Symmetry breaking and the emergence of path-dependence

Abstract

Path-dependence offers a promising way of understanding the role historicity plays in explanation, namely, how the past states of a process can matter in the explanation of a given outcome. The two main existing accounts of path-dependence have sought to present it either in terms of dynamic landscapes or branching trees. However, the notions of landscape and tree both have serious limitations and have been criticized. The framework of causal networks is both more fundamental and more general that that of landscapes and trees. Within this framework, I propose that historicity in networks should be understood as symmetry breaking. History matters when an asymmetric bias towards an outcome emerges in a causal network. This permits a quantitative measure for how path-dependence can occur in degrees, and offers suggestive insights into how historicity is intertwined both with causal structure and complexity.

Appendix

Theorem 1

A coarse-graining of the explanandum makes an explanation increasingly convergent and a coarse-graining of the explanans makes an explanantion increasingly divergent.

Proof

We will prove it for an explanation that is purely parallel, thus neither convergent nor divergent. The generalization for a random explanation holds analogously.

Assume a deterministic explanation (O, I, f), so that f is a bijection $f:I\rightarrow O$. Define an equivalence relation $\sim$ on O such that $o_1 \sim o_2$ iff $o_1, o_2 \in A$ for some A (dependent on theoretical interests) with $\#A >1$. Because f is a bijection there exists a uniquely defined $B \in I$ such that $f(B) = A$. Call B the ‘basin’ and A the ‘attractor’ of f on I.

Then O / A represents a coarse-graining of the explanandum and I / B a coarse-graining of the explanans. So define an associated function $R_c: I \rightarrow \#O/A: i \mapsto f(i)$ and relation $R_d \in I/B \times O = {(f^{-1}(o),o)|o\in O}$. Because f is a bijection, $\#I = \#O > \# O/A$ and $\#O = \#I > \# I/B$, and hence $R_c$ will be a non-injective surjection, and $R_d$ a non-function. Hence the number convergent structures has increased in explanation $(O/A, I, R_c)$, and the number of divergent structures has increased in $(O,I/B,R_d)$. $\square$

Theorem 2

Let (O, I, R) be symmetrical at some instant in time. Then (O, I, R) is symmetric at all prior instants.

Proof

Assume (O, I, R) is symmetric at time t, corresponding to the set of intermediate states S. Let $S'$ represent some earlier generation of states. From the local symmetry of (O, I, R) at S we can deduce that $P(o|s^*) = p \in [0,1]$ for all $s^* \in S$.

Take a random predecessor state $s' \in S'$. Assume it branches out to a number of states $s^* \in S$. Then

$$\begin{aligned} P(o|s')= & {} \sum _{s^*} P(o|s^*)P(s^*|s')\\= & {} p \sum _{s^*}P(s^*|s')\\= & {} p \end{aligned}$$

since the sum of the probabilities of all paths leaving $s'$ is 1. Thus the network is symmetric at $S'$.

This also means that the bias p towards outcome o is preserved as long as the network remains symmetric. $\square$