††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Spreader events and the limitations of projected networks for capturing
dynamics on multipartite networks

Hyojun A. Lee  Department of Chemical and Biological Engineering, Northwestern
University, Evanston, Illinois, United States of America  Luiz G. A. Alves
Department of Chemical and Biological Engineering, Northwestern University,
Evanston, Illinois, United States of America  Luís A. Nunes Amaral
<EMAIL_ADDRESS>Department of Chemical and Biological Engineering,
Northwestern University, Evanston, Illinois, United States of America
Department of Physics and Astronomy, Northwestern University, Evanston,
Illinois, United States of America  Northwestern Institute on Complex Systems,
Northwestern University, Evanston, Illinois, United States of America

###### Abstract

Many systems of scientific interest can be conceptualized as multipartite
networks. Examples include the spread of sexually transmitted infections,
scientific collaborations, human friendships, product recommendation systems,
and metabolic networks. In practice, these systems are often studied after
projection onto a single class of nodes, losing crucial information. Here, we
address a significant knowledge gap by comparing transmission dynamics on
temporal multipartite networks and on their time-aggregated unipartite
projections to determine the impact of the lost information on our ability to
predict the systems’ dynamics. We show that the dynamics of transmission
models can be dramatically dissimilar on multipartite networks and on their
projections at three levels: final outcome, the magnitude of the variability
from realization to realization, and overall shape of the temporal trajectory.
We find that the ratio of the number of nodes to the number of active edges
over the time aggregation scale determines the ability of projected networks
to capture the dynamics on the multipartite network. Finally, we explore which
properties of a multipartite network are crucial in generating synthetic
networks that better reproduce the dynamical behavior observed in real
multipartite networks.

## Introduction

The theoretical study of transmission processes, whether infectious diseases,
innovations, or mores, has typically followed one of two approaches:
compartment models Daley and Gani (2001) and network models Pastor-Satorras
and Vespignani (2001); Liljeros  _et al._ (2003); Barrat  _et al._ (2008). In
compartment models, individuals belong to a small number of compartments, and
they transit in and out of compartments according to specific rates.
Compartment models, are mean-field models in which all members of a
compartment are equally likely to interact with members of another
compartment. In network models, individuals interact pairwise according to the
connections defined in the underlying social network. Both approaches have
produced significant advances both separately and when integrated as can be
seen from the studies and forecasting of the current SARS-CoV-2 pandemic and
of other recent pandemics Chinazzi  _et al._ (2020); Maier and Brockmann
(2020).

Research published in the past two decades has demonstrated that network
representations provide a fruitful abstraction of many complex systems Amaral
and Ottino (2004); Newman (2010). Despite the remarkable advances, the task of
determining the simplest network representation able to fully capture the
characteristics of the system remains more of an art than a science. Indeed,
although limitations in data collection procedures and less developed
theoretical frameworks lead many scholars to settle for static, unipartite,
unweighted, and undirected network representations, recent research has
demonstrated that such representations may be inadequate Barrat  _et al._
(2004); Stouffer  _et al._ (2005); Buldyrev  _et al._ (2010); Kivelä  _et al._
(2014).

For example, in the context of human mobility or international trade, one
cannot disregard the weights (i.e., number of passengers in transportation
networks, or exchange volume in international trade flows) of the connections
between system components Fagiolo  _et al._ (2008); Alves  _et al._ (2020). In
cases such as these, the system is more accurately conceptualized through
weighted networks Barrat  _et al._ (2004). Similarly, in the context, for
example, of the spread of information within social systems, one cannot ignore
the multiple media — email, Twitter, WhatsApp, and so on – that are used by
individuals to communicate. In cases such as these, the system is more
accurately conceptualized through multilayer networks Buldyrev  _et al._
(2010). As a result, in recent years there has been an important and thriving
research stream focused on multiplexed systems Kivelä  _et al._ (2014);
Boccaletti  _et al._ (2014).

Another significant characteristic of many non-equilibrium complex systems is
that interactions among components are not immutable Gorochowski and
Richardson (2017); Li  _et al._ (2017); Miritello  _et al._ (2011); Liu  _et
al._ (2014). For example, direct flight connections to a city can be added or
dropped over time; friendships flourish and wane; trophic relations in an
ecosystem change with the seasons. In cases such as these, the system is more
accurately conceptualized through temporal networks Buldyrev  _et al._ (2010);
Holme and Saramäki (2012). The temporal aggregation of evolving networks can
significantly affect processes such as transmission phenomena Holme and
Saramäki (2012); Pastor-Satorras  _et al._ (2015). As a result, a significant
research thrust has focused on the study of temporal networks and its
implications to dynamical processes Boccaletti  _et al._ (2006); Perra  _et
al._ (2012); Takaguchi  _et al._ (2013); Karimi and Holme (2013); Perotti  _et
al._ (2014); Holme (2018); Petri and Barrat (2018); De Domenico  _et al._
(2016); Krings  _et al._ (2012); Clauset and Eagle (2012); Ribeiro  _et al._
(2013); Darst  _et al._ (2016).

In contrast to the deserved attention provided by these important and
flourishing research streams to weighted, multilayer and temporal networks,
multipartite networks remain overlooked (see Ref. Benson  _et al._ (2018) for
exceptions and Refs. Bisanzio  _et al._ (2010); Wen and Zhong (2012); Han  _et
al._ (2018); Pavlopoulos  _et al._ (2018); Cao  _et al._ (2011) for epidemics
on bipartite networks). However, many important biological and social systems
are most naturally conceptualized as multipartite networks. Examples include
cellular metabolism Maslov and Sneppen (2002), ecosystems Pilosof  _et al._
(2017), collaborations Börner  _et al._ (2004), heterosexual romantic
relationships Liljeros  _et al._ (2003); Helleringer and Kohler (2007); Rocha
_et al._ (2011), research collaborations Guimera  _et al._ (2005), or product
recommendation systems Leskovec  _et al._ (2006); Zhou  _et al._ (2007);
Godoy-Lorite  _et al._ (2016). These systems comprise multiple classes of
components — for example, movies, actors, directors, producers, agents,
studios, and so on in movie-production networks — and edges can only connect
nodes of different types — producers connect through movies or through
studios.

Despite the diversity of ways complex systems can be represented and projected
onto complex networks, our understanding of how modeling choices affect the
dynamical process remains limited. Presumably because of the difficulty in
obtaining detailed data, or for simplicity of analysis Rivera  _et al._
(2010). In particular, multipartite networks are typically studied after
projection onto a single type of node considered to be the most important,
such as actors in movie-production networks. However, by construction, the
unipartite projection of a multipartite network is dramatically less
information rich than the real network. For this reason, there is growing
interest and effort in the community to collect open-access, large-scale, and
information rich datasets on different network projections with time
dependence to better understand real networks (see, for instance, Ref. Aaron
Clauset and Sainz (2016)).

