Multinomial Adjusted Rand Index (MARI)

• MARI is a similarity measure that adjusts the Rand Index using a multinomial null model, providing an unbiased baseline for comparing clusterings.
• It assigns objects to clusters based on empirical proportions, allowing seamless application to both hard and fuzzy clustering scenarios with efficient computation.
• MARI is ideal for contexts with unbalanced or uncertain cluster definitions, offering a conservative alternative to the classical ARI with clearer discrimination of true associations.

The Multinomial Adjusted Rand Index (MARI) is a similarity measure for assessing the correspondence between two clusterings of a finite set, distinguished by its adjustment for chance under a multinomial (categorical) assignment model. Unlike the classical Adjusted Rand Index (ARI), which relies on a hypergeometric (permutation) null model that fixes cluster sizes, MARI employs an independent assignment mechanism where objects are assigned to clusters in proportion to their empirical sizes, yielding a more interpretable and conservative adjustment. This approach is particularly well-suited for comparisons in contexts with uncertain or fuzzy cluster definitions, dependent partitions, or pronounced cluster size imbalance. MARI consistently offers unbiased baselining under conditions of independent draws and admits transparent generalization from hard to fuzzy clusterings (DeWolfe et al., 2023, Sundqvist et al., 2020).

1. Definition and Mathematical Framework

Let $C_1$ and $C_2$ denote two clusterings of $N$ objects. The unadjusted Rand Index (RI) between $C_1$ and $C_2$ is defined as the proportion of unordered pairs $\{i,j\}$ that are concordant in both clusterings, i.e., either both are grouped together in both or separated in both: $$RI(C_1, C_2) = \frac{|\text{concordant pairs}|}{\binom{N}{2}}$$ Since random clusterings often result in RI significantly above zero, the ARI incorporates an expectation term under a specified random model: $ARI = \frac{RI - E[RI]}{1 - E[RI]}$ Where $E[RI]$ is the expected value of RI under the chosen null model.

MARI arises when one selects the multinomial (categorical) random model for $E[RI]$, in which each object is independently assigned to cluster $\ell$ with probability $p_1^\ell$ (for $C_1$) and $p_2^k$ (for $C_2$); $p_1^\ell = c_1^\ell/N$ and $p_2^k = c_2^k/N$ with $c_1^\ell$ and $c_2^k$ being cluster sizes. This leads to the closed-form expectation: $$E_{\mathrm{Mult}}[RI] = \left( \sum_\ell (p_1^\ell)^2 \right) \left( \sum_k (p_2^k)^2 \right) + \left( 1 - \sum_\ell (p_1^\ell)^2 \right) \left( 1 - \sum_k (p_2^k)^2 \right)$$ MARI is then computed as: $$\mathrm{MARI} = \frac{RI - E_{\mathrm{Mult}}[RI]}{1 - E_{\mathrm{Mult}}[RI]}$$ By construction, $\mathrm{MARI}=1$ for identical clusterings and is unbiased (expected value zero) under the multinomial null model (DeWolfe et al., 2023, Sundqvist et al., 2020).

2. The Multinomial Random Assignment Model

The multinomial model prescribes independent assignment of each object to clusters according to observed size proportions. Cluster sizes are not fixed, but their expected frequency matches the empirical proportions. This randomization accurately reflects sampling variability and does not enforce hard constraints, rendering it preferable when the independence of assignments is assumed or when partition dependencies exist (Sundqvist et al., 2020).

Correspondingly, in the infinite-population context of (Sundqvist et al., 2020), (cluster, label) pairs are i.i.d. draws from a discrete distribution with joint cell probabilities $\pi_{k\ell}$, so contingency table cell counts $(n_{k\ell})$ follow a multinomial distribution. The marginal probabilities are $\pi_{k\cdot}$ and $\pi_{\cdot\ell}$, and the expectation is given by $\sum_{k,\ell} (\pi_{k\cdot}\pi_{\cdot\ell})^2$ under independence.

