=Paper= {{Paper |id=Vol-2917/paper1 |storemode=property |title=On the Effect of Complex Network Topology in Managing Epidemic Outbreaks |pdfUrl=https://ceur-ws.org/Vol-2917/paper1.pdf |volume=Vol-2917 |authors=Yulian Kuryliak,Michael Emmerich,Dmytro Dosyn |dblpUrl=https://dblp.org/rec/conf/momlet/KuryliakED21 }} ==On the Effect of Complex Network Topology in Managing Epidemic Outbreaks== https://ceur-ws.org/Vol-2917/paper1.pdf

On the Effect of Complex Network Topology in
Managing Epidemic Outbreaks
Yulian Kuryliak1 , Michael Emmerich2 and Dmytro Dosyn1
1
Institute of Computer Science and Information Technologies, ICSIT of Lviv Polytechnic National University, 12 Stepan
Bandera street, 79000 Lviv, Ukraine
2
Leiden Institute of Advanced Computer Science, LIACS Leiden University, Niels Bohrweg 1, 2333CA Leiden, The
Netherlands

Abstract
This paper investigates the effect of the structure of the contact network on the dynamics of the epi-
demic outbreak. In particular, we focus on the peak number of critically infected nodes (PCIN), deter-
mining the maximum workload of intensive healthcare units that should be kept low when managing
an epidemic. As a realistic model and simulation method, we developed a continuous-time Markov
chain model, and an open-source available efficient simulation software based on Gillespie’s Stochas-
tic Simulation Algorithm (SSA). Virus propagation is compared on random graph models featuring a
selected range of complex network topologies: Erdős–Rényi, Watts-Strogatz, Barabási–Albert and com-
plete graph (Clique). Continuous-time Markov chains are used to simulate the infection process. The
simulation was performed in networks with 200 nodes and different numbers of edges. In addition, we
study age- and gender-determined and weighted characteristics of nodes on the PCIN as well as the cor-
relation of macroscopic graph characteristics such as the clustering coefficient and the average shortest
path length. The analysis used the data of the demographic distribution of Ukraine as of 2020 and data
on mortality from COVID-19 in Ukraine, as of December 16, 2020. It is proved that the deterministic
characteristics slightly lower values of critically infected, in small networks for the used dataset. The
simulations show that the increase of the average shortest path length is significant on the reduction of
the PCIN, whereas other characteristics such as clustering and age distribution, are of lesser importance.

Keywords
epidemic outbreak, complex networks, network topology, contact process, continuous time Markov
chain

1. Introduction
Predicting the dynamics of the spread of infectious diseases in society under different cir-
cumstances is an urgent scientific and practical social problem. The recent outbreak of the
COVID-19 pandemic has increased interest in epidemiology and, as a result, the science of
complex networks that can be used to approximate social networks.
It is important to start the fight against the virus as soon as it is detected, but it usually takes a
long time to invent a vaccine[1], approve it, and distribute it. It is therefore important to contain

