Analysis and Control of the Linear Threshold Model of Cascades in Large-Scale Networks. A Local Mean-Field Approach.

The spread of new ideas, behaviors and technology may exhibit cascading effects in social, economic and technological networks. These phenomena generally depend on the topology of the network as well as the nature of the local agents' dynamics. In this thesis we consider the Linear Threshold Model, deployed on random graphs. The model describes a binary activation process in a network of agents. At every iteration, each agent compares the number of active neighbors with a personal activation threshold, which determines the subsequent active or inactive state of the agent. The threshold condition can also be interpreted as a graphical game with coordination structure. In representing processes of technology adoption, it is more suitable a Permanent Activation variant of the Linear Threshold Model where active agents can never deactivate. We proved a sufficient condition under which the two version of the model coincide. We analyzed the linear threshold model on a large random network, specifically the directed configuration model with heterogeneous agents. The tree-like local structure of the random networks allows to approximate the evolution of the expected fractional activation with a recursive equation. This equation, called Local Mean-Field dynamic, describes the evolution of the expected activation on an infinite tree with the same statistical properties of the original network. We proved a concentration theorem: for a generic instance of the network, the probability that the activation process and the Local Mean Field dynamic are close converges to one exponentially fast in the network size. If the activation thresholds are constant, the analysis reduces to the study of the fixed point of a scalar autonomous system and the corresponding trajectories. This analysis gives the asymptotic extension of the activation: we observed that in networks with sufficiently heterogeneous thresholds selective activation may occur. With constant thresholds the approach can be extended to study the Permanent Activation dynamic. Remarkably, the Local Mean Field dynamic equation and the concentration theorem continue to hold when the thresholds are dynamically adjusted, making the approach amendable to the design of control strategies. We formulated an optimal control problem and we considered a simplified version on a regular network. We compared the optimal solution with two sub-optimal strategies, developed with the aim to identify an heuristics for the problem's solution. Several aspects of the research discussed in the this Dissertation can be further investigated and generalized. To mention one, the comparison of the analysis presented here with other network topologies and possibly real network data.