Graph-Based Coordination Complexity Metric

• Graph-Based Coordination Complexity Metric is a measure that converts multi-agent trajectories into a weighted graph to assess coordination difficulty.
• It aggregates agent dependency entropy, spatial interference, and goal overlap into a single scalar score to enable curriculum ordering in MARL.
• Empirical validation with a Spearman rank correlation of ρ = 0.952 confirms its effectiveness in predicting task difficulty.

Searching arXiv for the cited papers and related coordination-graph work. The graph-based coordination complexity metric is a task-difficulty measure for cooperative multi-agent reinforcement learning (MARL) that represents a coordination task as a weighted interaction graph and aggregates three components—agent dependency entropy, spatial interference, and goal overlap—into a scalar score used for curriculum ordering from “easy” to “hard” (Ebadulla et al., 9 Jul 2025). It was introduced to address the absence of validated task complexity metrics for multi-agent coordination, and it is empirically validated by a Spearman rank correlation of $\rho = 0.952$ with $p < 0.001$ between predicted complexity and empirical difficulty determined by random agent performance evaluation (Ebadulla et al., 9 Jul 2025).

1. Coordination-graph formulation

The metric models a cooperative coordination task as an undirected weighted graph $G=(V,E)$, where $V=\{1,\dots,N\}$ is the set of agents and an edge $(i,j)\in E$ exists if agents $i$ and $j$ come into “coordination proximity” at least once during a trajectory (Ebadulla et al., 9 Jul 2025). The graph is therefore not an abstract communication topology; it is induced from agent trajectories.

Edge weights encode how often agents interact. For each time step $t\in\{1,\dots,T\}$, the formulation defines

$$D_{ij}(t) = \begin{cases} 1, & \text{if } \|p_i(t)-p_j(t)\|_2 < \theta \ 0, & \text{otherwise} \end{cases}$$

with $\theta=0.5$ in normalized units, and then sets

$p < 0.001$0

These weights are later normalized to probabilities $p < 0.001$1 for the entropy term (Ebadulla et al., 9 Jul 2025).

This representation is tailored to cooperative coordination environments in which interaction frequency, spatial packing, and objective contention jointly determine difficulty. A plausible implication is that the graph acts as a trajectory-derived sufficient summary for curriculum ordering, rather than as a replacement for the underlying MARL state or policy representation.

2. Core components of the metric

The scalar complexity score is built from three normalized components. Each captures a different aspect of cooperative difficulty: diversity of interaction structure, collision-avoidance pressure, and competition over goals (Ebadulla et al., 9 Jul 2025).

Component	Definition	Intended role
Agent dependency entropy	$p < 0.001$2	Diversity and unpredictability of coordination links
Spatial interference index	$p < 0.001$3 with $p < 0.001$4	Tight packing and collision-avoidance difficulty
Goal overlap score	$p < 0.001$5	Shared or contested objectives

The agent dependency entropy is defined on the set of edges with positive weight. If $p < 0.001$6 denotes the set of such edges, then

$p < 0.001$7

and

$p < 0.001$8

Its stated intuition is to measure unpredictability and diversity of coordination links (Ebadulla et al., 9 Jul 2025). In practice, higher entropy indicates that coordination is distributed across multiple interaction channels rather than concentrated in a small subset of agent pairs.

The spatial interference index begins with the per-agent average minimum distance,

$p < 0.001$9

followed by the inverse-distance transformation

$G=(V,E)$0

with scaling factor $G=(V,E)$1 so that higher $G=(V,E)$2 means tighter packing (Ebadulla et al., 9 Jul 2025). This term directly targets collision-avoidance difficulty in tasks where agents must move in close proximity.

The goal overlap score is defined by

$G=(V,E)$3

where $G=(V,E)$4 is the number of agents, $G=(V,E)$5 is the number of goals, and $G=(V,E)$6 is the fraction of goals that are privately assigned, with $G=(V,E)$7 meaning all goals are independent (Ebadulla et al., 9 Jul 2025). The score grows when agents outnumber goals and the local-cooperation ratio is low.

3. Aggregation and empirical validation

The three components are linearly combined using empirically tuned weights:

$G=(V,E)$8

This yields a scalar $G=(V,E)$9 for each task, which is used to order tasks from “easy” to “hard” (Ebadulla et al., 9 Jul 2025).

Validation was carried out on 15 validation tasks spanning a range of coordination patterns. For each task, a purely random policy was run to estimate empirical difficulty, for example by average success rate or cumulative reward over many episodes. The predicted complexity ranking induced by $V=\{1,\dots,N\}$0 was then compared with the empirical difficulty ranking using the Spearman rank correlation $V=\{1,\dots,N\}$1, producing

$V=\{1,\dots,N\}$2

The reported interpretation is that this high, statistically significant correlation confirms that $V=\{1,\dots,N\}$3 reliably predicts task difficulty (Ebadulla et al., 9 Jul 2025).

A common misconception is to treat any graph-derived scalar as interchangeable with this metric. The reported validation is specific: the score is validated against empirical difficulty determined by random agent performance evaluation in cooperative coordination settings, not against generic graph-theoretic hardness, not against communication cost, and not against worst-case planning complexity. This distinction matters when comparing it with other graph-based complexity notions.

4. Use in curriculum learning with MADDPG

The metric is embedded into a curriculum learning framework by sorting generated tasks in increasing order of $V=\{1,\dots,N\}$4 and training agents sequentially along that ordering (Ebadulla et al., 9 Jul 2025). The training rule is explicit: agents train on the current task until they either achieve $V=\{1,\dots,N\}$5 success rate or exceed 300 episodes, then advance to the next task.

