Two-Terminal Reliability in Network Design

• Two-terminal reliability is the probability that two designated nodes remain connected after random edge failures in a network.
• The analysis relies on the reliability polynomial and lexicographic comparisons of connected subgraph counts to classify optimal network structures.
• Recent research constructs unique locally most reliable graphs (LMRTTGs) that optimize network robustness under low edge-survival probabilities.

Two-terminal reliability refers to the probability that, in a network modeled as a graph where edges independently fail with a fixed probability, two designated terminal vertices remain connected by a path after failures. This notion is foundational in reliability engineering, combinatorics, and applied network science, serving as a measure of network robustness under random edge removals. Recent research has sharply focused on the classification and construction of networks that maximize two-terminal reliability under various constraints, notably in the regime of "local optimality," i.e., maximizing reliability in the low edge-survival probability limit (Romero, 20 Sep 2025).

1. Formal Definition and Fundamental Properties

Let $G = (V, E)$ be a simple, undirected graph with $n$ vertices and $m$ edges, and let $s, t \in V$ be distinguished terminals. Under the standard model, each edge $e \in E$ is independently retained ("up") with probability $p \in [0,1]$ and deleted ("down") with probability $1-p$. The two-terminal reliability of $G$ at $p$ is defined as

$$R_G(p) = \Pr [ G_p \text{ has a path joining } s \text{ and } t ],$$

where $G_p$ is the random subgraph containing those edges that survived. Equivalently,

$$R_G(p) = \sum_{E' \subseteq E} I_{E'} \cdot p^{|E'|} (1-p)^{m - |E'|},$$

where $I_{E'}$ is the indicator that $G_{E'}$ contains an $s$–$t$ path.

$R_G(p)$ is a polynomial in $p$ of degree at most $m$, and for fixed $(G,s,t)$, its coefficients encode fundamental combinatorial data about $s$–$t$ connectivity structures.

2. Local and Uniform Optimality: Existence and Uniqueness

A two-terminal graph $G \in T(n,m)$ (the class of all $n$-vertex graphs with $m$ edges and labeled terminals) is termed locally most reliable (LMRTTG) if, for every $H \in T(n,m)$, there exists $\delta>0$ such that $R_G(p) \geq R_H(p)$ for all $p \in (0, \delta)$. This focuses on the "low-survival" regime—networks particularly robust when edge reliability is poor.

Recent results establish:

• For all $n \geq 4$ and $5 \leq m \leq \binom{n}{2}$, there exists a unique LMRTTG in $T(n,m)$, denoted $G(n,m)$ (Romero, 20 Sep 2025).
• For "small-edge" cases ($5 \leq m \leq 2n-3$), $G(n,m)$ has an explicit construction: its edge set consists of the edge $st$ and a set of edges connecting the terminals to a collection of auxiliary vertices, maximizing the number of minimal $s$–$t$ paths [Definition 2.4, (Romero, 20 Sep 2025)]. For odd $m$, $E(G_{n,m}) = \{st\} \cup \{sv_i, v_it\}$ for $i$ up to $(m-1)/2$; for even $m$, an additional edge $v_1v_2$ is included.
• For "large-edge" cases ($m>2n-3$), the LMRTTG is constructed by adjoining universal terminals to a unique "H-optimal" graph on $n-2$ vertices and $m-(2n-3)$ edges, where optimality is attained with respect to the invariant $H(G) = M_2(G) - 6k_3(G)$ (the second Zagreb index minus six times the triangle count).
• Uniformly most reliable graphs (maximizing $R_G(p)$ over all $p$) are significantly rarer. For many classes, such graphs do not exist (Romero, 20 Sep 2025), a consequence of the competing combinatorial requirements as $p$ varies.

3. Lexicographic Comparison and Polynomial Coefficients

The uniqueness proof for LMRTTGs exploits a lexicographic comparison of reliability polynomial coefficients. If, for two graphs $G$ and $H$, the minimal index $j$ where $N_j(G) > N_j(H)$ (with $N_j(G)$ the number of $s$–$t$ connected subgraphs with $j$ edges) determines that $R_G(p) > R_H(p)$ for sufficiently small $p$ (by Lemma 2.1, elementary calculus). The implication is that maximizing the lowest coefficients (connected by minimal sets of edges) determines optimality in the low-$p$ regime. This analytic principle underlies all local optimality classifications.

For $m > 2n-3$, the invariants $H(G)$, based on graph degrees and triangle counts, enumerate potential topologies and their impact on minimal path structure. The unique maximum of $H(G)$ selects the optimal inner structure for $G(n,m)$.

4. Construction Methods for LMRTTGs

The construction of LMRTTGs proceeds in two cases:

• Case 1: $5 \leq m \leq 2n-3$. Direct: $G(n,m)$ has two terminals, auxiliaries, and edges forming a bi-star centered at $s$ and $t$ plus possibly one inter-auxiliary edge.
• Case 2: $m > 2n-3$. Construct a graph $H_{n-2, m-(2n-3)}$ with maximum $H$ in the class of graphs with $n-2$ vertices and $m-(2n-3)$ edges (optimized for minimal path redundancy and triangle minimization). Then, append two universal terminals $s$ and $t$ that connect to all other vertices.

Both constructions are explicit and algorithmic; the latter may require searching for $H$-optimal graphs via combinatorial invariants or enumeration. This method covers all parameter ranges $(n,m)$ in $T(n,m)$.

5. Implications for Network Design and Reliability Analysis

The theory of LMRTTGs provides actionable insight for network designers:

• For any desired size $(n,m)$, the optimal topology for worst-case reliability (small $p$ regime) is now classified.
• The prevalence of local but not uniform optimality suggests that network optimality for reliability is parameter sensitive; robust design should specify the operating $p$ regime.
• For communication networks, the structure of the optimal LMRTTG is akin to a "bi-star" with terminal-centric connections, minimizing minimal path length and maximizing the number of edge-disjoint $s$–$t$ paths.

A plausible implication is that the complexity of network redundancy must be balanced with triangle minimization to avoid excessive clustering that can degrade reliability in certain parameter regimes.

6. Open Questions and Further Directions

Although all LMRTTG cases in $T(n,m)$ have been settled (Romero, 20 Sep 2025), extensions to multi-terminal reliability, weighted networks, or directed settings remain open. The combinatorics of H-optimal graphs, connections to tools such as the Tutte polynomial and generalized Zagreb indices, and computational algorithm design for large networks continue to receive attention. Analysis of uniform optimality versus local optimality in broader classes—especially for variable node reliability or dynamic edge models—constitutes a promising direction.

This classification also interfaces with reliability ranking poset theory for minimal matchstick networks (Dragoi et al., 2019), where explicit combinatorial orders are used to structure the search for optimal networks. Future work may deepen the connections between poset theory, polynomial invariants, and reliability optimization in diversified network contexts.