Osaka et al. Comput Soc Netw (2017) 4:2
DOI 10.1186/s40649-017-0038-2

Open Access

RESEARCH

Effect of direct reciprocity and network
structure on continuing prosperity of social
networking services
Kengo Osaka1, Fujio Toriumi2 and Toshihauru Sugawara1**Correspondence:sugawara@waseda.jp
1
Departmentof Computer Science
and Communications
Engineering, Waseda
University, 3‑4‑1 Okubo,
Shinjuku, Tokyo 169‑8555,
Japan
Full list of author information
is available at the end of thearticleAbstract
Background: Social networking services (SNSs) are widely used as communicative
tools for a variety of purposes. SNSs rely on the users’ individual activities associatedwith some cost and effort, and thus it is not known why users voluntarily continue
to participate in SNSs. Because the structures of SNSs are similar to that of the public
goods (PG) game, some studies have focused on why voluntary activities emerge as
an optimal strategy by modifying the PG game. However, their models do not include
direct reciprocity between users, even though reciprocity is a key mechanism thatevolves and sustains cooperation in human society.
Proposed methods: We developed an abstract SNS model called the reciprocity
rewards and meta-rewards games that include direct reciprocity by extending theexisting models. Then, we investigated how direct reciprocity in an SNS facilitates
cooperation that corresponds to participation in SNS by posting articles and comments and how the structure of the networks of users exerts an influence on the strategies of users using the reciprocity rewards game.
Experimental results: We run reciprocity rewards games on various complex networks and an instance network of Facebook and found that two types of stable cooperation emerged. First, reciprocity slightly improves the rate of cooperation in completegraphs but the improvement is insignificant because of the instability of cooperation.
However, this instability can be avoided by making two assumptions: high degree offun, i.e. articles are read with high probability, and different attitudes to reciprocal andnon-reciprocal agents. We then propose the concept of half free riders to explain whatstrategy sustains cooperation-dominant situations. Second, we indicate that a certain
WS network structure affects users’ optimal strategy and facilitates stable cooperation
without any extra assumptions. We give a detailed analysis of the different characteristics of the two types of cooperation-dominant situations and the effect of the memoryof reciprocal agents on cooperation.
Keywords: Social networking services, Meta-norms game, Reciprocity, Evolutionary
game, Complex networks

Background
Social networking services (SNSs), such as Twitter, Facebook, and Google+, are frequently used by many users around the world not only to share and exchange local information among limited specialized and close-friend groups but also to publish/obtain© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium,provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.

Osaka et al. Comput Soc Netw (2017) 4:2

public information for the purposes of exchanging opinion, advertising, marketing, andpolitics [1, 2]. SNSs are usually run by companies and non-profit organizations but cannot persist without huge amounts of up-to-date content being continually posted byindividual users. Such user activities incur various costs in terms of creating and submitting the content, and why users continue to post articles and comments is not wellunderstood. In addition, free riders (or lurkers) exist, that is, users who just read the content and never post articles. To provide incentives to individual users to keep submitting
content, many SNSs have introduced specific mechanisms, such as providing comments
on articles, comments on comments, indication of the number of followers, signs showing articles have been read, and “Like” buttons. These mechanisms can provide quantitative rewards (e.g. showing the numbers of readers of articles and users who clicked on
the “Like” buttons) as well as psychological rewards that provide feelings of connection
to people and a sense of belonging [3]. However, these incentive mechanisms also relyon users’ voluntary behavior and, thus, incur some cost and time.
As the variety of social media on the Internet continues to grow, it is becoming important to identify the conditions, mechanisms, and/or design methodologies inherent tomaintaining an active and thriving SNS. Thus, a number of approaches, such as network
analysis [4, 5], social psychology [6], analysis of exchange patterns [7], and evolutionary
game theory [8], have been used to tackle this issue. Here, we focus on the evolutionary
game theoretic approach since the mechanisms maintaining for thriving of SNSs havenot been fully investigated from this viewpoint.
In this approach, SNSs are assumed to have the characteristics of a public goods
game, because SNS are shared resources sustained by many users. For example, Toriumi
et al. [8] and Hirahara et al. [9] modeled an SNS as a rewards game (RG) and a metarewards game (MRG), which are dual parts of Axelrod’s meta-norms game, and theirown extension, called an SNS-norms game, to identify evolved behaviors of agents thatmodel SNS users. They then analyzed the conditions under which a cooperation-dominant situation arises, where cooperation in this game corresponds to posting an article and a comment and a cooperation-dominant situation corresponds to a situation inwhich most users post them, so SNSs are active. They found that meta-rewards such ascomments on article comments [8] and a simple (so, low-cost) response mechanism forrewards such as “Like” buttons for articles [10] play an important role in SNSs. However,these studies did not consider social and personal relationships between peers. Furthermore, some SNSs have no mechanism to provide meta-rewards, so another mechanism
and/or interactive structure also seems to affect SNS activities.
Cooperation is a key activity to maintain public goods games. Nowak [11] pointed outthat at least one of five mechanisms—kin selection, direct and indirect reciprocity, network reciprocity, and group selection—is necessary for cooperation to evolve in humansociety. Rand and Nowak [12] subsequently showed empirical evidence of human cooperation sustained by these mechanisms. Other studies have also reported that thesemechanisms, especially reciprocity, exist and that reciprocity plays a crucial role inonline networks [7, 13, 14]. We believe that reciprocity, especially direct reciprocity, isessential in an SNS because connections between users are usually established throughdirect interaction such as “reading articles” and “commenting on articles”.

