[Paper Review] Modeling wireless network routing using sheaves

Michael Robinson|arXiv (Cornell University)|Jun 28, 2016

Data Management and Algorithms1 references3 citations

TL;DR

This paper introduces a sheaf-theoretic framework to model routing and interference in single-channel wireless networks using CSMA/CD protocols. By constructing an activation sheaf on a time-dependent link complex, it characterizes non-interfering transmission schedules as global sections, enabling the derivation of an upper bound on network throughput based on data payload sheaves and transmission patterns.

ABSTRACT

This article explains how to construct a sheaf model for passing traffic through a wireless network with a single channel carrier sense multiple access/collision detection (CSMA/CD) media access model.

Motivation & Objective

• To model wireless network routing and interference using algebraic topology, specifically sheaf theory.
• To formalize non-interfering transmission schedules as global sections of an activation sheaf on a simplicial complex.
• To derive an upper bound on network throughput based on transmission patterns and data payload sheaves.
• To provide a mathematical framework for analyzing time-varying network activity and data flow constraints.
• To enable protocol-level modeling through layered sheaf constructions and spectral sequences for future protocol design.

Proposed method

• Construct a link complex $ L $ as the clique complex of a graph where nodes are connected if their mutual signal strength exceeds a threshold $ T $.
• Define an activation sheaf $ \mathcal{A} $ that assigns to each cell the set of nodes whose transmissions are decodable or $ \perp $ if none.
• Use restriction maps between cells to model transmission dependencies and interference constraints.
• Define global sections of the activation sheaf as valid, non-interfering transmission schedules across the network.
• Introduce a time-dependent data payload sheaf $ \mathcal{D} $ and project it via surjections to obtain an active payload sheaf $ \mathcal{P} $ restricted to valid transmission patterns.
• Derive an upper bound on network throughput as the dimension of the space of global sections of $ \mathcal{P} $, assuming vector space payloads.

Experimental results

Research questions

• RQ1 How can wireless network routing and interference be modeled using sheaf theory to capture non-overlapping transmission constraints?
• RQ2 What is the mathematical characterization of a valid, non-interfering transmission schedule in a wireless network with CSMA/CD?
• RQ3 How can the throughput of a time-varying wireless network be bounded using sheaf-theoretic constructions?
• RQ4 In what way do data payload sheaves and activation sheaves interact to model end-to-end data flow?
• RQ5 Can layered sheaf constructions and spectral sequences be used to model increasing protocol complexity in wireless networks?

Key findings

• Global sections of the activation sheaf correspond exactly to non-interfering transmission schedules, where only one node transmits per active region.
• The active region of a transmitting node is a connected, closed subcomplex containing the node, ensuring consistent interference management.
• The dimension of the space of global sections of the projected payload sheaf $ \mathcal{P} $ provides an upper bound on network throughput for a given transmission pattern.
• Data flow through the network is modeled by constructing a global section of $ \mathcal{P} $ that tracks a packet’s path across nodes and timeslices.
• The model ensures that interference is captured via the sheaf structure, with incompatible transmissions blocked by the restriction maps.
• The framework allows for decoupling of timeslices in data payload modeling, though actual data dependencies are preserved through sheaf morphisms.