Trackable Graphs

Abstract

Abstract. In this chapter we present the notion of trackable graph. We show how results presented in this monograph allow to efficiently recognize trackable graphs.

Imagine you are responsible for a network on which an agent is moving. The network can be modeled as a (directed) graph, and at each step the agent choses a node among all neighbors of its current node, and jumps to it. A number of recent contributions deal with the problem of “tracking” such an agent, that is, in some sense localize the agent on the network [24, 30, 87, 119, 120]. This network is endowed with sensors that give you information about the node in which the agent is. In practical applications however, the information is rarely sufficient to determine uniquely the current node of the agent: for instance, the network can be subject to noise, or two nodes that are too close can be activated together. Also, the sensors data can be transmitted in real time through a channel that only allows you to receive at each step a limited information about the sensors activations. Clearly, in general, the longer the experience lasts, the more trajectories will be possible. How to compute the set of all possible trajectories, given a sequence of observations?What are the possible growths of the number of trajectories when the observation length increases? How to determine the worst growth for a particular network? In Section 8.1 we formalize this problem and present the notion of trackable graphs recently introduced by Crespi et al. [30], and we give practical motivations for it. In Section 8.2 we answer to the above questions.We then conclude and raise some possible future work.