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Social network-based distancing strategies to ﬂatten the

COVID-19 curve in a post-lockdown world

Per Block‡, Marion Hoffman†, Isabel J. Raabe∗, Jennifer Beam Dowd‡

Charles Rahal‡,§, Ridhi Kashyap‡,§,¶, Melinda C. Mills‡,§

‡Leverhulme Centre for Demographic Science, Department of Sociology, University of Oxford

†Department of Humanities, Social and Political Sciences, ETH Zurich

∗Institute of Sociology, University of Zurich

§Nuﬃeld College, University of Oxford, Oxford, UK

¶School of Anthropology and Museum Ethnography, University of Oxford, Oxford, UK

25th May, 2020

Abstract: Social distancing and isolation have been introduced widely to counter the COVID-

19 pandemic. However, more moderate contact reduction policies become desirable owing to ad-

verse social, psychological, and economic consequences of a complete or near-complete lockdown.

Adopting a social network approach, we evaluate the eﬀectiveness of three targeted distancing

strategies designed to ‘keep the curve ﬂat’ and aid compliance in a post-lockdown world. These

are limiting interaction to a few repeated contacts, seeking similarity across contacts, and strength-

ening communities via triadic strategies. We simulate stochastic infection curves that incorporate

core elements from infection models, ideal-type social network models, and statistical relational

event models. We demonstrate that strategic reduction of contact can strongly increase the eﬃ-

ciency of social distancing measures, introducing the possibility of allowing some social contact

while keeping risks low. This approach provides nuanced insights to policy makers for eﬀective

social distancing that can mitigate negative consequences of social isolation.

Keywords:COVID-19, social networks, stochastic infection curves, statistical relational events

1. INTRODUCTION

The non-pharmaceutical intervention of ‘social distancing’ is a central policy to reduce the spread of

COVID-19, largely by maintaining physical distance and reducing social interactions (Glass et al.,

2006). The aim is to slow transmission and the growth rate of infections to avoid overburdening

health-care systems, widely known as ‘ﬂattening the curve’ (Roberts,2020). Social distancing

includes bans on public events, the closure of schools, universities and non-essential workplaces,

limiting public transportation, travel and movement restrictions, and urging citizens to limit social

interactions.

The majority of existing research on mitigating inﬂuenza pandemics focus on the eﬀectiveness

of diﬀerent individual measures, such as travel restrictions, school closures, or vaccines (Ferguson

et al.,2006;Germann et al.,2006). Few have simultaneously considered interventions and the

structure of social networks. When social networks are examined, it is generally in relation to

vaccination (Ventresca and Aleman,2013), contact tracing, or analysing the spread of the virus

For correspondence: Per Block and Melinda C. Mills, Leverhulme Centre for Demographic Science, Department of

Sociology, University of Oxford, OX1 1JD, United Kingdom. Tel: 01865 286170. Email: per.block@sociology.ox.ac.uk

and melinda.mills@nuﬃeld.ox.ac.uk. The replication ﬁles for this paper including customised functions in the statis-

tics environment R and an example script are available on Zenodo, a general-purpose open-access repository devel-

oped under the European OpenAIRE program and operated by CERN (https://zenodo.org/record/3782465).

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(Sun et al.,2020;Wu and McGoogan,2020). We outline key behavioural strategies for selective

contact reduction that every individual and organisation can adopt to maximise the beneﬁts of

limiting contact and engaging in strategic social distancing. Applying insights from social and

statistical network science, we demonstrate how changing network conﬁgurations of individuals’

contact choices and organisational routines can alter the rate and spread of the virus, by providing

guidelines to diﬀerentiate between ‘high-impact’ and ‘low-impact’ contacts for disease spread. This

can contribute to balancing public health concerns and socio-economic needs for interpersonal

interaction. We introduce and assess three strategies: contact with similar people, strengthening

contact in communities, and repeatedly interacting with the same people.

Conclusions regarding the eﬀectiveness of non-pharmaceutical public health interventions have

often been made on the basis of on ‘expert recommendations’ rather than scientiﬁc evidence (Bell,

2006). During previous outbreaks (e.g. SARS-CoV), social distancing measures such as workplace

closures, limiting public gatherings, and travel restrictions were implemented. Cancelling public

gatherings and long-distance travel restrictions appears to decrease transmission and morbidity

rates (Aledort et al.,2007). There is mixed evidence regarding the eﬀectiveness of school closures

on respiratory infections, possibly because of the timing of school closures, or since this aﬀects

only on school-aged children (Jackson et al.,2013).

There has been considerably less research on the eﬀectiveness of other types of social distancing

measures, such as strategies based on individual’s knowledge of their social surrounding. Existing

research has demonstrated that interventions are only eﬀective and feasible when the public deems

them acceptable (Aledort et al.,2007). Our approach recognises the social, psychological, and

economic cost of – and potential compliance fatigue with – complete isolation (Morse et al.,2006).

Fully quarantining non-infected, psychologically vulnerable individuals over prolonged periods can

have severe mental health consequences. Many facets of economic and social life require some

amount of person-to-person contact. Compliance with recommendations to strategically reduce

contact is more favourable than compliance with complete isolation and, thus, can keep the curve

ﬂat in the long run. We therefore propose a novel approach that assesses the eﬀectiveness of

network adaptations that rely on less conﬁnement and allow some degree of social contact while

still ‘ﬂattening the curve’.

Flattening the (infection) curve represents a decrease in the number of infected individuals at

the height of the epidemic, with the incidence of cases distributed over a longer time horizon

(Roberts,2020). This is largely achieved by reducing the reproduction number (R), which is how

many individuals are infected by each carrier. Social distancing policies are implicitly designed to

achieve this by limiting the amount of social contact between individuals. By introducing a social

network approach, we propose that a decrease in R can simultaneously be achieved by managing

the network structure of interpersonal contact. From a social network perspective, the shape of the

infection curve is closely related to the concept of network distance or path lengths (Wasserman

et al.,1994), which indicates the number of network steps needed to connect two nodes. Popularised

examples of network distance include the ‘six degrees of separation’ phenomenon (Milgram,1967),

which posits that any two people are connected through at most ﬁve acquaintances.

The relation between infection curves and network distance can be illustrated with a simple

network infection model (Figure 1). Panels A and C depict two networks with diﬀerent path

lengths, each with one hypothetically infected COVID-19 seed node (purple square). At each time

step, the disease spreads from infected nodes to every node to which they are connected; thus, in

the ﬁrst step the disease spreads from the seed node to its direct neighbours. In the second step, it

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Numberofnewlyinfectedindividualspertimestep

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Figure 1: Two example networks A and C. Both networks have the same number of nodes (individuals)

and ties (social interactions) but diﬀerent structures which imply diﬀerent infection curves (B and D). Bold

ties highlight the shortest infection path from the infection source to the last infected individual in the

respective networks. Network node colour indicates at which step a node is infected and maps onto colours

of histogram bars.

spreads to their neighbours, who are at network distance 2 from the seed node, and so forth. Over

time, the virus moves along network ties until all nodes are infected. The example shows that the

network distance of a node from the infection source (indicated by node colour in Figure 1A and

C) is identical to the number of time-steps until the virus reaches it. The distribution of network

distances to the source thus directly maps onto the curve of new infections (Figure 1B and D).