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Osaka et al. Comput Soc Netw (2017) 4:2

Thus, our study was aimed at investigating the effect on direct reciprocity between
users on cooperation-dominant situations and how network structures affect the users’interaction strategy. For this purpose, we extended an existing abstract model of an
SNS [8] to include direct reciprocity. The extended model is called a reciprocity (meta-)
rewards game whose structure is similar to the (M)RG, but agents tag peer agents anddecide their behaviors on the basis of recent reciprocal behaviors of these peers. We
then tried to determine the circumstances in which the rates of cooperation increase
and when the established cooperation collapses in this game. Note that we particularly
focused on the reciprocity rewards game that has no mechanism for giving a metareward because a number of thriving SNSs do not have meta-rewards as mentioned
above; thereby, we wanted to find another mechanism or interaction structure that promotes a thriving SNS.
We experimentally show that in complete graphs, the Watts–Strogatz (WS)model [15], the connecting nearest neighbor CNN model [16], and Barabasi–Albert(BA) model [17] networks, users do not cooperate with all neighbor agents but rather
with a few close friends established on the basis of past reciprocal behavior. To explain
this phenomenon, we propose the concept of half free riders (or partial lurkers) and discuss the interaction structure needed to maintain a cooperation-dominant situation inwhich the SNS continues to prosper. This is one main difference from the previous studies [8, 10]. Our experimental results suggest that user behavior like a half free rider sustains cooperative activity in all types of network. However, this phenomenon raises incomplete graphs under the additional conditions that (1) agents seldom miss the posted
articles and (2) they do not comment on articles posted by non-reciprocal agents. We
also conducted experiments using WS, BA, and CNN networks and tried to understandhow the network structure affected reciprocity, and thus the prosperity of the SNS. We
found that the network structure strongly affects continuation of a cooperation-dominant situation without any additional conditions. We also examined the (non-)reciprocal
relationships and the effect of varying the term of memory of the reciprocal agents.

Related work
A number of studies have attempted to understand what factors affect social media
by analyzing the network structures of social media. For example, Karamon et al. [4]devised an algorithm that can analyze important network-based features of huge social
networks for better understanding the user behavior therein. Saito and Matsuda [5] analyzed network structures to identify two key types of user who draw the attention ofmany other users on Twitter and showed that one type has higher link reciprocity. In
the field of social psychology, Lin and Lu [6] empirically studied reasons people havefor joining SNSs and found that enjoyment is the most influential factor for people tocontinue using an SNS. They also found a notable difference between genders, i.e. thenumber of active peers is an influential factor for women’s enjoyment, and results intheir continued use of social media, whereas the number of members has an insignificant effect on men’s enjoyment. Qasem et al. [2] attempted to detect influential actors inSNSs in order to identify who can influence others and to improve information diffusion
and market efficiency.

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Osaka et al. Comput Soc Netw (2017) 4:2

A number of studies have paid attention to reciprocity between users by conducting
empirical analyses. For example, Surma [18] focused on reciprocity because it is crucial
in social exchanges. He analyzed reciprocity on Facebook and showed strong empiricalevidence that reciprocity messages sent from a user on online social networks increase
reciprocity reactions from her/his audience. Faraj and Johnson [7] found that network
exchange patterns in an online community are characterized by reciprocity patterns andare different from those characterized by preferential attachment [17]. Takano et al. [14]analyzed player action logs and found that cooperation based on reciprocity could beobserved in a network game. These papers suggest that reciprocity is a key factor in
keeping social media thriving.
A number of studies proposed abstract models of social media and investigated theirproperties. Anderson [19] proposed a game theoretic approach aimed at understanding how social media emerge as a driving force in contemporary marketing and how
this would affect future marketing. Toriumi et al. [8] revealed that SNSs have similar
properties to public goods games, but they correspond to the dual part of the metanorms game [20] because SNSs seem to lack a means of punishing non-cooperators andonly give (psychological) rewards to cooperators. Their model includes two games, therewards game (RG) and the meta-rewards game (MRG), and they indicated that metarewards facilitate cooperation, resulting in active use of an SNS. Hirahara et al. [9, 10]
subsequently extended this model to the SNS-norms game that includes the characteristics of the interaction patterns in SNSs. They ran this game in a variety of complex
networks and found that users at network hubs facilitate posting articles to some degree
even if no meta-reward is provided. However, their study did not take into accountreciprocal relationships between peer agents.
We have already presented a model including reciprocity in an SNS [21], but described
only a few experimental results on it. This paper is an extended version of our previous
workshop paper [21], and it details more thorough experiments using a variety of complex networks and discusses the characteristics of our model through a detailed analysis.

Proposed model for social networking services
Reciprocity rewards and meta‑rewards games

Social networking services are sustainable only when many articles and comments onthem are posted by and shared among many anonymous participants. Although somecost personal time and effort, users can obtain information by reading such posts andcan receive responses that provide feelings of connectivity, empathy, and contentment.
On the other hand, there are many free riders who only read content. Therefore, SNSs
have the properties of public goods that are produced and sustained by continuous
cooperative activities in the SNS community. The game structure of this mutual interference is essentially an n-person prisoner’s dilemma (PD) game. Toriumi et al. [8] proposed RG and MRG as dual games of the norms and meta-norms games proposed by
Axelrod [20] (Fig. 1) and attempted to explain the mechanism of continuous voluntary
participation in SNSs. Although they showed that a meta-reward, which corresponds to“comments on a comment” in an SNS, for example, can provide an incentive to a user
for voluntary participation, they ignored reciprocity, which is crucial to characterizing
activities in SNSs [7, 14, 18]. Hence, we devised the reciprocity rewards game (RRG) andPage 4 of 20

Osaka et al. Comput Soc Netw (2017) 4:2

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sees
rewards
gets
gets

sees

rewards

does not reward
does not see

gets
gets

cooperates

gets
gets

does not reward
does not see

defects

Rewards Game
Meta-Rewards Game

Fig. 1 Meta-rewards and rewards games

the reciprocity meta-rewards game (RMRG) by incorporating reciprocal relationships
among agents into the RG and MRG (see Fig. 2).
Let A = {1, . . . , n} be the set of agents. Agents are connected with a graph G = (A, E),
where E is the set of links between agents. The set of neighbor agents of i ∈ A is denotedby Ai (⊂A). Agents in an R(M)RG game select the strategy of either cooperation or defection. Cooperation indicates posting articles and comments, and defection indicates justreading them. A user who almost always selects defection is defined as a free rider. Agenti ∈ A has three learning parameters: the probability of cooperation (i.e. posting a new
article) Bi, the probability of giving rewards (e.g. posting a comment on the article) toreciprocal agents LCi, and the probability of giving rewards to other (normal) agents LNi.
We call Bi, LCi, and LNi the article posting rate, the reciprocal comment rate, and the normal comment rate, respectively. We also call LCi and LNi the comment rates hereafter. To
apply the genetic algorithm (GA), we express each of these parameters as three bits, sothey take on a discrete, i.e. value 0/7, 1/7,…, or 7/7. This expression is identical to the oneused in the meta-norms game [20]. Agent i has a memory for reciprocal agents Wi (⊂Ai),
which is the set of reciprocal agents that are defined as neighbor agents that posted comments on i’s articles in the most recent TW (>0) rounds. Positive integer TW is called theseesrewards
sees

