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%%\unnumbered% uncomment this for unnumbered level heads
\begin{document}
\title{Limitation of time promotes cooperation in temporal games}
% Use letters for affiliations, numbers to show equal authorship (if applicable) and to indicate the corresponding author
\author[a,b,1]{Jiasheng Wang}
\author[1,2]{Jiasheng Wang}
%\email{1510478@tongji.edu.cn}
\affil[a]{Department of Computer Science and Technology, Tongji University, 4800 Cao'an Road, Shanghai 201804, China}
\affil[b]{Key Laboratory of Embedded System and Service Computing (Tongji University), Ministry of Education, Shanghai 200092, China}
\affil*[1]{Department of Computer Science and Technology, Tongji University, 4800 Cao'an Road, Shanghai 201804, China}
\affil[2]{Key Laboratory of Embedded System and Service Computing (Tongji University), Ministry of Education, Shanghai 200092, China}
\author[a,b,1,2]{Yichao Zhang}
\author*[1,2]{Yichao Zhang}
%\affil[1]{Department of Computer Science and Technology, Tongji University, 4800 Cao'an Road, Shanghai 201804, China \\ Key Laboratory of Embedded System and Service Computing (Tongji University), Ministry of Education, Shanghai 200092, China}
\author[c,d,2]{Guanghui Wen}
\affil[c]{Department of Systems Science, School of Mathematics, Southeast University, Nanjing 210016, China}
\affil[d]{School of Engineering, RMIT University, Melbourne VIC 3000, Australia}
\author*[3,4]{Guanghui Wen}
\affil[3]{Department of Systems Science, School of Mathematics, Southeast University, Nanjing 210016, China}
\affil[4]{School of Engineering, RMIT University, Melbourne VIC 3000, Australia}
\author[a,b,2]{Jihong Guan}
\author*[1,2]{Jihong Guan}
%\email{jhguan@tongji.edu.cn}
\author[e,f]{Shuigeng Zhou}
\author[5,6]{Shuigeng Zhou}
%\email{sgzhou@fudan.edu.cn}
\affil[e]{Shanghai Key Laboratory of Intelligent Information Processing, Shanghai 200433, China}
\affil[f]{School of Computer Science, Fudan University, 220 Handan Road, Shanghai 200433, China}
\affil[5]{Shanghai Key Laboratory of Intelligent Information Processing, Shanghai 200433, China}
\affil[6]{School of Computer Science, Fudan University, 220 Handan Road, Shanghai 200433, China}
\author[g]{Guanrong Chen}
\author[7]{Guanrong Chen}
%\email{gchen@ee.cityu.edu.hk}
\affil[g]{Department of Electrical Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Hong Kong SAR, China}
\affil[7]{Department of Electrical Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Hong Kong SAR, China}
\author[h]{Krishnendu Chatterjee}
\affil[h]{Institute for Science and Technology, A-3400 Klosterneuburg, Austria}
\author[8]{Krishnendu Chatterjee}
\affil[8]{Institute for Science and Technology, A-3400 Klosterneuburg, Austria}
\author[i,j,k]{Matja{\v z} Perc}
\affil[i]{Faculty of Natural Sciences and Mathematics, University of Maribor, Koro{\v s}ka cesta 160, 2000 Maribor, Slovenia}
\affil[j]{Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan}
\affil[k]{Complexity Science Hub Vienna, Josefst{\"a}dterstra{\ss}e 39, 1080 Vienna, Austria}
\author[9,10,11]{Matja{\v z} Perc}
\affil[9]{Faculty of Natural Sciences and Mathematics, University of Maribor, Koro{\v s}ka cesta 160, 2000 Maribor, Slovenia}
\affil[10]{Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan}
\affil[11]{Complexity Science Hub Vienna, Josefst{\"a}dterstra{\ss}e 39, 1080 Vienna, Austria}
% Please give the surname of the lead author for the running footer
\leadauthor{Wang, Zhang}
% Please add here a significance statement to explain the relevance of your work
\significancestatement{The coupling between temporal interactions and rational decision making can be seen everywhere in real life. But the existing theoretical framework is insufficient to explain the widespread cooperation in such temporal games. We therefore conducted a series of online game experiments, which reveal a significant correlation between the high level of cooperation among individuals and the uncertainty of reestablishing collaborative relationships over time. This correlation, on the one hand, urges us to reconsider why the dynamic nature of the networks has an impact on human cooperation, and on the other hand, highlights the aptness of temporal games to explain prosocial behavior in collaborative systems.}
% Please include corresponding author, author contribution and author declaration information
\authorcontributions{J.W. and Y.Z. designed the research; J.W. and Y.Z. analyzed the data; Y.Z. and G.W. organized the online experiments; J.W., Y.Z., and M.P. wrote the paper; J.G., S.Z, G.C., K.C., and M.P. reviewed and revised the paper.}
\authordeclaration{The authors declare that they have no conflict of interest.}
\equalauthors{\textsuperscript{1}J.W. and Y.Z contributed equally to this work.}
% \subsection*{Data and materials availability}
% All the data related to this paper may be requested from the corresponding author according to the material transfer agreement.
\correspondingauthor{\textsuperscript{2}To whom correspondence should be addressed. Corresponding authors: Yichao Zhang, E-mail: yichaozhang@tongji.edu.cn; Guanghui Wen, E-mail: wenguanghui@gmail.com; Jihong Guan, E-mail: jhguan@tongji.edu.cn}
% Keywords are not mandatory, but authors are strongly encouraged to provide them. If provided, please include two to five keywords, separated by the pipe symbol, e.g:
\keywords{temporal networks $|$ non-cooperative game $|$ human subjects $|$ cooperation}
\begin{abstract}
Temporal networks are obtained from time-dependent interactions between individuals. The interaction can be an email, a phone call, a face-to-face meeting, or a collaboration. We propose a temporal game framework where interactions between rational individuals are embedded into two-player games with a time-dependent aspect of interaction. This allows studying the time-dependent complexity and variability of interactions and how they affect prosocial behavior. Based on a simple mathematical model, we find that the level of cooperation is promoted when the time of collaboration is limited and identical for every individual. We confirm and validate this with a series of systematic human experiments that forms a foundation for comprehensively describing human temporal interactions in collaborative environments. Our research reveals an important incentive for human cooperation, and it lays the foundations for better understanding this fascinating aspect of our nature in realistic social settings.
\end{abstract}
\dates{This manuscript was compiled on \today}
%\doi{\url{www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX}}
\begin{document}
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\abstract{Temporal networks are obtained from time-dependent interactions between individuals. The interaction can be an email, a phone call, a face-to-face meeting, or a collaboration. We propose a temporal game framework where interactions between rational individuals are embedded into two-player games with a time-dependent aspect of interaction. This allows studying the time-dependent complexity and variability of interactions and how they affect prosocial behavior. Based on a simple mathematical model, we find that the level of cooperation is promoted when the time of collaboration is limited and identical for every individual. We confirm and validate this with a series of systematic human experiments that forms a foundation for comprehensively describing human temporal interactions in collaborative environments. Our research reveals an important incentive for human cooperation, and it lays the foundations for better understanding this fascinating aspect of our nature in realistic social settings.}
\keywords{temporal networks, non-cooperative game, human subjects, cooperation}
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%\dropcap{T}his PNAS journal template is provided to help you write your work in the correct journal format. Instructions for use are provided below.
\section{Introduction}\label{sec1}
\dropcap{M}any complex collaborative systems in nature, society, and engineering can be modeled through networks. In a network, nodes represent collaborating individuals, and links represent their friendships~\cite{albert_rmp02}. In the early stage of network modeling, links are simplified to be weightless, undirected, and static. In order to improve the ability to depict real systems, weighted~\cite{shen_rsos18}, directed~\cite{Pagan2019game}, and dynamic~\cite{melamed_pnas18} network models have been put forward successively. The application of these network models in various fields has fully proved that the closer the framework is to reality, the stronger its ability to explain behaviors.
Many complex collaborative systems in nature, society, and engineering can be modeled through networks. In a network, nodes represent collaborating individuals, and links represent their friendships~\cite{albert_rmp02}. In the early stage of network modeling, links are simplified to be weightless, undirected, and static. In order to improve the ability to depict real systems, weighted~\cite{shen_rsos18}, directed~\cite{Pagan2019game}, and dynamic~\cite{melamed_pnas18} network models have been put forward successively. The application of these network models in various fields has fully proved that the closer the framework is to reality, the stronger its ability to explain behaviors.