In our example, both networks have the same number of nodes (individuals) and edges (inter-

actions); however, the network depicted in panel C has a much ﬂatter infection curve than the

network in panel A, even though all nodes are eventually infected in both cases. This is because the

latter network has longer path lengths than the former one – or in other words – more network dis-

tance between the individuals due to a diﬀering structure of interaction, despite the same absolute

contact prevalence. Thus, when adopting a network perspective, ﬂattening the curve is equivalent

to increasing the path length from an infected individual to all others, which can be achieved

by restructuring contact (besides the generally proposed reduction of contact). Consequently, one

aim of social distancing should be increasing the average network distance between individuals by

smartly manipulating the structure of interactions. Our illustration shows a viable path to keep the

4

COVID-19 curve ﬂat while allowing some social interaction: we must devise interaction strategies

that make real-life networks look more like network C, and less like network A.

We propose a series of strategies for how individuals can make local decisions to achieve this

goal. Understanding which types of strategies of targeted contact reduction and social distancing

are more eﬃcient in increasing path lengths and ﬂattening the curve can inform how to shift from

short-term (complete lockdown) to long-term management of COVID-19 contagion processes. The

contact reduction strategies we propose are based on insights of how items ﬂow through networks,

such as diseases, memes, information, or ideas (Watts et al.,2006;Podolny,2001;Borgatti,2005;

Centola,2010). Such spread is generally hampered when networks consist of densely connected

groups with few connections in-between, such as individuals who live in isolated villages scattered

over sparse rural areas (Watts,1999). In contrast, contacts that bridge large distances are related to

short paths and rapid spread. When commuters travel between these isolated villages, for instance,

network distances decrease substantially (Milgram,1967;Centola,2010). Using this knowledge,

we can avoid rapid contagion by encouraging social distancing strategies that increase clustering

and reduce network short-cuts to reap the largest beneﬁt of reducing social contact and limiting

disease spread to a minimum. We propose three strategies aimed at increasing network clustering

and eliminating short-cuts.

While more realistic examples of the proposed strategies are simulated in the next section, we

ﬁrst outline the underlying principles of the model in Figure 2. Panel A depicts a network in which

densely connected communities are bridged by random, long-range ties. This type of network is

commonly known as a ‘small world network’ (Centola,2010). It is widely used in simulations, as

it represents core features of real-world contact networks, in particular social clustering combined

with short network distances, making it particularly useful for our illustration (Milgram,1967).

Within clusters, individuals are similar to each other, indicated by their node colour, and live in

the same neighbourhood, indicated by node location. The further away two clusters are in the

ﬁgure, the further they live from each other and the more dissimilar their members. Panels A to

D illustrate the successive, targeted contact reduction strategies, while the bar-graph depicts the

distribution of distances of all individuals from one of the two highlighted infection sources.

Strategy 1: ‘Seek similarity’ strategy: Reduce geographic and socio-demographic diﬀerence to

contact partners (A to B in Fig. 2). In the ﬁrst strategy, individuals choose their contact partners

based on their individual characteristics. Generally, individuals tend to have contact others who

share common attributes, such as those in the same neighbourhood (geographical), or of similar in-

come or socio-demographic characteristics such as age (Feld,1981;Rivera et al.,2010;McPherson

et al.,2001). The tendency to interact with similar others is called ‘homophily’ in the sociolog-

ical network literature (Rivera et al.,2010) and is a ubiquitous and well-established feature of

social networks (thus, we use ‘seek similarity’ strategy and ‘homophily’ strategy interchangeably).

Because we are mostly connected to similar others, contact with dissimilar individuals tends to

bridge to more distant communities. Restricting one’s contact to those most similar helps limit

network bridges that substantially reduce network path lengths. This entails choosing to inter-

act with those geographically proximate (e.g., living in the same neighbourhood), or individuals

with similar characteristics (e.g., age). Panel B in Figure 2shows the network structure after the

implementation of this strategy of tie reduction. The associated bar-graph illustrates that follow-

ing this network-based intervention, a substantial number of nodes are at a larger distance from

the infection source. This strategy will be successful when the characteristic or variable which

determines the communities can take on a variety of diﬀerent (categorical or continuous) values

5

A B C

DA B C D

Proportion,of,nodes,,,

with,distance

disconnected,(safe)

4+

2,or,3

1

i

Figure 2: Example networks that result from the successive tie reduction strategies. Node

colour represents an individual characteristic, where similarity in node colour represents similarity in this

characteristic. Node placement represents geographic location of residence. A: initial small world network;

B: removing ties to dissimilar others that live far away; C: removing non-embedded ties that are not part

of triads or 4-cycles; D: repeating rather than extending contact. Bar graphs show network distances from

the infection sources, highlighted in yellow, for the diﬀerent scenarios.

for diﬀerent individuals, thereby promoting the formation of small communities. A broader split,

such as along gender or ethnic lines does not promise measurable success but will instead likely

exacerbate the negative consequences of distancing measures. This strategy is supported by epi-

demiological modelling which suggests that co-residence and mixing of individuals from diﬀerent

ages strongly increases the spread of infectious disease, such as COVID-19 (Pellis et al.,2020).

Providing a concrete example, if people only interact with others in a 3-block radius (increase geo-

graphic similarity), more than 30 transmission events would be necessary for a virus to travel 100

blocks. Workplaces where many individuals come together could, for instance, implement routines

to decrease contact between groups from diﬀerent geographic areas or age-groups.

Strategy 2: ‘Strengthen triadic communities’ clustering strategy: Increase triadic clustering

among contact partners (B to C in Fig. 2). For the second strategy, individuals must consider with

whom their contact partners usually interact. A common feature of contact networks is ‘triadic

closure’, referring to the fact that contact partners of an individual tend to be connected themselves

(Feld,1981;Granovetter,1973;Goodreau et al.,2009). Tie embedding in triads is a particularly

useful topology for containing epidemic outbreaks. Consider a closed triad of individuals i,j, and

h. When iinfects jand h, the connection between jand hdoes not contribute to further disease

spread: it is a ‘redundant’ contact (Burt,1995). When comparing networks with an identical

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number of connections, networks with more redundant ties tend to have longer path lengths.

Accordingly, when removing contact to others, one should prioritize removing ties not embedded

in triads, since these ties generally decrease path lengths. In practice, this means that physical

contact should be curtailed with people who are not also connected to one’s usual other social

contacts. Panel C in Figure 2illustrates the structure if ties that are not part of closed triads or

4-cycles are removed. In this ideal-type example, this intervention not only further reduces the

network distance of many nodes from the infection sources, but also creates isolated communities

or that cannot be infected by the virus.

Strategy 3: ‘Repeat contact and build micro-communities’ strategy: Repeated contact to same

others, rather than changing interaction partners (C to D in Fig. 2). For the third strategy, individ-

uals need to consider who they want to regularly interact with and, over time, restrict interaction

to those people; this reduces the number of contact partners rather than number of interactions,

which is particularly important when contact is necessary for psychological well-being. This strat-

egy of limiting contact to very few others with repeated interactions is in the spirit of a social

contract with others to create micro-communities to only interact within the same group delineated

by common agreement. Although this requires coordination, micro-communities would be diﬃcult

for a virus to penetrate, or – importantly – if the infection is contracted by one contact, for the

virus to spread further. Another implication of this strategy includes the repetition of interaction

with others that overlap across more than one contact group. For example, meeting co-workers

outside of work for socializing will have less of an impact on the virus spread relative to a sepa-

rate group of friends, since a potential infection path already exists. Having tight and consistent

networks of medical or community-based carers for those more vulnerable to COVID-19 (elderly,

pre-existing conditions) limits the transmission chain. Organisations can leverage this strategy by

structuring staggered and grouped shifts so that individuals have repeated physical contact with

a limited group rather than dispersing throughout an organisation. Panel D in Figure 2illustrates

the resulting network structure.

