Edge Factor Metric

• Edge Factor Metric is a suite of measures that quantifies edge-criticality and balance using geometric analysis and graph theory to assess both continuous and discrete structures.
• In geometric contexts, edge metrics involve constructing normal forms and using singular differential equations to model metric degeneracies near manifold boundaries.
• In graph theory, concepts like edge metric dimension and related invariants uniquely characterize edge distances and assess network cohesion through combinatorial and spectral methods.

The edge factor metric encompasses distinct but related mathematical constructs in geometric analysis and graph theory, each aimed at quantifying the criticality, identifiability, or “balance” of edges with respect to the underlying structure. In geometric settings, it refers to canonical forms of Riemannian metrics, particularly “edge metrics,” near singular or fibred boundaries. In discrete contexts, it refers predominantly to the edge metric dimension—encoding the minimal size of a landmark set that uniquely identifies all edges by distance vectors—and, more generally, to edge-based invariants such as the edge quasi-$\lambda$-distance-balance and link cohesion in networks. The following sections provide a comprehensive technical analysis of these concepts, their precise definitions, principal results, analytic frameworks, and implications in both mathematics and applied network sciences.

1. Edge Metrics and Normal Form: Differential Geometric Setting

Edge metrics are singular geometric structures defined on the interior of a compact manifold-with-boundary %%%%1%%%%, equipped with a boundary fibration. The defining feature is a metric $g$ whose degeneracy at the boundary $\partial X$ is controlled so that $g$ behaves analogously to a product metric of the interior with a singular factor encoding the fibration.

A central result is the derivation of a canonical “normal form” for these metrics, which amounts to constructing local coordinates $(x, y, z)$ near $\partial X$ and a diffeomorphism $\psi$ such that:

• $\psi|_{\partial X} = \operatorname{Id}$,
• In these coordinates, the metric reads

$g = dx^2 + k,$

where $k$ is a symmetric 2-tensor, smooth up to the boundary, satisfying $x\partial_x k = 0$.

The derivation of this normal form requires two geometric conditions:

• Normalization: The “horizontal” part of the dual metric $|dx|^2_{g_H} = 1$ on $\partial X$, establishing a canonical scale transverse to the fibers.
• Exactness: For every defining function $x$, an associated 1-form $\alpha_x$ defined on the fibers is globally exact (i.e., $\alpha_x = d_V f$ for some $f$). This ensures that the “gauge freedom” in the choice of defining function can be controlled.

These conditions facilitate the solution of the singular eikonal equation

$|dx|^2_g = 1,$

which is achieved by reducing to a nonlinear first-order PDE for a conformal factor $\omega$ (where $x = x_0 e^\omega$) and then constructing the normal form using a characteristic Hamiltonian flow-out formalism in the first jet bundle. The resulting normal forms are in bijection with the representative metrics for a reduced conformal infinity class, encoding boundary data analogous to the conformal infinity in the context of Poincaré–Einstein metrics (Graham et al., 2012).

2. Edge Metric Dimension: Discrete Graph Theoretic Foundation

The edge metric dimension (also known as the edge factor metric in some literature) is a parameter for a connected graph $G = (V, E)$ defined as follows:

• For a vertex $v \in V$ and an edge $e = uw \in E$, the distance is $d_G(e, v) = \min\{d_G(u, v), d_G(w, v)\}$.
• A set $S \subseteq V$ is an edge metric generator if for every pair of distinct edges $e_1, e_2 \in E$ there exists a $w \in S$ such that $d_G(w, e_1) \neq d_G(w, e_2)$.
• The edge metric dimension, denoted $\operatorname{edim}(G)$, is the smallest cardinality of such a set.

Table: Comparison of edge metric dimension and vertex metric dimension in selected families (Kelenc et al., 2016, Kartelj et al., 2018)

Graph family	$\operatorname{edim}(G)$	$\dim(G)$
Path $P_n$	1	1
Cycle $C_n$	2	2
Complete $K_n$	$n-1$	$n-1$
Complete bipartite $K_{r,t}$	$r + t - 2$	$r + t - 2$
General Petersen $GP(n,1)$	3	2 or 3 (parity dependent)

This parameter demonstrates highly nontrivial behavior: in most classical families its value coincides with the standard metric dimension, but for wheel graphs, fan graphs, certain toroidal and Petersen-type constructions, or explicitly constructed families, $\operatorname{edim}(G)$ and $\dim(G)$ can be arbitrarily far apart (Kelenc et al., 2016, Kartelj et al., 2018, Knor et al., 2020, Sedlar et al., 2021, Zhu et al., 2021).

3. Analytic and Algorithmic Properties

• Complexity: Computing $\operatorname{edim}(G)$ is NP-hard, with the decision form (“Is $\operatorname{edim}(G) < r$?”) being NP-complete via reductions from classical SAT (Kelenc et al., 2016).
• Bounds: $1 \leq \operatorname{edim}(G) \leq |V| - 1$ for any connected graph; for an $n$-dimensional hypercube $Q_n$, $\operatorname{edim}(Q_n) \leq n$.
• Approximation: The problem admits a polynomial-time $O(\log |E|)$-approximation via transformation to set cover (Kelenc et al., 2016).
• Closed Formulas: Numerous exact formulas exist for standard classes. For instance, for trees $T$:

$$\operatorname{edim}(T) = \sum_{\text{vertices } v, \ l_v > 1} (l_v - 1),$$

where $l_v$ is the number of legs (pendant paths) at $v$ (Kelenc et al., 2016).

