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Chaos 32, 023107 (2022); https://doi.org/10.1063/5.0070358 32, 023107
© 2022 Author(s).
Dynamical analysis of evolutionary public
goods game on signed networks
Cite as: Chaos 32, 023107 (2022); https://doi.org/10.1063/5.0070358
Submitted: 06 September 2021 • Accepted: 18 January 2022 • Published Online: 04 February 2022
Xiaowen Zhong, Guo Huang, Ningning Wang, et al.
Chaos ARTICLE scitation.org/journal/cha
Dynamical analysis of evolutionary public goods
game on signed networks
Cite as: Chaos 32, 023107 (2022); doi: 10.1063/5.0070358
Submitted: 6 September 2021 ·Accepted: 18 January 2022 ·
Published Online: 4 February 2022
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Xiaowen Zhong, Guo Huang, Ningning Wang, Ying Fan,a)and Zengru Di
AFFILIATIONS
School of Systems Science, Beijing Normal University, 100875 Beijing, China
a)Author to whom correspondence should be addressed: yfan@bnu.edu.cn
ABSTRACT
In evolutionary dynamics, the population structure and multiplayer interactions significantly impact the evolution of cooperation levels.
Previous works mainly focus on the theoretical analysis of multiplayer games on regular networks or pairwise games on complex networks.
Combining these two factors, complex networks and multiplayer games, we obtain the fixation probability and fixation time of the evolution-
ary public goods game in a structured population represented by a signed network. We devise a stochastic framework for estimating fixation
probability with weak mistrust or strong mistrust mechanisms and develop a deterministic replicator equation to predict the expected density
of cooperators when the system evolves to the equilibrium on a signed network. Specifically, the most interesting result is that negative edges
diversify the cooperation steady state, evolving in three different patterns of fixed probability in Erdös–Rényi signed and Watts–Strogatz
signed networks with the new “strong mistrust” mechanism.
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0070358
Evolutionary games originate from genetic ecology’s conflict and
cooperation between animals and plants. In most cases, genetic
ecologists find that the evolutionary results can be studied by
the game theory without relying on any rationality hypothesis.
Economists use this theory to analyze social habits, norms, or
institutions in human society and explore a variety of mech-
anisms. As we all know, common resources require partial-
group contribution, and everyone in the society has access to
use resources. Because of human psychology and resource con-
straints, it is impossible for everyone to adopt a cooperative
strategy. However, the defective strategy, which involves enjoying
benefits without any investment, is advantageous to an individ-
ual. Hence, facilitating cooperation in the game becomes the
focus of the public goods game research. In the classical research
framework of fixation probability and fixation time, we find that
strong mistrust from human psychology will make the system
change qualitatively with negative edges. With the strong mistrust
mechanism, negative edges diversify the cooperation steady state
and make the game more challenging to predict, which is quite
different from a situation involving only positive edges.
I. INTRODUCTION
Cooperation has become a recurring theme in the evolu-
tionary game theory since Darwin’s study.1–7Numerous studies
show that evolutionary game dynamics are performed in a finite,
homogeneous, or heterogeneous population.8–10 The main chal-
lenge is identifying the selective differences, i.e., natural selection
tends to evolve into a situation where one strategy is entirely
dominant. Thereinto, fixation probability has been studied using
random walks,11 the coalescent theory,12 the pairwise comparison
process,13 etc.
Recently, the fixation probability in complex networks, pro-
viding a convenient framework for characterizing the population
structure, has been renewed.14–16 For instance, Su et al.17 propose
a multiplayer game model on a regular weighted network, and its
payoff depends on the strategy of interacting players and the edge
diversity connecting with its neighbors. When generalizing spa-
tial public goods games (PGGs) in a multilayer weighted network,
Ref. 18 provides an analytical formula to predict when coopera-
tion is favored over defection under weak selection. Unlike other
networks, the signed networks provide a convenient framework
Chaos 32, 023107 (2022); doi: 10.1063/5.0070358 32, 023107-1
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for simultaneously analyzing positive and negative social relations.
These signed networks, which are weighted networks with positive
links indicating friendship or trust and negative links indicating hos-
tile relationships or distrust between participants, have been widely
employed in a variety of social contexts.19 Researchers have increas-
ingly attracted attention to opinion dynamics20,21 and information
spreading22 on signed networks. The presence of negative edges
causes the system to take longer to evolve to the consensus state in
opinion dynamics, inhibiting the spread of information.
