-- We show how graphons can be used to model and analyze open multi-agent systems, which are multi-agent systems subject to arrivals and departures, in the specific case of linear consensus. First, we analyze the case of replacements, where under the assumption of a deterministic interval between two replacements, we derive an upper bound for the disagreement in expectation. Then, we study the case of arrivals and departures, where we define a process for the evolution of the number of agents that guarantees a minimum and a maximum number of agents. Next, we derive an upper bound for the disagreement in expectation, and we establish a link with the spectrum of the expected graph used to generate the graph topologies. Finally, for stochastic block model (SBM) graphons, we prove that the computation of the spectrum of the expected graph can be performed based on a matrix whose dimension depends only on the graphon and it is independent of the number of agents. Open multi-agent systems are a framework used to analyze networks subject to arrivals, departures or replacements of agents at a rate similar to the scale time of the process [1], [2]. This type of systems are essentially characterized by the agent internal dynamics, the evolution of the network and the arrivals and departures [3]. This is usually done by considering trivial dynamics like complete graphs [4]-[6], bounds on the algebraic connectivity or diameter [7], [8] or just connectivity at all time instants [9].

Modern artificial intelligence relies on networks of agents that collect data, process information, and exchange it with neighbors to collaboratively solve optimization and learning problems. This article introduces a novel distributed algorithm to address a broad class of these problems in "open networks", where the number of participating agents may vary due to several factors, such as autonomous decisions, heterogeneous resource availability, or DoS attacks. Extending the current literature, the convergence analysis of the proposed algorithm is based on the newly developed "Theory of Open Operators", which characterizes an operator as open when the set of components to be updated changes over time, yielding to time-varying operators acting on sequences of points of different dimensions and compositions. The mathematical tools and convergence results developed here provide a general framework for evaluating distributed algorithms in open networks, allowing to characterize their performance in terms of the punctual distance from the optimal solution, in contrast with regret-based metrics that assess cumulative performance over a finite-time horizon. As illustrative examples, the proposed algorithm is used to solve dynamic consensus or tracking problems on different metrics of interest, such as average, median, and min/max value, as well as classification problems with logistic loss functions.

Current trust and reputation models continue to have significant limitations, such as the inability to deal with agents constantly entering or exiting open multi-agent systems (open MAS), as well as continuously changing behaviors. Our study is based on CA, a previously proposed decentralized computational trust model from the trustee's point of view, inspired by synaptic plasticity and the formation of assemblies in the human brain. It is designed to meet the requirements of highly dynamic and open MAS, and its main difference with most conventional trust and reputation models is that the trustor does not select a trustee to delegate a task; instead, the trustee determines whether it is qualified to successfully execute it. We ran a series of simulations to compare CA model to FIRE, a well-established, decentralized trust and reputation model for open MAS under conditions of continuous trustee and trustor population replacement, as well as continuous change of trustees' abilities to perform tasks. The main finding is that FIRE is superior to changes in the trustee population, whereas CA is resilient to the trustor population changes. When the trustees switch performance profiles FIRE clearly outperforms despite the fact that both models' performances are significantly impacted by this environmental change. Findings lead us to conclude that learning to use the appropriate trust model, according to the dynamic conditions in effect could maximize the trustor's benefits.

We propose a method for analyzing the distributed random coordinate descent algorithm for solving separable resource allocation problems in the context of an open multiagent system, where agents can be replaced during the process. In particular, we characterize the evolution of the distance to the minimizer in expectation by following a time-varying optimization approach which builds on two components. First, we establish the linear convergence of the algorithm in closed systems, in terms of the estimate towards the minimizer, for general graphs and appropriate step-size. Second, we estimate the change of the optimal solution after a replacement, in order to evaluate its effect on the distance between the current estimate and the minimizer. From these two elements, we derive stability conditions in open systems and establish the linear convergence of the algorithm towards a steady-state expected error. Our results enable to characterize the trade-off between speed of convergence and robustness to agent replacements, under the assumptions that local functions are smooth, strongly convex, and have their minimizers located in a given ball. The approach proposed in this paper can moreover be extended to other algorithms guaranteeing linear convergence in closed system.

The resource allocation problem consists of the optimal distribution of a budget between agents in a group. We consider such a problem in the context of open systems, where agents can be replaced at some time instances. These replacements lead to variations in both the budget and the total cost function that hinder the overall network's performance. For a simple setting, we analyze the performance of the Random Coordinate Descent algorithm (RCD) using tools similar to those commonly used in online optimization. In particular, we study the accumulated errors that compare solutions issued from the RCD algorithm and the optimal solution or the non-collaborating selfish strategy and we derive some bounds in expectation for these accumulated errors.