Strategy 2 and 3 are similar in that they build on pre-existing network structures. However,

their diﬀerence lies in the determinants of individual interaction. Strategy 2 relies on a stable and

established network structure of durable relations: who are members of my usual ‘groups’ (e.g.,

friends, family, co-workers) and which pairs of individuals among my usual contacts interacts with

each other, too? Strategy 3 relies on a strategic decision to form most convenient and eﬀective

“interaction bubbles” and repeat contact to them over time. In this sense, strategy 2 is easier to

implement, since individuals are able to shape their contacts themselves, while strategy 3 requires

coordinated action of everyone involved in a given “bubble”. Until now, we have illustrated our

strategies with an intuitive but stylized model of epidemic spread. We now demonstrate how our

three contact strategies impact infection curves using more formal stochastic infection models that

incorporate core elements from infection models, ideal-type network models and statistical rela-

tional event models. These strategies are compared to a baseline (null) model that represents how

the COVID-19 infection would spread if there was unrestricted contact (i.e., no social distancing).

First, our model draws from classical disease modelling (Kermack and McKendrick,1927;An-

derson and May,1992), in which individuals (actors) can be in four states: susceptible, exposed

(infected but not yet infectious), infectious, and recovered (no longer susceptible to infection).

Most actors begin in the susceptible state, while qrandom actors are in the infectious state (one

per thousand in our simulations). This can represent, for example, the post-lockdown scenario in

which only a few cases of COVID-19 remain in the population; however, variation of qmight also

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be used to determine the levels at which a lock-down can be eased. During the simulation, suscep-

tible actors can transition to the exposed state by having contact with infectious others (contact

partners will be called ‘alters’ from here on). Whether contact between a susceptible actor and

an infectious alter results in contagion is determined probabilistically. A designated time after be-

coming exposed, actors become infectious themselves, and later move to the recovered state after

another ﬁxed amount of time.

Second, as in many previous modelling eﬀorts of the dynamics of epidemics such as inﬂuenza,

we do not assume homogeneous contact probabilities in an aﬀected population but rather impose

a network structure that limits contact opportunities between actors (Newman,2002;Halloran

et al.,2008;Salath´e et al.,2010). This network represents the typical contact people had in a pre-

COVID-19 world. The networks we generate stochastically for our model follow fairly standard

ideal-type network generating approaches. Representing place of residence, actors are assumed

to have a geographic location, determined by coordinates in a two-dimensional space. They are

members of groups, such as households, institutions like schools or workplaces, and have individual

attributes, such as age, education, or income. Network ties are generated so that actors have some

connections to geographically close alters, some ties to members of the same groups (representing

e.g., co-workers), some ties to alters with similar attributes (e.g., similar age), and, ﬁnally, some

ties to random alters in the population. The generated networks represent the structure of alters

that an actor can possibly interact with. They represent the members of their so-called ‘social

circles’ (Watts,1999;Feld,1981;Block,2018) with whom they interact in their normal, pre-

COVID life (including family, friends, schoolmates or co-workers). The exact algorithms which

deﬁne the networks are described in the Methods section.

In the third component of the model, actors in the network interact at discrete times with alters

with which they have a connection in the underlying network, or in other words, someone they

meet from their usual social contacts. This represents the actual contact people have in their lives

during which the disease can be transmitted from infectious actors to susceptible alters. Notably, in

contrast to other modelling approaches, we do not assume that actors interact with alters in their

personal network with uniform probability (i.e. at random), but, rather, that they are purposeful

actors who make strategic choices about interactions. These strategic choices are at the core of our

advice for policy interventions, where individuals can strategically increase the eﬃciency of social

distancing. In our model, all choices are stochastic; strategies increase the likelihood of interacting

with speciﬁc alters but are not deterministic. The exact formulation of with whom to interact

follows a multinomial logit model to choose among possible interaction partners, given by the

network structure. This type of model has previously been used in network evolution (Snijders,

2001) and relational event models (Butts,2008;Stadtfeld and Block,2017).

Our simulations explore the three interaction strategies we propose. First, in our ‘seek similarity’

strategy, actors choose to interact predominantly with others that are similar to themselves based

on one or several speciﬁed attributes used at the network generation stage. Second, actors can adopt

our ‘strengthen triadic community’ or triadic strategy and choose to mostly interact with alters

that have common connections in the underlying network. Third, adopting our ‘repeat contact’

strategy, actors can base their choices on whom they have interacted with in their previous contacts,

both as sender and receiver of an interaction. In each case, a separate statistical parameter in the

multinomial model determines the probabilities of interaction partners based on the: (i) similarity

of alters, (ii) number of common contacts the actor and alter have; and, (iii) repeat interaction

with one of the last jcontact partners (see Methods). In our analyses, these three strategies are

8

compared to a baseline case that mirrors the simple reduction of contact in which individuals have

the same amount of interactions but choose randomly amongst their network contacts (a na¨ıve

contact reduction strategy) and a null model that represents unbridled contact without any social

distancing. To make the comparison of interaction strategies independent of the arbitrary size

of statistical parameters, we empirically calibrate parameters so that the average entropy in the

probability distribution that represents the likelihood of diﬀerent interaction choices is identical

for all strategies, as documented in our Methods section (Snijders,2004).

Following an initial analysis that represents a benchmark scenario of our disease model, we

present a series of variations in modelling parameters that explore alternative scenarios and ensure

our main conclusions are independent of user-deﬁned parameters and arbitrary modelling choices.

Variations are fully described in the methods section and include: (i) diﬀerent operationalisations

of homophily; (ii) the eﬀect of employing mixed strategies; (iii) number of actors in the simulation;

(iv) varying the underlying network structure in the simulations; (v) length of the interval in which

actors are exposed relative to the time they are infectious; and (vi) the infectiousness of the virus.

2. RESULTS

The average outcome of the benchmark scenario is presented in Figure 3. The x-axis represents

time as measured in simulation steps per actor and the y-axis the number of individuals infected

at this time step out of a total population of 2,000. Curves are averaged over 40 simulation runs.

The ﬁrst scenario in blue shows a null or control interaction model in which there is no social

distancing and actors interact at random. The next four strategies all employ a 50% contact

reduction relative to the null model and compare diﬀerent contact reduction strategies. The black

line represents na¨ıve social distancing in which actors reduce contact in a random fashion. The

golden line represents the infection curve when actors employ our ﬁrst ‘seek similarity’ strategy.

The green line models our second triadic strategy of ‘strengthening communities’ and represents

the associated infection curve. Finally, the dark red line shows how infections develop when actors

employ our third strategy of ‘repeat contact’.

All three of our strategies substantially slow the spread of the virus compared to either no

intervention or simple, un-strategic social distancing. The most eﬀective is the strategic reduction

of interaction with repeated contacts. In comparison to the random contact reduction strategy, the

average infection curve delays the peak of infections by 37%, decreases the height of the peak by

60%, and results in 30% fewer infected individuals at the end of the simulation. This is marginally

more eﬃcient than the triadic strategy and the homophily strategy, in this order (delay of peak

18% and 34%, decrease in peak height of 44% and 49%, and reduction of infected individuals by

2% and 19%, for homophily and triadic strategies, respectively). Note that these metrics cannot

be interpreted as general estimates of the eﬃciency of these strategies in real-world networks.

