Smooth Signed Graph Signals

• Smooth signed graph signals are functions on network nodes that model paired similarity and dissimilarity, capturing both positive and negative interactions.
• They extend classical graph signal processing by using the net Laplacian to enforce similarity on positive edges and contrast on negative ones.
• Efficient ADMM-based optimization and empirical validations demonstrate robust recovery of signed network structures in diverse applications.

Smooth signed graph signals are functions defined on the nodes of a signed network—where edges encode both positive (similarity) and negative (dissimilarity or antagonism) interactions—exhibiting regularity or structured variation as quantified by a generalized notion of smoothness. This concept extends classical graph signal processing to settings where relationships are not only cooperative but may also be adversarial or inhibitory, providing a principled framework for learning, analysis, and inference across a broad array of domains, including biology, social science, and machine learning.

1. Fundamental Concepts: Smoothness in Signed Graphs

In classical unsigned graphs, signal smoothness is commonly assessed using the Laplacian quadratic form, measuring low variation between strongly connected nodes. For signed graphs, smoothness must simultaneously reflect similarity along positive edges and dissimilarity (contrast) along negative edges.

Let $G = (V, E, W, S)$ represent a signed graph with node set $V$, edge set $E$, weight matrix $W \in \mathbb{R}^{n \times n}$, and a sign matrix $S \in \{-1, +1\}^{n \times n}$ indicating positive or negative edges. A graph signal $x \in \mathbb{R}^n$ is defined as smooth if, for positive-weighted edges, $x_i \approx x_j$ and for negative-weighted edges, $x_i \approx -x_j$ (or at least encourages strong contrast). This property can be captured via the net Laplacian: $L_n = D - A,$ where $A$ is the signed adjacency matrix and $D$ is the diagonal degree matrix, $D_{ii} = \sum_{j} |A_{ij}|$. The quadratic form

$$x^\top L_n x = \frac{1}{2}\sum_{i \neq j} s_{ij}w_{ij} (x_i - x_j)^2$$

biases $x$ to be similar across positive edges ($s_{ij}=+1$) and dissimilar across negative edges ($s_{ij}=-1$) (Karaaslanli et al., 13 Jul 2025).

2. Mathematical Framework and Model Formulation

The most direct approach to quantifying and leveraging smoothness in signed graphs uses the above quadratic form as a signal prior. In practical inference or learning contexts, smooth signed graph signals are often modeled as outputs of low-pass signed graph filters using the net Laplacian as the shift operator: $x = h(L_n)z,$ where $z$ is a random vector (e.g., i.i.d. Gaussian) and $h(\cdot)$ is a graph filter that attenuates high-frequency components. Since $L_n$ can be indefinite for general signed graphs, a scalar shift is applied: $\tau = L_n + \gamma I,$ with $\gamma$ large enough to ensure $\tau$ is positive semidefinite, allowing use of the quadratic smoothness penalty $x^\top \tau x$ for optimization and learning (Karaaslanli et al., 13 Jul 2025).

3. Graph Learning from Smooth Signed Signals

Given observations $X = [x^{(1)}, \dots, x^{(m)}]$ assumed to be smooth signed graph signals, the central learning problem is to infer the underlying signed graph (i.e., the net Laplacian) that best explains the data. This is formulated as a constrained minimization: $$\min_{L^+, L^-} \ \operatorname{tr}(X^\top [L^+ - L^-] X) + \alpha_1 \|L^+\|_F^2 + \alpha_2 \|L^-\|_F^2$$ subject to:

• Laplacian structure: $L^+$ and $L^-$ are valid Laplacians (symmetric, off-diagonal non-positivity, zero row sums, and fixed trace $\operatorname{tr}(L^+)=2n$, $\operatorname{tr}(L^-)=2n$),
• Complementarity: $L^+_{ij}$ and $L^-_{ij}$ cannot both be nonzero (at most one type of connection per edge),
• Non-positivity: Off-diagonal entries are non-positive.

The net Laplacian is $L_n = L^+ - L^-$, encoding both attraction and repulsion (Karaaslanli et al., 13 Jul 2025).

