# On the mod $k$ chromatic index of graphs

Oothan Nweit111School of Mathematical Sciences, Zhejiang Normal University,
Jinhua, Zhejiang 321004, China.$~{}^{,}$222E-mail<EMAIL_ADDRESS>,
Daqing Yang11footnotemark: 1$~{}^{,}$333E-mail:
dyang@zjnu.edu.cn.$~{}^{,}$444Corresponding author, grant numbers: NSFC
12271489, U20A2068.

###### Abstract

For a graph $G$ and an integer $k\geq 2$, a $\chi^{\prime}_{k}$-coloring of
$G$ is an edge coloring of $G$ such that the subgraph induced by the edges of
each color has all degrees congruent to $1~{}({\rm mod}\ k)$, and
$\chi^{\prime}_{k}(G)$ is the minimum number of colors in a
$\chi^{\prime}_{k}$-coloring of $G$. In [“The mod $k$ chromatic index of
graphs is $O(k)$”, J. Graph Theory. 2023; 102: 197-200], Botler, Colucci and
Kohayakawa proved that $\chi^{\prime}_{k}(G)\leq 198k-101$ for every graph
$G$. In this paper, we show that $\chi^{\prime}_{k}(G)\leq 177k-93$.

Keywords: edge colorings, modulo, orientation, maximum average degree

##  1 Introduction

All graphs considered here are simple. Let $G=(V,E)$ be a graph, and
$v(G):=|V(G)|$ and $e(G):=|E(G)|$. If $X\subseteq V(G)$, then $G[X]$ is the
subgraph of $G$ induced by $X$. For an integer $k\geq 2$, a
$\chi^{\prime}_{k}$-coloring of $G$ is a coloring of the edges of $G$ such
that the subgraph induced by the edges of each color has all degrees congruent
to $1~{}({\rm mod}\ k)$, and the ${\rm mod}\ k$ chromatic index of graph $G$,
denoted by $\chi^{\prime}_{k}(G)$, is the minimum number of colors in a
$\chi^{\prime}_{k}$-coloring of $G$. Pyber [4] proved that
$\chi^{\prime}_{2}(G)\leq 4$ for every graph $G$ and asked whether
$\chi^{\prime}_{k}(G)$ is bounded by some function of $k$ only. Scott [5]
proved that $\chi^{\prime}_{k}(G)\leq 5k^{2}\log k$ for any graph $G$, and in
turn asked if $\chi^{\prime}_{k}(G)$ is in fact bounded by a linear function
of $k.$ Botler, Colucci, and Kohayakawa [1] answers Scott’s question
affirmatively by proving the following theorem.

######  Theorem 1.1 ([1])

For every graph $G$ we have $\chi^{\prime}_{k}(G)\leq 198k-101$.

Also in [1], Botler, Colucci, and Kohayakawa proposed the following
conjecture:

######  Conjecture 1.2 ([1])

There is a constant $C$ s.t. $\chi^{\prime}_{k}(G)\leq k+C$ for every graph
$G$.

In this paper, we improve the upper bound of the mod $k$ chromatic index of
graphs by proving the following theorem.

###### Theorem 1.3

For every graph $G$ we have $\chi^{\prime}_{k}(G)\leq 177k-93$.

In the proof of Theorem 1.1, Botler et al. in [1] applies the following two
lemmas.

######  Lemma 1.4 (Mader [3])

If $k\geq 1$, $G$ is a graph with $e(G)\geq 2kv(G)$, then $G$ contains a
$k$-connected subgraph.

######  Lemma 1.5 (Thomassen [6])

If $k\geq 1$ and G is a $(12k-7)$-edge-connected graph with an even number of
vertices, then $G$ has a spanning subgraph in which each vertex has degree
congruent to $k~{}({\rm mod}\ 2k)$.

A graph $G$ is $k$-divisible if $k$ divides the degree of each vertex of the
graph $G$. By applying Lemma 1.4 and Lemma 1.5, Botler et al [1] proved the
following lemma.

######  Lemma 1.6 ([1])

If graph $G$ does not contain a nonempty $k$-divisible subgraph, then
$e(G)<2(12k-6)v(G)$.

For a graph $G$, let $N_{G}(v)$ denote the neighbors of $v$, $E_{G}(v)$ denote
the edges that are incident to $v$, and let $d_{G}(v)$ be the degree of $v$,
i.e., $d_{G}(v)=|E_{G}(v)|$. Let $\vec{G}=(V,\vec{E})$ be an orientation of
$G$, for $v\in V$, let $N_{\vec{G}}^{+}(x)$ denote the out-neighbor(s) of $x$,
i.e., $N_{\vec{G}}^{+}(x)=\\{y:x\rightarrow y\\}$, let $d_{\vec{G}}^{+}(x)$ be
the out-degree of $x$, i.e., $d_{\vec{G}}^{+}(x)=|N_{\vec{G}}^{+}(x)|$; if $y$
is an out-neighbor of $x$, then we say edge $\overrightarrow{xy}$ an out-edge
of $x$. Let $N_{\vec{G}}^{-}(x)$ denote the in-neighbor(s) of $x$, i.e.,
$N_{\vec{G}}^{-}(x)=\\{y:x\leftarrow y\\}$, let $d_{\vec{G}}^{-}(x)$ be the
in-degree of $x$, i.e., $d_{\vec{G}}^{-}(x)=|N_{\vec{G}}^{-}(x)|$; if $y$ is
an in-neighbor of $x$, then we say edge $\overleftarrow{xy}$ an in-edge of
$x$. Let $\Delta^{+}\left(\vec{G}\right)=\max_{v\in V}d^{+}_{\vec{G}}(v)$,
$\Delta^{-}\left(\vec{G}\right)=\max_{v\in V}d^{-}_{\vec{G}}(v)$. We drop the
subscripts $G$ or $\vec{G}$ in the above notations when $G$ or $\vec{G}$ is
clear from the context.

The maximum average degree of a graph $G$, denoted by ${\rm mad}(G)$, is
defined as

${\rm mad}(G)=\max_{H\subseteq G}\frac{2e(H)}{v(H)},$

which places a bound on the average vertex degree in all subgraphs. It has
already attracted a lot of attention and has a lot of applications. The
following theorem is well-known (cf. [2], Theorem 4), we use it in our proof
of Theorem 1.3.

###### Theorem 1.7

Let $G$ be a graph. Then $G$ has an orientation $\vec{G}$ such that
$\Delta^{+}(\vec{G})\leq d$ if and only if ${\rm mad}(G)\leq 2d$.

##  2 Proof of Theorem 1.3

In this section, we prove Theorem 1.3. The proof uses the following lemma,
which was available in [7]. For the completeness of this paper, we present its
short proof here.

###### Lemma 2.1

([7, Lemma 1.7], adapted) Let $d\geq 0$ be an integer. If an oriented graph
$\vec{G}$ has $\Delta^{+}(\vec{G})\leq d$, then there exists a linear order
$\sigma$ of $V(\vec{G})$, such that for any vertex $u\in V(\vec{G})$, the
number of vertices that are the in-neighbors of $u$, and precede $u$ in
$\sigma$ is at most $d$.

