Estimation of Laplacian spectra of direct and strong product graphs

Calculating a product of multiple graphs has been studied in mathematics, engineering, computer science, and more recently in network science, particularly in the context of multilayer networks. One of the important questions to be addressed in this area is how to characterize spectral properties of a product graph using those of its factor graphs. While several such characterizations have already been obtained analytically (mostly for adjacency spectra), characterization of Laplacian spectra of direct product and strong product graphs has remained an open problem. Here we develop practical methods to estimate Laplacian spectra of direct and strong product graphs from spectral properties of their factor graphs using a few heuristic assumptions. Numerical experiments showed that the proposed methods produced reasonable estimation with percentage errors confined within a +/-10% range for most eigenvalues.Comment: 14 pages, 7 figures; to be published in Discrete Applied Mathematic

Evolution of Fairness in the Not Quite Ultimatum Game

The Ultimatum Game (UG) is an economic game where two players (proposer and responder) decide how to split a certain amount of money. While traditional economic theories based on rational decision making predict that the proposer should make a minimal offer and the responder should accept it, human subjects tend to behave more fairly in UG. Previous studies suggested that extra information such as reputation, empathy, or spatial structure is needed for fairness to evolve in UG. Here we show that fairness can evolve without additional information if players make decisions probabilistically and may continue interactions when the offer is rejected, which we call the Not Quite Ultimatum Game (NQUG). Evolutionary simulations of NQUG showed that the probabilistic decision making contributes to the increase of proposers' offer amounts to avoid rejection, while the repetition of the game works to responders' advantage because they can wait until a good offer comes. These simple extensions greatly promote evolution of fairness in both proposers' offers and responders' acceptance thresholds.Comment: 14 pages, 3 figure

Predicting stock market movements using network science: An information theoretic approach

A stock market is considered as one of the highly complex systems, which consists of many components whose prices move up and down without having a clear pattern. The complex nature of a stock market challenges us on making a reliable prediction of its future movements. In this paper, we aim at building a new method to forecast the future movements of Standard & Poor's 500 Index (S&P 500) by constructing time-series complex networks of S&P 500 underlying companies by connecting them with links whose weights are given by the mutual information of 60-minute price movements of the pairs of the companies with the consecutive 5,340 minutes price records. We showed that the changes in the strength distributions of the networks provide an important information on the network's future movements. We built several metrics using the strength distributions and network measurements such as centrality, and we combined the best two predictors by performing a linear combination. We found that the combined predictor and the changes in S&P 500 show a quadratic relationship, and it allows us to predict the amplitude of the one step future change in S&P 500. The result showed significant fluctuations in S&P 500 Index when the combined predictor was high. In terms of making the actual index predictions, we built ARIMA models. We found that adding the network measurements into the ARIMA models improves the model accuracy. These findings are useful for financial market policy makers as an indicator based on which they can interfere with the markets before the markets make a drastic change, and for quantitative investors to improve their forecasting models.Comment: 13 pages, 7 figures, 3 table