[Paper Review] An anti-incursion algorithm for unknown probabilistic adversaries on connected graphs

Jesse Geneson|arXiv (Cornell University)|Jan 4, 2017

Advanced Malware Detection Techniques1 references3 citations

TL;DR

This paper proposes a modified depth-first search algorithm for a cop pursuing an unknown probabilistic gambler on connected graphs, where the gambler's movement distribution is unknown. By strategically pausing at randomly selected leaves for an extra turn, the cop reduces the expected capture time to approximately 1.95n, improving upon the prior upper bound of 1.97n for unknown adversaries.

ABSTRACT

A gambler moves on the vertices $1, \ldots, n$ of a graph using the probability distribution $p_{1}, \ldots, p_{n}$. A cop pursues the gambler on the graph, only being able to move between adjacent vertices. What is the expected number of moves that the gambler can make until the cop catches them? Komarov and Winkler proved an upper bound of approximately $1.97n$ for the expected capture time on any connected $n$-vertex graph when the cop does not know the gambler's distribution. We improve this upper bound to approximately $1.95n$ by modifying the cop's pursuit algorithm.

Motivation & Objective

• To reduce the expected capture time for a cop pursuing an unknown probabilistic gambler on connected graphs.
• To improve upon the prior upper bound of approximately 1.97n established by Komarov and Winkler.
• To investigate whether the conjectured bound of 1.5n (3n/2) is achievable, particularly in worst-case graphs like the star.
• To analyze the impact of strategic pauses at selected leaves on the evasion probability and capture efficiency.
• To refine the upper bound through probabilistic analysis of evasion under constrained visitation patterns.

Proposed method

• The cop performs a depth-first search (DFS) on a spanning tree T of the connected graph G.
• The cop selects a random subset U of ⌈0.72912n⌉ vertices and pauses for an extra turn at each leaf in U.
• The direction of the DFS (forward or backward) is chosen uniformly at random at the start of each cycle.
• The expected number of turns per DFS is bounded at ≤1.36456n due to the random direction choice.
• Evasion probability is analyzed in two cases: when two high-probability vertices are visited multiple times, and otherwise.
• A reduction technique using truncated searches S and S′ is used to upper-bound evasion probability, leading to an exponential decay bound.

Experimental results

Research questions

• RQ1 Can the expected capture time for an unknown probabilistic adversary on a connected n-vertex graph be improved below the prior upper bound of 1.97n?
• RQ2 What is the optimal number of strategic pauses (extra turns at leaves) that minimizes the expected capture time?
• RQ3 Does the star graph remain the worst-case scenario for the expected capture time under the unknown gambler model?
• RQ4 How does the distribution of vertex visitation frequencies affect the evasion probability in a DFS-based pursuit strategy?
• RQ5 Can tighter analytical bounds on evasion probability be derived using probabilistic inequalities and symmetry arguments?

Key findings

• The expected capture time is bounded above by approximately 1.95335n, improving the prior upper bound of 1.97n.
• The evasion probability in a single DFS is at most 0.17745, derived from the bound (1−1/n)^1.72912n ≤ e^−1.72912.
• The expected number of DFS cycles until capture is less than 1.21574, based on the evasion probability.
• The expected number of turns in unsuccessful DFS cycles is bounded at <0.58879n, and in the successful cycle at ≤1.36456n.
• The constant 0.72912 was selected to minimize the expression x/(1−e^(1−x)) − x/2 at x=2.72912, optimizing the bound.
• The upper bound is not tight, as the known best bounds for specific graphs (e.g., cycle C_n) range from 1.082n to 1.953n.