We consider the problem of diffusing information in networks that contain malicious nodes. We assume that each normal node in the network has no knowledge of the network topology other than an upper bound on the number of malicious nodes in its neighborhood. We introduce a topological property known as r-robustness of a graph, and show that this property provides improved bounds on tolerating malicious behavior, in comparison to traditional concepts such as connectivity and minimum degree. We use this topological property to analyze the canonical problems of distributed consensus and broadcasting, and provide sufficient conditions for these operations to succeed. Finally, we provide a construction for r-robust graphs and show that the common preferential-attachment model for scale-free networks produces a robust graph.
A core question in the study of large networks (both natural and engineered) is: how do the actions of a small subset of the population affect the global behavior of the network? For instance, the fields of sociology and epidemiology examine the spread of ideas, decisions and diseases through populations of people, based on the patterns of contact between the individuals in the population [1], [2], [3]. In this context, one can ask whether a few stubborn individuals (who do not change their beliefs) are able to affect the decisions reached by the rest of the population [4], [5]. Similarly, the efficacy of engineered networks (such as communication networks, or multi-agent systems) is often predicated on their ability to disseminate information throughout The authors are with the Department of Electrical and Computer Engineering at the University of Waterloo. E-mail for corresponding author: ssundara@uwaterloo.ca.
the network [6], [7]. For example, the ‘broadcast’ operation is used as a building block for more complex functions, allowing certain nodes to inform all other nodes of pertinent information [6].
Another important operation is that of ‘distributed consensus’, where every node in the network has some information to share with the others, and the entire network must come to an agreement on an appropriate function of that information [8], [9], [10], [11], [12].
The ability of a few individuals to affect the global behavior of the system is clearly a doubleedged sword. When the network contains legitimate leaders or experts, it is beneficial to ensure that the innovations introduced by these small groups spread throughout the population. On the other hand, networks that facilitate diffusion are also vulnerable to disruption by individuals that are not acting with the best interests of the society in mind. In engineering applications, these individuals could correspond to faulty or malicious nodes that do not follow preprogrammed strategies due to malfunctions or attacks, respectively. Thus, a fundamental challenge is to identify network properties and diffusion dynamics that allow legitimate information to propagate throughout the network, while limiting the effects of illegitimate individuals and actions.
The problem of transmitting information over networks (and specifically, reaching consensus) in the presence of faulty or malicious nodes has been studied extensively over the past several decades (e.g., see [8], [6] and the references therein). It has been shown that if the connectivity of the network is 2f or less for some nonnegative integer f , then f malicious nodes can conspire to prevent some of the nodes from correctly receiving the information of other nodes in the network. Conversely, when the network connectivity is 2f + 1 or higher, there are various algorithms to allow reliable dissemination of information (under the wireless broadcasting model of communication) [13], [14]. However, these methods require that all nodes have full knowledge of the network topology, along with the specific parameters of the algorithm applied by all other nodes. Furthermore, the computational overhead for these methods is generally quite high [8], [13].
It is not surprising that there is a tradeoff between how much each node knows about the overall network and the conditions required for those nodes to overcome malicious adversaries.
The objective of this paper is to analyze information dissemination strategies in networks with adversaries when each normal node only has access to its neighbors’ values, and does not know anything about the rest of the network (i.e., the topology, number of nodes, location and behavior of malicious nodes, etc.); it only knows that the total number of adversaries in its own neighborhood is bounded by some known quantity. We introduce the concept of rrobust graphs, and show that such graphs provide resilience to malicious nodes. We focus on the particular applications of fault-tolerant broadcast and distributed consensus, and similarly to [15], we consider a locally bounded fault model where there is an upper bound on the number of adversarial nodes in the neighborhood of any reliable node, but there is no other a priori bound on the total number of adversaries in the network. In the case of fault-tolerant broadcast, our conditions can be applied to show that broadcast will succeed in certain networks that do not meet the conditions provided in [15]. For distributed consensus, our conditions provide separate sufficient and necessary conditions for all normal nodes to reach consensus while limiting the ability of locally-bounded malicious nodes to influence the final value.
Consider a network modeled by the directed graph G = {V, E}, where V = {1, …, n} is the set of nodes and E ⊆ V × V is the set of edges in the network. An edge (j, i) ∈ E indicates that node i can be influenced by (or receive information from) node j. The set of neighbors of node i is defined as
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