Regularization via Frobenius norms ensures control over graph density, and the complementarity constraint guarantees unique edge labeling. This methodology is specifically tailored to signed networks, setting it apart from general (unsigned) graph learning frameworks.

4. Optimization and Algorithmic Approach

The described learning problem is nonconvex due to the complementarity constraint but can be efficiently approached via an Alternating Direction Method of Multipliers (ADMM) framework. The variables $L^+$ and $L^-$ are vectorized, and slack variables introduced, allowing the augmented Lagrangian to be minimized with block-wise updates:

• $\theta^+, \theta^-$: Variables for the strictly upper-triangular parts of $L^+, L^-$,
• $z^+, z^-$: Slack variables enforcing equality with $\theta^+, \theta^-$,
• Updates are performed by minimizing over $z^+, z^-$ with respect to all constraints (including complementarity) using elementwise projections.

To improve scalability, a fast version (fastSGL) restricts edge consideration to a candidate set (e.g., $k$ nearest-neighbors per node), reducing per-iteration complexity from $O(n^2)$ to $O(nk)$ while maintaining empirical accuracy (Karaaslanli et al., 13 Jul 2025).

5. Theoretical Guarantees and Error Analysis

The methodology provides theoretical results on algorithmic convergence and estimation error. Under appropriate conditions (choice of penalty parameter, sub-Gaussian signal assumptions, bounded variable norms), it is shown that the ADMM iterates converge to a stationary point of the nonconvex objective. The estimation error of the learned net Laplacian has the bound: $$\|\hat{L} - L_0\|_F \leq \frac{C n}{\alpha_m \sqrt{m} + (\frac{1}{\alpha_m})(\|\Sigma_0\|_F + 2\hat{\alpha}_m M)}$$ where $\alpha_m$ and $\hat{\alpha}_m$ are regularization terms determined by algorithmic parameters and data, $m$ is sample size, $n$ is the number of nodes, and $\Sigma_0$ is the true signal covariance (Karaaslanli et al., 13 Jul 2025).

This provides insight into the required sample complexity and regularization to obtain accurate signed graph estimation as a function of network size, signal smoothness, and edge structure.

6. Empirical Validation and Applications

The proposed framework is validated on synthetic and real-world data, including:

• Signed variant random graphs (Erdős–Rényi, Barabási–Albert, random geometric),
• Synthetic signals filtered by low-pass signed graph filters (Gaussian, Heat, Tikhonov),
• Gene regulatory network (GRN) inference using simulated single-cell RNA-seq data.

Performance is measured using macro F1 scores (treating edge inference as a three-way classification: positive, negative, or no edge). The method (SGL) consistently outperforms signed Laplacian learning (SLL) and proximal ADMM (pADMM) in recovering signed graph topologies, achieving higher F1 scores and more accurate recovery of both positive and negative edges.

In GRN inference, where edge negativity denotes inhibition, the net Laplacian approach avoids problematic signal flipping found in SLL—yielding biologically meaningful signed network reconstructions when observed gene expressions are nonnegative (Karaaslanli et al., 13 Jul 2025).

The fastSGL variant demonstrates substantially reduced runtime scaling (linear in $n$ for fixed $k$), with robust empirical accuracy.

7. Significance and Broader Impact

The formalism of smooth signed graph signals and the corresponding graph learning algorithms provide foundational tools for domains where duality of relationship type (cooperative vs. antagonistic, activation vs. repression) is essential for modeling and inference. Applications include:

• Social networks with trust-distrust relationships,
• Biological regulatory networks (activator-inhibitor dynamics),
• Signed consensus and opinion dynamics models,
• Any system where signals or behaviors propagate with mixed reinforcement (attraction/repulsion).

The combination of the net Laplacian-based smoothness model, complementarity-constrained nonconvex optimization, efficient ADMM-based solvers, and strong empirical performance establishes a rigorous pathway for structure discovery in signed networks. This approach marks a critical advancement over unsigned graph learning, directly addressing the complexities and subtleties inherent to antagonistic and heterogeneous-interaction systems (Karaaslanli et al., 13 Jul 2025).