However, evolutionary games on signed networks23 have not
addressed the numerical estimation of fixation probability and fixa-
tion time. In this study, we compute the fixation probability and fix-
ation time of evolutionary multi-agents game dynamics in a signed
network with weak selection and investigate the effects of negative
edges on fixation probability. We propose a stochastic framework
to calculate the fixation probability and fixation time based on the
Markov process with the absorbing state.24–26 By introducing the
negative edges, the strategy updating probability with negative edges
is different from that with positive edges, which brings a new chal-
lenge to compute transition probability in a multi-agent cooperative
game.
Most studies focused on strategy updating via genetic repro-
duction or cultural imitation. However, strategy updates that
consider human psychology in social networks are common in
behavioral experiments. Human psychology is one of the key ele-
ments leading to the limited rationality of humans, resulting in
“mutation” in the strategy updating process. Previous studies27–29
found that an individual’s offspring with a specific type mutates
into another type of individual with the probability U. The system
then guarantees that all types are constantly represented in a pop-
ulation through modifying the mutation rate U and the extinction
of one type does not lead to a prolonged absence of that type. It has
been previously observed that high mutation rate reduces cooper-
ators’ clustering, hindering their evolutionary success.30 However,
introducing negative edges into the evolutionary game, we provide
a new type of strategy updating process, i.e., strong mistrust mech-
anism, based on relationship. Individuals can produce an offspring
of the opposite type to replace the dead individual when connected
with negative edges, allowing individuals to explore new strategies.
Hence, the absorbing Markov process is not applicable in this con-
dition. Here, we aim to investigate the effect of the strong mistrust
mechanism on the frequency of cooperators, and then we present
an analytical approximation to the evolutionary dynamics by using
an ergodic Markov chain with a fixed probability distribution.
Unlike the weak mistrust mechanism, two critical values can emerge,
dividing the population system into three stages on signed networks.
This study is organized as follows. After describing some basic
properties of the evolutionary PGG in Sec. II, we analyze the evo-
lutionary PGG’s fixation probability and fixation time on signed
networks with a weak mistrust mechanism based on the absorbing
Markov chain. Additionally, we use the pair approximation method
in Sec. III to derive a deterministic equation of the expectation of
the cooperation level. In Sec. IV, we used the ergodic Markov chain
to predict a fixed probability of the evolutionary PGG on signed
networks with strong mistrust mechanisms. In this section, we
discover that some interesting results can be obtained after
introducing negative edges.
II. EVOLUTIONARY PGG ON SIGNED NETWORKS WITH
A WEAK MISTRUST MECHANISM
A. Game model and its strategy updating mechanism
On finite structured populations, we consider an evolutionary
multiplayer game. Each node represents an individual on the net-
work. The individuals interact with their neighbors via a signed
network in which the weight of the links is 1 or −1, with posi-
tive links representing friendship or trust and negative representing
hostile relationships or distrust between them.
For each generation of the multiplayer game, an individual i
will participate in ki+1 games organized by its neighbors and one
game organized by itself, and then accumulate its benefit from all
games it involved, where kidenotes the degree of individual i. Every
individual can choose strategy C (cooperate) or D (defect). The pay-
off for these two strategies is different and also related to the choices
of other players in each round; where aiand bidenote the payoffs
obtained by players adopting strategy C and D when facing iplay-
ers with strategy C, respectively,31 shown in Table I. Additionally,
we focus on a simple PGG with non-exclusive and non-competitive
properties. Thus, aiand bican be expressed as
ai=(i+1)m−kc,
bi=im,(1)
where the term krefers to the number of the games that an indi-
vidual participates. mdenotes the income and cdenotes the cost.
Income needs to be higher than the cost, then people are willing to
contribute to the public goods game.32
Strategy updating rules—Moran process.33 At each time step,
one individual is randomly chosen to die and one of his neighbors
will reproduce an offspring with a probability proportional to its
fitness.34 Additionally, the identical offspring (clone) will replace the
dead individual.
If piis individual i’s payoff, his fitness can be expressed as
fi=1−ω+ω×pi, (2)
where ωdetermines the intensity of selection, indicating how much
the payoff affects fitness. When ω=0, the payoff does not affect
fitness, and strategies C and D are neutral variables. When ω=1,
while the intensity of selection is extremely strong, fitness is entirely
determined by the payoff. In this study, we set ω=0.02 when
referring to weak selection and ω=0.8 when referring to strong
selection.
