Locally Directed Graphs

• Locally directed graphs are structures that assign local orientations to vertex-edge incidences, enabling complex combinatorial constraints beyond traditional directed graphs.
• They bridge local properties with global outcomes, underpinning advances in spectral analysis, connectivity theories, and distributed optimization methods.
• Applications span probabilistic modeling, signal processing, and network algorithms, offering actionable insights into both theoretical and practical graph analyses.

Locally directed graphs encompass a broad family of structures in modern graph theory, combining local orientation, directionality, and often additional local data to encode complex dependencies and combinatorial constraints. The concept appears across spectral graph theory, combinatorics, stochastic modeling, algebraic graph theory, and algorithmic designs. This article gives a rigorous overview of locally directed graphs, emphasizing definitions, structural results, spectral frameworks, algorithmic methods, and key applications as synthesized in recent research.

1. Definitions and Core Properties

A locally directed graph generalizes the standard notion of a directed graph by incorporating directionality at the level of local neighborhoods or incidences. In many constructions, a locally directed graph (or "local digraph"; Editor's term) is a multigraph without self-loops in which each vertex-edge incidence carries an extra orientation or sign (sgn(v,e) ∈ {+,–}). This assignment may differ from classical directed graphs, where each edge canonically has a single orientation from head to tail. Here, both endpoints of an edge may independently assign local orientation, allowing, for instance, bidirectionality or more complex local flow. Locally directed graphs often encode additional constraints or properties—such as net-degree, local symmetry, or structural decomposition—in their combinatorial formulation (Chao et al., 28 Sep 2025, Gellert et al., 2015).

In spectral theory, for a directed graph Γ, the normalized Laplace operator Δ acts on functions v:V→ℝ via

$$\Delta v(i) = v(i) - \frac{1}{d_i^{(\text{in})}} \sum_j w_{ij} v(j)$$

where $w_{ij}$ is the weight from j to i and $d_i^{(\text{in})}$ is the in-degree of vertex i. This generalizes the symmetric Laplacian on undirected graphs to the directed setting and, when restricted to strongly connected or locally directed regions, captures local flow and directionality (Bauer, 2011).

Formally, locally directed graphs include:

• Bidirected graphs: Edges have local orientation at each endpoint; net-degree sequences generalize indegree–outdegree statistics.
• Local independence graphs: Edges encode asymmetric, dynamic dependencies (e.g., local Granger-causality) among stochastic processes, allowing cycles and bidirectional effects (Didelez, 2012).
• Locally symmetric graphs: Neighborhoods possess local automorphism groups with specified ranks (e.g., locally rank 3) and special geometric features (Bamberg et al., 2014).

2. Local-to-Global Structural Implications

Locally induced structures frequently enforce strong global properties. For example, graphs where every neighborhood satisfies the Dirac condition (minimum degree ≥ n⁄2 within the neighborhood) will have high overall connectivity and pancyclicity (Kubicka et al., 2015). Theorems in the literature establish that every connected locally Ore graph (with the Ore condition holding locally) of sufficiently large order is 3-connected; locally Dirac graphs have edge-connectivity equal to their minimum degree. These properties are derived via local reasoning extended through induction on neighborhoods and global coloring arguments.

For graphs with local symmetries—such as locally triangular graphs with locally rank 3 automorphism groups—global classification is feasible. The only such connected graphs are halved graphs of n-cubes, folded cubes of even dimension (n≥8), or coset graphs of binary Golay codes; their automorphism groups are determined completely and yield precise orbit structure within neighborhoods (Bamberg et al., 2014).

In distributed optimization and algorithmic settings, local rules (e.g., message passing in sparse random graphs or local subgradient projections) govern global convergence, consensus, or performance bounds (Xi et al., 2016, Gamarnik et al., 2014). Correct handling of directionality—often via additional state variables or surplus consensus—ensures effective computation over asymmetric networks.

3. Spectral and Comparison Frameworks

Spectral analysis of locally directed graphs is grounded in the properties of the normalized Laplacian Δ. The spectrum of Δ encodes both local and global structure. For any eigenvalue λ ≠ 0, the extremal bounds

$|1-\lambda| \leq r$

with

$$r(i) = \frac{\sum_j |w_{ij}|}{d_i^{(\text{in})}}, \quad r = \max_i r(i),$$

tie extremal eigenvalues to maximal, isolated, strongly connected components—regions where local directionality determines spectral extremality (Bauer, 2011).

Comparison theorems bridge directed and undirected settings:

• For balanced Γ, the Laplacian eigenvalues of the associated undirected graph U(Γ) are majorized by the real parts of the eigenvalues of Δ: $$\lambda(U(\Gamma)) \prec \mathrm{Re}[\lambda(\Gamma)]$$
• For strongly connected graphs with nonnegative weights, one constructs an undirected "modified" graph $\tilde{\gamma}$ via Perron–Frobenius theory with edge weights

$$\tilde{w}_{ij} = \frac{w_{ij}}{d_i^{(\text{in})}} \varphi(i) + \frac{w_{ji}}{d_j^{(\text{in})}} \varphi(j)$$

and degree

$\tilde{d}_i = 2 \varphi(i)$

where $\varphi$ is the Perron eigenvector of the transition operator. Spectral bounds then follow: $$\min_{i\neq 0}\lambda_i(\tilde{\gamma}) \le \min_{i\neq 0}\mathrm{Re}(\lambda_i(\Gamma)),\,\,\,\max_{i}\mathrm{Re}(\lambda_i(\Gamma)) \le \max_{i}\lambda_i(\tilde{\gamma})$$ These tools allow the transfer of Cheeger-type inequalities and variational guarantees from undirected graphs to locally directed settings.

