We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

https://gisc.uc3m.es/~cuesta/PDFs/PLR06208.pdf

Evolutionary game theory: temporal and spatial effects beyond replicator dynamics

Evolutionary game dynamics is one of the most fruitful frameworks for studying evolution in different disciplines, from Biology to Economics. Within this context, the approach of choice for many researchers is the so-called replicator equation, that describes mathematically the idea that those individuals performing better have more offspring and thus their frequency in the population grows. While very many interesting results have been obtained with this equation in the three decades elapsed since it was first proposed, it is important to realize the limits of its applicability. One particularly relevant issue in this respect is that of non-mean-field effects, that may arise from temporal fluctuations or from spatial correlations, both neglected in the replicator equation. This review discusses these temporal and spatial effects focusing on the non-trivial modifications they induce when compared to the outcome of replicator dynamics. Alongside this question, the hypothesis of linearity and its relation to the choice of the rule for strategy update is also analyzed. The discussion is presented in terms of the emergence of cooperation, as one of the current key problems in Biology and in other disciplines.

/pdf/evolutionary-game-theory-temporal-and-spatial-effects-beyond-4hbhqjijrq.pdf

Evolutionary game theory: Temporal and spatial effects beyond replicator dynamics

A survey on the fusion process in opinion dynamics

We develop a behavioral model for liquidity and volatility based on empirical regularities in trading order flow in the London Stock Exchange. This can be viewed as a very simple agent based model in which all components of the model are validated against real data. Our empirical studies of order flow uncover several interesting regularities in the way trading orders are placed and cancelled. The resulting simple model of order flow is used to simulate price formation under a continuous double auction, and the statistical properties of the resulting simulated sequence of prices are compared to those of real data. The model is constructed using one stock (AZN) and tested on 24 other stocks. For low volatility, small tick size stocks (called Group I) the predictions are very good, but for stocks outside Group I they are not good. For Group I, the model predicts the correct magnitude and functional form of the distribution of the volatility and the bid-ask spread, without adjusting any parameters based on prices. This suggests that at least for Group I stocks, the volatility and heavy tails of prices are related to market microstructure effects, and supports the hypothesis that, at least on short time scales, the large fluctuations of absolute returns are well described by a power law with an exponent that varies from stock to stock.

An empirical behavioral model of liquidity and volatility

Based on the daily data of the American and Chinese stock markets, the dynamic behavior of a financial network with static and dynamic thresholds is investigated. Compared with the static threshold, the dynamic threshold suppresses the large fluctuation induced by the cross correlation of individual stock prices and leads to a stable topological structure in the dynamic evolution. Long-range time correlations are revealed for the average clustering coefficient, average degree and cross correlation of degrees. The dynamic network shows a two-peak behavior in the degree distribution.

https://iopscience.iop.org/article/10.1088/1367-2630/12/4/043057/pdf

Financial networks with static and dynamic thresholds

We investigate the probability distribution of the volatility return intervals τ for the Chinese stock market. We rescale both the probability distribution P q ( τ ) and the volatility return intervals τ as P q ( τ ) = 1 / τ ¯ f ( τ / τ ¯ ) to obtain a uniform scaling curve for different threshold value q . The scaling curve can be well fitted by the stretched exponential function f ( x ) ∼ e − α x γ , which suggests memory exists in τ . To demonstrate the memory effect, we investigate the conditional probability distribution P q ( τ | τ 0 ) , the mean conditional interval 〈 τ | τ 0 〉 and the cumulative probability distribution of the cluster size of τ . The results show clear clustering effect. We further investigate the persistence probability distribution P ± ( t ) and find that P − ( t ) decays by a power law with the exponent far different from the value 0.5 for the random walk, which further confirms long memory exists in τ . The scaling and long memory effect of τ for the Chinese stock market are similar to those obtained from the United States and the Japanese financial markets.

Scaling and memory effect in volatility return interval of the Chinese stock market

Cooperation in the snowdrift game on directed small-world networks under self-questioning and noisy conditions

Redundant correlation effect on personalized recommendation

In a static environment, optional participation and a local agglomeration of cooperators are found to be beneficial for the occurrence and maintenance of cooperation. In the optional public goods game, the rock–scissors–paper cycles of different strategies yield oscillatory cooperation but not stable cooperation. In this paper, by incorporating population density and individual mobility into the spatial optional public goods game, we study the coevolutionary dynamics of strategy updating and benefit-seeking migration. With low population density and slow movement, an optimal level of cooperation is easy to be reached. An increase in population density and speed-up of free-floating of competitive agents will suppress cooperation. A log–log relation between the levels of cooperation and the free-floating probability is found. Theoretical analysis indicates that the decrease of cooperator frequency in the present model should result from the increased interactions between different agents, which may originate from the increased cluster size or the speed-up of random-movement.

Tian Qiu

Papers

Financial networks with static and dynamic thresholds

Scaling and memory effect in volatility return interval of the Chinese stock market

Cooperation in the snowdrift game on directed small-world networks under self-questioning and noisy conditions

Redundant correlation effect on personalized recommendation

Coupled dynamics of mobility and pattern formation in optional public goods games