Walking around in a changing world: Understanding random walks over dynamic graphs

The growing interest in understanding how things connect and the importance of connectedness on various processes has lead to a large and multidisciplinary body of work over the past decade. A fundamental aspect is its dynamic nature as all networks constructed by nature or man suffer structural changes over some time-scale. For example, consider online social networks, the Web, a mobile wireless network or the neural network of an individual. Random walks are a fundamental building block for understanding networks and have been applied to problems related to clustering, ranking, searching and routing. Its relatively well-understood behavior on a static network and algorithmic simplicity support its prominent role as a building block for different mechanisms. However, very little is known about their behavior on dynamic networks. In this talk we present a general modeling framework for dynamic networks and continuous time random walks. We then analyze the long-term behavior (steady state) of the walker on dynamic networks and show it to be non-trivial, in striking contrast to the static case. However, we characterize its steady state behavior for three general special cases: (i) walker rate is much faster or slower than network dynamics (time-scale separability); (ii) walker is proportional to the degree of the node it resides on (coupled dynamics); (iii) degrees of nodes in the same connected component are identical (structural restriction). Finally, we apply our framework to mobile wireless networks and show interesting properties in this scenario that could be explored in the design of algorithms. The theoretical findings presented in this talk were obtained in collaboration with other researchers and will be published this year in ACM SIGMETRICS 2012.