As an intriguing behavior in human collaborative systems, the emergence of cooperation has attracted researchers from social and natural sciences for half a century~\cite{boyd2005solving, gachter2009reciprocity, rand_tcs13, perc_pr17}. Although we are certainly not exempt from selfishness and the fundamental principles of Darwinian evolution, cooperation is nevertheless ubiquitous across human societies~\cite{nowak_11}. While the impetus for our strong cooperative drive has been linked to the difficulties of the genus \textit{Homo} in rearing offspring that survived and to the emergence of alloparental care~\cite{hrdy_11}, and to the formation of alliances in times of conflicts~\cite{bowles_11}, it is still puzzling as to why we have, as a species, achieved such high levels of cooperation. Our altruistic behavior distinguishes us markedly from other mammals, and they indeed form the bedrock for our astonishing evolutionary success.
The studies of human cooperation in $n$-person games begin with population games, also known as mean-field games~\cite{maynard_n73,hofbauer_98,cressman_03}. In such a well-mixed population, cooperation can hardly prevail with imitative update rules when individuals play non-cooperative games such as the prisoner's dilemma~\cite{szabo_pr07}. If the population exhibits a relatively stable social structure, the consequence may be different~\cite{santos_prl05, ohtsuki_n06, santos_pnas06, santos_n08, tanimoto_pre07, fu_pre09, lee_s_prl11, rand_pnas14, fu2017leveraging, allen2017evolutionary, fotouhi_rsif19} -- a finding with roots in the seminal paper by Nowak and May~\cite{nowak_n92b}, who observed clusters of cooperators on a square lattice that protected them from invading defectors. Nevertheless, social networks are seldom static. We disconnect, reconnect, and then form connections with new people over time. This realization has revealed new mechanisms for cooperation that may sustain cooperative behavior under extremely adverse conditions, when the temptation to defect is high and where on static networks cooperation would long perish~\cite{perc_bs10}. An individual also does not interact with all his friends all the time but likely does so only occasionally.
@ -108,8 +104,8 @@ One of our key contributions is a detailed online experiment for the theoretical
%Demonstration of the assumption
To clarify the impact of the limited time, we invited 183 human subjects and designed a set of comparative online experiments. In a match, the participants are allocated to the nodes of pre-generated networks. We test two classes of networks, the Barab\'{a}si and Albert's scale-free network~\cite{BA} and Watts and Strogatz's small-world networks, since they are the most well-known social network models. We show that the limitation to the individuals' time resources statistically promotes the participants' level of cooperation, which aligns with the theoretical prediction presented below.
\section*{Theoretical framework of temporal games}\label{FTG}
\subsection*{Temporal game model}\label{Tgm}
\section{Theoretical framework of temporal games}\label{FTG}
\subsection{Temporal game model}\label{Tgm}
In a two-strategy (i.e., only two moves are allowed) game, define $i$'s strategy as
$
\Omega_i=\left(
@ -166,9 +162,9 @@ In a mean-field view, Eq.~\ref{LambdaiP} can be written as
\end{equation}
where $P(k_i,k_j)$ is the probability that a link exists between $i$ and $j$, dependent on the topology of the collaborative network. We show an illustration of such a collaborative network in Fig.~\ref{fig:tdnc}A. To clarify the generating procedure of the network, we provide the communication log among the individuals in this round in Fig.~\ref{fig:tdnc}B. In the log, Alice tried collaborating with Tom for $\mathfrak{T}$, while Tom had agreed to work with Jerry and Frank when he received Alice's request. Thus, Alice turned to Frank and Jerry, but it was a bit late to make appointments with them as they were partially engaged. As a result, Alice took $0.8\mathfrak{T}$ to play with Frank and Jerry and wasted $0.2\mathfrak{T}$ in this round.
%
\begin{figure*}[ht]%[tbhp]
\begin{figure*}[!htbp]%[tbhp]
\centering
\includegraphics[]{vis.eps}
\includegraphics[width=\textwidth]{vis.eps}
\caption{Illustration of the temporal games in a two-player collaborative system. (A) One round of the temporal game on a social network. The blue circle is Jerry's neighborhood. Alice, Bob, and Tom are Jerry's partners in this round. The color of a time slot represents a partner; for instance, yellow represents Frank. $C$ or $D$ in the time slot denotes the move from the individual at the tail of a directed dashed line to the indicated specific partner. (B) The generating procedure of the circumstance presented in (A). In the communication log, the records are sorted by their sequence numbers in ascending order. Only if both players agree to collaborate (the response to a request is OK) will their colors appear in each other's collaboration schedule, i.e., a time slot in (A).}
\label{fig:tdnc}
\end{figure*}
@ -185,11 +181,11 @@ For a heterogeneous network as the Barab\'{a}si-Albert (BA) networks~\cite{barab
%\end{equation}
%where $S_{k_j}\left(r\right)$ denotes the average available time of a neighbor with degree $k_j$ at round $r$. When $\theta > 0$, trading partners may bring $i$ more payoff, since $\mathcal{T}>\mathcal{R}$ in the PD game. Interestingly, one can see that $\theta_{k_i,r} = S\left(r\right)$ in the WS networks. Therefore, one can use $S\left(r\right)$ to measure the available time resource in the neighborhood. Although $\theta_{k_i,r} \neq S\left(r\right)$ in the BA networks, $S\left(r\right)$ is applicable to depict the average available time resource in the neighborhood. Therefore, we will adopt $S\left(r\right)$ to represent the available time resource in the system.
\subsection*{Proportion of cooperation in the temporal game}
\subsection{Proportion of cooperation in the temporal game}
In the temporal game, each game between partners is coupled with a duration. Therefore, the level of cooperation should be measured by the duration and their moves. We define the proportion of cooperation as ${P_c} = \frac{{{T_C}}}{T_G}$, where $T_G$ is the total duration of the moves and $T_C$ is the total duration of cooperation in the games.
%\subsection*{Dissipative system in the temporal game}
%\subsection{Dissipative system in the temporal game}
%
%In our experiment, two types of the temporal social dilemma, dissipative scenario and classical scenario, are tested. In the dissipative scenario, individuals are provided with an initial resource. In each round, individuals play the game with some of their friends to earn more payoffs. Specifically, the aggregated payoff of an individual $i$ in round $r$ ($r\in\mathds{N}$) can be written as
%\begin{equation}
@ -205,7 +201,7 @@ Note that current studies on decision time~\cite{evans2019cooperation, evans2015
%In the following, we will propose a model to reproduce the statistical results in our empirical experiments. As modeling the level of cooperation is a rather challenging task, we adopt a mean-field method and make some necessary assumptions. Note that the assumptions are not the components of the temporal games but only the tools to modeling the level of cooperation.
\subsection*{Mathematical modeling the available time of individuals}\label{MMATI}
\subsection{Mathematical modeling the available time of individuals}\label{MMATI}
As is known, for each game between two players, each player has to experience one of the four possible cases, namely, cooperating with a cooperator (CC), cooperating with a defector (CD), defecting a cooperator (DC), and defecting a defector (DD). We define a state vector $\mathbf{\Phi}$ by $(\Phi_{CC},\Phi_{CD},\Phi_{DC},\Phi_{DD})$, in which each entry corresponds to the probability of experiencing the respective outcome. Generally, a memory-one strategy can be written as $\mathbf{p}=(p_{CC},p_{CD},p_{DC},p_{DD})$, corresponding to the probabilities of cooperating under each of the previous outcomes. Since players update their moves with the memory-one strategies in each time step, the update can be considered a Markov process. One can find a Markov transition matrix $M_i$ to realize the update. For two players, $i$ and $j$, we have
\begin{equation}
\tiny{
@ -224,7 +220,7 @@ where the vectors $\mathbf{p}=(p_{CC},p_{CD},p_{DC},p_{DD})$ and $\mathbf{s}=(s_
\mathbf{\Phi}_i(r)=\mathbf{\Phi}_i(r-1)M_i.
