_D(\tau)$, the probability that the broadcast delay is equal to $\tau$, is given by

\begin{align}
p_{D}^\phi(\tau)&=\sum_{\Xi\in 2^{\Phi_N}\setminus\{\emptyset\}}p(\Xi[\tau-1])\\&\cdot\underbrace{\mathbb{P}(\cup_{J\in 2^{\Xi[\tau-1]}\setminus\{\emptyset\}}\{J\subseteq\cap_{i=0}^{\tau-2}\Psi[i]\})}_{(*)}\\
&=\sum_{\Xi\in 2^{\Phi_N}\setminus\{\emptyset\}}p(\Xi[\tau-1])\\
&\quad\cdot\underbrace{\left(\sum_{n=1}^{|\Xi|}(-1)^{n+1}\sum_{J\in[\Xi]_n}\prod_{u=0}^{\tau-2} w^{\phi_I}_{\mathcal{O}J}[u]\right)}_{\text{(\textdagger)}}\,,
\end{align}

where $2^{\{\cdot\}}$ denotes the power set of some set $\{\cdot\}$, and (†) is the inclusion-exclusion formula applied to $(*)$.  Finally, we use <Ref> to find EAoB given by

\begin{align}
\mathbb{E}\left[B^{\phi}_\mathcal{O}[t]\right] &=\mathbb{E}\left[D^{\phi}_\mathcal{O}\right]\\
&=\mathbb{E}\left[D^{\phi}_\mathcal{O}[0]\right]=\sum_{k=1}^\infty k\cdot p_D^\phi(k)\,.
\end{align}

While this representation of average broadcast age is complete, a more intuitive characterization can be developed in the form of bounds on the average broadcast delay. We begin with an empirical observation. Based on simulation results, we observe that the average broadcast age is bounded above by the average broadcast age of an alternate packet reception process in which all packet reception indicators were independent random variables, albeit preserving the same distribution (see <Ref>). Recall that $X^{\phi_I}_{\mathcal{O}i}[t]$ denotes the time elapsed since time $t$ until the next packet reception. The conjecture based on this observation is formalized as follows.

The EAoB is bounded above by

\begin{align*}
\mathbb{E}\left[B^{\phi}_{\mathcal{O}}\right]\leq\mathbb{E}\left[\Tilde{D}^{\phi}_{\mathcal{O}}\right]=\mathbb{E}\left[\max_{i\in\phi_N}\Tilde{X}^{\phi_I}_{\mathcal{O}i}[0]\right]\,,
\end{align*}

where $\Tilde{X}^{\phi_I}_{\mathcal{O}i}[0]\stackrel{d}{=}X^{\phi_I}_{\mathcal{O}i}[0]$ and $\left\{\Tilde{X}^{\phi_I}_{\mathcal{O}i}[0]\right\}_{i\in\phi_N}$ are independent.

Simulation of Broadcast age compared against the maximum of independent random variables. The actual simulation's EAoB is consistently less than the independent random variable counterpart.

<Ref> Illustrates the conjecture stated, plotting the simulated broadcast delay and the broadcast delay expected if all packet reception processes were independent, maintaining the same packet reception distributions.

Having established an explicit formulation of EAoB as well as a conjectured upper bound, we may turn to the more general, instance-independent regime. Upper bounds are pursued in the instance-independent scenario, as outlined in the following subsection.

§.§ Instance-independent (BI)

In <Ref> we considered a particular instantiation of the nodes. Here, we take the expectation with respect to node and interferer positions. An upper bound can be found with a differential equation approach. Consider the squared ordered distances of the $i$ nearest receivers to $\mathcal{O}$ denoted $ R_{1}^2\leq R_2^2\leq\,\hdots\,\leq R_{i}^2$.
In two-dimensional Poisson Point Processes such as $\Phi_N$, the squared ordered distances have the same distribution as that of the arrival times in a one-dimensional Poisson Process $\Phi_N'\subset\mathbb{R}^+$ of intensity $\lambda'=\lambda\pi$ <cit.>

