A risk-security tradeoff in graphical coordination games

TL;DR

This article characterizes the operator's fundamental tradeoff between security against worst-case broad attacks and vulnerability from focused attacks, and shows that this tradeoff significantly improves when the operator selects a decision-making process at random.

Abstract

A system relying on the collective behavior of decision-makers can be vulnerable to a variety of adversarial attacks. How well can a system operator protect performance in the face of these risks? We frame this question in the context of graphical coordination games, where the agents in a network choose among two conventions and derive benefits from coordinating neighbors, and system performance is measured in terms of the agents' welfare. In this paper, we assess an operator's ability to mitigate two types of adversarial attacks - 1) broad attacks, where the adversary incentivizes all agents in the network and 2) focused attacks, where the adversary can force a selected subset of the agents to commit to a prescribed convention. As a mitigation strategy, the system operator can implement a class of distributed algorithms that govern the agents' decision-making process. Our main contribution characterizes the operator's fundamental trade-off between security against worst-case broad attacks and vulnerability from focused attacks. We show that this tradeoff significantly improves when the operator selects a decision-making process at random. Our work highlights the design challenges a system operator faces in maintaining resilience of networked distributed systems.

Figures (4)

• Figure 1: (Left) An example three-node line network under a broad adversarial attack. The imposter nodes are depicted as the labelled smaller circles and agents in the network are the bigger circles. The color of each circle indicates the node's action - green for $x$, blue for $y$. In this example, maximum welfare is $\max_{a\in\mathcal{A}} W(a) = 4(1+\alpha_{\text{sys}})$, achieved when all three agents play $x$. The adversary's target set $S$ attaches an $x$-imposter to node 1 and $y$-imposters to nodes $2$ and $3$. For operator gains $\alpha \leq \frac{1}{2}$, $a=(a_1,a_2,a_3)=(x,y,y)$ is the welfare-minimizing SSS, i.e. it satisfies $a=\underset{a\in\text{LLL}(\mathcal{A},\alpha,S;G)} {\operatorname{arg}\,\operatorname{min}}\; W(a)$. This gives a risk of $R_{\text{b}}(\alpha,S;G) = 1 - \frac{1}{2(1+\alpha_{\text{sys}}) }$. For $\alpha > \frac{1}{2}$, the welfare-minimizing SSS is $(x,x,x)$. This gives optimal efficiency, i.e. a risk of $0$. (Right) An example of a four node star network under a focused attack where a subset $F$ of three nodes are targeted to be fixed (squares). Only the center node is unfixed. In this example, the maximum welfare is $\max_{a\in\mathcal{A}_F} W(a) = 4$, achieved when the center plays $y$. This is because the alternative action (when center plays $x$) gives the suboptimal welfare $2(1+\alpha_{\text{sys}}) < 4$ due to $\alpha_{\text{sys}} < 1$. For operator gains $\alpha < 1$, the center node plays $y$ in the SSS. This yields optimal efficiency, i.e. the risk is $R_{\text{f}}(\alpha,F;G) = 0$. For $\alpha \geq 1$, the center node plays $x$, giving a risk of $R_{\text{f}}(\alpha,F;G) =1 - \frac{1 + \alpha_{\text{sys}}}{2}$. The methods to calculate stochastically stable states under both types of attacks follow standard potential game arguments and are detailed in Section \ref{['sec:analysis']}.
