Analytical comparison of cross impact steady state, DEMATEL, and page rank for analyzing complex systems

Active and passive sums of direct, powered, total, and stable state impact analyses are the different criteria for studying the influence of variables in complex social, economic, and technological systems. The three methods that are applied for studying complex systems are DEMATEL (total impacts analysis), cross impact analysis (stable state), and Page rank (stable state of left normalized stochastic matrix). These approaches are applied to select influential and dependent variables of complex systems. These methods notably differ in rank order of row- and column-sums of variables as the models are based on different assumptions. However, mathematical relationships can be drawn, and several mathematical properties can be investigated and analyzed between different methods to better understand models, their interpretations, applications, and numerical accuracy. In this paper, we analytically compare the three methods, develop analytical formulas and approximations of the steady states and their differences in active and passive rank orders, and show how sparsity and the problem dimension influence the differences in rank orders and quality of approximation by numerical simulation. The research contributes by exploring the analytical relationships and approximations between normalized and unnormalized cross-impact matrices and their steady states and comparing the outcomes of different models. Its findings can help understand the application of different models in studying complex systems in business and marketing research to identify influential and dependent variables. The results can support managerial and strategic planning decision-making for selecting an appropriate method, combining methods for robust decision-making, applying them in the proper context, and accurately evaluating and analyzing the results.