Proof. We recursively construct a linear ordering $\sigma=v_{1}v_{2}\ldots
v_{n}$ of $V=V(\vec{G})$ as follows. Suppose that we have constructed the
final sequence $v_{i+1}\ldots v_{n}$ of $L$. (If $i=n$ then this sequence is
empty.) Let $M=\left\\{v_{i+1},\ldots,v_{n}\right\\}$ be the set of vertices
that have already been ordered and $U=V-M$ be the set of vertices that have
not yet been ordered. Let $\vec{G_{U}}\subseteq\vec{G}$ be the subgraph of
$\vec{G}$ induced by $U$. If we have not yet finished constructing $\sigma$,
we choose $v_{i}\in U$ so that $d_{\vec{G_{U}}}^{-}(v_{i})$ is minimal in
$\vec{G_{U}}$. Since $\sum_{v\in U}d_{\vec{G_{U}}}^{-}(v)=\sum_{v\in
U}d_{\vec{G_{U}}}^{+}(v)$, and
$\Delta^{+}\left(\vec{G_{U}}\right)\leq\Delta^{+}\left(\vec{G}\right)\leq d$,
we have $d_{\vec{G_{U}}}^{-}(v_{i})\leq d$. This proves the lemma.

Combining Lemma 2.1 and the techniques used in [1], we prove the following
lemma, which is the key to the proof of Theorem 1.3.

###### Lemma 2.2

Let $d\geq 1$ be an integer. If a graph $G$ has an orientation $\vec{G}$ such
that $\Delta^{+}(\vec{G})\leq d$, then $\chi^{\prime}_{k}(G)\leq 7d+2k-3$.

Proof. If a graph $G$ has an orientation $\vec{G}$ such that
$\Delta^{+}(\vec{G})\leq d$, by applying Lemma 2.1, we can suppose linear
ordering $\sigma:=v_{1}v_{2}\ldots v_{n}$ of $V(G)$ satisfying that for any
vertex $u\in V(\vec{G})$, the number of vertices that are the in-neighbors of
$u$, and precede $u$ in $\sigma$ is at most $d$.

Following the above linear ordering $\sigma$, we give a
$\chi^{\prime}_{k}$-coloring of $G$ by coloring the edges incident with
$v_{i}$ for each $v_{i}\in\\{v_{1},\ldots,v_{n-1}\\}$ in turn. At step
$v_{i}$, we name this procedure as processing vertex $v_{i}$, which means that
we color all the edges incident with $v_{i}$ that are not colored yet at this
time. After we have finished processing vertex $v_{i}$, we shall maintain that
we have a $\chi^{\prime}_{k}$-coloring of the graph spanned by the edges
incident with $v_{1},\ldots,v_{i}$, we call this a good partial
$\chi^{\prime}_{k}$-coloring after step $v_{i}$.

To define the coloring method, we partition all the colors into two sets
$C_{1}$ and $C_{2}$ such that $|C_{1}|=3d-1$ and $|C_{2}|=4d+2k-2$, note that
$|C_{1}|+|C_{2}|=7d+2k-3$. For each $1\leq i\leq n-1$, we use the colors in
$C_{1}$ to color the uncolored out-edges of $v_{i}$ and use the colors in
$C_{2}$ to color the uncolored in-edges of $v_{i}$. Equivalently, for any
directed edge $uv$ $(u\rightarrow v$ in $\vec{G})$, if $u$ is processed before
$v$, then edge $uv$ is colored with a color in $C_{1}$; if $v$ is processed
before $u$, then edge $uv$ is colored with a color in $C_{2}$.

By induction on $i$, we give a good partial $\chi^{\prime}_{k}$-coloring of
$G$ after processing $v_{i}$. For the induction hypothesis, suppose when we
begin to process vertex $v_{i}$, all the edges that are incident with a vertex
$v$ that precedes $v_{i}$ in $\sigma$ have already been colored.

For each $1\leq i\leq n-1$, when $v_{i}$ is processed, let $U^{+}(v_{i})$
denote the unprocessed out-neighbor(s) of $v_{i}$, i.e.,
$U^{+}(v_{i})=N_{\vec{G}}^{+}(v_{i})\cap\\{v_{i+1},\ldots,v_{n}\\}$; let
$U^{-}(v_{i})$ denote the unprocessed in-neighbor(s) of $v_{i}$, i.e.,
$U^{-}(v_{i})=N_{\vec{G}}^{-}(v_{i})\cap\\{v_{i+1},\ldots,v_{n}\\}$. When we
process $v_{i}$, we color the uncolored out-edges $\\{v_{i}v_{j}:v_{j}\in
U^{+}(v_{i})\\}$ such that all out-edges of $v_{i}$ are colored with distinct
colors; to color the uncolored in-edges $\\{v_{j}v_{i}:v_{j}\in
U^{-}(v_{i})\\}$, we use colors in $C_{2}$ that are different than having been
used in the out-edges of $v_{i}$ (refer $X(v_{i})$ in the following
paragraph); and we do the coloring in this order.

For the induction step, suppose we process vertex $v_{i}$, where
$i\in\\{1,2,\ldots,n-1\\}$. Suppose
$N_{\vec{G}}^{+}(v_{i})\cap\\{v_{1},\ldots,v_{i-1}\\}=\\{x_{1},\ldots,x_{\ell}\\}=X(v_{i})$.
Since $\Delta^{+}(\vec{G})\leq d$, $|X(v_{i})|\leq d$. Suppose
$N_{\vec{G}}^{-}(v_{i})\cap\\{v_{1},\ldots,v_{i-1}\\}=\\{y_{1},\ldots,y_{r}\\}=Y(v_{i})$,
then by Lemma 2.1, $|Y(v_{i})|\leq d$. For the induction hypothesis, we
suppose each edge $y_{j}v_{i}$, where $y_{j}\in Y(v_{i})$, is colored with a
color in $C_{1}$; and each edge $v_{i}x_{j}$, where $x_{j}\in X(v_{i})$, is
colored with a color in $C_{2}$. When we process vertex $v_{i}$, i.e., color
the remaining uncolored edges incident with $v_{i}$, we do this in two steps.

