Group-Differentiated Growth

• Group-differentiated growth is a multidisciplinary framework that defines growth through internal structural differences, subgroup embeddings, and differentiated constituents.
• It integrates approaches from geometric group theory, social network analysis, and cell biology, employing methods such as relative growth, rank–index rigidity, and diffusion models.
• The framework reveals that internal organization, ambient geometry, and interaction topology shape growth dynamics, offering precise predictive rules across diverse systems.

Group-differentiated growth denotes a family of research perspectives in which growth is resolved through differences between groups, subgroup embeddings, decomposed factors, or differentiated constituents, rather than reduced to a single aggregate law. In finitely generated group theory, this perspective appears in the study of word growth, relative subgroup growth, distortion, rank–index relations, and automorphism growth; in social and biological systems, it appears in the study of heterogeneous group sizes, platform-specific diffusion mechanisms, and collective growth produced by internal specialization. The shared theme is structural dependence: growth is governed by ambient geometry, decomposition, heterogeneity, or interaction topology, and not only by a scalar rate (Davis et al., 2012, Grigorchuk et al., 2018, Vranić et al., 2022, Yamagishi et al., 2015).

1. Conceptual range of the term

In geometric group theory, growth is classically defined from a finite generating set $A$ by the ball-counting function

$$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$

with the equivalence class of $\gamma_G^A$ independent of the generating set. This yields the familiar polynomial, exponential, and intermediate regimes, but several of the cited works replace this coarse invariant by finer ones: the relative growth $g_H(r)=\#(B_G(r)\cap H)$ of a subgroup inside an ambient group, the distortion $\Delta_H^G(r)$, the rank $d(H)$ of finite-index subgroups as a function of $[G:H]$, and the growth of elements under endomorphisms or automorphisms (Grigorchuk, 2011, Davis et al., 2012, Falconer et al., 2011).

In the social-group literature, the basic unit is not a finitely generated algebraic group but a social group as a mesoscopic object inside a larger social system. The central variables are group size $S_i(t)$, new members $N_i(t)$, and logarithmic growth rate $R_i(t)=\log\frac{S_i(t)}{S_i(t-1)}$. Here “universal growth” refers to the invariance of normalized size distributions, while “group-differentiated growth” enters through differences in mechanisms, particularly the balance between social diffusion and random diffusion, or through persistent group-specific growth rates $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$0 (Vranić et al., 2022, Grabowicz et al., 2011).

In multicellular aggregates, the same expression is used in a still different sense: a collective of initially identical cells differentiates into distinct types, and the aggregate achieves a higher growth rate than isolated cells by division of labor and exchange of diffusible products. Growth is therefore a property of the differentiated group composition, not merely of the isolated constituent (Yamagishi et al., 2015).

This suggests that the phrase is best read as a cross-disciplinary interpretive label for structurally resolved growth: the relevant “groups” may be algebraic groups, social communities, or differentiated cell collectives, but in each case growth is explained by internal differentiation or by the way subunits are embedded in a larger structure.

2. Geometric group theory: from word growth to relative and spectral differentiation

The classical background is Milnor’s problem on the growth of finitely generated groups. Polynomial growth is characterized by Gromov’s theorem as equivalent to virtual nilpotence, while Grigorchuk’s construction showed that finitely generated groups of intermediate growth do exist, thereby refuting the dichotomy “polynomial or exponential” (Grigorchuk, 2011). This coarse trichotomy is only a first layer. A more differentiated picture emerges once growth is indexed by generating sets, subgroups, or subsemigroups.

For subgroup embeddings, the relative growth

$$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$1

and the distortion

$$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$2

measure how the subgroup $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$3 appears inside the ambient geometry of $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$4. For infinite cyclic subgroups, the classification is especially sharp: every nonzero superadditive function of at most exponential growth can occur as the relative growth of an embedding $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$5, and every increasing superadditive function can occur as the distortion of an embedding of $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$6 into a finitely generated solvable group (Davis et al., 2012). In the cyclic case, linear relative growth is equivalent to being undistorted, whereas exponential distortion forces relative growth at least $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$7. The same paper also constructs pathological finitely generated subgroup embeddings whose distortion is not equivalent to any superadditive function and is not bounded above by any recursive function, while the relative growth can still be $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$8 in the non-effective sense (Davis et al., 2012). Group-differentiated growth here is literally ambient-dependent growth.