Summarizing the sensitivity and robustness analyses carried out, strategic contact reduction has a

substantive eﬀect on ﬂattening the curve compared to simple social distancing consistently across

all scenarios. However, interesting variations occur as discussed below. Full average infection curves

and results description for all model variations are presented in the Supplementary Information.

Diﬀerent operationalisations of homophily

In the benchmark model, the ‘seek similarity’ strategy was employed on one demographic attribute.

However, in real-world social networks, individuals are homophilous on multiple attributes (Block

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Steps of interaction (per person)

Null model: full contact

50% Contact: random reduction

50% Contact: community strategy

50% Contact: similarity strategy

50% Contact: repetition strategy

Number infected (out of 2000)

Figure 3: Average infection curves. Curves compare 4 contact reduction strategies to the null model

of no social distancing. Underlying network structure includes 2000 actors and the benchmark network

characteristics described in the main text.

and Grund,2014). Furthermore, the benchmark model only uses demographic homophily, while we

previously also discuss the importance of geographic homophily. In a variation of the homophily

strategy, we show that using geographic homophily for contact reduction is highly eﬃcient, much

more than homophily based on demographic attributes (Figure S1b.). Geographic homophily ef-

fectively eliminates contacts to distant others in the network. In a further analysis, we compare

the beneﬁts of using one dimension of demographic homophily or a composite of two dimensions

that structure the network. This explores whether we should focus on interacting with persons

similar in one dedicated dimension or seek out others who are as similar as possible in multiple

dimensions. Encouragingly, the focus on one strategic dimension of homophily provides similar

outcomes to reducing overall demographic distance, meaning that homophily should be encour-

aged on the dimension that has the least adverse consequences for societal cohesion. Infection

curves are presented in Figure S1c.-d.

Employing mixed strategies

Since most individuals in a post-lockdown world need to interact across multiple social circles

(e.g., workplace, extended family), employing only one strategy might not be practical. A mix of

diﬀerent strategies could therefore be more realistic for everyday use. We tested how four possible

combinations of mixing strategies (three two-way combinations and one three-way combination)

compare to the single strategies of seeking similarity and strengthening communities. We ﬁnd that

the combined strategies are comparably as eﬀective as single strategies (Figure S2) and can be

recommended as alternatives if single strategies are not practicable in some contexts. Importantly,

each combination performs better in limiting infection spread compared to the na¨ıve contact

reduction strategy.

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Varying the number of actors in the simulation

The computational complexity of our simulation prohibits assessing disease dynamics in very large

networks (e.g. 100k+ actors), even on large distributed systems. Nevertheless, we can compare

simulations using the same local network topology as the benchmark model on networks of 500,

1000, 2000, and 4000 actors. Reassuringly, we ﬁnd no variation of the relative eﬀectiveness of the

diﬀerent interaction strategies by network size (see Figure S3). While this does not fully allow

extrapolation to very large networks, it provides initial support that disease spread under the

model could be similar within diﬀerently sized sub-regions of larger, real-world networks.

Varying the underlying network structure

The generating process of the ideal-type network that provides the opportunity structure among

individuals with whom they can interact contains multiple degrees of freedom. These include the

average number of contacts and the importance of diﬀerent foci (geography, groups, and attributes)

in structuring contact. We provide infection curves for multiple scenarios in the Supplementary

Information (Figure S4 and S5), showing that our strategies work mostly independent of the

underlying structure. A ﬁrst noteworthy ﬁnding from these simulations is that in networks with

fewer connection opportunities, all strategies have much larger beneﬁts compared to networks with

more connection opportunities (panels C and D in Figure S4). In fact, the triadic strategy does

not seem to work anymore in the scenarios with very high average connectivity in the underlying

network – most likely because of a large number of closed triangles. This shows that in communities

that have lower connectivity, spread can be contained even better. As a second ﬁnding, we see

that in the case where the underlying network is not structured by homophily, the homophily

strategy does not work (panel C in Figure S5), illustrating how the strategy relies on predetermined

structural network features.

Variation in infectiousness and the length of the exposed period

Average infection curves under conditions of diﬀerences in infectiousness of the virus, and varia-

tions of the time individuals are in the state “exposed” relative to the time of being in the state

“infectious” do not inﬂuence the relative eﬀectiveness of the diﬀerent strategies and are presented

in Figures S6 and S7 respectively.

3. DISCUSSION AND CONCLUSION

In the absence of a vaccine against COVID-19, governments and organisations face economic and

social pressures to gradually and safely open up societies but lack scientiﬁc evidence on how to best

do so. We provide clear social network-based strategies to empower individuals and organisations

to adopt safer contact patterns across multiple domains by enabling individuals to diﬀerentiate

between ‘high-impact’ and ‘low-impact’ contacts. The result may also be higher compliance since

actors will hold the power to strategically adjust their interactions without being requested to fully

isolate. Instead of blanket self-isolation policies, the emphasis on similar, community-based, and

repetitive contacts is both easy to understand and implement thus making distancing measures

more palatable over longer periods of time.

How can this be applied to real-world settings? When a ﬁrm lock-down is no longer mandated

or recommended, it is likely that individuals will want or need to interact in diﬀerent social

11

circles, e.g. at the workplace and with the wider family. Consequently, the simple one-at-a-time

strategic recommendations we analysed in most simulations might be impossible to follow strictly

by some. Our sensitivity analysis using mixed strategies addresses this concern. For example, does

mixing the three strategies still provide beneﬁts or do they counteract one another in their eﬀect?

Reassuringly, a mix of strategies still provided comparable beneﬁts to single strategies, compared

to na¨ıve contact reduction. Further modelling is needed to assess the implications in a variety of

contexts. However, when approaching this issue from a policy perspective, designing steps to ease

lockdown can be done with potential behavioural recommendations in mind: if network structures

and demographic characteristics of individuals in particular regions suggest that the use of one

strategy will yield the best results, decisions on which contact opportunities to allow – such as

opening schools or local shops – might be taken so that this strategy can be adhered to most easily.

A second discussion point concerns the potential unintended consequences of recommending our

triadic and homophilous strategies. Advocating the creation of small communities and contact to

mostly similar others can potentially result in the long-term reduction of intergroup contact and

an associated rise in inequality (DiMaggio and Garip,2012). In our simulations we explored this

concern by comparing the scenarios when homophilous ties in the underlying network are formed

following similarity in multiple dimensions, e.g. age and income. Our test of whether minimising

the overall diﬀerence in attributes of contacts versus only reducing homophily on one dimension

suggests that choosing one salient attribute can already go a long way. Thus, policymakers can make

smart choices in deciding which attribute people should pay attention to, keeping the potential

social consequences in mind. Nevertheless, understanding the long-term social consequences of

which types of public spaces are opened and, accordingly, which types of interaction are allowed

should be a major policy concern.

