Similarity-based Population Diversity Indicator (SPDI)

• SPDI is a diversity metric that computes the mean pairwise Euclidean distance between chromosome encodings in evolutionary graph neural architecture search.
• It regulates genetic algorithm operations by adaptively switching crossover and mutation rates to balance exploration and exploitation in a vast discrete search space.
• SPDI’s integration within MACC-MGNAS improves convergence speed, solution quality, and computational efficiency, as shown by detailed ablation studies.

Searching arXiv for the cited paper and closely related diversity-indicator work to ground the article in current literature. arxiv_search(query="(Wang et al., 23 Sep 2025) A Modality-Aware Cooperative Co-Evolutionary Framework for Multimodal Graph Neural Architecture Search", max_results=5, sort_by="relevance") arxiv_search(query="Similarity-based Population Diversity Indicator SPDI genetic algorithm diversity arXiv", max_results=10, sort_by="relevance") Similarity-based Population Diversity Indicator (SPDI) is a diversity-control mechanism introduced in the modality-aware cooperative co-evolutionary framework MACC-MGNAS for multimodal graph neural architecture search. In that setting, SPDI quantifies how similar the current population of architecture chromosomes is by computing the mean pairwise Euclidean distance between their numeric encodings, and then uses the resulting scalar as a control signal for adaptive adjustment of crossover and mutation rates. Its stated purpose is to balance exploration and exploitation in a large, discrete, multimodal search space, thereby reducing slow convergence and premature convergence to suboptimal solutions (Wang et al., 23 Sep 2025).

1. Conceptual role and problem setting

SPDI is designed for GA-based graph neural architecture search in multimodal graph neural networks, where chromosomes encode discrete choices such as modality-specific message functions, aggregators, update rules, and fusion operators. The search space is described as huge and highly discrete, with architecture search characterized as NP-hard and comparable to a multi-choice knapsack problem. Within MACC-MGNAS, cooperative co-evolution decomposes chromosomes into modality-specific and fusion blocks; this divide-and-conquer structure accelerates search, but also increases the risk that the global population collapses into a narrow region of the architecture space (Wang et al., 23 Sep 2025).

The mechanism addresses a limitation identified by the authors: “conventional implementations lack adaptive mechanisms to balance exploration and exploitation, leading to slow convergence or premature convergence to suboptimal solutions.” In this formulation, premature convergence corresponds to a population that becomes too similar too early, so recombination mostly mixes nearly identical chromosomes. Conversely, excessively high diversity can cause inefficient wandering, with the algorithm continuing to explore without sufficiently refining promising regions. SPDI provides a quantitative indicator of population diversity at the coordinator level and switches the search regime accordingly.

The term “similarity-based” refers here to geometric similarity in chromosome-encoding space. Chromosomes are mapped to numeric vectors, pairwise Euclidean distances are computed, and small distances are interpreted as architectural similarity while large distances are interpreted as architectural dissimilarity. This makes SPDI a population-level statistic derived from pairwise similarity proxies rather than from fitness alone.

2. Formal definition

Each MGNN architecture is encoded as a chromosome

$\mathbf{C} = (g_1, g_2, \ldots, g_K),$

where each gene $g_i$ is a discrete component choice. At generation $t$, the coordinator maintains a global population

$$\mathcal{P}_t = \{\mathbf{C}_1, \mathbf{C}_2, \ldots, \mathbf{C}_N\},$$

with population size $N$. To evaluate similarity, each chromosome is mapped by an encoding function

$\pi(\mathbf{C}) \in \mathbb{R}^K.$

For two distinct chromosomes $\mathbf{C}_i, \mathbf{C}_j \in \mathcal{P}_t$, SPDI uses the pairwise Euclidean distance

$$d_{ij} = \big\| \pi(\mathbf{C}_i) - \pi(\mathbf{C}_j) \big\|_2, \quad i \neq j. \tag{24}$$

The population diversity indicator is the mean pairwise distance: $$\mathrm{SPDI}(\mathcal{P}_t) = \frac{2}{N(N-1)} \sum_{i<j} d_{ij}. \tag{25}$$

This definition yields a single scalar per generation. By construction, it increases when chromosomes are more dispersed in encoding space and decreases when the population clusters.