In the first step, we use the colors in $C_{1}$ to color the uncolored out-
edges $\\{v_{i}v_{j}:v_{j}\in U^{+}(v_{i})\\}$, such that all the out-edges of
$v_{i}$ have distinct colors. We show that we can do this for any edge
$v_{i}v_{j}$ with $v_{j}\in U^{+}(v_{i})$. For vertex $v_{j}$, since $v_{j}\in
U^{+}(v_{i})$, $v_{j}$ has not been processed yet. Therefore, by Lemma 2.1 and
the induction hypothesis, the in-edges of $v_{j}$ that have been colored with
colors in $C_{1}$ is at most $d-1$ (note that in the counting we removed the
in-edge $v_{i}v_{j}$ of $v_{j}$). For vertex $v_{i}$, when we begin to process
vertex $v_{i}$, by the induction hypothesis, the edges incident with $v_{i}$
and colored with colors in $C_{1}$ are $y_{j}v_{i}$, where $y_{j}\in
Y(v_{i})$. During processing vertex $v_{i}$, for the edge $v_{i}v_{j}$ with
$v_{j}\in U^{+}(v_{i})$, at most $|U^{+}(v_{i})|-1$ edges incident with
$v_{i}$ are colored with colors in $C_{1}$. Note that,

$|C_{1}|-|Y(v_{i})|-(|U^{+}(v_{i})|-1)-(d-1)\geq(3d-1)-d-(d-1)-(d-1)\geq 1.$

This proves that there is a color left for $v_{i}v_{j}$ with $v_{j}\in
U^{+}(v_{i})$.

In the second step, we color the uncolored in-edges
$R(v_{i})=\\{v_{j}v_{i}:v_{j}\in U^{-}(v_{i})\\}$ of $v_{i}$. Note that
$v_{i}$ has at most $|X(v_{i})|\leq d$ processed out-neighbors before $v_{i}$
is processed. Observe that for any edge $x_{i}x_{\ell^{\prime}}$ with
$x_{\ell^{\prime}}\in X(v_{i})$, $x_{i}x_{\ell^{\prime}}$ is an in-edge of
$x_{\ell^{\prime}}$. By the induction hypothesis, $x_{i}x_{\ell^{\prime}}$ is
colored when $x_{\ell^{\prime}}$ is processed, and is colored with a color in
$C_{2}$. After removing the colors that used by edges $x_{i}x_{\ell^{\prime}}$
with $x_{\ell^{\prime}}\in X(v_{i})$, there are at least
$|C_{2}|-|X(v_{i})|\geq 4d+2k-2-d=3d+2k-2$ colors left in $C_{2}$ that can be
used to color edges in $R(v_{i})$.

We partition these left colors in $C_{2}$ arbitrarily into sets $A(v_{i})$ and
$B(v_{i})$ so that $|A(v_{i})|=d+k$ and $|B(v_{i})|\geq 2d+k-2$. For each
$v_{j}\in U^{-}(v_{i})$ is an unprocessed in-neighbor of $v_{i}$. We say that
a color $c$ is $forbidden$ at $v_{j}$ if there is out-edge of $v_{j}$ is
colored with $c$, and we call the colors in $A(v_{i})$ that are not forbidden
at $v_{j}$ $available$ at $v_{j}$. Note that at most $d-1$ out-edges of
$v_{j}$ are colored (removing the out-edge $v_{j}v_{i}$ of $v_{j}$ in the
counting). This implies that at least $k+1$ colors in $A(v_{i})$ are available
at $v_{j}$.

Let $R^{*}(v_{i})$ be the maximal subset of $R(v_{i})$ that can be colored
with colors in $A(v_{i})$ in a way such that:

  * (a)

each in-edge $v_{j}v_{i}\in R^{*}(v_{i})$ of $v_{i}$ is colored with a color
available at $v_{j}$;

  * (b)

the number of edges in $R^{*}(v_{i})$ colored with any color is congruent to
$1~{}({\rm mod}\ k)$.

Let $\bar{R}(v_{i})=R(v_{i})\setminus R^{*}(v_{i})$ be the set of the
remaining edges in $R(v_{i})$. We claim that $|\bar{R}(v_{i})|<|A(v_{i})|$.

Assume otherwise that $|\bar{R}(v_{i})|\geq|A(v_{i})|$, and suppose
$A(v_{i})=\\{a_{i}:1\leq i\leq d+k\\}$,
$\bar{R}(v_{i})=\\{e_{j}=w_{j}v_{i}:w_{j}\in U^{+}(v_{i}),~{}1\leq j\leq
t,\mbox{ and }t\geq d+k\\}$. We define an auxiliary bipartite graph $T$ with
vertices bipartition $A(v_{i})$ and $\bar{R}(v_{i})$, edges
$E(T)=\\{a_{i}e_{j}:\mbox{where $e_{j}=w_{j}v_{i}$, $a_{i}$ is available at
}w_{j}\\}$.

Since, for each $w_{j}\in U^{+}(v_{i})$, there are at least $k+1$ colors in
$A(v_{i})$ available at $w_{j}$, we have $d_{T}(e_{j})\geq k+1$ for every
$e_{j}\in\bar{R}(v_{i})$. Therefore,

$\sum_{a_{i}\in
A(v_{i})}d_{T}(a_{i})=|E(T)|=\sum_{e_{j}\in\bar{R}(v_{i})}d_{T}(e_{j})\geq(k+1)t\geq(k+1)(d+k).$

Since $|A(v_{i})|=d+k$, we concluded that there exists a color $a_{i}$ in
$A(v_{i})$, $d_{T}(a_{i})\geq k+1$, which means that color $a_{i}$ is
available on at least $k+1$ edges in $\bar{R}(v_{i})$.

If some edge in $R^{*}(v_{i})$ is already colored with $a_{i}$, then we color
$k$ edges in $\bar{R}(v_{i})$ with color $a_{i}$. If no edge in $R^{*}(v_{i})$
is colored with $a_{i}$, then we color $k+1$ edges in $\bar{R}(v_{i})$ with
color $a_{i}$. Both of these cases contradict with the maximality of
$R^{*}(v_{i})$. This proves that $|\bar{R}(v_{i})|<|A(v_{i})|=d+k$.

Finally we show that we can color all the edges in $\bar{R}(v_{i})$ with
distinct colors in $B(v_{i})$. For this, it suffices to note that, for each
$w_{j}v_{i}\in\bar{R}(v_{i})$, there are at most $d-1+|\bar{R}(v_{i})|-1\leq
2d+k-3<|B(v_{i})|$ colors of $B(v_{i})$ that are either forbidden at $w_{j}$,
or were used on previous edges of $\bar{R}(v_{i})$.

We prove our main result by using Lemma 1.6, Theorem 1.7, Lemma 2.1, and Lemma
2.2. The proof is similar to Theorem 5 in [1], the differences are
applications of Theorem 1.7 and Lemma 2.1 here, and Lemma 2.2 is stronger than
the corresponding one in [1].

Theorem 1.3. For every graph $G$ we have $\chi^{\prime}_{k}(G)\leq 177k-93$.

Proof. Let $H$ be a maximal subgraph of $G$ such that $d_{H}(v)\equiv
1~{}({\rm mod}\ k)$ for every $v\in V(H),$ and let $G^{\prime}=G\backslash
E(H)$. Then $V(G)\setminus V(H)$ is independent. Since otherwise, there exists
an edge $e$ with both ends in $V(G)\setminus V(H)$; then $H^{\prime}=H+e$
would be a graph for which $d_{H^{\prime}}(v)\equiv 1~{}({\rm mod}\ k)$; but
this contradicts the maximality of $H$.

