# The effect of co-location on human communication networks

Daniel Carmody<EMAIL_ADDRESS>,  Martina Mazzarello  ,  Paolo Santi  ,
Trevor Harris  ,  Sune Lehmann  ,  Timur Abbiasov  ,  Robin Dunbar  and  Carlo
Ratti  Senseable City Lab, Massachusetts Institute of Technology, Cambridge,
MA, USA  Department of Statistics, Texas A&M University, College Station, TX,
USA  Technical University of Denmark, Kgs Lyngby, Denmark  Department of
Experimental Psychology, Oxford University, Oxford OX2 6GG, UK  Instituto di
Informatica e Telematica del CNR, Pisa, Italy

###### Abstract.

The ability to rewire ties in communication networks is vital for large-scale
human cooperation and the spread of new ideas. We show that lack of researcher
co-location during the COVID-19 lockdown caused the loss of more than 4,800
weak ties – ties between distant parts of the social system that enable the
flow of novel information – over 18 months in the email network of a large
North American university. Furthermore, we find that the re-introduction of
partial co-location through a hybrid work mode led to a partial regeneration
of weak ties. We quantify the effect of co-location in forming ties through a
model based on physical proximity, which is able to reproduce all empirical
observations. Results indicate that employees who are not co-located are less
likely to form ties, weakening the spread of information in the workplace.
Such findings could contribute to a better understanding of the spatio-
temporal dynamics of human communication networks, and help organizations that
are moving towards the implementation of hybrid work policies evaluate the
minimum amount of in-person interaction necessary for a productive work
environment.

###### Key words and phrases:

Communication networks, email networks, weak ties, co-location, working
environments, COVID-19

##  1\. Introduction

The ability to establish and leverage communication networks to share
information and collaboratively accomplish sophisticated tasks is a
distinguishing feature of humans[1]. While the ability to form social ties
with others was originally developed when individuals were in close proximity,
technological improvements have allowed increasingly remote forms of
communication and collaboration, that now include tele- and video-
conferencing, email, and chats[2]. These changes reinvigorated longstanding
debates about the extent to which social relationships are predicated upon
physical proximity[3]. While earlier studies in sociology and organizational
science discuss the role of spatial propinquity in producing interpersonal
ties [4, 5], the causal mechanisms through which co-location affects social
networks remain understudied[6, 7, 8, 9].

Addressing this question is all the more relevant today. First, in planning
the transition towards the post-COVID-19-pandemic “new normal”, institutions
and policy-makers world-wide are wondering about the best way to reshape work
environments following COVID-19[10, 11, 12, 13]. Second, the massive shift to
remote work during the past two years has produced a trove of Big Data that
promises to elucidate what happens when we remove physical presence as a main
conduit of communication[14]. While recent work has started to highlight
changes in the communication networks of information workers due to mandatory
remote work, the data was collected across several campuses in the continental
US – making it difficult to link these network changes to a lack of physical
proximity [15]. Hence, the following question remains open: what is the effect
of co-location on human communication networks?

In this study, we explore the mechanism via which the complete removal and
subsequent partial re-introduction of physical co-location at a large North
American university – the MIT campus – affects the structure of its digital
communication network. We find that, despite the robustness of many network
measures to the shift to remote work, physical co-location plays a crucial
role in the formation of weak ties.

Since Mark Granovetter’s seminal work in 1973[16], weak ties have been
identified as fundamental microscopic structures that enable the spread of
ideas and opportunities in social networks. Our hypothesis is that weak ties
form due to chance encounters in and around the office, so that removing the
possibility for chance encounters should affect the formation of weak ties. By
hindering new weak tie formation, the removal of physical co-location leads to
increased redundancy in email networks – more information is spread between
fewer people. Said differently, physical proximity is vital for updating the
people one communicates with over time. This is in line with earlier work on
the notion of propinquity, used to denote the tendency to form connections
among proximate individuals[3]. Propinquity can be modeled with a modification
of the link-central preferential attachment model which includes a co-location
factor $\tau$, accurately reproducing the dynamics of formation and stability
of weak ties caused by long-term removal and partial re-introduction of
physical co-location on the MIT campus.

##  2\. Results

###  2.1. Forming the daily email network

We build and analyze a large email network of research workers at MIT in
Cambridge, Massachusetts. Despite the wide-scale adoption of synchronous
video-conferencing technology, email remains a universal mode of digital
communication used by researchers to exchange information and organize
meetings. In order to study changes in communication behavior due to fully
remote work, we study the email habits of 2,834 MIT faculty and postdocs over
18 months starting on December 26, 2019. Each researcher belongs to a
“research unit” which describes their campus affiliation (see Supplementary
Information for a complete list – the partition of the MIT research community
into research units is in general finer than the partition into departments).
During March 2020 MIT started implementing COVID-19 contingency plans, which
led to a progressive decrease in campus attendance and culminated on Monday,
March 23, 2020 with the halting of in-person research activities. For each
day, the number of emails sent between each pair of (anonymized) individuals
is determined exactly for $>66\%$ pairs of individuals from randomized,
aggregated data and used to form the edge weights of an undirected network.
For the remaining pairs we estimate the number of emails sent using non-
negative matrix factorization (see Methods).

###  2.2. Robustness of network metrics to the remote work transition

Our initial analysis did not show many changes in the network: directly
comparing connected components and the number of intra/inter-research unit
connections in the email network in February 2020 and February 2021 using a
paired test on the logarithms of these network metrics (see Methods)
highlights few significant differences (Fig 1 panel a). However, due to the
seasonal nature of academic work, paired testing comparing only February is
not sufficient to estimate the short- and cumulative long-term effects of
fully remote working on the communication network.

To provide a statistically robust estimation of long-term effects, we design a
methodology based on the Bayesian Structural Time Series (BSTS) approach. The
approach is based on the construction of a synthetic counterfactual time
series from a covariate unaffected by the treatment. A well-constructed
synthetic counterfactual can capture fluctuations in the time series of
interest due to seasonality and confounding factors other than the treatment.
If the predictive power of the covariate wanes over time (which can happen
because e.g. the model is only fit on data before the treatment is applied),
then the width of the posterior predictive interval around the synthetic
counterfactual will increase over time, representing our increased uncertainty
about the distant future. As the treatment in this case corresponds to the
termination of campus access for researchers, we use email data from weekends
when most researchers were not entering the office to construct our
counterfactual (see Methods). When studying these cumulative effects, there
are also no significant differences in the number of intra/inter-research unit
connections, number of connected components, and the size of the giant
component due to remote work (Figure 1 panel b). A detailed plot highlighting
the evolution of the uncertainty of the BSTS model over time can be found in
Supplementary Information. The fact that the sign (loss vs gain) of the change
in network metrics like connected components, inter-unit connections, and size
of giant component differs in panels a and b is indicative of the fact that
choosing a single month in 2020 and 2021 for comparison does not accurately
capture cumulative effects.

Figure 1. Robustness of global network topology in daily reciprocated email
networks. a, The change from the beginning of the Spring 2020 semester
(weekdays between February 5 and March 5, 2020) to the Spring 2021 semester
(weekdays between February 17 and March 19, 2021) in the number of users in
the largest component, the number of components, the number of connections
(edges) between users who are in the same research unit, the number of
connections (edges) between users in distinct research units. 95% confidence
intervals from two-sided $z$ test on the coefficients of a generalized least
squares model fit to the log difference in means ($n=16$ days for all
variables). Full results are in Supplementary Tables 1-4. b, Predicted
cumulative change in network metrics compared to a synthetic counterfactual
from March 23, 2020 to July 15, 2021. Posterior predictive intervals with 95%
coverage computed using Bayesian structural time series ($n_{\mathrm{pre}}=8$
weeks, $n_{\mathrm{post}}=72$ weeks). Fitted values/intervals use the mean as
the measure of central tendency.

###  2.3. Weak tie formation is impeded by remote work

To study the effect of remote work on the sorts of connections which might
arise from serendipitous encounters on campus, we investigate the structure of
weak ties in the email network. Because local bridges can be identified
knowing only the topology of a social network, Granovetter suggested using the
notion of a local bridge as an accessible proxy for the notion of a weak
tie[16] (see Supplementary Information for detailed definitions). To check
that our results are robust to alternative definitions of weak ties, we repeat
our analysis (obtaining very similar results) using low contact frequency ties
in Supplementary Information. Figure 7 panel a provides an illustration of the
way in which chance encounters led to the formation of new local bridges.

The removal of physical co-location (as a consequence of mandatory fully
remote work) caused an immediate and persistent drop in the number of weak
ties formed in the MIT email network. Figure 7 panel b shows the causal effect
of a lack of co-location estimated with a piecewise polynomial regression
discontinuity design on both the number of weak ties and new (not previously
seen) weak ties. There is a statistically significant 6.2% drop in the number
of weak ties and a 38.7% drop in the number of new weak ties coinciding with
the sudden absence of co-location. Because we study a fixed population of
users, as we see more ties the number of new weak ties will naturally decrease
– thus the downward trend of new weak ties is expected. However, the
significant jump discontinuity on March 23, 2020 indicates that the absence of
physical co-location is negatively associated with the ability to form new
weak ties. The decrease in the addition of new weak ties hints at a stagnation
effect – researchers are not updating their pool of weak ties as often as
would be expected.

Not only is the drop in weak ties sudden and statistically significant at the
onset of the transition to full remote working, but it is also cumulatively
significant over the course of more than one year. Using BSTS to estimate the
cumulative effect of a lack of co-location, we see in Figure 7 panel c a
significant predicted loss of more than 5,100 weak ties from March 23, 2020
until July 15, 2021 due to remote work – approximately 1.8 ties per person in
the 2,834 researchers we study. Thus we find that remote work leads to a long-
lasting, statistically significant drop in weak ties. Panel c also shows a
non-significant cumulative drop in the number of new weak ties formed in the
network – we explain this phenomenon in detail in the following section.

Finally, we identify a striking difference in the mechanism via which local
bridges disappear due to remote work: in the short-term through the end of the
Spring 2020 semester, local bridges become embedded in triangles, while in the
long-term they are dropped from the network (Figure 7 panel d). We also
confirm in the Supplementary Information that ego networks become more
stagnant is the absence of co-location – the social contacts of researchers
become more similar from week to week after remote work. Specifically, by
computing the intersection over union of the edges in the daily reciprocated
email networks on day $d$ and day $d+7$, we see a lasting, significant
increase in the stability of network edges from week to week after remote work
(Supplementary Figure 1 panel h).