A particularly important aspect of some multipartite networks is the presence
of strong temporal discontinuities for, at least, one type of node. These
nodes present discrete timestamps and confined temporal existence: Movies have
production schedules and romantic relationships span finite periods [Fig.
1(a)]. For this reason, these temporal multipartite networks have a stronger
and more complex temporal structure than the typical temporal network and
detailed temporal information is frequently unavailable [Fig. 1(b)]. Thus, it
remains unclear whether the approach of temporal fine-graining used in the
context of temporal networks to obtain static snapshots Holme (2005) would be
sufficient to generate an accurate conceptualization of multipartite networks
that include a class of nodes with heterogeneous and discontinuous event
dynamics.

Figure 1: Temporal multipartite network, unipartite projections, and time-
aggregated networks. (a) Temporal multipartite network (left) comprised of a
class of agents nodes (larger blue circles) and a class of event nodes (pink
circles) and its temporal agent projection (right). Event nodes have specific
creation times and finite temporal duration (such as a day or a week). (b)
Time-aggregated unipartite projection over the agent nodes shows simpler
structures and dynamics. When the network is projected onto the agent network,
nodes with temporal properties can be aggregated over different periods (a
week, a month, and so on), losing crucial information on the ordering of
events.

To address this question, we investigate the extent to which unipartite
projections of a multipartite network can be used to accurately describe and
predict transmission dynamics Kivelä  _et al._ (2014); Shakarian  _et al._
(2015); Myers  _et al._ (2012); Nematzadeh  _et al._ (2014); Masuda  _et al._
(2017) taking place within multipartite networks in which one class of nodes
has discontinuous dynamics. As we mentioned earlier, we focus on transmission
dynamics because they have been used to model a broad class of important
phenomena such as epidemic outbreaks, opinion formation, and the spread of
innovations Darst  _et al._ (2016); Valdano  _et al._ (2015a, b). Because of
the broad range of applications of interest, we will consider model
predictions for the full range of infectability values from the low infection
rate of viruses, such as HIV Rocha  _et al._ (2011) to the high infection rate
of “unavoidable” memes Gleeson  _et al._ (2016).

We organize our analysis around two systems with very different sizes and
characteristics — but for which we have high-quality data — and that have been
studied in the context of the adoption of innovations among small teams of
highly trained professions Weiss  _et al._ (2014), and in the context of the
assembly of creative teams Wasserman  _et al._ (2015). We simulate
transmission dynamics on these networks and on synthetic networks derived from
them using two well-known transmission models — susceptible infectious (SI)
and general contagion (GC) Anderson and May (1992); Dodds and Watts (2004,
2005).

Significantly, we find under real-world conditions that the time-aggregated
unipartite projections are broadly unable to predict the dynamics observed on
the full temporal multipartite networks. Using model networks, we show that
when the ratio of number of agents to the number of events/teams is less than
one, unipartite projections will fail to reproduce the final number of
infected agents, the magnitude of the variability from different realizations,
and the overall shape of the temporal trajectories observed on multipartite
networks. Furthermore, to address the cases in which unipartite projections
are inadequate, we investigate the minimal amount of information that must be
retained from a multipartite network in order to generate synthetic
multipartite networks that properly reproduce the dynamics observed in the
real networks.

## Empirical multipartite networks and their projections

A multipartite network is a network whose nodes are partitioned into $K$
distinct non-overlapping sets. The nodes in each set have edges can connect
only to nodes in the other $K-1$ sets [Fig. 1(a)]. We consider two contexts
where connectivity is well described by multipartite networks, physician
coverage teams in a hospital and movie-producing teams.

For the former, we consider the multipartite networks formed by the physicians
providing high-intensity critical care coverage in the medical intensive care
unit (MICU) of a Chicago hospital that we studied previously Weiss  _et al._
(2014). Specifically, we obtained the MICU coverage schedules of 76 physicians
from July 2012 to August 2016. Each team is composed of one attending and one
fellow and has confined temporal existence (see the Supplemental Material SM
and Ref. Weiss  _et al._ (2014) for details). We construct a temporal
multipartite network for each academic year comprised of three sets of nodes:
coverage teams, attendings, and fellows. Attendings and fellows are only
connected to team nodes. Edges between nodes are created when a fellow and an
attending are together in the same team. Since the team has finite temporal
existence, the edges exist only while the teams are active, typically a week.

For the latter, we construct a network of producer collaborations on U.S.
movies, that we obtained by crawling the Internet Movie Database (IMDb,
https://imdb.com). We focus here on production teams for the 2,009 US-produced
movies released in the period 1990-2000. We identify 5,758 active producers
during the considered period. Unlike the physician teams, we lack information
on the exact production period for each movie.

Because the temporal aggregation time scale is known to affect static network
structure Krings  _et al._ (2012); Holme and Saramäki (2012); Pastor-Satorras
_et al._ (2015), we compare the temporal multipartite physician network with
different unipartite projected networks aggregated over different periods of
time, spanning from 7 days to 12 months. In the unipartite projected networks,
the edges are static during the respective time period and therefore lose
information about the temporal ordering of interactions Zhou  _et al._ (2007).
In addition to the unweighted unipartite network projection, we consider two
weighting approaches for defining edge weight: the number of shared teams and
the number of days in shared teams. For the movie producers, we aggregate time
over four time scales 1, 2, 5, and 10 years. We model the edges as unweighted
for the producer projected network because the vast majority of producers
participate in only one movie per year.

## Transmission dynamics

Here, we focus on transmission dynamics and use the language of infectious
diseases to investigate the limitations of time-aggregated unipartite
projected networks for capturing the dynamics of temporal multipartite
networks. The physicians’ network is a tripartite network composed of two
classes of agent nodes (attendings and fellows) and one class of event nodes
(teams). At time zero, two agent-nodes are randomly chosen to become
infectious (for the multipartite network, we chose the two nodes that are
connected to the first event node), whereas all other nodes are deemed
susceptible and to have no prior exposure (see Methods for details). We then
track the number of infectious individuals over time.

Figure 2(a) shows the mean number of infectious agents over time for two
different parameter values for the SI model on multipartite and projected
networks. It is visually apparent that the dynamics on the projected networks
yield different dynamics from those observed on the multipartite network [Fig.
2(a)].

Figure 2: Spreading dynamics on temporal multipartite networks and on their
time-aggregated unipartite projections. (a) Mean fractions of infectious
agents (full lines) and 95% confidence interval for the dynamics (shaded
regions) on the physicians’ multipartite networks and unweighted unipartite
projected networks using 7 days and 1 month temporal aggregation (initial
infectious agents: 2). The results for the projected network are obtained for
the asynchronous status update. (b) Mean fractions of infectious agents (full
lines) and 95% confidence interval for the dynamics (shaded regions) on the
producers’ multipartite networks and unweighted unipartite projected networks
using 2 and 5 year temporal aggregation (initial infectious agents: 5% of
producers appearing in the first year). The results for projected networks are
obtained for the asynchronous status update. The right side of each panel
shows the distributions of the final fraction of infectious agents for each
set of parameters and conditions.