\end{equation}
To model the the available time of individuals in the temporal games, we first assume that no players at round $r-1$ reject the requests from an individual $i$ if they are available. The time left for him to make use of in round $r$ can be denoted by $S_{i}\left(r\right)=\mathfrak{T} - \sum_{j\in{P_i}}\tau_{u_{ij}\left(r-1\right)}$, where $\mu_{ij}\left(r-1\right)$ denotes the random portion of time in the request from $i$ or $j$ in round $r-1$ and takes a random real number between 0 and 1. If $i$ applies for playing with $j$ for $S_{i}\left(r\right)\mu_{ij}\left(r\right)$, the successful probability of the request is
To model the the available time of individuals in the temporal games, we first assume that no players at round $r-1$ reject the requests from an individual $i$ if they are available. The time left for him to make use of in round $r$ can be denoted by $S_{i}\left(r\right)=\mathfrak{T} - \sum_{j\in{P_i}}\tau_{u_{ij}\left(r-1\right)}$, where $\mu_{ij}\left(r-1\right)$ denotes the random portion of time in the request from $i$ in round $r-1$. If $i$ applies for playing with $j$ for $S_{i}\left(r\right)\mu_{ij}\left(r\right)$, the successful probability of the request is
\begin{equation}
{\omega_{i,j}\left(r,\mu_{ij}\left(r\right)\right)} = \left\{ {\begin{array}{*{20}{cl}}
{1}, & {S_{j}\left(r\right)\geq S_{i}\left(r\right)\mu_{ij}\left(r\right)},\\
@ -236,7 +232,7 @@ assuming $j$ wish to play. Therefore, the expectation of difference in individua
&\varrho_{i}\left(r\right)=-\sum_{j\in {N_i-P_i\left(r-1\right)}}\omega_{i,j}\left(r,\mu_{ij}\left(r\right)\right)\left(S_{i}\left(r\right)\right. \\\nonumber
&\left.+\sum_{l\in P_i\left(r-1\right)}\alpha_{il}\left(r-1\right) \left( \mathbf{\Phi}_{il}\left(r\right) \cdot \begin{bmatrix}\chi_{i,CC} \\ \chi_{i,CD} \\ \chi_{i,DC} \\ \chi_{i,DD}\end{bmatrix}\right)\right)\mu_{ij}\left(r\right),
\end{eqnarray}
where $\mathbf{\chi_i}$ denotes $i$'s probabilities of reassigning time after experiencing the four outcomes. $\alpha_{il}\left(r\right)$ denotes the time share which $i$ assigns to $j$ at round $r$.
where $\mathbf{\chi_i}$ denotes $i$'s probabilities of reassigning time after experiencing the four outcomes. $\alpha_{il}\left(r\right)$ denotes the time share which $i$ assigns to $l$ at round $r$.
Note that
\begin{equation}
\sum_{l\in {P_i}}\alpha_{il}\left(r\right)+S_{i}\left(r\right)=1.
@ -322,8 +318,8 @@ As the evolution procedure of $S_i\left(r\right)$ in the system can not be model
%Summarizing the iterative procedure, the backup to defect $\theta$ in Eqn.~\ref{eqn:theta} is substituted into Eqn.~\ref{eqn:pc} to derive $P_{c,r}$ first; second, $P_{c,r}$ is used to derive $P_{CC,r}$, $P_{CD,r}$, $P_{DC,r}$, and $P_{DD,r}$ in Eqn.~\ref{eqn:probabilities}; with the probabilities of the four outcomes, one has $E\left(\Lambda_r\right)$ in Eqn.~\ref{Elambda_r} and $\phi_r$ in Eqn.~\ref{eqn:phir}; $\phi_r$ is then used to derive $\omega(G, r+1)$ in Eqn.~\ref{eqn:omega} and $P_{c,r+1}$ in Eqn.~\ref{eqn:pc}. The initial payoff $\phi_0$s of both modes are given at the beginning of each match. Therefore, one can iterate the procedure to derive $P_{c,r}$ and optimize $\alpha$, $\beta$, $\gamma$, and $\xi$ in Eqn.~\ref{eqn:alpha} finally. Notably, the mathematical model is based on the statistical results and simulation results obtained by our experiments. We adopt a series of approximations to model the decision-making procedure of humans only to provide a better understanding of the procedure. The accuracy of prediction is not our focus.
\section*{Results}\label{Er}
To show the impact of time redistribution, we first simulate the evolution of moves when agents play a traditional Prisoner's dilemma (PD) game with their neighbors in the BA and WS networks. In a network, a player starts a game with a gaming request to a neighbor. In our simulations, all the agents in the network are selected one by one, following a random sequence. For a selected agent, it evenly allocates the time left to its requests to the uncoordinated neighbors. If the requested neighbor has enough time to accept the gaming request, he will accept it. After one round of the game, agents will uniformly update their moves with the Zero-Determinant Extortionate strategy proposed in reference~\cite{stewart_pnas12}. The strategy will wipe the cooperators out in a few rounds. If an agent defects in a round, the pair will be taken apart with a certain probability. The separation means the time assigned to the pair will be redistributed next round. More details on the simulations will be provided in Section~\emph{Simulation on the social networks}.
\section{Results}\label{Er}
To show the impact of time redistribution, we first simulate the evolution of moves when agents play a traditional Prisoner's dilemma (PD) game with their neighbors in the BA and WS networks. In a network, a player starts a game with a gaming request to a neighbor. In our simulations, all the agents in the network are selected one by one, following a random sequence. For a selected agent, it evenly allocates the time left to its requests to the uncoordinated neighbors. If the requested neighbor has enough time to accept the gaming request, he will accept it. After one round of the game, agents will uniformly update their moves with the Zero-Determinant Extortionate strategy proposed in reference~\cite{stewart_pnas12}. The strategy will wipe the cooperators out in 100 rounds. If an agent defects in a round, the pair will be taken apart with a certain probability. The separation means the time assigned to the pair will be redistributed next round. More details on the simulations will be provided in Section~\emph{Simulation on the social networks}.
In Fig.~\ref{fig:APC}(a) and \ref{fig:APC}(b), the results show the level of cooperation decays with rounds for agents playing the `divide-and-conquer' (D\&C) games~\cite{zhang_yc_sr15,wang_js_sr17,melamed_pnas18} in both networks. After being affected by the temporal mechanisms, the rates of decay slow down in Fig.~\ref{fig:APC}(c) and \ref{fig:APC}(d). We show the difference in the level of cooperation between the temporal games and the D\&C games~\cite{zhang_yc_sr15,wang_js_sr17,melamed_pnas18} in Fig.~\ref{fig:APC}(e) and \ref{fig:APC}(f), which will be amplified when human subjects play. The amplification may originate from $S\left(r\right)$ shown in Fig.~\ref{fig:APC}(g) and \ref{fig:APC}(h), which will be much lower when humans play the temporal games.
@ -346,23 +342,25 @@ To test the validity of our theoretical results, we invite 183 volunteers to att
\begin{table*}[tbhp]
\caption{The basic information of matches.}
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{|c|c|c|c||c||c|}
\hline
Game Number & Game Type & Type of Network & Number of Participants & Number of Rounds & Corresponding Panel in Fig.~\ref{fig:HMBAWS} \\ \hline
G1224 & D\&C & BA & 39 & 13 & Fig.~\ref{fig:HMBAWS}(a) \\ \hline
G1230 & D\&C & BA & 17 & 16 & Fig.~\ref{fig:HMBAWS}(a) \\ \hline
G646 & Temporal Games & BA & 50 & 11 & Fig.~\ref{fig:HMBAWS}(b) \\ \hline
G903 & Temporal Games & BA & 44 & 28 & Fig.~\ref{fig:HMBAWS}(b) \\ \hline
G646 & Temporal Games & BA & 50 & 11 & Fig.~\ref{fig:HMBAWS}(b) and Fig.~\ref{fig:HMBAWS}(e) \\ \hline
G903 & Temporal Games & BA & 44 & 28 & Fig.~\ref{fig:HMBAWS}(b) and Fig.~\ref{fig:HMBAWS}(f) \\ \hline
G1228 & D\&C & WS & 34 & 13 & Fig.~\ref{fig:HMBAWS}(c) \\ \hline
G1234 & D\&C & WS & 21 & 15 & Fig.~\ref{fig:HMBAWS}(c) \\ \hline
G936 & Temporal Games & WS & 22 & 24 & Fig.~\ref{fig:HMBAWS}(d) \\ \hline
G933 & Temporal Games & WS & 22 & 28 & Fig.~\ref{fig:HMBAWS}(d) \\ \hline
G936 & Temporal Games & WS & 22 & 24 & Fig.~\ref{fig:HMBAWS}(d) and Fig.~\ref{fig:HMBAWS}(g)\\ \hline
G933 & Temporal Games & WS & 22 & 28 & Fig.~\ref{fig:HMBAWS}(d) and Fig.~\ref{fig:HMBAWS}(h)\\ \hline
\end{tabular}
}
\label{PRGNF}
\end{table*}
After comparing Fig.~\ref{fig:HMBAWS}(a) with Fig.~\ref{fig:HMBAWS}(b) and Fig.~\ref{fig:HMBAWS}(c) with Fig.~\ref{fig:HMBAWS}(d), one can see that the decay of $P_c\left(r\right)$ in the temporal games is slower than that in the D\&C games. The result confirms our theoretical prediction, indicating the limitation on gaming time promotes the level of cooperation in gaming social networks.
After comparing Fig.~\ref{fig:HMBAWS}(a) with Fig.~\ref{fig:HMBAWS}(b) and Fig.~\ref{fig:HMBAWS}(c) with Fig.~\ref{fig:HMBAWS}(d), one can see that the decay of $P_c\left(r\right)$ in the temporal games is slower than that in the D\&C games. The result confirms our theoretical prediction, indicating the limitation on time promotes the level of cooperation in gaming social networks.