Figure 2. Changes in weak ties in the MIT email network after the shift to
remote work. ***$:p<.001$, **$:.001\leq p<.01$, *$:.01\leq p<.05$. a, A
candidate mechanism for local bridge formation in a social network which
requires co-location. b, A drop of 55.70 (6.2 %) in the number of local
bridges (weak ties) after March 23, 2020 ($p=.002$, 95% CI: [-91.627,
-19.774]). There is a drop of -38.03 (38.7%) in the mean number of new (not
previously seen) weak ties appearing each weekday after March 23, 2020
($p<.001$, 95% CI: [-53.38, -22.68]). Statistics represent the results of a
two-sided $z$ test corresponding to a local polynomial regression
discontinuity design ($n_{\mathrm{pre}}=42$ days, $n_{\mathrm{post}}=188$
days). c, There is a cumulative loss of 5110 weak ties throughout an entire
year ($p<.001$, 95% PPI: [-4957,-5267]) and a non-significant loss of 2930 new
weak ties ($p=.241$, 95% PPI: [-11588, 5730]). Posterior predictive intervals
computed using Bayesian structural time series ($n_{\mathrm{pre}}=8$ weeks,
$n_{\mathrm{post}}=72$ weeks). d, The number of weak ties which become strong
(become embedded in triangles) or are churned (dropped from the network) in a
30 day rolling window. Fitted values/intervals use the mean as the measure of
central tendency.

###  2.4. Weak tie formation and physical proximity

As people are more likely to meet by chance on campus if their offices are
nearby [9], we expect to observe relatively more consistent changes in the
formation of new weak ties for co-located MIT personnel. For each week we
predict the mean value of the dependent variable (weak ties between
researchers in a fixed distance range) during business days. We use the mean
of weekend values as our covariate for the BSTS approach with treatment on
March 23, 2020 outlined previously. To have a consistent measure of distance
across all days in the data, we use distance between the campus offices of
researchers rather than the distance between their active work environments –
during the shift to remote work the distance between researchers’ campus
offices does not change. We use four distance thresholds: $0$m (researchers
working in the same lab), $0-150$m (close/nearby researchers in distinct
labs), $150-650$m (researchers at medium distance), $>650$m (far away
researchers). The distribution of distances between researcher offices can be
found in Supplementary Information.

Using BSTS with a synthetic counterfactual constructed from weekend email
network data, we find an immediate and lasting drop in the number of new weak
ties between researchers in distinct but nearby research labs (Figure 3 panel
b). This is in line with our expectation that propinquity contributes to weak
tie formation. Given this decrease, it may seem surprising that there is an
increase in the number of new weak ties between researchers in the same lab
(Figure 3 panel b). However, Yang et al.[15] discovered an increase in the use
of asynchronous communication (e.g. email) after the COVID-19 pandemic. A
plausible explanation for the increase in new weak ties between researchers in
the same lab is that after the shift to remote work, email was used to
schedule one-on-one meetings or ask small questions between same-lab
researchers formerly scheduled or asked in person. Figure 3 panels c and d
show a non-significant decrease in the number of new weak ties formed between
researchers in distinct labs at medium and far distances. This is also
compatible with our intuition, as we don’t expect researchers who work far
away from one another to have many chance encounters even when working in-
person.

Figure 3. Formation of new weak ties stratified by distance. a The estimated
effect between researchers in the same lab. b The estimated effect between
researchers in distinct labs within 150 meters. c The estimated effect between
researchers in distinct labs between 150 and 650 meters. d The estimated
effect between researchers in distinct labs further than 650 meters. Shaded
regions represent 95% posterior predictive intervals computed using Bayesian
structural time series with a synthetic counterfactual constructed from
weekend data ($n_{\mathrm{pre}}=8$ weeks, $n_{\mathrm{post}}=72$ weeks for all
panels). Fitted values/intervals use the mean as the measure of central
tendency.

###  2.5. The effect of hybrid work on the formation of weak ties

MIT re-opened its campus for the Fall 2021 semester starting on September 8,
2021. However, following MIT recommendation many research labs adopted a
hybrid mode of work with researchers only physically present for (at most) 3
out of 5 business days each week, implying that the chance of serendipitous
encounters was still lower than before the COVID-19 pandemic. Furthermore,
limitations on the number of people allowed to eat together at a time and
ongoing restrictions to international travel prevented departments from
hosting large-scale events where researchers might typically mix. To estimate
the causal effect of the end of remote work, we calculate a synthetic
counterfactual for 2021 weekday email data from 2020 weekday email data (see
Methods).

The _percentage_ of new weak ties at close distances is higher than expected
in Fall 2021 given the percentage at close distances in Fall 2020 (Figure 4
panel a), with no significant differences observed at other distance
thresholds. The number of weak ties rises more sharply than expected on
September 8, 2020 (Figure 4 panel b) given the rise at the beginning of the
Fall 2020 semester. Despite this, the total number of new weak ties is lower
or similar to the predicted value after hybrid work (panel b). Taken together,
the results of Figure 4 hint at the partial but incomplete success of the
hybrid work model at allowing researchers to once more form new weak ties with
other proximal researchers.

Figure 4. The effect of re-introducing co-location between researchers.
***$:p<.001$, **$:.001\leq p<.01$, *$:.01\leq p<.05$. a, The predicted
cumulative change due to hybrid work in the _percentage_ of new weak ties
between researchers in the same lab, researchers in distinct labs at distance
less than 150m, researchers in distinct labs at distance between 150m and
650m, and researchers in distinct labs at distance larger than 650m. 95%
posterior predictive intervals computed using Bayesian structural time series
($n_{\mathrm{pre}}=29$ days, $n_{\mathrm{post}}=35$ days). b, A RDD on the
_difference_ of the number of weak ties in September 2020 and September 2021
using a two-sided $z$ test $(\mathrm{effect}=67.5,p=0.022,95\%$ CI:
$[9.740,125.266],n_{\mathrm{pre}}=34$ days, $n_{\mathrm{post}}=26$ days). c,
The changes in the _total number_ of new ties between researchers at the
distance thresholds detailed above with 95% posterior predictive intervals
computed using Bayesian structural time series. Fitted values/intervals use
the mean as the measure of central tendency.

###  2.6. Modeling the effect of distance on tie formation

Our empirical results are consistent with the existence of some kind of
mechanism via which co-location promotes weak tie formation. Here we seek to
address the following question: given a collection of potential communicating
pairs of researchers, how can we choose which pairs of researchers communicate
each weekday in order to accurately capture the topological structure and
temporal dynamics of real world networks?

Previous work has identified at least four factors relevant to tie
formation[17, 18, 19]: focal closure, triadic closure, link-centric
preferential attachment, and physical co-location. Figure 5 gives a
diagramatic depiction of these four factors. However, determining a simple
functional form via which these factors combine in order to govern the
evolution of a dynamic communication network is still an open problem. Here we
describe a simple network evolution model via which co-location
multiplicatively scales the effect of homophily in order to determine which
pairs of people communicate on a given day (see Methods for details). We
simulate the formation of email networks on weekdays by creating an edge
memory dictionary from the last two weeks of February 2020, then generating
new graphs each day using our model. Previous work on the effect of distance
on tie formation has primarily focused on distance and homophily as separate,
additive factors which both contribute positively to tie formation[3, 20, 21].
However, bringing people with clashing personalities close together is more
likely to make them enemies than friends, and hence unlikely to make them
contact one another via email. For this reason we view co-location as a
multiplicative factor which scales the effects of homophily in order to form
new ties. See Supplementary Information for an ablation study of our model
together with a discussion of other random graph models.

Figure 5. Illustration of model mechanisms. In our tie choice model,
illustrated above, the probability of two people interacting depends on four
key mechanisms: Co-location, whether the pair belongs to the same research
unit, whether the pair have a mutual connection, and their pattern of past
communication.

Our goal is to reproduce the dynamics of weak tie formation in networks with a
simple network evolution model. To reproduce the qualitative features observed
in the data, we set the co-location variable $\tau$ for each pair of
individuals to zero (corresponding to no physical co-location) starting from
March 23, 2020 then back to 1 on September 8, 2021. Upon the removal of co-
location, our model produces a drop in the number of weak ties (Figure 6 panel
a) and new weak ties (Figure 6 panel b) which is qualitatively similar to what
we observe in the empirical data. It also reproduces the increase in edge
stability (Figure 6 panel c), as well as the robustness of long-distance ties
to a sudden absence of co-location (Figure 6 panel d). Here the weekly
periodicity is measured as the intersection over union of the edge sets of the
networks on days $d$ and $d-7$. By looking at the number of weak ties, the
clustering coefficient, and the week to week periodicity (panels a,b) produced
by our model after September 8, 2021, we see that our model predicts that
complete reintroduction of co-location results in a complete recovery of weak
ties. The signs of the log of the distance interaction coefficients allows us
to identify the following potential mechanism for the drop in weak ties: two
researchers are more likely to form new weak ties when they are co-located. To
further confirm this hypothesis, we have simulated a scenario without the
sudden transition to fully remote work modeled through the change in the value
of the physical co-location variable on March 23, 2020. The results, reported
in the Supplementary Information, show no observable drop in weak ties,
providing further evidence in support of our explanatory hypothesis.

Figure 6. A tie choice model in which distance multiplicatively scales the
effects of homophily. The output of our model when artificially setting office
distances to a large fixed constant after March 23, 2020, then returning
researcher offices to their initial positions on September 8, 2021.
***$:p<.001$, **$:.001\leq p<.01$, *$:.01\leq p<.05$. a, A simulated drop in
number of local bridges on March 23, 2020 $(\mathrm{effect}=-65.2,p<.001,95\%$
CI: $[-76.109,-54.316],n_{\mathrm{pre}}=42,n_{\mathrm{post}}=238))$ followed
by a rise in the fall $(\mathrm{effect}=51.9,p<.001,95\%$ CI:
$[42.091,61.693],n_{\mathrm{pre}}=129,n_{\mathrm{post}}=38)$. b, The number of
new weak ties entering the network $(\mathrm{effect}=-47.0,p<.001,95\%$ CI:
$[-60.566,-33.510],n_{\mathrm{pre}}=42,n_{\mathrm{post}}=238))$. All
statistics are two-sided $z$ tests. c, The weekly periodicity and daily
clustering coefficient, d, The number of simulated weak ties between users in
distinct research units first appearing between February 4 and March 23, 2020
versus between March 23, 2020 and May 22, 2020. Fitted values/intervals use
the mean as the measure of central tendency.

