Spreading and Structural Balance on Signed Networks

TL;DR

This work studies dynamics on weighted signed networks by classifying graphs into structurally balanced, structurally antibalanced, and strictly unbalanced categories and linking each class to the spectrum of the signed adjacency matrix $\mathbf{W}$ and its unsigned counterpart $\bar{\mathbf{W}}$. A key result is that the spectral radius contracts, $\rho(\mathbf{W})<\rho(\bar{\mathbf{W}})$, if and only if the graph is strictly unbalanced, explaining diminished long-run activity in that regime. The paper then analyzes two dynamical frameworks—the linear adjacency dynamics with coupling matrix $\mathbf{W}$ and the extended linear threshold (ELT) model—and shows consistent patterns across balance classes, including fixed-point behavior in balanced networks, alternating dynamics in antibalanced networks, and vanishing activity in strictly unbalanced networks; it also introduces two perturbation-based measures $d_b(G)$ and $d_a(G)$ to quantify proximity to balance and antibalance. Numerical experiments on synthetic signed stochastic block models and a real Highland tribes network corroborate the theoretical predictions and illustrate how balance structure shapes diffusion and contagion-like processes on signed graphs.

Abstract

Two competing types of interactions often play an important part in shaping system behavior, such as activatory or inhibitory functions in biological systems. Hence, signed networks, where each connection can be either positive or negative, have become popular models over recent years. However, the primary focus of the literature is on the unweighted and structurally unbalanced ones, where all cycles have an even number of negative edges. Hence here, we first introduce a classification of signed networks into balanced, antibalanced or strictly unbalanced ones, and then characterize each type of signed networks in terms of the spectral properties of the signed weighted adjacency matrix. In particular, we show that the spectral radius of the matrix with signs is smaller than that without if and only if the signed network is strictly unbalanced. These properties are important to understand the dynamics on signed networks, both linear and nonlinear ones. Specifically, we find consistent patterns in a linear and a nonlinear dynamics theoretically, depending on their type of balance. We also propose two measures to further characterize strictly unbalanced networks, motivated by perturbation theory. Finally, we numerically verify these properties through experiments on both synthetic and real networks.

Figures (9)

• Figure 1: Example of signed regular lattices of degree $4$ with different structurally balanced configurations, where positive edges are in black, negative edges are dashed in orange, and the whole neighbourhood of node $v_0$ is in different colour(s) from the others (in grey), with the ones that are positively activated in red and the others that are negatively activated in green.
• Figure 2: Example of signed regular lattices of degree $4$ with different structurally antibalanced configurations, where positive edges are in black, negative edges are dashed in orange, and the whole neighbourhood of node $v_0$ are in different colour(s) from the others (in grey), with the ones that are positively activated in red and the others that are negatively activated in green.
• Figure 3: Signed networks from SSBM that are balanced ($\eta = 0$, upper left), close to being balanced ($\eta = 0.05$, upper right) with $d_b(G) = 0.077,\, d_a(G) = 0.503$ and $3$ edges disturbing the balanced structure, antibalanced ($\eta=1$, bottom left), and close to being antibalanced ($\eta = 0.95$, bottom right) with $d_b(G) = 0.525,\, d_a(G) = 0.047$ and $2$ edges disturbing the antibalanced structure, where the node colour indicates the bipartitions of the relevant balanced or antibalanced structures, and the edge colour indicates the sign (black: positive; orange: negative).
• Figure 4: Evolution of the state values from the linear adjacency dynamics on the networks in Figure \ref{['fig:exp-ssbms']} that are balanced ($\eta = 0$, upper left), close to being balanced ($\eta = 0.05$, upper right), antibalanced ($\eta=1$, bottom left), and close to being antibalanced ($\eta = 0.95$, bottom right), where, in the same row, the results on the right are expected to be close to those on the left, to some extent, since the underlying networks are close to each other by our proposed measures.
• Figure 5: Evolution of the state values from the signed random walks on the networks in Figure \ref{['fig:exp-ssbms']} that are balanced ($\eta = 0$, upper left), close to being balanced ($\eta = 0.05$, upper right), antibalanced ($\eta=1$, bottom left), and close to being antibalanced ($\eta = 0.95$, bottom right), where the difference between the results in the same row is expected to be larger in this case.

Theorems & Definitions (37)

• Theorem 2.1: structure theorem for balance harary_1953_balance
• Theorem 2.2: structure theorem for antibalance harary_1957_duality
• Definition 3.1: strict unbalance
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Theorem 3.4: spectral theorem of balance and antibalance
• proof
• Remark 3.5