3. Computational Methods and Complexity

MARI can be calculated using straightforward combinatorial manipulations for both hard and fuzzy clusterings, with cost $O(N + n_1 n_2)$ for contingency construction and sum computations. For hard clusterings, contingency tables are built by a single pass over the data; in fuzzy contexts, cluster counts $c_1^\ell$ are replaced by the sum of memberships, preserving computational scalability (DeWolfe et al., 2023).

The algorithm presented in (Sundqvist et al., 2020) leverages sparse representation of contingency tables. Contingency cell pairs are extracted efficiently using linear-time bucket or radix sorting, which reduces space and time complexity to $O(N)$. The aricode package implements these routines without explicit construction of full contingency matrices.

4. Comparative Properties: MARI vs. ARI

ARI (classical) employs a hypergeometric null model, which constrains marginals and presumes independence via sampling without replacement. MARI, in contrast, uses a multinomial null that (i) does not hold cluster sizes fixed, (ii) permits dependencies, and (iii) incorporates sampling randomness. The difference in centering is non-negligible for small $N$ or unbalanced partitions: ARI demonstrates an $O(1/N)$ bias under the multinomial model, with possible deviations exceeding 0.01–0.05 for small $N$ ($N<64$), but vanishing for large $N$ ($N\geq200$) (Sundqvist et al., 2020).

On benchmark datasets (UCI, simulated outliers), permutation-model ARI may yield values $>0.5$ for random labelings, while MARI centers around $0$ and typically stays below $0.3$, signaling a more stringent baseline when cluster assignments are random or partitions are independent (DeWolfe et al., 2023).

5. Application to Hard and Fuzzy Clusterings

MARI is naturally generalized from partitions (hard clusterings) to fuzzy clusterings. In the latter, each object $i$ possesses membership vectors, e.g., $C_1(i) \in \Delta_{n_1}$, and counts $c_1^\ell$ are replaced by $\sum_i [C_1(i)]_\ell$. The expectation formula remains unchanged structurally, with all sums substituting counts by accrued memberships. This equivalence is formalized algebraically and supports efficient computation for both clustering types (DeWolfe et al., 2023).

Empirically, MARI yields lower adjusted indices than permutation-model ARI when clusters are fuzzy or uneven, and penalizes size imbalance more strongly than ARI. For mixture model scenarios, MARI under the categorical null has been observed to be more conservative, providing clearer discrimination between true associations and random baseline (DeWolfe et al., 2023).

6. Illustrative Examples and Practical Guidance

Examples contrasting MARI and ARI for balanced and skewed clusterings highlight that MARI more severely penalizes extreme size imbalance. In toy cases such as $C_1 = (1,1,2,2,3)$ vs. $C_2 = (A,A,A,B,B)$ with $N=5$, ARI approximates $0.04$ while MARI computes to $-0.18$, reflecting the difference in null model assumptions (Sundqvist et al., 2020).

Practical recommendations are:

• Prefer MARI over ARI when cluster sizes are unknown, partitions may be dependent, or $N$ is small.
• For large $N$ and balanced clusters, ARI and MARI are nearly equivalent.
• Use MARI’s expectational formula as a default when “chance” is best seen as i.i.d. multinomial assignment, particularly in fuzzy or highly imbalanced contexts (DeWolfe et al., 2023, Sundqvist et al., 2020).

7. Summary and Implications

The Multinomial Adjusted Rand Index provides an unbiased, closed-form chance correction built on a transparent i.i.d. categorical mechanism. It guarantees meaningful centering under independence and exhibits conservative behavior when clusters are uneven or fuzzy. MARI’s computational efficiency and clarity of assumptions render it preferable for a wide range of clustering comparison tasks, especially where permutation-model ARI biases may obscure true similarity relationships. A plausible implication is that MARI should be the baseline similarity measure when random assignment better reflects the baseline structure than fixed-size permutations (DeWolfe et al., 2023, Sundqvist et al., 2020).