##  3\. Discussion

Several sociologists have argued that the lack of connections during the
COVID-19 pandemic negatively impacted mental and physical well-being as well
as innovation, collaboration, and creativity [22, 23]. However, the mechanism
via which such effects have occurred has yet to be explicitly identified. As
businesses and universities make crucial decisions about the amount of in-
person work after the COVID-19 pandemic, understanding the lasting effects of
remote work on research communities is of paramount importance.

Our study shows that the transition to fully remote work on the MIT campus –
with consequent complete removal of physical co-location between co-workers –
had notable effects on the email communication network: while some common
topological features were preserved, the formation of weak ties was hindered,
causing weak tie deterioration and network stagnation in the long term.
Employees who are not co-located are less likely to form ties, weakening the
spread of information in the network [24, 25, 26]. The mechanism of weak tie
formation can be successfully reproduced using a link formation model through
which co-location multiplicatively scales the effects of homophily between
researchers.

Our findings have implications for the design of future research campuses and
work environments, as well as for the development of new virtual technologies
which seek to recreate interactions that happen in physical offices. Today it
is of the utmost importance to identify what is the “minimum amount” of in-
presence work that enables the formation of weak ties, so that individual and
societal benefits related to remote work can be preserved without impacting
the generation of new ideas and innovation in general. While previous studies
have documented the effects of reduced in-person collaboration in the short
term[15], we show that the shift to remote communication produces a long-
lasting impact on the formation of local bridges in collaborative networks,
with effects accumulating over time. Expanding on the existing methodology, we
demonstrate that paired testing estimates, conventionally used to evaluate the
short-term effects of stay-at-home restrictions, do not accurately represent
the long-term changes in the social network. We provide an alternative
estimation strategy that produces more robust inference about the long-term
effects.

We also would like to highlight several important considerations on our causal
inference strategy. We use the Bayesian structural time series method to
estimate the long-term effects of the mandatory shift to remote work by
constructing a counterfactual prediction of weekday email exchanges using
weekend emails data. The validity of our methodology is supported by three
main arguments. First, at a weekly level, weekend email exchanges are
sufficiently predictive of the weekday emails due to common unobserved
seasonality and the fact that work communication tends to spillover into
weekends. Second, even if weekend email exchanges are affected by the shift to
mandatory remote work, this is not due to changes in co-location of
researchers since people typically do not come to the office on weekends.
Finally, while there are other potential pandemic-related confounding factors
such as changes in childcare which could a priori contribute to changes in the
MIT email network, it is reasonable to expect such confounding factors to be
distributed independently of the distance between researcher offices. Thus by
stratifying connections by the distance between researcher offices, we
directly attribute the observed loss in new weak tie formation to a lack of
co-location. Still, our approach is not without its limitations. In
particular, the predictive power of the weekend email exchanges is limited by
the fact that we only observe the network for three months prior to
intervention and could benefit from adding data for prior years.

Our results suggest that the loss of social connections that otherwise
spontaneously emerge in shared spaces can not be immediately restored by
simply returning back to offices. When designing work-from-home policies,
firms and organizations should consider ways to promote serendipitous
interactions across organizational units if they want to retain efficient
discovery and transmission of novel information. Still, our initial findings
on the hybrid work model that followed the reintroduction of partial in-person
collaboration at MIT show a slight recovery in the number of weak ties –
especially between researchers who are once again co-located. This hints at
the possibility of establishing a work balance trade-off by combining in-
person and remote interactions among colleagues, which could inform the
transition to a hybrid, post-COVID-19 ‘new normal’.

##  4\. Methods

###  4.1. Ethical Review

This research was reviewed and classifed as exempt by the Massachusetts
institute of Technology (MIT) Committee on the Use of Humans as Experimental
Subjects (MIT’s Institutional Review Board), because the research was
secondary use research involving the use of de-identifed data.

###  4.2. Data Preprocessing

Data was analyzed using python 3.7.9, numpy 1.21.5, pandas 1.1.5, statsmodels
0.12.1, osmnx 1.0.1, python-flint 0.3.0, scipy 1.6.0, and geopandas 0.8.1.

We start with a fixed set of anonymized researchers $\mathcal{R}$ (research
staff, faculty, and postdocs) from 112 different research units. As required
by MIT staff for privacy reasons, these researchers are grouped via 10 random
partitions (no researchers are shared between the groups in a partition) of
$\mathcal{R}$, $\mathcal{R}_{i}=\coprod_{j}G_{j}^{i}$, $1\leq i\leq 10$. We
describe below how to estimate individual-level data from aggregated data
which was necessary in order to ensure that the studied users were active
through the entire time period of interest. Each group $G_{j}^{i}$ contains at
least 5 researchers, and all researchers in $G_{j}$ belong to the same
research unit. Let $g_{ij}\in\mathcal{R}$ denote the collection of researchers
in $G^{i}_{j}$. For a pair of groups $G^{i}_{j}$,$G^{i}_{k}$, and a day $d$
our data contains the sum $W^{i}_{jk}$ of all emails sent between groups $j$
and $k$ for randomization $i$:

(1) $W^{i}_{jk}=\sum_{g_{ij}\in G^{i}_{j}}\sum_{g_{ik}\in
G^{i}_{k}}\operatorname{Emails}_{d}(g_{ij},g_{ik})$

For each research unit $U$, let $\mathcal{R}^{U}$ denote the collection of
researchers in the research unit. Because each group contains only researchers
from a single research unit, each equation (1) is a sum over researchers from
at most two research units. Grouping the equations (1) by pairs of research
units $U,V$ on each day $d$ we obtain a collection of constrained linear
systems of Diophantine equations

(2) $\displaystyle W^{U}_{V}=$ $\displaystyle A^{U}_{V}x^{U}_{V}$ (3)
$\displaystyle x^{U}_{V}\geq$ $\displaystyle 0$

where $x$ is the column vector whose entries are
$\operatorname{Emails}_{d}(g_{ij},g_{ik})$, $1\leq i\leq 10$,
$g_{ij}\in\mathcal{R}^{U}$, $g_{jk}\in\mathcal{R}^{V}$ and $A^{U}_{V}$ is the
matrix of coefficients of the equations (1). Each of these linear systems is
guaranteed to have at least one solution (the actual number of emails sent),
but may be underdetermined. If there is a unique solution, we use the Hermite
normal form [27] of $A^{U}_{V}$ to find it. If the system is underdetermined,
we use non-negative matrix factorization [28] with an $\ell^{1}$ penalty in
order to quickly estimate a sparse non-negative solution, as the algorithms
which compute exact sparse integer solutions are slow. This procedure yields
an estimate for each day $d$ and each pair $u,v\in\mathcal{R}$ of
$\operatorname{Emails}_{d}(u,v)$. In the Supplementary Information we show
that our approximate solutions are very close to true solutions.

###  4.3. Network formation

To rule out changes in the data due to departures from the university or new
hires, we ensure that each user sent at least one email over the university
network before the end of the 2020 spring semester _and_ at least one email
after May 20, 2021. Denote this set of active users by $\mathcal{A}$.

We are missing data from December 23, 2020 and January 19,20, and 21 of 2021;
because these days are during the winter holiday at MIT this does not
significantly affect our analysis. For each of the remaining 562 days from
December 26, 2019 through July 15, 2021, we obtain a weighted, undirected
network whose nodes represent (anonymized) individuals. Fix a day $d$, for
each user $u$ in $\mathcal{A}$, let $\operatorname{Nb}_{d}(u)$ denote the
number of people whom $u$ emailed on day $d$. For a pair of users
$u,v\in\mathcal{A}$, let $\operatorname{Emails}_{d}(u,v)$ denote the
(estimated as in the previous section) number of emails sent from $u$ to $v$
on day $d$. To rule out massmails, let $\mathcal{A}_{d}\subseteq\mathcal{A}$
be the subset

$\mathcal{A}_{d}=\\{u\in\mathcal{A}\mid 1\leq\operatorname{Nb}(u)<100\\}.$

Define a weighted, undirected network $N_{d}$ with nodes $\mathcal{A}_{d}$.
For two nodes $u,v$, there is an edge $(u,v)$ if
$\operatorname{Emails}(u,v)\geq 1$ and $\operatorname{Emails}(v,u)\geq 1$. The
weight of the edge is defined to be
$\min(\operatorname{Emails}_{d}(u,v),\operatorname{Emails}_{d}(v,u))$.
Although the email data is partially estimated due to randomization and
aggregation, for $>66$ % of the edges with non-zero weight in the estimated
network $N_{d}$ we were able to recover the true number of emails sent. If we
include all edges between users contributing to a non-zero weight edge in at
least one random aggregation of the network (but which may have weight zero in
the estimated network), $>99.9$% of edges have the ground truth number of
emails. When building the undirected network $N_{d}$, we consider four
possible time-windows during which emails can be reciprocated: the same day
(daily), within 5 business days (weekly), within 10 business days (bi-weekly),
or within 21 business days (monthly). For results on weekly, bi-weekly, and
monthly networks see Supplementary Information. Previous studies have found
that more than 90% of emails are replied to the same day that they are sent,
with more than half being replied to within 47 minutes [29]. Requiring emails
to be reciprocated the same day hides interactions between users who typically
respond slowly to emails; however, it is useful for filtering out massmails,
observing sharp discontinuities in the data, and for increasing the power of
hypothesis tests. Allowing longer periods of reciprocation captures weaker
ties missed in the daily reciprocated email network, but we are forced to
sacrifice some statistical power either to autocorrelation or lower sample
size; additionally the networks become more saturated, destroying some
topological features of interest. Examples of daily, weekly, bi-weekly, and
monthly email networks are reported in the Supplementary Information.

To examine whether this hybrid mode of work returned tie formation to pre-
pandemic levels, we first restrict to a collection of 2,206 researchers who
sent at least 5 emails after September 2021 and before May 2020, then proceed
as above to form networks from December 23, 2020 to October 31, 2021. We
choose a stricter requirement for inclusion in the network than previously as
we observe many users becoming inactive starting in summer 2021.

When comparing February 2020 to February 2021, we pair the days in the two
months as follows: the first Tuesday of the MIT semester in February 2020 is
paired with the first Tuesday of the MIT semester in February 2021, etc. For
each pair of days $(d_{2020}^{i},d_{2021}^{i})$, we consider the set of users
$\mathcal{A}_{d_{2020}^{i}}\cap\mathcal{A}_{d_{2021}^{i}}$ who were active on
both days and, as above, form a pair of undirected networks
$(G_{2020}^{i},G_{2021}^{i})$ whose edges $(u,v)$ correspond to reciprocated
emails on $d_{2020}^{i}$ (for $G_{2020}^{i}$) or $d_{2021}^{i}$ (for
$G_{2021}^{i}$). Directly comparing these networks allows us to completely
remove the effects of seasonality or any difference in makeup of active users
in February 2020 and February 2021.