Assuming individual ias a focal player, his/her neighbor jis
selected to reproduce the offspring with a probability
Wsj→si=fj
Pk∈Vifk
, (3)
TABLE I. The payoffs of the two strategies.
Opposing C players 0 1 ·· · i·· · ki...
Ca0a1·· · ai·· · aki...
Db0b1·· · bi·· · bki...
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FIG. 1. Strategy updating process with the weak mistrust mechanism. Eac h node
represents a site and is occupied either by a cooperator (blue) or by a defector
(red). Each edge can be negative (dashed line) or positive (solid line). Under the
weak mistrust mechanism, a random individual dies, and then it is replaced by a
neighbor sampled proportionally to fitness with higher probability if they connected
with a positive edge or with lower probability if they connected with a negative
edge.
in which Viis the set of individual i’s neighbors.35 Considering the
relationship between individuals, individuals tend to assume that
their friend’s behavior is correct and imitate it. However, when there
is a hostile relationship, individuals are more resistant to learning
from the other one’s behavior, even if the other person’s income is
higher than his.36 Therefore, if there is a positive link(friend/trust)
between iand j, the strategy updating probability is W∗µ+,µ+>1,
where µ+denotes the friendliness factor. Whereas if a negative link
connects iand j,iwill learn the strategy of jwith the probability of
W∗µ−, 0 < µ−<1, where µ−denotes the hostile factor, shown in
Fig. 1. Individual transition rates can be used to formulate dynamics
in terms of a master equation. This is especially convenient in the
mean-field case, i.e., for population games.
B. Evolutionary PGG on homogeneous networks with
weak selection in a finite population
Pair approximation35 can be used to capture the evolution of
strategies in a population described using a homogeneous struc-
ture. Let pCand pDdenote the frequency of strategies C and D in
the population. pCD denotes the frequency of CD pairs. For an indi-
vidual with strategy D, the probability for him/her to find someone
with strategy C is qC|D. To calculate the payoff of a specific strat-
egy individual obtained from the PGG, we should be aware of the
number of his/her neighbors and neighbors’ neighbors. Based on the
mean-field approximation assumption, we compute the mean pay-
off πD
C—the player adopting strategy C, who is the neighbor of the
selected individual with strategy D, who has kCneighbors adopting
strategy C, see details in Appendix 1 in supplementary material.37
Because the system contains two absorbing states: a population full
of strategy C or D, the analytical method of the absorbing Markov
chain can be used to calculate the fixation probability and fixa-
tion time.38 In the absorbing Markov chain, we transform the state
transition matrix to its canonical form, that is,
M→Q R
0I,
see more details in Appendix 2 in supplementary material.
According to the definition of absorbing Markov chain, the
element nij of the fundamental matrix N=(I−Q)−1gives the
expected number of times that the process is in the transient state
φjif started in the transient state φi. Therefore, the mean fixation
time tthat the system is attracted to the absorbing state from any
initial state φiis
t=Ns,
where sis a column vector with each entry equal to 1. Furthermore,
using the theorem, we can compute the fixation probability,
B=NR,
where Ris the same as in the canonical form. The element in the
Bmatrix describes the probability that the system starts from the
transient state and, finally, enters the absorbing state.
C. Deterministic equation on homogeneous networks
with strong selection in an infinite population
Consider the deterministic replicator dynamics in a structured
population when N→ ∞ with a strong selection.3,39 When the
population size goes to infinity, the population structure can be con-
sidered a well-mixed population. When the pair approximation is
used, the system can be described by only two variables, pCand
pCC. Consider the dynamics of the expectation value of pCand pCC ,
which is the frequency of cooperator and CC pairs, respectively. By
deriving the deterministic equation of pCand pCC, a cooperator is
selected to be replaced by his defective neighbor, the frequency pC
of cooperators in the population decreases by 1
Nwith the proba-
bility Pr 1pC= − 1
Nand the proportion of CC pairs pCC will also
be decreased by 2kC
kN with the probability Pr 1pCC = − 2kC
kN . Other-
wise, if a defector is selected to be replaced by his cooperative neigh-
bors, then pCincreases by 1
Nwith the probability Pr 1pC=1
Nand
pCC will be increased by 2kC
kN with the probability Pr 1pCC =2kC
kN ,
see more details in Appendix 3 in the supplementary material. Thus,
in the limit N→ ∞, the dynamical mean-field equations of pCcan
be written as
˙
pC=1
NPr 1pC=1
N+−1
NPr 1pC= − 1
N. (4)
Chaos 32, 023107 (2022); doi: 10.1063/5.0070358 32, 023107-3
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Regarding pairs, the time derivative of pCC in Eq. (5) is given by
˙
pCC =
k
X
kc=0
2kC
kN Pr 1pCC =2kC
kN
+
k
X
kc=0−2kC
kN Pr 1pCC = − 2kC
kN . (5)
D. Numerical results
This section describes the numerical simulation of the study,
which is consistent with the above-mentioned theories. Note that
the results are shown using 100 realizations on 100 signed networks
with 100 individuals. Unless otherwise specified, the friendliness fac-
tor µ+is set to be 1.5 while the hostile factor µ−is set to be 0.5.