gets
gets

does not reward

does not reward
does not see

defects

Reciprocity Rewards Game
Reciprocity Meta-Rewards Game

Fig. 2 Reciprocity (meta-)rewards gamerewards

does not see

gets
gets

cooperates

gets
gets

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memory term. We define a round as the period in which all agents perform R(M)RG or(M)RG once with the neighbor agents.
An RRG or RMRG proceeds as follows. For ∀i ∈ A, the parameter Sit (0 ≤ Sit ≤ 1) isdefined randomly or with a certain method in the t-th round (t is a positive integer)
when i is going to post an article. Intuitively, Sit expresses the degree of fun of (or the
degree of attention to) the contents of the article that i is going to post and the associated comments after that. Thus, if Sit ≥ 1 − Bi, i posts a new article with cost F and agent∀j ∈ Ai reads the article posted by i and gains reward M by reading it. Then, j proceeds
to the next phase with probability Sit. Then, agent j comments on the article with probability Lji, where Lji = LCj if i ∈ Wj; otherwise Lji = LNj. Then, j pays a cost C, and i gains
a reward R through j’s comment. The game chain so far is referred to as the RRG.
Subsequent to the RRG, k ∈ Ai reads j’s comment and proceeds to the next phase withprobability Sit. If this happens, k posts a response to the comment with probability Lkj,
where Lkj = LCk if j ∈ Wk and Lkj = LNk if j � ∈ Wk . When k posts it, k pays a cost C ′′ and jgains a reward R′′. The RMRG ends here. Note that, because reciprocity is not taken into
account in RG or MRG, agent i has only two parameters, Bi and LNi (LNi is denoted by
Li in RG) and Wk = ∅. Li in (M)RG is called the comment rate after this. The glossary ofvariables used in the RRG and RMRG is shown in Table 1.
From the definition of reciprocal agents, a reciprocal relationship is mutual or one
sided, and expresses the internal cognitive state of each agent. We think that the two
types of reciprocity exist in the human relationships and thus in SNSs. This will be discussed using the experimental results in the "Analysis of reciprocal relationship" section.
Evolution by genetic algorithm

Reciprocity rewards game and RMRG are evolutionary games, as are the (meta-)norms
game and (M)RG, and we define the evolutionary parameter setting in accordance with
their experiments [8, 20]. One generation of the game is defined as a period in which allagents have four chances to post articles and write/receive comments related to them.1
At the end of one generation, each agent selects two agents as parents from its neighborson the basis of fitness values, which are defined as the cumulative rewards received
minus the cumulative costs incurred during the current generation. This process is continued up to a certain generation.
Each of three learning parameters, Bi, LCi, and LNi, is represented as a three-bit gene,and each agent has a nine-bit gene.2 The initial values of the nine-bit genes are set randomly. The evolution consists of three phases: (1) parent selection, (2) crossover, and (3)mutation. A child agent of i for the next generation is generated as follows. First, in theparent selection phase, i selects two parent agents from i and i’s neighbor agents on thebasis of a probability distribution {j | j ∈ Ai ∪ {i}} defined as�j = (vj − vmin )2 /(vk − vmin )2 ,(1)k∈Ai ∪{i}

1

The length of one generation is determined according to the previous studies [8, 20]. We also conducted our experiments in which the length of generation was set to six, eight, or ten as well as four, but we could not see a clear difference.
2

Likewise, two learning parameters in RG and MRG are encoded into a six-bit gene.

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Table 1 Parameters/variables, descriptions in RRG and RMRG, and values in the experiments below
Parameter

Description

Value

A

Set of agents|A| = 20 or 1000

Bi

i’s probability of posting an article

VariableLCi

i’s probability of posting a comment to a reciprocal agent

VariableLN ii’s probability of posting a comment to a normal agent

Variable

Sit

Degree of fun of i’ article

0 < Sit < 1(random)−3.0

F

Cost of posting article

M

Reward for reading article

1.0

C

Cost of comment

−2.0

R

Reward for receiving comment

9.0

Wi

i’ memory of reciprocal agents

VariableTW

Memory term

1 (if nothing mentioned)

where vk is the fitness value of agent k ∈ A, and vmin = mini∈Ai ∪{i} vi . Then, two newgenes are generated using uniform crossover from the genes in the selected parent
agents and one of them is randomly selected in the crossover phase. In the mutation
phase, each bit of the gene of the child agent is inverted with a probability of 0.005. Thismeans that if there are 20 agents in the network, 0.9 bits will mutate on average. After
that, the derived gene is used for the child agent of i.

Experiments and discussion
Experimental setting

We focused on the reciprocity rewards game (RRG) in the experiments because thereward had by commenting on a comment seems small and thereby insignificant in
an SNS. Furthermore, simple response mechanisms (rewarding mechanisms), such as“Like” buttons and “read” icons, cannot be used to give meta-rewards for these simple
responses. In the first experiment (Exp. 1), we compared the results of the RRG withthose of the RG [8] and investigated the features of the RRG. The agent network (A,E)
was a complete graph; we chose it because it was used in Toriumi [8] and also seems
to be a basis of comparison with those when (A,E) are other types of network. The purpose of Exp. 1 was to investigate how reciprocity affected user behavior in an SNS bycomparing the transitions of the average rates of cooperation, that is, posting an article
or a comment, in the RG or RRG. Then, we also tried to ascertain the causes of sustainability and collapse of cooperation in the RG and RRG, as well as the reciprocal interaction structure in the RRG. The second experiment (Exp. 2) investigated how network
structures affect cooperation and the reciprocal structure in RRG. For this purpose, we
ran RRGs on the WS, CNN, and BA networks and compared their results with those ofcomplete graphs. In the third experiment (Exp. 3), we clarified the effect of the memory
term, TW, on the evolution of cooperation in various networks. Note that the experimental data described below are the average values of 100 independent experimental
runs based on the different random seeds.