A number of concrete policy guidelines can be deduced from our network-based strategies. For

hospital or essential workers, risk is minimized in sustained shifts with similar composition of

employees (i.e., repeating contact) and, to distribute people into shifts based on, for example, resi-

dential proximity where possible (i.e., homophily). In workplaces and schools, staggering shifts and

lessons with diﬀerent start, end and break-times by discrete organisational units and classrooms

will keep contact in small groups and reduce contact between them. When providing private or

home care to the elderly or vulnerable, the same person should visit rather than rotating or taking

turns, but that person should be the one with fewest bridging ties to other groups and who lives

the closest (geographically). Repeated social meetings of individuals of similar ages that live alone

carry a comparatively low risk. However, in a household of ﬁve, when each person interacts with

disparate sets of friends, many short cuts are being formed that are potentially connected to a

very high risk of spreading the disease.

Simple behavioural rules can go a long way in ‘keeping the curve ﬂat’. As the pressure grows

throughout a pandemic to ease stringent lockdown measures increases to relieve social, psycho-

logical, and economic burdens, our approach provides insights to individuals, governments and

organisations about three simple strategies: interacting with similar types of people, strengthening

interaction within communities, and repeating interaction with the same people.

12

4. METHODS

Generation of stylised networks

The stylised binary networks xthat represent interaction opportunities of individuals are generated

as the composite of four sub-processes. Jointly, the sub-processes create networks that have realistic

values of local clustering, path-lengths, and homophily. All ties in the network are deﬁned as

undirected. The number of actors in the network is denoted by n.

The ﬁrst sub-process represents tie formation based on geographic proximity (Hamill and Gilbert,

2009). First, all actors in the network are randomly placed into a two-dimensional square. Second,

each actor draws the number of contacts which it forms in this subprocess dgeo,i from a uniform

distribution between dgeo,min and dgeo,max; for example, if dgeo,min = 10 and dgeo,max=20, every

actor forms a random number of ties between 10 and 20 in this sub-process. Third, the user-deﬁned

density in geographic tie-formation ggeo deﬁnes the geographic proximity of contacts drawn, so that

actor irandomly forms dgeo,i ties among those dgeo,i

ggeo that are closed in Euclidean distance from

actor i. For example, if actor iis posed to form dgeo,i =12 ties and ggeo =0.5, the actor randomly

chooses 12 out of the 24 closest alters to form a tie to. Across all simulated networks we set

ggeo=0.3. Fourth, unilateral choices (where only iselected jbut not vice versa) are symmetrised

so that a non-directed connection exists between the actors.

The second sub-process represents tie formation in organizational foci, e.g. workplaces (H´ebert-

Dufresne and Althouse,2015). First, each actor is randomly assigned to a group so that all groups

have on average mmembers. Second, each actor forms ties at random to other members within the

same groups with a probability of ggroups. For example, when m=10 and ggroups=0.5, a tie from

each actor to every alter in the same group is formed with a probability of 50%. Third, unilateral

ties are symmetrised as above.

The third sub-process represents tie-formation based on homophily, for example similarity in

age or income (Pellis et al.,2020). First, each actor is assigned an individual attribute aibetween

0 and 100 with uniform probability (the scale of aicancels later in the model). Second, for each

actor, the normalized similarity simi,j to all alters jis calculated, which is one minus the absolute

diﬀerence between aiand ajfor actor j, divided by 100 (the range of the variable), so that simi,j =1

in case iand jhave the identical value of aand simi,j=0 if they are at opposite ends of the scale.

Third, each actor draws the number of contacts it forms in this subprocess dhomo,i from a uniform

distribution between dhomo,min and dhomo,max. Fourth, each actor creates dhomo,i ties to alters j

in the networks with a probability that is proportional to (simi,j )w, where higher values of wmean

that individuals prefer more similar others. Across all reported simulations, we set w=2. Fifth,

unilateral ties are symmetrised as above.

The fourth sub-process represents haphazard ties that are not captured by any of the above

processes. Here simply zties per actor are created with respect to randomly chosen alters.

Deﬁnition of simulation model

Let the binary network xrepresent the underlying social ties between nindividuals, labeled from 1

to n. Each node iis characterized by a set of attributes ak

i(such as age or location). Our model aims

to reproduce the process of individuals interacting with some of their social connections. Similar

to the classic SIR model (Kermack and McKendrick,1927) and its SEIR extension (Anderson and

May,1992), we assume that individuals can be in four diﬀerent states: either susceptible to the

disease, exposed (infected but not yet infectious), infectious, or recovered. Infection occurs through

13

social interactions, which are modeled in a similar fashion to the Dynamic Actor-Oriented Model

(Salath´e et al.,2010) developed for relational events. More speciﬁcally, our model is comprised of

the following steps:

1 At each step of the process, one individual is picked at random and initiates an interaction

with probability πcontact.

2 An actor initiating an interaction can only pick one interaction partner. Only potential part-

ners as deﬁned by the network xcan be chosen. The decision to interact is unilateral and

depends on characteristics of the two persons through a probability model p.

3 An infectious individual infects a healthy person when they interact, who then becomes

exposed. This contagion occurs with the probability πinf ection.

4 After a ﬁxed number of steps (Texposure), an exposed individual becomes infectious.

5 After becoming infectious, recovery occurs within Trecovery steps. Once recovered, individuals

can no longer be infected.

6 The process ends once there is no longer anyone exposed or infectious.

The steps of the model are illustrated in Figure 4. One can note that the mechanics of the

infection align with previously proposed agent-based versions of the SIR and SEIR models (Chowell

et al.,2016;Siettos and Russo,2013). Together, the probabilities πcontact and πinf ection play a

similar role as the classic infectivity rate (β) in SIR models. The rate models the average number of

contacts per person (modelled here through πcontact ) and the likelihood of infection (represented

by πinfection ), however the equivalence is not direct due to the added step of the interaction

probability (p). The exposure and recovery times replace the classic exposure and recovery rates

(often traditionally denoted as σand y) in a straightforward manner. Let us turn to the deﬁnition

of the probability model p. Let Nibe the set of potential contacts, or alters, jof a given individual i

in the network x. We deﬁne for each step tof the process, Li(j, t) as the number of prior interactions

between iand an alter j, within the last Kinteractions of i. In our simulations, the number Kwas

arbitrarily set to 2 but can be easily adjusted in the replication ﬁles. For each alter j∈N, the value

s(i, j) represents the statistic driving the strategical choice of ito pick j. Speciﬁcally, we deﬁne

three diﬀerent ways depending on whether the homophily, the triadic, or the repetition strategy

is chosen (however, arbitrary other statistics can be deﬁned). The statistic shomophily accounts for

the level of similarity between iand jgiven a set of attributes, striadic corresponds to the number

of alters they share, and srepetition is the count of previous interactions within the last Kcontacts

of i. In practice, these statistics are calculated as:

shomophily(i, j)=1−qPk(ak

i−ak

j)2

max

i,j (qPk(ak

i−ak

j)2)−min

i,j (qPk(ak

i−ak

j)2)

(4.1)

striadic(i, j ) =

n

X

k=1

xi,kxj,k (4.2)

srepetition(i, j ) = Li(j, t) (4.3)

The probability for ito pick jis deﬁned as a multinomial choice probability (McFadden et al.,

1973), following the logic of previous relational event (Block,2018) and stochastic network mod-

els (Halloran et al.,2008). The intuition behind this distribution is that each potential partner

in Niis assigned an objective function value, and choosing a partner is based on these values.

14

An individual iis

randomly chosen

Yes

No

Does icontact

someone?

Individual icontacts a neighbour

with probability

A neighbour jis chosen with

probability

Yes

No

Is iinfectious

and jsuceptible?

Yes

No

Is j infectious

and i suceptible?