To explain the behavior, we measure the average available time $S\left(r\right)$ in the four time-involved matches. The evolution of $S\left(r\right)$ for the two BA networks are shown in Fig.~\ref{fig:HMBAWS}(e) and Fig.~\ref{fig:HMBAWS}(f), respectively. The corresponding results for the two WS networks are shown in Fig.~\ref{fig:HMBAWS}(g) and Fig.~\ref{fig:HMBAWS}(h), respectively. One can see that $S\left(r\right)$ fluctuates around a small positive value in the four panels, revealing the difficulty of finding new partners when humans play the temporal games is more significant than our theoretical prediction. The difference in $P_c\left(r\right)$ between the theoretical prediction and human behavior suggests that the rising of the difficulty of finding new partners may lead to the promotion of $P_c\left(r\right)$, which to some extent explains why the limited time promotes the level of cooperation in a social network.
To explain the behavior, we measure the average available time $S\left(r\right)$ in the four time-involved matches. The evolution of $S\left(r\right)$ for the two BA networks and two WS networks are shown in Fig.~\ref{fig:HMBAWS}(e)-Fig.~\ref{fig:HMBAWS}(h), respectively. For conciseness, the basic information of matches is listed in Table~\ref{PRGNF}. One can see that $S\left(r\right)$ fluctuates around a small positive value in the four panels, revealing the difficulty of finding new partners when humans play the temporal games is more significant than our theoretical prediction. The difference in $P_c\left(r\right)$ between the theoretical prediction and human behavior suggests that the rising of the difficulty of finding new partners may lead to the promotion of $P_c\left(r\right)$, which to some extent explains why the limited time promotes the level of cooperation in a real social network.
The other behavior which should be noted is that the level of cooperation generally decays with rounds in Fig.~\ref{fig:HMBAWS}.
@ -372,7 +370,7 @@ The behavior is caused by the number of rounds for each match being limited, alt
\begin{figure}[ht]
\centering
\includegraphics[height=4in,clip,keepaspectratio,trim=0 0 0 0]{Human_Merged_BA_WS.eps}
\caption{Evolution of the average proportion of cooperation $P_c\left(r\right)$ and the average proportion of cooperation $S\left(r\right)$ in the temporal games played by human subjects. (a) and (c) show the results of the D\&C games in the BA networks and WS networks, respectively. (b) and (d) show the results of the temporal games in the BA networks and WS networks, respectively. Horizontal coordinates denote the number of rounds. (e) and (f) show the results of two temporal games in the BA networks. (g) and (h) show the results of two temporal games in the WS networks.}\label{fig:HMBAWS}
\caption{Evolution of the proportion of cooperation $P_c\left(r\right)$ and the average available time $S\left(r\right)$ in the temporal games played by human subjects. (a) and (c) show the results of the D\&C games in the BA networks and WS networks, respectively. (b) and (d) show the results of the temporal games in the BA networks and WS networks, respectively. Horizontal coordinates denote the number of rounds. (e) and (f) show the results of two temporal games in the BA networks. (g) and (h) show the results of two temporal games in the WS networks.}\label{fig:HMBAWS}
\end{figure}
%\begin{figure}[ht]
@ -448,7 +446,7 @@ The behavior is caused by the number of rounds for each match being limited, alt
%
%Finally, people are making targeted choices. In our experiment, an individual is allowed to choose their neighbors to play with and to move for a specific reason. Our result shows that 7.71\% of the individuals (96 out of 1,244) are identified as `divide-and-conquer' ($D\&C$) individuals~\cite{zhang_yc_sr15,wang_js_sr17,melamed_pnas18}, who may cooperate with some partners to stabilize their partnerships and defect the other partners so as to pursue higher payoffs in the current round. The rest individuals are uncertain, since they have only one partner or their partners' moves are identical.
\section*{Discussion}\label{Dis}
\section{Discussion}\label{Dis}
As a theoretical framework closer to realistic scenarios, the temporal game has demonstrated its capacity to illuminate complex behaviors in our social experiment. The human behaviors revealed from the human temporal games were rarely reported previously in the literature. When the available time resources of individuals in the gaming network are scarce, the individuals are more likely to maintain the current relationships through cooperation. The underlying mechanism is that interactions are not obligated but spontaneous. If an individual's time resource cannot afford the requested duration of the interaction, he will have no choice but to abandon it, which makes it much harder to find new partners. The accordance of empirical and simulation results confirms the significance of the mechanism. Our finding reveals a fundamental reason for lasting altruistic behaviors in real human interactions, providing a novel perspective for understanding the prevailing of human cooperative behaviors in temporal collaboration systems.
@ -461,16 +459,15 @@ Note that the limitation on time is an objective fact in human collaboration sys
Our work considers the temporal game framework and presents some surprising results. There are several interesting future directions, both in terms of theoretical and experimental results. However, the basic theoretical model and the key experimental results we present in this work for temporal games are the first steps to modeling realistic networks with time-dependent interactions. Such realistic modeling will allow better analysis, prediction, and design principles for the emergence of cooperation in network models, profoundly impacting disciplines from preserving natural resources to designing institutional policies.
\section*{Materials and Methods}
\small{
\subsection*{Experimental design}\label{ED}
\section{Materials and Methods}
\subsection{Experimental design}\label{ED}
In order to build an experimental environment as close as possible to natural temporal two-player collaborative systems, two realistic factors are considered in our empirical study. First, the interactive time is determined by negotiation. The setting restores the temporal property of a game in reality. A dynamic reconnection is implemented in the network by rejecting a friend's request and then proposing a game with another friend~\cite{rand_pnas11,wang_j_pnas12}. Second, a `divide-and-conquer' ($D\&C$) framework, also referred to as targeted decision, is adopted, in which the individuals who propose a game or accept a gaming request have to decide whether to cooperate ($C$) or to defect ($D$) in each round of the game~\cite{zhang_yc_sr15,wang_js_sr17,melamed_pnas18}. Most existing research on gaming networks is performed under a framework where individuals choose the same move to interact with all their neighbors~\cite{rand_pnas11,fehl_el11,wang_j_pnas12}. On the contrary, in real-world scenarios, people do not normally defect their long-term partners after being defected by other partners. In a realistic social network, they would choose a specific move to play with a partner, referred to as the D\&C game in the literature~\cite{zhang_yc_sr15,wang_js_sr17}. When the diffuse decision scheme is replaced by the D\&C or targeted decision scheme, the impact of dynamic reconnection on promoting cooperation will become negligible~\cite{melamed_pnas18}.
The coupling between temporal interaction and rational decision-making can be seen everywhere in real life. Still, the existing theoretical frameworks seem insufficient to explain the widespread cooperation in such temporal games. Under the framework of temporal games, we designed a series of online game experiments. With the experimental data, we present a surprising finding: limitation of time promotes cooperation in temporal games. This finding, on the one hand, urges us to reconsider how much the dynamic nature of networks can impact human cooperation. On the other hand, it implies the potential of the temporal game framework to explain various collective behaviors in real two-player collaborative systems.
%
Our results have a profound impact on the study of pro-social behavior. By accounting for the time-dependent aspect to model a realistic network, we present an interesting finding which can improve our understanding of widespread cooperation in time-dependent collaborations.
\subsection*{Experimental setup and game rules}\label{ESGR}
\subsection{Experimental setup and game rules}\label{ESGR}
A series of online human subject experiments were designed to build a two-player collaborative system of rational individuals. A total of 183 human subjects participated in 8 matches in the experiment. The majority of subjects are students from Tongji University and Southeast University in China. To implement the designed scenario, a novel online gaming platform was developed, called the \emph{War of Strategies} (http://strategywar.net, see~(Section \textbf{Experimental Platform and Interface} of \textbf{SI} for the details of the platform).
In the online experiments, participants played a traditional Prisoner's dilemma (PD) game, where $C$ and $D$ were the only available actions. Each participant interacted with the individuals who had agreements with him in one round, after which the agreements need to be redrafted.
@ -481,7 +478,7 @@ In each round of the game, individuals can make requests for interactions with t
During the match, the individual IDs are randomly generated. The individuals can only see their own game records, where each record includes the moves of both sides and the time durations. The topological structures beyond their immediate neighbors are invisible to them. Besides, individuals are shown their aggregated payoff, time resources, number of rounds played, and their decision time remaining.
\subsection*{Simulation on the social networks}\label{SSN}
\subsection{Simulation on the social networks}\label{SSN}
Here, we will present the process of the simulation.
Step 1, Generate a structured population such as the Barab\'{a}si and Albert's scale-free network~\cite{BA} with degree $m_0=m=3$ or Watts and Strogatz's small-world network with $P_{rewire}=0.1$ and $K=6$. Randomly assign the agents to be cooperators with a probability of 0.5. The size of the population is set to 1,024.
