Signed Weighted Graphs

• Signed weighted graphs are graphs where each edge carries a sign (+1 or –1) and a real-valued weight, capturing both the nature and strength of interactions.
• Spectral analyses using signed adjacency and Laplacian matrices uncover unique balance properties and inertia dynamics critical for network stability.
• Applications span social and biological networks to graph learning, with advanced optimization and homomorphism methods enhancing modeling and inference.

A signed weighted graph is a generalization of a graph in which each edge is assigned both a sign (typically +1 or –1) and a real-valued weight. This structure models not only the existence of relationships (edges) but explicitly captures both the polarity (friend/foe, similarity/dissimilarity, attraction/repulsion) and the strength of interactions between nodes. Signed weighted graphs are foundational in a range of scientific domains, including the paper of social balance, spectral properties, graph learning, network dynamics, and graph embeddings.

1. Structural Definitions and Foundational Properties

Formally, a signed weighted graph is denoted as $G = (V, E, \omega, \sigma)$, where $V$ is the set of vertices, $E$ is the set of edges, $\omega : E \to \mathbb{R}\setminus\{0\}$ is a weight function, and $\sigma : E \to \{-1, +1\}$ is a signature (or sign) function. Often, the edge "weight" is encoded as a single real value, with its sign encapsulating both $\omega$ and $\sigma$, so that $a_{ij} = \omega_{ij}\sigma_{ij}$ for the adjacency or Laplacian matrices.

Signed graphs, introduced by Harary in the 1950s, represent phenomena like friendship/enmity in social networks and activation/inhibition in gene regulatory systems. Signed weights encode not just the qualitative nature (positive/negative) but also the quantitative intensity of relationships. Signed structures often require new concepts: balance (every cycle has a positive sign product), switching (vertex-based local sign-flip operations), forbidding subgraphs, and specialized matrix formulations.

The algebraic and combinatorial properties of signed weighted graphs differ fundamentally from those of unsigned or unsigned weighted graphs due to the signs’ influence on spectral characteristics, balance, and network consensus dynamics (Ramezani, 2015, Akbari et al., 2017, Monfared et al., 2019).

2. Spectral and Matrix Theory

Signed Adjacency and Laplacian

The (signed) weighted adjacency matrix $A_{s}$ is defined as

$$A_s(i, j) = \begin{cases} \omega_{ij} \sigma_{ij} & \text{if } (i, j) \in E \ 0 & \text{otherwise} \end{cases}$$

Spectral properties—especially the spectrum of $A_s$ and various Laplacians—are used to characterize combinatorial and dynamical features:

• The signed Laplacian for a signed weighted graph is $L = D - A_s$, where $D$ is the diagonal degree matrix (with $D_{ii} = \sum_j |a_{ij}|$ or variants); the spectrum of $L$ is not necessarily positive semidefinite, even for connected graphs, due to negative edge contributions (Monfared et al., 2019, Karaaslanli et al., 13 Jul 2025).
• The net Laplacian $L_n = D_n - A_s$, with $D_n=\operatorname{diag}(A_s \mathbf{1})$, is used especially in signal processing and learning, as it naturally differentiates attractive and repulsive interactions while enabling smoothness-based inference (Karaaslanli et al., 13 Jul 2025).

Inertia and Perturbation Analysis

The inertia of a signed Laplacian is the triple $(n_+, n_-, n_0)$, counting positive, negative, and zero eigenvalues, respectively. The inertia can change dramatically as weights (and signs) vary. A key combinatorial parameter is the "flexibility" $\tau$ of a signed graph, providing an upper bound on the possible distinct Laplacian inertias as $\binom{\tau+2}{2}$, and tight connections exist between the block structure (maximal subgraphs without a cut vertex) and inertia uniqueness (Monfared et al., 2019). Perturbation theory demonstrates that generic (arbitrarily small) adjustments of weights can ensure simple spectra.

Spectral Characterizations and Orthogonality

Special families of signed weighted graphs have adjacency matrices with just two eigenvalues—often achieved using block constructions and orthogonal signed matrices. These cases link to the construction of Ramanujan graphs via 2-lifts and Hadamard or Conference matrices, highlighting deep interconnections between algebraic graph theory, signal processing, and combinatorics (Ramezani, 2015).