We further investigate the dynamic differences using the GC model to compare
the temporal trajectories of the four multipartite networks. Our benchmark
multipartite networks (i.e., the networks for the different academic years)
show similar temporal trajectories for a wide range of parameters, $0.1<d<1$
and $0.1<p<1$; Fig. S1 of the Supplemental Material SM . In contrast, we can
observe that the dynamics of time-aggregated unipartite projected networks
compared with our benchmark multipartite network can overestimate the number
of infectious agents over the course of time for a wide range of parameters
for asynchronous dynamics simulations (see Figs. S2 to S4 in the Supplemental
Material SM ), and it can overestimate or underestimate the number of infected
agents depending on the time-aggregation interval and parameter range for
synchronous dynamics simulations (see Figs. S5 to S7 in the Supplemental
Material SM  for synchronous dynamics). We will provide a quantification of
these differences in the following sections.

Transmission dynamics on projected networks also lack the trajectory
“diversity” displayed by the dynamics on real networks [compare 95% confidence
intervals in Fig. 2(a)], pointing at deep differences of the dynamics.

The producers’ network is a bipartite network where there are one class of
agent nodes (producers) and one class of event nodes (movies). Using the SI
model to simulate the transmission process on temporal multipartite networks
and compare with their projections, we find that the movie producers network
exhibit less variable temporal trajectories than the physicians’ networks
[Fig. 2(b)]. Nonetheless, it is visually apparent that the dynamics on the
projected movie-producers’ networks yield different dynamics from those
observed on the multipartite network. Transmission dynamics taking place on
projected networks consistently overestimate the number of infected agents
over the course of time. These results are less striking than those obtained
for the physicians’ network, but still consistent.

### Factors controlling stability of unipartite projections

The movie producers’ network and physicians’ network differ significantly in
terms of the number of agents, the team sizes, the aggregation time scale, and
the number of teams per aggregation cycle. Thus, we investigate which of these
factors can explain the ability of projected networks to make better
predictions. Additionally, we study a simple bipartite network model with
community structure Guimerà  _et al._ (2007); Colizza  _et al._ (2006) in
order to account for the possibility of modular structure in real-world
multipartite networks (see Methods).

We find that, as the degree of temporal coarse graining increases, the
transmission dynamics of unipartite projected networks become more similar to
those of temporal bipartite networks (Fig. 3). This is to be expected because
the smaller the temporal graining is, the more the projected networks become
an accurate snapshot of the bipartite networks.

We also find that the number of agent $N$ increases whereas the number of
teams per aggregation cycle $T_{c}$ remain fixed and the difference between
the transmission dynamics of the bipartite networks and their projected
networks also increases (Fig. 3).

Our analysis, thus, reveals that the determining factor is the ratio of the
number of agents to the number of teams per aggregation cycle. It is important
to note, however, that the trajectory of the transmission dynamics for the
projected and the bipartite networks have distinct functional forms. Thus,
even the temporal fine-grained networks used in the context of temporal
networks to obtain static snapshots Holme (2005) would not be sufficient to
generate an accurate conceptualization of temporal multipartite networks that
include a class of nodes with discontinuous dynamics.

Figure 3: Transmission dynamics on bipartite networks and their unipartite
projections. Average and spread of transmission dynamics for the GC model
using infection probabilities $p=0.1$ (left column) and $p=0.5$ (right column)
and dose $d=0.5$. Note that as the ratio agents and teams (i.e.,
$r\equiv\frac{N}{T_{c}}$) increases, the difference in the dynamics becomes
less pronounced.

## Building realistic multipartite networks

Since simulation on unipartite projections cannot, for many conditions,
replicate the transmission dynamics observed on multipartite networks, we ask
whether there are suitable null models such that the dynamics on the generated
networks better replicate the dynamics on real temporal multipartite networks.
Specifically, we aim to identify the crucial information needed for creating
adequate synthetic multipartite networks and we focus here on the physicians’
networks because they show the largest dynamical differences between projected
and multipartite networks.

In building our multipartite network models, we focus on three types of
aggregate information: the number of agents in each agent class of nodes;
distribution of the number of teams that an agent participates in; and
distribution of gap durations between consecutive participations in a team by
an agent. We construct three null models by using increasing amounts of
information about the real network [Fig. 4(a)]. The first generative model,
denoted N, makes use only of the number of agents $N$ in the system. The
second model, denoted NT, adds also information on the distribution of the
number of teams $T$ in which agents participate [Fig. 4(b) top]. The third
model, denoted NTG, adds information on the distribution of gap $G$ durations
between participation in consecutive teams as well [Fig. 4(b) bottom].

Figure 4(c) shows the level of information that is used in each model and the
steps for generating a synthetic multipartite network. We show the
distributions of the number of teams and gap durations for the three types of
synthetic networks and the real networks in Fig. S8 of the Supplemental
Material SM .

Figure 4: Generation of synthetic multipartite networks. (a) One year schedule
for attendings and fellows in the MICU. Each row displays the entire year
schedule for a given physician. Each square represents one week and filled
squares indicate participation in a coverage team by a given physician.
Consecutive white squares show gap durations between participation in coverage
teams for each physician. (b) Distribution of the number of coverage teams a
physician is part of during a year and distribution of the duration of gaps
between participation in consecutive coverage teams. We first note that
whereas coverage teams are reportedly assigned randomly by the hospital based
on the number of teams that each physician has to be assigned to, we find that
the distribution of the number of teams in which physicians participate does
not obey a Poisson distribution Masuda and Holme (2017). Therefore,
physicians’ team assignments cannot be purely random. The distributions of gap
durations are also not exponential, which is the expected distribution for
inter-event times that occur at a constant rate Keeling and Eames (2005).
These two findings suggest that there are other constraints that govern how
physicians are assigned to teams. (c) Algorithm for generating synthetic
multipartite networks according to levels of information incorporated. We
create $N$ physician nodes as in the multipartite network we are trying to
replicate (the N model). We can then either draw the number of coverage teams
assigned to a physician from a Poisson distribution or from the distribution
of the number of teams in the multipartite network without replacement (the NT
model). Note that the total number of a team must be equal to the number in
the multipartite network. We assign physicians’ team participation at random.
If we are including information about the distribution of gap durations, then
we calculate the gap duration distribution for the synthetic network and
compare it to that of the multipartite network. The teams are adjusted to
match the gap duration distribution until the difference between the synthetic
network and the multipartite network is not statistically significant (the NTG
model).

