Convergence analysis of distributed subgradient methods over random networks

A decentralized event-triggered approach is developed for leader–follower systems with time-invariant and time-varying network topologies in the absence of full-state feedback. Continuous interagent communication is eliminated by communicating information only during events. At the event times, an observer communicates with the neighboring agents to obtain update information. During the interevent intervals, an open-loop estimate is used as a substitute for local feedback. The advantages of our novel approach are that: 1) the amount of communication required can be reduced and determining the event times requires no interagent communication, with the event-triggered condition designed using stability analysis so that bounded consensus is guaranteed; 2) bounded state consensus can be achieved without receiving full-state information from the followers; and 3) the switching sequence of the time-varying network topologies can be arbitrary, since a switching controller is developed based on the common-Lyapunov-function approach to ensure state consensus. Intermittent feedback is present and the dynamics of the time-varying network topologies are discrete, which together prove that Zeno behavior never occurs. The stability of the closed loop is analyzed to guarantee bounded convergence for the designed control methodologies. The results of simulations are presented that demonstrate the efficacy of our novel controllers.

This paper studies the problem of adaptively searching for an unknown target using multiple agents connected through a time-varying network topology. Agents are equipped with sensors capable of fast information processing, and we propose an asynchronous decentralized algorithm for controlling their search based on noisy observations. We propose asynchronous decentralized algorithms for adaptive query-based search that combine the Bayesian bisection method and social learning. Under standard assumptions on the time-varying network dynamics, we prove convergence to correct consensus on the value of the parameter as the number of iterations grow. Our results establish that stability and consistency can be maintained even with one-way updating and randomized pairwise averaging, thus providing a scalable low complexity alternative to the synchronous decentralized estimation algorithms studied in previous works. We illustrate the effectiveness and robustness of our algorithm for random network topologies.

This study investigates amplitude death in delayed coupled oscillators with a time-varying network topology. The local stability of amplitude death is governed by a linear time-varying system. This system is difficult to analyze; however, if the time-varying period is sufficiently short compared with the oscillation period, the linear time-varying system can be regarded as the linear time-invariant system. Thus, we can easily analyze the local stability of amplitude death in the fast time-varying network. Our analytical results are verified by some numerical simulations.

We consider a min-max optimization problem over a time-varying network of computational agents, where each agent in the network has its local convex cost function which is a private knowledge of the agent. The agents want to jointly minimize the maximum cost incurred by any agent in the network, while maintaining the privacy of their objective functions. To solve the problem, we consider subgradient algorithms where each agent computes its own estimates of an optimal point based on its own cost function, and it communicates these estimates to its neighbors in the network. The algorithms employ techniques from convex optimization, stochastic approximation and averaging protocols (typically used to ensure a proper information diffusion over a network), which allow time-varying network structure. We discuss two algorithms, one based on exact-penalty approach and the other based on primal-dual Lagrangian approach, where both approaches utilize Bregman-distance functions.We establish convergence of the algorithms (with probability one) for a diminishing step-size, and demonstrate the applicability of the algorithms by considering a power allocation problem in a cellular network.

This paper presents greedy gossip with eavesdropping (GGE), a novel randomized gossip algorithm for distributed computation of the average consensus problem. In gossip algorithms, nodes in the network randomly communicate with their neighbors and exchange information iteratively. The algorithms are simple and decentralized, making them attractive for wireless network applications. In general, gossip algorithms are robust to unreliable wireless conditions and time varying network topologies. In this paper we introduce GGE and demonstrate that greedy updates lead to rapid convergence. We do not require nodes to have any location information. Instead, greedy updates are made possible by exploiting the broadcast nature of wireless communications. During the operation of GGE, when a node decides to gossip, instead of choosing one of its neighbors at random, it makes a greedy selection, choosing the node which has the value most different from its own. In order to make this selection, nodes need to know their neighbors' values. Therefore, we assume that all transmissions are wireless broadcasts and nodes keep track of their neighbors' values by eavesdropping on their communications. We show that the convergence of GGE is guaranteed for connected network topologies. We also study the rates of convergence and illustrate, through theoretical bounds and numerical simulations, that GGE consistently outperforms randomized gossip and performs comparably to geographic gossip on moderate-sized random geometric graph topologies.