Focusing on this one-dimensional point process, consider a small interval in $\mathbb{R}^+$ given by $(x,x+\Delta]$ for very small $\Delta$. A receiver $y\in\Phi_N$ exists in the point process $\Phi_N'$ in the interval $(x,x+\Delta]\subset \mathbb{R}^+$ with probability $\lambda'\Delta$. We define the EAoB over the set of receivers in $\Phi_N$ that map to $(0,u]\in\mathbb{R}^+$ to be ${\overline{B}(u)}$. If a receiver does not exist in $(x,x+\Delta]\subset \mathbb{R}^+$, then the EAoB $\overline{B}(x+\Delta)$ would be the same as $\overline{B}(x)$. If a receiver $y$ does exist in the interval, either $y$ receives a packet after all the other receivers in $(0,x]$ with probability $\varsigma$ or it does not with probability $1-\varsigma$.
By setting the probability of $y$ getting a packet after the rest of the receivers to $\varsigma=1$, we upper bound the time to broadcast to all receivers that are in $(0,x+\Delta]$, as shown in the following:

\begin{align}
\overline{B}(x+\Delta)&=(1-\lambda'\Delta)\overline{B}(x)\\
&+(\lambda'\Delta)\left( \overline{B}(x)+\varsigma\frac{1}{\mu\left(\sqrt{x+\Delta}\right)}\right)\\
&\leq \overline{B}(x)+(\lambda'\Delta)\frac{1}{\mu\left(\sqrt{x+\Delta}\right)}\,,\label{eq:diff_ub}
\end{align}

where $\mu(\cdot)$ is given by <Ref>. As mentioned before, we upper bound in <Ref> by setting $\varsigma$ to 1. The average packet reception delay  $\frac{1}{\mu\left(\sqrt{x+\Delta}\right)}$ is readily found by taking the reciprocal of $\mu(\sqrt{x+\Delta})$ as given by <Ref> since the packet reception process is i.i.d Bernoulli. By bringing $\overline{B}(x)$ over to the left-hand side of the equation, dividing both sides by $\Delta$ and taking the limit as $\Delta\to 0$, we arrive at the following:

\begin{align}
&\lim_{\Delta\to 0}\frac{\overline{B}(x+\Delta)-\overline{B}(x)}{\Delta}\leq \lim_{\Delta\to 0}\frac{\lambda'}{\mu\left(\sqrt{x+\Delta}\right)}\\
&\frac{d\overline{B}}{dx}\leq \frac{\lambda'}{\mu\left(\sqrt{x}\right)}=\frac{\lambda\pi}{p}\exp\left(p\lambda\pi C x\right)\\
&\frac{d\overline{B}}{dr}=\frac{d\overline{B}}{dx}\frac{dx}{dr}\leq  \frac{\lambda'}{\mu\left(r\right)} \cdot 2r=\frac{2\lambda\pi r}{p}\exp\left(p\lambda\pi C r^2\right) \label{eq:diff_intermediate}
\end{align}

Solving the differential equation in <Ref> with the initial condition $B(0)=0$, we obtain

\begin{align}
B(r)\leq \frac{1}{p^2C}\left(e^{p\lambda\pi C r^2}-1\right)
\end{align}

Instance-Independent AoB scaling

<Ref> compares the upper bound through the differential analysis above against simulation.

In the next section we characterize the collection problem and find bounds on the performance.

§ COLLECTION

In the collection problem, the base station acts as a receiver situated at the origin, with a set of transmitters $\phi_N$ positioned within $b_2(0,r)$ sending updates to the base station. We characterize the expected age of collection (EAoC), defined as $\mathbb{E}\left[C^{\phi}_\mathcal{O}[t]\right]$. As in the broadcast section, the collection age is characterized in both instance-dependent and instance-independent regimes.

