Threshold games and cooperation on multiplayer graphs

Objective: The study investigates the effect on cooperation in multiplayer games, when the population from which all individuals are drawn is structured - i.e. when a given individual is only competing with a small subset of the entire population. Method: To optimize the focus on multiplayer effects, a class of games were chosen for which the payoff depends nonlinearly on the number of cooperators - this ensures that the game cannot be represented as a sum of pair-wise interactions, and increases the likelihood of observing behaviour different from that seen in two-player games. The chosen class of games are named "threshold games", and are defined by a threshold, $M > 0$, which describes the minimal number of cooperators in a given match required for all the participants to receive a benefit. The model was studied primarily through numerical simulations of large populations of individuals, each with interaction neighbourhoods described by various classes of networks. Results: When comparing the level of cooperation in a structured population to the mean-field model, we find that most types of structure lead to a decrease in cooperation. This is both interesting and novel, simply due to the generality and breadth of relevance of the model - it is likely that any model with similar payoff structure exhibits related behaviour. More importantly, we find that the details of the behaviour depends to a large extent on the size of the immediate neighbourhoods of the individuals, as dictated by the network structure. In effect, the players behave as if they are part of a much smaller, fully mixed, population, which we suggest an expression for.Comment: in PLOS ONE, 4th Feb 201

On the Keyhole Hypothesis:High Mutual Information between Ear and Scalp EEG

We propose and test the keyhole hypothesis—that measurements from low dimensional EEG, such as ear-EEG reflect a broadly distributed set of neural processes. We formulate the keyhole hypothesis in information theoretical terms. The experimental investigation is based on legacy data consisting of 10 subjects exposed to a battery of stimuli, including alpha-attenuation, auditory onset, and mismatch-negativity responses and a new medium-long EEG experiment involving data acquisition during 13 h. Linear models were estimated to lower bound the scalp-to-ear capacity, i.e., predicting ear-EEG data from simultaneously recorded scalp EEG. A cross-validation procedure was employed to ensure unbiased estimates. We present several pieces of evidence in support of the keyhole hypothesis: There is a high mutual information between data acquired at scalp electrodes and through the ear-EEG “keyhole,” furthermore we show that the view—represented as a linear mapping—is stable across both time and mental states. Specifically, we find that ear-EEG data can be predicted reliably from scalp EEG. We also address the reverse view, and demonstrate that large portions of the scalp EEG can be predicted from ear-EEG, with the highest predictability achieved in the temporal regions and when using ear-EEG electrodes with a common reference electrode

Sleep EEG Derived From Behind-the-Ear Electrodes (cEEGrid) Compared to Standard Polysomnography: A Proof of Concept Study

Electroencephalography (EEG) recordings represent a vital component of the assessment of sleep physiology, but the methodology presently used is costly, intrusive to participants, and laborious in application. There is a recognized need to develop more easily applicable yet reliable EEG systems that allow unobtrusive long-term recording of sleep-wake EEG ideally away from the laboratory setting. cEEGrid is a recently developed flex-printed around-the-ear electrode array, which holds great potential for sleep-wake monitoring research. It is comfortable to wear, simple to apply, and minimally intrusive during sleep. Moreover, it can be combined with a smartphone-controlled miniaturized amplifier and is fully portable. Evaluation of cEEGrid as a motion-tolerant device is ongoing, but initial findings clearly indicate that it is very well suited for cognitive research. The present study aimed to explore the suitability of cEEGrid for sleep research, by testing whether cEEGrid data affords the signal quality and characteristics necessary for sleep stage scoring. In an accredited sleep laboratory, sleep data from cEEGrid and a standard PSG system were acquired simultaneously. Twenty participants were recorded for one extended nocturnal sleep opportunity. Fifteen data sets were scored manually. Sleep parameters relating to sleep maintenance and sleep architecture were then extracted and statistically assessed for signal quality and concordance. The findings suggest that the cEEGrid system is a viable and robust recording tool to capture sleep and wake EEG. Further research is needed to fully determine the suitability of cEEGrid for basic and applied research as well as sleep medicine

Measured <i>α</i>_<i>crit</i> for various realizations of the social networks.

<p>We see that in particular the <i>N</i>-based variation is strong. The coloured region coresponds to the (0, 15) × (0, 35)-region in (<i>f</i>₁, <i>f</i>₂)-space; see the appendix for an explanation of <i>f</i>₁, <i>f</i>₂.</p

Depictions of the different network types studied.

<p>In each case the connections of a single individual are highlighted. <i>L</i> = 20, <i>N</i> = 6. a) Fully mixed network with no stable connections. b) Random network. c) Ring-like network with no long-range connections. d) Ring-like network with long-range connections. For odd <i>N</i>, two long connections are made, to preserve symmetry.</p

The probability of having insufficient cooperators as a function of <i>N</i>, for <i>M</i> = 2, assuming that 〈<i>x</i>〉 is equal to the upper root of Eq (2), as predicted by the mean field theory.

<p>Note that the different extents of the lines (i.e. that <i>N</i> = 9 covers a smaller part of the <i>α</i>-axis than <i>N</i> = 3) stem from the limitations of the mean field model.</p

(Color online) Population average of <i>x</i> for various systems with threshold dynamics, for <i>M</i> = 2, <i>dt</i> = 0.01, <i>L</i> = 300.

<p>Note that in a-c, the value plotted is , such that a value of 1 means that on average, exactly <i>M</i> players are cooperating in each match. Example behaviour of mean field, full mixing and random networks, <i>N</i> = 7. a)Example behaviour of rings without long connections. The outlier is <i>N</i> = 3. b) Example behaviour of rings with few long connections. c) Comparisons of critical <i>α</i>-values for different models, as a function of <i>N</i>. “Fully Mixed” and “Random Topology” overlap for part of the range.</p

<i>α</i>_<i>crit</i>, both predicted from the mean field theory as well as measured from the model implemented on regular, random networks.

<p>We see that for both static <i>M</i> as well as relative, there is an <i>N</i>-dependence. However, in the static case, when <i>M</i> < < <i>N</i> the dependence is not predicted by the mean field theory.</p

Threshold Games and Cooperation on Multiplayer Graphs - Fig 7

<p>a) Comparison of finite size effects to clustering effects. b) Scatter-plot of <i>L</i>_<i>eff</i> vs. <i>L</i>* (defined in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0147207#pone.0147207.e008" target="_blank">Eq (6)</a>). We see an approximatively linear relation. In sum, we see a good evaluation of the finite-size explanation of the topology dependence in 〈<i>x</i>〉 vs. <i>α</i>.</p

An example of a “social” network, as described in [25], for <i>L</i> = 300, <i>f</i>₁ = <i>f</i>₂ = 0.

<p>See the appendix for further details on the generation and <i>f</i>₁, <i>f</i>₂. All connections between individuals are drawn. Proximity in space reflects connectedness. Plot created using Gephi [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0147207#pone.0147207.ref026" target="_blank">26</a>].</p