MoMLeT+DS 2021: 3rd International Workshop on Modern Machine Learning Technologies and Data Science, June 5,
2021, Lviv-Shatsk, Ukraine
" yulian.kuryliak.kn.2017@lpnu.ua (Y. Kuryliak); m.t.m.emmerich@liacs.leidenuniv.nl (M. Emmerich);
dmytro.h.dosyn@lpnu.ua (D. Dosyn)
0000-0002-6600-2637 (Y. Kuryliak); 0000-0002-7342-2090 (M. Emmerich); 0000-0003-4040-4467 (D. Dosyn)
© 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org)
the spread of the virus until the time when vaccination becomes available and make treatment
of the disease available to those in need. The recent COVID-19 pandemic has shown that it
is of paramount importance to reduce the number of patients that have to be treated at the
same time in intensive care units (ICUs) to avoid the risks that hospitals run out of capacity. To
reduce this number, non-pharmaceutical means, such as contact restrictions, will be effective[2].
The question, which contact restrictions are most effective, is a topic of ongoing research[1].
Classical epidemiology has mainly focused on idealized homogeneous network structures such
as complete graphs or networks where each person has about the same number of contacts.
However, a more detailed look at how changes in the contact network structure will affect the
spread of an epidemic is necessary. To gain insights into methods for effectively slowing down
the spread of the virus in the network, it will be useful to conduct a study into the sensitivity
of the rate of spread of the virus to the topology of the network. The existing literature on
this topic is mainly focused on asymptotical analysis[3] or the early stage of the spread of an
epidemic, where the reduction of the largest eigenvalue of the contact network (adjacency
matrix) plays a crucial role[4,5]. However, once an epidemic is already spread out across a
network, other dynamics need to be taken into account. This topic, however, received relatively
small attention in the literature[3].
Another downside of the classical studies is that they focus mainly on the epidemic threshold
(or reproduction number, which is inversely proportional to it), whereas in a real pandemic
other factors deserve more attention when it comes to managing the outbreak: According to
research[6], forecasting and limiting the peak loads on the hospitals is one of the main tasks in
a pandemic. An important indicator that depends on the basic epidemic parameters (topology
of the contact network and the rate of infection) is the peak number of simultaneously infected
nodes (PCIN). In[7], a cholera epidemic outbreak was simulated using Continuous Time Markov
Chains (CTMCs) for a SIR model, with nodes of the network that may be susceptible, infected
with symptoms, infected without symptoms, and excluded. In[8], CTMCs were used to model
the distribution of COVID-19 based on already known statistics. A similar simulation was
performed in[9]. Both studies use the SIR model and all three do not take into account network
topology and population demographics.
The objective of this paper is to provide a realistic, yet efficient, method for simulation of the
spreading process, and first results on the effects of network topology, with a focus on the peak
number of infected individuals. Instead of asymptotic analysis (such as differential equations
and mean-field models), we propose using the stochastic simulation algorithm (or Gillespie’s
algorithm[10]) for simulating CTMCs. This method is realistic, encompasses all stages of an
outbreak, flexible and efficient in assessing the effect of network topology on contact networks
of moderate to large size. In contrast to mean-field methods such as differential equations, the
stochastic simulation-based analysis also has the advantage that error margins of the model
can easily be assessed, which allows for a robust risk assessment.
2. Methodology
2.1. Epidemiological Model
To solve problems related to the analysis of the dynamics of the spread of infectious diseases in
a population, usually are considering certain generalized models according to which individuals
of such a population are in one of three main possible states: 1) susceptible to infection (S), 2)
infected and able infect others (I) and 3) removed from the list of susceptible and infected due
to the acquisition of immunity or death due to the fatal course of the disease (R). According to
this division, the main epidemiological models are the SI, SIS, and SIR models[3]. For diseases
with a high rate of spread and fast occurrence of symptoms, models are usually used that do
not take into account mortality and fertility, as well as population aging, i. e., it is assumed that
the demographic distribution is stable throughout the epidemic.
Due to the need to model the spread of COVID-19 among the population, the SIR model was
selected as the most appropriate. Although COVID-19 has a certain incubation period, it is
difficult to determine, therefore, we consider a person contagious immediately after infection.
The possibility of returning the removed persons to a state of susceptibility due to the gradual
loss of acquired immunity is also not taken into account. The main parameter is the infection rate
𝜆. The infection rate depends on the intensity of the contact, its type, and the contagiousness
of the virus itself. Since the virus can affect people in different ways depending on their
gender and age, as well as depending on comorbidities and other factors, in order to take
into account these features in the model of disease spread in Ukraine, it was decided to use
the demographic distribution as of 2020. (Age structure of the population of Ukraine https:
//www.lv.ukrstat.gov.ua/dem/piramid/all.php). Data on the number of infected and mortality
as of December 16, 2020, were also used, see operational monitoring of the situation around
COVID-19: https://nszu.gov.ua/e-data/dashboard/covid19.

