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Highlights • Price dynamics of carbon pilots are studied as a whole by visibility graph networks. • The networks are similar in topology properties but differ in degree of measures. • Jaccard similarity coefficient shows a low similar in price dynamics of pilots. • A price-dependence mode for the linkage of pilots is proposed. • Visibility graph method is more reliable for measuring weak coupling of time series. Abstract Similarity and heterogeneity analysis of price dynamics across the region provide experiences for China's national carbon emission trading scheme. This study examines the dynamic behavior of carbon prices from the visibility graph network point of view, notably introducing the Jaccard coefficient to measure the similarity. By mapping daily carbon trading prices onto visibility graph networks, we analyze the characteristics of the carbon prices by topological measures of the networks. For the next step, we evaluate the similarity between pilots by the Jaccard similarity coefficient of the constructed visibility graph networks. Consequently, we cluster the pilots based on topology measures of the networks. Results show that the seven carbon markets in China have different patterns of similarity in visibility network properties: They are all small-world networks and are scale-free except for Chongqing pilot. Only Hubei and Shenzhen pilots are assortative, indicating a weak degree of market efficiency. At the same time, the two pilots have the highest value of Jaccard similarity coefficients and Shenzhen pilot presents the highest similarity to the supposed integrated market. According to the structure measures of the networks, the pilots are clustered into four groups which can be seen as sub-markets. Findings in this study indicate that inequalities across those groups should be sufficiently considered in the future national carbon market with particular emphasis on the Hubei pilot. Some sub-markets are suggested to be built according to the similarity in price dynamics. [ABSTRACT FROM AUTHOR]

• Study Fractal Market Hypothesis by Multifractal Detrended fluctutation analysis. • Carbon price returns are multifractal in both the short and long terms. • The Hurst exponent indicates the inefficiency of all pilot markets. • Quantify inefficiency and rank the pilot markets in different time periods. • Market activity has a greater influence on inefficiency in the long term. An efficient carbon market is important for mitigating carbon emissions and many other energy issues. In this study we investigate the multifractality and efficiency of carbon market using Multifractal Detrended Fluctuation Analysis, and the data are based on the carbon price returns of seven pilots in the carbon market of China. First, we find the price return series is multifractal for both the short term and the long term. Asymmetric distributions of carbon prices and the general Hurst exponents in the return series are detected, which suggest a lack of efficiency in the pilot markets under study. Then, we measure the efficiencies of these carbon market pilots using an indicator that we develop based on the generalized Hurst exponents, and the results suggest ubiquitous inefficiencies among them. But it is also found that the efficiencies of most of the pilots have been improving over time. The ranking of the pilot markets' efficiencies depends on the time window under study. For time scales shorter than a month, Chongqing pilot generally ranks the highest and Hubei pilot the lowest. Beijing pilot performs the best while Tianjin the worst when the time scale is longer. Lastly, regression results show that the carbon market pilot's inefficiency value is positively correlated with market activity in the short run while negatively in the long run. Based on the empirical results, effect of the inefficiency on low-carbon development and the policy implications are discussed. [ABSTRACT FROM AUTHOR]