### Quantification of the similarity of two ensembles of dynamical
trajectories

Next, we address how to quantify the differences in the outcomes between the
transmission dynamics on the synthetic networks and that on the real networks.
To this end, we define a new metric, the dynamic overlap of two ensembles of
dynamical trajectories, which we denote as $\Omega$. We focus on the GC model
since the SI model is recovered for $D=d$. We compare both the final number of
infectious nodes as well as the similarity of their trajectories when using
the same parameters, on either synthetic or empirical networks. Quantifying
such a comparison is not trivial and different methods have been introduced to
compare the outcomes of transmission dynamics in networks de Arruda  _et al._
(2020), however, there has been less focus on comparing the transmission
trajectories over the entire period. Inspired by the weak Fréchet distance Alt
and Godau (1995), we first define the distance between two dynamic
trajectories as the area between the curves for an observation period $T$

$A_{j}^{i}({\cal N}_{1},{\cal N}_{2})\equiv\int_{0}^{T}~{}|~{}f_{i}(t|{\cal
N}_{1})-f_{j}(t|{\cal N}_{2})~{}|~{}dt\,,$ (1)

where $f_{i}(t|{\cal N}_{1})$ is the $i^{th}$ realization of the dynamics on
network ${\cal N}_{1}$. We control for the potentially different number of
agents in each network by scaling the trajectories by the number of agents in
the network. We demonstrate how individual realizations of the dynamics on the
same network can yield quite distinct trajectories in the top panel of Fig.
5A(a). Thus, our aim must be to quantify whether a particular trajectory on a
network falls within the expected range for trajectories obtained for
simulations of the dynamics on a set of reference networks, ${\cal R}=\\{{\cal
R}_{1},{\cal R}_{2},\dots\\}$. To this end, we define the distance of
trajectory $i$ to the mean trajectory $R$ on a set of reference networks as

$A_{R}^{i}({\cal N}_{k},{\cal R})\equiv\int_{0}^{T}~{}|~{}f_{i}(t|{\cal
N}_{k})-\overline{f_{R}(t|{\cal R})}~{}|~{}dt\,,$ (2)

where $\overline{f_{R}(t|{\cal R})}$ represents an average over realizations
of the simulations and over the set of networks in ${\cal R}$ [middle panel,
Fig. 5(a)]. By calculating $A_{R}^{i}({\cal N}_{k},{\cal R})$ for the $i^{th}$
trajectory, we can estimate the percentiles for the distances to the mean
trajectory, $P_{2.5}\\{A_{R}^{i}({\cal N}_{k},{\cal R)}\\}$ and
$P_{97.5}\\{A_{R}^{i}({\cal N}_{k},{\cal R)}\\}$, that encompass 95% of the
dynamical trajectories generated on the set of reference networks [bottom
panel, Fig. 5(a)]. We define the dynamic overlap $\omega$ for a given network
${\cal N}_{k}$ as

$\omega({\cal N}_{k})\equiv\int_{P_{2.5}}^{P_{97.5}}~{}A_{R}~{}p(A_{R}|{\cal
N}_{k})~{}dA_{R}\,,$ (3)

where $p(A_{R}|{\cal N}_{k})$ is the probability of observing a specific value
of $A_{R}^{i}({\cal N}_{k},{\cal R})$. The dynamic overlap for a set of
multipartite networks is $\Omega\equiv P_{50}\\{\omega({\cal N}_{k})\\}\,$,
which takes values in $[0,1]$. In practice, we define ${\cal R}$ for
multipartite networks as the set of networks for 2012, 2013, and 2015 and use
the real network for 2014 as a control for estimating the similarity of the
trajectories for different parameter values. We chose ${\cal R}_{2014}$ as the
control because its number of physicians takes a value close to the mean for
the four networks. We then compare the trajectories of the GC model for the
control network with results obtained for synthetic multipartite network
generated using information from ${\cal R}_{2014}$ [Fig. 5(b)] .

Figure 5: Systematic quantification of the similarity of transmission dynamics
on different networks. (a) The top panel shows the simulation of 1,000
independent stochastic realizations of the GC model dynamics for each network:
the 2014 control temporal multipartite network (red), the reference set of
multipartite networks for 2012, 2013, and 2015 (blue), and 25 synthetic
networks (green). The middle panel shows the areas between the mean of
realizations from the reference networks (reference curve, i.e., thick black
line in the top panel) and each realization from the reference networks. The
bottom panel shows the area distribution $p(A_{R}|{\cal N}_{k})$ where we
identify the 95% confidence boundaries for the observed overlapping. The
percentages indicate the overlapping between the areas from the simulations
compared with those from the reference set. (b) The panel shows the box-plot
of the area overlapping for the control multipartite network, the reference
set, and the synthetic network for a specific set of parameters, as indicated
in the figure. We calculate $\omega({\cal N}_{k})$ for each set of networks in
multipartite networks and synthetic networks. Because there are 25 synthetic
networks, we define $\Omega$ as the median of $\omega({\cal N}_{k})$ for the
25 networks. (c) The heat maps show the dynamic range overlap $\Omega$ for the
entire parameter space for different networks. Each cell shows the $\Omega$
value of the GC model dynamics compared with those obtained on the reference
set ${\cal R}$ for the multipartite network (left), for projected networks
using synchronous (middle), and asynchronous (right) update.

We show $\Omega$ for the control multipartite network ${\cal R}_{2014}$ —
multipartite network from the hospital for the year 2014 in Fig. 5(c). Similar
results are obtained for $\Omega$ when using ${\cal R}_{2012},{\cal R}_{2013}$
and ${\cal R}_{2015}$ as control networks (see Figs. S9 to S11 in the
Supplemental Material SM ). Even though ${\cal R}_{2014}$ is not structurally
different from the networks in ${\cal R}$, we still found that $\Omega$ is
close to zero when $p$ is large. This is due to the fact that the
stochasticity of the transmission dynamics becomes negligible when almost all
interactions lead to successful transmission. Under these conditions, even
small differences in the network lead to statistically significant differences
in the dynamics (Fig. S1 in the Supplemental Material SM ).

In contrast, for the projected networks, $\Omega$ is generally close to zero
for nearly all regions of parameter space [Fig. 5(c), and Figs. S12 to S15 in
the Supplemental Material SM ]. The exceptions are those parameter values for
which the variance in the trajectories for the multipartite networks is very
large. Because the trajectory diversity for the dynamics on the projected
networks are usually small (Fig. 2), it enables all the trajectories on the
projected networks to fall within the 95% confidence interval of the dynamics
on the multipartite networks (see the case of $p=0.1$ in Fig. 2 and Figs. S1
to S6 in Supplemental Material SM ). This interpretation is supported by the
fact that when we take the time-aggregated unipartite networks as the
reference set and the multipartite networks as the ones being compared, then
$\Omega$ drops to zero (Figs. S16 and S17 in the Supplemental Material SM ).
If the transmission dynamics were truly similar, $\Omega$ would take similar
values even when ${\cal R}$ and $\\{{\cal N}_{k}\\}$ are switched so that
$\\{{\cal N}_{k}\\}$ are used as reference networks, as indeed is the case for
synthetic multipartite networks (Fig. S18 in the Supplemental Material SM ).