3. Combinatorial Constructions and Homomorphisms

Switching, Balance, and Vector-Valued Generalizations

Switching is a key operation: for $\eta: V \rightarrow \{-1, +1\}$, edge signs are flipped according to $o_{sw}(uv) = o(uv) \cdot \eta(u) \eta(v)$. The classification of balance (all cycles positive) is switching-invariant; this is of central importance in both spectral and combinatorial problems (Ghorbani et al., 2020, K et al., 2022).

Vector-valued switching extends this notion, allowing $\eta: V \to \{-1, 0, 1\}^k$ and using $\mathrm{sgn}(\langle\eta(u), \eta(v)\rangle)$ to flip edge signs. This allows finer quantification of imbalance (balancing dimension and strong balancing dimension) with tight lower and upper bounds, connected to clique numbers and negative inner product set properties (K et al., 2022).

Homomorphisms, Extended Double Covers, and Weighted Extensions

Homomorphism theory for signed weighted graphs requires both structural (adjacency, incidence) and sign-preserving constraints on image mappings. Extensions such as the Extended Double Cover (EDC) construction "double" a signed graph and manipulate its sign geometry to establish optimal homomorphism bounds for large graph classes. The introduction of weighted signed graphs as explicit objects enables algebraic distance-based arguments, leading to new results in chromatic number research and bounding classes of graphs (e.g., planar-complete graphs, $K_4$-minor-free graphs) (Foucaud et al., 2021).

Weighted signed graphs necessitate homomorphisms preserving absolute weight and cyclic sign products. Algebraic distance notions (positive path exists vs. not) are used to encode generalized walk constraints.

4. Algorithms and Learning from Data

Smooth Signal-Based Signed Graph Learning

A central recent development is the learning of signed graphs from observed graph signals (Karaaslanli et al., 13 Jul 2025). Unlike unsigned approaches, which cannot capture antagonistic or dissimilar relations, signed graph learning methods:

• Use net Laplacian-based smoothness: For a signal $x$, smoothness is $$x^\top L_n x = \frac{1}{2}\sum_{i \neq j} \sigma_{ij} \omega_{ij}(x_i - x_j)^2$$.
• Model observed data as outputs of a low-pass graph filter defined by the net Laplacian.
• Pose the signed graph learning problem as the minimization of signal total variation, constrained by Laplacian structure and complementarity constraints ensuring that each edge is assigned only one sign: for each pair $(i, j)$, at most one of $[L^+]_{ij}$, $[L^-]_{ij}$ is nonzero.
• Use nonconvex optimization (e.g., ADMM with slack variables) and a fast version with $O(nk)$ per-iteration cost (substantially lower than the generic $O(n^2)$ for $n$ nodes), making the approach scalable. Theoretical convergence guarantees and non-asymptotic error bounds are provided.

This framework is empirically validated on networks generated by multiple random graph models and real gene regulatory networks, where it outperforms signed Laplacian-based methods—especially in recovering inhibition/activation links from nonnegative data—and provides precise control over the estimated fractions of positive, negative, and absent connections.

Table: Comparison of Signed Graph Learning Methods

Method	Graph Operator	Handles Signed Edges	Handles Weights	Optimization Approach
SGL (Karaaslanli et al., 13 Jul 2025)	Net Laplacian (+ shift)	Yes	Yes	ADMM (fast version O(nk))
SLL	Signed Laplacian	Yes	Yes	Proximal-ADMM (higher complexity)
wsGAT (Grassia et al., 2021)	Attention GNN	Yes	Yes	Neural Attention (MLP, deep learn)

Spectral, Domination, and Circuit Cover Algorithms

Combinatorial algorithms exist for embedding signed graphs in low-dimensional spaces with friend/enemy constraints (Kermarrec et al., 2014), determining full-rank sign assignments (Akbari et al., 2017), enumerating switching isomorphism classes (Sehrawat et al., 2018), and constructing signed circuit covers with tight bounds (e.g., 6-covers for K$_4$-minor-free graphs) (Lu et al., 2023). For instance, embedding algorithms in 1D use chordal decomposition and perfect elimination ordering of the positive subgraph, yielding $O(n^2)$ polynomial time embedding and recognition (Kermarrec et al., 2014).