2.2. Models of the topology of contacts in a social network
The dynamics of viral spread depend on the network topology, including local characteristics
(e.g., local clustering and degrees of nodes) and global characteristics (e.g., eigenvalue spectrum,
shortest path characteristics). Real social networks do not have a clear structure but may have
certain patterns, so to describe them approximately it is common practice to use random graph
models of complex networks, where networks are generated according to certain rules and
probability distributions. Within a random graph model, certain properties of networks are
typically shared, such as small world or clustering characteristics, and so on. Networks of human
contacts are displayed in the form of graphs where an edge connects individuals (nodes) that
are in contact with each other. Because human interaction is often bidirectional, we consider
here undirected graphs, noting that the simulation methods in this paper can be easily adapted
to directed graphs.
The most common network topology discussed in classical epidemiology is that of a Complete
Graph, where each node is connected to every other node. However, in real networks, other
topologies are more common, such as small-world networks, and scale-free networks. Small-
world models assume a small graph distance between people in a social network (an example is
a rule of "6 handshakes"). It is scale-free networks that are closest to real networks, including
social ones. Scale-free networks are subject to the power law, where the probability 𝑃 (𝑘) of the
degree 𝑘 of the nodes follows the law 𝑃 (𝑘)∼𝑘 −𝛾 . Each of the topologies is described in detail
in [11].

2.2.1. Erdős–Rényi model
The Erdős–Rényi graph model is a random graph where 𝐿 links are randomly distributed across
a set of 𝑁 nodes. The degree of a node in an ER graph follows a binomial distribution, and not
a power law. It is also known, that for 𝐿 > 𝑁 log 𝑁 the network tends to be fully connected,
whereas if 𝐿 < 𝑁 the network tends to be fragmented into many isolated components.

2.2.2. Watts-Strogatz model
The Watts-Strogatz model is a small-world model. The construction of the Watts-Strogatz model
begins with a grid in which each node is connected strictly to 𝑚 neighbors, after which each of
the edges can reconnect to a randomly selected node with a probability 𝑝, this process is called
reconnection. As a result of reconnection, the average distance between nodes decreases. The
Watts-Strogatz model is characterized by high clustering, and also by a small average shortest
path 𝑙 which decreases with increasing probability of reconnection of node 𝑝 (to a certain value).

2.2.3. Barabási–Albert model
The Barabási–Albert model is a scale-free network with exponent 𝛾 = 3, and it is built on the
principle of growth and preferential attachment, that is, at each step a node with 𝑚 edges is
added. New nodes are linked to others by preferential attachment to nodes with a probability
that is proportional to the degree of the other node. The preferential attachment process leads
to the creation of hubs, i. e. a few nodes to which are connected many other nodes with, on
average, smaller degrees. These hubs serve as shortcuts for information or diseases spreading
through the networks. Therefore the average shortest distance in such networks tends to be
small.

2.3. Infection Model and Stochastic Simulation Algorithm
For modeling the infection process in this work we use Continuous Time Markov chains
(CTMCs)[12]. In contrast to other models such as discrete Markov chains and cellular automata,
it features realistic modeling of time. Moreover, it is not based on asymptotical simplifications
and stability of mean values as do the classical epidemiological models based on differential
equations. To tame the state-space explosion we make use of the underlying principles of
Gillespie’s stochastic simulation algorithm: (1) simulation of time between two state transitions
and the simulation of the next state can be separated, and (2) only a small number - linear in
the number of nodes - of state transitions can occur in a single step of the simulation[10].
In the CTMC model of the contact process, all rates at which transitions occur between
network states are described in a generator matrix = 𝑄. This matrix is a 2𝑁 × 2𝑁 matrix and
𝑞𝑖𝑗 is the rate at which the system changes from network state 𝑗 given it is in state 𝑖, and
(a) Complete graph for n=8 (b) Erdős–Rényi model for 𝑁 = 50, 𝐿 = 50

𝑞𝑖𝑖 = − 𝑖̸=𝑗 𝑞𝑖𝑗 is the rate of leaving state 𝑖 (on the diagonal). Using a state space of size 2𝑁
∑︀
becomes however computationally prohibitive for larger 𝑁 .
In the specific case of simulating the SIR epidemic process, the only non-zero transition
rates are those where a single additional node gets infected or where a node that is infected is
removed from the network. Let 𝑖 denote the current state and 𝑖+𝑗 denote a network state where
node 𝑗 ∈ [1,𝑁 ] is a node that potentially gets infected in addition to the previously infected
Figure 2: Example of expected time

nodes in state 𝑖. Then

if node 𝑗 is susceptible;
⎧
⎨𝜆𝑣(𝑗),
⎪
𝑙,𝑙̸=𝑖 𝑞𝑖+𝑙 , if 𝑖 == 𝑖+𝑗 ;
∑︀
𝑞𝑖,𝑖+𝑗 =
0, if node 𝑗 is not susceptible.
⎪
⎩

