Pseudo-Undirected Graphs

• Pseudo-undirected graphs are generalizations of undirected graphs that relax symmetry constraints by allowing asymmetric weights, multiple edges, and self-loops.
• They are analyzed through combinatorial, geometric, and spectral frameworks, using polytope encodings and modified Laplacians for deeper structural insights.
• Applications span from network control and consensus protocols to advanced graph sampling and machine learning, supporting robust modeling of complex systems.

A pseudo-undirected graph is a generalization of the standard undirected graph paradigm that systematically relaxes one or more canonical symmetry or edge multiplicity constraints, while preserving a bidirectional connection structure. Pseudo-undirected graphs may allow asymmetric or distinct weights on opposing directions of each undirected edge, multiple edges (bundles) between the same pair of nodes, self-loops, negative edge weights (within admissible bounds), or algebraic generalizations capturing partial symmetries. These models arise in combinatorics, algebraic geometry, Markov chain sampling, network dynamics, spectral theory, and control systems, each context giving rise to distinct structural and analytical consequences.

1. Structural Variants of Pseudo-Undirected Graphs

Pseudo-undirected graphs subsume distinct generalizations, each relaxing different regularities of standard undirected graphs:

• Pseudographs: Allow both multiple parallel edges (bundles) between the same pair of vertices and self-loops. The combinatorial structure is explicitly constructed in the theory of pseudograph associahedra by encoding multi-edges as bundles and loops as additional rays in a product polytope, providing a combinatorial-geometric representation for connectivity data (Carr et al., 2010).
• Graphs with Self-Loops: The definition of pseudo-connected graphs requires each vertex to participate in a loop or be otherwise connected, and every connected subgraph to be anchored by at least one self-loop. This guarantees the positive definiteness of the Laplacian (&&&1&&&).
• Mixed or Hermitian Graphs (Algebraic Pseudo-Undirected Graphs): In spectral theory, edges may be “mixed” (some undirected, some directed with paired orientations), and/or weighted with complex phases, preserving Hermitian symmetry in the adjacency or incidence matrices (Abudayah et al., 2022).
• Bidirectionally-Connected Weighted Digraphs: In consensus dynamics, each undirected connection is replaced by two directed edges with potentially distinct (even negative) weights—yielding a “pseudo-undirected” Laplacian that is generally non-symmetric yet guarantees consensus and control properties under appropriate constraints (Sinha et al., 24 Sep 2025).
• Graph Spaces with Multiplicities or Loops: Intermediate models such as loopy-multigraphs (multiedges allowed, but only single self-loops) and multiloop-graphs (multiple self-loops allowed, but no multiedges) are critical in the context of Markov chain Monte Carlo (MCMC) sampling on degree-sequence- or triangle-constrained spaces (Nishimura, 2017).

These variants enable the modeling of systems where purely undirected symmetry does not hold, or where multiplexing, feedback, or negative interactions play essential roles.

2. Combinatorial and Geometric Encoding

The theory of pseudograph associahedra (Carr et al., 2010) systematically encodes the connectivity of pseudo-undirected graphs into the face poset of a simple polytope (or a polytopal cone when loops are present). The construction begins with a product of base polytopes:

$$\Delta_{n-1} \times \left( \prod_{B_i \in G} \Delta_{b_i-1} \right) \times \mathbb{R}_+^{\lambda}$$

where $\Delta_{n-1}$ is the simplex on $n$ nodes, $B_i$ are the bundles (non-loop parallel edges) with multiplicities $b_i$, and each loop contributes a ray. Iterated truncations then correspond to full tubes—subgraphs containing all incident bundles and loops—resulting in a polytope whose faces correspond to compatible sets of connected subgraphs (tubes).