Similarly, by the maximality of $H$, $G^{\prime}[V(H)]$ has no nonempty
$k$-divisible subgraph. By Lemma 1.6, for every nonempty $J\subseteq
G^{\prime}[V(H)]$, we have $e(J)<2(12k-6)v(J)$. Thus,

${\rm mad}(G^{\prime}[V(H)])=\max_{J\subseteq
G^{\prime}[V(H)]}\frac{2e(J)}{v(J)}<\max_{J\subseteq
G^{\prime}[V(H)]}\frac{2(24k-12)v(J)}{v(J)}=2(24k-12).$

By Theorem 1.7, $G^{\prime}[V(H)]$ has an orientation
$\overrightarrow{G^{\prime}[V(H)]}$ such that
$\Delta^{+}(\overrightarrow{G^{\prime}[V(H)]})\leq 24k-12$.

For every vertex $u\in V(H)$, by the maximality of $H$, $u$ has at most $k-1$
neighbors in $V(G)\setminus V(H)$. For every edge $e=uv$ in $G^{\prime}$ with
$u\in V(H)$ and $v\in V(G)\setminus V(H)$, we orient $e$ from $u$ to $v$.

Thus there exists an orientation $\vec{G^{\prime}}$ of $G^{\prime}$, such that
$\Delta^{+}(\vec{G^{\prime}})\leq 25k-13$. By Lemma 2.2, there exists a
$\chi^{\prime}_{k}$-coloring of $G^{\prime}$ using at most $177k-94$ colors.
Then color all $E(H)$ with a new color, this proves the theorem.

Remark. In the above proof of Theorem 1.3, for all the edges $e=uv$ in
$G^{\prime}$ with $u\in V(H)$ and $v\in V(G)\setminus V(H)$, we can orient $e$
from $v$ to $u$. Then define a linear ordering $\sigma^{\prime}$ beginning
with vertices in $V(G)\setminus V(H)$, and concatenating a linear ordering of
$V(H)$ that has been proved existing by Lemma 2.1. By using this
$\sigma^{\prime}$ and some adjusted arguments of Lemma 2.2, we may have a
slightly better upper bound of $\chi^{\prime}_{k}(G)$. But to do this, we
shall have to tweak all the proofs of Lemma 2.2. As the authors in [1] have
mentioned, we think we would be far from the truth still (refer Conjecture
1.2), we skip this small improvement here for the readability of this paper.

## References

  * [1] F. Botler, L. Colucci, and Y. Kohayakawa, _The mod $k$ chromatic index of graphs is $O(k)$_, J. Graph Theory. 2023; 102: 197-200.
  * [2] S. L. Hakimi, _On the degree of the vertices of a directed graph_ , J. Franklin Inst. 279 (1965), 290-308. MR31 #4736.
  * [3] W. Mader, _Existenz n-fach zusammenhängender Teilgraphen in Graphen genügend grosser Kantendichte_ , Abh. Math. Semin. Univ. Hambg. 37 (1972), 86-97.
  * [4] L. Pyber, _Covering the edges of a graph by …_ , Sets, Graphs and Numbers (Budapest, 1991), vol. 60, Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam, 1992, pp. 583-610.
  * [5] A. D. Scott, _On graph decompositions modulo $k$_, Discrete Math. 175 (1997), no. 1-3, 289-291.
  * [6] C. Thomassen, _Graph factors modulo $k$_, J. Combin. Theory Ser. B. 106 (2014), 174-177.
  * [7] D. Yang, _Generalization of transitive fraternal augmentations for directed graphs and its applications_ , Discrete Math. 309 (2009) no. 13, 4614-4623.


# A Survey of Passive Sensing in the Workplace

Subigya Nepal1  Dartmouth College
Hanover, NH, USA
<EMAIL_ADDRESS>Gonzalo J. Martinez1  University of Notre Dame
Notre Dame, IN, USA
<EMAIL_ADDRESS>Shayan Mirjafari  Dartmouth College
Hanover, NH, USA
<EMAIL_ADDRESS>Koustuv Saha  Georgia Institute of Technology
Atlanta, GA, USA
<EMAIL_ADDRESS>Vedant Das Swain  Georgia Institute of Technology
Atlanta, GA, USA
<EMAIL_ADDRESS>Xuhai Xu  University of Washington
Seattle, WA, USA
<EMAIL_ADDRESS>Pino G. Audia  Dartmouth College
Hanover, NH, USA
<EMAIL_ADDRESS>Munmun De Choudhury  Georgia Institute of
Technology
Atlanta, GA, USA
<EMAIL_ADDRESS>Anind K. Dey  University of Washington
Seattle, WA, USA
<EMAIL_ADDRESS>Aaron Striegel  University of Notre Dame
Notre Dame, IN, USA
<EMAIL_ADDRESS>Andrew T. Campbell  Dartmouth College
Hanover, NH, USA
<EMAIL_ADDRESS>
###### Abstract

As new technology inches into every aspect of our lives, there is no place
more likely to dramatically change in the future than the workplace. New
passive sensing technology is emerging capable of assessing human behavior
with the goal of promoting better cognitive and physical capabilities at work.
In this article, we survey recent research on the use of passive sensing in
the workplace to assess wellbeing and productivity of the workforce. We also
discuss open problems and future directions related to passive sensing in the
future workplace.

**footnotetext: These authors contributed equally to this work.

##  I Introduction

With the growing adoption of technologies in the workplace, Future of Work
research is investigating how to intelligently shape the workspace to expand
on both on- and off-balance-sheet talent. Ubiquitous sensing with the use of
passive sensors is one of the technologies on the rise for the Future of Work.
Passive sensors refer to various methods to collect data from individuals in
situ without requiring their direct interaction with any artifact or person
[1]. These technologies have the potential to advance the understanding and
support of workers passively, pervasively and unobtrusively. Sensing
technologies offer the advantage of passively collected context based in-situ
data, which can also be used to make a variety of inferences about workers’
behaviors.

Sensing technologies offer the ability to draw empirical contextualized
insights from naturalistic situations. A number of studies find associations
between sensing and behavioral markers of mental wellbeing [2], personality
[3] and productivity [4]. Within the context of Future of Work, sensing
technologies are particularly valuable to assess workers’ experiences inside
and outside the workplace over time and at scale. For instance, the evaluation
of workers’ performance and satisfaction levels currently depends on
subjective reports from workers themselves, colleagues and/or supervisors.
However, passive and quantitative information from sensing technologies can
enable us to build systems to understand personal factors that contribute to
the worker’s wellbeing and performance objectively. This might help to
introduce new hands-off assessment methods for the Future of Work.

We present a survey of existing research on the use of passive sensing
technologies in the workplace to assess and promote wellbeing and productivity
of the workforce. In this work, we consider “workplace” as the setting or
place of employment where individuals perform tasks for their employer without
regards to whether the space is private or shared. The results point out open
problems and possible future directions in the field and suggest challenges to
furthering the Future of Work using passive sensing technologies.

