Community Consensus: Converging Locally despite Adversaries and Heterogeneous Connectivity

TL;DR

This work characterize cases where, although global consensus is not reached, some subsets of agents will still converge to the same values $\mathcal{M}_{i}$ among themselves, and defines more relaxed requirements in terms of the number of malicious agents in each community, and the number of edges connecting an agent in a community to agents external to the community.

Abstract

We introduce the concept of community consensus in the presence of malicious agents using a well-known median-based consensus algorithm. We consider networks that have multiple well-connected regions that we term communities, characterized by specific robustness and minimum degree properties. Prior work derives conditions on properties that are necessary and sufficient for achieving global consensus in a network. This, however, requires the minimum degree of the network graph to be proportional to the number of malicious agents in the network, which is not very practical in large networks. In this work, we present a natural generalization of this previous result. We characterize cases where, although global consensus is not reached, some subsets of agents $V_i$ will still converge to the same values $M_i$ among themselves. To reach this new type of consensus, we define more relaxed requirements in terms of the number of malicious agents in each community, and the number $k$ of edges connecting an agent in a community to agents external to the community.

Figures (5)

• Figure 1: Example of graph where two subsets of agents $V_1$ and $V_2$ are visibly more connected within themselves. Lower connectivity between $V_1$ and $V_2$ is represented by the orange edges.
• Figure 2: Flow of the proofs.
• Figure 3: Example of Case1 (left) and Case2 (right) in \ref{['thm:rsconstruction']}. With respect to $S$ and selecting the subset $V_i$, agent $u$ is $3-2 = 1$-excess reachable. When considering $V_i^*$, one agent more is outside $S$ and $u$ becomes $2$-excess reachable. In Case2, $u$ becomes $3-3 = 0$-excess reachable.
• Figure 4: Plots of our simulation results (cf. \ref{['sec:simulations']}). (a) - Example 1: Constraints from \ref{['thm:mainresult']} are respected, (b) - Example 2: Robustness constraint is violated: agents in $G[V_2]$ converge to two different values, (c) - Example 3: Minimum degree constraint is violated. Only in (a) do legitimate agents reach community consensus. The value of 60 belongs to malicious agents and that the x-axis uses logarithmic scale.
• Figure 5: Graph used to simulate the case where robustness constraints are not met. The dashed line separates $V_2$ in two subsets of agent: $S_1 = \{u_1, \dots, u_5\}$ and $S_2 = \{u_6, \dots, u_9\}$. Recall that, in this case, it is $f_2 = 1$ and $k_2 = 1$. Therefore it has to be $d_{min}^{V_2} \ge 2f_2 + k_2 + 1 = 4$. In this example, no agent in $V_2$ is $1$-excess reachable under the partition $\{S_1, S_2\}$, thus $G[V_2]$ is not $(1, 2)$-excess robust.

Theorems & Definitions (11)

• Definition 1: $r$-excess reachable set
• Definition 2: $r$-excess robust graph
• Definition 3: $(r, s)$-excess robustness
• Definition 4: $(k_i, f_i)$-community
• Definition 5: Resilient Asymptotic Consensus
• Theorem 1
• Remark 1
• Remark 2
• Definition 6: Community Isolation
• Proposition 1