A second refinement studies how the exponential growth rate depends on the generating set itself. For a non-elementary hyperbolic group $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$9, the set

$\gamma_G^A$0

is well-ordered, and every real number can occur as $\gamma_G^A$1 for only finitely many generating sets up to $\gamma_G^A$2 (Fujiwara et al., 2020). The corresponding set for finitely generated non-elementary subgroups is also well-ordered, and the same remains true after passing to finitely generated non-elementary subsemigroups. For non-abelian limit groups over a free group, including free groups and closed hyperbolic surface groups, the growth ordinal is exactly $\gamma_G^A$3 (Fujiwara et al., 2020). Thus, even inside a fixed hyperbolic group, admissible growth rates form a discrete, highly structured spectrum rather than a continuum.

A third axis appears in finite-group product growth. For finite subsets $\gamma_G^A$4, one studies $\gamma_G^A$5 rather than ball growth. The survey by Helfgott emphasizes that the qualitative behaviour of $\gamma_G^A$6 depends strongly on the ambient group: simple linear groups, solvable linear groups, and permutation groups require different tools, and “small growth” is best understood through approximate subgroup structure and product theorems (Helfgott, 2013). In this setting, group-differentiated growth means that the same quantitative question—how large $\gamma_G^A$7 must be—has sharply different answers according to the structural class of the ambient group.

3. Intermediate growth and rank–index rigidity

A particularly fine form of group-differentiated growth is the relation between the rank $\gamma_G^A$8 of a finite-index subgroup and its index $\gamma_G^A$9. For a finitely generated residually finite group, the global rank gradient is

$g_H(r)=\#(B_G(r)\cap H)$0

and along a descending chain $g_H(r)=\#(B_G(r)\cap H)$1 one studies

$g_H(r)=\#(B_G(r)\cap H)$2

The paper “On the rigidity of rank gradient in a group of intermediate growth” introduces the stronger notion of $g_H(r)=\#(B_G(r)\cap H)$3-RG rigidity, in which $g_H(r)=\#(B_G(r)\cap H)$4 and $g_H(r)=\#(B_G(r)\cap H)$5 are related by two-sided inequalities after applying scaling functions $g_H(r)=\#(B_G(r)\cap H)$6 and $g_H(r)=\#(B_G(r)\cap H)$7 (Grigorchuk et al., 2018).

For the original Grigorchuk group $g_H(r)=\#(B_G(r)\cap H)$8, which is an infinite finitely generated 2-group of intermediate word growth, just-infinite, and a branch self-similar automorphism group of the rooted binary tree, the main theorem states that $g_H(r)=\#(B_G(r)\cap H)$9 is normally $\Delta_H^G(r)$0-RG-rigid with

$\Delta_H^G(r)$1

More precisely, there exists $\Delta_H^G(r)$2 such that for every nontrivial normal subgroup $\Delta_H^G(r)$3,

$\Delta_H^G(r)$4

Since $\Delta_H^G(r)$5 is just-infinite, every nontrivial normal subgroup has finite index, so the theorem covers all nontrivial normal subgroups (Grigorchuk et al., 2018).

The result is substantially finer than ordinary word-growth classification. Word growth places $\Delta_H^G(r)$6 somewhere between polynomial and exponential. Rank–index rigidity says that, for normal finite-index subgroups, $\Delta_H^G(r)$7 and $\Delta_H^G(r)$8 are equivalent up to multiplicative constants. In this sense, the growth of subgroup rank is tuned to a $\Delta_H^G(r)$9-scale that is invisible to the coarse intermediate-growth label (Grigorchuk et al., 2018).

The explicit model sequence is provided by vertex stabilizers. If $d(H)$0 is a vertex at level $d(H)$1 of the rooted binary tree and $d(H)$2, then for $d(H)$3,

$d(H)$4

Hence along this chain,

$d(H)$5

showing that the main inequalities are essentially sharp (Grigorchuk et al., 2018).

The proof depends on the branch structure of $d(H)$6, the decomposition of level stabilizers and rigid stabilizers into direct products of copies of a fixed subgroup $d(H)$7, and nilpotent and elementary abelian 2-group quotients used to estimate ranks. A key proposition shows that every nontrivial normal subgroup lies between rigid stabilizers at nearby levels, allowing both index and rank to be estimated in terms of the same level parameter (Grigorchuk et al., 2018).

This section also clarifies a possible misconception. The broader literature relates rank gradient to cost and $d(H)$8-Betti numbers, and for amenable groups rank gradient along Farber chains is $d(H)$9. The Grigorchuk-group result shows that vanishing rank gradient in the usual asymptotic sense does not preclude a highly rigid global relation between rank and index for normal subgroups (Grigorchuk et al., 2018).

4. Endomorphism and automorphism growth under decomposition

For an endomorphism $[G:H]$0 of a finitely generated group with generating set $[G:H]$1, Bowen’s growth rate is

$[G:H]$2

Falconer, Fine, and Kahrobaei show that this can be computed from generators alone: if

$[G:H]$3

then $[G:H]$4, the limit $[G:H]$5 exists, and

$[G:H]$6

They also prove the generator bound $[G:H]$7, where $[G:H]$8, and the power rule $[G:H]$9 (Falconer et al., 2011).

