Global value chains (GVCs) are formed through value-added trade, and some regions promote economic integration by concluding regional trade agreements to promote these chains. However, there is no way to quantitatively assess the scope and extent of economic integration involving various sectors in multiple countries. In this study, we used the World Input--Output Database to create a cross-border sector-wise trade in value-added network (international value-added network (IVAN)) covering the period of 2000--2014 and evaluated them using network science methods. By applying Infomap to the IVAN, we confirmed for the first time the existence of two regional communities: Europe and the Pacific Rim. Helmholtz--Hodge decomposition was used to decompose the value flows within the region into potential and circular flows, and the annual evolution of the potential and circular relationships between countries and sectors was clarified. The circular flow component of the decomposition was used to define an economic integration index, and findings confirmed that the degree of economic integration in Europe declined sharply after the economic crisis in 2009 to a level lower than that in the Pacific Rim. The European economic integration index recovered in 2011 but again fell below that of the Pacific Rim in 2013. Moreover, sectoral analysis showed that the economic integration index captured the effect of Russian mineral resources, free movement of labor in Europe, and international division of labor in the Pacific Rim, especially in GVCs for the manufacture of motor vehicles and high-tech products.