[Paper Review] A Multi-Urn Model for Network Search

Christopher E. Marks, Tauhid Zaman|arXiv (Cornell University)|Aug 24, 2016

Advanced Bandit Algorithms Research10 references3 citations

TL;DR

This paper proposes a multi-urn model for efficiently searching for a target vertex in a social network by sequentially querying neighbors of known vertices. Using dynamic programming, it proves that a block policy—fully examining one known vertex’s neighbors before switching—is always optimal, regardless of correlation in connection probabilities, with explicit optimality conditions for independent and single-connection cases.

ABSTRACT

We consider the problem of finding a specific target individual hiding in a social network. We propose a method for network vertex search that looks for the target vertex by sequentially examining the neighbors of a set of "known" vertices to which the target vertex may have connected. The objective is to find the target vertex as quickly as possible from amongst the neighbors of the known vertices. We model this type of search as successively drawing marbles from a set of urns, where each urn represents one of the known vertices and the marbles in each urn represent the respective vertex's neighbors. Using a dynamic programming approach, we analyze this model and show that there is always an optimal "block" policy, in which all of the neighbors of a known vertex are examined before moving on to another vertex. Surprisingly, this block policy result holds for arbitrary dependencies in the connection probabilities of the target vertex and known vertices. Furthermore, we give precise characterizations of the optimal block policy in two specific cases: (1) when the connections between the known vertices and the target vertex are independent, and (2) when the target vertex is connected to at most one known vertex. Finally, we provide some general monotonicity properties and discuss the relevance of our findings in social media and other applications.

Motivation & Objective

• To model network vertex search as a sequential multi-urn problem where each urn represents a known vertex and marbles its neighbors.
• To minimize the expected number of queries to locate a target vertex (represented as a red marble) among neighbors of known vertices.
• To determine optimal search policies under general probabilistic dependencies between target and known vertices.
• To characterize optimal strategies in two specific cases: independent connections and at-most-one-connection scenarios.
• To provide insights applicable to social media, law enforcement, and email forensics where search costs are non-trivial.

Proposed method

• Models each known vertex as an urn containing marbles representing its neighbors, with one red marble indicating the target.
• Applies dynamic programming to the multi-urn search problem to derive optimal search policies.
• Uses a recursive value function to evaluate expected search cost from any state of remaining marbles.
• Derives necessary and sufficient conditions for block policy optimality using inequalities involving marginal gains in success probability per query.
• Analyzes the impact of correlation in connection probabilities on policy structure, showing block policies remain optimal.
• Considers extensions involving non-uniform query costs, imperfect information, and non-random drawing order.

Experimental results

Research questions

• RQ1 Is there a universally optimal search policy for the multi-urn network search model, regardless of dependencies in connection probabilities?
• RQ2 Under what conditions is a block policy—fully querying one known vertex before moving to another—optimal?
• RQ3 How does the optimal policy change when connections between the target and known vertices are independent versus when the target connects to at most one known vertex?
• RQ4 Can a greedy policy be suboptimal in the independent-urn case, and if so, by how much?
• RQ5 How do non-uniform query costs or non-random drawing order affect the structure of optimal search policies?

Key findings

• A block policy is always optimal for the multi-urn network search model, regardless of correlation in the connection probabilities between the target and known vertices.
• In the case of independent urns, the optimal policy is determined by a threshold condition based on the ratio of success probability to query cost per marble.
• For the single-red-marble case, where the target connects to at most one known vertex, the optimal policy is to prioritize urns with the highest connection probability.
• The paper shows that a purely greedy policy is not optimal in the independent-urn case, demonstrating that local gains do not guarantee global optimality.
• The model remains robust under various generalizations, including non-uniform query costs and imperfect information, suggesting broader applicability to real-world search problems.
• The dynamic programming framework enables precise characterization of optimal policies and provides a foundation for analyzing performance bounds of heuristic strategies.