To ensure that social networks (e.g. opinion consensus, cooperative estimation, distributed learning and adaptation etc.) proliferate and efficiently operate, the participating agents need to collaborate with each other by repeatedly sharing information. However, sharing information is often costly for the agents while resulting in no direct immediate benefit for them. Hence, lacking incentives to collaborate, strategic agents who aim to maximize their own individual utilities will withhold rather than share information, leading to inefficient operation or even collapse of networks. In this paper, we develop a systematic framework for designing distributed rating protocols aimed at incentivizing the strategic agents to collaborate with each other by sharing information. The proposed incentive protocols exploit the ongoing nature of the agents' interactions to assign ratings and through them, determine future rewards and punishments: agents that have behaved as directed enjoy high ratings -- and hence greater future access to the information of others; agents that have not behaved as directed enjoy low ratings -- and hence less future access to the information of others. Unlike existing rating protocols, the proposed protocol operates in a distributed manner, online, and takes into consideration the underlying interconnectivity of agents as well as their heterogeneity. We prove that in many deployment scenarios the price of anarchy (PoA) obtained by adopting the proposed rating protocols is one. In settings in which the PoA is larger than one, we show that the proposed rating protocol still significantly outperforms existing incentive mechanisms such as Tit-for-Tat. Importantly, the proposed rating protocols can also operate efficiently in deployment scenarios where the strategic agents interact over time-varying network topologies where new agents join the network over time.

Distributed formation control of multi-UAV systems using relative information

The evolution toward 6G Computing Power Networks (CPN) aims to deeply integrate multi-tier computing resources across Cloud, Edge, and end devices. However, the significant heterogeneity of computing resources, characterized by varying hardware architectures such as CPUs, GPUs, and NPUs, coupled with the time-varying network topology caused by terminal mobility, poses severe challenges to realizing efficient integrated scheduling that satisfies Quality of Service (QoS). To address spatiotemporal mismatches between task requirements and hardware architectures, this paper proposes an integrated scheduling method combining Discrete Time-Varying Graph (DTVG) construction with Multi-Agent Reinforcement Learning (MARL). Specifically, we model the dynamic interaction between mobile tasks and heterogeneous nodes as a DTVG to capture spatiotemporal evolution and employ a QMIX-based algorithm to enable collaborative decision-making among distributed agents. Simulation results demonstrate that the proposed approach effectively solves the joint optimization problem of heterogeneous resource matching and dynamic path planning, significantly outperforming traditional baselines in terms of resource utilization and average latency. This study confirms that incorporating graph-theoretic modeling with reinforcement learning offers a robust solution for the complex coupling of communication and computation in dynamic 6G networks.

In this article, we propose a novel four-stage distributed controller for a dc microgrid that achieves proportional power sharing and average voltage regulation for the voltages at actuated and unactuated buses. The controller is presented for a dc microgrid comprising multiple distributed generation units (DGUs) with time-varying actuation states, dynamic <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$RLC$</tex-math> </inline-formula> lines, nonlinear constant impedance, current, and power (ZIP) loads, and a time-varying network topology. The controller comprising a nonlinear gain, proportional–integral (PI) controllers, and two dynamic distributed averaging stages is designed for asymptotic stability. This constitutes deriving passivity properties for the dc microgrid, along with each of the controller subsystems. Thereafter, design parameters are found through a passivity-based optimization using the worst-case subsystem properties. The resulting closed loop is robust against DGU actuation changes, network topology changes, and microgrid parameter changes. The stability and robustness of the proposed control are verified via simulations.

In the two-layer space communication network (TLSCN), communications can be performed to obtain higher throughput and lower latency by introducing various types of spatial nodes. However, the mobility of spatial nodes and the connectivity of spatial links result in the time-varying network topology, and intermittent link connection. This further leads to the lack of continuous contact, unreliable transmission, and high transmission cost. In this article, contact plan design (CPD) is employed to address the above problems by increasing the contact reliability while decreasing the contact cost. First, a directional space-time graph (DSTG) is constructed by considering the motion trajectory of spatial nodes and the time evolution nature of TLSCN. Afterwards, based on DSTG, we propose three CPD methods with greedy algorithm by considering the limited computing power of nodes. The three methods can optimize the objective functions of the total contact reliability, total contact cost, and invalid probability cost, respectively. The simulation results show that the proposed CPD methods can effectively improve the contact reliability, reduce the contact cost of TLSCN, and perform well with the increase of network density.

