The Impact of Complex and Informed Adversarial Behavior in Graphical Coordination Games

TL;DR

The paper investigates how adversaries can degrade the performance guarantees of log-linear learning in graphical coordination games, focusing on ring graphs and a budgeted influence model. It develops a rigorous framework using potential games and resistance-tree analysis to bound the worst-case efficiency under four policy classes (static/dynamic and informed/uninformed), showing ring graphs with minimal connectivity are most vulnerable. The authors derive tight lower bounds for static and dynamic policies, reveal a phase transition where sophistication dominates at low budgets ($\gamma<\alpha$) and information dominates at high budgets ($\gamma>\alpha$), and demonstrate saturation phenomena where increasing budget yields no further damage. Complementary simulations corroborate the theoretical results, offering insights into protecting distributed systems against adversarial interventions.

Abstract

How does system-level information impact the ability of an adversary to degrade performance in a networked control system? How does the complexity of an adversary's strategy affect its ability to degrade performance? This paper focuses on these questions in the context of graphical coordination games where an adversary can influence a given fraction of the agents in the system, and the agents follow log-linear learning, a well-known distributed learning algorithm. Focusing on a class of homogeneous ring graphs of various connectivity, we begin by demonstrating that minimally connected ring graphs are the most susceptible to adversarial influence. We then proceed to characterize how both (i) the sophistication of the attack strategies (static vs dynamic) and (ii) the informational awareness about the network structure can be leveraged by an adversary to degrade system performance. Focusing on the set of adversarial policies that induce stochastically stable states, our findings demonstrate that the relative importance between sophistication and information changes depending on the the influencing power of the adversary. In particular, sophistication far outweighs informational awareness with regards to degrading system-level damage when the adversary's influence power is relatively weak. However, the opposite is true when an adversary's influence power is more substantial.

Figures (4)

• Figure 1: This figure highlights the interplay between an adversary's informational awareness (informed vs uninformed), strategic sophistication (static vs dynamic), budget, and the minimum efficiency it can induce on the system. The green and red lines characterize minimum efficiencies induced from four different adversarial models on ring networks of sufficiently large size, as a function of fractional budget $\gamma \in [0,1]$ (the fraction of agents the adversary can influence). At $\gamma = 0$, neither adversarial model can induce any damage on efficiency (black circle). For low budgets (i.e. $\gamma < 0.5$), strategic sophistication is more valuable than having system-level information about the network. The converse holds true for higher budgets (i.e. $\gamma > 0.5$): system-level information is more valuable than the ability to implement dynamic policies.
• Figure 2: An illustration of the constructed influence set given by \ref{['eq:W1']}, \ref{['eq:W2']} to stabilize an isolated $y$ segment. The $y$ adversaries belonging to $S_y$ are depicted as the smaller circles attaching to agents (larger circles) in the network. In this example, $\alpha = \frac{1}{4}$ and $|L_y| = 9$. The necessary and sufficient number of adversaries to stabilize the segment is 5.
• Figure 3: We illustrate here both defensive and offensive strategies in an aggressive policy. (Left) Defensive $y$ strategies are applied the first and third segments from the left. The fourth agent from the left transitioning from $x$ to $y$ at time $t+1$ activates a defensive $x$ strategy in the second. No adversaries are deployed to $x$ segments until only two neighboring agents playing $x$ remain. (Right) A defensive and offensive $y$ strategy are applied simultaneously to the first segment. The offensive strategy attaches a $y$ adversary to the $x$ agent that has a $y$ neighbor.
• Figure 4: (Left) The graph formed by connecting recurrent classes through the minimum resistance edge leaving each class, is composed of disconnected subgraphs $\Sigma_u$ with the structure illustrated above (Lemma \ref{['lem:subgraphs']}). All paths in $\Sigma_u$ lead to a "head node" $u$ that has minimal disagreement among all other nodes. Disagreement decreases along any path. (Right) The resistance tree rooted at $a$ of minimum stochastic potential. A minimum resistance edge leaving each subgraph exits via the head node (Lemma \ref{['lem:head_leave']}).

Theorems & Definitions (27)

• Definition 1: Log-Linear Learning
• Theorem 2.1
• Theorem 2.2
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Definition 2
• Lemma 5.1: from Young1993
• Definition 3