§.§ Instance-dependent (CD)

We begin the instance-dependent analysis by connecting AoC with a metric we denote as collection delay. For each transmitter $i\in\phi_N$, define $Y^{\phi}_{i\mathcal{O}}[t]$ to be the time elapsed between the current time $t$ and the successful reception of the next packet transmitted by $i$ to the base station after time $t$. The collection delay $\mathcal{K}^{\phi}_{\mathcal{O}}[t]$ is defined as

\begin{align*}
\mathcal{K}^{\phi}_{\mathcal{O}}[t]\stackrel{\Delta}{=}\max_{i\in\phi_N}Y^{\phi}_{i\mathcal{O}}[t]
\end{align*}

Since the packet reception process is i.i.d. over time and the AoC process is stationary since the network began at ${t=-\infty}$, the time index can be dropped. By an identical line of reasoning as that in <Ref>, we conclude that

\begin{align}\label{eq:coll_delay}
\mathbb{E}\left[C^\phi_{\mathcal{O}}\right]=\mathbb{E}\left[ \mathcal{K}^{\phi}_{\mathcal{O}}\right]\,.
\end{align}

Since $\theta >1$, the event of a packet reception at time $t$ at the base station from transmitter $x$ is disjoint from the event of a packet reception in the same time slot from transmitter $j\neq x$. That is, due to the threshold setting being larger than $1$, only a single packet can be received at the receiver in a single time slot. When packet reception events from different transmitters are disjoint and the packet reception process is time invariant, we observe that the update collection process resembles a coupon collection process.

In the classical Coupon Collector Problem, there are $n$ distinct coupons that are to be collected. Coupons are drawn randomly at each time step. At any time, the probability of drawing any one of $n$ coupons is uniformly $\frac{1}{n}$, independent of all other time steps, and so the resulting average time it takes to draw all $n$ distinct coupons at least once is $nH_n$, where $H_n$ denotes the $n$th harmonic number $H_n=\sum_{k=1}^n\frac{1}{k}$.

The variant of the CCP in the collection scenario is one in which $|\phi_N|$ distinct coupons need to be drawn but do not have a uniform probability of being drawn by the base station <cit.>. Additionally, there is the possibility of drawing an unwanted NULL coupon – the event where no packet is successfully received – which occurs if either no transmitter attempts a transmission or all attempted transmissions failed to exceed $\theta$.

The expression for the average collection can be found, expressed in <Ref>.

Given knowledge of the node and interferer locations $\phi$ the EAoC is given by

\begin{align}\mathbb{E}\left[C_\mathcal{O}^\phi\right]&=\mathbb{E}\left[\max_{i\in\phi_N} Y^\phi_{i\mathcal{O}}[0]\right]\\&=\sum_{i=1}^n(-1)^{i+1}\sum_{A\in [\phi_N]_k}\frac{1}{\sum_{u\in A}\mu^\phi_{u\mathcal{O}}}\, .\end{align}

Since we have established the equivalence of EAoC and expected collection delay in <Ref>, we focus on the expected collection delay. We invoke the maximum-minimums identity to represent $\max_{i\in\phi_N}Y^{\phi}_{i\mathcal{O}}$ as a sum of the minima of the non-empty subsets of $\phi_N$. The maximum-minimums identity states that for a finite set of numbers $A$ with cardinality $n$,

\begin{align}\label{eq:max_min_id}
\max A=\sum_{k=1}^n (-1)^{k+1}\sum_{L\in [A]_k}\min {L}\,.
\end{align}

Applying the identity to $\max_{i\in\phi_N}Y^{\phi}_{i\mathcal{O}}$,

\begin{align}
\mathbb{E}\left[\max_{i\in\phi_N}Y^{\phi}_{i\mathcal{O}}\right]=\sum_{k=1}^{|\phi_N|}(-1)^{k+1}\sum_{A\in[\phi_N]_k}\mathbb{E}\left[\min_{j\in A}Y^{\phi}_{j\mathcal{O}}\right]\, .
\end{align}

Due to the disjointedness of packet reception events between any transmitters in a time slot $t$, the random variable $\min_{j\in A}Y^{\phi}_{j\mathcal{O}}$ is a geometric random variable with parameter $\sum_{j\in A}\mu_{j\mathcal{O}}^\phi$. Thus,

\begin{align}
\mathbb{E}[\max_{i\in\phi_N}Y^{\phi}_{i\mathcal{O}}]=\sum_{k=1}^{|\phi_N|}(-1)^{k+1}\sum_{A\in[\phi_N]_k}\frac{1}{\sum_{j\in A}\mu^\phi_{j\mathcal{O}}}\,,
\end{align}

and the proof is complete.