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Table 2 Glossary of parameters defined in the "Experiment and discussion" section
Parameter

Description

B

Average of Bi in RG and RRG

LC

Average of LCi in RRG

LN

Average of LNi in RRG

LAverage of Li in RG

S′

Fixed value of the degree of fun of articles

LNmax

Maximum value of LNi

p

Re-wiring probability in the WS modelu

Conversion probability in the CNN modelm

Number of links when a node is added in the BA modelra

Average number of reciprocal agents, riarm

Average number of mutual reciprocal agents, rimroAverage number of one-sided reciprocal agents, rio

Role of reciprocity on cooperation—complete graphs

The parameter values in Exp. 1 are listed in Table 1. These values were determined on
the basis of the experiments of Axelrod [20] and Toriumi et al. [8]. With this setting,
Toriumi et al. [8] found that if the reward of comments was larger than the cost of posting articles and comments, cooperation emerged, but only in the MRG, and cooperation
did not emerge if there was no meta-reward mechanism. Figure 3 indicates the probabilities of posting articles and comments over the generations. Note that we defined theaverage posting article rate as B = i∈A Bi /|A|, the average reciprocal comment rate as
LC = i∈A LCi /|A|, and the average normal comment rate as LN = i∈A LNi /|A| in the
RRG. We also investigated B and the average comment rate L = i∈A Li /|A| in the RG.
Note that the parameters that are and will be defined in this section are summarized inTable 2.
In RG on the complete graph (Fig. 3a), B and L undergo transitions at approximately
0.11 and 0.05, respectively; this result is consistent with that of Toriumi et al. [8] as mentioned above. On the other hand, in the RRG B and LC make transitions at approximately
0.33 and 0.40, respectively, and LN transitions at approximately 0.11 in RRG (Fig. 3b).
These results indicate that the values of B, LC , and LN are larger in the RRG than in
the RG. These figures suggest that by taking into account reciprocity when deciding thea Reward game (RG)

1

L

0.8

0.7

0.7

0.6

0.6

0.5
0.4
0.3

0.3
0.2
0.1

1000
Generation

1500

2000

Fig. 3 Article posting and comment rates in RG and RRG

LN

0.4

0.1
500

LC

0.5

0.2

0

B

0.9

0.8

0

b Reciprocity reward game (RRG)

1
B

Posting rate

Posting rate

0.9

0

0

500

1000
Generation

1500

2000

Osaka et al. Comput Soc Netw (2017) 4:2

1

a RG

0.9

B

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5
0.4
0.3

0.1
1000
Generation

1500

2000

LC
LN

0.3
0.2

500

B

0.4

0.1
0

C

0.5

0.2
0

b RRG

1

L

Posting rate

Posting rate

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0

0

500

1000
Generation

1500

2000

Fig. 4 Article posting and comment rates in RG and RRG (an experimental run)behavior, the activity in the SNS improves, although the amount of improvement is quite
limited. Another observation is that the dispersion of B and LC were large (Fig. 3b).
To see more clearly why B, LC , and LN increased to a limited extent, we investigated
one experimental run of RG and RRG (Fig. 4). Figure 4a indicates that the article posting and comment rates, B and L, in the RG, rose intermittently (but never reached 1.0)
but then quickly dropped off. Such temporary cooperation was caused by mutation.
However, the RG cannot maintain a cooperative situation; for example, when B and L
increased in some agents, these agents temporarily gained rewards. However, because
the free riders were benefited more in such a situation, cooperation soon disappeared.
On the other hand, in the RRG (Fig. 4b), B or LC sometimes reached (near) 1.0 andlasted much longer than in the RG (although they also dropped rapidly afterward). Thismeans that almost all agents cooperate (by posting articles) and post comments on cooperators’ articles during this period. Furthermore, we can see that LC rarely fell to zero.
The difference between the RG and RRG is that agents in the RRG distinguish reciprocal
agents from other agents and so behave differently towards them. That is, agent i with alarge LC comments only on articles posted by reciprocal agents who commented on pastarticles posted by agent i. Such selective comments prevent the collapse of cooperation
by reducing the cumulative cost of making comments. However, they prevent collapse
only when LC > LN ; otherwise, when many agents begin to comment on arbitrary articles without enough rewards and free riders gain high total rewards, cooperation collapses like in the RG.
Now let us explain what the above phenomena correspond to in an actual SNS. WhenSNS users do not consider direct reciprocity (that is, they are in an RG), users who
would otherwise often comment must stop commenting because the RG has no incentive for it and it is more beneficial for agents to behave as free riders. On the other hand,
if individual users consider direct reciprocity when making comments, they preferentially comment on posts by reciprocal users preferentially by referring to their memory,
Wi. Thus, when LC > LN , such selective comment behavior for receiving comments inthe subsequent rounds sustains the norm for cooperation. We also believe that LC > LN
is a reasonable assumption in an actual SNS. We discuss this in the "Discussion" section.
Sustainment or collapse of cooperation

We attempted to identify why the average rates of B, LC , and LN declined. The first factor
that affects sustainability of cooperation is the relative values of LC and LN , as mentioned