Figure 1: Wearables and smartphones are two common passive sensing
technologies, offering physiological data such as heart rate, skin
temperature, physical activity and sleep and contextual data such as location,
proximity and phone usage.

##  II SURVEY METHODOLOGY

To narrow our search and to accommodate relevant work, we searched for key
terms such as, “passive sensing & future of work”, “wearables & worker
productivity” and “wearables & employee health”. We obtained over 25,000
articles on database corpora like Google Scholar. To further condense the
number of articles, we mostly consider papers published since 2015. However,
we also include seminal earlier work. We mostly selected papers whose venue
had a Google h5 index of 20 or higher, with a few exceptions.

Because there is an abundance of papers that use different ubiquitous
technologies to study behavior of different sets of people, we had further
restrictions in place. First, the studies had to consider real workforce while
they are performing their everyday job. This means, for instance, that
articles where the participants were students and they simulated being in an
office environment did not satisfy our selection criteria. Second, the sensing
technology used had to be truly passive in the sense that they should be
unobtrusive, portable and should make people feel comfortable to go about
their regular daily work routine without the hassle to carry it around or
interact with it. The technology should blend with the workers’ routine,
rather than the workers changing their everyday routine to satisfy the study
requirements. As such, studies that used large and clunky head mounts (e.g.,
EEG headsets), gaze trackers, cameras, chest straps, laptops and other
cumbersome scientific devices were excluded. We focus on devices that we
expect to have a higher acceptance in the office environment. This allows us
to paint a real picture on the current state of the art, the progress made and
the challenges faced, which we hope will be useful to promote and further
situate research in the field.

##  III PASSIVE SENSING

As ubiquitous devices become more embedded in our lives, they offer an
unprecedented ability to passively capture our daily behavior at a high
resolution via multiple sensors. Researchers have been leveraging passive
sensing techniques to model users’ behavior and their environment from various
perspectives. Before the emergence of smartphones and wearable devices,
sensors built in cell phones (such as GPS and Bluetooth) were employed to
passively sense users’ context. RealityMining [5] uses location signals and
Bluetooth log data to recognize social patterns in daily user activity,
identify socially significant locations and model organizational rhythms.
CenceMe [6] uses a wider range of sensors – including embedded cameras,
microphones, accelerometer, GPS, temperature, light, humidity, magnetometer
and button clicks – to detect users’ activities and habits and injected these
into popular social networking applications to promote social interaction. In
addition to capturing personal behavior, passive sensing devices at a large
scale enabled people-centric urban sensing. MetroSense [7] combines mobile
sensors (carried by people) and static sensors (installed among a city) to
establish a large-scale passive sensing network that could support a wide
variety of applications such as enterprises and hospitals. Recently, the
advancement of mobile and wearable technology has further enhanced the
capability of ubiquitous devices through longitudinal passive sensing. For
example, StudentLife [2] uses smartphones to track college students’ daily
behaviors for more than two months to assess their mental health, academic
performance and behavioral trends.

Passive sensing is increasingly involved in various aspects of our daily
lives. Within the workplace, it has been used to monitor physiological factors
of workers, promote work safety, enhance efficiency among other things [8].
Recently, the Tesserae [9] project involved over 700 information workers to
investigate how such technologies can be leveraged to measure workplace
performance such as organizational citizenship behavior, as well as
psychological traits and physical characteristics. In the rest of the article,
we focus on applications of passive sensing technologies in the workplace to
assess wellbeing and productivity of the workforce as the transition to the
Future of Work takes place.

TABLE I: Studied behavior and sensors used. The following table lists all the
behaviors studied in the papers we surveyed and the corresponding sensor data
used.

Studied behavior | Sensor/data
---|---
Stress/anxiety |  | HR, step count [10]; HR, electrodermal activity, skin temperature [11];
---
Skin conductance, skin temperature, acceleration [12];
HR, stress, physical activity, context, user state, phone usage, Bluetooth
[13];
HR, HRV, step count, sleep, acceleration, fat burn, cadence, breathing depth,
sitting time, Bluetooth [14]
Sleep |  | HR, sleep duration, sleep efficiency [15]; phone usage, movement, HR [16];
---
HR, stress, physical activity, context, user state, phone usage, Bluetooth
[13]
Affect/mood |  | Skin conductance, skin temperature, acceleration [12];
---
Sleep, activity duration [17]; speech activity, Bluetooth [18];
HR, stress, physical activity, context, user state, phone usage, Bluetooth
[13];
physical activity, HR, skin response, skin temperature, respiration [19]
Focus/awakeness | Physical activity, HR, skin response, skin temperature, respiration [19]
Task performance |  | HR, step count, Bluetooth [20]; Bluetooth [21];
---
Physical activity, speech activity, face to face interaction, proximity [22];
Distance, location, still duration, sleep duration, sleep dept, phone usage,
desk sessions [23];
HR, stress, physical activity, sleep, phone usage, Bluetooth [4];
Commute based physiology, weather, commute duration, commute variability [24]
Citizenship behavior |  | HR, stress, physical activity, sleep, phone usage, Bluetooth [4]; HR, step count [10];
---
Distance, location, still duration, sleep duration, sleep dept, phone usage,
desk sessions [23]
Commute based physiology, weather, commute duration, commute variability [24]
Deviance |  | HR, step count, Bluetooth [20]; HR, step count [10];
---
HR, stress, physical activity, sleep, phone usage, Bluetooth [4];
Distance, location, still duration, sleep duration, sleep dept, phone usage,
desk sessions [23]
Promotion | Physical activity, HR, stress, phone usage, Bluetooth, sleep, distance [25]
Cognitive load/interruptibility |  | HR, HRV, EDA, skin temperature, wrist movements [26];
---
HR, HRV, step count, sleep, circadian rhythm [27]
Workplace activity |  | EDA, blood volume pulse (BVP), acceleration, phone usage [28];
---
User activity, application usage, location, ringer mode [29]

##  IV FACTORS IN FUTURE OF WORK

###  IV-A WELLBEING

Over the years, there has been an increase in studies that explore the sensing
capabilities of smartphones and wearables to assess as well as improve
worker’s health and wellbeing. Among the papers selected, mental health issues
such as stress, anxiety, affects are important dimensions studied. Most of
these studies ask participants to self-report their health and wellbeing using
validated instruments. While some studies use one-off survey scores, others
employ experience sampling to study worker’s wellbeing in a longitudinal
fashion. We list the key studies in Table I.

Assessing worker’s stress and/or anxiety is perhaps the most studied topic
pertaining to worker’s wellbeing. As stress and anxiety are known to have an
impact on heart rate (HR), most of the studies rely on HR based signals
obtained from wrist worn wearables that employ photoplethysmography (PPG) [30]
sensors. For instance, Feng et al. [10] show that PPG based HR and step counts
from a Fitbit can be used to classify the anxiety level of nursing
professionals. The authors introduce a pipeline for discovering behavioral
consistency features, the inclusion of which leads to an overall improvement
in the predictive model, with an accuracy of about 58%. For this work, the
authors use the median-splitted score of self-reported State-Trait Anxiety
Inventory (STAI) [31] as the ground truth. Similarly, in a study of 10
construction workers, Jebelli et al. [11] use a wrist wearable capable of not
only capturing the participant’s HR through PPG signals, but also their
electrodermal activity (EDA) and peripheral skin temperature (ST). The study
uses Salivary cortisol as the ground truth to assess baseline stress levels of
the workers. Authors obtain an accuracy of about 73% when distinguishing
between low, medium and high stress level of the workers.