We proceed to find bounds on the EAoC in the instance-independent case in the following subsection.

§.§ Instance-independent (CI)

In this scenario, the locations of the transmitters are no longer assumed to be known, distributed according to the Poisson Point Process $\Phi_N$. We begin with an upper bound on EAoC. Conditioning on the size of $\Phi_N$ to be $n$, the nodes are distributed i.i.d. uniform in the disk $b_2(\mathcal{O},r)$. Based on this conditioning, and assuming no nodes are present within a small distance $\epsilon$ of the base station, an upper bound for EAoC is outlined in the following claim:

Conditioned on the number of transmitters $|\Phi_N|=n$ in the disk $b_2(\mathcal{O},r)$ the EAoC is bounded above as given by

\begin{align}
\mathbb{E}\left[C^\phi_\mathcal{O}\,\big|\, |\Phi_N|=n\right]\leq \left(\overline{\mu}\right)^{-1}H_n\, ,
\end{align}

\begin{align}
\overline{\mu}=\left(1-\frac{p}{1+\theta
\left(\frac{\epsilon}{r}\right)^{\beta}}\right)^{n-1}\cdot \mu(r)
\end{align}

The most disadvantaged transmitter in terms of successful delivery probability is one situated at a distance $r$ from the base station, while the remaining $n-1$ transmitters are close enough to the base station to observe no path loss, i.e. at distance $\epsilon$. The success probability of this disadvantaged transmitter is

\begin{align}
\left(1-\frac{p}{1+\theta \left(\frac{\epsilon}{r}\right)^{\beta}}\right)^{n-1}\cdot\mu(r)\,,
\end{align}

since each of the other transmitters contribute equally to the interference observed at the base station.

If all transmitters have this same pessimistic delivery probability, the EAoC would be that of a classical CCP, the resulting EAoC given in <Ref>.

We now present numerical simulations that highlight the interplay between broadcast and collection.

§ NUMERICAL RESULTS

2cSimulation Parameter Settings
Parameter     Default Value
$\lambda$     x]<EMAIL_ADDRESS>
$\theta$     x]@l@5

$r$     x]@l@$10$

$\beta$     x]<EMAIL_ADDRESS>
$p$     x]<EMAIL_ADDRESS>
Table of default parameter values when held constant as part of the numerical simulation

We examine EAoB and EAoC with different network parameter settings using numerical simulation. Unless stated otherwise, the default network parameter settings for simulation are provided in <Ref>. The EAoB and EAoC for each parameter settings is determined using Monte Carlo simulation, simulating $250000$ time slots per trial.

Instance-independent Age scaling with radius $r$ going from $10$ to $15.5$

Instance-independent EAoB scaling with radius $r$ going from $10$ to $14$

Instance-independent EAoC scaling with radius $r$ going from $10$ to $14$

Instance-independent EAoB and EAoC scaling, with intensity $\lambda$ going from $1e-3$ to $1e-1$.

In <Ref>, the scaling behavior of EAoB and EAoC combined is shown as the radius of $b_2(\mathcal{O},r)$ is increased. The scaling behavior is super-exponential in both cases, with EAoC consistently larger than EAoB for the same radius. This is expected, since in the collection scenario at most one packet can be delivered to the base station in a given time slot, whereas when broadcasting, it is possible for multiple receivers to get a packet simultaneously. Moreover, the interference observed at the base station in the collection case is, in expectation, larger since both $\Phi_I$ and $\Phi_N$ are sources of interference when collecting. <Ref> plot EAoB and EAoC versus radius for different values of the medium access probability, showing that a greater value of $p$ result in larger age for these parameter settings. In <Ref>, the density is varied on a logarithmic scale. The figure depicts the exponential growth of EAoB and EAoC with respect to node and interferer intensity $\lambda$.