Here 𝜆 is the infection rate of the virus, 𝑣(𝑗) is the number of infected neighbors of the vertex
𝑗 in state 𝑖. The transition time from the state is exponentially distributed:
{︃
𝜆𝑒−𝜃𝑥 , if 𝑥 ≥ 0;
𝐹 (𝑥) =
0, if 𝑥 < 0

where 𝜃 = 𝑞𝑖𝑖 . The expected value is given by 1/𝜃, and in the CTMC by 1/𝑞𝑖𝑖 . The probability
of transition from state 𝑖 to state 𝑗, that is 𝑝𝑖𝑗 is determined by the formula:
𝑞𝑖𝑗
𝑝𝑖𝑗 = ∑︀
𝑙,𝑙̸=𝑖 𝑞𝑖𝑙

and it can be simulated by “roulette wheel” simulation[13]. Generate a uniform
∑︀ random number
∑︀𝑘 𝑧
in [0,1), and choose 𝑘 (the vertex which will be infected) for which holds 𝑘−1
𝑙 𝑝𝑙 < 𝑧 < 𝑙 𝑝𝑙

3. Results
We have developed open-source simulation software

https://github.com/YulianKuryliak/EpidemicOutbreak

. A slightly modified SIR model was used, in which there are two types of infected - infected
and critically infected (ie, people which need medical care). We assume that an infected person
becomes contagious immediately after infection. Infectiousness in the case of a normal infection
lasts 10 days, after which the person ceases to be considered contagious, according to the SIR
model, and is removed from the study network. Based on the current statistics taken from the
death statistics in Ukraine from COVID-19 as of December 16, 2020, a certain probability of
death due to disease was chosen, namely the weighted average mortality in case of mortality by
age and number of people of this age. The weighted probability of death 𝑝𝑑 is calculated by the
formula: ∑︀
𝑎𝑚𝑖 * 𝑝𝑚𝑖 + 𝑎𝑤𝑖 * 𝑝𝑤𝑖
𝑝𝑑 = 𝑖 ,
𝑡𝑜𝑡𝑎𝑙
where 𝑎𝑚𝑖 - number of men in age range 𝑎𝑤𝑖 - number of women in age range 𝑝𝑚𝑖 - probability
of death for men of age range 𝑝𝑤𝑖 - probability of death for women of age range 𝑡𝑜𝑡𝑎𝑙 -
total number of people 𝑖 - age range It is important to predict the occupancy of hospitals,
which depends on PCIM. According to the WHO (https://www.who.int/indonesia/news/detail/
08-03-2020-knowing-the-risk-for-covid-19), there are about 20% of people who develop disease
symptoms in the case of COVID-19. Thus, the probability of critical infection will be considered
a weighted mortality value, namely, 0.018 increased by 0.2. We will also assume that a critically
infected individual spreads the disease within 14 consecutive days, after which they are removed
from the network.
Description of input parameters. The experiments used the following parameters.

• network size: 200 was selected for all experiments;
• infection rate;
• network type: can be a complete graph, Erdős–Rényi model, Barabási–Albert model,
Watts-Strogatz model;
• type of probability of critical infection: weighted by the number and age for all nodes or
separately for each node;
• the number of edges (in the case of the Barabási–Albert and Watts-Strogatz models) or
the total number of edges in the network (for the Erdős–Rényi model, initial for the node);
• term of spread of the disease (in the usual case): set to 10 consecutive days;
• period of spread of the disease by critically infected persons: 14 consecutive days are set.

3.1. Study of the influence of individual and weighted probability of critical
infection
The importance of taking into account the individual age for each of the nodes was tested.
For this purpose, 36 simulations were used with the same and different probability of critical
infection, which has a characteristic age dependence. Since all network topologies are generated
regardless of age and gender (in this numerical experiment they were assigned randomly with
probabilities corresponding to the demographic distribution of Ukraine in 2020), an experiment
was conducted for the worst case of the epidemic - a full graph with 200 nodes, the initial
number of patients - 1 and a virus infection rate of 0.002. The number of infected in each of the
following points of time is shown in Fig. 3.
When modeling the spread of infection, taking into account the dependence of the probability
of infection on the age and sex of each person in the group, or without such a dependence, we
obtained two similar curves of the dynamics of spread.
The median values of the number of infected and critically infected persons at each time
point were further investigated. For simulations with the same probability of critical infection,
the ratio of the number of critically infected to the number of all infected is equal to 0.218
(critically infected / all infected) at any time.
For simulations with a differentiated probability of critical infection, the average value of the
ratio was 0.212, the median value was 0.212, and the standard deviation was 0.013 for the interval
0 ... 45 consecutive days; and a mean of 0.214, a median of 0.213, and a standard deviation of
Figure 3: Comparison of simulations with the same and different probabilities of critical infection (CI),
with an error of ± 25% of the median values.