Step 2, Shuffle the agent list and iteratively ask an agent to broadcast gaming requests to its neighbors. In each request, the agent evenly allocates its time left to its uncoordinated neighbors, i.e., $\mu_{ij}\left(r\right)=\frac{1}{\left | N_i-P_i\left(r-1\right) \right | }$, where $j\in N_i-P_i\left(r-1\right)$. If a neighbor has enough time to accept the request, he will accept it.
@ -489,28 +486,12 @@ Step 3, Each pair of the matched agents game for one round and update their move
Step 4, If an agent defects in the round, the pair will be taken apart with a probability of 0.5, that is, $\mathbf{\chi}=[0,0.5,0.5,0.5]$.
Step 5, Repeat Steps 2, 3, and 4 until the preset number of rounds.
\showmatmethods % Display the Materials and Methods section
\bmhead{Acknowledgments}
\acknow{Y. Z. was supported by the National Natural Science Foundation of China (Grant No. 61503285) and by the Municipal Natural Science Foundation of Shanghai (Grant No. 17ZR1446000). J. G. was supported by the National Natural Science Foundation of China (Grant No. 61772367) and by the Program of Shanghai Science and Technology Committee (Grant No. 16511105200). S. Z. was supported by the Program of Science and Technology Innovation Action of the Science and Technology Commission of Shanghai Municipality (STCSM) (Grant No. 17511105204). G. C. was supported by the Hong Kong Research Grants Council (Grant No. CityU-11200317). K. C. was supported by ERC Consolidator Grant 863818 (FoRM-SMArt). M. P. was supported by the Slovenian Research Agency (Grant Nos. J1-2457, J1-9112, and P1-0403).
}
}%small
\showacknow % Display the acknowledgments section
Y. Z. was supported by the National Natural Science Foundation of China (Grant No. 61503285) and by the Municipal Natural Science Foundation of Shanghai (Grant No. 17ZR1446000). J. G. was supported by the National Natural Science Foundation of China (Grant No. 61772367) and by the Program of Shanghai Science and Technology Committee (Grant No. 16511105200). S. Z. was supported by the Program of Science and Technology Innovation Action of the Science and Technology Commission of Shanghai Municipality (STCSM) (Grant No. 17511105204). G. C. was supported by the Hong Kong Research Grants Council (Grant No. CityU-11200317). K. C. was supported by ERC Consolidator Grant 863818 (FoRM-SMArt). M. P. was supported by the Slovenian Research Agency (Grant Nos. J1-2457, J1-9112, and P1-0403).
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% \readytosubmit %% Uncomment this line before submitting, so that the instruction page is removed.
\title{Limitation of time promotes cooperation in temporal games}
\author{Jiasheng Wang, Yichao Zhang, Guanghui Wen, Jihong Guan, Shuigeng Zhou, \\ \normalsize{\textbf{Guanrong Chen, Krishnendu Chatterjee, and Matja{\v z} Perc}} }
\correspondingauthor{Yichao Zhang\\E-mail: yichaozhang@tongji.edu.cn\\
Guanghui Wen\\E-mail: wenguanghui@gmail.com\\
Jihong Guan\\E-mail: jhguan@tongji.edu.cn}
\leadauthor{Wang, Zhang}
\begin{document}
\title{Supplementary Information for Limitation of time promotes cooperation in temporal games}
\author{Jiasheng Wang}
\author*{Yichao Zhang}\email{yichaozhang@tongji.edu.cn}
\author*{Guanghui Wen}\email{wenguanghui@gmail.com}
\author*{Jihong Guan}\email{jhguan@tongji.edu.cn}
\author{Shuigeng Zhou}
\author{Guanrong Chen}
\author{Krishnendu Chatterjee}
\author{Matja{\v z} Perc}
% \instructionspage
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%% Adds the main heading for the SI text. Comment out this line if you do not have any supporting information text.
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\begin{appendices}
\section*{Volunteers Recruitment and Experimental Setup}
@ -68,7 +78,7 @@ To implement the experimental scenarios of the temporal divide-and-conquer games
\subsection*{Architecture}
The platform is developed based on several open source software, composed of three components: Portal, Distributor, and Worker. The architecture is shown in Fig.~\ref{fig:arch}.
\begin{figure}[ht]%[tbhp]
\begin{figure}[!htbp]%[tbhp]
\centering
\includegraphics[width=.7\linewidth]{architecture}
\caption{Platform architecture of the WoS.}
@ -1006,53 +1016,50 @@ Round 14:
7. He/She had to deny to play with player 19884 due to lack of time resource.
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\affil*[1]{\orgdiv{Department}, \orgname{Organization}, \orgaddress{\street{Street}, \city{City}, \postcode{100190}, \state{State}, \country{Country}}}
\affil[2]{\orgdiv{Department}, \orgname{Organization}, \orgaddress{\street{Street}, \city{City}, \postcode{10587}, \state{State}, \country{Country}}}
\affil[3]{\orgdiv{Department}, \orgname{Organization}, \orgaddress{\street{Street}, \city{City}, \postcode{610101}, \state{State}, \country{Country}}}
%%==================================%%
%% sample for unstructured abstract %%
%%==================================%%
\abstract{The abstract serves both as a general introduction to the topic and as a brief, non-technical summary of the main results and their implications. Authors are advised to check the author instructions for the journal they are submitting to for word limits and if structural elements like subheadings, citations, or equations are permitted.}
%%================================%%
%% Sample for structured abstract %%
%%================================%%
% \abstract{\textbf{Purpose:} The abstract serves both as a general introduction to the topic and as a brief, non-technical summary of the main results and their implications. The abstract must not include subheadings (unless expressly permitted in the journal's Instructions to Authors), equations or citations. As a guide the abstract should not exceed 200 words. Most journals do not set a hard limit however authors are advised to check the author instructions for the journal they are submitting to.
%
% \textbf{Methods:} The abstract serves both as a general introduction to the topic and as a brief, non-technical summary of the main results and their implications. The abstract must not include subheadings (unless expressly permitted in the journal's Instructions to Authors), equations or citations. As a guide the abstract should not exceed 200 words. Most journals do not set a hard limit however authors are advised to check the author instructions for the journal they are submitting to.
%
% \textbf{Results:} The abstract serves both as a general introduction to the topic and as a brief, non-technical summary of the main results and their implications. The abstract must not include subheadings (unless expressly permitted in the journal's Instructions to Authors), equations or citations. As a guide the abstract should not exceed 200 words. Most journals do not set a hard limit however authors are advised to check the author instructions for the journal they are submitting to.
%
% \textbf{Conclusion:} The abstract serves both as a general introduction to the topic and as a brief, non-technical summary of the main results and their implications. The abstract must not include subheadings (unless expressly permitted in the journal's Instructions to Authors), equations or citations. As a guide the abstract should not exceed 200 words. Most journals do not set a hard limit however authors are advised to check the author instructions for the journal they are submitting to.}
\keywords{keyword1, Keyword2, Keyword3, Keyword4}
%%\pacs[JEL Classification]{D8, H51}
%%\pacs[MSC Classification]{35A01, 65L10, 65L12, 65L20, 65L70}
\maketitle
\section{Introduction}\label{sec1}
The Introduction section, of referenced text \cite{bib1} expands on the background of the work (some overlap with the Abstract is acceptable). The introduction should not include subheadings.
Springer Nature does not impose a strict layout as standard however authors are advised to check the individual requirements for the journal they are planning to submit to as there may be journal-level preferences. When preparing your text please also be aware that some stylistic choices are not supported in full text XML (publication version), including coloured font. These will not be replicated in the typeset article if it is accepted.
\section{Results}\label{sec2}
Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text.
\section{This is an example for first level head---section head}\label{sec3}
\subsection{This is an example for second level head---subsection head}\label{subsec2}
\subsubsection{This is an example for third level head---subsubsection head}\label{subsubsec2}
Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text.
\section{Equations}\label{sec4}
Equations in \LaTeX\ can either be inline or on-a-line by itself (``display equations''). For
inline equations use the \verb+$...$+ commands. E.g.: The equation
$H\psi = E \psi$ is written via the command \verb+$H \psi = E \psi$+.
For display equations (with auto generated equation numbers)
one can use the equation or align environments:
\begin{equation}
\|\tilde{X}(k)\|^2 \leq\frac{\sum\limits_{i=1}^{p}\left\|\tilde{Y}_i(k)\right\|^2+\sum\limits_{j=1}^{q}\left\|\tilde{Z}_j(k)\right\|^2 }{p+q}.\label{eq1}
\end{equation}
where,
\begin{align}
D_\mu &= \partial_\mu - ig \frac{\lambda^a}{2} A^a_\mu \nonumber \\
F^a_{\mu\nu} &= \partial_\mu A^a_\nu - \partial_\nu A^a_\mu + g f^{abc} A^b_\mu A^a_\nu \label{eq2}
\end{align}
Notice the use of \verb+\nonumber+ in the align environment at the end
of each line, except the last, so as not to produce equation numbers on
lines where no equation numbers are required. The \verb+\label{}+ command
should only be used at the last line of an align environment where
\verb+\nonumber+ is not used.
\begin{equation}
Y_\infty = \left( \frac{m}{\textrm{GeV}} \right)^{-3}
\left[ 1 + \frac{3 \ln(m/\textrm{GeV})}{15}
+ \frac{\ln(c_2/5)}{15} \right]
\end{equation}
The class file also supports the use of \verb+\mathbb{}+, \verb+\mathscr{}+ and
\verb+\mathcal{}+ commands. As such \verb+\mathbb{R}+, \verb+\mathscr{R}+
and \verb+\mathcal{R}+ produces $\mathbb{R}$, $\mathscr{R}$ and $\mathcal{R}$
respectively (refer Subsubsection~\ref{subsubsec2}).
