Pacemaker effects on online social rhythms on a social network

The dynamics of coupled oscillators in a network are a significant topic in complex systems science. People with daily social rhythms interact through social networks in everyday life. This can be considered as a coupled oscillator in social networks, which is also true in online society (online social rhythms). Controlling online social rhythms can contribute to healthy daily rhythms and mental health. We consider controlling online social rhythms by introducing periodic forcing (pacemakers). However, theoretical studies predict that pacemaker effects do not spread widely across mutually connected networks such as social networks. We aimed to investigate the characteristics of the online social rhythms with pacemakers on an empirical online social network. Therefore, we conducted an intervention experiment on the online social rhythms of hundreds of players (participants who were pacemakers) using an avatar communication application (N = 416). We found that the intervention had little effect on neighbors’ online social rhythms. This may be because mutual entrainment stabilizes the neighbors’ and their friends’ rhythms. That is, their online social rhythms were stable despite the disturbances. However, the intervention affected on neighbors’ rhythms when a participant and their neighbor shared many friends. This suggests that interventions to densely connected player groups may make their and their friends’ rhythms better. We discuss the utilization of these properties to improve healthy online social rhythms.


Introduction
Social physics, a study that uses physical methods to explain social phenomena [1], has been applied to understand the dynamics of human online social rhythms [2].These rhythms are influenced by social cues [3] and can be modeled as coupled oscillators in complex social networks [2,4,5].The synchronization of coupled oscillators occurs when connections exceed a certain threshold, leading to global patterns [6][7][8][9].Online social rhythms, which reflect everyday lifestyle rhythms, are often driven by online communication and influenced by friends' rhythms.One study [2] showed that the entrainment of people's online social rhythms emerges if their closeness reaches a threshold.This entrainment spreads via densely connected clusters; that is, having many shared friends facilitates the entrainment of online social rhythms.Consequently, long-range correlations of online social rhythms emerge.
We conducted an intervention for online social rhythms to investigate their nature and control further.We considered this system a coupled oscillator on a network with periodic forcing.Theoretical studies [10][11][12][13][14][15][16] modeled the control of coupled oscillators in a network by periodic forcing as the entrainment of other nodes by a pacemaker node (periodic forcing).A study [11] showed that the pacemaker effect exponentially decreases with the distance from the pacemaker in mutually connected networks.Long-range correlations with a pacemaker emerge if the networks are directionally biased, that is, closer to feed-forward networks [12].The characteristics of the coupled oscillations and pacemakers in such unidirectional networks explain the neural mechanisms underlying the circadian rhythms of organisms [12,13,[17][18][19].A previous study [14] also demonstrates the limitations of a pacemaker on a one-dimensional torus network.The pacemaker does not influence the rhythms of distant nodes.This implies that controlling the rhythms of oscillators necessitates the small-world nature of the network.However, even in a network possessing small-world characteristics (like a scale-free network), if the interaction between nodes becomes overly strong, the network's inherent characteristics are lost, making rhythm control by a pacemaker unachievable [15].Thus, the interaction of oscillators on the network is a significant subject in the study of dynamics on a network [15,20].Addressing this issue with new data (human online social rhythms) carries substantial meaning in the field of network science.
Insights for controlling online social rhythms using a pacemaker can contribute to research in clinical psychology because regulating circadian [5,21,22] and online social rhythms [23][24][25][26] contributes to improved mental health.These findings are supported by Interpersonal and Social Rhythm Therapy (IPSRT) [27].This reveals the potential usefulness of the online IPSRT [28].
In this study, we investigated the effect of pacemakers on online social rhythms in online social networks.Theoretical studies [11,12] predicted that the pacemaker effect does not spread widely in mutually connected networks.However, clarification about actual human online social rhythms on a social network is essential, which is significant in clinical psychology.It is also crucial to elucidate the relaxation time of the pacemaker effect, effect size, and correlation length.Therefore, we conducted an intervention experiment on online social rhythms using the avatar communication application 'Pigg Party4 ' and analyzed participants' and their neighbors' online social rhythms.

Pigg Party
Pigg Party is a Japanese avatar communication application, in which players communicate with personally designed avatars in virtual spaces (figure 1).This is available for iOS and Android devices.A previous study [2] indicated the entrainment of online social rhythms among Pigg Party players.
They can communicate synchronously using text through their avatars in virtual spaces.In addition to sending text messages, players can respond to dozens of avatar animations, known as avatar actions.
Players can send a lightweight and asynchronous message 'like' similar to Facebook's.They send 'likes' by choosing a friend's or stranger's avatar or icon 5 .