SPDI is coupled to a threshold $\tau_t$ that selects between exploit and explore modes. The crossover rate is set by

$g_i$0

and the mutation rate is set by

$g_i$1

The interpretation is explicit. High diversity triggers exploit mode, with high crossover and low mutation. Low diversity triggers explore mode, with low crossover and high mutation. In the former regime, recombination consolidates promising patterns across already diverse architectures; in the latter, mutation injects novel genes when recombination among near-identical parents is no longer productive (Wang et al., 23 Sep 2025).

3. Placement within MACC-MGNAS

Within MACC-MGNAS, SPDI is computed at the coordinator level on the full global population, not on modality-local subpopulations. It is evaluated once per global generation, before the population is split into modality and fusion blocks. The coordinator first computes all $g_i$2, then computes $g_i$3, and then sets $g_i$4 and $g_i$5 according to the threshold rule. Only after this step does the MACC decomposition proceed (Wang et al., 23 Sep 2025).

The decomposition of a chromosome is written as

$g_i$6

with modality blocks $g_i$7 and a fusion block $g_i$8. Local workers evolve these blocks independently, whereas the coordinator reassembles full chromosomes, evaluates global fitness, and applies the global GA operators. SPDI therefore measures diversity over full architectures rather than within any single modality or the fusion block alone. The paper explicitly states that there is no separate SPDI per modality.

This placement makes SPDI complementary to the other two major components of MACC-MGNAS. MACC supplies the structural decomposition and cooperative co-evolution protocol. MADTS, the modality-aware dual-track surrogate, accelerates worker-side local evolution through Gaussian-process surrogates and variance-aware fusion of local and global fitness signals. SPDI, by contrast, regulates global search dynamics. The relation is described as decoupled but synergistic: MADTS improves efficiency and local proposal quality, while SPDI prevents global population collapse or unproductive drift (Wang et al., 23 Sep 2025).

Operationally, the generation cycle can be summarized as follows. The algorithm initializes the population, archives, and incumbent architecture; at each generation it computes pairwise distances and SPDI, adjusts crossover and mutation rates, decomposes chromosomes, evolves blocks locally with MADTS, merges elites, evaluates global fitness, applies variation with the SPDI-governed rates, and updates the incumbent. SPDI thus appears at the beginning of every generation as a feedback signal that controls the subsequent variation step.

4. Exploration–exploitation regulation and convergence behavior

The rationale for SPDI is that architecture search requires both sufficient heterogeneity and sufficient consolidation. When diversity is high, strong crossover and weak mutation exploit the fact that chromosomes already differ enough for recombination to be useful. When diversity is low, weak crossover and strong mutation counteract the loss of search breadth and reintroduce novelty into the population. In the MACC-MGNAS workflow, this is especially relevant because local workers may converge within modality-specific subspaces even while the globally assembled architectures remain suboptimal (Wang et al., 23 Sep 2025).

The empirical convergence plots reported in the paper associate a distinctive mid-run behavior with this adaptive mechanism. MACC-MGNAS reaches high validation F1 quickly, around generation 7, and then shows a second improvement around generation 17. The paper attributes this second lift directly to SPDI’s adaptive regulation of crossover and mutation, stating that the “diversity injection reactivates exploration without losing stability.” By contrast, GA, EDA, PSO, BO, and DC-NAS either plateau early or progress more slowly. PSO and EDA are reported to exhibit temporary drops due to uncontrolled stochastic exploration, and BO converges to a lower ceiling (Wang et al., 23 Sep 2025).

The reported quantitative pattern is consistent with that interpretation. MACC-MGNAS reaches approximately $g_i$9 F1 by generation 7, exceeds GA, EDA, PSO, and DC-NAS, and attains approximately $t$0 F1 around generation 20, while BO is around $t$1 and the remaining baselines are at or below $t$2. Final convergence is reported as approximately $t$3 with the lowest variance, while BO approaches approximately $t$4. This suggests that SPDI acts not merely as a stagnation detector, but as a stabilizing controller for diversity injection.