§ CONCLUSION

We defined AoB and AoC as information freshness metrics suitable for the cases of broadcast and collection, respectively, in spatially-distributed wireless networks. We characterized the expected AoB and AoC when the locations of nodes and interferers are known and unknown. When the locations are known and the packet transmission process is stationary, we showed that expected AoB and AoC were equivalent to the expected broadcast delay and collection delay, respectively. Upper-bounds were found in the instance-independent scenario: the AoB upper-bound is a solution to a differential equation, and the AoC upper-bound uses the solution to the worst-case packet delivery success probability given a small exclusion radius $\epsilon$. We demonstrated through numerical simulation the relation between AoB and AoC and network parameters such as density and medium access probability.

§ PROOF OF <REF>

For any time $t$ the packet reception vector $\Vec{{\mathbbm{1}}}^{\phi_I}_{\mathcal{O}}[t]=\left\{\mathbbm{1}^{\phi_I}_{\mathcal{O}i}\right\}_{i=1}^{|\phi_N|}$ can take values from a finite set of states $\mathcal{N}$ of cardinality $2^{|\phi_N|}$.  Therefore, the packet reception process over time is a discrete-time Markov Chain. Since the packet reception process is stationary and time-invariant by virtue of the ALOHA transmissions and i.i.d fading, this Markov Chain is irreducible, aperiodic, and positive recurrent. We denote the probability that the process is in state $u\in\mathcal{N}$ to be $p_u$.  WLOG, we denote the probability that the process transitions from state $j\in\mathcal{N}$ at time $t$ to $k\in\mathcal{N}$ at time $t+1$ to be $p_{jk}$ for any time $t\geq 0$. By Kolmogorov's criterion for Markov chain time-reversibility, the process is time-reversible if and only if the condition

\begin{align}\label{eq:kolmogorov}
p_{{j_1}{j_2}}p_{{j_2}{j_3}}\hdots p_{{j_{n-1}}{j_n}}p_{j_{n}{j_1}}=p_{j_{1}{j_n}}p_{{j_{n}}{j_{n-1}}}\hdots p_{{j_3}{j_2}}p_{{j_2}{j_1}}
\end{align}

is satisfied for all finite sequences of states $j_1,j_2,\hdots,j_n\in\mathcal{N}$.

Since the event the process is in state $j$ at time $t$ is independent of the event it is in state $k$ at time $t+1$ for all time $t\geq 0$, the condition in <Ref> holds, establishing that the packet reception process is time-reversible. As shown  in <Ref>, for any time $t$, the AoB $B^{\phi}_{\mathcal{O}}[t]$ is shown to be equivalent to $D^{\phi}_{\mathcal{O}}(t')$, the broadcast delay of the time-reversed process starting from the same time slot. Since time-reversibility implies that expectations with respect to forward time are the same as that over reverse time, coupled with the fact that  $\mathbb{E}\left[D^{\phi}_{\mathcal{O}}[t]\right]=\mathbb{E}\left[D^{\phi}_{\mathcal{O}}\right]$, we may conclude that

\begin{align*}
\mathbb{E}\left[B^{\phi}_{\mathcal{O}}\right]=\mathbb{E}\left[D^{\phi}_{\mathcal{O}}\right]\,.
\end{align*}


rightsretained

# Understanding Stakeholders’ Perceptions and Needs Across the LLM Supply
Chain

Agathe Balayn

Lorenzo Corti

Fanny Rancourt

Fabio Casati

Ujwal Gadiraju

ServiceNow, Delft University of Technology, Trento University
<EMAIL_ADDRESS>Delft University of Technology<EMAIL_ADDRESS>ServiceNow<EMAIL_ADDRESS>ServiceNow
<EMAIL_ADDRESS>Delft University of Technology
<EMAIL_ADDRESS>
(2024)

###### Abstract

Explainability and transparency of AI systems are undeniably important,
leading to several research studies and tools addressing them. Existing works
fall short of accounting for the diverse stakeholders of the AI supply chain
who may differ in their needs and consideration of the facets of
explainability and transparency. In this paper, we argue for the need to
revisit the inquiries of these vital constructs in the context of LLMs. To
this end, we report on a qualitative study with 71 different stakeholders,
where we explore the prevalent perceptions and needs around these concepts.
This study not only confirms the importance of exploring the “who” in XAI and
transparency for LLMs, but also reflects on best practices to do so while
surfacing the often forgotten stakeholders and their information needs. Our
insights suggest that researchers and practitioners should simultaneously
clarify the “who” in considerations of explainability and transparency, the
“what” in the information needs, and “why” they are needed to ensure
responsible design and development across the LLM supply chain.