###  4.4. Link-centric preferential attachment

The goal of our model is not to serve as a tool for prediction, but to
understand the mechanism via which distance impacts link formation. Through
experimentation, we found that using link-centric preferential attachment
alone to propagate a dynamic network produced daily networks which had too
many local bridges. Thus we use a two step approach which first produces an
intermediate network using link-centric preferential attachment, then adds
edges in a way which increases the clustering coefficient of the network. This
is analogous to the reverse of the Watts-Strogatz method [30], where the
intermediate network has high clustering coefficient and local bridges are
added afterwards.

Our link formation model has the following parameters:

  * •

$P$, the periodicity of the model. $P$ controls how much the graph on day $d$
looks like the graphs on days $d-7$. In other words, the higher the value of
$P$, the closer the dynamic network is to being 7-periodic (or 5-periodic if
weekends are removed).

  * •

$O$, the tendency to connect with old links. The higher the value of $O$, the
more likely it is that a given link will connect with a previous partner
rather than someone new. This is one of the standard parameters from a vanilla
link-centric preferential attachment model, and this parameter decays
exponentially in the number of days between contact: $O=ce^{-d}$ where $c$ is
a constant and $d$ is the number of days since the link last appeared.

  * •

$N$, the tendency to reach out to new people. This is typically the complement
of $O$ in vanilla link-centric attachment models (we have more parameters than
the standard two).

  * •

$D$, the tendency to connect with people in the same department.

  * •

$F$, the tendency to be introduced to a mutual friend.

The parameters $O,N$, and $P$ rely on a memory dictionary which stores the
days on which a given edge has appeared. For all of the above parameters
$\\{P,O,N,D,F\\}$, we include interaction terms

$\\{C_{P},C_{O},C_{N},C_{D},C_{F}\\}$

controlling the extent to which co-location amplifies or dampens the effect.
For example, from the empirical data we conclude that co-location should
dampen periodicity while amplifying the probability of reaching out to new
partners.

Let $e=(u,v)\in\mathcal{A}\times\mathcal{A}$ be a pair of nodes. For each
parameter $Q\in\\{P,O,N,D,F\\}$ let $\mathbbm{1}_{Q}$ denote the associated
indicator variable. Specifically,

$\displaystyle\mathbbm{1}_{P}(e)=$ $\displaystyle\begin{cases}1&\text{if $u,v$
connected 7 days in the past}\\\ 0&\text{otherwise}\end{cases}$
$\displaystyle\mathbbm{1}_{N}(e)=$ $\displaystyle\begin{cases}1&\text{if $u,v$
have never connected}\\\ 0&\text{otherwise}\end{cases}$
$\displaystyle\mathbbm{1}_{O}(e,d_{c})=$
$\displaystyle\begin{cases}\sum_{d_{p}}e^{-(d_{c}-d_{p})/562}&\text{if $u,v$
have previously connected}\\\ 0&\text{otherwise}\end{cases}$
$\displaystyle\mathbbm{1}_{D}(e)=$ $\displaystyle\begin{cases}1&\text{if $u,v$
are in the same research unit}\\\ 0&\text{otherwise}\end{cases}$
$\displaystyle\tau(e)=$ $\displaystyle\begin{cases}1&\text{if $u,v$ have
offices within 150 meters}\\\ 0&\text{otherwise}\end{cases}$

where $d_{c}$ is the current day, and $d_{p}$ are the past days on which the
tie $e$ was present in the network. The 150 meter cutoff is chosen based on
the empirical results.

Consider the set $E$ of all edges which appear on at least one day in the
empirical data. Note that the use of $E$ rather than the set of all possible
edges makes this model unsuitable for prediction tasks. On each day $d$, we
start by adding an edge $e\in E$ to the random network $G^{d}_{1}$ with
probability proportional to

$\displaystyle p_{e}\propto$
$\displaystyle\tau(e)\sum_{Q\in\\{P,O,N,D\\}}C_{Q}\mathbbm{1}_{Q}(e)Q$
$\displaystyle+$
$\displaystyle(1-\tau(e))\sum_{Q\in\\{P,O,N,D\\}}\frac{1}{C_{Q}}\mathbbm{1}_{Q}(e)Q$

If $C_{Q}>1$ then co-location amplifies the effect of parameter $Q$, while if
$C_{Q}<1$ it dampens the effect. In total, in the first step we add
$\varepsilon_{1}$ edges to $G^{d}_{1}$, where
$\varepsilon_{1}\sim\mathcal{N}(1000,10)$.

In the second step, we add the parameter $F$,

$\mathbbm{1}_{F}(e)=\begin{cases}1&\text{if $d_{G_{1}}(u,v)=2$ }\\\
0&\text{otherwise}\end{cases}$

where $G^{d}_{1}$ is the random network constructed in step 1, and $d_{G}$ is
the usual (unweighted) shortest path distance in the graph $G$. In words,
$\mathbbm{1}_{F}(e)=1$ if adding $e$ will close a triangle in the network. A
new edge $e$ is added to the network in step two with probability

$\displaystyle p_{e}\propto$
$\displaystyle\tau(e)\sum_{Q\in\\{P,O,N,D,F\\}}C_{Q}\mathbbm{1}_{Q}(e)Q$
$\displaystyle+$
$\displaystyle(1-\tau(e))\sum_{Q\in\\{P,O,N,D,F\\}}\frac{1}{C_{Q}}\mathbbm{1}_{Q}(e)Q$

In total we add $\varepsilon_{2}$ edges to $G^{d}_{1}$ in the second step,
where $\varepsilon_{2}\sim\mathcal{N}(500,10)$. Including the parameter $F$
has the effect of raising the expected number of triangles in the random
graph, and hence lowering the percentage of edges which are local bridges.

With parameters fixed, we proceed to form networks one day at a time, adding
the edges from the current network to a memory dictionary after formation. To
model the effect of remote work, for each day $d$ after March 23, 2020, we set
the distance between researcher offices to be a fixed constant larger than 650
meters (all other parameters remain fixed). For parameter values we set:

$P=80000$ | $N=2000$ | $O=1$ | $D=90000$ | $F=200000$
---|---|---|---|---
$C_{P}=\frac{2}{3}$ | $C_{N}=2$ | $C_{O}=1$ | $C_{D}=\frac{10}{11}$ | $C_{F}=\frac{2}{3}$

###  4.5. Regression Discontinuity

Figure 7 (panels a and b) (respectively Figure 4 panel c) show the drop in
weak tie and new weak tie formation (respectively increase in weak ties) due
to the policy change on March 23rd, 2020. We used Regression Discontinuity
Designs (RDD) [31, 32, 33] to estimate the causal impact of the policy change.
RDDs are a classic, quasi-experimental procedure for estimating treatment
effects in observational studies. In an RDD, treatment assignment is
determined by an assignment variable rather than through randomization.

For an RDD to be valid, we need only assume that the response is continuous
with the assignment variable near the cutoff and that subjects cannot
precisely manipulate the assignment variable [32, 33]. Panels a and b in
Figure 7 and panel c in Figure 4 show that the responses (weak ties and new
weak ties) are continuous with the assignment variable time, albeit observed
with noise. The assignment variable, time, is not precisely manipulable by
subjects since the announced policy was not known far in advance. Furthermore,
there would be little reason to manipulate assignment since subjects are free
to send emails at the same rate before and after the policy change.

In Figure 7 panel a and Figure 4 panel c, we model the weekly mean number of
weak ties with the discontinuous linear regression

(4) $Y=\alpha+\eta D+\beta_{2}D(X-c)+\epsilon$

where $c$ is the cutoff date (either March 23rd, 2020 or September 8, 2021)
and $D$ is the binary variable

$D=\begin{cases}1&X\geq c\\\ 0&X<c\end{cases}$

that indicates if the date $X$ is before or after the policy change date $c$.
The error term $\epsilon$ is assumed to be heteroskedastic white noise. The
coefficient $\eta$ is the impact of the policy, and measures the gap between
the two sides of the regression. We estimate $\eta$ and the other coefficients
with generalized least squares with AR(n) structured covariance matrix with
$n=5$ and report the value of $\hat{\eta}$ and its p-value in Figure 7. We use
heteroskedasticity-robust estimators for standard errors, so that in total
standard errors are robust to autocorrelation (from the AR(5) GLS) and
heteroskedasticity [34].

In panel a, we assumed a discontinuous order one polynomial trend line because
the data did not display any apparent higher-order non-linear behavior. The
data were subset to January 3rd, 2020 to October 1st, 2020, to semi-localize
our regression around the discontinuity, which reduces bias in $\hat{\eta}$
[33], and to avoid influence from the two outlying regions (before January
3rd, 2020 and during December 2020). These outliers correspond to winter break
at MIT and represent a natural and expected decrease in weak ties not due to
the policy change.

In Figure 2 panel b, we similarly model the rate of new weak tie formation
over time. We assumed a second-order discontinuous polynomial trend (Equation
5) due to the observed parabolic behavior before the cutoff point. Using the
same notation as in Equation 4, our linear regression is given by

(5) $Y=\alpha+\eta
D+\beta_{1}(X-c)+\beta_{2}(X-c)^{2}+\beta_{3}D(X-c)+\beta_{4}D(X-c)^{2}+\epsilon.$

The coefficient $\eta$ is, again, the causal impact of the policy change. We
report the value of $\widehat{\eta}$ and its p-value in Figure 7.

###  4.6. Bayesian Structural Time Series

We stress that when using Bayesian methods, reported CIs are credible
intervals of the predicted dependent variable (also called posterior
predictive intervals), and $p$-values are posterior tail probabilities.
Bayesian structural time-series (BSTS) combines a state-space model for time-
series data and Bayesian model averaging for parameter selection and
estimation [35].