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above. Thus, we limited the maximum value of the normal comment rate, LNi , to LNmax(0 ≤ LNmax ≤ 1). Note that LNi is expressed by three bits (it is a value ranging from 0 to7), so the probability of the normal comment rate is (LNi × LNmax )/7. Another factor that
affects the rewards of agents is the degree of fun of the posted articles and the associatedcomments, Sit. Thus, we set Sit to a certain positive constant value 0 ≤ S ′ ≤ 1, and investigated how the evolution of cooperation changed as a result of varying S ′.
Figure 5 plots the changes in B, LC , and LN over generations in the RRG when LNmax
was 0.1 (Fig. 5a) and when S ′ was 1.0 (Fig. 5b). By comparing these figures with Fig. 3b,
we can see that B and LC slightly increased in both cases, and no significant differences
existed between them. However, when we set LNmax = 0.1 and S ′ = 1.0, we observed theemergence of cooperation, as shown in Fig. 6. To investigate the influence of LNmax andS ′ , we conducted a more detailed experiment. The results, plotted in Fig. 7, indicate how
the posting rates varied in accordance with S ′ (LNmax = 0.1, Fig. 7a) and LNmax (S ′ = 1.0,
Fig. 7b). Figure 7b shows that when S ′ = 1.0, if LNmax ≤ 0.55, B was nearly 1.0, but it
gradually decreased to around 0.5 as LNmax increased from 0.5 to 1.0. Conversely, Fig. 7a
indicates that when LNmax = 0.1, if S ′ ≥ 0.8, B remained almost 1.0, but as S ′ decreased
from 0.8 to 0, B gradually reached zero. Note that we will denote the RRG manipulatedwith parameters S ′ and LNmax by RRG(S ′, LNmax), respectively, e.g. RRG (0.8, 0.1).

B

LC

LN

b S’ = 1 (RRG)
B

0.8

0.8

0.7

0.7

Posting rate

Posting rate

1
0.9

a LNmax = 0.1 (RRG)

1
0.9

0.6
0.5
0.4
0.3

LN

0.6
0.5
0.4
0.3

0.2

0.2

0.1

0.1

0

LC

0

500
1000
Generation

0

1500

0

500

1000
Generation

Fig. 5 Article posting and comment rates in RRG (LNmax = 0.1or S′ = 1.0)

1
0.9

Posting rate

0.8
0.7

B

LC

LN

0.6
0.5
0.4
0.3
0.2
0.1
0

0

500

Generation

1000

Fig. 6 Article posting and comment rates in RRG (LNmax = 0.1and S′ = 1.0)

1500

1500

Osaka et al. Comput Soc Netw (2017) 4:2

a LNmax = 0.1 (fixed, RRG)

1
0.8

B

LC

0.8

LN

0.6
0.4

B

LC

LN

0.6
0.4
0.2

0.2
0

b S’ = 1 (fixed, RRG)

1

Posting rate

Posting rate

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0

0.2

0.4

0.6

0.8

1

0

Awareness of articles, S’

0

0.2

0.4

0.6

0.8

1

Upperlimit of normal comment rate, L Nmax

Fig. 7 Change in article posting and comment rates in RRG

These results indicate that LNmax and S ′ strongly affect the emergence of cooperationin the RRG: a smaller LN corresponds to the situation where agents may read articles
from non-reciprocal agents but do not comment on them that much. A larger S ′ corresponds to the situation where agents rarely miss articles posted by agents. Thus, while
agents received more rewards from non-reciprocal agents by behaving like free riders,
they commented on the articles posted by reciprocal agents. Both conditions must be
satisfied in order to remove the causes of collapse from the RRG.
Effect of network structure

To investigate the effect of network structure on the emergence of cooperation in RRG,
we ran the RRGs on WS networks [15] whose average degree was 20 (since the completegraph in Exp. 1 was 20), on CNN networks [16] and BA networks. We set the numberof nodes (agents) to 1000 in each network. Note that we did not use S ′ and LNmax in the
RRG, and so did not fix the degree of fun of posts, S, or limit the comment rates to theusual neighbors, LNi, in Exp. 2.
We observed quite different phenomena in the WS networks. As shown in Fig. 8,
the rates of B, LC , and LN remained relatively high and the dispersion of B was small
when 0 ≤ p ≤ 0.1, where p is the re-wiring probability in the WS model. However, whenp = 0.3 and 0.5, B decreased and fluctuated more. Figure 9 shows how the average ratesof B, LC , and LN varied with p with which we can investigate the relationship between
the re-wiring probability and the agent’s activity. Note that the WS model generates aregular graph when p = 0, whereas it generates a random network when p = 1 [22].
The cluster coefficients are small when p > 0.1, so the small-world property with a large
cluster coefficient only appears when p ≤ 0.1. Figure 9 indicates that B was around 0.94
when p ≤ 0.1, but as the re-wiring probability increased ( p > 0.1), cooperation became
weak, and eventually B reached around 0.3, which was smaller than the B of the complete graphs (Fig. 3b). This result suggests that cooperation is dominant in WS networks
with the small-world property and large-cluster coefficients. Note that B was small in the
RG on the WS networks [23].
Figure 10 shows how B, LC , and LD varied with the conversion probability, u, i.e. theprobability for converting a potential link to a link [16] in the CNN model, and Fig. 11
shows the same in the BA networks in which we varied m, which is the number of links
when a new node is added. These figures indicate that B ranged between 0.6 and 0.8,

Osaka et al. Comput Soc Netw (2017) 4:2

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a p=0

1

B

0.5
0.4

0.5
0.4
0.3
0.2

0.1

0.1

1

c p = 0.05

600
900
Generation

1200

0

1500

B

LC

0.4

0.2
0.1
1200

0

1500

e p = 0.3
B

0.8

LC

0.5
0.4

600
900
Generation

1200

1500

f p = 0.5
B

LC

LN

0.6
0.5
0.4
0.3

0.2

0.2

0.1

0.1
1200

LN

0.7

0.3

600
900
Generation

LC

0.8
Posting rate

0.6

300

300

0.9

LN

0.7

0

0

1

0.9

0

B

0.4

0.1

1

d p = 0.1

0.5
0.3

600
900
Generation

1500

0.6

0.2

300

1200

0.7

0.3

0

600
900
Generation

0.8
Posting rate

0.5

300

0.9

LN

0.6

0

0

1

0.7

LN

0.6

0.2

300

LC

0.7

0.3

0

B

0.8

0.6

0.8
Posting rate

LN

0.7

0.9

Posting rate

LC

0.9

Posting rate

Posting rate

0.8

0b p = 0.01

1

0.9

0

1500

0

300

600
900
Generation

1200

1500

Fig. 8 Article posting and comment rates in RRG (WS networks)