This framework explicitly distinguishes multiple edges and self-loops. The face lattice is isomorphic to the poset of “tubings” (collections of compatible tubes), with vertices corresponding to maximal tubings and lower-dimensional faces corresponding to tubings with more tubes. The dimensionality satisfies:

$$\dim \mathcal{K}_G = n - 1 + (\text{number of redundant edges}),$$

where redundant edges are the excess in each bundle over a single edge, quantifying the expansion in combinatorial complexity due to pseudo-undirected features.

3. Algebraic and Spectral Frameworks

Pseudo-undirected graphs play a significant role in spectral and algebraic generalizations:

• Laplacians with Self-Loops: The spectrum of the Laplacian for a (pseudo-connected) graph with self-loops is analyzed using a decomposition

$$\mathcal{L}(G) = \mathcal{L}(G^o) + \sum_{(i,i) \in G} e_i e_i^T,$$

where $G^o$ removes the self-loops and the sum adds positivity to the diagonal, shifting the spectrum away from zero and ensuring positive definiteness (Acikmese, 2015). The lifting construction maps $G$ to a larger graph without self-loops such that

$$\sigma(\mathcal{L}(G)) \subseteq \sigma(\mathcal{L}(\hat{G})) \cap [0, 2 d_{\max}(G^o) + 1].$$

• Hermitian Incidence and Adjacency Matrices: In mixed graphs, the α-Hermitian adjacency matrix $H^{\alpha}(D)$ and β-incidence matrix $B^\beta(D)$ are parameterized by unimodular complex numbers carefully chosen to preserve key spectral relations:

$B^* B = H_\gamma(L(D)) + 2I,$

extending the classical line graph incidence-adjacency relation to pseudo-undirected (mixed) graphs (Abudayah et al., 2022). The algebraic structure admits nontrivial orientations and edge-types while maintaining real spectra and other familiar properties.

• Pseudo-Euclidean Embeddings: Graph embeddings in pseudo-Euclidean space assign each node both an “attract” and a “repel” vector, using an inner product of the form

$e_{ij} = a_i \cdot a_j - r_i \cdot r_j,$

enabling the modeling of intransitive and heterophilic structures not possible with purely symmetric dot-product models (Peysakhovich et al., 2021). Such embeddings efficiently describe social, co-occurrence, or biological graphs with forbidden triads or nonmetric similarity.

4. Dynamics and Control on Pseudo-Undirected Graphs

In networked systems and consensus protocols, pseudo-undirected graphs permit robust extensions of classical average consensus to more flexible, robust, or designable consensus behavior:

• Weighted, Possibly Negative Interactions: The pseudo-undirected framework models each undirected link as a pair $(i\to j, j\to i)$ with possibly distinct and sign-varying weights, leading to a Laplacian $\mathcal{L}=E_{\text{out}} \mathcal{W} E^T$ that is non-symmetric. The system

$\dot{x} = -\mathcal{L} x$

achieves consensus at

$x^* = \frac{\sum_i p_i x_i(0)}{\sum_i p_i},$

where $p$ is the positive left eigenvector of $\mathcal{L}$, enabling consensus values outside the convex hull of initial states—crucial for applications such as simultaneous interception with actuator constraints (Sinha et al., 24 Sep 2025).

• Admissibility of Negative Weights: The introduction of negative weights is controlled via transfer-function gain margins; consensus is preserved if the magnitude of the perturbation

$|\Delta| < \frac{1}{|M(j\omega_{pc})|},$

where $M(s)$ is the edge transfer function and $\omega_{pc}$ a phase-crossover frequency. Bounds depend on the edge’s network position, with central edges in paths tolerating more negative perturbation.

• Applications: This framework is applied in multi-agent guidance law design, for example enabling time-to-go consensus at values beyond those achievable with simple averaging, thus satisfying physical constraints not accommodated by standard models.