Another health related issue that is of frequent interest is sleep. In one
study of 138 individuals working as full-time nursing professionals, Feng et
al. [15] use PPG based HR from Fitbit, sleep duration, sleep efficiency, REM
Sleep duration features to gauge sleep quality of the participants. The study
uses the Pittsburgh Sleep Quality Index (PSQI) [32] survey responses from
participants as the ground truth. A score above 7 is taken as one class
(potential sleep disorders) and a lower score is associated with the remaining
class. The authors propose an optimization problem to identify the key
meaningful motifs. Motif features obtained using the proposed pipeline results
in higher performance while predicting sleep quality – while Fitbit summary
results in 61.90% F1 score, the motif extracted from the proposed pipeline
results in 77.47% F1 score. As wearable-based sleep detection is increasingly
being used and relied on, one research work investigates its accuracy while
tracking 700 information workers throughout a year using wearables and phones
[16]. Martinez et al. report that the wearables can overestimate sleep and
they propose fusing phone usage activity (i.e., whether the phone is being
used) with wearable data to mitigate this. Using the proposed approach,
authors generate better models of self-reported sleep for the information
workers than using either stream alone.

Worker’s affect, emotions and mood have also been studied in this domain. For
example, Bin Morshed et al. [17] explore mood instability of 603 information
workers and find that the sleep and activity duration obtained from a Garmin
wearable are negatively correlated with mood instability scores. The authors
compute mood instability in the study using self-reported positive and
negative affect schedule (PANAS) [33]. Umematsu et al. [12] use wrist
wearables that capture skin conductance, skin temperature and acceleration
from 39 workers in a Japanese company over a 30-day period. Stress, mood and
health scores are collected every morning, using self-reported scores from 0
to 100. The authors then try forecasting the scores for the next day using the
previous day’s physiological data. They obtain a mean absolute error of 13.47,
14.09 and 18.51 while predicting stress, mood and health score respectively.

In another study of 50 hospital workers which also uses PANAS, Nadarajan et
al. [18] find that speech activity can explain some variance in predicting
positive affect measure. Employees wear a specifically designed audio badge
during their work-shift hours. The authors extract several features from the
audio to identify foreground speech. They then use a linear mixed effects
model to estimate positive and negative affect from foreground activation
(i.e., the percentage of recording time that foreground speech is present).
Similarly, in another study, Robles-Granda et al. [13] utilize multiple
sensing modalities to assess anxiety, sleep and affect of 757 information
workers. Sensing modalities include wearable, phone application, Bluetooth
beacons and social media. Models trained on the fusion of all the features
from different sensing modalities leads to up to 13.9% improvement in the
symmetric mean absolute percentage error (SMAPE) when predicting affect,
anxiety and sleep quality scores.

In yet another study, Booth et al. employ multiple sensing modalities to study
stress, anxiety and affect of hospital workers [14]. The ground truth for
stress and anxiety are obtained as self-reports on a 5-point Likert scale,
whereas PANAS is used for assessing affect. The work investigates how patterns
in movement data in the workplace can relate with mental wellness. The
hospital staffs carry a Bluetooth tag with them which communicates with
several Bluetooth hubs placed in different rooms of the hospital. The nearest
hub which has the highest RSSI is taken as the location estimate. For each
consecutive data in which the participant is in the same location, authors
record the entry time, the location and the duration spent there. The authors
extract motifs from each time series of locations where participants linger
and cluster these motifs to capture temporal relationships between rooms.
Authors obtain 58% accuracy while classifying the mental wellness of the
participants (between a binary label with one value representing high stress,
negative affect, and anxiety, and the other value representing low stress and
positive affect) using the combination of extracted motif features with Fitbit
data (step count, heart rate and sleep) and smart shirt data (heart rate,
breathing, acceleration etc.).

Other health and wellbeing related topics that have been studied using passive
sensing among workers include focus and awakeness. Soto et al. utilize
biometric data from an arm-wear (viz., physical activity, HR, skin response,
skin temperature and respiration) to estimate worker’s stress, focus and
awakeness [19]. The study ran for 8 weeks with 14 knowledge workers working at
a large power and automation company. The ground truth for awakeness and focus
along with stress were obtained as self reports from the participants. The
authors report that personalized models outperformed generic model, predicting
baselines ranging from 3% improvement to up to 52% improvement in the
precision, recall and accuracy score. There are also several other works that
fuse passive sensing data with some other modality to assess wellbeing. For
instance, Saha and Grover et al. [34] propose contextualizing on offline
behaviors as obtained from passive sensors to make models better adapted to
the social media signals. Authors first cluster participants based on the
passive sensing data obtained from their wearable, Bluetooth beacons and
smartphone application. Thereafter, they use social media derived features to
predict different constructs within each cluster, obtaining an improvement on
the baseline generic model by up to 5.43% in predicting self-reported anxiety,
affect and sleep quality of information workers.

Passive sensing technologies are also being embraced by organizations to
promote employees’ wellbeing. Organizations make use of gamification,
personalized recommendations or even offer incentive programs to encourage
employees to be more active in their day-to-day life [8]. Researchers studying
the use of wearable technologies in corporate wellness programs report that
their usage has a positive impact on employee wellbeing and health [35].
Passive sensing technologies, thus, can not only be used to monitor and assess
health and wellbeing of workers, but they may also be utilized for health
interventions and promoting a healthy lifestyle for the workers.

###  IV-B PRODUCTIVITY

Driven by the early success in the assessment of wellbeing in organizations,
researchers have turned their focus to the use of passive sensing in the
assessment of workplace performance. By tracking behaviors, recent work has
attempted to characterize workplace performance with the long-term goals of
identifying the behaviors that support productivity as well as generating
interventions that might improve it. Nevertheless, quantifying workplace
performance in an objective yet generalizable manner remains a challenge in
the area. For instance, Amazon uses passive sensing on smartphones to measure
the performance of their drivers and make sure that they avoid potentially
dangerous behaviors like checking the phone while driving. However, phone
usage could be the desired behavior in a different job, e.g., consulting.

Therefore, researchers in the pervasive computing community have often turned
to job performance inventories developed by psychologists as ground truth to
measure perceived workplace performance across organizations and industries in
a generalizable manner. They have used passive sensing to predict participant
scores for these inventories, or to classify individuals into higher and lower
performers within the workplace. The inventories usually measure either task
performance such as Individual Task Proficiency (ITP), In-Role Behavior (IRB);
or assess behaviors that promote the effectiveness of organizations and its
members such as Interpersonal and Organizational Deviance (IOD-ID, IOD-OD),
Organizational Citizenship Behavior (OCB).