The same paper develops a layered comparison theory. Growth rate is invariant under restriction to finite-index $S_i(t)$0-invariant subgroups; on quotients it cannot increase; and on nilpotent groups the contribution of deeper lower-central-series layers is discounted by a factor $S_i(t)$1. The main nilpotent formula is

$S_i(t)$2

correcting Bowen’s statement by the essential $S_i(t)$3 exponent (Falconer et al., 2011). The Heisenberg example

$S_i(t)$4

exhibits the point sharply: the central layer grows by $S_i(t)$5, but the global growth rate is $S_i(t)$6, not $S_i(t)$7, because central growth contributes with weight $S_i(t)$8 (Falconer et al., 2011).

The 2026 paper “Automorphism growth and group decompositions” systematizes this decomposition principle. For direct products

$S_i(t)$9

with $N_i(t)$0 free abelian and $N_i(t)$1, one has

$N_i(t)$2

and similarly for conjugacy length. The dominant exponential rate is therefore the maximum of the rates on the factors and on the abelianization (Fioravanti, 11 Mar 2026).

For $N_i(t)$3-invariant graphs of groups, if all vertex groups are at most polynomial-growing under the restriction, then the whole group has at most polynomial growth; if some vertex group has docile exponential growth, then the whole outer automorphism is docile, with exponential rate equal to the maximum of the relevant vertex-group rates (Fioravanti, 11 Mar 2026). For free products

$N_i(t)$4

the fully irreducible case introduces an additional source of growth: the Perron number $N_i(t)$5 of the train-track transition matrix. The global exponential rate is

$N_i(t)$6

and the polynomial degree can increase by at most $N_i(t)$7 relative to the vertex-group contributions (Fioravanti, 11 Mar 2026).

Taken together, these results make group-differentiated growth a decomposition-sensitive phenomenon. Growth on the whole group is not arbitrary: it is assembled from growth on factors, graded pieces, vertex groups, or the free part, with precise “assembly rules” determined by the algebraic decomposition (Fioravanti, 11 Mar 2026).

5. Social groups: universal statistical laws and heterogeneous mechanisms

The paper “Universal growth of social groups: empirical analysis and modeling” studies Meetup groups in London and New York and Reddit subreddits as mesoscopic social units. Group size is defined by

$N_i(t)$8

where $N_i(t)$9 is the number of new members in month $R_i(t)=\log\frac{S_i(t)}{S_i(t-1)}$0, and logarithmic growth rate by

$R_i(t)=\log\frac{S_i(t)}{S_i(t-1)}$1

The datasets comprise $R_i(t)=\log\frac{S_i(t)}{S_i(t-1)}$2 Meetup groups and $R_i(t)=\log\frac{S_i(t)}{S_i(t-1)}$3 members in London, $R_i(t)=\log\frac{S_i(t)}{S_i(t-1)}$4 Meetup groups and $R_i(t)=\log\frac{S_i(t)}{S_i(t-1)}$5 members in New York, and $R_i(t)=\log\frac{S_i(t)}{S_i(t-1)}$6 subreddits with $R_i(t)=\log\frac{S_i(t)}{S_i(t-1)}$7 active members on Reddit (Vranić et al., 2022).

A central empirical result is that all three systems exhibit log-normal size distributions. Maximum-likelihood fits yield

$R_i(t)=\log\frac{S_i(t)}{S_i(t-1)}$8

so the Meetup systems are very similar, whereas Reddit has the same functional form but a much broader distribution (Vranić et al., 2022). The authors also normalize final group size by cohort means,

$R_i(t)=\log\frac{S_i(t)}{S_i(t-1)}$9

and show that, within each system, yearly cohort distributions collapse onto a single log-normal curve. Thus the output distribution is universal within each system, even though total system size and information and communication technology usage change over time (Vranić et al., 2022).

The mechanism is nevertheless differentiated. The microscopic model couples a bipartite affiliation network $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$00 with a social network $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$01. Active members create groups with probability $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$02, otherwise join an existing group either by social diffusion with probability $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$03, choosing proportionally to

$$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$04

or by random choice with probability $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$05 (Vranić et al., 2022). Using empirical $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$06, median estimates $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$07 for London and New York, $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$08 for Reddit, and $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$09 for all three systems, the model is calibrated by Jensen–Shannon divergence. The best matches are $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$10 for London, $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$11 for New York, and $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$12 for Reddit (Vranić et al., 2022).