Specifically, we ran simulations on multiple homogeneous networks,
including Erdös–Rényi (ER) signed and Watts–Strogatz (WS) net-
works. It is critical to define the ER signed and WS signed networks.
FIG. 2. Fixation probabilities Bto pure cooperators and expected fixation time hti/Nwith dif ferent initial states for (a) ER signed networks and (b) WS signed networks. The
dots are numerical simulations, and the solid lines are from the theory. Network size N=100, mean degree hki = 6, and the proportion of negative edges is 0.2. Income
mis 2 and cost cis 1.5. Here, ρ0is the proportion of cooperators in an initial state.
Chaos 32, 023107 (2022); doi: 10.1063/5.0070358 32, 023107-4
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FIG. 3. Heatmaps of the expected fraction of C strategy in terms of mand propor tion of negative edges pnunder the weak mistrust mechanism. (a) Theoretical results
derived from the deterministic equation [Eq. (4)]. (b) Simulation result on the ER signed network. (c) Simulation result on the WS signed network. The initial fraction of C
strategy ρ0=0.5 and the cost c=1.5.
ER signed network is generated with two steps. First, an unweighted
and undirected network is generated based on the random net-
work generation algorithm.40 Second, for each edge, we re-define
its weight as −1 with probability pn, and otherwise 1. Similarly, WS
signed network is generated in the same way except for the small-
world network generation algorithm.41 We investigate the fixation
probability Band expected fixation time hti/Nin weak selection
by numerical simulations on ER signed networks and WS signed
networks to compare the simulation results with theoretical results.
Moreover, we calculate the mean square error(MSE) between the
theoretical curves and simulation results of each sub-figure in Fig. 2.
In this study, we discover that both as homogeneous networks, ER
signed networks are better than WS signed networks in terms of
predicting fixation probability because the MSE of fixation prob-
ability in an ER signed network (0.071) is less than that in a WS
signed network (0.29). However, in terms of the expected fixation
time, the WS signed network (MSE of expected fixation time=7.65)
has better prediction than the ER signed network (MSE of expected
fixation time=39.42). A specific error can be observed between
the prediction curves and the simulation results, which is related
to the mean-field assumption that the degrees of their neighbors
are approximate to their degrees when calculating the payoffs of
individuals.
Furthermore, it is essential to look into N→ ∞ in detail when
investigating the deterministic equation of the expected density of
cooperators pCin strong selection (w=0.8). We mapped the sys-
tem’s phase space to fully understand evolutionary dynamics as a
function of the enhancement factor mand a fraction of negative
edges pn. Moreover, we compare theoretical results with simula-
tion results for both ER and WS signed networks. Figure 3 shows
three distinct phases: cooperators only (C), cooperators and defec-
tors (C+D), and defectors only (D). When the enhancement factor
mis significantly small, the system evolves to the D phase irrespec-
tive of the fraction of negative edges. As mincreases, the C+D phase
emerges in the system in a narrow window of intermediate m. For
a relatively high m, the only C phase dominates with an extensive
regime of a fraction of negative edges. The enhancement factor m
and fraction of negative edges pnaffect the steady state of the coop-
erator frequencies in this updating mechanism. However, it is m
that largely determined the direction that the system evolves. The
expected density of the cooperators pCon the ER signed network
and the WS signed network is highly consistent with the prediction
of theoretical results except that the area of the C+D phase is more
significant than that in theoretical results.