Task performance refers to duties or actions that are formally recognized and
rewarded by management [36]. Several studies attempt to predict ITP and IRB as
measures of task performance. In a study of 84 full-time nursing professionals
over a ten-week period, Feng et al. [20] use Fitbit and Bluetooth proximity
data to study how an individual’s physiological responses vary according to
the work environment (room and function of the room). The authors compute a
measure of mutual dependency of time in each room in the hospital and
physiological responses and find this measure along with baseline aggregated
features (e.g., min, mean, max, SD, etc.) leads to predictions of ITP with an
adjusted R2 score of .11 and .12 for IRB. Another study [22] recruits 67
nurses and instruments them with a sociometric badge capable of measuring
physical activity, speech activity, face to face interaction and social
network inferred through the sociometric badge proximity sensor. Multiple
linear regression models with sensed features as predictors are able to
predict 49% of the variance in groups’ perception of workload as measured in
an ad-hoc survey and 63% of the variance in groups’ perception of
productivity.

In studies of information workers, several works stand out. The study by Das
Swain et al. [21] recruits 249 individuals over 62 days and used Bluetooth
beacons to analyze desk and away from desk sessions per hour, time at work and
time at home. The authors produce a measure of organization fit by analyzing
the convergence of individual routines in the aforementioned behaviors and
their organization’s normative behavior. They find that a linear model with
Big Five Personality [37] measures along with routine fit could predict up to
28% of the variance in IRB, with routine fit having a significant (p $<$ 0.05)
positive association with IRB. In another study of 603 information workers
over 110 days, Das Swain et al. [23] propose the concept of organizational
personas, a clustering-based approach in which personality facets are
constructed over Big Five Personality data and activity facets are constructed
from multi-modal data from a fitness tracker, a phone agent (location, phone
usage) and Bluetooth beacons (desk sessions). The authors report that the
activity facets account for a small but statistically significant part of the
variance in ITP. In a similar study of 554 information workers using a phone
agent, beacons and a Garmin fitness tracker, Mirjafari et al. [4] use K-Means
clustering on the participant averages of ITP, IRB, OCB, IOD-ID and IOD-OD
scores to classify them into higher and lower performers in their
organizations. Results indicate that features from the fitness tracker alone
can discriminate performers with an AUROC of .72; features from the phone
agent can discriminate performers with an AUROC of .65 and features from both
can do so with an AUROC of .83; showing how multi-modality can better explain
these complex behaviors.

In a follow-up study of 298 information workers, Mirjafari et al. [38] used
auto-encoder generated features based on passive sensing data from mobile
phones and a Garmin fitness tracker to predict day-to-day job performance
dynamics, i.e., detecting whether there was an improvement, decline, or no
change in the job performance for individuals in the study as assessed by ITP,
IRB, OCB, IOD-ID and IOD-OD. The final model used XGBoost to achieve an F1
score of 75%, surpassing the authors’ baseline of 33%. The authors also show
the use of Gradient Analysis, a technique popular in the computer vision
domain [39, 40], to provide user-actionable interpretation of the model,
demonstrating the potential to inform individuals of what behaviors should be
modified in order to improve their performance.

In the behavioral side of workplace performance inventories, IOD-ID and IOD-OD
are measures of “bad” conduct in the workplace. Behaviors that indicate IOD-ID
can involve cursing a co-worker, playing pranks, or making fun of someone.
Behaviors that indicate IOD-OD can be tardiness or absenteeism, leaving work
early without permission, putting little effort into work, among others [41].
On the other hand, OCB is a measure of “good” behavior. OCB is composed of
behaviors that reflect altruism, conscientiousness, sportsmanship, courtesy
and civic virtue [42]. Several studies have focused on these measures.

Feng and Narayanan [10] propose a method for capturing behavioral consistency
in wearable data using the activity curve model. They find that consistency
features improve accuracy by up to 6% when compared to using only summary
features from the Fitbit fitness tracker in a study of 97 hospital workers
throughout 10 weeks. Nepal et al. [24] use commute based passive sensing data
(viz. location-related context, physiology, variability) from phones and
wearables to predict the OCB and IOD score of workers. Authors collect passive
data of 275 workers throughout a year while the workers are commuting to and
from work. They train a stacked machine learning model on the commute data and
achieve MAEs of less than 10% of each performance ground truths, suggesting
that passive sensors may be capable of measuring an individual’s commuting
experience and its impact on their job performance. In a previously mentioned
study by Feng et al. [20], the mutual dependency between physiological
response and time spent in different environments at the workplace improves
the prediction of IOD-ID (adjusted R2 = 0.065) and IOD-OD (adjusted R2 =
0.092) as well. Mirjafari et al. [4] find in their classification of higher
and lower performers that passive sensing data is associated with IOD-ID, IOD-
OD and OCB. Das Swain et al. [23] also report that the activity facet created
from passively sensed data is associated with IOD-OD, IOD-ID and OCB. They
also find that a better fit between individual and organizational desk
routines measured through beacon proximity implied lower IOD [21]. However,
maximum accuracy was still low with 62.4% for IOD-ID, 60.6% for IOD-OD, 56.6%
for OCB, and 57.8% for STAI.

The work by Nepal et al. [25] follows a different line of research. Instead of
attempting to predict subjective performance measures from objective data, it
intends to find what objective data is related to individual success in
organizations, i.e., getting promoted. The authors note that if promotion can
be detected and it produces objective physiological signals, researchers could
learn from these signals and through the study of job promotion – which could
be inferred – what behaviors could be behind the high performance that granted
the promotion to measure them objectively, foster them and replicate them.

In the modern era, nurturing collaboration through open offices has become the
norm and workers face necessary interruptions throughout their workday to
efficiently tackle problems. However, there are studies demonstrating that the
nature and timing of interruptions negatively affect workplace productivity
[43, 44] and increase worker stress [45]. Therefore, evaluating
interruptibility has significant value to companies in terms of cost and time.
To continuously measure a person’s interruptibility, Zuger et. al [27] study
13 software engineers over two weeks using wearable sensors in addition to
keyboard and mouse interaction data. The ground truth in this study was
collected using a pop-up question during work which asked the user to rate
their interruptibility on a 1-7 Likert scale, and these ratings are further
grouped into three states (12—345—67) for classification. Ultimately, they
predicted interruptibility accurately (68.3%) by training a random forest on
several features extracted from sensor modalities like heart rate, activity,
and sleep.

