Dynamic Stabilization and Control of Material Flows in Networks and its Relationship to Phase Synchronization Reik Donner, Aude Hofleitner, Johannes H¨ ofener, Stefan L¨ ammer, and Dirk Helbing Abstract— We study the self-organization and optimization of conflicting material flows on complex networks as it may take place in the case of vehicular traffic or the supply of goods in a production network. A decentralized control is used to approach a demand-driven switching between ”on” and ”off” states of the flow in a particular direction at intersections or merges represented by nodes in the corresponding networks. Whereas intrinsic oscillatory instabilities of material flows in networks usually have negative effects on the performance of the overall system, these self-organized oscillations allow to optimally use the available transportation capacity of the network. Under rather general conditions, our control approach leads to phase synchronization of the switching dynamics at the respective nodes, which is studied using a new framework for measuring the strength and homogeneity of frequency locking in networks of oscillatory components. I. I NTRODUCTION Network structures can be found in numerous complex systems in nature and society. In many of these networks, the flows of material or information between the respective nodes represent the essential dynamics of the system [1]. This applies in particular to traffic, production, telecom- munication, supply, or biological networks [2][3]. However, such flows are typically characterized by a large number of interacting transportation processes. Therefore, the aim of an efficient organization and control of the network dynamics is to minimize the total amount of time required for all these processes. Typically, such an optimization is difficult and demanding, as the topology of the underlying networks is composed of a potentially large number of merges and intersections at which there are conflicts between the flows on different routes [4][5]. To avoid physical collisions, these flows have to be controlled by devices like traffic lights, which lead to an oscillatory switching between “service” and “no-service” states for different links at the respective node. The operation strategy of these devices is decisive for maximizing the system performance. One traditional strategy for the optimization of material flows is a central control of the flows in the whole network. However, in large networks, there is a large amount of (particularly delayed) information which makes a central organization of all flows a complicated problem. In contrast to this, decentralized control strategies [5][6] are much more The authors of this paper gratefully acknowledge financial support of the German Research Foundation (project no. He 2789/5-1,8-1). The presented research has been carried out at the Institute of Traffic and Economy, Dresden University of Technology, Andreas-Schubert-Str. 23, 01062 Dresden, Germany. D. Helbing works as Professor of Sociology, in particular of Modeling and Simulation at the Swiss Federal Institute of Technology, Universit¨ atstr. 41, 8092 Z¨ urich, Switzerland. A. Hofleitner is a master’s student at ´ Ecole Polytechnique, 91128 Palaiseau Cedex, France. flexible to react to local changes of the current state of the system and may therefore lead to an even better performance than centralized ones. Unlike the controlled oscillations which are ”triggered” by the action of traffic lights and similar devices, material flows in many real-world networks show already an intrinsic oscillatory behavior. There are different possible reasons for this kind of dynamical behavior, including a varying external demand or supply of material (forced oscillations) as well as symmetry breaking bifurcations or instabilities due to the presence of feedback loops [7] (self-sustained oscillations). In the latter case, the instabilities of the stationary system may have severe consequences for the overall performance of the entire network. The resulting oscillations may be further amplified by phenomena like the Bullwhip effect in production systems, which may finally lead to a highly irregular system behavior [7]. In this paper, we demonstrate how a suitable decentral- ized control can be used to reach controlled oscillations of material flows at the intersections within a transportation network. For this purpose, we study if the self-organized oscillations yield an optimal control of the material flows, and try to understand the importance of phase synchro- nization (a) between neighboring nodes and (b) within the entire network for an optimization of the overall capac- ity of the system. In Sec. 2, we present a basic model for a self-organized control of conflicting material flows. Possible approaches to phase synchronization analysis in complex networks are summarized in Sec. 3. In Sec. 4, we present some examples for the occurrence of frequency locking of the material flow dynamics in the presence of periodic external forcings. Subsequently, the occurrence of self-sustained phase synchronization phenomena in networks in the presence of a decentralized control is studied in Sec. 5. Finally, we discuss the general relationship between the self- organized optimization of material flows in networks and the occurrence of phase synchronization. II. SELF-ORGANIZED NETWORK FLOWS To describe the behavior of material flows at intersections, we assume that the effects of acceleration or deceleration near intersections are small enough to be negligible (corre- sponding to an adiabatic speed adjustment). Moreover, all links in the considered networks have an infinite (or at least sufficiently high) storage capacity, i.e., there is no possibility of complete congestion in our simplified model. In this contribution, we will use the following notations: N i is the amount of queued material on the incoming link