Thus, we find that as we increase the real-world information built into the
null models, the more closely the infection trajectories tend to mirror those
observed for the control network [Fig. 6(a)]. In fact, calculating the overlap
$\Omega$ of the transmission dynamics in synthetic multipartite networks with
the reference set of network ${\cal R}$ for different values of parameter of
the GC model, there is a strong agreement between the transmission
trajectories and the control network for a wide range of parameters [Fig.
6(b), and Figs. S9 to S11 in the Supplemental Material SM  for comparisons
using other years than 2014 as control network]. Thus, even simplistic
synthetic networks can produce $\Omega$ values that are quite similar to the
values obtained for the control network and can capture real dynamic much
better than the projected networks (Fig. S19 in the Supplemental Material SM
).

Furthermore, to verify that this model is generalizable to other cases, we
generate synthetic multipartite networks using the characteristics of the
producers’ network. We randomly selected 5% of the producers active in the
first year as initial infectious agents. We found that the transmission
dynamics on even the simplest synthetic network are much more similar to the
transmission dynamics on the original multipartite network than on projected
networks which is consistent with the results from the physicians’ network.
However, we observed that the infection rates are not as similar to the real
multipartite network compared to the physicians’ case, although the overall
shape of the infection curves is quite similar (Fig. S20 of the Supplemental
Material SM ). This difference may be due to the fact that unlike for the
physicians’ networks, producer networks are more heterogeneous. An alternative
or even complementary explanation is that physician and producer teams have
different formation mechanisms. In the physicians’ networks, one attending and
one fellow form a team, whereas teams vary in size and composition for the
producers’ network (Figs. S21 and S22 in the Supplemental Material SM ).

Figure 6: Model dynamics on multipartite networks and on their synthetic
networks. (a) Mean fraction of infectious agents over time (lines) and 95%
confidence interval for the dynamics (shaded regions) on the multipartite and
synthetic multipartite networks using SI model with infection probabilities
$p=0.1$ and $p=0.5$. The histograms show the final fraction of infectious
agents for each network. (b) The heat maps show the dynamic range overlap
$\Omega$ for the entire parameter space for different networks. Each cell
shows the $\Omega$ value of the GC model dynamics compared with those obtained
on the reference set ${\cal R}$ for the multipartite network (top left) and
the three synthetic multipartite model networks.

## Discussions

Transmission dynamics on systems best described by temporal multipartite
networks show much greater variability across realization than one is led to
believe from simulating dynamics on time-aggregated unipartite projection
networks. This difference means that studies conducted on projected networks
are inadequate both for obtaining estimates of expected outcomes and deciding
whether an observed outcome is consistent with a given model. We show that the
ratio of the number of agents to the number of teams per aggregation cycle has
a significant impact on the resulting differences of simulating the
transmission dynamics on multipartite networks versus their unipartite
projections. This means that there is the need to accomplish a trade-off
between decreasing the number of teams per aggregation cycle whereas
maintaining a small ratio of the number of agents to the number of teams per
aggregation cycle.

Additionally, we show that simulations on very simplistic multipartite
synthetic networks can capture real dynamics much better than simulations on
projected networks. Our results inform about the types of information to focus
on when there are limited resources for collecting data on complex
multipartite networks.

Finally, because of the growing body of information-rich networks datasets
Aaron Clauset and Sainz (2016), our results open new venues to explore a
variety of systems that can be represented as temporal multipartite networks,
including very large networks with different topological heterogeneity that
could result in new behaviors that cannot be unveiled by simple time-
aggregated unipartite projected networks.

## Methods

### Transmission models

We study the transmission dynamics as a contagion process diffusing either on
the actual temporal multipartite network, or on different projections obtained
for different levels of temporal aggregation, or on synthetic multipartite
networks. We consider two contagion models: the SI model and the general GC
model.

SI model: This model agents can be in one of two states, susceptible ($S$) and
infectious ($I$), which we represent as 0 and 1, respectively. At each event
or time step $t\geq 1$, a susceptible node in contact with an infectious node
becomes infected with probability $p$.

GC model: This model interpolates between two classes of transmission
mechanisms: threshold Granovetter (1978) and infection Anderson and May
(1992). When a susceptible agent is exposed to infectious agents, they receive
an infection dose $d$ with probability $p$. Susceptible agents keep in memory
the infection doses received over the previous $\tau$ time steps. Susceptible
agents become infected when their cumulative remembered infection doses exceed
a threshold, $D$ (see algorithm 1 in the Supplemental Material SM ). For
simplicity, we restrict our attention to the case where $\tau=\infty$. For
both transmission models, each simulation is stochastic and starts with two
infectious nodes chosen randomly from a fully susceptible population. Note
that the SI model is captured by the case where $d=D$ in the GC model.

### Simulation of dynamics

We systematically study the transmission dynamics for $p\in\\{0.1,...,1.0\\}$
(SI) and for 100 pairs of values $p\in\\{0.1,...,1.0\\}\times
d\in\\{0.1,...,1.0\\}$ (GC) and for four different time scales of aggregation
in the unipartite projection. For each set of parameter values and each
network, we simulate 1,000 independent realizations.

Unipartite network dynamics: In order to assure the robustness of our results
for unipartite projected networks, we consider two methods to update the state
of the agents — asynchronous update using a random walk procedure Tang  _et
al._ (2009); Aggarwal and Ryoo (2011), and synchronous update Granovetter
(1978); Shakarian  _et al._ (2015); Chen  _et al._ (2010); Valente (1996).

In the asynchronous update, at the start of transmission, $t=0$, we randomly
select two nodes as initially infected nodes $n_{1},n_{2}$. At each time step,
a node $n_{1}$ is selected and one of its neighbors is randomly choose to
calculate the transmission dynamic. Then we repeat the previous step $t$ times
according to the time interval the network was projected and the number edges
in the network. The probability of choosing the neighbor and the size of the
influence two active nodes have on each other is proportional to the edge
weights. At each step only the chosen nodes interact (see algorithm 2 in the
Supplemental Material SM ). When one node is in the susceptible state and the
other is in the infected state, the susceptible node receives a dose $d$ with
probability $p$ for the GC model or is infected with probability $p$ for the
SI model (see algorithm 1 in the Supplemental Material SM ). If both nodes are
susceptible or infected (when nodes are in the same state), no exchange
occurs. Asynchronous update behaves as if it has temporal properties because
the edges are selectively active for a certain duration.