• Figure 2: (a) The worst-case risk from broad attacks $R_{\text{b}}^*(\alpha)$\ref{['eq:RA_WC']} is a piecewise constant function defined over countably infinite half-open intervals. The graphs and their corresponding target set which attain each level of worst-case broad risk are illustrated for $\alpha < 1$. Here, the $x,y$ labels indicate the type of imposter influence on the agents (circles) in the network, and the color of the circles depict the action played in the welfare-minimizing SSS (green=$x$, blue=$y$). If $\alpha \in I_k$, $k=1,2,\ldots$(recall \ref{['eq:Ik']}), the worst-case risk is achieved on a star graph of $k+2$ nodes where all nodes but one are targeted with a $y$ imposter. The one leaf node has an $x$ imposter attached, giving a single miscoordinating link in the network. (b) The worst-case risk from focused attacks $R_{\text{f}}^*(\alpha)$\ref{['eq:RE_WC']}. The graphs and their corresponding fixed sets which attain the worst-case focused risks are illustrated for $\alpha = \frac{1}{2}, 1$, and $2$. The nodes' color represents the worst-case SSS at $\alpha$ (blue $=y$, green $=x$). The targeted fixed agents are represented as squares and the unfixed agents as circles. Here $\frac{1}{2}< \alpha_{\text{sys}} < 1$. The proofs establishing all worst-case graphs are detailed in Section \ref{['sec:analysis']}.
• Figure 3: Security-risk tradeoffs are depicted by the achievable worst-case risk levels from deterministic gains (blue) and randomized gains (red, green, black). The Pareto frontiers for three different randomized strategies $\bm{\alpha}^1,\bm{\alpha}^2 \in \mathbb{R}_+^5$, and $\bm{\alpha}^3 \in \mathbb{R}_+^{300}$, are shown in increasing order of improvement. The strategies $\bm{\alpha}^1$ and $\bm{\alpha}^2$ randomize over the highest three broad risk levels in addition to the lowest two. The strategy $\bm{\alpha}^3$ randomizes over the highest 298 broad risk levels and the lowest two. We chose the values as follows. For $k = 1,2$, we set $\alpha_1^k = \alpha_{\text{sys}}$, $\alpha_j^k = (1-\epsilon_k) \frac{j-1}{j} + \epsilon_k \frac{j}{j+1}\in I_j$ for $j = 2,3$, $\alpha_4^k = 1+\epsilon_k$, and $\alpha_5^k = \frac{3}{2} + \epsilon_k$. We have set $\epsilon_1 = 0.5$ and $\epsilon_2 = .01$. Hence, Par($\bm{\alpha}^2$) improves upon Par($\bm{\alpha}^1$) via Claim \ref{['pareto_incr']}. For $k=3$, we set $\alpha_1^3 = \alpha_{\text{sys}}$, $\alpha_j^3 = (1-\epsilon_3) \frac{j-1}{j} + \epsilon_3 \frac{j}{j+1}\in I_j$, $j = 2,3,\ldots,298$, $\alpha_{299}^3 = 1+\epsilon_3$, and $\alpha_5^3 = \frac{3}{2} + \epsilon_3$. Claim \ref{['pareto_more']} ensures Par($\bm{\alpha}^3$) improves upon Par($\bm{\alpha}^2$). We chose $\epsilon_3 = .01$ and $\alpha_{\text{sys}} = 1/4$.
• Figure 4: An illustration of the constructive process (proof of Lemma \ref{['star_reduction']}) that generates a member $(S',G')\in\mathbb{S}_m$ from any graph $(S,G)$ with one $y$-partition, and $\alpha < 1$. Here, the labels on each node indicate the type of imposter influence. Green (blue) nodes play $x$ ($y$) in the SSS. (Left) Start with an arbitrary graph-adversary pair $(S,G)$. (Center) The partitions of $(S,G)$ are re-cast as star subgraphs with the same number of edges. When there is more than one edge between the $y$ and an $x$-partition, new nodes are created for the excess outgoing edges. This re-casting preserves the original efficiency $J_{\text{b}}(\alpha,S;G)$. (Right) The active $x$-links are converted into $y$-links by redirecting them to the center of the $y$-partition. This results in a graph $(S',G') \in \mathbb{S}_m$.

Theorems & Definitions (37)

• Definition 1
• Theorem 1
• Definition 2
• Theorem 2
• Remark 1
• Corollary 1
• proof
• Corollary 2
• proof
• Theorem 3