Why does social complexity need compression?

First, we define a social network $G\left( {N,E} \right)$ , where $N$ is group size and $E$ is the number of edges. In a fully connected bidirectional network, the connectivity increases with group size as $E = N\left( {N - 1} \right) = O\left( {{N^2}} \right)$ . In small groups where the number of individuals remains relatively low, each member can explicitly track all relationships. The storage and retrieval costs at this scale remain modest, making social navigation relatively straightforward. However, as group size increases, the all-to-all tracking strategy becomes infeasible due to both memory constraints and retrieval efficiency. The cognitive cost of storing relationships grows quadratically with the number of individuals, making explicit tracking computationally intractable. The number of all possible pathways among the group grows even worse at $N!$ . Furthermore, social decision-making requires quick access to relevant ties, but a flat social network structure lacks efficiency in lookup and recall.

Fractal networks as a computationally efficient solution

As shown by Dunbar and others, the solution that appears consistently across primate species, including human societies, is the fractal organization of social organization (Hamilton et al., Reference Hamilton, Milne, Walker, Burger and Brown2007; Hamilton et al., Reference Hamilton, Walker, Buchanan and Sandeford2020; Zhou et al., Reference Zhou, Sornette, Hill and Dunbar2005; Dunbar, Carron, & Shultz, Reference Dunbar, Carron and Shultz2018; Hill, Bentley, & Dunbar, Reference Hill, Bentley and Dunbar2008; Sandeford, Reference Sandeford2018). As we will see, this hierarchical, self-similar structure effectively reduces both cognitive and behavioral overhead. Instead of tracking all individual relationships, primates encode higher-order summaries that organize social connections into nested level, $k \in {\rm{\Omega }}$ , where ${\rm{\Omega }}$ is the depth of the network. And so, let ${N_K}$ be the group size at the $k$ th level.

Using Dunbar’s numbers, at the most intimate level, individuals maintain strong ties with a small core group, consisting of approximately five members, ${N_1} = 5$ , with whom they have frequent, high-bandwidth interactions; ${E_1} = 20$ . Beyond this core, a secondary group of trusted allies, generally around fifteen individuals, ${N_2} = 15$ , is maintained with bidirectional and strong connections; ${E_2} = 210$ . Expanding further, an extended network of approximately fifty individuals, ${N_3} = 50$ , where ${E_3} = 2,450$ relationships are inferred rather than explicitly tracked, often using transitive closure principles, meaning that “friend-of-a-friend” logic helps individuals navigate these indirect ties. Beyond this, individuals belong to a wider community of approximately 150 members, ${N_3} = 150$ , where relationships are stored as categorical groups rather than explicitly maintained as individual ties, where ${E_4} = 22,350$ . Out of interest, the total number of all possible paths between 150 individuals is about $5.71 \times {10^{262}}$ , far greater than the number of atoms in the observable universe.

Mathematically, instead of maintaining a fully connected graph, social cognition compresses networks using a fractal scaling law:

$${E_k} \approx CN_k^\beta, 0 \lt \beta \lt 1$$

and so, summing across all $k$ hierarchical levels we have ${E_{\rm{\Omega }}} \approx CN_{\rm{\Omega }}^\beta, $

where $C$ is the cognitive load of an individual, and $\beta $ controls the efficiency of compression. This scaling law is ultimately derived from the fact ${N_k} \approx {B^k}$ , where $B$ is the branching ratio, often between 3 and 4 empirically in all kinds of social networks. A value of $\beta $ closer to 1 implies minimal compression, while a value closer to 0 implies maximal compression. Note that in Dunbar’s paper, the network degree of group sizes—a reasonable estimate of ${E_k}\sim {N^\beta }$ —he reports is $\beta \approx 0.6 - 0.8$ . It can then be shown that the total storage and retrieval complexity of this system then shifts from the brute-force quadratic growth of ${N^2}$ to a dramatically more efficient $N\log N$ . This hierarchical compression model ensures that instead of every individual requiring explicit memory of every relationship, they rely on group-based abstractions and inferred social ties, substantially reducing the computational load while maintaining network cohesion. In other words, the cost of keeping track of the connectivity of an individual reduces from $N$ to $\log N$ .

Fractal networks as efficient coordination systems

The advantages of a hierarchical, fractal network extend beyond the reduction in memory requirements. Such network structures have the small-world properties, where local clustering is high, yet the presence of long-range weak ties ensures efficient connectivity across the broader network (Hamilton, Reference Hamilton2022). This then vastly improves navigability, as weak ties function as shortcuts, reducing the search cost in large social groups.

Additionally, fractal networks enhance error tolerance, as the redundancy in layers ensures that if some social ties are lost or broken, the structure remains resilient and retains overall integrity. Furthermore, as groups grow beyond Dunbar’s number, ${N_k} \gt 150$ , the same fractal organization extends into formal institutions, where social roles and organizational structures take over cognitive limitations, enabling large-scale human societies to function effectively.

Social cognition as an evolved compression algorithm

If we accept that one of the key selective gradients to which the brain evolved in response to was optimizing social computation, then a fractal social network architecture is a natural collective solution. The human brain does not brute-force social coordination by storing all relationships explicitly. Instead, it organizes relationships hierarchically to reduce memory overhead. It uses heuristics to index them—kinship norms or friend-of-a-friend rules—to infer indirect ties, which minimizes retrieval costs and ensures that even individuals without direct interactions can be socially navigated through intermediaries. This balance between redundancy and efficiency provides resilience under social stress and environmental changes, allowing groups to adapt without overwhelming cognitive load.

The core question for future research

If fractality solves the problem of exploding complexity in social networks, a fundamental question remains: Are there universal scaling laws governing social cohesion and information compression across species and societies? To investigate this, researchers could apply information-theoretic measures of network compression to different social species and institutional structures. Additionally, simulations of scaling thresholds in social systems may provide insights into phase transitions in coordination, shedding light on why certain social structures persist while others collapse. Comparisons with artificial intelligence models, particularly how machine learning architectures handle large-scale social inference, could reveal parallels between biological social cognition and digital network optimization. It is intriguing to note scaling laws abound throughout the biological and social sciences (West, Reference West2017; Bettencourt, Reference Bettencourt2021; Barenblatt & Isaakovich, Reference Barenblatt and Isaakovich1996) though the explicit link to lossy compression has yet to be made, as far as I am aware.