As a state-space model, BSTS combines three components of state: a local
linear trend:

$\displaystyle\mu_{t+1}$ $\displaystyle=\mu_{t}+\delta_{t}+\eta_{\mu,t}$
$\displaystyle\delta_{t+1}$ $\displaystyle=\delta_{t}+\eta_{\delta,t},$

with $\eta_{\mu,t}\sim\mathcal{N}(0,\sigma_{\mu}^{2})$,
$\eta_{\delta,t}\sim\mathcal{N}(0,\sigma_{\delta}^{2})$; a seasonality
component:

$\gamma_{t+1}=-\sum_{s=0}^{S-2}\gamma_{t-s}+\eta_{\gamma,t}$

with $S$ the number of seasons and $\eta_{\gamma,t}$ again an independent
error; and (static) covariates which are predictive of the time series in
question before the intervention:

$Z_{t}=\beta^{T}\mathbf{x}_{t}.$

For the local linear trend and seasonality components, we use the default
priors of the CausalImpact library:

$\frac{1}{\sigma}\sim\mathcal{G}(10^{-2},10^{-2}s_{y}^{2})\qquad
s_{y}^{2}=\sum_{t}\frac{(y_{t}-\overline{y})^{2}}{n-1}$

where $\mathcal{G}(-,-)$ denotes a Gamma distribution. For the covariates (the
weekend data), in general a spike-and-slab prior is used with the spike
defined by

$p(\xi)=\prod_{i=1}^{J}\pi_{j}^{\xi_{j}}(1-\pi_{j})^{1-\xi_{j})}$

with $\pi_{j}$ initialized to $\frac{M}{J}$ where $M$ is the expected model
size. The slab part of the spike-and-slab prior is

$\displaystyle\beta_{\xi}\mid\sigma_{\epsilon}^{2}$
$\displaystyle\sim\mathcal{N}(0,\sigma^{2}_{\epsilon}(\Sigma_{\xi})^{-1})^{-1}$
$\displaystyle\frac{1}{\sigma^{2}_{\epsilon}}$
$\displaystyle\sim\mathcal{G}\left(\frac{\nu_{\epsilon}}{2},\frac{s_{\epsilon}}{2}\right)$
$\displaystyle\Sigma^{-1}$
$\displaystyle=\frac{1}{n}\left\\{\frac{1}{2}X^{T}X+\frac{1}{2}\mathrm{diag}(X^{T}X)\right\\}$

where $X$ is the covariate data. Because we include only one covariate (the
weekly minimum) the spike-and-slab prior collapses to just a normal-inverse
Gamma distribution. We use 1000 iterations of Markov Chain Monte Carlo (MCMC)
to compute posterior predictive distributions.

Consider the binary variable $X(r,d)$ defined by

$X(r,d)=\begin{cases}1&\text{researcher $r$ is in their office on day $d$}\\\
0&\text{otherwise}.\end{cases}$

For us, “treatment” consists of setting $X(r,d)$ to zero for $d$ greater than
March 23, 2020 by not permitting researchers to enter their campus office. The
time series whose counterfactual we want to estimate is the weekday maximum of
the network measure while the covariate is the weekend minimum. As most
employees are not physically present in their office on weekends, $X(r,d)=0$
for most $r$ when $d$ is a weekend so that the treatment has little effect. We
verify this assumption by looking at the number of distinct MAC addresses
connected to routers in on-campus research labs on the weekday and weekend
(see Supplementary Information).

When studying the effect of hybrid work, we construct a counterfactual using
weekday email data spanning July 22 2020 through October 14, 2020 as a
covariate for email data spanning July 28, 2021 through October 20, 2021,
aligning so that the start of the Fall 2020 and Fall 2021 semesters coincide.
We also remove Memorial Day (a university holiday) from both the 2020 and 2021
data.

##  5\. Data Availability

A subset of the data containing 1000 (anonymized) users is available on
Zenodo[36]. Because of privacy concerns for MIT employees, the entire dataset
of emails and pairwise distances cannot be publicly released. Source data for
Figures 1,2,3,4,6 is available with this manuscript.

##  6\. Code Availability

The code used to analyze the data can be obtained from Code Ocean[37].

##  7\. Acknowledgments

P.S. and C.R. thank FAE Technology, MipMap, Samoo Architects & Engineers,
GoAigua, DAR Group, Ordinance Survey, RATP, Anas S.p.A., ENEL Foundation and
all of the members of the MIT Senseable City Laboratory Consortium for
supporting this research. S.L. was supported by the Carlsberg Foundation
(CF20-0044) and the Villum Foundation (Nation-scale Social Networks).

##  8\. Author Contributions

D.C., M.M., C.R., and P.S. designed the research. M.M. provided data access.
D.C. and T.A.H. processed and analysed the data. D.C., M.M., T.A.H., T.A., and
P.S. performed the interpretation and writing. S.L., T.A., and R.D provided
theoretical expertise. P.S. and C.R. continuously advised the project. C.R.
framed initial hypothesis.

##  9\. Competing Interests

The authors declare no competing financial interests.

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##  10\. Research Units

Here we list the research units used in the study. Only a subset of
researchers from each unit appear in the email networks. Note that in addition
to standard denominations corresponding to departments (e.g. Chemistry), there
are also subgroups corresponding to department heads and institute professors
(distinguished professors).

Abdul Latif Jameel Poverty Action Lab, Aeronautics and Astronautics,
Anthropology Program, Archaeology, Architecture, Architecture and Planning -
Depart Heads, Biology, Brain and Cognitive Sciences, Center for Advanced
Urbanism, Center for Biomedical Innovation, Center for Collective
Intelligence, Center for Environmental Health Sciences, Center for Global
Change Science, Center for Information Systems Research, Center for
International Studies, Center for Real Estate, Center for Transportation and
Logistics, ”Chancellors Office”, Chemical Engineering, Chemistry, Civil and
Environmental Engineering, Comp Sci and Artificial Intel Lab HQ, Comp Sci and
Artificial Intelligence Lab, Comparative Media Studies/Writing, Cybersecurity
at MIT Sloan, D-Lab, DAPER Administration, DAPER Intercollegiate Sports, DLC
Heads Science, Dean for Student Life - Dept Heads, Dean of Humanities, Arts,
and Social Sci, Dean of Science, Department of Biological Engineering, Dept
Administrators and Lab Directors, Dept Heads Vice President for Research,
Division of Comparative Medicine, Earth, Atmospheric and Planetary Sciences,
Economics, Electrical Engineering-Computer Science, Global Studies and
Languages, Haystack Observatory, Health Sciences and Technology Program,
History Section, Industrial Performance Center, Inst for Data Systems and
Society Profs, Inst for Medical Eng. and Science Prof, Institute Professors,
Institute for Data, Systems, and Society, Institute for Medical Eng. and
Science, Institute for Soldier Nanotechnologies, J-WAFS, Jameel Clinic for ML
in Health, Kavli Inst for Astrophysics and Space Rsrh, Koch Inst - Integrative
Cancer Research, Lab for Information and Decision Systems, Laboratory for
Nuclear Science, Leaders for Global Operations Program, Libraries, Linguistics
and Philosophy, Literature Section, MIT Energy Initiative, MIT Environmental
Solutions Initiative, MIT Open Learning, ”MIT Program in Womens and Gender
Studies”, MIT Quest for Intelligence, MIT Sea Grant College Program, MIT Sloan
Human Resources, MIT-SUTD Collaboration, MIT.nano, MITii, Materials Research
Laboratory, Materials Science and Engineering, Mathematics, McGovern Institute
for Brain Research, Mechanical Engineering, Media Lab, Microsystems Technology
Laboratories, Music and Theater Arts Section, Nuclear Reactor Laboratory,
Nuclear Science and Engineering, OSATT-Technology Licensing Office, Office of
the President, Office of the Provost, Open Learning, J-WEL, Open Learning,
J-WEL Research, Projects, Open Learning, J-WEL, Higher Ed, Open Learning,
Playful Journey Lab, Physics, Picower Institute for Learning and Memory,
Plasma Science and Fusion Center, Political Science, Prog in Science,
Technology, and Society, Program in Art, Culture and Technology, Program in
Media Arts and Sciences, ROTC Air Force, ROTC Army, ROTC Navy, Research
Laboratory of Electronics, SCC Dept/Lab/Center/ Director Org, SHASS Department
Heads, School of Architecture and Planning, School of Engineering, School of
Engineering Professors, Schwarzman College of Computing, Sloan School of
Management, SoE Dept/Lab/Center/Director Org, Sociotechnical Systems Research
Center, Supply Chain Management Program, System Design and Management Program,
Urban Studies and Planning, VP for Research, World Wide Web Consortium

##  11\. Local bridges

######  Definition 11.1.

Given an undirected network $N$, an edge $(u,v)$ in $N$ is a _local bridge_ if
there is no node $n$ such that there are edges $(u,n)$ and $(v,n)$. In other
words, the endpoints of a local bridge have no common neighbors.

As the definition of a local bridge given above requires an undirected
network, we will restrict to reciprocated communications when studying local
bridges. The link between weak ties and local bridges comes via the following
definition from Granovetter [16].

######  Definition 11.2.

Fix an undirected network $N$ together with a labeling of each edge in $N$ as
either “strong” or “weak”. We say that a node $A$ violates the Strong Triadic
Closure Property if it has strong ties to two other nodes $B$ and $C$, and
there is no edge between $B$ and $C$. We say a node $A$ satisfies the Strong
Triadic Closure Property if it does not violate it.

From here, a well-known argument by contradiction implies that local bridges
and a weak ties are intimately related in networks which satisfy Strong
Triadic Closure.

######  Theorem 11.3 (Granovetter [16]).

If a node $A$ in an undirected network $N$ satisfies the Strong Triadic
Closure Property and is involved in at least two strong ties, then any local
bridge it is involved in must be a weak tie.

There are two distinct ways to lose local bridges in a network: first, an edge
which is a local bridge on day $d$ can fail to exist in the network on day
$d+k$ for some $k>0$; second, an edge which is a local bridge on day $d$ can
be embedded in a triangle in the network on day $d+k$.

##  12\. Cumulative plots and robustness of network formation

Here we plot more detailed versions of results from the main text which show
the evolution of the structural time series model uncertainty over time. Note
that if our covariate (weekend data) becomes less predictive of weekday data,
then the width of the posterior predictive interval will expand over time to
compensate. Specifically, Supplementary Figure 7 corresponds to Figure 2 in
the main text, showing additionally the intersection over union of the network
edge sets on days $d$ and $d+7$ each week in panel h. If people are not
forming new weak ties, then assuming a constant number of ties over time
(supported by the non-significant difference in number of emails sent
before/after COVID), people must be holding onto old ties for longer or
forming ties which already have a large overlap with their ego network. In
both cases there would be increased redundancy in information spread. Panel h
directly demonstrates a statistically significant increase in week to week
redundancy, measure as periodicity, in email networks due to the shift to
remote work. Supplementary Figure 8 shows the evolution of uncertainty over
time for Figure 4 in the main manuscript detailing the change in new weak ties
with a partial re-introduction of co-location. Supplementary Figure 9 is
identical to Supplementary Figure 7 except that we do not prune nodes with
more than 100 outgoing connections from the network in order to show that our
results are robust to this pruning. Similarly Supplementary Figure 10 is
identical to Figure 3 in the main manuscript but computed without first
pruning network nodes. We again note that there are no qualitative differences
from the results in the main text.

