We analyze Japanese inter-firm network data showing scale-free properties as an example of a real complex network. The data contains information on money flow (annual transaction volume) between about 7000 pairs of firms. We focus on this money-flow data and investigate the correlation between various quantities such as sales or link numbers. We find that the flux from a buyer to a supplier is given by the product of the fractional powers of both sales with different exponents. This result indicates that the principle of detailed balance does not hold in the real transport of money; therefore, random walk type transport models such as PageRank are not suitable.

We analyzed nonlinear transport as defined for directed complex networks, where the flux from one node to a neighboring node is given preferentially according to the scalar quantities at the neighbor nodes. This is known as the generalized gravity interaction. In our research, we discovered a novel phase transition type. In the diffusion phase, the scalar quantity is scattered over the whole system, whereas in the localization phase, the flow tends to form localized confluence patterns owing to nonlinearity, resulting in the appearance of special nodes that irreversibly attract huge amounts of flow. We analytically considered the transition for selected network configurations, demonstrating that the transition point depends on the network topology. We also demonstrated that the diffusion phase of this transport model fits well with data from business firms, implying that the whole network structure can be used to model money flow in the real world.

The world economy consists of highly interconnected and interdependent commercial and financial networks. Here, we develop temporal and structural network tools to analyze the state of the economy and the financial markets. Our analysis indicates that a strong clustering can be a warning sign. Reduction in diversity, which was an essential aspect of the dynamics surrounding the financial markets crisis of 2008, is seen as a key emergent feature arising naturally from the evolutionary and adaptive dynamics inherent to the financial markets. Similarly, collusion amongst construction firms in a number of regions in Japan in the 2000s can be identified with the formation of clusters of anomalous highly connected companies. V C 2013 Wiley Periodicals, Inc. Complexity 19: 22-36, 2013 Key Words: economics; evolutionary dynamics; network theory; quantitative finance. ANALYSIS 1.Dynamics of Financial MarketsI n order to investigate the dynamics of financial markets we have developed a simple multi agent network model for a basic financial system, comprising of three fundamental types of agents: Banks, Investors and Borrowers (see section 2 for details). Our approach to modeling this system is inspired by the modeling of societies and ecosystems, in which a key role is played by the This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.virtual intra and interdependence of species [1][2][3][4]. This translates in our model into a focus on: (i) the dynamics of infection of business strategies within the banking sector and of culture dissemination within the investment and fund management community, and (ii) the topological aspects of the network of interactions. In order to focus more clearly on the influence of the collective action of agents, and their interaction amongst FIGURE 1Crisis Mapping from Evolutionary Dynamics: Plot A shows the frequency of crisis and the relative number of times that each type of crisis scenario occurs. The front floor (dark blue) indicates the distributions when evolutionary dynamics are present, with realizations resulting in 1 or 2 crises being by far the most common. The back floor (green) showing the results without dynamics is entirely distributed into the first block (no crisis), indicating that evolutionary dynamics are an essential feature in order to see crises occur. Plot B illustrates the time line of crises as predicted by the model including the evolutionary dynamics for 1 crisis (light purple) and 2 crises (light yellow) simulations. Time is shown vertically, increasing downwards, while the horizontal axis denotes different realizations of the model. A crisis is defined when >2% of the Bank agents fail or require financial assistance over a year, which corresponds to the historical average registered in the first and second US banking crisis over the simulation period. 4 The fir...

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We introduce a method to extract main-stream structures for a given complex network flow by trimming less effective links. As the resulting main streams generally have an almost loopless treelike structure, we can define the stream basin size for each node, which characterizes the importance of the node with regard to the flow. As a real-world example, we apply this method to an interfirm trading network, both for the money flow and its conjugate-the material or service flow-confirming that both basin size distributions follow a similar power law that differs significantly from the basin size distributions of rivers in nature. We theoretically analyze the process of trimming and derive a consistent statistical formulation between the original link number and the basin size.