Distributed Saddle-Point Dynamics in Multilayer Networks

TL;DR

This work extends distributed optimization from simple graphs and multiplex networks to general multilayer networks using a tensor-Laplacian framework. By formulating a tensor-based augmented Lagrangian and deriving saddle-point dynamics, the authors prove convergence to the global optimizer under standard convexity assumptions and Lipschitz gradients, while revealing how interlayer diffusion shapes convergence speed. The approach unifies diffusion, consensus, and convex optimization in a multilayer setting, and is validated through numerical experiments that reveal phase-transition–like behavior in consensus times. The framework paves the way for scalable, decentralized optimization in complex interdependent systems such as power grids, brain networks, and multi-energy infrastructures, and suggests directions for extending to hypergraphs and higher-order topologies.

Abstract

Multilayer networks provide a more advanced and comprehensive framework for modeling real-world systems compared to traditional single-layer and multiplex networks. Unlike single-layer models, multilayer networks have multiple interacting layers, each with unique topological features. In this paper, we generalize previously developed results for distributed optimization in multiplex networks to the more general case of multilayer networks by employing a tensor formalism to represent multilayer networks and their tensor-Laplacian diffusion dynamics. Although multiplex networks are a special case of multilayer networks, where each layer has the same number of replica nodes connected one-to-one, this generalized framework removes the need for replica nodes, allowing variability in both topology and number of nodes across layers. This approach provides a fully generalized structure for distributed optimization in multilayer networks and enables more complex interlayer connections. We derive the multilayer combinatorial Laplacian tensor and extend the distributed gradient descent algorithm. We provide a theoretical analysis of the convergence of algorithms. Numerical examples validate our approach, and we explore the impact of heterogeneous layer topologies and complex interlayer dynamics on consensus time, underscoring their implications for real-world multilayer systems.

Figures (9)

• Figure 1: Example of a multiplex network with two layers. Each layer has replica nodes, but a different topology. The connections between layers are one-by-one. On the side, we depict the Laplacians, $L_{1}$, $L_{2}$, the supra-Laplacian $\mathcal{L}$, and the optimal consensus dynamics for this system. The vector $y$ contains the state of each node within the two layers, while $\lambda$ is the vector of Lagrange multipliers. Further developments can be found in rodriguezcamargo2023consensusRODRIGUEZCAMARGO20231217.
• Figure 2: Example of a multilayer network with three layers. Each layer has different number of nodes and different topology. The connections between layers may not be one-by-one.
• Figure 3: Multilayer network with $M=2$ layers and $N(L_{1})=3$ nodes in layer $L_{1}$ and $N(L_{2})=5$ nodes in layer $L_{2}$. The node 1 of the layer $L_{1}$ has a connection with node 2 of layer $L_{2}$ and the node 3 of the layer $L_{1}$ has a connection with node 4 of layer $L_{2}$.
• Figure 4: Consensus dynamics for the distributed optimization problem for multilayer network with $M=2$ layers and $N(L_{1})=3$ nodes in layer $L_{1}$ and $N(L_{2})=3$ nodes in layer $L_{2}$. The interlayer diffusion constant was set to $D_{x}^{1 \rightarrow 2 } = 0.5$.
• Figure 5: Multilayer network with $M=4$ layers and $N(L_{1})=3$, $N(L_{2})=4$, $N(L_{3})=5$, $N(L_{6})=6$.

Theorems & Definitions (2)

• Lemma 1
• Theorem 1