To demonstrate the effect of the enhancement factor mon pC,
we consider the variation of pCwhen system reached the equilibrium
in different m. When m≥1.15, the final Steady state of the coop-
erators’ frequencies is essentially the same, regardless of the initial
cooperator density (Fig. 4). By comparing the result obtained when
m<1.15, the evolution of the cooperation level depends on the
initial fraction of cooperators. Therefore, m=1.15 will be a critical
point to separate the abovementioned two different stages.
FIG. 4. Expected fraction of C strategy (cooperators) in equilibrium in depen-
dence on the initial fraction of C strategy ρ0for different values of m. Interestingly,
above a critical mvalue, the fraction of C strategy in equilibrium is independent
of ρ0.
Chaos 32, 023107 (2022); doi: 10.1063/5.0070358 32, 023107-5
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As previously discussed, negative edges have small effect on
evolutionary PGG with a weak mistrust mechanism—the expected
fraction of cooperators depends only on payoffs. Furthermore, the
weak mistrust mechanism does not produce complex phenomena in
the population system. Negative edges decrease the imitation proba-
bility between individuals, which can cause a rare mutation rate that
is justified under genetic reproduction; however, it is less appropri-
ate to model social learning. Nevertheless, a high mutation rate is
beneficial for maintaining population diversity, and the system is
more likely to evolve into complexity. The effect of negative edges
on the evolutionary PGG with a strong mistrust mechanism should
be investigated.
III. EVOLUTIONARY PGG ON SIGNED NETWORKS
WITH STRONG MISTRUST MECHANISM
A. Evolutionary PGG on a homogeneous network with
weak selection in finite population
The Moran process of evolutionary PGG on a signed network
yields simple analytical results, but the method is not entirely appro-
priate for the strong mistrust mechanism. When two people are
friends, there is a strong sense of trust between them, and they do
not question each other’s strategic choices. However, if the two peo-
ple are in a hostile relationship, there is a low level of trust between
them, and one of them will aggressively doubt the other’s approach,
by assuming that the other is using a reverse strategy to obtain a
more significant payoff. Therefore, the model is modified as fol-
lows. The Moran process assumes that an individual X is chosen
uniformly at random to be replaced, and its neighbors compete for
the vacancy proportionally to his fitness. If an individual X and
Y are connected by a positive edge, Y will reproduce an offspring
of the same strategy to replace X. If a negative edge connects an
individual X to an individual Y, an offspring with the opposite strat-
egy is propagated to replace X, shown in Fig. 5. This evolutionary
process can also be considered as a Markov chain. This type of
strategy updating mechanism allows for the absence of an absorb-
ing state in the system. Unlike the weak mistrust mechanism, by
introducing negative edges in the strong mistrust mechanism, the
transition probabilities T+(n−1),T0(n),T−(n+1)can be varied
not only among pairs of nodes holding different strategies but also
among pairs of nodes holding the same strategy, see more details in
Appendix 4 in supplementary material.
The state transition matrix is
M=
1−pn
N
pn
N0··· 0
T−(1)T0(1)T+(1)··· 0
0T−(2)T0(2)··· 0
.
.
..
.
..
.
.....
.
.
0 0 ··· pn
N1−pn
N
.
The state transition matrix Mshows that this cannot be calculated
using the previously proposed method because it is not an absorb-
ing Markov chain.42 However, it is possible to go from one state to
another, and as pii >0, i=0, 1, 2, ...,N, the period of each state is
1. For these above-mentioned reasons, the Markov chain is ergodic.
FIG. 5. Strategy updating process with a strong mistrust mechanism. Each node
represents a site and is occupied either by a cooperator (blue) or by a defector
(red). Each edge can be negative (dashed line) or positive (solid line). Under this
strong mistrust mechanism, a random individual dies, and then a neighbor sam-
pled proportionally to fitness generates an offspring with the same strategy to
replace the dead one if they connected with a positive edge or with an opposite
strategy to replace the dead one if they connected with negative edge.
For an ergodic Markov chain, there is a fixed probability distribution
g=g0,g1,...,gNsuch that g=gM and gis strictly positive.
B. The deterministic equation on homogeneous
networks with a strong selection in an infinite
population
We used a modified form of abovementioned replicator equa-
tions to describe the evolutionary process to analyze the determin-
istic equation of the expectation density under the strong mistrust
mechanism. Compared with the weak mistrust mechanism, the
probability of a defector being replaced by a cooperator is, whether
his/her connecting link is positive or negative,
0=kC1−ω+ωπ D
C·1−pn+kD1−ω+ωπ D
D·pn
kC1−ω+ωπ D
C+k−kC1−ω+ωπD
D. (6)
Here, a defector can become a cooperator not only from cooperative
neighbors connected to him via the positive link but also from his
defective neighbors connected to him via the negative link. Similarly,
the probability of a cooperator being replaced by a defector is
3=k−kC1−ω+ωπC
D·1−pn+kC·1−ω+ωπ C
C·pn
kC1−ω+ωπ C
C+k−kC1−ω+ωπC
D.