5. Sampling, Enumeration, and Algorithmic Implications

Pseudo-undirected models intervene in random graph sampling and Markov chain enumeration:

• Edge-Swap MCMC and Graph Spaces: In studying the swap connectivity of graph spaces, distinct intermediate classes—such as loopy-multigraphs and multiloop-graphs—are considered. These admit only a subset of possible edge multiplicities or self-loops compared to full pseudographs (Nishimura, 2017). Double edge-swap MCMC is irreducible (i.e., the graph-of-graphs is connected) for loopy-multigraphs for any degree sequence, but only for multiloop-graphs if specific parity and degree constraints are satisfied. Additional structural constraints—such as triangle counts—can fragment the space, blocking the sampler’s ability to reach all graph realizations with standard swaps.

Graph Space	Multiedges Allowed	Multiple Self-Loops Allowed	Swap Connectivity Criteria
Multiloop-Graph	No	Yes	Requires odd-degree and other conditions
Loopy-Multigraph	Yes	No (single)	Always connected for any degree sequence

• Algorithmic Considerations: Additional topology constraints (e.g., triangle count) can disconnect feasible spaces for swaps of size ≤8, implying that even high-order swaps may fail to ensure uniform sampling.

6. Theoretical and Algorithmic Properties

Pseudo-undirected graphs unify and extend critical theoretical notions:

• Face Poset Encoding and Polytope Deformations: In the associahedral framework, the face poset repairs the breakdown of connectivity encoding occurring when allowing loops and bundles, while deformation maps (Φₑ for contractions, Θₑ for deletions) keep track of how combinatorial invariants change as the network is simplified (Carr et al., 2010).
• Unification of Intersection Graphs and Geometric Structures: The notion of co-strongly pseudo transitive orientation provides a framework to unify intersection graph classes (axis-parallel rectangles, half segments, filament graphs), linking algorithmic solvability (e.g., $O(n^3)$ maximum independent set computation) to deeper combinatorial structure (Shahrokhi, 2018).
• Markov Properties and Dualities in Graphical Models: Duality between conditional independence structures (in undirected/concentration and bidirected/covariance graphical models) is naturally formalized in the pseudo-undirected paradigm, exploiting (reverse) pseudographoid closure under weaker conditions than full intersection and composition (Malouche et al., 2013).
• Random Orientation and Percolation: Randomly oriented graphs preserve certain positive correlation properties reminiscent of undirected percolation, even under bias—informing connectivity analysis in networks exhibiting pseudo-undirected character due to stochastic or externally induced orientations (Narayanan, 2016).

7. Applications and Future Directions

Pseudo-undirected graphs are emerging as a versatile modeling tool in network theory, enumerative combinatorics, optimization, spectral analysis, sampling, control, and learning:

• Network Control and Coordination: The ability to modulate consensus value placement transcendently or robustly using asymmetrically weighted, bidirected edges enables the design of distributed algorithms with expanded operating regimes (Sinha et al., 24 Sep 2025).
• Graph Embedding and Machine Learning: Pseudo-Euclidean attract-repel embeddings exploit latent bidirectional/intransitive geometries to improve compression, interpretability, and link prediction in real-world graphs (Peysakhovich et al., 2021).
• Compactification and Moduli Spaces: Extensions from simple graphs to pseudographs in the construction of associahedra are motivated by problems in the compactification of spaces associated with moduli of curves, matroids, and beyond (Carr et al., 2010).
• Markov Chain Theory and Null Model Construction: Robust characterization of the connectivity of sampling spaces is critical for hypothesis testing and the construction of network null models, with pseudo-undirected constraints being essential for certain degree or motif-preserving ensembles (Nishimura, 2017).
• Spectral and Dynamical Generalizations: Hermitian-adjacency frameworks and associated incidence matrices suggest promising avenues in the spectral theory of quantum graphs, and in the paper of energy, stability, and expansion properties in mixed or multiplexed network systems (Abudayah et al., 2022).

The paper of pseudo-undirected graphs continues to reveal new connections between combinatorics, geometry, algebra, dynamics, and computation, providing a flexible language and rigorous toolkit for modeling and analyzing networked systems with structural features beyond classical symmetry.