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We present a continuous threshold model (CTM) of cascade dynamics for a network of agents with realvalued activity levels that change continuously in time. The model generalizes the linear threshold model (LTM) from the literature, where an agent becomes active (adopts an innovation) if the fraction of its neighbors that are active is above a threshold. With the CTM we study the influence on cascades of heterogeneity in thresholds for a network comprised of a chain of three clusters of agents, each distinguished by a different threshold. The system is most sensitive to change as the dynamics pass through a bifurcation point: if the bifurcation is supercritical the response will be contained, while if the bifurcation is subcritical the response will be a cascade. We show that there is a subcritical bifurcation, thus a cascade, in response to an innovation if there is a large enough disparity between the thresholds of sufficiently large clusters on either end of the chain; otherwise the response will be contained.

Emergency vehicles (EMVs) play a crucial role in responding to time-critical events such as medical emergencies and fire outbreaks in an urban area. The less time EMVs spend traveling through the traffic, the more likely it would help save people's lives and reduce property loss. To reduce the travel time of EMVs, prior work has used route optimization based on historical traffic-flow data and traffic signal pre-emption based on the optimal route. However, traffic signal pre-emption dynamically changes the traffic flow which, in turn, modifies the optimal route of an EMV. In addition, traffic signal pre-emption practices usually lead to significant disturbances in traffic flow and subsequently increase the travel time for non-EMVs. In this paper, we propose EMVLight, a decentralized reinforcement learning (RL) framework for simultaneous dynamic routing and traffic signal control. EMVLight extends Dijkstra's algorithm to efficiently update the optimal route for the EMVs in real-time as it travels through the traffic network. The decentralized RL agents learn network-level cooperative traffic signal phase strategies that not only reduce EMV travel time but also reduce the average travel time of non-EMVs in the network. This benefit has been demonstrated through comprehensive experiments with synthetic and real-world maps. These experiments show that EMVLight outperforms benchmark transportation engineering techniques and existing RL-based signal control methods.

The linear threshold model (LTM) has been used to study spread on single-layer networks defined by one inter-agent sensing modality and agents homogeneous in protocol. We define and analyze the heterogeneous multiplex LTM to study spread on multi-layer networks with each layer representing a different sensing modality and agents heterogeneous in protocol. Protocols are designed to distinguish signals from different layers: an agent becomes active if a sufficient number of its neighbors in each of any a of the m layers is active. We focus on Protocol OR, when a = 1, and Protocol AND, when a = m, which model agents that are most and least readily activated, respectively. We develop theory and algorithms to compute the size of the spread at steady state for any set of initially active agents and to analyze the role of distinguished sensing modalities, network structure, and heterogeneity. We show how heterogeneity manages the tension in spreading dynamics between sensitivity to inputs and robustness to disturbances.

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