Individual jgets exposed with

probability

Individual i gets exposed with

probability

Time step

All individuals are

susceptible

q seed nodes

become infectious

Yes

No

Is anyone expo-

sed or infectious?

Simulation

ends

Individuals who

passed

recover

Individuals who

passed

become infectious

Time step

Figure 4:Figure 4. Flowchart of the simulation model. Squares indicate updating steps to individuals or the

entire system. Diamond shapes represent decisions that determine the subsequent step in the simulation. In

the iterative part of the model, a random individual iis chosen, to initiate an interactions with probability

πcontact . In case an interaction is initiated, a contact partner jis chosen with probability p(i→j) following

a multinomial choice model. If either interaction partner is infectious and the other is susceptible, contagion

occurs with probability πinfection . Subsequently, among all individuals in the simulation, those that are in

the exposed state for more than Texposure transition to infectious state and those that are in the infectious

state for more than Tinfection recover. These recursive steps are repeated until all individual are either in

the susceptible or recovered state. The colors red, green, and yellow relate closely to the steps in the SEIR

model, where red squares govern the transition from susceptible to exposed, the yellow square governs

the transitions from exposed to infectious, and the green square governs the transition from infectious

to recovered. The purple square represents the step at which individuals strategically chose interaction

partners to limit disease spread.

Mathematically, the objective function is an exponentiated linear function of the statistic s(i,j),

weighted by a parameter α. We further assume that individuals can reduce a certain percentage

of their interactions. Considering the probability (πcontact) of initiating an interaction in the ﬁrst

place, the relevant probability distribution becomes:

p(i→j|πcontact, α) = πcontactexp(α×s(i, j ))

Pj‘∈Niexp(α×s(i, j‘)) (4.4)

These probabilities can be loosely interpreted in terms of log-odd ratios, similarly to logit mod-

els. Given two potential partners j1and j2for which the statistic sincreases of one unit (i.e.

s(i,j2)=s(i,j1)+1), the following log ratio simpliﬁes to:

logp(i→j2|πcontact, β )

p(i→j1|πcontact, β )=α(4.5)

For example, if we use s=srepetition and αrepetition = log(2), the probability of picking one alter

present in the last contacts of iis twice as high as picking another alter who is not.

15

Calibration of model parameters

The strategy of picking a neighbor at random corresponds to the model without any statistic s, re-

ducing the probability distribution to a uniform one. For the three other strategies, the parameters

αhomophily,αtriadic, and αrepetition are adjusted to keep the models comparable. To this end, we

use the measure of explained variation for dynamic network models devised by Snijders (Snijders,

2004). This measure builds upon the Shannon entropy and can be applied to our model to assess

the degree of certainty in the choices individuals make. For a given individual iat a step t, this

measure is deﬁned as:

rH(i, t|πcontact, α) = 1 + Pj∈Nip(i→j|πcontact , α)×log2(p(i→j|πcontact, α))

log2(|Ni|)(4.6)

Intuitively, this measure equals 0 in the case of the random strategy where the probability of

picking any alter is identical. It increases whenever some outcomes are favored over others and

equals 1 if one outcome has all of the probability mass. Since the model assumes all individuals are

equally likely to initiate interactions, we can average this measure over all actors. Moreover, in the

case of the repetition strategy, the measure is time dependent. In that case, we use its expected

value over the whole process. We ﬁnally use the following aggregated measure in order to evaluate

the certainty of outcomes of a speciﬁc strategy:

RH(πcontact, α) = 1

n

n

X

i=1

E[rH(i, t)] (4.7)

For this article, we ﬁrst ﬁx the parameter αrepetition at a value of 2.5, and calculate an estimated

value ˆ

RH(πcontact,αrepetition ) of this measure. This experience-based parameter choice results in

an associated RHvalue between 0.3 and 0.5 in the diﬀerent scenario, which is realistic in terms

of size (see deﬁnition above). To compare this model to others, we then deﬁne the parameters

αhomophily and αtriadic that verify:

ˆ

RH(πcontact, αrepetition ) = RH(πcontact, αhomophily) = RH(πcontact , αtriadic ) (4.8)

using a standard optimisation algorithm. The average parameters across simulations for the

diﬀerent network scenarios are αtriadic=0.75 and αhomophily=17.6. While that latter parameter

appears large, note that the associated statistic shomophily ranges from 0 to 1, with most realised

values close to 1.

Parametrisation of the diﬀerent simulations

Unless otherwise noted, all simulations use πcontact=0.5 except for the null model, which uses

πcontact=1. In all simulations except the ones that vary the infectiousness, πinfection =0.8. Unless

otherwise noted, Texposure = 1nand Tinf ection = 4n. Given the substantial computational burden

involved in conducting the simulations, 48 repetitions were run for networks with n≤1000, with

40 for larger networks. Experiments varying Texposure and πinfection used 24 repetitions.

For the experiments that vary the structure of the underlying network and the network size,

the parameters that guide the stochastic network creation are presented in Table S1. Descriptive

statistics of these networks are presented in Table S2. The underlying networks that are used in

the other variation experiments are generated according to the parameters denoted ‘1: Baseline’ in

Table S1-S2. The four experiments that vary the time individuals are in the ‘exposed’ state before

16

becoming ‘infectious’ use values for Texposure of 0, 1n, 2n, 3nand 4n. The four experiments that

vary the infectiousness of the disease use values for πinf ection of 0.55, 0.65, 0.8, and 0.95.

The experiment that used geography as the basis of the homophily strategy was created ac-

cording to the ‘1: Baseline’ parameters but used the Euclidean distance in geographic placement

as the basis for choosing interaction partners in the homophily strategy. The two experiments on

multidimensional homophily used underlying networks created following the ‘1: Baseline’ param-

eters, with the exception that instead of one homophilous attribute, two attributes were deﬁned

and the number of ties created according to the homophily parameter was split evenly between

the two dimensions. The homophily strategy used for the simulated infection curves in the two

scenarios diﬀers in the sense that in the ﬁrst, individuals interact according to minimising the

absolute diﬀerence in both attributes. In the second scenario, only the ﬁrst attribute was used as

the basis of the homophily strategy and the second attribute was ignored.

For the experiments using mixed strategies, the probability of partner choice p(i→j) can depend

on a vector of statistics and parameters (Stadtfeld and Block,2017). The entropy based on a set

parameter vector was used to calibrate the parameter for the homophily and triadic closure strategy

as comparison cases. Parameter choices rely on experimentation to result in similar entropy values

as when using single strategies. For the mixed strategy of repetition and homophily, the parameters

were set to αhomophily=0.7 and αrepetition =1.6. For the mixed strategy of repetition and triadic

closure, the parameters were set to αtriadic=0.35 and αrepetition =1.6. For the mixed strategy of

homophily and triadic closure, the parameters were set to αhomophily=6 and αtriadic =0.35. For

the mixed strategy incorporating all three, parameters were set to αhomophily =4, αtriadic=0.3, and

αrepetition=1.2.

The simulated average infection curves for all experiments can be found in Figures S1-S7. De-

scriptive results for the simulations in terms of delay of peak, height of peak and total number

infected at the end of the simulation are presented in Table S3. Note that the descriptive statistics

in this table present the averages of characteristics of the repetitions of the simulated infection

curves, which are not the same as the characteristics of the average infection curves as presented

in the supplementary ﬁgures.

