This chapter opens many avenues for potential future research, including the following.
• DIVER targets optimization of the scalar asymptotic consensus value, being the sum of the user opinions weighted by the corresponding users’ eigenvector centralities. It may be fruitful to generalize DIVER to the optimization of the whole asymptotic opinion distribution, rather than its scalar summary.
• For the purposes of efficiently computing potential impact of candidate edge addition, we have studied the question of how to efficiently estimate mean first passage times to/from top-centrality states in a Markov chain. While our answer to the latter question was based on an empirical study on scale-free networks, providing formal convergence bounds for MFPT estimation would benefit many scientific areas dealing with Markov processes.
• Finally, while DIVER was proposed as a method to defend a social network against social control, it clearly can be used as an influence tool. Thus, it is important to study the ways to identify whether an edge recommendation process in a social network targets strategic change of the opinion distribution, as well as identify the goal of that process.
Chapter 5 Conclusion
In this thesis, we outline the scope of the emerging cross-discipline of dynamic processes in networks, and make contributions to the development of that field along the three dimensions of analysis, modeling, and control. In terms of analysis, relying upon combinatorial network algorithms, we devise a general method to analyze an observed process of polar opinion for-mation in social networks, with applications to anomaly detection in and extrapolation of a series of the process’ states. With respect to modeling, we design a class of non-linear models capturing complex polar opinion formation behavior in social networks as well as connected to such physical processes as non-linear heat diffusion, and provide a general approach towards theoretical analysis of such models’ behavior relying on the dynamical systems theory and non-smooth analysis. Finally, we address the problem of fighting social control via strategically augmenting social networks, building upon the theory of Markov chains, with our theoretical and algorithmic results’ generally contributing to the area of the networked control of Markov processes.
This thesis opens up a number of avenues for future research, particularly, in dealing with processes in socio-economic networks.
5.1 Open Problems in Network Process Analysis
Below, we provide two general problems dealing with the analysis of observed network processes.
Predicting Future Evolution of Network Process: One open problem within the scope of the general network process analysis framework is that of the efficient network process state series extrapolation. Consider a situation when, having observed a network process’ dynamics, we need to predict how it will evolve in the near future. For example, having observed how the political opinions of a social network’s users have evolved in the past, we want to predict how these opinions will change at the time of the upcoming elections. In other words, we want to predict the future state of a network process based on a series of its past states. While the method developed in Chapter 2 of this thesis can estimate the likelihood of a specific change in a network process’ state, here we are faced with the need to search for the most plausible future state of the process. One approach is to extrapolate the series of distances between the adjacent past states of the process to estimate the expected distance to the process’ future state, and, then, among all future state candidates take as the prediction the state the distance to which is closest to the made distance estimate. In Chapter 2, we demonstrated that this distance-based search works for predicting partially observed future states of a network process, but its fundamental bottleneck is the need to enumerate a possibly exponential number of future state candidates. To solve the problem of the efficient exploration of the state space of a network process, we need to utilize the domain knowledge about the structure of this state space, and exploit semi-metricity of the underlying distance measure, aiming at drastically reducing the size of the fraction of the state space that we actually need to traverse.
Combinatorial Network Process Identification: Above, we have advocated the model-based network process analysis approach, having assumed that we have a model for an
ob-served network process. In many cases, we may indeed have a reliable model for the process, as in the case with the network version of the heat equation for the process of heat diffusion in a material network, or Kuramoto model for the process of neuronal synchronization between pairs of neurons in a brain network. However, we may have a process whose nature and, hence, model are unclear, or a process for which there are many alternative models and it is not ob-vious which model fits the process best. Having observed the evolution of a network process, the key question here is what is the (best) model for a given process. While there are methods for identification of non-linear dynamical systems, there are no such methods for combinato-rial processes, such as those characterized by combinatocombinato-rial models of information diffusion through a social network. An extension of a network process identification problem is a prob-lem of disentangling a mixture of processes occurring in the same network simultaneously.
Clearly, there is a need for the development of new theories and algorithms that would enable combinatorial network process identification.