Natural algorithms in a networked world: technical perspective

hoW do Birds flock and fish school? How do individuals in a social network reach agreement, even though they are often only influenced by other like-minded individuals? How do fire-flies flash synchronously? How can one engineer a swarm of robots to behave like bird flocks? These are some of the questions that have occupied the minds of many researchers across a diverse set of disciplines over the past two decades, from computer graphics to statistical physics to evolutionary biology, and from control theory to robotics, mathematics , and computer science. Most of the early research, which happened in computer graphics and statistical physics , was on modeling and simulation of collective behavior. Over the past decade the focus has shifted to rigorous theory, leading to what one might call a theory of collective phenomena in mul-tiagent systems. This theory blends dy-namical systems, graph theory, Markov chains, and algorithms. The collective phenomena are often modeled as many-degrees-of-freedom (discrete-time or continuous time) dynamical systems with an additional twist that the interconnection structure between individual dynami-cal systems change, since the motion of each node in a flock (or opinion of an individual) is affected primarily by those in the node's local neighborhood. The twist here is that the local neighborhood is not fixed: neighbors are defined based on the actual state of the system: For example, in case of opinion dynamics, sociological models indicate that individuals are often influenced by those whose opinions are close to them. In other words, as opinions evolve, neighborhood structures change as the function of the evolving opinion, resulting in a catch 22. Similarly, as birds aggregate their directions with their neighbors, the set of their neighbors change. When the neighborhood structures stay intact throughout the evolution of the dynamics, the behavior of the model is easy to analyze. Using Per-ron-Frobenius theory, one can easily show that connectivity of the network induced by the local neighborhood relations is in fact enough to result in global agreement (in opinions, direction of motion in a flock, or frequency of oscillations in coupled os-cillators). One can extend the analysis to the case of time varying or switching networks. If the network change is modeled exogenously, in order to get agreement all that is needed is to make sure the underlying graphs that encode the neighborhood structure stay connected over time, that is, over time the union of the edges of the …