This leads to a precise formulation of differentiated growth: the output class is log-normal across systems, but the relative importance of social versus random diffusion differs, with social interactions more critical for online groups such as Reddit than for offline groups such as Meetup (Vranić et al., 2022). The paper explicitly rejects a simple Gibrat interpretation: log-normal group sizes do not arise from size-independent proportional growth, because the observed logarithmic growth rates are themselves log-normal and depend on current group size (Vranić et al., 2022).

A complementary mechanism appears in the Flickr study “Heterogeneity shapes groups growth in social online communities.” There each group $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$13 grows approximately linearly,

$$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$14

with group-specific growth rate $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$15 held constant over time and drawn from a lognormal distribution with

$$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$16

The data consist of $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$17 groups tracked daily for $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$18 days and more than $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$19 public groups observed in two large snapshots; about half of the tracked groups have $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$20 under linear regression, and over $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$21 of groups with $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$22 have $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$23 (Grabowicz et al., 2011). Together with an inhomogeneous birth rate that grows approximately linearly in time, this heterogeneity reproduces the heavy-tailed empirical size distribution and produces an ensemble-level relation

$$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$24

without imposing preferential attachment (Grabowicz et al., 2011).

The Flickr model therefore supplies a second social meaning of group-differentiated growth. Groups grow differently because they possess different persistent $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$25 and different ages $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$26, not because growth is explicitly proportional to current size. A common misconception addressed by the paper is that heavy-tailed size distributions necessarily require rich-get-richer dynamics; the authors show that intrinsic heterogeneity plus linear growth and an inhomogeneous birth process suffice (Grabowicz et al., 2011).

6. Differentiation, division of labor, and cooperative group growth in cell aggregates

In “Symbiotic Cell Differentiation and Cooperative Growth in Multicellular Aggregates,” group-differentiated growth is formulated as a dynamical-systems problem for interacting cells with identical catalytic networks. Each cell contains chemicals $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$27, with reactions

$$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$28

and intracellular concentrations satisfy

$$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$29

The growth rate is

$$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$30

and volume evolves by

$$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$31

Cells divide when their volume doubles, and interact through a well-mixed medium with fixed volume $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$32 (Yamagishi et al., 2015).

Under low nutrient concentration $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$33 and strong coupling, measured by a small ratio $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$34, the homogeneous state becomes unstable and initially identical cells spontaneously differentiate. In the representative networks 1–3, two types emerge. Type-1 has high concentration of $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$35, almost no $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$36, and produces $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$37; type-2 has high concentration of $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$38, almost no $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$39, and produces $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$40. The two types exchange diffusible products and each uses only a subnetwork of the full catalytic network (Yamagishi et al., 2015).

The paper distinguishes four dynamical phases in a reduced two-cell model: phase (I), no differentiation; phase (II), pitchfork bifurcation from a symmetric fixed point to two asymmetric fixed points; phase (III), oscillation death, where isolated-cell oscillations are replaced by differentiated steady states under coupling; and phase (IV), bistability between synchronized oscillation and oscillation-death differentiation (Yamagishi et al., 2015). Differentiation therefore arises through nonlinear symmetry breaking rather than imposed cell-type labels.

The collective benefit is measured by

$$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$41

and $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$42 precisely in the parameter regions where differentiation occurs and is stable (Yamagishi et al., 2015). The mechanism is division of labor. If an isolated generalist cell has $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$43 components and $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$44 nutrient-reaction paths, while a differentiated specialist effectively uses $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$45 components and $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$46 nutrient-reaction paths, then the approximate ratio is

$$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$47

For networks 1 and 2, the values $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$48, $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$49, $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$50, $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$51 give

$$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$52

which exceeds $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$53 for $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$54 (Yamagishi et al., 2015). The higher the nonlinear reaction order, the greater the benefit from concentrating catalytic effort on fewer tasks.

The paper also analyzes robustness of the cell-type composition. If $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$55 is the volume fraction of type-1 and $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$56 its growth rate, then stability of the symmetric state $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$57 is determined by the sign of $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$58. The derived criterion is

$$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$59

For network 1, $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$60, so the majority type gains a further growth advantage and coexistence is unstable. For network 2, $$\gamma_G^A(n)=|B_A(n)|=\bigl|\{g\in G:|g|_A\le n\}\bigr|,$$61, so deviations are self-correcting and coexistence is stable (Yamagishi et al., 2015). The difference comes from whether key products are retained as non-diffusible internal reserves or shared by diffusion.

This biological usage gives perhaps the most literal meaning of group-differentiated growth: the growth rate of the group is an emergent function of internal differentiation. The aggregate grows faster than undifferentiated isolated cells only because it contains multiple specialized, mutually dependent types (Yamagishi et al., 2015). Across the mathematical, social, and biological literatures, this is the common structural lesson: growth is not merely a rate but a relation between scale and organization.