Cascading dynamics on complex networks

Frequently, complex systems in nature as well as in human society suffer massive catastrophes triggered from only a small fraction of their constituents. Unexpected epidemic spread of diseases, the power outage, and the congestion of packet transport in the Internet are prototypical examples of such avalanche phenomena. Such a cascading dynamics is not always harmful to us. The information cascades making popular hits of books, movies, and albums are good to writers, actors, and singers, respectively. Thus, it is interesting to understand and predict how those cascades propagate in complex systems. Recently, the network approach, by which a system is viewed as a network consisting of nodes representing its constituents and links interactions between them, simplifies complicated details of complex systems. In my talk, I present how cascading dynamics in complex systems can be modeled on complex networks. In many cases, cascading dynamics spreads along the path that is a tree. When the network is scale-free in the degree distribution, the tree can be a critical branching tree. In this case, the avalanche dynamics can be understood the multiplicative branching process. Using this method, we can set up the self-consistent equation for the avalanche size distribution, which behaves in a power law fashion. This method can be modified to study the fad propagation problem. it is important to choose a triggering node. Such theoretical results are confirmed by numerical simulations. I will introduce a recent model to prevent such a cascading dynamics on the complex network. For the problem of packet congestions, we study the transition between the free-flow to congested state as the number of packets increases. At the transition point, there exists a synchronized phase, in which the traffic flow changes in jammed or relaxed state, alternatively. In this case, the congested nodes (routers) can spread out over the entire systems, and recover to the normal state, alternatively. The congested area can fluctuate over time. As a result, the power spectrum of the traffic amount on the network exhibits 1/f-type behavior.