Robust Distributed Averaging: When are Potential-Theoretic Strategies Optimal?

Abstract

We study the interaction between a network designer and an adversary over a dynamical network.The network consists of nodes performing continuous-time distributed averaging. The adversary strategically disconnects a set of links to prevent the nodes from reaching consensus. Meanwhile, the network designer assists the nodes in reaching consensus by changing the weights of a limited number of links in the network. We formulate two Stackelberg games to describe this competition where the order in which the players act is reversed in the two problems. Although the canonical equations provided by the Pontryagin's maximum principle seem to be intractable, we provide an alternative characterization for the optimal strategies that makes connection to potential theory. Finally, we provide a sufficient condition for the existence of a saddle-point equilibrium for the underlying zero-sum game. 2 optimization, and flocking [2]-[4]. Besides engineering, linear distributed averaging finds applications in other fields as well. For instance, social scientists use averaging to describe the evolution of opinions in networks [5]. In practice, communication among agents is prone to different types of non-idealities which can affect the convergence properties of the associated distributed algorithms. Transmission delays [6], noisy links [7], [8], and quantization [9] are some examples of non-idealities that are due to the physical nature of the application. In addition to physical restrictions, researchers have also studied averaging dynamics in the presence of malicious nodes in the network [10], [11].In [12], we explored the effect of an external adversary who attempts to prevent the nodes from reaching consensus by launching network-wide attacks. When the adversary is capable of disconnecting certain links in the network, we derived the optimal strategy of the adversary and demonstrated that it admits a potential-theoretic analogy. In this paper, we also introduce a network designer that attempts to counter the effect of the adversary and help the nodes reach consensus. The designer is capable of changing the weights of certain links. Both the adversary and the designer are constrained by their physical capabilities, e.g., battery life and communication range. To capture such constraints, we allow the adversary and the designer to affect only a fixed number of links at any point in time. The conflicting objectives of the designer and the adversary calls for a game-theoretic formulation to study their interaction.Such a competition between a network designer and an adversary can occur in various practical applications. For example, in a wireless network, the link weights in such a network represent the capacities of the corresponding links. The designer can modify the capacity of a certain link using various communication techniques such as introducing parallel channels between two nodes as in orthogonal frequency division multiple access (OFDMA) networks [13]. In OFDMA networks, the number of parallel links between two nodes is us...

Consensus under Network Interruption and Effective Resistance Interdiction

“…Robustifying network operations in the presence of nonidealities of physical nature (such as switching topologies and transmission delays [1], [2]; noisy links [3]; quantization [4]; random link failures [5]), as well as securing them against misbehavior of nodes and malicious adversarial attacks have been important areas of research in network security [6], [7]. Most research in this arena has been focused on applications such as consensus formation, or distributed averaging, and selected behavior of the adversary that leads to disruption of the underlying operation [8], [9]. In this paper, our goal is to analyze network interdiction's influence on the operations of the consensus dynamics from a computational perspective.…”

“…Our work is based on the modeling framework and analysis described in the recent paper [8], which adopted the setting of distributed averaging (as network operation). Without the presence of an adversary, distributed averaging involves the computation of the average of the initial values stored at each node of a graph in a distributed way, where, in an evolving process, each node (agent) updates its value as a linear combination of the values of its neighbors.…”

“…Now, if we bring into the picture an adversary who wants to disrupt that process, we can endow the adversary with the capability to break a limited number of links to prevent convergence to the true average or slow down the convergence. To measure the convergence speed, the paper [8] introduces the objective function…”

“…where k(•) is a positive weighting function, and the adversary wants to maximize it by deciding what links to break at each time instance t. 1 It was shown in [8] that even under a more elaborate setting, where there is a network designer who can repair a limited number of edges, the game between the adversary and the network designer admits a saddle-point strategy. Further, it was erroneously shown that for a fixed network designer's strategy, the adversary's optimal strategy exhibits a low worst-case complexity, and one can efficiently compute the adversary's optimal strategy using potentialtheoretic analogy [8, Theorem 1].…”

“…As we show in this work, finding the adversary's optimal strategy even for a simpler static setting and without a network designer is strongly NPhard. We also provide a counterexample and point out the source of the error in [8]. On the positive side, we provide a quadratic program formulation for the adversary's optimal strategy.…”