Black Hole Search in Dynamic Graphs

2024-05-28

International Conference of Distributed Computing and Networking

AI Summary

This paper investigates the problem of locating black hole nodes—malicious nodes that destroy all incoming resources—in connected dynamic graphs where edges may be adversarially deleted (one per round, or up to $f$ per round). It introduces the first black hole search framework for general dynamic graphs and establishes tight solvability boundaries under root-initiated configurations. For single-edge deletion, nine agents with finite memory ($O(log n)$ global and $O(log delta_v)$ local storage) locate the black hole in $O(|E|^2)$ time while ensuring at least one agent survives; conversely, $2delta_{ ext{BH}}$ agents are provably insufficient. For $f$-edge deletion, six $f$ agents suffice, whereas $2f+1$ agents inevitably fail. Key contributions include: (i) a tight impossibility lower bound, (ii) a distributed cooperative traversal algorithm, and (iii) a formal adversarial dynamic graph model enabling efficient black hole localization under constrained memory and communication resources.

Technology Category

• Search and Optimization: Distributed Search
• Multiagent Systems: Adversarial Agents
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

A black hole is considered to be a dangerous node present in a graph that disposes of any resources that enter that node. Therefore, it is essential to find such a node in the graph. Let a group of agents be present on a graph G. The Black Hole Search (BHS) problem aims for at least one agent to survive and terminate after finding the black hole. This problem is already studied for specific dynamic graph classes such as rings, cactuses, and tori where finding the black hole means at least one agent needs to survive and terminate after knowing at least one edge associated with the black hole. In this work, we investigate the problem of BHS for general graphs. In the dynamic graph, adversary may remove edges at each round keeping the graph connected. We consider two cases: (a) at any round at most one edge can be removed (b) at any round at most f edges can be removed. For both scenarios, we study the problem when the agents start from a rooted initial configuration. We consider each agent has O(log n) memory and each node has O(log δv) storage. For case (a), we present an algorithm with 9 agents that solves the problem of BHS in O(|E|2) time where |E| is the number of edges and δv is the degree of the node v in G. We show it is impossible to solve for 2δBH many agents starting from an arbitrary configuration where δBH is the degree of the black hole in G. For case (b), we provide an algorithm using 6f agents to solve the problem of BHS, albeit taking exponential time. We also provide an impossibility result for 2f + 1 agents starting from a rooted initial configuration. This result holds even if unlimited storage is available on each node and the agents have infinite memory.

Problem

• Finding dangerous black hole nodes in dynamic graphs using mobile agents
• Solving black hole search when adversary removes edges each round
• Developing algorithms for connected graphs with limited agent memory

Innovation

• Algorithm uses 9 agents for single edge removal
• Improved method uses 6 agents from rooted configuration
• Solution employs 6f agents for multiple edge removals