On the Price of Anarchy for Flows over Time

Dynamic network flows, or network flows over time, constitute an important model for real-world situations in which steady states are unusual, such as urban traffic and the internet. These applications immediately raise the issue of analyzing dynamic network flows from a game-theoretic perspective. In this paper, we study dynamic equilibria in the deterministic fluid queuing model in single-source, single-sink networks—arguably the most basic model for flows over time. In the last decade, we have witnessed significant developments in the theoretical understanding of the model. However, several fundamental questions remain open. One of the most prominent ones concerns the price of anarchy, measured as the worst-case ratio between the minimum time required to route a given amount of flow from the source to the sink and the time a dynamic equilibrium takes to perform the same task. Our main result states that, if we could reduce the inflow of the network in a dynamic equilibrium, then the price of anarchy is bounded by a factor, parameterized by the longest path length that converges to [Formula: see text], and this is tight. This significantly extends a result by Bhaskar et al. (SODA 2011). Furthermore, our methods allow us to determine that the price of anarchy in parallel-link and parallel-path networks is exactly 4/3. Finally, we argue that, if a certain, very natural, monotonicity conjecture holds, the price of anarchy in the general case is exactly [Formula: see text].

Settlement Schemes and their Implication on Eastern Mau Water Catchment, Kenya

Settlement schemes are aimed at settling landless people and those displaced by disasters to support socio-economic and environmental development of a country. Eastern Mau Forest Reserve is an important water catchment that has settlement schemes established, which has led to encroachments and degradation of the catchment. This paper, therefore, assesses the implications of human settlements on Eastern Mau water catchment by examining the trends in land use/cover change and river flows for four decades. Primary data was collected from key informant interviews based on purposive sampling whereas secondary data was derived from Landsat satellite images over a 10-year period and analysed using Maximum Likelihood Function from ArcGIS. Data on river flows from River Njoro was obtained from Water Resources Authority Office in Nakuru. Rainfall and temperature data were obtained from Kenya Meteorological Station in Nakuru. Time series analysis is used to understand the trend in river flows over time while Pearson correlation is used to determine relationship between farmlands and river flows. The results indicate a sharp decline in forest cover by 42.7% and an increase in farmlands by 41%. Dense vegetation and farmlands have an inverse relationship as an increase in farmlands lead to a decrease in forest cover. People have settled beyond the established settlement schemes boundaries leading to encroachment and drying up of some rivers. There is also an increase in rainfall and river flows over the years, with monthly river flows increasing in peak flows and declining during low seasons. There is a positive correlation between farmlands and river flows between 1989 and 2020. There is need for regeneration of encroached areas and defining boundary of Eastern Mau to allow initiatives and interventions that help with sustainable management of the catchment area.

Algorithms for flows over time with scheduling costs

Flows over time have received substantial attention from both an optimization and (more recently) a game-theoretic perspective. In this model, each arc has an associated delay for traversing the arc, and a bound on the rate of flow entering the arc; flows are time-varying. We consider a setting which is very standard within the transportation economic literature, but has received little attention from an algorithmic perspective. The flow consists of users who are able to choose their route but also their departure time, and who desire to arrive at their destination at a particular time, incurring a scheduling cost if they arrive earlier or later. The total cost of a user is then a combination of the time they spend commuting, and the scheduling cost they incur. We present a combinatorial algorithm for the natural optimization problem, that of minimizing the average total cost of all users (i.e., maximizing the social welfare). Based on this, we also show how to set tolls so that this optimal flow is induced as an equilibrium of the underlying game.

Computation of Dynamic Equilibria in Series-Parallel Networks

We consider dynamic equilibria for flows over time under the fluid queuing model. In this model, queues on the links of a network take care of flow propagation. Flow enters the network at a single source and leaves at a single sink. In a dynamic equilibrium, every infinitesimally small flow particle reaches the sink as early as possible given the pattern of the rest of the flow. Although this model has been examined for many decades, progress has been relatively recent. In particular, the derivatives of dynamic equilibria have been characterized as thin flows with resetting, which allows for more structural results. Our two main results are based on the formulation of thin flows with resetting as a linear complementarity problem and its analysis. We present a constructive proof of existence for dynamic equilibria if the inflow rate is right-monotone. The complexity of computing thin flows with resetting, which occurs as a subproblem in this method, is still open. We settle it for the class of two-terminal, series-parallel networks by giving a recursive algorithm that solves the problem for all flow values simultaneously in polynomial time.

Clinically Localized Seizure Focus Maybe Not Exactly the Position of Abating Seizures: A Computational Evidence

Seizure focus localization is the key to control seizures. However, in this paper, we show that the clinically localized seizure focus may be not exactly the positions to abate seizures. Firstly, the reliability of a previously proposed methodology employed to estimate the synchronicity and directionality of information flows over time between EEG signals, is numerically assessed with a coupled mass neural model. Then 10 channels' EEG signals from a patient with focal epilepsy are used to reconstruct the dynamical complex network of pathological seizure. This may facilitate to identify the evolution paths of information flows and localize the potential seizure foci. What's more, based on the controllability and observability principles of complex systems, we can focus on the key nodes which is effective to control the network seizure behaviors and the key ones that can allow us to estimate the state of all other variables. Results show that to fully control the epileptic network may not just be related to the focus zone, it may also involves in other non-focus nodes. In addition, we use the spatiotemporal neural network model connected by our modeled dynamical adjacent matrix to successfully reproduce the original EEG signals which can be effectively abated by applying the normal distribution noise stimulation with cathodic phase pulses (cNDNs) on the identified key nodes or resecting them. Our results enrich the clinical results and provide new insights into the seizure resection and electronic stimulation therapies.

Long-Term Behavior of Dynamic Equilibria in Fluid Queuing Networks

Steady State in Equilibrium for Flows over Time

New Monetarism in Continuous Time: Methods and Applications*

We develop a New Monetarist model in continuous time where agents trade continuously in competitive markets and infrequently in pairwise meetings. Agents can produce and consume both in flows over time intervals and in discrete quantities at points in time. We detail the methodology to solve individual optimisation problems and characterise the full set of perfect foresight equilibria. We illustrate the role of continuous time and the tractability of our approach with three applications related to monetary policy: (i) forward guidance with policy announcements; (ii) aggregate demand management and firm entry; (iii) open market operations with partially liquid assets.