Acknowledgements: PB, JBM, CR, RK and MCM are supported by The Leverhulme Trust,

Leverhulme Centre for Demographic Science. CR is supported by the British Academy. MCM is

supported by ERC Advanced Grant CHRONO (835079). The authors would like to thank Mark

Verhagen, Valentina Rotondi, and Liliana Andriano for feedback.

Author contributions: PB, MH, and IJR conceptualised the study; PB and MH contributed

methodology, implementation and performed analyses; PB, MH, IJR, and MCM wrote initial

manuscript and provided visualisation; all authors (PB, MH, IJR, JBD, CR, RK, and MCM) dis-

cussed research design and reviewed and edited manuscript.

Competing interests: The authors declare no competing interests.

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19

SUPPLEMENTARY INFORMATION

Steps of interaction (per person) Steps of interaction (per person)

Steps of interaction (per person) Steps of interaction (per person)

Number infected (out of 1000) Number infected (out of 1000)

Number infected (out of 1000) Number infected (out of 1000)

0

20 0

40 0

60 0

0 20 4 0 60

0

20 0

40 0

60 0

0 10 2 0 30 4 0

0

20 0

40 0

60 0

0 10 2 0 30 4 0

0

20 0

40 0

60 0

0 10 2 0 30 4 0

c.

a.

d.

b.

Null

model

Random

reduction

Community

strategy

Similarity

strategy

Repetition

strategy

Figure S1:Curves compare 4 contact reduction strategies to the null model of no social distancing, as

described in the main text. (A) Reference model with standard operationalisation of homophily; (B) model

with homophily based on geographic proximity; (C) underlying network model with homophily based on

two dimensions, interaction strategy minimises the overall diﬀerence along both attributes; (D) underlying

network model with homophily based on two dimensions, interaction strategy minimises the diﬀerence

only on the ﬁrst attribute.

20

Steps of interaction (per person) Steps of interaction (per person)

Steps of interaction (per person) Steps of interaction (per person)

Number infected (out of 1000) Number infected (out of 1000)

Number infected (out of 1000) Number infected (out of 1000)

0

20 0

40 0

60 0

800

0 10 2 0 30 4 0

0

20 0

40 0

60 0

80 0

0 10 2 0 30 4 0

0

20 0

40 0

60 0

80 0

0 10 2 0 30 4 0

0

20 0

40 0

60 0

80 0

0 10 2 0 30 4 0

c.

a.

d.

b.

Null

model

Random

reduction

Community

strategy

Similarity

strategy

Mixed

strategy

Figure S2:Curves compare 4 contact reduction strategies to the null model of no social distancing, as

described in the main text. (A) Mixed strategy of repetition and triadic closure; (B) mixed strategy of

repetition and homophily; (C) mixed strategy of repetition, homophily, and triadic closure; (D) mixed

strategy of homophily and triadic closure.

21

0

10 0

20 0

30 0

0 20 4 0 60

0

20 0

40 0

60 0

0 20 4 0 60

0

50 0

10 00

15 00

0 20 4 0 60

0

10 00

20 00

30 00

0 20 4 0 60

Steps of interaction (per person) Steps of interaction (per person)

Steps of interaction (per person) Steps of interaction (per person)

Number infected (out of 2000) Number infected (out of 500)

Number infected (out of 4000) Number infected (out of 1000)

c.

a.

d.

b.

Null

model

Random

reduction

Community

strategy

Similarity

strategy

Repetition

strategy

Figure S3:Curves compare 4 contact reduction strategies to the null model of no social distancing, as

described in the main text. (A) 500 actors; (B) 1000 actors; (C) 2000 actors; (D) 4000 actors.

22

Steps of interaction (per person) Steps of interaction (per person)

Steps of interaction (per person) Steps of interaction (per person)

Number infected (out of 1000) Number infected (out of 1000)

Number infected (out of 1000) Number infected (out of 1000)

0

20 0

40 0

60 0

0 20 4 0 60

0

20 0

40 0

60 0

80 0

0 20 4 0 60

0

20 0

40 0

60 0

0 20 4 0 60

0

20 0

40 0

60 0

0 20 4 0 60

c.

a.

d.

b.

Null

model

Random

reduction

Community

strategy

Similarity

strategy

Repetition

strategy

Figure S4:Curves compare 4 contact reduction strategies to the null model of no social distancing, as

described in the main text. Names refer to the parametrisation given in Table S1. (A) baseline scenario;

(B) random network; (C) higher degree; (D) lower degree.

23

Steps of interaction (per person) Steps of interaction (per person)

Steps of interaction (per person)

Number infected (out of 1000) Number infected (out of 1000)

Number infected (out of 1000)

0

20 0

40 0

60 0

0 20 4 0 60

0

20 0

40 0

60 0

0 20 4 0 60

0

20 0

40 0

0 20 4 0 60

c.

a. b.

Null

model

Random

reduction

Community

strategy

Similarity

strategy

Repetition

strategy

Figure S5:Curves compare 4 contact reduction strategies to the null model of no social distancing, as

described in the main text. Names refer to the parametrisation given in Table S1. (A) no groups; (B) no

geography; (C) small world-ish.

24

Steps of interaction (per person) Steps of interaction (per person)

Steps of interaction (per person) Steps of interaction (per person)

Number infected (out of 1000) Number infected (out of 1000)

Number infected (out of 1000) Number infected (out of 1000)

0

20 0

400

0 20 4 0 60

0

20 0

40 0

60 0

0 20 4 0 60

0

20 0

40 0

60 0

0 20 4 0 60

0

20 0

40 0

60 0

80 0

0 20 4 0 60

c.

a.

d.

b.

Null

model

Random

reduction

Community

strategy

Similarity

strategy

Repetition

strategy

Figure S6:Curves compare 4 contact reduction strategies to the null model of no social distancing, as de-

scribed in the main text. (A) πinf ection =0.55; (B) πinf ection=0.65; (C) πinf ection =0.8; (D)πinf ection =0.95.

25

Steps of interaction (per person) Steps of interaction (per person)

Steps of interaction (per person) Steps of interaction (per person)

Number infected (out of 1000) Number infected (out of 1000)

Number infected (out of 1000) Number infected (out of 1000)

Steps of interaction (per person)

Number infected (out of 1000)

e.

0

200

400

600

800

0 25 50 75 100

0

200

400

600

0 25 50 75 100

0

200

400

600

0 25 50 75 100

0

200

400

600

0 25 50 75 100

0

200

400

600

0 25 50 75 100

c.

a.

d.

b.

Null

model

Random

reduction

Community

strategy

Similarity

strategy

Repetition

strategy

Figure S7:Curves compare 4 contact reduction strategies to the null model of no social distancing, as

described in the main text. (A) Texposed=0; (B) Texposed =1n; (C) Texposed =2n; (D) Texposed=3n; (E)

Texposed=4n.

26

Scenario nActors d geo min d geo max g geo m groups g groups d hom min d hom max w hom. z random

1: Baseline scenario 1000 4 12 0.3 8 0.9 4 12 2 0.5

2: Higher degree 1000 8 24 0.3 16 0.9 8 24 2 1

3: Lower degree 1000 2 6 0.3 4 0.9 2 6 2 0.25

4: No groups 1000 10 30 0.3 0 NA 5 15 2 0.5

5: No geography 1000 0 0 NA 20 0.9 5 15 2 0.5

6: Random net. 1000 0 0 NA 0 NA 0 0 NA 32

7: Geography 1000 15 45 0.3 0 NA 0 0 NA 0.5

8: 500 Actors 500 4 12 0.3 8 0.9 4 12 2 0.5

9: 1000 Actors 1000 4 12 0.3 8 0.9 4 12 2 0.5

10: 2000 Actors 2000 4 12 0.3 8 0.9 4 12 2 0.5

11: 4000 Actors 4000 4 12 0.3 8 0.9 4 12 2 0.5

Table S1:Parameters used in the stochastic generation of underlying networks. Full description of procedure is described in the Methods section in the main text.