1
B

Posting rate

0.8

LC

LN

0.6

0.8

0.6
0.4
0.2
0

0

0.2

0.4

Re-wiring probability, p

1

Fig. 9 Article posting rates in WS network together with approximate polynomial curves

and gradually increased with u and m, although LC and LN decreased. The increase in
B was due to the existence of hub agents with extremely high degrees: Even if LN is low,such hub agents receive comments from non-reciprocal agents. Thus, a strategy that

Osaka et al. Comput Soc Netw (2017) 4:2

Page 13 of 20

1
B

Posting rate

0.8

LC

LN

0.6
0.4
0.2
0

0

0.2

0.4
0.6
0.8
Conversion probability, u

1Fig. 10 Article posting rates in CNN networks together with approximate polynomial curves

1
B

LC

LN

Posting rate

0.8
0.6
0.4
0.2
0

0

5

10
15
Number of initial links, m

20

Fig. 11 Article posting rates in BA networks together with approximate polynomial curves

makes LC and LN as small as possible is more optimal. However, such cooperation of
hub agents was sustained by the neighbor agents in a non-cooperative strategy. We will
discuss this phenomenon in the "Discussion" section.
Stability/instability of agents’ strategies

The most different phenomenon observed in Exps. 1 and 2 was the dispersion of B. For
example, B in Figs. 3b and 8e, f were fluctuated although they were the average valuesof 100 experimental runs. However, B was stable in Fig. 8a–d. The similar dispersion
in B could be seen in CNN and BA networks whereas we do not show the graphs. To
see the changes in the agents’ strategies, we randomly selected one experimental run
and plotted the changes in posting rate (Fig. 12). In the RRG on the WS network with
p = 0.01, B was almost always larger than 0.9 and seemed stabler than in the other networks (Fig. 12a). However, on the WS network with p = 0.5, B fluctuated with many
sudden rises and falls (Fig. 12b). These sudden rises and falls appeared when p wasaround 0.2, and they gradually increased in height and numbers with p. Similar fluctuations appeared in the RRG on the CNN and BA networks (Fig. 12c, d) and in the RRG on
the complete graph in Exp. 1 (Fig. 4b).
These data indicate that although the average values of B were comparatively a bit
higher (around 0.7 and sometimes around 0.8) on the CNN and BA networks, as shownin Figs. 10 and 11, they were unstable; B was stable observed only on the WS networks

Osaka et al. Comput Soc Netw (2017) 4:2

Page 14 of 20

a WS, p = 0.01

1

B

LN

0.6
0.5
0.4

0.5
0.4
0.3
0.2

0.1

0.1
600
900
Generation

1200

0

1500

c CNN, u = 0.9
B

0.8

LC

Posting rate

0.5
0.4

d BA, m = 20
B

LN

0.4
0.2

0.1

0.1
1500

LC

0.5
0.3

1200

1500

0.6

0.2

600
900
Generation

1200

0.7

0.3

300

600
900
Generation

0.8

0.6

0

300

0.9

LN

0.7

0

0

1

0.9

LN

0.6

0.2

300

LC

0.7

0.3

0

B

0.8

0.7

1

Posting rate

LC

0.9

Posting rate

Posting rate

0.8

0b WS, p = 0.5

1

0.9

0

0

300

600
900
Generation

1200

1500

Fig. 12 Article posting rates in WS, CNN, and BA networks (an experimental run)with low p (≤0.1) values. Although the similar stability was observed in the RRG (1.0,0.1) in the complete graphs (Fig. 6), it had different characteristics; we will discuss thismore in the "Discussion" section.
Analysis of reciprocal relationshipTo express how many reciprocal neighbor agents and what types of reciprocal
structure sustain cooperative behavior in agent networks, we should first introduce some notation. For agent i ∈ A and for the set of i’s reciprocal neighbors Wi ,we define the number of reciprocal neighbors as ria = |Wi |. There are two types ofreciprocity: Wim = {j ∈ Wi | i ∈ Wj } is the set of mutual reciprocal agents of i, and
Wio = {j ∈ Wi | i �∈ Wj } is the set of one-sided reciprocal agents of i. If we express
rim = |Wim | and rio = |Wio |, ria = rim + rio obviously holds. Furthermore, we will express
the average values of these parameters by eliminating the subscript i, e.g. r a = i∈A ria.
Table 3 lists the relationship between the number of reciprocal agents and the network
type. Note that the networks in this table have their average degrees equal or close to 20.
The average number of reciprocal agents, r a, in the RRG was small on complete graphs,
CNN networks, BA networks, and WS networks with high p values [marked with anasterisk (*) in Table 3]. Because LN was small, agents rarely received comments, and they
eventually stopped posting. In this situation, a free rider was the optimal strategy. Nevertheless, B remained slightly higher than that on complete graphs (see the "Discussion"section).
On the other hand, the values of r a were around 4 in RRG (0.8, 0.1) on the completegraphs and in the RRG on the WS networks with a low re-wiring probability, p. These
values were not large but large enough to maintain cooperation, i.e. posting articles by

Osaka et al. Comput Soc Netw (2017) 4:2

Page 15 of 20

Table 3 Number of reciprocal agents
Network type

ra

rm

ro

Complete graph (RRG)*

1.46

0.61

0.85

Complete graph RRG (0.8, 0.1)

4.65

3.66

0.99

Complete graph RRG (1, 0.1)

17.53

17.48

0.05

WS network (p = 0.01, RRG)

3.70

1.41

2.29

WS network (p = 0.5, RRG)*

0.34

0.07

0.27

CNN (u = 0.9, RRG)*

0.72

0.23

0.49

BA (m = 10, RRG)*

0.92

0.35

0.57

Facebook network (RRG)