0.0018 for a period of 10 ... 30 consecutive days (at this time, according to the data obtained
from the simulations, a large number of infected people appear).
The curve of critically infected with the individual probability of critical infection is below the
curve with the same probability of critical infection. The number of critically infected people
varies quantitatively but not qualitatively depending on the individual probability of becoming
critically ill, which may be due to fewer older people in the network, as the selected number of
nodes is not enough to accurately reproduction the demographic distribution.

3.2. Study of the influence of network topology on the number of
simultaneously infected nodes
For comparison, the most common variants of complex networks with different topologies
were selected, namely, a random network model Erdős–Rényi, a small-world network model
Watts–Strogatz, and a scale-free network model Barabási–Albert. For an objective comparison
and compared them with the complete graph. Similar average nodes are selected, and hence the
number of edges in the networks.
The values in this and the following tables are averaged for a sample of 100 networks.
As can be seen from the data contained in Table 1, a similar average degree of node network
properties (namely, the clustering factor and the average shortest path) differ significantly. To
study the influence of parameters, 32. . . 40 simulations were performed for each of the networks.
With the same infection rate for different networks, we have a different number of infected at
each point in time, and therefore a different number of critically infected at the same time(Fig.4).
As mentioned earlier, the number of infected in the same time is important for predicting hospital
occupancy, therefore, experiments were performed for different infection rate to estimate the
maximum number of co-patients for these rates in each of the networks.
As demonstrated in Fig.5, infection in the complete graph occurs much faster than in other
Table 1
Parameters of the network. * : total number of edges in the network. ** : reconnection factor of 0.1
Network Initial Clustering Network Mean Median Diameter Average
model number of coefficient average degree degree of net- shortest
edges for (global) clustering of node of node work path
node coefficient length
Erdős – 400* 0.0198 0.021 4 4 8 3.91
Rényi
Watts - 2 0.2567 0.28 4 4 10 5.09
Strogatz**
Barabási – 2 0.0348 0.058 3.97 3 6 3.5
Albert
Complete 199 1 1 199 199 1 1
graph

Figure 4: Infection curves for the infection rate 𝜆 = 0.2 in different topologies

models. Infection in the Erdős–Rényi model occurs faster than in the Watts-Strogatz model, but
the Erdős–Rényi model has an asymptote close to 190 nodes because there are nodes that are
not part of the giant cluster. Infection in the Barabási–Albert model occurs faster than in the
Erdős–Rényi model.
According to Table 1, the Barabási–Albert model has a lower mean shortest distance and an
average clustering factor. The Erdős–Rényi model has a higher mean shortest distance from
the Barabási–Albert model, but a lower one than the Watts-Strogatz model, and the lowest
clustering coefficient. The Watts-Strogatz model has the highest clustering coefficient and the
average shortest distance.
Based on the simulation results and the data from Table 1, it can be concluded that for virus
(a)

(b)
Figure 5: Curves of the number of simultaneously infected for different network topologies depending
on the infection rate 𝜆 (a) - on a linear scale; (b) - on a logarithmic scale for the x axis.
spread in networks with a similar average node level, the average shortest distance has a much
greater effect on the virus spread rate than the clustering coefitient.

3.3. Comparison of the influence of the number of connections in the
Erdős–Rényi and Barabási–Albert networks on the peak value of
infected
Since the average values of the number of contacts per person are often used, it is worth paying
attention to the random Erdős–Rényi model. It was compared with the Barabási–Albert model,
which is closer to real networks. And also because these models have a similar average shortest
distance. According to the data shown in Tables 2 and 3, the networks generated according
to the model have significantly higher Barabási–Albert clustering coefficients than randomly
generated networks as predicted by the Erdős–Rényi model.

