Topology Efficiency Index (TEI)

• Topology Efficiency Index (TEI) is a metric that quantifies the efficiency of network topologies by balancing task success against communication or hardware cost.
• TEI is derived from rigorous mathematical formulations in both multi-agent systems and hardware network designs, capturing trade-offs like success rate versus message count and resource expenses.
• TEI offers practical design guidance by benchmarking protocols and topologies for optimal performance in constrained environments.

The Topology Efficiency Index (TEI) is a rigorously defined metric used to evaluate the efficiency of topological structures in networks and multi-agent systems. TEI quantifies the trade-off between a system’s performance and the resource cost (either in communication acts or hardware complexity) imposed by a specific connectivity topology. It has been introduced and analyzed in resource-centric network evaluation (Wei et al., 26 Jan 2026) as well as in the context of communication-efficient multi-agent reinforcement learning (Zhang et al., 12 Nov 2025), where its interpretation and mathematical formulation vary to match domain requirements.

1. Mathematical Formulation

In multi-agent communication settings, TEI is defined as the ratio of the empirical success rate to the total number of communication acts over an epoch:

$\Phi_{\mathrm{TEI}_t} = \frac{\mathscr{S}_t}{C_t}$

where:

• $\mathscr{S}_t \in [0,1]$ is the task success rate (fraction of successfully completed episodes),
• $C_t$ is the total number of directed communication acts, computed as the sum of all non-self adjacency matrix entries across $L$ communication rounds and $N$ agents:

$C_t = \sum_{l=1}^{L} \;\sum_{i=0}^{N-1} \sum_{\substack{j=0\j\neq i}}^{N-1} G_{i,j}^{\,t(l)}$

In hardware network design, TEI quantifies the normalized cost required to maintain a non-blocking, full-throughput topology:

$$\mathrm{TEI} = \frac{H}{L_{\mathrm{host}}} \cdot [k + \gamma k^2] \cdot \frac{N}{M}$$

where:

• $H$ is the mean hop count per packet,
• $L_{\mathrm{host}}$ is the connectivity to hosts,
• $k$ is the router radix,
• $\mathscr{S}_t \in [0,1]$0 is the ratio of crossbar to interface cost,
• $\mathscr{S}_t \in [0,1]$1 is the router-to-host ratio (Wei et al., 26 Jan 2026).

Both contexts share the principle that TEI is minimized when high system-level performance is achieved with minimal connectivity cost—either in messages or hardware.

2. Theoretical Properties

The properties of TEI are dictated by its mathematical construction:

• Domain: In multi-agent systems, $\mathscr{S}_t \in [0,1]$2, $\mathscr{S}_t \in [0,1]$3, giving $\mathscr{S}_t \in [0,1]$4 or higher if $\mathscr{S}_t \in [0,1]$5 is small and $\mathscr{S}_t \in [0,1]$6 nonzero.
• Extremal Behavior: $\mathscr{S}_t \in [0,1]$7 for zero success rate, while very large TEI values appear if $\mathscr{S}_t \in [0,1]$8 with extremely low communication.
• Monotonicity: For fixed $\mathscr{S}_t \in [0,1]$9, increasing $C_t$0 decreases TEI, while for fixed $C_t$1, increasing $C_t$2 increases TEI (Zhang et al., 12 Nov 2025).
• Hardware context: TEI increases with hop count $C_t$3 or required router complexity $C_t$4, and is sensitive to the technological ratio $C_t$5 between link and router costs (Wei et al., 26 Jan 2026).

3. TEI in Multi-Agent Communication Protocols

TEI serves as an explicit post-hoc communication-efficiency metric to evaluate and compare protocols in multi-agent reinforcement learning. Given communication adjacency matrices (indicating message passing) and binary success indicators per episode, TEI can be computed as follows:

$N$9

In empirical studies on the Traffic Junction environment, TEI values allow performance-to-cost comparison across MAGIC, CommNet, TarMAC, GA-Comm, and IC3Net algorithms, with typical TEI in the $C_t$6 range (Zhang et al., 12 Nov 2025). Increasing communication rounds generally decreases TEI, while efficiency-augmented loss functions significantly improve it.

4. TEI in Network Hardware Design

In network architecture, TEI is characterized as the normalized hardware cost per unit host-throughput required for non-blocking operation. The explicit formula,

$C_t$7

endows TEI with the following comparative properties across common topologies:

Topology	Router Radix $C_t$8	TEI Expression (normalized by $C_t$9)
2D Torus	$L$0	$L$1
Hypercube	$L$2	$L$3
Flat Butterfly	$L$4	$L$5
Fat Tree	const $L$6	$L$7, $L$8
n-plane Star	$L$9	$N$0

For large scales, indirect topologies (Fat Trees) minimize TEI by capping router radix at the technology-optimal value, while direct networks become inefficient as $N$1 dominates cost (Wei et al., 26 Jan 2026).

5. Applications and Comparative Insights

In multi-agent learning, TEI is not used as a training objective but rather as a principal metric for post-hoc comparative evaluation. When protocol topologies are fixed (e.g., fully connected), $N$2 is nearly constant, so TEI is analytic and not amenable as a learning signal. Higher TEI values signal protocols that are more parsimonious with bandwidth and, therefore, better suited for constrained environments.

In network hardware, TEI facilitates selection among candidate topologies by expressing the absolute resource efficiency under strict performance guarantees. Topologies optimizing TEI under cost model parameters ($N$3, $N$4) and problem scale ($N$5, $N$6) are identified via explicit formulas, enabling engineering design with clear efficiency trade-offs.

6. Practical Design Guidance

From the resource-centric analysis, clear regimes emerge:

• For small networks ($N$7), low-radix direct topologies or parallel star planes may win in absolute cost.
• For medium networks ($N$8), hypercube-like logarithmic-diameter networks typically yield minimal TEI.
• For very large networks, indirect Fat Tree topologies with fixed, technology-optimal router radix minimize TEI (Wei et al., 26 Jan 2026).

A key design rule is that redundancy is most efficiently implemented by parallel network planes (multi-plane stars) rather than by intrinsic path diversity within a single graph.

7. Significance and Limitations

TEI offers a compact, dimensionless score enabling incisive comparisons of efficiency across protocols, algorithms, or hardware designs. Its interpretability at the extremes, analytic tractability, and empirical correlation with deployment cost (whether message or hardware) have made it a central metric in both communication protocol design (Zhang et al., 12 Nov 2025) and network architecture analysis (Wei et al., 26 Jan 2026). However, TEI does not capture emergent behavioral properties beyond its numerator and denominator and assumes homogeneity in communication or traffic; deployment contexts with highly variable loads or heterogeneous success definitions may require complementary analysis.

A plausible implication is that the widespread adoption of TEI (or resource-centric analogs) could standardize efficiency benchmarking across domains involving structured connectivity and resource-constrained coordination.