In the synchronous update, at time step $t$, the transmission of infection can
happen across all connected nodes, and all nodes update their state
simultaneously (see algorithm 3 in the Supplemental Material SM ). This
updating strategy is inspired by a modified linear threshold model, where a
susceptible node is influenced by all of its infected neighbors at each time
step Valente (1996); Granovetter (1978); Shakarian  _et al._ (2015); Chen  _et
al._ (2010). The influence a node exerts on another depends on the edge
weights. In the synchronous update, it is assumed that edges always existed
during the interval of the simulation, therefore, in a network of finite size
and non-zero transmission rate, all nodes will eventually become infected
Gross  _et al._ (2006).

Multipartite network dynamics: For the multipartite networks, the transmission
can progress through the active members of the coverage team; physicians who
are part of a team at each time point interact and influence each other. When
one node is in the susceptible state and the other is in the infected state in
an active team, the susceptible node receives a dose $d$ with probability $p$
for the GC model or is infected with probability $p$ for the SI model.
However, because a certain team can be active during different periods, for
each day, the infection process is repeated until the team is no longer active
(see algorithm 4 in the Supplemental Material SM ).

For the producers’ multipartite network, because we do not have data on exact
production periods, we randomized the order of the movie production during
each year to simulate the transmission dynamics. The number of movies produced
per year stays approximately constant (Fig. S22 in the Supplemental Material
SM ), therefore we can assume the gap between produced movies remains
approximately constant.

### Generating the synthetic bipartite network with a modular structure

We use the generative model for bipartite networks introduced by Guimerà et
al. Guimerà  _et al._ (2007), because it yields an ensemble of random
bipartite networks with a prescribed modular structure. To create a network
instance, we first dived $N$ agents equally into five groups. We then
sequentially create $N_{T}$ teams. Each time, we first select at random a
primary group. Then, with probability $p_{p}$ (team homogeneity) we select an
agent at random from the primary group. With probability $1-p_{p}$, we select
an agent at random from one of the four non-primary groups. Thus, we use the
generated bipartite networks to systematically investigate the properties of
the bipartite networks that affect the dynamics of the projected networks.

### Acknowledgements

We would like to thank M. Gerlach and J. Poncela-Casasnovas for their
thoughtful consideration and feedback on an early version of this work. We
would like to acknowledge financial support from the Department of Defense
Army Research Office, Grant W911NF-14-1-0259 and John, and Leslie McQuown. The
funder had no role in study design, data collection, and analysis, decision to
publish or preparation of the paper.

### Author contributions

H.A.L and L.G.A.A. contributed equally to this work. H.A.L, L.G.A.A., and
L.A.N.A. conceived and designed the study. H.A.L and L.G.A.A. performed the
numerical simulations and statistical analysis, created the figures, and wrote
the first draft of the paper. H.A.L, L.G.A.A., and L.A.N.A. wrote, read and
approved the final version of the paper.

### Conflicts of interest

The authors declare no competing interests.

