Constructing directed networks with a desired minimum balanced coloring

TL;DR

The paper addresses the problem of constructing directed networks with prescribed symmetry structures to enable systematic studies of symmetry-driven dynamics. It introduces a quotient-network expansion approach that, given a quotient network $\mathcal{Q}$ and target cluster sizes $n_1,\dots,n_P$, yields expanded adjacency $A$ subject to feasibility constraints. Key contributions include precise feasibility inequalities $n_l \ge \max_{k\neq l} A^\mathcal{Q}_{kl}$ and $n_l > A^\mathcal{Q}_{ll}$, a minimal-expansion rule $n_l=\max(\max_{k\neq l} A^\mathcal{Q}_{kl}, A^\mathcal{Q}_{ll}+1)$, and two implementation modes (Case I and Case II) with demonstrated scalability to tens of thousands of nodes. The work provides open-source code and a framework for generating realistic, symmetry-aware directed networks suitable for studying cluster synchronization and related dynamics.

Abstract

This work introduces a systematic algorithm for generating directed networks with prescribed symmetries by constructing expansions from a given quotient network. The method enables researchers to synthesize realistic network models with controllable symmetry structure, facilitating studies of symmetry-driven dynamics such as cluster synchronization in biological, social, and technological systems.

Figures (5)

• Figure 1: Example of a directed network and of the associated quotient network. Different colors are used to label nodes in different clusters.
• Figure 2: Example diagram of input tree representations of an expanded network which form clusters. The clusters enable the transformation of the expanded network into its quotient network.
• Figure 3: Example network for the quotient network expansion algorithm: (\ref{['Ex. Network Expansion']}a) Simple $P=3$-cluster quotient network example. Some zero-weight edges are added for clarity as dashed lines. (\ref{['Ex. Network Expansion']}b) The adjacency matrix of the quotient network contained in panel \ref{['Ex. Network Expansion']}a
• Figure 4: Case I and II example of network expansion utilizing input $\mathcal{R}$: (\ref{['Minimal Expansion Example']}a) A basic quotient network example in which $\mathcal{R}$ is composed of the values given in the corresponding paragraph of Section \ref{['Results']}. Cluster one is highlighted in red, then expanded into two red nodes as shown in panel \ref{['Minimal Expansion Example']}b. The same style follows for the remaining two clusters. (\ref{['Minimal Expansion Example']}b) A random minimal network expansion generated from $\mathcal{R}$ using the proposed algorithm. The expanded network is generated under Case I, leading to one of 72 possible minimal network expansions. (\ref{['Minimal Expansion Example']}c) A Case II example of random network expansion of $\mathcal{R}$. Given a user-set number of nodes per cluster where $n=10,10,10$ we generate a 30-node expanded network. The color of each edge matches that of its source node, e.g., an edge generated by a node in red cluster will also be colored red.
• Figure 5: A Case II $P=10$-cluster high-load network example of over 10,000 nodes, generated in under 20 seconds. The color of each edge matches that of its source node,

Theorems & Definitions (5)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5