Abstract-This paper addresses the problem of resilient innetwork consensus in the presence of misbehaving nodes. Secure and fault-tolerant consensus algorithms typically assume knowledge of nonlocal information; however, this assumption is not suitable for large-scale dynamic networks. To remedy this, we focus on local strategies that provide resilience to faults and compromised nodes. We design a consensus protocol based on local information that is resilient to worst-case security breaches, assuming the compromised nodes have full knowledge of the network and the intentions of the other nodes. We provide necessary and sufficient conditions for the normal nodes to reach asymptotic consensus despite the influence of the misbehaving nodes under different threat assumptions. We show that traditional metrics such as connectivity are not adequate to characterize the behavior of such algorithms, and develop a novel graph-theoretic property referred to as network robustness. Network robustness formalizes the notion of redundancy of direct information exchange between subsets of nodes in the network, and is a fundamental property for analyzing the behavior of certain distributed algorithms that use only local information.

Given a network of interconnected nodes, each with its own value (such as a measurement, position, vote, or other data), we develop a distributed strategy that enables some or all of the nodes to calculate any arbitrary function of the node values, despite the actions of malicious nodes in the network. Our scheme assumes a broadcast model of communication (where all nodes transmit the same value to all of their neighbors) and utilizes a linear iteration where, at each time-step, each node updates its value to be a weighted average of its own previous value and those of its neighbors. We consider a node to be malicious or faulty if, instead of following the predefined linear strategy, it updates its value arbitrarily at each time-step (perhaps conspiring with other malicious nodes in the process). We show that the topology of the network completely characterizes the resilience of linear iterative strategies to this kind of malicious behavior. First, when the network contains 2 or fewer vertex-disjoint paths from some node to another node , we provide an explicit strategy for malicious nodes to follow in order to prevent node from receiving any information about 's value. Next, if node has at least 2 + 1 vertex-disjoint paths from every other (non-neighboring) node, we show that is guaranteed to be able to calculate any arbitrary function of all node values when the number of malicious nodes is or less. Furthermore, we show that this function can be calculated after running the linear iteration for a finite number of time-steps (upper bounded by the number of nodes in the network) with almost any set of weights (i.e., for all weights except for a set of measure zero).

Online learning has become increasingly popular on handling massive data. The sequential nature of online learning, however, requires a centralized learner to store data and update parameters. In this paper, we consider online learning with distributed data sources. The autonomous learners update local parameters based on local data sources and periodically exchange information with a small subset of neighbors in a communication network. We derive the regret bound for strongly convex functions that generalizes the work by Ram et al.[1] for convex functions. More importantly, we show that our algorithm has intrinsic privacypreserving properties, and we prove the sufficient and necessary conditions for privacy preservation in the network. These conditions imply that for networks with greater-than-one connectivity, a malicious learner cannot reconstruct the subgradients (and sensitive raw data) of other learners, which makes our algorithm appealing in privacy-sensitive applications.

We investigate the vulnerabilities of consensus-based distributed optimization protocols to nodes that deviate from the prescribed update rule (e.g., due to failures or adversarial attacks). We first characterize certain fundamental limitations on the performance of any distributed optimization algorithm in the presence of adversaries. We then propose a resilient distributed optimization algorithm that guarantees that the non-adversarial nodes converge to the convex hull of the minimizers of their local functions under certain conditions on the graph topology, regardless of the actions of a certain number of adversarial nodes. In particular, we provide sufficient conditions on the graph topology to tolerate a bounded number of adversaries in the neighborhood of every non-adversarial node, and necessary and sufficient conditions to tolerate a globally bounded number of adversaries. For situations where there are up to F adversaries in the neighborhood of every node, we use the concept of maximal F -local sets of graphs to provide lower bounds on the distanceto-optimality of achievable solutions under any algorithm. We show that finding the size of such sets is NP-hard.

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