Figure 7. Changes in the structure of weak ties in the email network.
***$:p<.001$, **$:.001<p<.01$, *$:.01<p<.05$. a, There is a drop of 55.70 (6.2
%) in the mean number of local bridges (weak ties) caused by the switch to
fully remote work ($p=.002$, 95% CI: [-91.627, -19.774]). b, A statistically
significant drop of -38.03 (38.7%) in the mean number of new (not previously
seen) weak ties appearing each weekday ($p<.001$, 95% CI: [-53.38, -22.68]).
Statistics represent two-sided $z$ tests. c Local bridges in a social network
can be formed from chance encounters. d, A predicted loss of 75.16 weak ties
($p<.001$, 95% CI: [-77.44, -72.75]). e, A predicted loss of 43.09 new weak
ties ($p=0.23$, 95% CI: [-171.04, 79.08]). f, A cumulative drop of 5110 in the
number of weak ties throughout an entire year. g, A cumulative drop of 2930 in
new weak ties. h, A 0.02 (10.81%) increase in the week to week similarity of
network edges ($p<.001$, 95% CI: [7.75%,13.9%]). i, The number of weak ties
which become strong or are churned in a 30 day rolling window. Shaded regions
show a 95% posterior predictive interval computed using Bayesian structural
time series. $n_{\mathrm{pre}}=42$ days, $n_{\mathrm{post}}=188$ days for
panels a, b. $n_{\mathrm{pre}}=8$ weeks, $n_{\mathrm{post}}=72$ weeks for
panels d-h. Fitted values/intervals use the mean as the measure of central
tendency. Figure 8. The effect of re-introducing co-location between
researchers. ***$:p<.001$, **$:.001\leq p<.01$, *$:.01\leq p<.05$. a,d,f,h,
The change due to hybrid work in the _percentage_ of new weak ties between
researchers in the same lab, researchers in distinct labs at distance less
than 150m, researchers in distinct labs at distance between 150 and 650m, and
researchers in distinct labs at distance larger than 650m. b,e,g,i, The
changes in the _total number_ of new ties between researchers at the distance
thresholds detailed above. $n_{\mathrm{pre}}=29$ days, $n_{\mathrm{post}}=35$
days for the panels listed above. c, A discontinuity design (two-sided $z$
test) on the _difference_ of the number of weak ties in September 2020 and
September 2021 $(\mathrm{effect}=67.5,p=0.022,95\%$ CI:
$[9.740,125.266],n_{\mathrm{pre}}=34$ days, $n_{\mathrm{post}}=26$ days).
Fitted values/intervals use the mean as the measure of central tendency.
Figure 9. Changes in weak ties in the MIT email network with no pruned users
after the shift to remote work. ***$:p<.001$, **$:.001<p<.01$, *$:.01<p<.05$.
a, There is a drop of 52.0 in the mean number of local bridges (weak ties)
caused by the switch to fully remote work ($p=.005$, 95% CI: [-88.334,
-15.670]). b, A statistically significant drop of -39.8 in the mean number of
new (not previously seen) weak ties appearing each weekday ($p<.001$, 95% CI:
[-56.602, -22.947]). Statistics represent two-sided $z$ tests. c Local bridges
in a social network can be formed from chance encounters. d, A predicted loss
of 77.25 weak ties ($p<.001$, 95% CI: [-94.44, -60.4]). e, A predicted loss of
38.1 new weak ties ($p=0.29$, 95% CI: [-166.4, 86.51]). f, A cumulative drop
of 5253 in the number of weak ties throughout an entire year. g, A cumulative
drop of 2590 in new weak ties. h, A 0.02 (11.9%) increase in the week to week
similarity of network edges ($p<.001$, 95% CI: [8.9%,15.0%]). i, The number of
weak ties which become strong or are churned in a 30 day rolling window.
Shaded regions show a 95% posterior predictive interval computed using
Bayesian structural time series. $n_{\mathrm{pre}}=42$ days,
$n_{\mathrm{post}}=188$ days for panels a, b. $n_{\mathrm{pre}}=8$ weeks,
$n_{\mathrm{post}}=72$ weeks for panels d-h. Fitted values/intervals use the
mean as the measure of central tendency. Figure 10. Formation of new weak ties
stratified by distance in a network where no users are pruned. a, The
estimated effect between researchers in the same lab. b, The estimated effect
between researchers in distinct labs within 150 meters. c, The estimated
effect between researchers in distinct labs between 150 and 650 meters. d, The
estimated effect between researchers in distinct labs further than 650 meters.
Shaded regions represent 95% posterior predictive intervals computed using
Bayesian structural time series with a synthetic counterfactual constructed
from weekend data $n_{\mathrm{pre}}=8$ weeks, $n_{\mathrm{post}}=72$ weeks for
all panels. Fitted values/intervals use the mean as the measure of central
tendency.

##  13\. Estimating the number of researchers on campus

To justify the use of weekend email data to generate a synthetic
counterfactual, we estimate the number of distinct researchers co-located on
weekends and weekdays before and after the pandemic lockdown using WiFi data.
Specifically, we count the number of unique MAC addresses connected to a given
MIT campus WiFi router each hour from February 3, 2020 until October 10, 2020.
Supplementary Figure 11 shows that, as expected, far fewer people are on
campus during the weekend. To confirm that this effect is not due to a lack of
classes on the weekends, Supplementary Figure 12 shows the number of unique
MAC addresses in the Senseable City Lab on weekends and weekdays – a room in
which no classes are taught.

Figure 11. The number of users in the MIT math building on weekdays and
weekends. Counts are estimated using the number of distinct MAC addresses
connected to access points in MIT building 2. Figure 12. The number of users
in the Senseable City Lab on weekdays and weekends. Counts are estimated using
the number of distinct MAC addresses connected to access points in MIT
building 9 room 216.

##  14\. Robustness of the tie formation model

It is well known that standard random graph models like Erdős-Renyi, Watts-
Strogatz, and Barabási-Albert do not fully capture the topological features of
real social networks [10.5555/1809753]. For example, the Erdős-Renyi and
Barabási-Albert models have unrealistically low clustering coefficients, while
the Watts-Strogatz model has an unrealistic degree distribution.

To verify that a lack of co-location is what causes our model to reproduce the
drop in weak ties seen in the empirical data, we check that with no changes in
co-location our model produces no changes in weak ties in Supplementary Figure
13. Supplementary Figures 14 through 18 show the results of ablation studies
on our model from removing parameters. For each parameter except $O$, we see
changes to the output of the simulation which makes it inconsistent with
empirical results. Because the $O$ parameter is small and unaffected by co-
location, this is unsurprising.

Finally, Supplementary Figure 19 shows the distribution of distances between
researcher offices at MIT in order to demonstrate that our choices of distance
bins each contain a non-trivial number of researchers. Almost all researchers’
offices are within 2 km of one another.

Figure 13. Simulation results when leaving the distance between researchers
unchanged after March 23, 2020. a, The number of weak ties. b, The number of
new weak ties entering the network. c, The weekly periodicity and daily
clustering coefficient. d, The number of simulated weak ties between users in
distinct research units first appearing between February 4 and March 23, 2020
versus between March 23, 2020 and May 22, 2020. Figure 14. Simulation results
when removing the focal closure (same research unit) parameter. a, The number
of weak ties. b, The number of new weak ties entering the network. c, The
weekly periodicity and daily clustering coefficient. d, The number of
simulated weak ties between users in distinct research units first appearing
between February 4 and March 23, 2020 versus between March 23, 2020 and May
22, 2020. Figure 15. Simulation results when removing the new tie parameter.
a, The number of weak ties. b, The number of new weak ties entering the
network. c, The weekly periodicity and daily clustering coefficient. d, The
number of simulated weak ties between users in distinct research units first
appearing between February 4 and March 23, 2020 versus between March 23, 2020
and May 22, 2020. Figure 16. Simulation results when removing the old tie
parameter. a, The number of weak ties. b, The number of new weak ties entering
the network. c, The weekly periodicity and daily clustering coefficient. d,
The number of simulated weak ties between users in distinct research units
first appearing between February 4 and March 23, 2020 versus between March 23,
2020 and May 22, 2020. Figure 17. Simulation results when removing the
periodicity parameter. a, The number of weak ties. b, The number of new weak
ties entering the network. c, The weekly periodicity and daily clustering
coefficient. d, The number of simulated weak ties between users in distinct
research units first appearing between February 4 and March 23, 2020 versus
between March 23, 2020 and May 22, 2020. Figure 18. Simulation results when
removing the triadic closure parameter. a, The number of weak ties. b, The
number of new weak ties entering the network. c, The weekly periodicity and
daily clustering coefficient. d, The number of simulated weak ties between
users in distinct research units first appearing between February 4 and March
23, 2020 versus between March 23, 2020 and May 22, 2020. Figure 19. The
distribution of distances between researcher offices.

##  15\. Descriptive statistics for weekly, biweekly, and monthly emails
networks

Requiring emails to be reciprocated the same day is useful to increase the
number of independent pre-lockdown data points for statistical purposes, but
we lose email exchanges which are stretched over the course of a week or
longer. The mean response time in an email exchange is closely related to the
frequency of communication: if the mean response time between two people in
the 562 days of data is $d$ days, then the total number of reciprocated
exchanges (ignoring the number of emails sent per exchange) is at most
$\frac{562}{d}$. Thus by restricting to daily reciprocated emails, we may miss
the contributions of some infrequent ties. To expand the breadth of ties under
consideration and study how incorporating lagged communications affects our
results, we construct email networks whose edges represent the number of
reciprocated emails exchanged within 5 business days, 10 business days, and 21
business days. Supplementary Figures 20-23 show sample weekly, biweekly, and
monthly reciprocated email networks from April 4, 2020. Supplementary Figures
24 through 26 show standard network measures for emails networks whose edges
represent emails reciprocated within one week, two weeks, or one month. In
contrast to the daily email networks, there are seemingly long-lasting changes
in the size of the giant component in the networks; we are unable to quantify
this statistically as our construction of a synthetic counterfactual from
weekend email data breaks down when we allow emails to be reciprocated over
any period longer than 2 days.