The underlying learning algorithm is Multi-Agent Deep Deterministic Policy Gradient with centralized critics and decentralized actors. The reported hyperparameters are replay buffer 10 000, batch size 64, learning rate $V=\{1,\dots,N\}$6, $V=\{1,\dots,N\}$7, and $V=\{1,\dots,N\}$8 (Ebadulla et al., 9 Jul 2025). Within this setup, the graph-based metric serves as the curriculum scheduler rather than as a replacement for the MARL backbone.

This use clarifies the operational role of the score. It is not merely descriptive: it induces a task ordering strategy. The stated contribution is therefore twofold—a validated complexity metric for multi-agent curriculum design, and an application of that metric within a concrete MADDPG curriculum pipeline (Ebadulla et al., 9 Jul 2025).

5. Empirical behavior and curriculum-design implications

The framework was evaluated in two cooperative environments with different coordination regimes. In MultiWalker, a tight-coordination task, 2–4 bipeds must synchronize steps to transport a load, and small positional errors cause collapse; the paper identifies spatial interference and dependency entropy as critical in this environment (Ebadulla et al., 9 Jul 2025). In Simple Spread, $V=\{1,\dots,N\}$9 agents must each occupy $(i,j)\in E$0 landmarks; the environment exhibits loose coordination over space, but collision avoidance and goal assignment matter (Ebadulla et al., 9 Jul 2025).

In MultiWalker, curriculum agents were reported to converge in approximately 1 200 episodes versus more than 3 000 for random task sampling, and the curriculum yielded a 56× speed-up in sample efficiency (Ebadulla et al., 9 Jul 2025). In Simple Spread, curriculum achieved 93% final completion, whereas random sampling never progressed at 0%; a parameter-based curriculum sorting by agent count alone reached 53% (Ebadulla et al., 9 Jul 2025).

From these experiments, the paper states empirical guidelines for curriculum design. “Coordination Tightness as a Predictor” is the central guideline: tasks with high agent-dependency entropy and high spatial interference, described as tight coupling, benefit most from structured progression, with gains of up to 56× (Ebadulla et al., 9 Jul 2025). For loosely coupled tasks with low interference and low goal overlap, simple parameter-based orderings can suffice, but a complexity-based curriculum still offers consistent improvement (Ebadulla et al., 9 Jul 2025). The reported success-rate threshold of 0.6 and episode timeout of 300 are described as striking a balance between thorough learning on each subtask and overall training efficiency (Ebadulla et al., 9 Jul 2025).

A plausible implication is that the metric is particularly useful when naïve scalar proxies, such as agent count alone, fail to capture the coordination structure actually responsible for learning difficulty.

6. Relation to adjacent graph-based complexity notions

The broader literature uses graph structure to quantify several distinct forms of “complexity,” and these should not be conflated with the curriculum-oriented coordination complexity score above.

In coordination-graph MARL, treewidth is the principal graph invariant controlling the complexity of greedy joint-action selection. For a coordination graph with treewidth $(i,j)\in E$1, the induced decentralized constraint optimization problem can be solved in time $(i,j)\in E$2, and forests with $(i,j)\in E$3 admit exact action selection in $(i,j)\in E$4 (Yang et al., 2021). This is a complexity notion for execution-time optimization in value factorization, not a predictor of task ordering for curriculum learning.

Metric dimension provides another graph-based notion, defined as the minimum size of a resolving set $(i,j)\in E$5 such that every pair of vertices is distinguished by graph distances to landmarks in $(i,j)\in E$6. It measures the minimum number of reference points needed so that agents can self-locate by distance measurements alone; on planar graphs of maximum degree 6 the problem is NP-complete, whereas on outerplanar graphs it admits a polynomial-time algorithm (Diaz et al., 2011). This is a localization and distinguishability metric.

For directed graphs, spectral complexity is defined from the spectrum of the recurrence matrix and a Wasserstein distance. Its essential property is that it accounts for directed cycles, which in engineered and software systems increase risk through positive feedback loops, instabilities, and infinite execution loops (Mezić et al., 2018). Matrix energy, by contrast, defines graph complexity as the nuclear norm of the adjacency matrix,

$(i,j)\in E$7

and is presented as a measure of topological complexity that satisfies Weyuker’s nine criteria (Sinha et al., 2016).

Graph structural complexity takes a renormalization-like multi-scale view. For each node and scale, it evaluates the variance term $(i,j)\in E$8 and averages the resulting node-level sums over the graph; the measure peaks near the emergence of a giant component in Erdős–Rényi graphs and at bond-percolation thresholds in 2D and 3D lattices (Snarskii, 2024). This is a metric of multi-scale structural irregularity.

The phrase “coordination complexity” is also used in an information-theoretic sense: the minimal number of bits a centralized coordinator must broadcast so that distributed agents can choose actions that form a nearly optimal solution (Cummings et al., 2015). In binary-action polymatrix games, a different graph-based coordination-complexity metric is tied to whether each edge-labelled matrix satisfies

$(i,j)\in E$9

with polynomial-time solvability when the condition holds for all edges and NP-hardness otherwise (Deligkas et al., 2023).

Taken together, these works show that graph-based coordination complexity is not a single canonical object. The metric of (Ebadulla et al., 9 Jul 2025) is specifically a validated predictor of task difficulty for cooperative MARL curriculum design, built from graph entropy, interference, and goal overlap, and distinguished from graph invariants that quantify computational hardness, localization burden, cycle-induced interdependence, topological density, multi-scale irregularity, or communication requirements.