††conference: CHI’24, HCXAI workshop Honolulu, HI, USA

##  1 A Case for XAI Across the Supply Chain

Research addressing explainability and transparency of AI systems seldom
clarifies the purpose and the target stakeholders, often assuming the
information needs of stakeholders with little or no validation [18, 13, 24].
There has been an increasing recognition that the development of AI systems
spans a long supply chain of stakeholders [27, 8, 21]. However, only AI
developers [4, 9] and end-users [15, 11] have received considerable attention
thus far. Other stakeholders such as legal teams or product managers remain
largely absent in existing explorations. ††margin:  S1 – Study participants.
We cover a plurality of domains in the supply chain, including engineering (AI
developers, AI researchers), user experience (user research, content design),
business (product managers, business analysts, customer service), and
governance (legal and risk teams, government relations). The practical tasks
of our participants involve, e.g., implementing, testing, and overseeing AI
systems, or strategizing around the requirements of the system and broader
organization (e.g., functional requirements, definition of risks, etc.). These
participants’ responsibilities span executives, managers, less senior
employees, etc. Finally, we also interviewed end-users of the AI systems
developed by our participants –they are internal to the companies developing
the AI systems or independent thereof, and none are AI experts.  Inspired by
prior work [10, 12, 14, 28], we argue for the necessity to broaden our
understanding of AI stakeholders, their engagement and needs pertaining to
explanations and transparency.

Particularly, in an era in which AI systems are increasingly closed-source and
concealed behind API access (e.g., LLM-based chatbots), stakeholders may
elicit new needs for explainability and transparency. Following is an example
of such information needs across stakeholders. An applied researcher within
organization A might be looking for an open-source foundation model of any
organization and might select the model of organization B that is pre-trained
on the dataset containing the most relevant data samples to the application
they are developing. Then, the legal team of organization A might have to
enquire about the data collection practices of organization B, e.g., to make
sure that they respect licensing agreements [20] and labor rights of data
annotators [2]. Finally, the quality team of organization C deploying the AI
system they would buy from organization A would need to inspect explanations
about the outputs of the system to make sure that the system performs well-
enough for their application.

It is complex and challenging, yet vital to investigate and surface such
underexplored aspects in the AI supply chain and relevant practices across
organizations. Aligned with the notion of interpretative flexibility [17],
stakeholders’ perceptions of AI systems directly affect their needs and
collaborative practices across the AI supply chain. It can therefore be
challenging to appropriately investigate known and unknown explainability and
transparency needs.
In this work, we take a formative step towards characterising “who” the
stakeholders of explainability and transparency are across the AI supply chain
and “what” their information needs are. We report early results from a
qualitative inquiry about the perceptions and needs for explainability and
transparency of 71 AI stakeholders across different organizations. Our results
underscore the importance of grounding studies in the who, what, and why of
explainability and transparency, reflect on methodological best practices, and
bear implications for the HCXAI community in establishing and shaping relevant
policies.

##  2 Study Methodology

We interviewed 71 participants, across 10 private organizations, and spanning
a wide variety of roles (see S1). Participants were recruited via our
professional network, with snowball and convenience sampling, to recruit a
diversity of stakeholders. All participants develop or use LLMs. In the
interviews (on average 45 mins.), we first asked our participants to describe
their day-to-day activities, and to discuss their primary concerns,
challenges, and needs concerning the harmful impact of AI systems. This
allowed us to identify information needs. We then probed their understanding
of explainability and transparency: what these concepts evoke, how important
they are, and what the entailing challenges might be. We first analysed the
interviews through deductive coding looking for the “what”, “who”, and
challenges of the informational needs across stakeholders. Then, we performed
a round of inductive coding to identify other recurrent themes.