\section{Tables}\label{sec5}
Tables can be inserted via the normal table and tabular environment. To put
footnotes inside tables you should use \verb+\footnotetext[]{...}+ tag.
The footnote appears just below the table itself (refer Tables~\ref{tab1} and \ref{tab2}).
For the corresponding footnotemark use \verb+\footnotemark[...]+
\begin{table}[h]
\caption{Caption text}\label{tab1}%
\begin{tabular}{@{}llll@{}}
\toprule
Column 1 & Column 2 & Column 3 & Column 4\\
\midrule
row 1 & data 1 & data 2 & data 3 \\
row 2 & data 4 & data 5\footnotemark[1] & data 6 \\
row 3 & data 7 & data 8 & data 9\footnotemark[2] \\
\botrule
\end{tabular}
\footnotetext{Source: This is an example of table footnote. This is an example of table footnote.}
\footnotetext[1]{Example for a first table footnote. This is an example of table footnote.}
\footnotetext[2]{Example for a second table footnote. This is an example of table footnote.}
\end{table}
\noindent
The input format for the above table is as follows:
%%=============================================%%
%% For presentation purpose, we have included %%
%% \bigskip command. please ignore this. %%
%%=============================================%%
\bigskip
\begin{verbatim}
\begin{table}[<placement-specifier>]
\caption{<table-caption>}\label{<table-label>}%
\begin{tabular}{@{}llll@{}}
\toprule
Column 1 & Column 2 & Column 3 & Column 4\\
\midrule
row 1 & data 1 & data 2 & data 3 \\
row 2 & data 4 & data 5\footnotemark[1] & data 6 \\
row 3 & data 7 & data 8 & data 9\footnotemark[2]\\
\botrule
\end{tabular}
\footnotetext{Source: This is an example of table footnote.
This is an example of table footnote.}
\footnotetext[1]{Example for a first table footnote.
This is an example of table footnote.}
\footnotetext[2]{Example for a second table footnote.
This is an example of table footnote.}
\end{table}
\end{verbatim}
\bigskip
%%=============================================%%
%% For presentation purpose, we have included %%
%% \bigskip command. please ignore this. %%
%%=============================================%%
\begin{table}[h]
\caption{Example of a lengthy table which is set to full textwidth}\label{tab2}
\begin{tabular*}{\textwidth}{@{\extracolsep\fill}lcccccc}
\toprule%
& \multicolumn{3}{@{}c@{}}{Element 1\footnotemark[1]} & \multicolumn{3}{@{}c@{}}{Element 2\footnotemark[2]} \\\cmidrule{2-4}\cmidrule{5-7}%
Project & Energy & $\sigma_{calc}$ & $\sigma_{expt}$ & Energy & $\sigma_{calc}$ & $\sigma_{expt}$ \\
\midrule
Element 3 & 990 A & 1168 & $1547\pm12$ & 780 A & 1166 & $1239\pm100$\\
Element 4 & 500 A & 961 & $922\pm10$ & 900 A & 1268 & $1092\pm40$\\
\botrule
\end{tabular*}
\footnotetext{Note: This is an example of table footnote. This is an example of table footnote this is an example of table footnote this is an example of~table footnote this is an example of table footnote.}
\footnotetext[1]{Example for a first table footnote.}
\footnotetext[2]{Example for a second table footnote.}
\end{table}
\vfill\eject
In case of double column layout, tables which do not fit in single column width should be set to full text width. For this, you need to use \verb+\begin{table*}+ \verb+...+ \verb+\end{table*}+ instead of \verb+\begin{table}+ \verb+...+ \verb+\end{table}+ environment. Lengthy tables which do not fit in textwidth should be set as rotated table. For this, you need to use \verb+\begin{sidewaystable}+ \verb+...+ \verb+\end{sidewaystable}+ instead of \verb+\begin{table*}+ \verb+...+ \verb+\end{table*}+ environment. This environment puts tables rotated to single column width. For tables rotated to double column width, use \verb+\begin{sidewaystable*}+ \verb+...+ \verb+\end{sidewaystable*}+.
\begin{sidewaystable}
\caption{Tables which are too long to fit, should be written using the ``sidewaystable'' environment as shown here}\label{tab3}
\begin{tabular*}{\textheight}{@{\extracolsep\fill}lcccccc}
\toprule%
& \multicolumn{3}{@{}c@{}}{Element 1\footnotemark[1]}& \multicolumn{3}{@{}c@{}}{Element\footnotemark[2]} \\\cmidrule{2-4}\cmidrule{5-7}%
Projectile & Energy & $\sigma_{calc}$ & $\sigma_{expt}$ & Energy & $\sigma_{calc}$ & $\sigma_{expt}$ \\
\midrule
Element 3 & 990 A & 1168 & $1547\pm12$ & 780 A & 1166 & $1239\pm100$ \\
Element 4 & 500 A & 961 & $922\pm10$ & 900 A & 1268 & $1092\pm40$ \\
Element 5 & 990 A & 1168 & $1547\pm12$ & 780 A & 1166 & $1239\pm100$ \\
Element 6 & 500 A & 961 & $922\pm10$ & 900 A & 1268 & $1092\pm40$ \\
\botrule
\end{tabular*}
\footnotetext{Note: This is an example of table footnote this is an example of table footnote this is an example of table footnote this is an example of~table footnote this is an example of table footnote.}
\footnotetext[1]{This is an example of table footnote.}
\end{sidewaystable}
\section{Figures}\label{sec6}
As per the \LaTeX\ standards you need to use eps images for \LaTeX\ compilation and \verb+pdf/jpg/png+ images for \verb+PDFLaTeX+ compilation. This is one of the major difference between \LaTeX\ and \verb+PDFLaTeX+. Each image should be from a single input .eps/vector image file. Avoid using subfigures. The command for inserting images for \LaTeX\ and \verb+PDFLaTeX+ can be generalized. The package used to insert images in \verb+LaTeX/PDFLaTeX+ is the graphicx package. Figures can be inserted via the normal figure environment as shown in the below example:
%%=============================================%%
%% For presentation purpose, we have included %%
%% \bigskip command. please ignore this. %%
%%=============================================%%
\bigskip
\begin{verbatim}
\begin{figure}[<placement-specifier>]
\centering
\includegraphics{<eps-file>}
\caption{<figure-caption>}\label{<figure-label>}
\end{figure}
\end{verbatim}
\bigskip
%%=============================================%%
%% For presentation purpose, we have included %%
%% \bigskip command. please ignore this. %%
%%=============================================%%
\begin{figure}[h]%
\centering
\includegraphics[width=0.9\textwidth]{fig.eps}
\caption{This is a widefig. This is an example of long caption this is an example of long caption this is an example of long caption this is an example of long caption}\label{fig1}
\end{figure}
In case of double column layout, the above format puts figure captions/images to single column width. To get spanned images, we need to provide \verb+\begin{figure*}+ \verb+...+ \verb+\end{figure*}+.
For sample purpose, we have included the width of images in the optional argument of \verb+\includegraphics+ tag. Please ignore this.
\section{Algorithms, Program codes and Listings}\label{sec7}
Packages \verb+algorithm+, \verb+algorithmicx+ and \verb+algpseudocode+ are used for setting algorithms in \LaTeX\ using the format:
%%=============================================%%
%% For presentation purpose, we have included %%
%% \bigskip command. please ignore this. %%
%%=============================================%%
\bigskip
\begin{verbatim}
\begin{algorithm}
\caption{<alg-caption>}\label{<alg-label>}
\begin{algorithmic}[1]
. . .
\end{algorithmic}
\end{algorithm}
\end{verbatim}
\bigskip
%%=============================================%%
%% For presentation purpose, we have included %%
%% \bigskip command. please ignore this. %%
%%=============================================%%
You may refer above listed package documentations for more details before setting \verb+algorithm+ environment. For program codes, the ``verbatim'' package is required and the command to be used is \verb+\begin{verbatim}+ \verb+...+ \verb+\end{verbatim}+.