A further qualitative observation comes from the parallel-coordinate visualization of best architectures across runs. The figure shows recurrent motifs such as multiplicative message operators and normalized fusion together with residual architectural variation. The authors interpret this as evidence that SPDI consolidates effective patterns while preserving diversity across runs, so the search does not collapse to a single architecture.

5. Empirical effect and ablation evidence

The ablation study isolates the contribution of SPDI by comparing four variants: a centralized GA without MACC, a MACC+MADTS system without SPDI, a MACC+SPDI system without MADTS, and the full MACC-MGNAS system. The comparison shows that removing SPDI reduces both accuracy and efficiency (Wang et al., 23 Sep 2025).

For the full system, the reported mean F1 is $t$5, best F1 is $t$6, and computation cost is $t$7 GPU-hours. For the variant without SPDI, the mean F1 falls to $t$8, best F1 to $t$9, and GPU-hours increase to $$\mathcal{P}_t = \{\mathbf{C}_1, \mathbf{C}_2, \ldots, \mathbf{C}_N\},$$0. The paper summarizes this as evidence that “SPDI plays a crucial role in stabilizing convergence and improving efficiency.” In the supplied interpretation, removing SPDI lowers mean F1 by $$\mathcal{P}_t = \{\mathbf{C}_1, \mathbf{C}_2, \ldots, \mathbf{C}_N\},$$1 points, lowers best F1 by $$\mathcal{P}_t = \{\mathbf{C}_1, \mathbf{C}_2, \ldots, \mathbf{C}_N\},$$2, increases GPU cost by $$\mathcal{P}_t = \{\mathbf{C}_1, \mathbf{C}_2, \ldots, \mathbf{C}_N\},$$3, and increases run-to-run variance.

At the system level, MACC-MGNAS is reported to achieve an F1-score of $$\mathcal{P}_t = \{\mathbf{C}_1, \mathbf{C}_2, \ldots, \mathbf{C}_N\},$$4 within only 3 GPU-hours on the VulCE dataset, outperforming the state-of-the-art competitor by $$\mathcal{P}_t = \{\mathbf{C}_1, \mathbf{C}_2, \ldots, \mathbf{C}_N\},$$5 F1 while reducing computation cost by $$\mathcal{P}_t = \{\mathbf{C}_1, \mathbf{C}_2, \ldots, \mathbf{C}_N\},$$6 (Wang et al., 23 Sep 2025). Those gains are system-level rather than SPDI-only gains, but the ablation indicates that adaptive diversity control contributes materially to both final accuracy and wall-clock efficiency.

The empirical record therefore supports three specific claims about SPDI in this context: it improves final performance, reduces computation cost relative to a fixed-rate ablation, and lowers variance across runs. The broader implication that adaptive diversity control stabilizes cooperative co-evolutionary NAS is suggested by the convergence and ablation evidence rather than isolated in a theorem.

6. Relation to broader similarity-based diversity indicators

Although SPDI is introduced in MACC-MGNAS as a mean pairwise Euclidean-distance controller over architecture chromosomes, the underlying pattern belongs to a broader family of similarity-based diversity indicators. In symbolic regression, for example, population diversity has been defined through pairwise tree similarity and distance, with both individual-level and population-level average pairwise distances functioning as diversity scores (Burlacu et al., 2019). That work uses a fast hash-based tree similarity measure, defines

$$\mathcal{P}_t = \{\mathbf{C}_1, \mathbf{C}_2, \ldots, \mathbf{C}_N\},$$7

and then aggregates distances across the population. The term SPDI is not used there, but the supplied exposition identifies the resulting average pairwise distance as exactly a similarity-based population diversity indicator.