##  3 Results

1) The “what": Explainability and Transparency
Generally, our participants viewed transparency as encompassing information
beyond AI models (whereas XAI was viewed as being concerned with these models)
such as training data, code, and reports about other supply chain
stakeholders. Some information needs related to the outputs of the models, be
it their trustworthiness (accuracy or fairness), surrounding information
(e.g., confidence scores), or explanations (as understood by the algorithmic
XAI community). Other information items related to the internal workings of
the LLM (e.g., details about its training dataset) or its ’meta-properties’
(e.g., organization developing it).

Our study points toward several information needs that have been discussed in
prior literature [18, 13, 3, 6]. Interestingly enough, more recent artefacts
such as the Foundation Model Transparency Index [6], cover elements that our
participants perceived as general concerns rather than specific to
explainability and transparency (see S2). These include data labor,
environmental impact, and broader ethical concerns. Regardless, participants
focused on elements that are often considered incidental for XAI and
transparency (e.g., confidence values for individual predictions) or “less-
technical” (e.g., developers’ discretionary choices, discussed by a few prior
works [11, 21]). Some participants perceived XAI and transparency as
information primarily useful for end-users and AI system deployers. Disclosure
on LLM-generated samples (18% of participants) and basic explanations
pertaining to LLMs (24% of participants) were of particular interest. Other
participants discussed their own information needs.

††margin:  S2 – Information items discussed by our AI stakeholders.
Information need % "LLM performance": fairness-metric related 2 limits of LLM
22 "LLM decisions": confidence for each LLM output 2 fact that output is LLM-
generated 18 log of LLM inputs & outputs 4 traceability of errors & outputs 9
Explainability: explanations of LLM output via LLM input & addit. info used by
LLM 20 LLM system rationales leading to an LLM output 47 "LLM internals": LLM
design decisions 53 LLM evaluation decisions 18 customer data handling 20
"Meta info": basic explanations about LLM 24 guidelines for good use of LLM by
customer 9 who are the creators 2 open source 2 what other teams do about LLM
or for a specific LLM 4

2) The “why": Connecting “what” and “who”
Given the broad range of AI stakeholders interviewed, we note several trends
concerning the purposes for explainability and transparency. Unsurprisingly,
such trends are shaped by a stakeholder’s role, prior knowledge, and network
of collaborators (see Figure 1). Developers, algorithmic researchers, and
quality assessors frequently mentioned technical information (e.g., prediction
confidence and design decisions) necessary to select which AI model to fine-
tune, debug the overall system, or identify the potential algorithmic harms.
On the other hand, UX designers and researchers provide guidelines to
customers for responsible use of the AI system. Together with product
managers, they emphasized disclosing essential information to customers or
end-users (e.g., to gain buy-in) or to ethics review boards to attest to
Responsible AI endeavors (e.g., appropriate handling of customer data). Such a
purpose appears to be met by end-users’ need for transparency from the makers
of AI-powered products (e.g., to gauge the trustworthiness of a company).
Finally, we note that XAI and transparency were attributed cross-functional
purposes around fostering Responsible AI practices and accountability
internally and externally across the AI supply chain.

Figure 1: Information needs per category of stakeholders (fraction of the
participants who mentioned the need). We excluded information needs mentioned
by a single category of stakeholders: developers - LLM confidence (14%);
customer-centered roles - log of LLM inputs/outputs (18%), fairness
performance (9%); UX researcher - open-source (25%), work of other teams
(50%); end-user - LLM creators (100%).

3) Challenges faced by AI stakeholders
Our study highlighted several challenges AI stakeholders face concerning
explainability and transparency both at an individual and collaborative level.

At an individual level, participants expressed feasibility tensions in
implementing explainability and transparency, e.g., between XAI and customers’
data confidentiality. For instance, the quality assessment team discussed
challenges with information access. They reported on their need to get to know
customers and end-users of the LLMs, to appropriately assess these LLMs (e.g.,
considering social biases). Yet, they emphasized that accessing any
information about such external stakeholders is impossible from their position
in the AI supply chain. Additionally, technical implementation details are
often difficult to accurately report due to (1) the fast-paced development of
LLMs, (2) potential exposure to adversarial attacks, and (3) business secrets.