Similarly, for \verb+listings+, use the \verb+listings+ package. \verb+\begin{lstlisting}+ \verb+...+ \verb+\end{lstlisting}+ is used to set environments similar to \verb+verbatim+ environment. Refer to the \verb+lstlisting+ package documentation for more details.
A fast exponentiation procedure:
\lstset{texcl=true,basicstyle=\small\sf,commentstyle=\small\rm,mathescape=true,escapeinside={(*}{*)}}
\begin{lstlisting}
begin
for $i:=1$ to $10$ step $1$ do
expt($2,i$);
newline() od (*\textrm{Comments will be set flush to the right margin}*)
where
proc expt($x,n$) $\equiv$
$z:=1$;
do if $n=0$ then exit fi;
do if odd($n$) then exit fi;
comment: (*\textrm{This is a comment statement;}*)
$n:=n/2$; $x:=x*x$ od;
{ $n>0$ };
$n:=n-1$; $z:=z*x$ od;
print($z$).
end
\end{lstlisting}
\begin{algorithm}
\caption{Calculate $y = x^n$}\label{algo1}
\begin{algorithmic}[1]
\Require $n \geq 0 \vee x \neq 0$
\Ensure $y = x^n$
\State $y \Leftarrow 1$
\If{$n < 0$}\label{algln2}
\State $X \Leftarrow 1 / x$
\State $N \Leftarrow -n$
\Else
\State $X \Leftarrow x$
\State $N \Leftarrow n$
\EndIf
\While{$N \neq 0$}
\If{$N$ is even}
\State $X \Leftarrow X \times X$
\State $N \Leftarrow N / 2$
\Else[$N$ is odd]
\State $y \Leftarrow y \times X$
\State $N \Leftarrow N - 1$
\EndIf
\EndWhile
\end{algorithmic}
\end{algorithm}
%%=============================================%%
%% For presentation purpose, we have included %%
%% \bigskip command. please ignore this. %%
%%=============================================%%
\bigskip
\begin{minipage}{\hsize}%
\lstset{frame=single,framexleftmargin=-1pt,framexrightmargin=-17pt,framesep=12pt,linewidth=0.98\textwidth,language=pascal}% Set your language (you can change the language for each code-block optionally)
%%% Start your code-block
\begin{lstlisting}
for i:=maxint to 0 do
begin
{ do nothing }
end;
Write('Case insensitive ');
Write('Pascal keywords.');
\end{lstlisting}
\end{minipage}
\section{Cross referencing}\label{sec8}
Environments such as figure, table, equation and align can have a label
declared via the \verb+\label{#label}+ command. For figures and table
environments use the \verb+\label{}+ command inside or just
below the \verb+\caption{}+ command. You can then use the
\verb+\ref{#label}+ command to cross-reference them. As an example, consider
the label declared for Figure~\ref{fig1} which is
\verb+\label{fig1}+. To cross-reference it, use the command
\verb+Figure \ref{fig1}+, for which it comes up as
``Figure~\ref{fig1}''.
To reference line numbers in an algorithm, consider the label declared for the line number 2 of Algorithm~\ref{algo1} is \verb+\label{algln2}+. To cross-reference it, use the command \verb+\ref{algln2}+ for which it comes up as line~\ref{algln2} of Algorithm~\ref{algo1}.
\subsection{Details on reference citations}\label{subsec7}
Standard \LaTeX\ permits only numerical citations. To support both numerical and author-year citations this template uses \verb+natbib+ \LaTeX\ package. For style guidance please refer to the template user manual.
Here is an example for \verb+\cite{...}+: \cite{bib1}. Another example for \verb+\citep{...}+: \citep{bib2}. For author-year citation mode, \verb+\cite{...}+ prints Jones et al. (1990) and \verb+\citep{...}+ prints (Jones et al., 1990).
All cited bib entries are printed at the end of this article: \cite{bib3}, \cite{bib4}, \cite{bib5}, \cite{bib6}, \cite{bib7}, \cite{bib8}, \cite{bib9}, \cite{bib10}, \cite{bib11}, \cite{bib12} and \cite{bib13}.
\section{Examples for theorem like environments}\label{sec10}
For theorem like environments, we require \verb+amsthm+ package. There are three types of predefined theorem styles exists---\verb+thmstyleone+, \verb+thmstyletwo+ and \verb+thmstylethree+
%%=============================================%%
%% For presentation purpose, we have included %%
%% \bigskip command. please ignore this. %%
%%=============================================%%
\bigskip
\begin{tabular}{|l|p{19pc}|}
\hline
\verb+thmstyleone+ & Numbered, theorem head in bold font and theorem text in italic style \\\hline
\verb+thmstyletwo+ & Numbered, theorem head in roman font and theorem text in italic style \\\hline
\verb+thmstylethree+ & Numbered, theorem head in bold font and theorem text in roman style \\\hline
\end{tabular}
\bigskip
%%=============================================%%
%% For presentation purpose, we have included %%
%% \bigskip command. please ignore this. %%
%%=============================================%%
For mathematics journals, theorem styles can be included as shown in the following examples:
\begin{theorem}[Theorem subhead]\label{thm1}
Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text.
Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text.
Example theorem text.
\end{theorem}
Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text.
\begin{proposition}
Example proposition text. Example proposition text. Example proposition text. Example proposition text. Example proposition text.
Example proposition text. Example proposition text. Example proposition text. Example proposition text. Example proposition text.
\end{proposition}
Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text.
\begin{example}
Phasellus adipiscing semper elit. Proin fermentum massa
ac quam. Sed diam turpis, molestie vitae, placerat a, molestie nec, leo. Maecenas lacinia. Nam ipsum ligula, eleifend
at, accumsan nec, suscipit a, ipsum. Morbi blandit ligula feugiat magna. Nunc eleifend consequat lorem.
\end{example}
Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text.
\begin{remark}
Phasellus adipiscing semper elit. Proin fermentum massa
ac quam. Sed diam turpis, molestie vitae, placerat a, molestie nec, leo. Maecenas lacinia. Nam ipsum ligula, eleifend
at, accumsan nec, suscipit a, ipsum. Morbi blandit ligula feugiat magna. Nunc eleifend consequat lorem.
\end{remark}
Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text.
\begin{definition}[Definition sub head]
Example definition text. Example definition text. Example definition text. Example definition text. Example definition text. Example definition text. Example definition text. Example definition text.
\end{definition}
Additionally a predefined ``proof'' environment is available: \verb+\begin{proof}+ \verb+...+ \verb+\end{proof}+. This prints a ``Proof'' head in italic font style and the ``body text'' in roman font style with an open square at the end of each proof environment.
\begin{proof}
Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text.
\end{proof}
Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text. Sample body text.
\begin{proof}[Proof of Theorem~{\upshape\ref{thm1}}]
Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text.
\end{proof}
\noindent
For a quote environment, use \verb+\begin{quote}...\end{quote}+
\begin{quote}
Quoted text example. Aliquam porttitor quam a lacus. Praesent vel arcu ut tortor cursus volutpat. In vitae pede quis diam bibendum placerat. Fusce elementum
convallis neque. Sed dolor orci, scelerisque ac, dapibus nec, ultricies ut, mi. Duis nec dui quis leo sagittis commodo.
\end{quote}
Sample body text. Sample body text. Sample body text. Sample body text. Sample body text (refer Figure~\ref{fig1}). Sample body text. Sample body text. Sample body text (refer Table~\ref{tab3}).
\section{Methods}\label{sec11}
Topical subheadings are allowed. Authors must ensure that their Methods section includes adequate experimental and characterization data necessary for others in the field to reproduce their work. Authors are encouraged to include RIIDs where appropriate.
\textbf{Ethical approval declarations} (only required where applicable) Any article reporting experiment/s carried out on (i)~live vertebrate (or higher invertebrates), (ii)~humans or (iii)~human samples must include an unambiguous statement within the methods section that meets the following requirements:
\begin{enumerate}[1.]
\item Approval: a statement which confirms that all experimental protocols were approved by a named institutional and/or licensing committee. Please identify the approving body in the methods section
\item Accordance: a statement explicitly saying that the methods were carried out in accordance with the relevant guidelines and regulations
\item Informed consent (for experiments involving humans or human tissue samples): include a statement confirming that informed consent was obtained from all participants and/or their legal guardian/s
\end{enumerate}
If your manuscript includes potentially identifying patient/participant information, or if it describes human transplantation research, or if it reports results of a clinical trial then additional information will be required. Please visit (\url{https://www.nature.com/nature-research/editorial-policies}) for Nature Portfolio journals, (\url{https://www.springer.com/gp/authors-editors/journal-author/journal-author-helpdesk/publishing-ethics/14214}) for Springer Nature journals, or (\url{https://www.biomedcentral.com/getpublished/editorial-policies\#ethics+and+consent}) for BMC.