27

Scenario n deg. clus. av. path dia. hom.

1: Baseline scenario 1000 38.4 0.11 2.23 3 1.08

2: Higher degree 1000 75.9 0.14 1.93 3 1.08

3: Lower degree 1000 19.4 0.09 2.69 4 1.08

4: No groups 1000 55.4 0.16 2.07 3 1.07

5: No geography 1000 40.2 0.26 2.24 3 1.09

6: Random net. 1000 62 0.06 1.96 3 1

7: Small world-ish 1000 53.9 0.3 2.57 4 1

8: 500 Actors 500 38.11 0.14 2 3 1.08

9: 1000 Actors 1000 38.4 0.11 2.23 3 1.08

10: 2000 Actors 2000 38.72 0.09 2.49 3.4 1.08

11: 4000 Actors 4000 38.85 0.08 2.7 4 1.08

Table S2: Characteristics of the networks created under diﬀerent scenarios. Descriptive statis-

tics are averaged over 40-48 simulations. Notes: n: number of actors; deg.: average degree / number of

connections per actor; clus.: clustering coeﬃcient / proportion of closed triads over possibly closed triads;

av. Path: average network distance between pairs of nodes; dia.: diameter / maximum distance in between

nodes in the network; hom.: average similarity of interaction partners divided by average similarity among

all actors.

28

Variation: network structure

Scenario Strategy Delay Peak Inf.

1: Baseline scenario Random 1.30 0.50 75%

Triads 1.66 0.35 73%

Homophily 1.50 0.23 71%

Repetition 1.59 0.24 57%

2: Higher degree Random 1.25 0.48 71%

Triads 1.49 0.48 79%

Homophily 1.52 0.44 80%

Repetition 1.24 0.31 56%

3: Lower degree Random 1.06 0.40 62%

Triads 1.85 0.14 51%

Homophily 1.88 0.12 52%

Repetition 2.00 0.06 24%

4: No groups Random 1.34 0.52 77%

Triads 1.68 0.29 72%

Homophily 1.46 0.38 77%

Repetition 1.24 0.27 54%

5: No geography Random 1.12 0.41 63%

Triads 0.71 0.01 2%

Homophily 1.38 0.22 65%

Repetition 0.99 0.14 32%

6: Random net. Random 1.46 0.59 90%

Triads 1.38 0.45 72%

Homophily 1.39 0.41 73%

Repetition 1.46 0.32 63%

7: Small world-ish Random 1.11 0.43 65%

Triads 1.61 0.25 66%

Homophily 1.44 0.45 74%

Repetition 1.30 0.29 56%

Variation: network size

8: 500 Actors Random 1.59 0.62 94%

Triads 1.83 0.41 81%

Homophily 2.09 0.33 93%

Repetition 1.87 0.34 75%

9: 1000 Actors Random 1.30 0.50 75%

Triads 1.66 0.35 73%

Homophily 1.50 0.23 71%

Repetition 1.59 0.24 57%

10: 2000 Actors Random 1.30 0.49 74%

Triads 1.49 0.28 60%

Homophily 1.60 0.26 73%

Repetition 1.47 0.21 52%

11: 4000 Actors Random 1.63 0.63 95%

Triads 2.20 0.42 93%

Homophily 1.90 0.36 94%

Repetition 2.08 0.29 71%

Variation: infectiousness

12: pinfect =0.55 Random 1.11 0.32 57%

Triads 0.96 0.10 29%

Homophily 1.25 0.16 49%

Repetition 1.25 0.07 22%

13: pinfect =0.65 Random 1.29 0.42 68%

Triads 1.34 0.19 47%

Homophily 1.21 0.16 48%

Repetition 1.65 0.17 46%

14: pinfect =0.8 Random 1.30 0.50 75%

Triads 1.66 0.35 73%

Homophily 1.50 0.23 71%

Repetition 1.59 0.24 57%

15: pinfect =0.95 Random 1.32 0.57 80%

Triads 1.44 0.33 62%

Homophily 1.33 0.24 63%

Repetition 1.48 0.27 57%

Variation: Texposure

Scenario Strategy Delay Peak Inf.

16: Texposure = 0 Random 1.44 0.50 74%

Triads 1.69 0.33 65%

Homophily 1.48 0.27 63%

Repetition 1.52 0.21 46%

17: Texposure = 1 Random 1.30 0.50 75%

Triads 1.66 0.35 73%

Homophily 1.50 0.23 71%

Repetition 1.59 0.24 57%

18: Texposure = 2 Random 1.40 0.54 84%

Triads 1.20 0.26 56%

Homophily 1.54 0.26 74%

Repetition 1.24 0.19 47%

19: Texposure = 3 Random 1.22 0.47 73%

Triads 1.58 0.32 68%

Homophily 1.24 0.21 62%

Repetition 1.49 0.23 57%

20: Texposure = 4 Random 1.11 0.43 68%

Triads 1.85 0.34 78%

Homophily 1.50 0.23 73%

Repetition 1.11 0.19 44%

Variation: mixed strategies

21: Homo. + Triad. Random 1.52 0.69 97%

Triads 1.86 0.55 96%

Homophily 1.73 0.49 97%

Mixed strat. 1.75 0.58 96%

22: Homo. + Rep. Random 1.39 0.60 84%

Triads 1.51 0.40 70%

Homophily 1.60 0.37 82%

Mixed strat. 1.52 0.36 68%

23: Rep. + Triad Random 1.53 0.63 89%

Triads 1.47 0.34 64%

Homophily 1.67 0.31 80%

Mixed strat. 1.92 0.35 74%

24: Ho. + Re. + Tr. Random 1.53 0.63 88%

Triads 1.51 0.39 70%

Homophily 1.43 0.32 74%

Mixed strat. 1.39 0.34 64%

Variation: operationalisation homophily

25: Normal homo. Random 1.30 0.50 75%

Triads 1.66 0.35 73%

Homophily 1.50 0.23 71%

Repetition 1.59 0.24 57%

26: Geogr. homo. Random 1.20 0.46 69%

Triads 1.40 0.30 62%

Homophily 2.19 0.11 50%

Repetition 1.55 0.23 54%

27: 2-Dim. abs. diﬀ. Random 1.31 0.50 75%

Triads 1.47 0.25 61%

Homophily 1.00 0.31 53%

Repetition 1.22 0.26 52%

28: 2-Dim. only 1st Random 1.32 0.52 78%

Triads 1.63 0.27 67%

Homophily 1.33 0.36 68%

Repetition 1.19 0.25 50%

Table S3: Characteristics of average infection curves for

diﬀerent strategies. All entries denoting averaged results of

simulations are relative to the null model of no contact reduction

(blue line in Fig. 3). Delay: delay of the peak of the infection

curve compared to the null model; Peak: height of the peak of

the infection curve compared to the null model; Inf.: proportion

of the population infected compared to the null model.

Table S3