Table 2
Parameters of the Erdős–Rényi network
Number of Clustering Network Mean Median Diameter Average
edges in coefficient average degree of degree of of network shortest
network (global) clustering node node path length
coefficient
200 0.00768 0.0068 2 2 16 6.55
400 0.0198 0.021 4 4 8 3.91
1000 0.05 0.05 10 10 4 2.54
2000 0.1 0.1 20 20 3 2.02
4000 0.2 0.2 40 40 3 1.8
Complete 1 1 199 199 1 1
graph

Table 3
Parameters of the Barabási–Albert network
Initial num- Clustering Network Mean Median Diameter Average
ber of edges coefficient average degree of degree of of network shortest
for node (global) clustering node node path length
coefficient
1 0 0 1.99 1 14 6.29
2 0.0348 0.058 3.97 3 6 3.5
4 0.084 0.1 7,9 6 4 2.65
10 0.187 0.193 19.45 14 3 2.05
20 0.3 0.3 37.9 29 3 1.81
Complete 1 1 199 199 1 1
graph

In Fig. 6 it can be seen that for arbitrarily large virus infectivity, the curves for 200 and 400
edges have asymptotes of about 150 and 190 simultaneously infected nodes. This is due to the
(a)

(b)
Figure 6: Curves of the peak number of simultaneously infected for different number of edges (PCIN)
depending on the infection rate 𝜆 for the Erdős–Rényi model (a) - on a linear scale; (b) - on a logarithmic
scale for x axis.
(a)

(b)
Figure 7: Curves of the peak number of simultaneously infected nodes (PCIN) for different number of
edges depending on the infection rate 𝜆 for the Barabási–Albert model (a) - on a linear scale; (b) - on a
logarithmic scale for x axis.
detached vertices from the giant component. Therefore, such a model should not be used if it is
important to connect all nodes to a giant cluster.
The Barabási–Albert model is a better network option for a small number of edges because,
unlike the Erdős–Rényi network, it cannot contain unconnected nodes to a giant cluster. With
the same number of edges, the Barabási–Albert model has a higher average of the shortest
distance at a small average degree of the node, so the virus in such a network spreads faster.
With a large number of edges in the network, the average shortest distance between nodes
decreases, so the difference in the spread of the virus in the Erdős–Rényi and Barabási–Albert
networks is minimized.
Although the Barabási–Albert model has higher clustering coefficients, as can be seen from
Fig. 6 and Fig. 7, the difference in virus spread at a similar mean shortest distance is insignificant,
and therefore the clustering coefficients do not significantly affect the PCIN.

4. Conclusion
An epidemic outbreak has been simulated in complex networks generated according to models
such as the Erdős–Rényi random model, the Watts-Strogatz small-world model, the Barabási–Albert
scale-free model, and the complete graph using continuous Markov chains and efficient stochas-
tic simulation techniques for computing their stochastic trajectories.
The importance of taking into account individual node characteristics is analyzed, and it is
proved that they are not very important for large networks and weighted values can be used,
but in small networks, it is difficult to maintain demographic distribution, so it is better to apply
individual node characteristics when they are important.
It has been found that the number of infected at the same time(ie, peak values) and, conse-
quently, the number of critically infected people, depends on the rate of virus transmission on
the network, which determines the workload of hospitals, so it is important to take quarantine
measures to reduce the height of this peak (PCIN).
The rate of virus spread changes at the same infection rate and average node level for different
network models. The virus was found to spread fastest in the Barabási–Albert scale-free model,
slower in the Erdős–Rényi random model, and slowest in the Watts-Strogatz small-world model.
This supports the hypothesis that the speed of virus spread increases with decreasing average
shortest distance. In the example of the Erdős–Rényi, and Barabási–Albert models, it is shown
that at a similar mean shortest distance between nodes, the clustering coefficient has a negligible
effect on the peak value of infected nodes.
Based on this finding, it is obvious that to slow down the spread of the virus, and as a
consequence, to reduce the PCIN, it is necessary to increase the average shortest distance, which
is provided by mass quarantine measures and also by travel restrictions and close monitoring
of network hubs (e.g., individuals or groups of individuals that have many contacts to different
parts of the network).
5. Acknowledgements
Michael Emmerich acknowledges funding from the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie grant agreement Nr. 823866.

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