4.30

1.55

2.75

receiving comments. Of course, posting comments also incurred costs, but the agents
behaved as free riders on the other non-reciprocal neighbor agents, and in so doingreceived sufficient rewards. We call such a strategy the half free rider, i.e. be cooperative with reciprocal agents, but otherwise behave as a free rider. Half free riders withappropriate numbers of reciprocal agents strike a balance between rewards and costs,
and thus are optimal in these environments. A larger r a facilitates comments on articles,
and, thus, it seems to sustain cooperation. However, we also have to consider the cost ofcommenting; we will discuss this aspect in the next subsection.
The characteristic that distinguishes the WS networks from the complete graphs is
the ratio of r m–r a: Table 3 indicates that cooperation was sustained by mostly mutual
reciprocal relationships in RRG (0.8, 0.1) and RRG (1.0, 0.1) on the complete graphs,
but in the RRG on the WS network with p = 0.01, it was sustained by both types ofreciprocal relationship. Introduction of S ′ and LNmax generated mutual reciprocity, andRRG (1.0, 0.1) on the BA and CNN networks showed cooperative behaviors and many
mutual reciprocal links (the graphs of this experiment are not shown). We will examine
the meaning of S ′ and LNmax in an SNS in the "Discussion" section.
Effect of memory term on cooperation

In the third experiment (Exp. 3), we investigated how the memory term, TW, affected theagents’ behavior. Figure 13 plots the average posting rates, B, LC , and LN , when TW wasvaried from 1 to 20. Here, we could intuitively say that cooperation should be facilitatedbecause agents with a longer memory do not forget past reciprocal behavior. However,Fig. 13 indicates that in RRG (1.0, 0.1) on complete graphs (Fig. 13a) and in the RRG
on the WS networks (Fig. 13b), a longer memory negatively affected the evolution ofcooperation.
We can think of two reasons for this phenomenon. First, when TW is larger, agents
do not forget the reciprocal behavior of neighbors, so they continue to comment on
posted articles for a longer time. However, the opposite phenomenon also occurs. When
TW = 1, if agent i commented on an article of another agent j but j did not comment
back on an article posted by i, i would eliminate j from Wi. However, when i has a longer
memory, i would continue to consider j to be a reciprocal agent. Then, if j did not comment on i’s articles in a few rounds, i would still assume that j is a reciprocal agent. Second, agent i with a higher ria must engage in costly activity, because LCi > LNi and i has

Osaka et al. Comput Soc Netw (2017) 4:2

B

LC

LN

0.6
0.4

LC

LN

0.6
0.4
0.2

0.2
0

B

0.8
Posting rate

0.8

b WS network ( p = 0.01, RRG)

1a Complete graph (RRG(1.0, 0.1))

1

Posting rate

Page 16 of 20

0

5

10
Memory length, TW

15

0

20

0

5

10
15
Memory length, TW

20

Fig. 13 Relationship between memory term and article posting rates

to comment on more articles. Agents that did not comment on the posts of others but
received comments would have an advantage. This situation continues for longer when
TW is large and, thus, is likely to lead to collapse of cooperation. Actually, Fig. 14a, b,
which plots the average number of reciprocal agents versus TW, shows that the numberof reciprocal agents decreased as TW increased because of the increase in cost, although
the larger TW enabled agents to memorize more reciprocal actions by their neighbors.
If we compare Fig. 13a, b, we can see that the WS networks were somewhat robust
to longer memory terms. The same phenomenon is apparent in Fig. 14a, b, which indicates that in RRG (1.0, 0.1) on the complete graphs, the number of reciprocal agents, r a ,quickly decreased as TW increased. On the other hand, in the RRG on the WS networks
with p = 0.01, r a increased when TW = 2, 3, and 4 and cooperation was sustained byboth mutual and one-sided reciprocal agents (Fig. 14b). However, the characteristics thatwere responsible for the robustness to memory in the WS networks are still unknown.
Games on a Facebook network

Finally, we conducted the same experiments using an instance of the Facebook network [24] whose features are listed in Table 4. Figure 15 plots the experimental results,
while Table 3 lists the number of reciprocal agents in the RRG. Figure 15 indicates thatthe article posting rate B was around 0.88, which is close to that of the WS networks

a Complete graph (RRG(1.0, 0.1))

15

Mutual (r m )12

One-side (r o )

9
6
3
0

0

5

10
15
Memory length, TW

b WS network ( p = 0.01, RRG)

7

Reciprocal (r a)Number of reciprocal agents

Number of reciprocal agents

18

20

Reciprocal (r a)6

Mutual (r m )5One-side (r o)

4
3
2
1
0

0

5

10
15
Memory length, TW

Fig. 14 Relationship between memory term and number of reciprocal agents20

Osaka et al. Comput Soc Netw (2017) 4:2

Page 17 of 20

Table 4 Network characteristics of Facebook network
Network characteristics

Value

Number of users (agents)

4039

Average number of friends (degree)

43.691Clustering coefficient

0.606

Characteristic path length

3.693

Power-law exponent−1.1801
0.9

Posting rate

0.8
B

0.7

LC

LN

0.6
0.5
0.4
0.3
0.2
0.1
0

0

300

600
900
Generation
Fig. 15 Average article posting rates (a Facebook network)

1200

1500

with p between 0.1 and 0.2, and that it was stable. In addition, the number of reciprocal agents, r a, in Table 3 was around four and cooperation evolved and was sustained byboth one-sided and mutual reciprocal agents, although r o was slightly larger than r m, as
in the WS networks. This instance of the Facebook network is more complicated than
the networks used in Exp. 2, as is shown in Fig. 16, which visualizes it; it seems to consist of clusters of communities that have their own sub-structures and a few hub agents.
However, our results suggest that the RRG on the Facebook network had characteristics similar to those of the RRG on the WS networks with p = 0.1–0.2, and cooperative
behavior was sustained without introducing the parameters for manipulation, S ′ and/or
LNmax.
Discussion

Here, let us discuss the experimental results from the viewpoint of an SNS. First, we
can say that cooperation did not seem to emerge in the normal RRG on the completegraph, but if we manipulated RRG by introducing the parameters S ′ and LNmax, cooperation (posting articles) emerged as shown in Figs. 6 and 7. If we look at these figures carefully, we see that B and LC had similar values, and this indicates that cooperation arosefrom LC , i.e. high comment rates by reciprocal agents, mostly mutual reciprocal friends.
For example, cooperation emerged when S ′ ≥ 0.8 and LNmax = 0.1, i.e. RRG (0.8,0.1). LNmax = 0.1 means that users did not often comment on the articles if they wereposted by non-reciprocal friends, while S ′ = 0.8 means that users had attention to morethan 80% of the posted articles by (reciprocal) users. We believe that this behavior isOsaka et al. Comput Soc Netw (2017) 4:2

