Inner Parliament and Network Modularity

• Inner Parliament is a theory that uses network modularity to determine optimal parliamentary size by identifying cohesive social communities within constituent networks.
• It refines traditional laws like the cubic root law by applying analytical models and empirical findings to capture sublinear scaling of legislative bodies.
• The framework employs a network-theoretic model with key parameters such as mobility and modularity to optimize constituency partitioning and representation.

The inner parliament concept formalizes the determination of a representative chamber's optimal size by leveraging network modularity within a growing random network of constituents. It reframes the question of how many parliamentarians best represent a population as a principled optimization over the underlying community structure of the social network, rather than relying on classical communication or efficiency arguments. The core principle is that the optimal number of constituencies (and thus representatives) is the partition size that maximizes the modularity of the graph representing social interactions among citizens. This approach enables analytical predictions and systematic empirical scrutiny of scaling laws relating chamber size to population.

1. Classical Foundations: The Cubic Root Law

The cubic root law, attributed to Taagepera and collaborators, posits that the optimal size of a parliament, $S$, should scale as $S \propto N^{1/3}$, where $N$ is the effective population of a country. This law was originally motivated by heuristic considerations and has received empirical support in diverse democracies. However, the law's exponent is not universally supported by all data: recent analyses of European parliamentary sizes yield a slightly larger exponent, specifically $\gamma \approx 0.44$, in fits of chamber size to population, indicating deviation from the strict cubic root scaling (Gamberi et al., 2020).

2. Network-Theoretic Model of Parliamentary Representation

To derive a more mechanistic foundation for parliament size, the inner parliament framework models a country's population as a growing random graph. In this model:

• Each node corresponds to a citizen;
• Edges represent stable social ties, established as the network grows;
• Each citizen is assigned to one of $D$ constituencies, $\sigma_i \in \{1,\ldots,D\}$, typically ensuring constituencies are approximately equal in size via a round-robin assignment.

Network growth occurs sequentially. Upon joining, each new node connects to existing nodes preferentially by degree, modulated by a constituency-dependent attachment bias. The connection probability to an existing node $i$ upon adding node $N$ is:

$$p_{iN} = m\,\frac{\sum_{\ell=1}^{N-1} A_{i,\ell}(N-1)}{L(N-1)/D}\,p(\sigma_i|\sigma_N)$$

where $L(N)$ is the total number of links, $S \propto N^{1/3}$0 is the number of stubs per new node, and $S \propto N^{1/3}$1 is the adjacency matrix. The constituency-based attachment is parameterized by the "mobility" parameter $S \propto N^{1/3}$2, via:

$S \propto N^{1/3}$3

With $S \propto N^{1/3}$4 there is perfectly assortative, within-constituency attachment; with $S \propto N^{1/3}$5 constituents ignore partitions and connect uniformly.

This model encodes the hypothesis that socially-cohesive groups should align with representational districts, providing a micro-foundation for the design of parliaments.

3. Modularity as Optimality Criterion

Adopting the Newman-Girvan modularity function as an objective criterion, the optimal number of constituencies is defined as that which maximizes the modularity of the network's partition into districts:

$S \propto N^{1/3}$6

Modularity compares the real density of intra-constituency ties with a null model preserving degrees but assigning edges at random. High positive modularity indicates that constituencies correspond to clusters with more internal connectivity than expected by chance.

The expected modularity can be analytically approximated as:

$S \propto N^{1/3}$7

where $S \propto N^{1/3}$8 is Euler's constant and $S \propto N^{1/3}$9 is the digamma function. Maximizing this with respect to $N$0 yields the optimal chamber size (Gamberi et al., 2020).

4. Non-Monotonicity and Optimal Partition Size

The modularity as a function of constituency number $N$1 is provably non-monotonic:

• For $N$2, everyone is in one constituency, so modularity is zero.
• For $N$3, each person forms their own constituency, again yielding zero modularity.

For intermediate $N$4, modularity reaches a peak. Small $N$5 leads to constituencies that are socially heterogeneous, degrading community quality. Large $N$6 produces many singleton or small groups, diluting community structure. The maximizer of $N$7 aligns with the finest partition that still captures genuine social cohesion.

Formally, the optimal $N$8 is found not at these extremes, but as an interior maximum—the solution to

$N$9

which generally requires numerical root-finding due to special functions.

5. Analytical Scaling Law and Empirical Evidence

Asymptotic analysis in the large network limit ($\gamma \approx 0.44$0) demonstrates that the optimal number of constituencies satisfies:

$\gamma \approx 0.44$1

More generally, the fit $\gamma \approx 0.44$2 produces $\gamma \approx 0.44$3, $\gamma \approx 0.44$4 for high within-constituency affinity ($\gamma \approx 0.44$5) (Gamberi et al., 2020). By contrast, empirical European parliament data exhibit $\gamma \approx 0.44$6, which is intermediate between the theoretical prediction and the longstanding cubic root law's $\gamma \approx 0.44$7.

This result is summarized in the following table:

Prediction/Observation	Exponent $\gamma \approx 0.44$8	Context
Theoretical, maximal modularity (large $\gamma \approx 0.44$9)	$D$0	Model with $D$1
Empirical, European parliaments	$D$2	Eurostat (2017) data
Classical cubic root law	$D$3	Taagepera's conjecture

The deviation between theory and observation is attributed to modeling assumptions, particularly homogeneous mobility $D$4 across the whole population, lack of spatial constraints, and the abstraction of constituencies as virtual rather than strictly geographical.

6. Implications and Interpretation

Maximizing modularity enforces that representation is strongly rooted in community structure: constituencies form the most cohesive possible clusters under the network's constraints. This approach reframes the design of legislative chambers as a variant of the community-detection problem in networks. The resulting square-root scaling prediction offers an analytically tractable and mechanistically clear explanation for sublinear growth in parliament size with population—a hallmark of effective representation.

The analysis suggests that further refinement of the model, including spatial heterogeneity in the mobility parameter or more elaborate rules for constituency formation, could reconcile the theoretical exponent with empirical values. It implies that the size of representative assemblies can be grounded in the micro-structure of social interaction, rather than remaining at the level of empirical regularity or political compromise (Gamberi et al., 2020).