\section{Discussion}\label{sec12}
Discussions should be brief and focused. In some disciplines use of Discussion or `Conclusion' is interchangeable. It is not mandatory to use both. Some journals prefer a section `Results and Discussion' followed by a section `Conclusion'. Please refer to Journal-level guidance for any specific requirements.
\section{Conclusion}\label{sec13}
Conclusions may be used to restate your hypothesis or research question, restate your major findings, explain the relevance and the added value of your work, highlight any limitations of your study, describe future directions for research and recommendations.
In some disciplines use of Discussion or 'Conclusion' is interchangeable. It is not mandatory to use both. Please refer to Journal-level guidance for any specific requirements.
\backmatter
\bmhead{Supplementary information}
If your article has accompanying supplementary file/s please state so here.
Authors reporting data from electrophoretic gels and blots should supply the full unprocessed scans for key as part of their Supplementary information. This may be requested by the editorial team/s if it is missing.
Please refer to Journal-level guidance for any specific requirements.
\bmhead{Acknowledgments}
Acknowledgments are not compulsory. Where included they should be brief. Grant or contribution numbers may be acknowledged.
Please refer to Journal-level guidance for any specific requirements.
\section*{Declarations}
Some journals require declarations to be submitted in a standardised format. Please check the Instructions for Authors of the journal to which you are submitting to see if you need to complete this section. If yes, your manuscript must contain the following sections under the heading `Declarations':
\begin{itemize}
\item Funding
\item Conflict of interest/Competing interests (check journal-specific guidelines for which heading to use)
\item Ethics approval
\item Consent to participate
\item Consent for publication
\item Availability of data and materials
\item Code availability
\item Authors' contributions
\end{itemize}
\noindent
If any of the sections are not relevant to your manuscript, please include the heading and write `Not applicable' for that section.
%%===================================================%%
%% For presentation purpose, we have included %%
%% \bigskip command. please ignore this. %%
%%===================================================%%
\bigskip
\begin{flushleft}%
Editorial Policies for:
\bigskip\noindent
Springer journals and proceedings: \url{https://www.springer.com/gp/editorial-policies}
\bigskip\noindent
Nature Portfolio journals: \url{https://www.nature.com/nature-research/editorial-policies}
\bigskip\noindent
\textit{Scientific Reports}: \url{https://www.nature.com/srep/journal-policies/editorial-policies}
\bigskip\noindent
BMC journals: \url{https://www.biomedcentral.com/getpublished/editorial-policies}
\end{flushleft}
\begin{appendices}
\section{Section title of first appendix}\label{secA1}
An appendix contains supplementary information that is not an essential part of the text itself but which may be helpful in providing a more comprehensive understanding of the research problem or it is information that is too cumbersome to be included in the body of the paper.
%%=============================================%%
%% For submissions to Nature Portfolio Journals %%
%% please use the heading ``Extended Data''. %%
%%=============================================%%
%%=============================================================%%
%% Sample for another appendix section %%
%%=============================================================%%
%% \section{Example of another appendix section}\label{secA2}%
%% Appendices may be used for helpful, supporting or essential material that would otherwise
%% clutter, break up or be distracting to the text. Appendices can consist of sections, figures,
%% tables and equations etc.
\end{appendices}
%%===========================================================================================%%
%% If you are submitting to one of the Nature Portfolio journals, using the eJP submission %%
%% system, please include the references within the manuscript file itself. You may do this %%
%% by copying the reference list from your .bbl file, paste it into the main manuscript .tex %%
%% file, and delete the associated \verb+\bibliography+ commands. %%
%%===========================================================================================%%
\bibliography{sn-bibliography}% common bib file
%% if required, the content of .bbl file can be included here once bbl is generated
%%\input sn-article.bbl
\end{document}

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%% Journal article
@article{bib1,
author = "Campbell, S. L. and Gear, C. W.",
title = "The index of general nonlinear {D}{A}{E}{S}",
journal = "Numer. {M}ath.",
volume = "72",
number = "2",
pages = "173--196",
year = "1995"
}
%% Journal article with DOI
@article{bib2,
author = "Slifka, M. K. and Whitton, J. L.",
title = "Clinical implications of dysregulated cytokine production",
journal = "J. {M}ol. {M}ed.",
volume = "78",
pages = "74--80",
year = "2000",
doi = "10.1007/s001090000086"
}
%% Journal article
@article{bib3,
author = "Hamburger, C.",
title = "Quasimonotonicity, regularity and duality for nonlinear systems of
partial differential equations",
journal = "Ann. Mat. Pura. Appl.",
volume = "169",
number = "2",
pages = "321--354",
year = "1995"
}
%% book, authored
@book{bib4,
author = "Geddes, K. O. and Czapor, S. R. and Labahn, G.",
title = "Algorithms for {C}omputer {A}lgebra",
address = "Boston",
publisher = "Kluwer",
year = "1992"
}
%% Item 8. Book, chapter
@incollection{bib5,
author = "Broy, M.",
title = "Software engineering---from auxiliary to key technologies",
editor = "Broy, M. and Denert, E.",
booktitle = "Software Pioneers",
pages = "10--13",
address = "New {Y}ork",
publisher = "Springer",
year = "1992"
}
%% Book, edited
@book{bib6,
editor = "Seymour, R. S.",
title = "Conductive {P}olymers",
address = "New {Y}ork",
publisher = "Plenum",
year = "1981"
}
%% Chapter in a book in a series with volume titles
@inproceedings{bib7,
author = "Smith, S. E.",
title = "Neuromuscular blocking drugs in man",
editor = "Zaimis, E.",
volume = "42",
booktitle = "Neuromuscular junction. {H}andbook of experimental pharmacology",
pages = "593--660",
address = "Heidelberg",
publisher = "Springer",
year = "1976"
}
%% Paper presented at a conference
@misc{bib8,
author = "Chung, S. T. and Morris, R. L.",
title = "Isolation and characterization of plasmid deoxyribonucleic acid from
Streptomyces fradiae",
year = "1978",
note = "Paper presented at the 3rd international symposium on the genetics
of industrial microorganisms, University of {W}isconsin, {M}adison,
4--9 June 1978"
}
%% Data citation example
@misc{bib9,
author = "Hao, Z. and AghaKouchak, A. and Nakhjiri, N. and Farahmand, A.",
title = "Global integrated drought monitoring and prediction system (GIDMaPS) data sets",
year = "2014",
note = "figshare \url{https://doi.org/10.6084/m9.figshare.853801}"
}
%% Preprint citation example
@misc{bib10,
author = "Babichev, S. A. and Ries, J. and Lvovsky, A. I.",
title = "Quantum scissors: teleportation of single-mode optical states by means
of a nonlocal single photon",
year = "2002",
note = "Preprint at \url{https://arxiv.org/abs/quant-ph/0208066v1}"
}
@article{bib11,
author = "Beneke, M. and Buchalla, G. and Dunietz, I.",
title = "Mixing induced {CP} asymmetries in inclusive {B} decays",
journal = "Phys. {L}ett.",
volume = "B393",
year = "1997",
pages = "132-142",
archivePrefix = "arXiv",
eprint = "0707.3168",
primaryClass = "gr-gc"
}
@softmisc{bib12,
author = "Stahl, B.",
title = "deep{SIP}: deep learning of {S}upernova {I}a {P}arameters",
version = "0.42",
keywords = "Software",
howpublished = "Astrophysics {S}ource {C}ode {L}ibrary",
year = "2020",
month = "Jun",
eid = "ascl:2006.023",
pages = "ascl:2006.023",
archivePrefix = "ascl",
eprint = "2006.023",
adsurl = "{https://ui.adsabs.harvard.edu/abs/2020ascl.soft06023S}",
adsnote = "Provided by the SAO/NASA Astrophysics Data System"
}
@article{bib13,
author = "Abbott, T. M. C. and others",
collaboration = "DES",
title = "{Dark Energy Survey Year 1 Results: Constraints on Extended Cosmological Models from Galaxy Clustering and Weak Lensing}",
eprint = "1810.02499",
archivePrefix = "arXiv",
primaryClass = "astro-ph.CO",
reportNumber = "FERMILAB-PUB-18-507-PPD",
doi = "10.1103/PhysRevD.99.123505",
journal = "Phys. Rev. D",
volume = "99",
number = "12",
pages = "123505",
year = "2019"
}
%%============================================================================%%
%% while using chicago reference style, both abbreviated and expanded form of %%
%% author name format is acceptable. Refer below example for expanded form %%
%%============================================================================%%
%% author = "{Cameron, Deborah}", - single author
%% author = "{Saito, Yukio} and {Hyuga, Hiroyuki}", - double author
%%======================================%%
%% Example for author names with suffix %%
%%======================================%%
%% author = "{Price, R. A. Jr} and {Curry, N. {III}} and McCann, K. E. and
%% Fielding, J. L. and {Abercrombie, E. Jr}",

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