Fig. 16 Facebook network structure

reasonable and frequently occurs in actual SNSs. Users behave as half free riders; theyreceive information (rewards) from non-reciprocal friends, but respond only to information from reciprocal close friends to sustain SNS (because the articles posted by closefriends are likely to be interested). This activity balances rewards and costs in the SNS.
Note that the phenomenon described above also occurred in the RRGs on the CNN andBA networks, that is, B and LC became about 1.0 when S ′ ≥ 0.8 and LNmax = 0.1.
On the other hand, even in a normal RRG, cooperation, i.e. posting articles and comments, became the optimal strategy on the WS networks with p ≤ 0.1 and the SNS
seemed thriving. In this condition, WS networks have the properties of small-worldand high cluster coefficient. Our experimental results also suggest that the instance ofFacebook network that was generated in a bottom-up manner via interaction is likely tomaintain cooperation. In addition, its average number of reciprocal agents was higher
than those in other networks in the normal RRG. Although this network consists of afew communities, it has also the same properties (Table 4). Therefore, it is possible thatthese properties foster cooperation, but we need more study to clarify it.
Another implication from our experiments is that the number of reciprocal agents isimportant because the balance between rewards and costs because of reciprocal relationships strongly affects the agents’ strategies in a normal RRG. Now let us focus on themutual and one-sided reciprocity in these networks. When TW = 1, agents remembered
the reciprocal behavior only in the next round, and the number of mutual reciprocal
agents, r m, was likely to be small, so r o > r m in the normal RRG (Fig. 14b). However,when TW ≥ 2, r m and r o had the similar values and they always coexist. This phenomenon was also true in the Facebook network. We believe that such a reciprocal structure
is common in human society and, thus, observed in the SNSs in the real world. On theother hand, r m ≫ r o in RRG (1.0, 0.1) on the complete graph; thus, manipulation by S ′and LNmax fostered cooperation by enhancing the effect of mutual reciprocity. This situation corresponds to a dense cluster that becomes active if all users are communicatedclosely each other.
On the other hand, Figs. 10 and 11 indicate that B in the normal RRG on the CNN
and BA networks was larger than that on the complete graphs. This result reflects the

Page 18 of 20

Osaka et al. Comput Soc Netw (2017) 4:2

existence of hub agents, i.e. agents that have extremely high degrees. Even on these networks, just reading articles (acting as a free rider) is optimal. However, a few agents comment on articles because of the mutation that corresponds to the vagaries of users. Thus,
hub agents receive sufficient comments, have some reciprocal agents, and, thereby, continue posting articles. Because the hub agents earn rewards, their neighbor agents arelikely to copy the hub’s gene for the next generation, and consequently B increases tosome degree. However, this behavior is, of course, not optimal for non-hub agents; theycannot earn sufficient rewards. Hence, B, LC , and LN stopped increasing in the experiments. By contract, the WS networks have no hub agents. Thus, the essential cause ofprosperity of the RRG resides in the network structure of the WS networks; this finding
is not so obvious and its explanation remains an open problem.

Conclusion
Social networking services are used by users around the world for purposes such as
chatting with close friends, exchanging opinions, advertising, marketing, and politics. Although SNSs are usually run by companies and (non-)profit organizations, they
depend on their users’ voluntary behaviors such as posting articles and comments. Sincesuch voluntary behavior must incur some cost and effort on the part of the user, it is
not obvious what makes them continue to participate in an SNS. Because SNSs havethe features of the public goods game, a number of evolutionary game theoretic studies
have attempted to understand the mechanism behind cooperation, which corresponds
to posting articles in an SNS, by using extending the public goods game in various ways.
However, conventional game theoretic models of SNSs ignore reciprocity, even though
it plays an important role in evolving cooperation in human society. Thus, in this study,
we devised a model, called the reciprocity rewards game (RRG), of an SNS that is basedon the public goods game by incorporating direct reciprocity between users into theconventional model. After that, we ran the RRG on a variety of networks to see how
reciprocity between users and the network structure affect cooperation. Our experimental results indicate that reciprocity in complete networks slightly raises the rate ofcooperation, but the rise is not significant. However, if we manipulate the RRG so thatusers hardly miss any articles posted by their neighbors but at the same time hardly
comment on articles posted by non-reciprocal agents, cooperation emerges. Second, in
RRGs on WS networks with a low re-wiring probability p, cooperation emerges without
any manipulation like in the RRG on the complete graph. We also analyzed the evolved
reciprocal structures and the effect of the memory term on cooperation. Finally, using an
instance network of Facebook, we conducted the same experiment, and its results indicate characteristics similar to those of WS networks with p between 0.1 and 0.2.
We found that a certain network structure facilitates cooperation in an RRG andmakes cooperation robust to the memory term, but its essential cause is still unknown:
this will be our future work. We also plan to add indirect reciprocity to our model andanalyze the relationship between costs and rewards in continuous use of an SNS.
Authors’ contributions
All authors participated in the discussion of designing model and planning experiments, and thus almost equallycontributed to the work. KO conceived the idea and implemented the code for experiments. FT and TS designed theexperiments. TS and KO composed the draft of the manuscript. All authors read and approved the final manuscript.

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Osaka et al. Comput Soc Netw (2017) 4:2

Author details
1
Department of Computer Science and Communications Engineering, Waseda University, 3‑4‑1 Okubo, Shinjuku,
Tokyo 169‑8555, Japan. 2 Graduate School of Engineering, The University of Tokyo, 7‑3‑1 Hongo, Bunkyo, Tokyo 113‑8656,
Japan.
Acknowledgements
This work was in part supported by KAKENHI (25280087).
Competing interests
The authors declare that they have no competing interests.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 5 February 2017 Accepted: 17 May 2017

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