Supplementary Figure 27 shows the number of local bridges and new local
bridges in weekly, biweekly, and monthly reciprocated email networks. The
number of new weak ties in each case seems to stabilize between 40-60 new weak
ties in agreement with the number of new weak ties in the daily email
networks. This can be partially explained by the fact that we use a sliding
window approach with a stride of one to construct each network. Curiously, the
rate of weak tie formation (number of new ties per day) seems to approach a
constant around 45 regardless of the reciprocation window.

Figure 20. An example daily email network. We allow emails to be reciprocated
only on the same business day. Local bridges are purple. Figure 21. An example
weekly email network. We allow emails to be reciprocated up to 5 business days
later. Local bridges are purple. Figure 22. An example biweekly email network.
We allow emails to be reciprocated up to 10 business days later. Local bridges
are purple. Figure 23. An example monthly email network. We allow emails to be
reciprocated up to 21 business days later. Local bridges are purple. Figure
24. Summary statistics for weekly email networks. a, The number of connections
(edges) between users who are in the same research unit. b, The number of
connections (edges) between users who are in distinct research units. c, The
number of nodes in the daily email network who are attached to at least one
edge. d, The number of connected components. e The number of nodes in the
largest connected component. f The average degree of each node in the weekly
email networks. Figure 25.  Summary statistics for biweekly email networks. a,
The number of connections (edges) between users who are in the same research
unit. b, The number of connections (edges) between users who are in distinct
research units. c, The number of nodes in the daily email network who are
attached to at least one edge. d, The number of connected components. e The
number of nodes in the largest connected component. f The average degree of
each node in the biweekly email networks. Figure 26. Summary statistics for
monthly email networks. a, The number of connections (edges) between users who
are in the same research unit. b, The number of connections (edges) between
users who are in distinct research units. c, The number of nodes in the daily
email network who are attached to at least one edge. d, The number of
connected components. e The number of nodes in the largest connected
component. f The average degree of each node in the monthly email networks.
Figure 27.  a, The number of local bridges each day in monthly reciprocated
email networks. b, The number of local bridges each day in biweekly
reciprocated email networks. c, The number of local bridges each day in weekly
reciprocated email networks. d, The number of new (not previously seen) local
bridges in monthly reciprocated email networks. e, The number of new (not
previously seen) local bridges in biweekly reciprocated email networks. f, The
number of new (not previously seen) local bridges in weekly reciprocated email
networks.

##  16\. Alternate definitions of weak tie

From the definition of local bridge, one can see that an isolated dyad – an
edge whose source and target have degree one – is a local bridge. However,
such edges do not correspond to the intuitive notion of a “bridge” as a link
between distinct communities. Supplementary Figure 28 panel a shows that the
observed drop in local bridges persists even when we ignore local bridges
which are isolated dyads.

A previous study on the effect of remote work on tech employee collaboration
patterns defined bridging connections as those connections in monthly
communication networks with a low local constraint [BurtHoles], and weak ties
as ties with below-median communication time [15]. Our definition of weak ties
(local bridges) is more similar to their definition of bridging tie than their
definition of weak tie. Supplementary Figure 28 panels b and c show Burt’s
measure of structural holes and a measure of structural holes commonly used in
topological data analysis, respectively. In both cases, there are no
significant long-term changes in the structure of holes in the network. Panel
d confirms that there is no significant change in the number of emails sent
along local bridges. Though this may seem to be in contrast to the results
found by Microsoft, it’s important to note that they primarily studied
communication media other than email.

The Jaccard coefficient of an edge $(u,v)$ is defined as $\frac{\|N(u)\cap
N(v)\|}{\|N(u)\cup N(v)\|}$ where $N(u)$ is the set of neighbors of $u$ in the
email network, and similarly for $v$. Local bridges are edges with Jaccard
coefficient 0. Supplementary Figures 29-32 show the number of edges with
different Jaccard coefficients in daily, weekly, biweekly, and monthly email
networks. The most noticeable prolonged changes in number of edges occurs in
sets of edges with low Jaccard coefficient.

Finally, another commonly considered dimension of tie strength is frequency of
contact. After computing the number of days on which each tie appears in the
network and obtaining a distribution of tie frequency, we choose four
different frequency cutoff thresholds to distinguish between “strong” and
“weak” ties: the 50th, 75th, 90th, and 95th percentile of the frequency
distribution. Supplementary Figure 33 shows the predicted percentage of weak
ties in the email network on each day for these four cutoffs, and we see that
by defining “weak ties” to be ties which occur with frequency below the 95th
percentile (ties which occur on less than 12.6% of the dates in our data) of
the frequency distribution, we recover a long-lasting drop similar to the drop
in local bridges described in the main manuscript. Supplementary Figures 34
and 35 show that infrequent ties and new infrequent ties respond the same way
to a lack of co-location as local bridges, making our results robust to the
definition of weak tie.

Figure 28. Other measures of topological holes in networks. a, The number of
local bridges each day, excluding those local bridges which are isolated dyads
(effect: -60.46, $p<.001$, 95% CI: [-70.78, -50.3]). b, The total persistence
of all generators of the first persistent homology with finite death (effect:
71.98, $p<.001$, 95% CI: [44.07, 99.4]). c, The average node constraingt, in
the sense of Burt (effect: 0.0, $p=.5$, 95% CI: [-0.13, 0.12]). d, The average
width of each local bridge each day measured as the total number of emails
sent along the bridge (effect: -0.03, $p=0.5$, 95% CI: [-0.85, 0.75]).
Posterior predictive intervals computed using Bayesian structural time series.
$n_{\mathrm{pre}}=8$ weeks, $n_{\mathrm{post}}=72$ weeks for all panels.
Fitted values/intervals use the mean as the measure of central tendency.
Figure 29. Number of edges with fixed Jaccard coefficients in daily email
networks. The Jaccard coefficient of an edge is defined as the intersection
over union of the neighbor sets of each edge. Data reported as edge counts
within fixed bins. Figure 30. Number of edges with fixed Jaccard coefficients
in weekly email networks. The Jaccard coefficient of an edge is defined as the
intersection over union of the neighbor sets of each edge. Data reported as
edge counts within fixed bins.  Figure 31. Number of edges with fixed Jaccard
coefficients in biweekly email networks. The Jaccard coefficient of an edge is
defined as the intersection over union of the neighbor sets of each edge. Data
reported as edge counts within fixed bins. Figure 32. Number of edges with
fixed Jaccard coefficients in monthly email networks. The Jaccard coefficient
of an edge is defined as the intersection over union of the neighbor sets of
each edge. Data reported as edge counts within fixed bins. Figure 33.
Infrequent edge counts measured for different frequency thresholds. a, The
number of infrequent edges at threshold 0.5 each day (effect: -0.0036,
$p<.01$, 95% PPI: [-0.0053, -0.0019]). b, The number of infrequent edges at
threshold 0.75 each day (effect: -0.016, $p=.18$, 95% PPI: [-0.049, 0.018]).
c, The number of infrequent edges at threshold 0.9 each day (effect: -0.02,
$p<.001$, 95% PPI: [-0.0230, -0.0216]). d, The number of infrequent edges at
threshold 0.95 each day (effect: -0.034, $p=0.5$, 95% PPI: [-0.041, -0.025]).
Posterior predictive intervals computed using Bayesian structural time series.
$n_{\mathrm{pre}}=8$ weeks, $n_{\mathrm{post}}=72$ weeks for all panels.
Fitted values/intervals use the mean as the measure of central tendency.
Figure 34.  Changes in infrequent edges in the MIT email network after the
shift to remote work. ***$:p<.001$, **$:.001\leq p<.01$, *$:.01\leq p<.05$. a,
A drop of 0.0175 in the percentage of infrequent edges after March 23, 2020
($p<.001$, 95% CI: [-0.049 -0.033]). There is a drop of 0.041 in the mean
percentage of new (not previously seen) infrequent edges appearing each
weekday after March 23, 2020 ($p<.001$, 95% CI: [-0.023 -0.012]). Statistics
represent the results of a two-sided $z$ test corresponding to a local
polynomial regression discontinuity design ($n_{\mathrm{pre}}=42$ days,
$n_{\mathrm{post}}=188$ days). There is a cumulative drop of .034 in the
percentage of infrequent edges throughout an entire year ($p<.001$, 95% PPI:
[-.042, -.026]) and a non-significant drop of 2930 new weak ties ($p=.36$, 95%
PPI: [-.064, .048]). Posterior predictive intervals computed using Bayesian
structural time series ($n_{\mathrm{pre}}=8$ weeks, $n_{\mathrm{post}}=72$
weeks). Fitted values/intervals use the mean as the measure of central
tendency. Figure 35. Formation of new infrequent edges stratified by distance
in the MIT email network. a The estimated effect between researchers in the
same lab. b The estimated effect between researchers in distinct labs within
150 meters. c The estimated effect between researchers in distinct labs
between 150 and 650 meters. d The estimated effect between researchers in
distinct labs further than 650 meters. Shaded regions represent 95% posterior
predictive intervals computed using Bayesian structural time series with a
synthetic counterfactual constructed from weekend data $n_{\mathrm{pre}}=8$
weeks, $n_{\mathrm{post}}=72$ weeks for all panels. Fitted values/intervals
use the mean as the measure of central tendency.

##  17\. Preprocessing error

Recall from the Methods section that we estimate the number of emails between
each pair of users on each day from randomized, aggregated data. In order to
quantify the quality of our estimations, we re-aggregate our individual
estimates and compare to each randomization of the aggregated data. The error
observed after aggregation is due to our use of non-negative matrix
factorization (NMF) in order to obtain approximate sparse solutions to the
number of emails sent per user when there is not a unique integer solution. Of
those pairs of people who possibly sent a non-zero number of emails on a given
day, we can and do solve for the number of emails exactly in approximately 66%
of cases; if we include the pairs which send no emails to one another in at
least one random aggregation (and hence send zero emails), we exactly solve
for the number of emails between 99.9% of users. When a system is
underdetermined, NMF is substantially faster than solving the constrained
system of linear Diophantine equations necessary to produce a sparse, non-
negative integer solution. Furthermore, any exact integer solution to an
underdetermined system of linear equations would itself be an approximation to
the true number of emails sent between users. This is a source of unmeasurable
error which is a consequence of working with aggregated data. Supplementary
Figure 36, panels b and c, show that of the tens of thousands of emails sent
each day, our errors are at most in the hundreds. Panel a shows that, per edge
in the aggregated network, we never exceed more than 3% error.