(7)
Chaos 32, 023107 (2022); doi: 10.1063/5.0070358 32, 023107-6
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Substitute Eqs. (6) and (7) into the formula of Pr 1pC= − 1
N,
Pr 1pCC = − 2kC
kN , Pr 1pC=1
N, Pr 1pCC =2kC
kN , respectively.
We can derive the ODE of pCand pCC by the following:
˙
pC=1
NPr 1pC=1
N+−1
NPr 1pC= − 1
N, (8)
˙
pCC =
k
X
kc=0
2kC
kN Pr 1pCC =2kC
kN
+
k
X
kc=0−2kC
kN Pr 1pCC = − 2kC
kN , (9)
which are very similar to those in Eqs. (4) and (5). See more details
in Appendix 5 in supplementary material.
C. Numerical results
This section summarizes and discusses the primary results
of evolutionary PGG with strong mistrust mechanism to compare
results with the weak one. Under this strong mistrust mechanism,
we compute the system’s fixed probability distribution, i.e. the
probability of each state occurring when the system reaches sta-
bility. Additionally, the entire fixed probability distribution shows
a single-peaked curve and the system reaches maximum probabil-
ity close to the state ¯ρ=0.5 (Fig. 6), where ¯ρis the proportion
of cooperators, indicating that the system is most likely to evolve
to state that half of the populations are cooperators. This result
may be explained by the fact that the introduction of negative
edges in the evolutionary mechanism with strong mistrust leads
to a high probability of switching strategies in individuals. Fur-
thermore, another type of strategy can emerge among individuals
using the same strategy connected to negative edges. Therefore,
population strategies change frequently, and the system eventu-
ally evolves to a state where cooperators and defectors coexist with
¯ρ=0.5.
To investigate the effect of the proportion of negative edges on
the evolution of cooperation, we conduct evolutionary PGG on ER
signed and WS signed networks with different proportions of nega-
tive edges (Fig. 7). It is observed that as the proportion of negative
edges pnincreases, the system evolves into three different stages.
In the ER signed network, when pn≤0.02 [Fig. 7(a1)], the system
will be likely to evolve to extreme states (state of all defectors or
all cooperators) just like the bi-polarization distribution of defec-
tors and cooperators shown by the blue lines, while the other states
is small and approaches zero. When 0.03 ≤pn≤0.05 [Fig. 7(a2)],
the probability of each state’s emergence is fluctuant, but they are all
contained within a small range shown by the red lines. Hence, few
differences push the system evolve to each state equiprobably. More-
over, the fixed probability distribution is in a transition phase, which
indicates that it is difficult to determine the final state of the system
from the fixed probability distribution. However, when the propor-
tion of negative edges is greater than 0.05 [Fig. 7(a3)], the fixed
probability distribution of the system will present a unimodal curve
and reach the maximum value around a cooperation level of 0.5.
This pulse-type distribution predicts that the increase of negative
edges, or even complete occupation, is conducive to the long-term
and matched coexistence between different strategies holders. A very
similar situation is observed in the WS signed network except that
in the first stage, when pn≤0.03 [Fig. 7(b1)], the system evolves to a
state of full defectors with the probability of almost 80%. This result
FIG. 6. Fixed probability under strong mistrust mechanism in the ER signed network and the WS signed network. The gray dots denote the simulation results, and solid
lines denote the theoretical results. The x-axis represents the states with different proportion of cooperators. (a) ER signed network; (b) WS signed network. The proportion
of negative edges pnis 0.2 and income mis 2, cost cis 1.5.
