By extending conventional graph-based models to include higher-order interactions among groups of agents, the Hodge Laplacian framework can distinguish gradient, cyclic, and harmonic components of large-scale system dynamics. The project will develop Hodge-Laplacian consensus algorithms, establish convergence and robustness conditions, and evaluate how higher-order topology affects information diffusion, synchronization, and resilience to disturbances.
- Task 1: Higher-order network modelling
- Represent the network as a simplicial complex and construct the associated incidence matrices and Hodge Laplacians for node-, edge-, and higher-order states.
- Task 2: Hodge consensus algorithm
- Develop a consensus algorithm based on the Hodge Laplacian and characterize its equilibria, convergence rate, and dependence on the topology of the simplicial complex.
- Task 3: Topological decomposition of the dynamics
- Use the Hodge decomposition to separate gradient, curl, and harmonic components of the system state and determine which components can be eliminated through local interactions.
- Task 4: Robustness and resilience analysis
- Study the effects of disturbances, measurement noise, communication failures, and changes in network topology. Establish conditions for stability, convergence, and bounded disagreement.
- Task 5: Numerical validation
- Validate the proposed methods on representative applications such as distributed sensing, flow networks, or multi-agent coordination, and compare them with conventional graph-Laplacian consensus algorithms.
- Task 6: Write the thesis
- Introduction, problem statement, literature review, methods, discussion, results.
Timeline
Master thesis projects can be taken in 10-20-30 credit formats (i.e., 10 credits in Semester 2, 20 credits in Semester 3, and 30 credits in Semester 4) or 30-30 credit (i.e., 30 credits in Semester 3 and 30 credits in Semester 4) formats. In my experience, the latter (30-30 credits) works better for students as you can devote your full attention to the thesis without coursework as a distraction. Nonetheless, here are proposed timelines for each system.
10-20-30 Credits
Semester 2
Begin and complete Task 1. Begin Tasks 2, 3, 4, and 5. Document results into a semester essay, along with the problem statement and literature review from Task 6.
Semester 3
Continue Tasks 2, 3, 4, and 5. Document the results in the Methods section of Task 6.
Semester 4
Complete Tasks 2, 3, 4, and 5. Write the results and critically analyze them in the discussion of Task 6. Complete Task 6.
30-30 Credits
Semester 3
Begin and complete Task 1. Begin Tasks 2, and 3. Document results into a semester essay, along with the problem statement and literature review from Task 7. Begin Tasks 4 and 5. Document the results in the Methods section of Task 6.
Semester 4
Complete Tasks 2, 3, 4, and 5. Write the results and critically analyze them in the discussion of Task 6. Complete Task 6.