In this short note I’d like to introduce a conceptual model for the emergence of higher-level abstractions in complex networks that allows us to approximately quantify the number of constraints on a complex system. By higher-level abstraction I mean a system whose dynamics are consistent with but not reducible to their elementary parts.

Let’s suppose we have a population of organisms capable of interaction and replication that is identified with $S$ so $\lvert S \rvert = N$ measures the population. Now, if every subset of $S$ may be identified with a clique of individual organisms we may say that:

\begin{equation} \begin{split} C_0 = \text{Pow}(S) \\ \lvert C_0 \rvert = 2^N - 1 \end{split} \end{equation}

where $\text{Pow}$ is used to define the space of possible undirected relations between organisms and this doesn’t include the empty set because nature abhors a vacuum.

We can also have cliques of cliques so:

\begin{equation} \begin{split} C_1 = \text{Pow} \circ \text{Pow} \circ S \\ \lvert C_1 \rvert = 2^{\lvert C_0 \rvert} - 1 = 2^{2^N - 1} - 1 \end{split} \end{equation}

where the elements of $C_1$ represent possible interactions between communities of organisms. So the elements of the power set represent higher-order objects with more complex interactions.

Furthermore, in general we have:

\begin{equation} \begin{split} C_n = \text{Pow}^n \circ S \\ \lvert C_n \rvert = 2^{\lvert C_{n-1} \rvert} - 1 \end{split} \end{equation}

and I think this idea formally captures to some degree what we mean by emergence in complex networks.

If each element of $C_n$ is identified with an equation we may say with some confidence that the number of constraints on a complex system grows super-exponentially in a manner that is most naturally expressed using tetration.