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 so measures the population. Now, if every subset of 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 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 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 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.