We consider the problem of tracking the minimum of a time-varying convex optimization problem over a dynamic graph. Motivated by target tracking and parameter estimation problems in intermittently connected robotic and sensor networks, the goal is to design a distributed algorithm capable of handling nondifferentiable regularization penalties. The proposed proximal online gradient descent algorithm is built to run in a fully decentralized manner and utilizes consensus updates over possibly disconnected graphs. The performance of the proposed algorithm is analyzed by developing bounds on its dynamic regret in terms of the cumulative path length of the time-varying optimum. It is shown that as compared to the centralized case, the dynamic regret incurred by the proposed algorithm over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> time slots is worse by a factor of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\log (T)$</tex-math></inline-formula> only, despite the disconnected and time-varying network topology. The empirical performance of the proposed algorithm is tested on the distributed dynamic sparse recovery problem, where it is shown to incur a dynamic regret that is close to that of the centralized algorithm.

In this paper, we propose a probabilistic consensus-based multi-robot search strategy that is robust to communication link failures, and thus is suitable for disaster affected areas. The robots, capable of only local communication, explore a bounded environment according to a random walk modeled by a discrete-time discrete-state (DTDS) Markov chain and exchange information with neighboring robots, resulting in a time-varying communication network topology. The proposed strategy is proved to achieve consensus, here defined as agreement on the presence of a static target, with no assumptions on the connectivity of the communication network. Using numerical simulations, we investigate the effect of the robot population size, domain size, and information uncertainty on the consensus time statistics under this scheme. We also validate our theoretical results with 3D physics-based simulations in Gazebo. The simulations demonstrate that all robots achieve consensus in finite time with the proposed search strategy over a range of robot densities in the environment.

ABSTRACTThis paper focuses on a distributed convex optimisation problem over networks with time-varying topologies. First, an event-triggered strategy is employed to reduce computation and communication load in the networked systems. Second, the exponential convergence of event-triggered zero-gradient-sum algorithm is guaranteed if the corresponding time-varying network topology satisfies a new connected condition, called cooperatively connected condition. This condition does not require topologies constantly connected or jointly connected but only requires the integral of the Laplacian matrix of the network topology over a period of time is connected. Hence, it is suitable for more general time-varying topologies. Third, a convergence analysis technique is developed which is based on the difference of the Lyapunov function rather than its differentiation. Finally, a simulation example is provided to verify the results obtained in this paper.

Multi-agent distributed optimization over a network minimizes a global objective formed by a sum of local convex functions using only local computation and communication. We develop and analyze a quantized distributed algorithm based on the alternating direction method of multipliers (ADMM) when inter-agent communications are subject to finite capacity and other practical constraints. While existing quantized ADMM approaches only work for quadratic local objectives, the proposed algorithm can deal with more general objective functions (possibly non-smooth) including the LASSO. Under certain convexity assumptions, our algorithm converges to a consensus within $\log_{1+\eta}\Omega$ iterations, where $\eta>0$ depends on the local objectives and the network topology, and $\Omega$ is a polynomial determined by the quantization resolution, the distance between initial and optimal variable values, the local objective functions and the network topology. A tight upper bound on the consensus error is also obtained which does not depend on the size of the network.

In current online economic dispatch problems of power systems, the cost function of each generation unit typically only includes time-varying generation costs, which does not account for the costs associated with current carbon emission market transactions and high-frequency communications. These factors need to be considered. Additionally, considering the lag in obtaining information about cost coefficients, feedback delay should be considered as a factor for the online optimization process of economic dispatch. This research addresses the challenge of online economic dispatch in the presence of delayed feedback and carbon emission expenses, proposing a decentralized, event-initiated algorithm for online optimization. To address these challenges, this paper proposes a decentralized, event-triggered online optimization algorithm tailored for Agentic AI in smart energy systems operating over an edge–cloud continuum. In each local optimization iteration, each node in the power system can access its local objective function with a delayed time sequence and updates its local decision-making behavior online based on information from multiple neighboring nodes, controlled by an event-triggering mechanism, to minimize the global cumulative generation cost and carbon emission cost. The research results show that, under the assumption that the time-varying balanced undirected communication topology remains connected, the designed online optimization algorithm ensures that the upper bound of static network regret grows sub-linearly, fundamentally related to feedback delays and event-triggering thresholds and scaled as O ( T ) . This work provides a foundational paradigm for Agentic AI in smart energy systems, bridging distributed optimization theory with practical constraints of modern edge-cloud infrastructures.