Participants in our study reported mixed needs for disclosure or concealment
of information. Broadly, at a collaborative level, many stakeholders appear to
lack knowledge about the functioning of AI and explainability and
transparency. For instance, several deployers of AI systems recognized not
being aware of potential harms caused by AI systems and hence not being able
to ask for relevant information in that regard. Other participants, such as
some algorithmic researchers, were only cognizant of algorithmic XAI concepts
and debates around AI open-source, which shows a narrow perspective on XAI.
This prevents them from identifying and understanding information relevant to
either their or others’ work.

As a consequence, we identified misconceptions (e.g., a participant argued
that making an AI system transparent necessarily leads to ensuring the
fairness of its outputs) and terminological confusion around XAI and
transparency (e.g., most participants use "transparency", "explainability" or
"accountability" assuming a certain meaning that they disentangle only when
prompted), which, more broadly, can cause miscommunication and mishandling of
AI systems. Finally, participants’ surroundings and time constraints (owing to
the ongoing LLMs race) affect their views on XAI and transparency, hampering
appropriate reflections.

##  4 Discussion & Implications

Our results emphasise the importance of adopting a broad view of the AI supply
chain when discussing XAI and transparency, illustrating an existing lack of
understanding of explainability and transparency across it. As prior works
(e.g., [5, 3, 14, 19, 22, 16]) had not identified as many stakeholders and
their purposes [26, 25, 7], nor all information items, we argue that there is
a strong need for future work to be more comprehensive. The supply chain view
also enabled us to surface concrete factors that impact information needs, and
diverse, complex challenges to be tackled in the future. Our preliminary
insights provide a range of methodological lessons for future work.

  * •

The complex relations between the “what", “who", and “why" echo the need to
investigate XAI and transparency in context, without solely focusing on the
information items (i.e., the “what"). This reaffirms a call for future work on
XAI and transparency to be grounded within the “who" and “why", as proposed,
e.g., by the AI FactSheet methodology [23, 21]. While many XAI and
transparency tools have been proposed (e.g., [13, 18, 6, 3]), they focus
solely on the “what" and require revisiting in light of our stakeholders’
needs and concomitant challenges.

  * •

Our study emphasizes the need for methodological triangulation to develop a
holistic understanding of stakeholders’ perceptions and needs across the AI
supply chain. XAI and transparency are complex constructs with a nuanced and
overlapping scope. They are perceived differently by those in different
positions within the supply chain, present great terminological ambiguity (not
to be mistaken with diversity), and evoke different information items,
stakeholders, and purposes. In a sense, they are boundary objects to discuss
information exchanges and needs across stakeholders. In our study, we used
these constructs as starting points for discussions. We observed that AI
stakeholders referred to different information needs for broader goals while
discussing their current work and concerns. E.g., knowledge of data access
rights per customer was discussed by AI developers for training or evaluating
the AI system but not in the context of XAI and transparency. Hence, we
believe that future studies would benefit from adopting complementary objects
of study: studying not only perceptions of XAI, transparency and the needs
thereof, but also in-depth information needs of the different stakeholders
while refraining from referring to the terms XAI and transparency.

  * •

We recommend bearing in mind stakeholders’ different interpretations of AI
“explainability” and “transparency” –often confused– and ensuring conceptual
clarity in discussions involving these terms. This can be done by asking
participants to clarify their use of the terms or providing clear definitions
depending on the goal of the studies.

  * •

Our findings can aid the development of a conceptual framework around AI
information needs, that could be used to carefully craft further studies. Such
a framework could start with the identified who, what, and why of AI
information needs, and also encompass personal obstacles, organizational
challenges, tensions and preferences.

Policy, Explainability, and Transparency.
Our results finally point out the impossibility of solely relying on
stakeholders’ perceived information needs to define XAI and transparency
obligations. Indeed, this topic was revealed to be highly dependent on
stakeholders’ knowledge and on their subjective judgment of the various
tensions highlighted – many information items found in prior work were not
mentioned by any participant, with or without a technical role or background,
despite all of them being decision makers across the supply chain and about
explainability and transparency considerations. The identified informational
challenges hint at the potential undesirability to answer any information
need. Hence, “who" should be responsible for making such decisions remains to
be explored both within organizations and on a regulatory level.

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