Other works target efforts to support the design of future interventions in
the workplace. For example, Kadoya et al. [46] use a wearable to derive
control measures to understand the relation of emotional states and
productivity in the workplace. Such understanding would help in the future to
focus interventions that can improve the emotional state of workers. Mark et
al. [47] found that affect balance correlates with concentration and workplace
productivity, and sleep measured by a fitness tracker was related to affect
balance. In another study, Schaule et al. [26] designed a proof of concept
system that integrated a smartwatch to predict office workers’ cognitive load
and suggest when users were available to be interrupted. Similarly, Di Lascio
et al. [28] show that wearables can predict workplace activity passively,
which has the potential to be used as the foundation of systems to
automatically block distractions during focus hours and to suggest breaks.
Kucukozer-Cavdar et al. [29] also create a model for predicting office
workers’ availability and inclination to take breaks using mobile sensing
features relevant to workers’ location, activity, application usage, ringer
mode. In summary, along with efforts to understand more about workplace
performance and behaviors that support productivity, these works show that the
pervasive computing community is also working to design the interventions that
support performance.

##  V DISCUSSION

The different work discussed in this article follow a wide variety of
approaches to solve the problem of assessing wellbeing and productivity
through passive sensing in-situ. In what follows, we discuss some
considerations to take into account for Future of Work research.

###  V-A Considerations for Ground Truth

While adapting to the Future of Work, researchers need to make additional
considerations beyond simply selecting validated survey instruments. For
instance, surveys at the workplace could bias participants from being candid
because of employer surveillance. However, remote work might mitigate such
concerns. Alternatively, many wellbeing constructs can be measured through
physiology (e.g., cortisol), but when the workforce is distributed across
diverse work spaces, such measures could include unforeseen artifacts which a
consistent workplace would otherwise insulate. Similarly, for productivity
measures, while task performance is important, for certain job roles deviance
measures or citizenship behavior could provide a more appropriate view of
performance. Apart from careful selection of ground truth instruments,
researchers should collaborate with industrial and organizational
psychologists and personnel management teams to learn which forms of
estimations are most relevant to them — whether that be categorical
classifications or continuous evaluations.

###  V-B Considerations for Sensor Deployment

Ideally, the set of constructs that researchers want to investigate drives the
choice of sensors. However, especially while estimating new measures, it can
be challenging to ascertain a finite combination of sensors to deploy.
Moreover, deployments are expensive and challenging. In longitudinal in-situ
studies, even if issues like privacy concerns and maintenance of the sensing
infrastructure can be mitigated, lack of participant compliance can
deteriorate the quality of data [48]. Also, it may not be practical for all
researchers to install a large suite of sensors and triage which signals are
actually meaningful. In such cases, researchers can consider forms of low-
burden sensing such as logging social media behavior [49]. Therefore, we need
to select sensors that sustain minimum burden on the user and still provide
data consistently.

###  V-C Considerations for Machine Learning Models

The workplace is a dynamic context and therefore passive sensing studies tend
to be in-situ studies over large periods of time. This approach comes with the
caveat of missing data that can make it challenging to train models,
especially when considering sensors that require constant skin contact that
can be affected by movement or simply forgetfulness, as is the case with
wearable devices. In addition, missing data is often not missing completely at
random, meaning that simply ignoring missing data as is done in several
studies can lead to a biased sample [50]. One way researchers can mitigate
this for behavioral data is by exploiting complementary streams of data to
impute the missing data [51, 16]. On the other hand, a large set of time
varying features opens an array of potential learning approaches, e.g., moment
statistics, regularity, or auto–encoders. Researchers can empirically compare
approaches to find the best model, but also consider other aspects like
interpretability and transferability [52] to select learning approaches.

###  V-D Privacy and Ethical Considerations

While we primarily highlighted the opportunities and potentials of these
technologies and methodologies, these also come at cost. For example, despite
the best intents, when instrumented at workplaces, these approaches can bear
ethical and privacy concerns. There are lingering questions regarding the best
practices of these algorithmic inferences for real-world measures and high-
risk decisions. These approaches can be misused in ethically questionable
ways, including compromised privacy, defying expectations and damaging the
trust between individuals and technologies. Table I shows that several
approaches rely on sensing that occurs most often outside the workplace, e.g.,
sleep and physical activity, sensing for which there is no conceivable way to
acquire consent from all involved parties such as speech activity, or sensing
that could be carried out using the sensors in devices that could be provided
by employers or fall into Bring-Your-Own-Device (BYOD) policies such as mobile
phones.

In fact, employer surveillance and employee’s subjective expectation of
privacy share a competing relationship [53] and only a thin line of difference
exists in perceiving the same technology as for surveillance or for supportive
intervention [54]. There is a need to ensure that future of work applications
benefit individuals without sacrificing their privacy or forcing them to adopt
behaviors solely on the need for increasing productivity in the workplace
according to arbitrary metrics. This calls for the need to design guidelines
that lead to responsible applications that balance the trade- offs of the
risks and benefits associated with these data-driven human-centered
assessments. In addition, these guidelines should prevent the ingraining of
biases in automated decisions and performance-evaluation systems, a danger
made obvious by early attempts at developing AI tools for recruiting that
resulted in biased result [55]. Considering the lack of diversity in the tech
industry in the US [56], there is a considerable risk that the training of
models based on existing employee data without special consideration of
disproportionate representation could result in AI-based tools encoding biased
behaviors that are not truly supporting task performance improvements for the
companies or well-being benefits for the employees. These aspects should not
be taken lightly and researchers should also incorporate metrics of AI
fairness [57] when conducting future of work studies that predict performance
outcomes.

###  V-E Limitations and Future Work

The existing body of work highlights several challenges with data-driven
human-centered assessments to facilitate workplace outcomes. Many of these
challenges suggest interesting future directions. In terms of
operationalization, there is a lack of availability as well as a consensus
about the definition of “ground-truth”. While self-reported measures are
considered to be the gold-standard source of data, self-reported data bear
their own limitations and biases. Moreover, from a methodological point of
view, most of the approaches are not generalizable across problem scenarios or
datasets — the same approach may not work on different populations or a
different outcome measure. This necessitates accounting for domain information
and problem-centric adjustments for model tuning.

Again, this field is evolving and most of the extant research is formative in
revealing the feasibility and efficacy of these technologies. However, real-
world and prospective adaptations of these approaches may bring in new
challenges. For instance, the accuracy metrics tend to reflect the model
performance at average, however, false positive on single instances (a single
individual) may bear larger risks related to hiring or firing employees. This
motivates thinking about novel measurement approaches that weigh in the risks
and qualitative assessments beyond quantitative accuracy measures. These
models not only need to be rigorously tested, but also proper guidelines need
to be set about the role of algorithmic inference and human judgment. All in
all, this literature survey shows us the potentials of passive and ubiquitous
sensing methodologies within the context of Future of Work. However, there are
several issues such as scalability, generalizability, privacy and ethical
concerns which hinder the broad adoption of such technologies in spite of the
widespread enthusiasm. Therefore, much work is needed to make workplace
sensing ubiquitous.

##  VI CONCLUSION

We presented a survey of the most recent relevant studies demonstrating the
current research trends and future potential of passive sensing technology in
the workplace. Many of the papers discussed are still in an exploratory phase,
where data-driven research is advancing what sensor streams and models can
best capture dimensions of the workplace and workforce. We highlighted some
open issues and potential directions for the use of passive sensing in Future
of Work. While research is still in its infancy, recent results shows a
promising future ahead.

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