## References

  * Daley and Gani (2001) Daryl J Daley and Joe Gani,  _Epidemic modelling: an introduction_ , 15 (Cambridge University Press, 2001).
  * Pastor-Satorras and Vespignani (2001) Romualdo Pastor-Satorras and Alessandro Vespignani, “Epidemic spreading in scale-free networks,” Physical Review Letters 86, 3200 (2001).
  * Liljeros  _et al._ (2003) Fredrik Liljeros, Christofer R. Edling, and Luís A. Nunes Amaral, “Sexual networks: Implication for the transmission of sexually transmitted infection,” Microbes Infect. 5, 189–196 (2003).
  * Barrat  _et al._ (2008) Alain Barrat, Marc Barthelemy, and Alessandro Vespignani,  _Dynamical Processes on Complex Networks_ (Cambridge University Press, 2008).
  * Chinazzi  _et al._ (2020) Matteo Chinazzi, Jessica T Davis, Marco Ajelli, Corrado Gioannini, Maria Litvinova, Stefano Merler, Ana Pastore y Piontti, Kunpeng Mu, Luca Rossi, Kaiyuan Sun, _et al._ , “The effect of travel restrictions on the spread of the 2019 novel coronavirus (covid-19) outbreak,” Science 368, 395–400 (2020).
  * Maier and Brockmann (2020) Benjamin F Maier and Dirk Brockmann, “Effective containment explains subexponential growth in recent confirmed covid-19 cases in china,” Science 368, 742–746 (2020).
  * Amaral and Ottino (2004) Luis A N Amaral and Julio M Ottino, “Complex networks,” The European Physical Journal B 38, 147–162 (2004).
  * Newman (2010) Mark Newman,  _Networks: An Introduction_ (Oxford University Press, 2010) pp. 1–784.
  * Barrat  _et al._ (2004) Alain Barrat, Marc Barthélemy, Ramualdo Pastor-Satorras, and Alessandro Vespignani, “The architecture of complex weighted networks.” Proceedings of the National Academy of Sciences 101, 3747–52 (2004).
  * Stouffer  _et al._ (2005) Daniel B. Stouffer, José Camacho, Roger Guimerà, C. A. Ng, and Luís A. Nunes Amaral, “Quantitative patterns in the structure of model and empirical food webs,” Ecology 86, 1301–1311 (2005).
  * Buldyrev  _et al._ (2010) Sergey V. Buldyrev, Roni Parshani, Gerald Paul, H. Eugene Stanley, and Shlomo Havlin, “Catastrophic cascade of failures in interdependent networks,” Nature 464, 1025–1028 (2010).
  * Kivelä  _et al._ (2014) Mikko Kivelä, Alex Arenas, Marc Barthelemy, James P. Gleeson, Yamir Moreno, and Mason A. Porter, “Multilayer networks,” J. Complex Networks 2, 203–271 (2014).
  * Fagiolo  _et al._ (2008) Giorgio Fagiolo, Javier Reyes, and Stefano Schiavo, “On the topological properties of the world trade web: A weighted network analysis,” Physica A: Statistical Mechanics and its Applications 387, 3868–3873 (2008).
  * Alves  _et al._ (2020) Luiz G A Alves, Alberto Aleta, Francisco A Rodrigues, Yamir Moreno, and Luis A Nunes Amaral, “Centrality anomalies in complex networks as a result of model over-simplification,” New Journal of Physics 22, 013043 (2020).
  * Boccaletti  _et al._ (2014) Stefano Boccaletti, Ginestra Bianconi, Regino Criado, Charo I Del Genio, Jesús Gómez-Gardenes, Miguel Romance, Irene Sendina-Nadal, Zhen Wang, and Massimiliano Zanin, “The structure and dynamics of multilayer networks,” Physics Reports 544, 1–122 (2014).
  * Gorochowski and Richardson (2017) Thomas E. Gorochowski and Thomas O. Richardson, “How behaviour and the environment influence transmission in mobile groups,” in  _Temporal Netw. Epidemiol._ , edited by Naoki Masuda and Petter Holme (2017) pp. 17–42.
  * Li  _et al._ (2017) Aming Li, Sean P. Cornelius, Yang-Yu Liu, Long Wang, and Albert-László Barabási, “The fundamental advantages of temporal networks,” Science 358, 1042–1046 (2017).
  * Miritello  _et al._ (2011) Giovanna Miritello, Esteban Moro, and Rubén Lara, “Dynamical strength of social ties in information spreading,” Physical Review E 83, 045102 (2011).
  * Liu  _et al._ (2014) Suyu Liu, Nicola Perra, Márton Karsai, and Alessandro Vespignani, “Controlling contagion processes in activity driven networks,” Physical Review Letters 112, 118702 (2014).
  * Holme and Saramäki (2012) Petter Holme and Jari Saramäki, “Temporal networks,” Physics Reports 519, 97–125 (2012).
  * Pastor-Satorras  _et al._ (2015) Romualdo Pastor-Satorras, Claudio Castellano, Piet Van Mieghem, and Alessandro Vespignani, “Epidemic processes in complex networks,” Reviews of Modern Physics 87, 925 (2015).
  * Boccaletti  _et al._ (2006) Stefano Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D. U. Hwang, “Complex networks: Structure and dynamics,” Physics Reports 424, 175 (2006).
  * Perra  _et al._ (2012) Nicola Perra, Bruno Gonçalves, Romualdo Pastor-Satorras, and Alessandro Vespignani, “Activity driven modeling of time varying networks,” Scientific Reports 2, 469 (2012).
  * Takaguchi  _et al._ (2013) Taro Takaguchi, Naoki Masuda, and Petter Holme, “Bursty communication patterns facilitate spreading in a threshold-based epidemic dynamics,” PloS ONE 8, e68629 (2013).
  * Karimi and Holme (2013) Fariba Karimi and Petter Holme, “Threshold model of cascades in empirical temporal networks,” Physica A: Statistical Mechanics and its Applications 392, 3476–3483 (2013).
  * Perotti  _et al._ (2014) Juan Ignacio Perotti, Hang-Hyun Jo, Petter Holme, and Jari Saramäki, “Temporal network sparsity and the slowing down of spreading,” arXiv:1411.5553 (2014).
  * Holme (2018) Petter Holme, “Probing empirical contact networks by simulation of spreading dynamics,” in  _Complex Spreading Phenomena in Social Systems_ (Springer, 2018) pp. 109–124.
  * Petri and Barrat (2018) Giovanni Petri and Alain Barrat, “Simplicial Activity Driven Model,” Physical Review Letters 121 (2018).
  * De Domenico  _et al._ (2016) Manlio De Domenico, Clara Granell, Mason A Porter, and Alex Arenas, “The physics of spreading processes in multilayer networks,” Nature Physics 12, 901–906 (2016).
  * Krings  _et al._ (2012) Gautier Krings, Márton Karsai, Sebastian Bernhardsson, Vincent D Blondel, and Jari Saramäki, “Effects of time window size and placement on the structure of an aggregated communication network,” EPJ Data Sci. 1, 4 (2012).
  * Clauset and Eagle (2012) Aaron Clauset and Nathan Eagle, “Persistence and periodicity in a dynamic proximity network,” arXiv:1211.7343 , 1–5 (2012).
  * Ribeiro  _et al._ (2013) Bruno Ribeiro, Nicola Perra, and Andrea Baronchelli, “Quantifying the effect of temporal resolution on time-varying networks,” Sci. Rep. 3, 3006 (2013).
  * Darst  _et al._ (2016) Richard K. Darst, Clara Granell, Alex Arenas, Sergio Gómez, Jari Saramäki, and Santo Fortunato, “Detection of timescales in evolving complex systems,” Scientific Reports 6, 39713 (2016).
  * Benson  _et al._ (2018) Austin R Benson, Rediet Abebe, Michael T Schaub, Ali Jadbabaie, and Jon Kleinberg, “Simplicial closure and higher-order link prediction,” Proceedings of the National Academy of Sciences 115, E11221–E11230 (2018).
  * Bisanzio  _et al._ (2010) Donal Bisanzio, Luigi Bertolotti, Laura Tomassone, Giusi Amore, Charlotte Ragagli, Alessandro Mannelli, Mario Giacobini, and Paolo Provero, “Modeling the spread of vector-borne diseases on bipartite networks,” PLoS One 5, e13796 (2010).
  * Wen and Zhong (2012) Luosheng Wen and Jiang Zhong, “Global asymptotic stability and a property of the sis model on bipartite networks,” Nonlinear Analysis: Real World Applications 13, 967–976 (2012).
  * Han  _et al._ (2018) She Han, Mei Sun, Benjamin Chris Ampimah, and Dun Han, “Epidemic spread in bipartite network by considering risk awareness,” Physica A: Statistical Mechanics and Its Applications 492, 1909–1916 (2018).