In runtime analysis of neutral evolutionary processes, the same general template appears with Hamming distance instead of Euclidean or tree distance. Population diversity is measured as the sum of pairwise Hamming distances,

$$\mathcal{P}_t = \{\mathbf{C}_1, \mathbf{C}_2, \ldots, \mathbf{C}_N\},$$8

or by its normalized variants, and the paper derives an exact drift equation and an equilibrium diversity value under unbiased mutation (Lengler et al., 2023). This shows that SPDI-like measures need not only be diagnostic; they can also admit analytic dynamical characterizations.

A second line of related work interprets similarity-based diversity through kernel matrices rather than direct averaging. Solow–Polasky diversity uses the exponential similarity matrix

$$\mathcal{P}_t = \{\mathbf{C}_1, \mathbf{C}_2, \ldots, \mathbf{C}_N\},$$9

and is mathematically identical to magnitude in the cited finite exponential-kernel setting (Emmerich, 29 Apr 2026). Comparative work in multiobjective diversity optimization places Solow–Polasky Diversity, Max–Min Diversity, and Riesz $N$0-Energy within a common indicator-based framework and studies properties such as monotonicity in species, twinning, monotonicity in distance, uniformity of maximizing sets, subset selection, and submodularity (Pereverdieva et al., 2024). This suggests that SPDI, while simpler than inverse-matrix or energy-based constructions, belongs to a broad design space of pairwise-similarity aggregators.

Other similarity-based effective-number constructions take abundance and similarity as joint inputs. The Leinster–Cobbold index

$N$1

generalizes Hill numbers through a similarity matrix and admits a decomposition into richness, balance, taxonomic similarity, and an equilibration factor (Chen et al., 2022). The DSN index extends this line further by incorporating both similarities and network connections into a Hill-type diversity operator (Kinjo, 2024). The Vendi family constructs diversity from the eigenvalues of a normalized similarity matrix and uses a Hill-number parameter $N$2 to tune sensitivity to rare versus common modes (Pasarkar et al., 2023). These are not direct ancestors of MACC-MGNAS SPDI, but they indicate that “similarity-based diversity indicator” spans several mathematically distinct aggregation schemes.

7. Scope, limitations, and extensions

The MACC-MGNAS paper presents SPDI as conceptually general: any GA-based NAS or evolutionary search over discrete architectures can, in principle, define a numeric encoding $N$3, compute average pairwise distances, and adapt crossover and mutation via a diversity threshold (Wang et al., 23 Sep 2025). This suggests applicability beyond multimodal MGNN search, including unimodal GNAS and other discrete evolutionary optimization problems.

The reported limitations arise at the level of the overall framework but bear directly on SPDI. Evolutionary NAS remains costly on very large graphs even with MADTS and SPDI. The chromosome length is predefined, which constrains openness of the search space and means that SPDI operates over fixed-length encodings. The evaluation is performed on one static vulnerability knowledge graph, so dynamic or continually evolving graphs are outside the tested scope. The paper also treats the threshold $N$4 abstractly, allowing it to be fixed or adapted, but does not specify a learned thresholding policy.

Several extensions are described as implicit or plausible rather than already implemented. A plausible extension is a more nuanced diversity indicator combining geometric distance with semantic or performance-based notions of architectural difference. Another plausible extension is adaptive or learned thresholding for $N$5. A third is modality-wise diversity control, so that worker-side operators would also be regulated rather than only the coordinator’s global operators. These directions are consistent with broader diversity-measure literature, which evaluates indicators in terms of properties such as monotonicity in individual varieties, twinning, and monotonicity in distance (Corriveau et al., 2015).

In that sense, SPDI occupies a specific position within the larger theory of diversity in evolutionary computation. It is not a universal diversity metric, nor is it an abstract biodiversity index transferred wholesale into NAS. It is a coordinator-level, similarity-derived control signal for a cooperative co-evolutionary GA, instantiated as mean pairwise Euclidean distance over full chromosome encodings and used to switch between exploration and exploitation. Its significance lies in turning population similarity into an explicit feedback variable within multimodal neural architecture search.