Intervention and participants
We conducted a four-week intervention experiment (26 July to 24 August 2022) on online social rhythms.In the intervention term, participants were motivated to use Pigg Party at specified times by incentives from the researchers.We call this the intervention on the online social rhythm of participants.At the specified times, if participants talk to their friends or participants' friends find that participants are online at the specified times, participants' friends may also use at the specified times, i.e. their social rhythms may be similar.
The participants of this experiment were recruited from the Pigg Party (N = 416; see table 1 for information on gender and age composition).
In the experiment, we randomly split the participants into four groups: morning, evening, all-day, and control (the number of participants in each group was 104).
Morning, evening, and all-day group participants are encouraged to send at least 5 'likes' to other players at the corresponding time for each group (morning: 7:00-8:00 AM; evening: 7:00-8:00 PM; and all-days: 0:00-23:59).The Pigg Party administrator sent reminder messages to the participants at the following times: morning group at 7:00 AM, evening group at 7:00 PM, and all-day group at 1:00 PM.Those who achieved 60 % (17 days) of their tasks in four weeks were rewarded with an Amazon gift card (300 JPY) and virtual coins from Pigg Party worth 500 JPY.The control group did not undergo any intervention.Randomly selected 23% of the control group participants received the same rewards as the other group participants for the lottery.

Participants' neighbors
We builted online social networks to study the intervention effects on participants' neighbors for each week, based on a previous study [2].Therefore, we acquired five networks for the weeks of intervention periods 6 .Table 2 shows the network features.
From these networks, we extracted and analyzed players who connected with the participants as neighbors.The numbers of neighbors in the morning, evening, all-day, and control groups were 804, 683, 784, and 797, respectively.
To construct these networks, we used the logs of users visiting other users' private rooms.As in previous research [2,25,30,31], we established a link between a visitor and the owner of a room because visitors are often friends with the owner [32].The time spent (in seconds) over a week was used to denote the weights of the links (w).The time spent per week was recorded.Although this network does not directly depict a communication network, it should positively correlate with the actual communication network.We adopted this network as a proxy because data regarding the actual communication routes between individuals are strictly limited owing to constitutionally protected communication privacy in Japan.
We created two variables expressing the relationships between the participants and their neighbors: 1) close-frined: A relationship with w >= 10 3.1 was considered a close friendship because the players' online social rhythms were synchronized when w >= 10 3.1 in a previous study [2].2) Number of shared friends: We counted the number of shared friends between the participants and their neighbors because having more shared friends facilitates the spread of online social rhythms [2].

Online social rhythms
We created vectors representing online social rhythms over a 24 h cycle according to a previous study [2].Each dimension of the vectors indicated the activity during a specific hour, that is, the dimensions of the vectors were 24.We constructed these vectors using weekly usage data, removing any fluctuations unrelated to the 24 h cycle.This allowed us to construct the social rhythm vectors.
To do this, we first built a time series from users' hourly usage data, then performed discrete Fourier transforms and extracted cycle factors, which were calculated by 24 h divided by integers i (i = 1, 2, . . ., 11), that is, 24, 12, 8, 6, 4.80, 4, 3.43, 3, 2.67, 2.40, and 2.18 h, and calculated the inverse Fourier transforms of these elements.This implies online social rhythm vectors were constructed based on 11 bandpass-filtered frequencies.
The results were normalized to create a weekly social rhythm vector for each participant.For example, we can evaluate the social rhythm similarity using the inner product of these vectors (cosine similarity).
The source code for online social rhythms is available on https://figshare.com/s/d27b9c4858af977ef81f.

Morning/evening activities
Each player's morning and evening activities were constructed using online social rhythm vectors.Morning activity was defined as the similarity between a player's online social rhythm vector and a morning vector, which had one in the seventh dimension (7:00 AM) and zero in other dimensions.The evening activity was also a similarity between a player's vector and the evening vector, which was the 19th dimension (7:00 PM) set to 1 and the others to 0.

Regressions to morning/evening activities
Regression analyses were conducted to evaluate the intervention effects on neighbors' morning and evening activities: where the objective variable y was morning or evening activities, µ was expected values of y, and σ was a standard deviation.µ was the weighted summation of the following explanatory and control variables.The explanatory variables x * included information on the participants connected to the analysis target players (the number of days in the morning task achievement or those in the evening; x 1 ) and the relationships between the players and the participants (a dummy variable expressing close friend relations x 2 and the number of shared friends x 3 ).These explanatory variables were set to zero when the group was not a morning group (or evening group).For the control, we used the neighbor's morning (evening) activity before the experiment (three-week averages; c 1 ), time-varying covariates for each week (weeks 2-5; week 1 was a reference variable), and the group of participants connecting with each neighbor (control group was a reference variable).We did not analyze these control variables.β * and γ * indicate coefficients of explanatory and control variables.