Neighborhood graph constructions aggregate local path information: $$w_{ij}[l] = \sum_{k_1,\dots,k_{l-1}} \prod_{s=1}^{l-1} \frac{1}{d_{k_s}^{(\text{in})}} w_{i k_1} w_{k_1 k_2} \dots w_{k_{l-1} j}$$ with Laplacian eigenvalues

$\lambda_i[l] = 1-(1-\lambda_i)^l$

The approach sharpens spectral bounds by amplifying local connectivity (Bauer, 2011).

4. Degree Sequences and Bidirected Generalizations

Locally directed graphs can be systematically analyzed through bidirected graphs, where local orientation at each incidence decouples indegree and outdegree. The net-degree of a vertex is

$$d_v(B) = \sum_{e} \tau(v, e),\quad \tau(v,e) \in \{-1, +1\}$$

A sequence $(d_1,\dots,d_n)$ is realizable if and only if

• The total sum is even.
• Each coordinate satisfies $-(n-1) \leq d_i \leq n-1$.

Compared to undirected graphs (Erdős–Gallai inequalities) or digraphs (zero-sum with subset constraints), the bidirected case is characterized solely by parity and coordinate bounds (Gellert et al., 2015). Degree-preserving operations now include local orientation swaps (Γ), two-switches (Σ), edge replacement (Λ), and addition/removal of cycles (Δ), constituting a full set for equivalence transformations.

Uniquely realizable net-degree sequences (those with only one realization) are precisely those where every vertex is forced to be a sink or a source, with at most one pair nonadjacent. Enumeration is explicit; there are

$2^n \cdot \binom{n}{2} + 2^n$

such sequences for n vertices.

5. Algorithmic and Dynamical Implications

Locality is central to graph algorithms for sparse and random networks. Local algorithms exploit the tree-like nature of neighborhoods in sparse graphs: decisions at each vertex depend only on its r–neighborhood. Recursive equations (as in belief propagation) for probabilities or marginal states have the form (on directed or locally directed graphs)

$P = \frac{1}{1 + \lambda \prod_{i} P_i}$

where P_i reflects the incoming messages or local decisions. In directed networks, finer separation of in-neighbors and out-neighbors is handled in recursion and message passing (Gamarnik et al., 2014).

Distributed convex optimization on directed graphs requires adaptation due to communication asymmetry. Algorithms such as Directed-Distributed Projected Subgradient (D-DPS) introduce auxiliary variables to compensate for the lack of doubly-stochastic weight matrices. Consensus and optimality are achieved with convergence rates

$O\left(\frac{\ln(k)}{\sqrt{k}}\right)$

where k is the iteration count. Surplus consensus mechanisms and careful selection of weight structures are essential in strongly connected directed settings (Xi et al., 2016, Xin et al., 2017).

6. Applications, Examples, and Open Problems

Locally directed graphs appear in structural combinatorics (classification of highly symmetric graphs (Bamberg et al., 2014)), probabilistic graphical models (local independence and dynamic dependencies (Didelez, 2012)), signal processing (transferability of filters and spectral matching via local distributions (Roddenberry et al., 2022)), descriptive set theory (Borel quasi-kernels and coloring bounds (Wang, 2 Sep 2025)), and optimization (differentially private distributed learning (Chen et al., 2023)).

In inducibility problems, local digraphs serve as a tight refinement of classical graph enumeration techniques. When a graph G admits a unique perfect matching M, construction of the associated local digraph ldg(G, M) leads to

$$\text{aind}(ldg(G, M)) \leq \frac{2^{|M|}|M|!}{|\text{Aut}(G)|}\epsilon_{\text{ind}}(G) \leq \text{ind}(ldg(G, M))$$

relating edge inducibility to vertex inducibility of the local digraph (Chao et al., 28 Sep 2025). Determining exact values or tightness of these bounds remains a challenging combinatorial task, with conjectures connecting local digraph properties to longstanding open problems in hypergraph theory.

7. Comparative View and Emergent Themes

The locally directed paradigm is richer than both undirected and standard directed graph settings:

• Flexibility in local orientation leads to a looser set of realizable degree sequences and more nuanced spectral properties.
• Structural results established for neighborhoods often induce strict global constraints (e.g., connectivity, cycle richness).
• Algorithmic designs leverage locality for scalable inference, learning, and optimization in large networks, often requiring new variables or balancing strategies in directed environments.
• Spectral and combinatorial connections illuminate correspondence and transfer—e.g., between the spectra of directed versus undirected and modified Laplacians, or between local symmetry and global automorphism.
• Ongoing questions concern the tightness of structural bounds, the precise role of local orientation in global optimization problems, and potential decidability results for approximations or equivalence classification (for example, in property testing or local structure matching (Rozantsev, 2020)).

Locally directed graphs therefore serve as a crucial organizing framework throughout modern graph theory, probabilistic modeling, and network computation, both for their combinatorial depth and their capacity to encode nuanced local-global phenomena.