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FIG. 7. The effect of the proportion of negative edges pnon fixed probability distribution. Each line corresponds to the fixed probability distribution with a specific value of
pn. The subgraphs above describe the fixed probability distribution on ER signed network. (a1)pn∈[0, 0.02] in the first stage; (a2)pn∈[0.03, 0.05] in the second stage;
(a3)pn∈(0.05, 1] in the third stage. The subgraphs below describe the fixed probability distribution on the WS signed network. (b1)pn∈[0, 0.03] in the first stage; (b2)
pn∈[0.04, 0.05] in the second stage; (b3)pn∈(0.05, 1] in the third stage. Income mis 2 and cost cis 1.5.
can be explained by that the introduction of negative edges may
make the individuals explore new strategy under our strong mistrust
mechanism, which can maintain and even prosper the cooperation
in the population.
To compare with the weak mistrust mechanism, we examine
evolutionary PGG’s cooperation level both derived from determin-
istic Eq. (8) and simulation results on ER and WS signed networks
with the same initial condition: ρ0=0.5 and c=1.5. The numeri-
cal results for different mand pnon homogeneous signed networks
are shown in Fig. 8. The simulation results on ER signed networks
[Fig. 8(b)] and WS signed networks [Fig. 8(c)] agree with theoret-
ical results in Fig. 8(a), which confirms that our method to predict
the expected density of cooperators is appropriate. But, it should be
noted that in the simulation results, the proportion of negative edges
pnin the defectors only phase is lower than that in the theoretical
result. For low mand pn, the population evolves to a low coopera-
tion level, whereas for high mbut small pn, the population evolves to
almost full cooperators state. Because of the strong mistrust causing
frequent changes in the strategy updating process, the population
will evolve to a state of coexistence of cooperators and defectors for
a wide range of pnand m, as shown in Fig. 8. When the population
is in equilibrium, the state of full defectors appears only in the top
left corner of the phase diagram with low m<1 and low pn<0.1.
When the enhancement factor mis low, negative edges are beneficial
to the cooperation, otherwise, inhibit the spreading of cooperation
in the population.
Chaos 32, 023107 (2022); doi: 10.1063/5.0070358 32, 023107-8
Published under an exclusive license by AIP Publishing
Chaos ARTICLE scitation.org/journal/cha
FIG. 8. Heatmap of expected fraction of C strategy in terms of m and propor tion of negative edges pnunder the strong mistrust mechanism. (a) Theoretical results derived
from Eq. (8). (b) Simulation result on ER signed network. (c) Simulation result on WS signed network. The initial fraction of C strategy p0=0.5 and the cost c=1.5.
IV. CONCLUSION
In this study, we propose a stochastic process framework for
studying the evolutionary process of PGG in homogeneous signed
networks (ER and WS signed networks) with weak mistrust and
strong mistrust mechanisms. Furthermore, we derive a system of
deterministic equations that describes how the expected frequency
of cooperative strategy changes over time through pair approxima-
tion on homogeneous signed networks. The results show that the
fixation probability and fixation time from the stochastic process
are consistent with the theoretical model under the weak mistrust
mechanism, as well as the fixed probability distribution under the
strong mistrust mechanism. We discover the effect of the proportion
of negative edges on the evolutionary process through deterministic
equations of the expected density of cooperators. Under the weak
mistrust mechanism, the negative edges did not affect the evolu-
tionary process. However, for the strong mistrust mechanism, one
unanticipated finding was that it made the evolutionary process
show three different patterns of fixed probability in ER signed and
WS signed networks. There are two critical points ρ1and ρ2(ρ1<ρ2)
to separate the system’s evolutionary directions. For a proportion
of negative value below ρ1, the population evolves toward extreme
states (full cooperators state or full defectors state). But when the
proportion of negative edges is ρ∈(ρ1,ρ2), it is difficult to predict
what state the system will eventually evolve to. However, for a pro-
portion of negative value above ρ2, the population will evolve to a
state that cooperators and defectors coexist.
Consequently, we show that the negative edge eliminates the
polarization of individual strategy and promotes the system to
evolve into a balanced state where two strategies coexist and have
the same number of advocates. However, the spatial distribution
of two kinds of strategies is often related to the structural balance
theorem of the signed networks. The match states between coopera-
tion and defection are an interesting phenomenon when the links of
the signed network are all negative. In addition, the aforementioned
results are conducted on homogeneous signed networks and further
research should be undertaken to investigate the heterogeneous
signed networks.
SUPPLEMENTARY MATERIAL
See the supplementary material for the mathematical deriva-
tion of evolutionary PGG model on signed networks with weak or
strong mistrust mechanism in detail.
ACKNOWLEDGMENTS
This work is supported by the National Natural Science Foun-
dation of China (NNSFC, Grant Nos. 61573065 and 71731002).
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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