Ethics statement
The Pigg Party application provider collects log data based on their terms of service 7 and privacy policy 8 .All Pigg Party users accepted the terms of service and privacy policy, which allowed analysis of their behavioral data for service improvements and academic studies.The data were pseudonymized, and identifying information was removed.Quantitative data outputs are presented at an aggregate level, meaning no identifying information was included.
The experiment in this study was approved by the Ethics Committee of Tokushima University.All procedures were conducted in accordance with the guidelines for studies involving human participants, the ethical standards of the institutional research committee, and the 1964 Helsinki Declaration and its later amendments.The participants provided informed consent to participate in our experiment and were allowed to stop at any time.They could also withdraw their responses after the completion of the experimental process.

Results
First, we confirmed the intervention effects on the participants' morning and evening activities.Figure 2 indicate that the interventions drove their activities according to their clock times.Therefore, they can be considered as pacemaker nodes.The intervention effect decreased in the latter half of the study period, particularly in the evening.All the intervention effects disappeared after the intervention.
Second, we checked neighbors' morning and evening activities (figure 3).These results did not indicate well-marked intervention effects on neighbors.In a previous study [2], shared friends and the closeness between players played crucial roles in the entrainment of online social rhythms.We analyzed these effects on the morning and evening activities of the neighbors from the participants.Tables 3 and 4 provide the regression analysis results.The number of friends shared indicated significance for morning and evening activities.The effect sizes for sharing friends were limited.The morning activities (mean: 0.143) increased by 0.34%, and evening activities (mean: 0.186) increased by 0.43% when a player and an intervention participant had one more friend.The effects of being close to the participants and the number of days of morning task achievement were not statistically significant.

Discussion
Despite the intervention in online social rhythms affecting the participants' rhythms, these effects were less widespread.A slight impact was observed only when players shared many friends.Therefore, the correlation length of the pacemaker effect was very short, and the effect size was small.After the experiments, the intervention effect on participants' rhythms rapidly disappeared.This indicates that the relaxation time was also short.These results support theoretical studies [11,12] that show that pacemaker effects do not spread across mutually connected networks.This is because mutually connected oscillator rhythms are insensitive to periodic forcing owing to mutual entrainment.This means that the online bandwagon effect (majority synching bias) [33] also plays out on social rhythms.This finding suggests that it is difficult to expect a spreading effect from a mere intervention in online social rhythms to enhance the spreading effects of IPSRT on online communication platforms such as social networking services.
This implies online social rhythms with mutual entrainment may remain stable despite disturbances.Therefore, if we create player clusters with good rhythms, for example, early to bed and early to rise, their rhythms may be stable from disturbances.Having shared friends drove entrainment between the intervention participants and their friends.This is consistent with a previous study [2], which showed that more densely connected clusters of players indicate more synchronization.
It may be possible to improve the correlation length, effect size, and relaxation time of intervention effects.For example, interventions with an entire group of closely interconnected nodes would solidify the intervention effect on them and improve the spillover effect on their friends (who inevitably have many mutual friends).This is because densely connected cluster [2] and strong interaction between nodes [15] stabilize rhythms.It would also be important to isolate the appropriate subgraphs (groups) because small-size networks enhance the effects of pacemaker [14].Interventions through highly synchronous communication, such as online chat, rather than asynchronous communication, such as 'like' communication, may also strengthen the spread effects.If we create a cluster in which people have healthy online social rhythms, they can maintain their healthy rhythms through mutual entrainment of their rhythms.The development of intervention methods to achieve these clusters will be a topic for future research.

Limitations and future work
We studied players on an avatar communication application, Pigg Party.Considering other online communication platforms, e.g.social networking services (SNS) and massively multiplayer online role-playing games (MMO-RPG) where players' demographic composition will be different and providing different user experiences, etc would broaden the scope of our findings.
As mentioned above, there is room for ingenuity in intervention.Various forms of intervention, such as involving synchronous communication, will need to be considered, although we considered the intervention to asynchronous communication in this study.The ratio of participants was tiny compared to the total number of active users (several hundred thousand active users of Pigparty over a six-month period [34]).Intervening with a more significant number of players or in density clusters may strengthen the ripple effect of the intervention.

Figure 1 .
Figure 1.Players chat with each other via their avatars in Pigg Party.

Figure 2 .
Figure 2. Mean activities of morning and evening of each group of partitipants.the periods between the black lines show the intervention terms.the interventions drove their online social rhythms, i.e. we can regard them as pacemaker nodes.

Figure 3 .
Figure 3. Mean activities of morning and evening of players who are participants' friends.these figures show that the intervention effect does not have a clear impact on them.

Table 1 .
Participants' gender and age.

Table 2 .
Mean values of the network features.

Table 3 .
Result of a regression for morning activity.only the number of sharing friends among the explanatory variables was statistically significant.