  * Pavlopoulos  _et al._ (2018) Georgios A Pavlopoulos, Panagiota I Kontou, Athanasia Pavlopoulou, Costas Bouyioukos, Evripides Markou, and Pantelis G Bagos, “Bipartite graphs in systems biology and medicine: a survey of methods and applications,” GigaScience 7, giy014 (2018).
  * Cao  _et al._ (2011) Lang Cao, Xun Li, Bing Wang, and Kazuyuki Aihara, “Rendezvous effects in the diffusion process on bipartite metapopulation networks,” Physical Review E 84, 041936 (2011).
  * Maslov and Sneppen (2002) Sergei Maslov and Kim Sneppen, “Specificity and stability in topology of protein networks,” Science 296, 910–913 (2002).
  * Pilosof  _et al._ (2017) Shai Pilosof, Mason A Porter, Mercedes Pascual, and Sonia Kéfi, “The multilayer nature of ecological networks,” Nature Ecology & Evolution 1, 1–9 (2017).
  * Börner  _et al._ (2004) Katy Börner, Jeegar T Maru, and Robert L Goldstone, “The simultaneous evolution of author and paper networks.” Proceedings of the National Academy of Sciences 101, 5266–73 (2004).
  * Helleringer and Kohler (2007) Stéphane Helleringer and Hans-Peter Kohler, “Sexual network structure and the spread of HIV in Africa: evidence from Likoma Island, Malawi,” Aids 21, 2323–2332 (2007).
  * Rocha  _et al._ (2011) Luis E.C. Rocha, Fredrik Liljeros, and Petter Holme, “Simulated epidemics in an empirical spatiotemporal network of 50,185 sexual contacts,” PLoS Computational Biology 7, e1001109 (2011).
  * Guimera  _et al._ (2005) Roger Guimera, Brian Uzzi, Jarrett Spiro, and Luis A Nunes Amaral, “Team assembly mechanisms determine collaboration network structure and team performance,” Science 308, 697–702 (2005).
  * Leskovec  _et al._ (2006) Jure Leskovec, Ajit Singh, and Jon Kleinberg, “Patterns of influence in a recommendation network,” in  _Pacific-Asia Conference on Knowledge Discovery and Data Mining_ (Springer, 2006) pp. 380–389.
  * Zhou  _et al._ (2007) Tao Zhou, Jie Ren, Matúš Medo, and Yi Cheng Zhang, “Bipartite network projection and personal recommendation,” Physical Review E 76, 046115 (2007).
  * Godoy-Lorite  _et al._ (2016) Antonia Godoy-Lorite, Roger Guimerà, Cristopher Moore, and Marta Sales-Pardo, “Accurate and scalable social recommendation using mixed-membership stochastic block models.” Proceedings of the National Academy of Sciences 113, 14207–14212 (2016).
  * Rivera  _et al._ (2010) Mark T Rivera, Sara B Soderstrom, and Brian Uzzi, “Dynamics of dyads in social networks: Assortative, relational, and proximity mechanisms,” Annual Review of Sociology 36, 91–115 (2010).
  * Aaron Clauset and Sainz (2016) Ellen Tucker Aaron Clauset and Matthias Sainz, “The Colorado Index of Complex Networks,”  (2016).
  * Holme (2005) Petter Holme, “Network reachability of real-world contact sequences,” Physical Review E 71, 046119 (2005).
  * Shakarian  _et al._ (2015) Paulo Shakarian, Abhinav Bhatnagar, Ashkan Aleali, Elham Shaabani, and Ruocheng Guo,  _Diffusion in social networks_ (Springer, 2015) pp. 1–101.
  * Myers  _et al._ (2012) Seth A Myers, Chenguang Zhu, and Jure Leskovec, “Information diffusion and external influence in networks,” in  _Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining_ (2012) pp. 33–41.
  * Nematzadeh  _et al._ (2014) Azadeh Nematzadeh, Emilio Ferrara, Alessandro Flammini, and Yong Yeol Ahn, “Optimal network modularity for information diffusion,” Physical Review Letters 113, 088701 (2014).
  * Masuda  _et al._ (2017) Naoki Masuda, Mason A Porter, and Renaud Lambiotte, “Random walks and diffusion on networks,” Physics Reports 716, 1–58 (2017).
  * Valdano  _et al._ (2015a) Eugenio Valdano, Luca Ferreri, Chiara Poletto, and Vittoria Colizza, “Analytical computation of the epidemic threshold on temporal networks,” Physical Review X 5, 021005 (2015a).
  * Valdano  _et al._ (2015b) Eugenio Valdano, Chiara Poletto, Armando Giovannini, Diana Palma, Lara Savini, and Vittoria Colizza, “Predicting epidemic risk from past temporal contact data,” PLoS Computational Biology 11, e1004152 (2015b).
  * Gleeson  _et al._ (2016) James P Gleeson, Kevin P O’Sullivan, Raquel A Baños, and Yamir Moreno, “Effects of network structure, competition and memory time on social spreading phenomena,” Physical Review X 6, 021019 (2016).
  * Weiss  _et al._ (2014) Curtis H Weiss, Julia Poncela-Casasnovas, Joshua I Glaser, Adam R Pah, Stephen D Persell, David W Baker, Richard G Wunderink, and Luís A Nunes Amaral, “Adoption of a high-impact innovation in a homogeneous population,” Physical Review X 4, 041008 (2014).
  * Wasserman  _et al._ (2015) Max Wasserman, Xiao Han T Zeng, and Luís A Nunes Amaral, “Cross-evaluation of metrics to estimate the significance of creative works,” Proceedings of the National Academy of Sciences 112, 1281–1286 (2015).
  * Anderson and May (1992) Roy M. Anderson and Robert M. May,  _Infectious Diseases of Humans: Dynamics and Control_ (Oxford University Press, 1992) p. 768.
  * Dodds and Watts (2004) Peter Sheridan Dodds and Duncan J Watts, “Universal behavior in a generalized model of contagion,” Physical Review Letters 92, 218701 (2004).
  * Dodds and Watts (2005) Peter Sheridan Dodds and Duncan J Watts, “A generalized model of social and biological contagion,” Journal of theoretical biology 232, 587–604 (2005).
  * (64) See Supplemental Material at https://journals.aps.org/pre/supplemental/10.1103/PhysRevE.103.022320/sm.pdf for data description, additional figures, supplementary text, and model algorithms.
  * Guimerà  _et al._ (2007) Roger Guimerà, Marta Sales-Pardo, and Luís A.Nunes Amaral, “Module identification in bipartite and directed networks,” Physical Review E 76, 036102 (2007).
  * Colizza  _et al._ (2006) V. Colizza, A. Flammini, M. A. Serrano, and A. Vespignani, “Detecting rich-club ordering in complex networks,” Nature Physics 2, 110–115 (2006).
  * Masuda and Holme (2017) Naoki Masuda and Petter Holme,  _Temporal Network Epidemiology_ (Springer, 2017).
  * Keeling and Eames (2005) Matt J. Keeling and Ken T.D. Eames, “Networks and epidemic models,” Journal of Royal Society Interface 2, 295–307 (2005).
  * de Arruda  _et al._ (2020) Guilherme Ferraz de Arruda, Giovanni Petri, Francisco A Rodrigues, and Yamir Moreno, “Impact of the distribution of recovery rates on disease spreading in complex networks,” Physical Review Research 2, 013046 (2020).
  * Alt and Godau (1995) Helmut Alt and Michael Godau, “Computing the Fréchet distance between two polygonal curves,” International Journal of Computational Geometry & Applications 05, 75–91 (1995).
  * Granovetter (1978) Mark Granovetter, “Threshold models of collective behavior,” American Journal of Sociology 83, 1420–1443 (1978).
  * Tang  _et al._ (2009) Jie Tang, Jimeng Sun, Chi Wang, and Zi Yang, “Social influence analysis in large-scale networks,” in  _Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining_ (2009) pp. 807–816.
  * Aggarwal and Ryoo (2011) Jake K Aggarwal and Michael S Ryoo, “Human activity analysis: A review,” ACM Computing Surveys (CSUR) 43, 1–43 (2011).
  * Chen  _et al._ (2010) Wei Chen, Yifei Yuan, and Li Zhang, “Scalable influence maximization in social networks under the linear threshold model,” IEEE international conference on data mining , 88–97 (2010).
  * Valente (1996) Thomas W. Valente, “Network models of the diffusion of innovations,” Computational & Mathematical Organization Theory 2, 163–164 (1996).
  * Gross  _et al._ (2006) Thilo Gross, Carlos J Dommar D’Lima, and Bernd Blasius, “Epidemic dynamics on an adaptive network,” Physical Review Letters 96, 208701 (2006).