Figure 36.  Error quantification for estimation of email edge weights using
non-negative matrix factorization. a, After re-aggregating the data, the mean
error in emails sent between each group of people averaged across the 10
randomizations. b, After re-aggregating the data, the total error in emails
sent per edge. c The total number of emails sent in the data on each day
before any preprocessing. d After re-aggregating the data, the cardinality of
the symmetric difference of estimated and actual edge sets.

##  18\. Statistics tables

Here we report statistics tables containing test results for each generalized
least squares regression. All null hypothesis tests are two-sided.

Dep. Variable: | Intra-unit connections | R-squared: | -0.000
---|---|---|---
Model: | GLSAR | Adj. R-squared: | -0.000
Method: | Least Squares | F-statistic: | 1.898
No. Observations: | 16 | AIC: | -6.145
Df Residuals: | 15 | BIC: | -5.372
Df Model: | 0 |  |
Covariance Type: | HC3 |  |

| coef | std err | z | P$>\|$z$\|$ | [0.025 | 0.975]
---|---|---|---|---|---|---
Intercept | 0.0345 | 0.025 | 1.378 | 0.168 | -0.015 | 0.083
c | 0.0345 | 0.025 | 1.378 | 0.168 | -0.015 | 0.083

Table 1. Statistics for intra-unit connections in Figure 1 panel a. The
difference in each pair of days is represented by the variable c.

Dep. Variable: | Inter-unit connections | R-squared: | 0.000
---|---|---|---
Model: | GLSAR | Adj. R-squared: | 0.000
Method: | Least Squares | F-statistic: | 2.877
No. Observations: | 16 | AIC: | -10.90
Df Residuals: | 15 | BIC: | -10.13
Df Model: | 0 |  |
Covariance Type: | HC3 |  |

| coef | std err | z | P$>\|$z$\|$ | [0.025 | 0.975]
---|---|---|---|---|---|---
Intercept | 0.0366 | 0.022 | 1.696 | 0.090 | -0.006 | 0.079
c | 0.0366 | 0.022 | 1.696 | 0.090 | -0.006 | 0.079

Table 2. Statistics for inter-unit connections in Figure 1 panel a. The
difference in each pair of days is represented by the variable c.

Dep. Variable: | Num connected comps. | R-squared: | 0.000
---|---|---|---
Model: | GLSAR | Adj. R-squared: | 0.000
Method: | Least Squares | F-statistic: | 8.333
No. Observations: | 16 | AIC: | -24.69
Df Residuals: | 15 | BIC: | -23.92
Df Model: | 0 |  |
Covariance Type: | HC3 |  |

| coef | std err | z | P$>\|$z$\|$ | [0.025 | 0.975]
---|---|---|---|---|---|---
Intercept | -0.0404 | 0.014 | -2.887 | 0.004 | -0.068 | -0.013
c | -0.0404 | 0.014 | -2.887 | 0.004 | -0.068 | -0.013

Table 3. Statistics for the number of connected components in Figure 1 panel
a. The difference in each pair of days is represented by the variable c.

Dep. Variable: | Giant comp. size | R-squared: | 0.000
---|---|---|---
Model: | GLSAR | Adj. R-squared: | 0.000
Method: | Least Squares | F-statistic: | 2.387
No. Observations: | 16 | AIC: | -4.477
Df Residuals: | 15 | BIC: | -3.704
Df Model: | 0 |  |
Covariance Type: | HC3 |  |

| coef | std err | z | P$>\|$z$\|$ | [0.025 | 0.975]
---|---|---|---|---|---|---
Intercept | 0.0407 | 0.026 | 1.545 | 0.122 | -0.011 | 0.092
c | 0.0407 | 0.026 | 1.545 | 0.122 | -0.011 | 0.092

Table 4. Statistics for giant component size in Figure 1 panel a. The
difference in each pair of days is represented by the variable c.

Dep. Variable: | Num. weak ties | R-squared: | 0.054
---|---|---|---
Model: | GLSAR | Adj. R-squared: | 0.045
Method: | Least Squares | F-statistic: | 7.481
No. Observations: | 225 | AIC: | 2685.
Df Residuals: | 222 | BIC: | 2696.
Df Model: | 2 |  |
Covariance Type: | HC3 |  |

| coef | std err | z | P$>\|$z$\|$ | [0.025 | 0.975]
---|---|---|---|---|---|---
Intercept | 891.5135 | 14.142 | 63.038 | 0.000 | 863.795 | 919.232
policy | -55.7007 | 18.330 | -3.039 | 0.002 | -91.627 | -19.774
policy:time | -0.0424 | 0.155 | -0.273 | 0.785 | -0.347 | 0.262

Table 5. Statistics for Figure 2 panel a. This table represents the output of
a local polynomial regression discontinuity design on the number of weak ties
with cutoff on March 23, 2020. The treatment effect is measured by the policy
variable.

Dep. Variable: | Num. new weak ties | R-squared: | 0.897
---|---|---|---
Model: | GLSAR | Adj. R-squared: | 0.895
Method: | Least Squares | F-statistic: | 321.2
No. Observations: | 385 | AIC: | 3195.
Df Residuals: | 379 | BIC: | 3219.
Df Model: | 5 |  |
Covariance Type: | HC3 |  |

| coef | std err | z | P$>\|$z$\|$ | [0.025 | 0.975]
---|---|---|---|---|---|---
Intercept | 128.8843 | 8.421 | 15.306 | 0.000 | 112.380 | 145.389
time | -0.3442 | 1.374 | -0.251 | 0.802 | -3.037 | 2.349
time2 | 0.0550 | 0.036 | 1.538 | 0.124 | -0.015 | 0.125
policy | -46.3961 | 8.661 | -5.357 | 0.000 | -63.371 | -29.421
policy:time | 0.0975 | 1.374 | 0.071 | 0.943 | -2.596 | 2.791
policy:time2 | -0.0547 | 0.036 | -1.529 | 0.126 | -0.125 | 0.015

Table 6. Statistics for Figure 2 panel b. This table represents the output of
a local polynomial regression discontinuity design on the number of new weak
ties with cutoff on March 23, 2020. The treatment effect is measured by the
policy variable.

Dep. Variable: | Num weak ties | R-squared: | 0.268
---|---|---|---
Model: | GLSAR | Adj. R-squared: | 0.240
Method: | Least Squares | F-statistic: | 14.30
No. Observations: | 55 | AIC: | 624.9
Df Residuals: | 52 | BIC: | 630.9
Df Model: | 2 |  |
Covariance Type: | HC3 |  |

| coef | std err | z | P$>\|$z$\|$ | [0.025 | 0.975]
---|---|---|---|---|---|---
Intercept | -177.6800 | 9.296 | -19.114 | 0.000 | -195.899 | -159.461
policy | 67.5030 | 29.471 | 2.290 | 0.022 | 9.740 | 125.266
policy:time | 0.8652 | 2.439 | 0.355 | 0.723 | -3.915 | 5.645

Table 7. Statistics for Figure 4 panel c. This table represents the output of
a local polynomial regression discontinuity design on the number of weak ties
with cutoff on September 8, 2021. The treatment effect is measured by the
policy variable.

Dep. Variable: | Simulated weak ties | R-squared: | 0.346
---|---|---|---
Model: | GLSAR | Adj. R-squared: | 0.341
Method: | Least Squares | F-statistic: | 88.49
No. Observations: | 285 | AIC: | 2730.
Df Residuals: | 282 | BIC: | 2741.
Df Model: | 2 |  |
Covariance Type: | HC3 |  |

| coef | std err | z | P$>\|$z$\|$ | [0.025 | 0.975]
---|---|---|---|---|---|---
Intercept | 1220.8649 | 4.313 | 283.042 | 0.000 | 1212.411 | 1229.319
policy | -65.2128 | 5.560 | -11.730 | 0.000 | -76.109 | -54.316
policy:time | 0.0255 | 0.026 | 0.982 | 0.326 | -0.025 | 0.076

Table 8. Statistics for Figure 6 panel a. This table represents the output of
a local polynomial regression discontinuity design on the number of weak ties
with cutoff on March 23, 2020. The treatment effect is measured by the policy
variable.

Dep. Variable: | Simulated weak ties | R-squared: | 0.392
---|---|---|---
Model: | GLSAR | Adj. R-squared: | 0.388
Method: | Least Squares | F-statistic: | 107.7
No. Observations: | 162 | AIC: | 1536.
Df Residuals: | 160 | BIC: | 1542.
Df Model: | 1 |  |
Covariance Type: | HC3 |  |

| coef | std err | z | P$>\|$z$\|$ | [0.025 | 0.975]
---|---|---|---|---|---|---
Intercept | 1160.2661 | 2.519 | 460.608 | 0.000 | 1155.329 | 1165.203
policy | 51.8918 | 5.000 | 10.377 | 0.000 | 42.091 | 61.693

Table 9. Statistics for Figure 6 panel a. This table represents the output of
a local polynomial regression discontinuity design on the number of weak ties
with cutoff on September 8, 2021. The treatment effect is measured by the
policy variable.

Dep. Variable: | Simulated new weak ties | R-squared: | 0.990
---|---|---|---
Model: | GLSAR | Adj. R-squared: | 0.989
Method: | Least Squares | F-statistic: | 716.8
No. Observations: | 275 | AIC: | 1760.
Df Residuals: | 269 | BIC: | 1781.
Df Model: | 5 |  |
Covariance Type: | HC3 |  |

| coef | std err | z | P$>\|$z$\|$ | [0.025 | 0.975]
---|---|---|---|---|---|---
Intercept | 67.5013 | 6.819 | 9.899 | 0.000 | 54.137 | 80.866
time | 2.6929 | 0.892 | 3.018 | 0.003 | 0.944 | 4.441
time2 | 0.2962 | 0.025 | 11.849 | 0.000 | 0.247 | 0.345
policy | -47.0380 | 6.902 | -6.815 | 0.000 | -60.566 | -33.510
policy:time | -2.7958 | 0.892 | -3.133 | 0.002 | -4.545 | -1.047
policy:time2 | -0.2959 | 0.025 | -11.837 | 0.000 | -0.345 | -0.247

Table 10. Statistics for Figure 6 panel b. This table represents the output of
a local polynomial regression discontinuity design on the number of new weak
ties with cutoff on March 23, 2020. The treatment effect is measured by the
policy variable.