Nodes Having a Major Influence to Break Cooperation Define a Novel Centrality Measure: Game Centrality

Cooperation played a significant role in the self-organization and evolution of living organisms. Both network topology and the initial position of cooperators heavily affect the cooperation of social dilemma games. We developed a novel simulation program package, called ‘NetworGame’, which is able to simulate any type of social dilemma games on any model, or real world networks with any assignment of initial cooperation or defection strategies to network nodes. The ability of initially defecting single nodes to break overall cooperation was called as ‘game centrality’. The efficiency of this measure was verified on well-known social networks, and was extended to ‘protein games’, i.e. the simulation of cooperation between proteins, or their amino acids. Hubs and in particular, party hubs of yeast protein-protein interaction networks had a large influence to convert the cooperation of other nodes to defection. Simulations on methionyl-tRNA synthetase protein structure network indicated an increased influence of nodes belonging to intra-protein signaling pathways on breaking cooperation. The efficiency of single, initially defecting nodes to convert the cooperation of other nodes to defection in social dilemma games may be an important measure to predict the importance of nodes in the integration and regulation of complex systems. Game centrality may help to design more efficient interventions to cellular networks (in forms of drugs), to ecosystems and social networks.

. The three worker groups of a former forest product manufacturing factory containing younger, English-speaking (yellow); older, English-speaking (blue); or younger, Spanish-speaking workers (green) were marked. Sam and Wendle (top right) were the union leaders, who failed to break the strike, while Bob and Norm (center, marked with diamonds) were the pair of workers, who successfully broke the strike. Tables   Table S1. List of consensus party hubs   Consensus party hub  ORFs a  YHR077C  YAR002W  YHR089C  YAR003W  YHR166C  YBL004W YHR200W

Supplementary
The open reading frame names of 63 consensus yeast party hubs were determined and listed as in [2] comparing the party hubs of the high fidelity yeast protein-protein interaction network [3] with those published in other 5 publications [4][5][6][7][8], and listing only those as 'consensus party hubs', which were never classified as a date hub.
The open reading frame names of 145 consensus yeast date hubs were determined and listed as in [2] comparing the date hubs of the high fidelity yeast protein-protein interaction network [3] with those published in other 5 publications [4][5][6][7][8], and listing only those as 'consensus date hubs', which were never classified as a party hub.

Description of the NetworGame algorithm
The 2.0 version of the NetworGame program is an updated version of the NetworGame 1.0 version published in a preliminary conference report [9]. NetworGame 2.0 is available in our web-site (www.linkgroup.hu/NetworGame.php). The 2.0 version utilizes our experiences gained with the 1.0 version. The NetworGame 2.0 program package is a cross-platform, generic tool to simulate repeated spatial games. This simulation program includes i.) options for pay-off matrices of any symmetric normal form games (with 2 strategies); ii.) several well-known, replicator-type strategy update rules, as well as the option for additional, user-defined strategy update rules in a 'plugin'-type format; iii.) synchronous, and semi-synchronous updating [10]; iv.) and the option for the inclusion of any real world networks in a Pajek format [11].
Here we provide the pseudocode for the algorithm, which describes the flow of the program and the effects of the configuration parameters. A User Guide of version 2.0 can be downloaded from here: www.linkgroup.hu/NetworGame.php.  Simulate m up to steps L -S i is the current strategy of node i -P i is the current payoff of node i -useWeights, x0 and x1 are weight parameters controlling the effect of edge weights -Neighbors(i) is the set of neighbors for node i -Payoff[i,j] is the payoff matrix value when strategy j plays against strategy j -payoffSchema is a configuration parameter for n = 1,2,...,L do simulating current round and calculating payoffs for i in Nodes do P i = 0 counter = 0 -the probability of a game is dependent on the weight parameters and edge weight W i,j for j in Neighbors(i) do if (not useWeights or random(0,1) <= (W i,j -x0)/(x1-x0)) then P i = P i + Payoff[S i ,S j ] counter = counter + 1 end if (payoffSchema = degree or payoffSchema = averaging) then P i = P i / counter (if counter > 0) end updating strategies (strategyUpdateRule can be implemented as a plugin, it may have memory, or might be one of the built-in rules: best takes over or proportional update for k in Nodes do S k = strategyUpdateRule(...) end end