Learning Trust Over Directed Graphs in Multiagent Systems (extended version)

TL;DR

The paper tackles learning the trustworthiness of agents in directed multiagent networks containing malicious actors by leveraging stochastic observations of trust. It introduces a two-stage protocol where legitimate agents first learn in-neighborhood trust and then propagate this information to identify all agents, achieving almost-sure and in-mean convergence of trust opinions. The analysis hinges on random finite-time learning of in-neighbors and convergence of weakly chained substochastic matrices, yielding finite-time bounds for learning times. Numerical simulations across cyclic and Erdős–Rényi topologies corroborate the theoretical guarantees, showing reliable identification of legitimate versus malicious agents and practical convergence times with varying network sizes and malicious counts.

Abstract

We address the problem of learning the legitimacy of other agents in a multiagent network when an unknown subset is comprised of malicious actors. We specifically derive results for the case of directed graphs and where stochastic side information, or observations of trust, is available. We refer to this as ``learning trust'' since agents must identify which neighbors in the network are reliable, and we derive a protocol to achieve this. We also provide analytical results showing that under this protocol i) agents can learn the legitimacy of all other agents almost surely, and that ii) the opinions of the agents converge in mean to the true legitimacy of all other agents in the network. Lastly, we provide numerical studies showing that our convergence results hold in practice for various network topologies and variations in the number of malicious agents in the network.

Figures (7)

• Figure 1: This schematic shows our problem setup with one malicious agent shown as a red node. Various stages of learning are depicted: (a) initial state (b) agents use their direct observations to learn the trustworthiness of other agents (c) agents indirectly learn the trustworthiness of the entire network by propagating their opinions.
• Figure 2: (a) Network with four legitimate and one malicious nodes. Legitimate nodes are black and the malicious node is red. (b) Learning dynamics for agent $q=2$: Both agents $2$ and $3$ directly observe agent $2$, so $2, 3\in \mathcal{D}_q$. $1$ and $4$ are in the set $\mathcal{C}_q$ since they do not directly observe agent $2$. (c) Learning dynamics for agent $q=5$: Both agents $1$ and $2$ directly observe agent $5$, so $1,2\in \mathcal{D}_q$. $3$ and $4$ are in $\mathcal{C}_q$ since they do not observe $5$.
• Figure 3: Three matrices with different contraction indices and the corresponding graphs. The path achieving the contraction index is given in yellow. (a) The only row that sums to less than one is row $1$. Therefore, the path with the maximum length from agent $1$ to another agent has a length of $3$. (b) Row $2$ also sums to less than one. Therefore, the longest path is the one from agent $2$ to $4$. (c) The only row that sums to less than one is row $1$. Since there is no path from agent $1$ to agent $3$ or $4$, the index of contraction is $\infty$.
• Figure 4: Example graph topologies with $|\mathcal{L}|=6$, $|\mathcal{M}|=9$ nodes.
• Figure 5: Convergence plots for three different cases where the number of malicious agents are chosen as $|\mathcal{M}|=1.5\times|\mathcal{L}|$. Solid lines represents the MSE. The shaded areas show the range of error among the legitimate agents. We see that in all cases, the MSE converges to zero eventually as predicted by our theory. Since the malicious agents can have influence on the other nodes in the beginning, we observe an increase in the error at first. This effect is higher in cyclic graphs since the information takes longer to propagate. Moreover, as we increase the size of the graph for cyclic graphs, the convergence time also increases. On the other hand, since Erdős–Rényi graph has good connectivity in all cases, the convergence time is not as sensitive to the graph size compared to the cyclic graphs.

Theorems & Definitions (19)

• Definition 1: Stochastic Observation of Trust $\alpha_{ij}$
• Definition 2: Opinion of Trust
• Lemma 1
• Corollary 1
• Lemma 2
• Remark 1
• Definition 3
• Definition 4
• Remark 2
• Lemma 3