Open computational questions remain: the embedding problem is NP-complete for general (non-complete) graphs, and efficient algorithms for domination and cover problems in large signed weighted graphs, or under additional constraints, are active research areas.

5. Applications and Specialized Contexts

Social and Biological Network Analysis

Signed weighted graphs are foundational in modeling antagonistic and affiliative relationships in social networks, e.g., trust/distrust, alliances, gene activation/inhibition, or economic competition. Node embeddings, clustering, and recommendation systems benefit from the interpretability and granularity offered by signed weights (Kermarrec et al., 2014) wsGAT:(Grassia et al., 2021).

Gene regulatory networks (GRNs) are naturally modeled as signed weighted graphs, with positive weights indicating activation and negative weights denoting inhibition. The SGL framework (Karaaslanli et al., 13 Jul 2025) demonstrates improved GRN inference from transcriptomic data, especially in single-cell datasets with dropout.

Spectral Design and Expander Construction

Families of signed weighted graphs with specified spectral properties facilitate the construction of Ramanujan and expander graphs using orthogonal signed matrices, Kronecker products, and 2-lifts, directly supporting network robustness and error-correcting code design (Ramezani, 2015).

Domination and Circuit Covering

Double domination and circuit cover problems, central in network resilience and redundancy planning, have analogs in signed weighted graphs, where the requirements must be balanced to accommodate signed structures and induced subgraph balance (Sehrawat et al., 2019, Lu et al., 2023).

Line Graph, Power Graph, and Product Graph Theory

Characterizations of signed line graphs (Cavaleri et al., 2021), n-th powers of signed graphs (V et al., 2020), and signed graph products (Shijin et al., 2020) require adaptation of classic theorems, forbidden subgraph lists, and eigenvalue bounds to retain sign-sensitive properties.

6. Open Problems and Future Research

Key directions are:

• Efficient algorithms for large-scale signed graph inference under more general noise and data models, or with dynamic, evolving, or multi-layered signed weighted graphs (Karaaslanli et al., 13 Jul 2025).
• Generalization of key spectral results and homomorphism bounds to non-bipartite, non-Euclidean, or highly structured signed weighted graphs (Ramezani, 2015, Foucaud et al., 2021).
• Quantitative understanding of the interplay between sign balance, combinatorial invariants, and spectral properties; especially balance criteria via matrix determinants (Roy et al., 2020).
• Incorporation of further structural constraints (e.g., balance, sign-symmetry, or forbidden patterns) in both statistical learning and optimization for real-world signed weighted networks (Ghorbani et al., 2020, K et al., 2022, Karaaslanli et al., 13 Jul 2025).
• Investigation of the minimal and maximal possible signed Wiener indices under variable sign assignments, and the structure of signed or weighted graphs that extremize distance-based indices (Spiro, 2021).

7. Theoretical Unification and Taxonomy

Signed weighted graphs serve as a nexus for graph-theoretic, algebraic, combinatorial, optimization, and data-driven approaches. By uniting sign information, arbitrary weights, and topological structure, they provide a universal mathematical infrastructure for understanding systems with antagonistic and cooperative interactions. Cutting-edge frameworks, such as the net Laplacian for learning, spectral matrix interlacing, balance criteria via signed distance Laplacians, and high-dimensional switching, provide the necessary toolkit to understand, analyze, and engineer complex networks across science and engineering.

---

Selected References:

• (Kermarrec et al., 2014) Signed graph embedding: when everybody can sit closer to friends than enemies
• (Ramezani, 2015) On the signed graphs with two distinct eigenvalues
• (Akbari et al., 2017) Some Criteria for a Signed Graph to Have Full Rank
• (Monfared et al., 2019) Inertias of Laplacian matrices of weighted signed graphs
• (Ghorbani et al., 2020) On sign-symmetric signed graphs
• (Grassia et al., 2021) wsGAT: Weighted and Signed Graph Attention Networks for Link Prediction
• (Foucaud et al., 2021) Extended Double Covers and Homomorphism Bounds of Signed Graphs
• (K et al., 2022) Vector Valued Switching in Signed Graphs
• (Lu et al., 2023) Signed circuit $6$-covers of signed $K_4$-minor-free graphs
• (Karaaslanli et al., 13 Jul 2025) Signed Graph Learning: Algorithms and Theory