• Hierarchical Products and Operations: Tight bounds exist for join, lexicographic, and corona products (Peterin et al., 2018), as well as for hierarchical products, and ILP formulations for exact computation have been proposed (Klavžar et al., 2020).
• Random Graphs: In $G(n, p)$,

$$\operatorname{edim}(G(n,p)) = (1 + o(1)) \frac{4 \log n}{\log(1/q)}, \quad q = 1 - 2p(1-p)^2(2-p),$$

indicating asymptotic minimal growth compared to the number of edges (Zubrilina, 2016).

4. Edge Factor Metrics Beyond Dimension: Cohesion, Balance, and Structural Sensitivity

a) Link Cohesion and Density

For complex, dense networks, edge factor metrics such as link cohesion quantify the strength of binding between nodes: $$L_{ij} = \frac{|N(i) \cap N(j)| + 1}{\sqrt{d_i \, d_j}},$$ where $N(i)$ is the neighbor set and $d_i$ is the degree of vertex $i$. Aggregated, this yields a density measure

$$D = \frac{1}{|V|} \sum_{i \in V} \frac{1}{|N(i)|} \sum_{j \in N(i)} L_{ij},$$

serving as a locally sensitive estimator of overall network cohesion (Savkli et al., 2020).

These metrics support scalable graph sparsification and fine-grained community detection, contrasting with global measures such as edge betweenness.

b) Structural Edge Importance in Networks

Higher-order metrics, such as the structural importance metric $\ell_e$, assess the impact of edge perturbations on global dynamics. For an adjacency matrix $A$ with principal eigenvalue $\lambda_0$ and eigenvector $\mathbf{v}_0$,

$\ell_e = 2 v_{0,i} v_{0,j}$

for edge $e = (i, j)$. This is the derivative of $\lambda_0$ with respect to $A_{ij}$ and captures the influence of $e$ on network resilience and dynamic processes. Its use in network evolution, especially temporal networks, enables linking edge-level changes with macroscopic behavior (Seabrook et al., 2020).

c) Edge Quasi-$\lambda$-Distance-Balanced Graphs (EQDBGs)

An EQDBG is defined using the counts $m_g^A(f)$ (number of edges closer to $g$ than $h$ for $f = gh$) and a constant $\lambda > 1$ such that $m_g^A(f) = \lambda^{\pm1} m_h^A(f)$. Such graphs are necessarily bipartite, and their edge-Szeged index satisfies a precise relation dependent on $\lambda$. This structure is preserved under certain product operations and admits extensions to “nicely” and “strongly” edge quasi-$\lambda$-distance-balanced graphs, enhancing the analytic apparatus for understanding global balance via local edge measures (Aliannejadi et al., 7 Jun 2024).

5. Edge Metric Dimension in Specific Structures and Applied Settings

• Chemical Graph Theory: The edge metric dimension is directly relevant for chemical networks where bonds are edges, e.g., silicate or coronoid–starphene networks (Sharma et al., 2021, Prabhu et al., 11 Jun 2024). In these cases, explicit constructions yield results such as $\operatorname{edim}(FCS_{a,b,c}) = 3$ for large $a, b, c$, and formulas for silicate networks depending on parity.
• Structured Graphs: For unicyclic graphs and cacti (graphs with edge-disjoint cycles), the edge metric dimension is characterized explicitly in terms of structural parameters (branch-resolving sets, configuration types) and is always one of at most two consecutive integers—distinguished by the presence of certain “obstructive” configurations (Sedlar et al., 2021, Zhu et al., 2021, Sedlar et al., 2021).
• Resilient Spanner Construction: In geometric and algorithmic settings, “edge factor metric” may refer to resilience of spanners against degree-bounded edge faults, i.e., for “$f$-faulty-degree $t$-spanner” constructions. Augmenting a $t$-spanner with the addition of $(4f-1)m$ edges yields an $f$-degree-resilient spanner with controlled stretch, optimal in the sense that such constructions are minimal up to leading constants (Biniaz et al., 28 May 2024).

6. Implications, Open Problems, and Future Research

The notion of the edge factor metric, whether as edge metric dimension or as a local/global edge-influence parameter, has wide-ranging implications:

• Network Verification and Security: Unique edge identification underlies efficient monitoring, localization, and detection of anomalies or transmission faults, especially in large-scale, dynamic or fault-prone networks.
• Complexity Theory and Graph Algorithms: Determining or approximating the edge metric dimension feeds into broader algorithmic studies; complexity barriers motivate research into heuristics for specialized graph classes or for dynamic/temporal networks.
• Spectral and Structural Analysis: Edge influence metrics such as $\ell_e$ connect combinatorics, dynamics, and spectral theory, motivating new mathematical frameworks for resilience, control, and intervention in engineered or social systems.
• Mathematical Physics and Geometric Analysis: In geometric models (edge metrics), constructing normal forms and understanding singularity structures underpin advances in inverse problems, scattering theory, and mathematical physics.

Open problems include:

• Complete characterization of graph classes where edge metric dimension is strictly less than, equal to, or more than the standard metric dimension.
• Extension of explicit dimension results to mixtures of cycles and trees or to graphs with “critical” attaching structures.
• Tight asymptotics for edge metric dimension in more general random graph or geometric models.
• Combinatorial and analytic classification of all EQDBGs and their relation to other global invariants.

This suite of concepts, collectively denoted “edge factor metric,” thus provides a unified foundation for quantifying and analyzing edge significance—topologically, geometrically, dynamically—across pure mathematics, applied sciences, and network engineering.