Yes. Percolation theory is actually a very useful way to think about system collapse, especially when the system is a network whose functioning depends on enough connections remaining intact.
The key idea is not simply “things gradually get worse.” It is:
a system can absorb a surprisingly large amount of local damage, and then suddenly lose global connectivity once a critical threshold is crossed.
Imagine a lattice of sites. Each site is functioning with probability p. When p is high, there is a giant connected cluster spanning the whole system. As failures accumulate and p falls, the system may still look largely intact locally. But at a critical value p_c, the spanning cluster disappears. Then what had been one functioning network breaks into disconnected islands.
So collapse can be pictured as:
\text{local failures}
\rightarrow
\text{shrinking connectivity}
\rightarrow
p \approx p_c
\rightarrow
\text{fragmentation}
\rightarrow
\text{loss of system-level function}.
This gives us a very important distinction between damage and collapse. If 10% of components fail, the system need not lose 10% of its function. It may lose almost nothing at first. Conversely, the failure of another 2% near the critical point may destroy the entire macroscopic network. The relation is highly nonlinear.
For example, take an electrical grid. Individual substations can fail while electricity is rerouted through alternative paths. But once enough important connections disappear, there is no longer a path connecting generators with consumers. The grid fragments into islands. Similarly, in transportation, banking, supply chains, ecosystems, communication networks, or bureaucracies, what matters is often not how many units remain, but whether the remaining units still form a connected functional cluster.
There is an especially interesting extension: interdependent-network percolation. Modern systems rarely consist of one network. Electricity depends on communications; communications depend on electricity; banking depends on telecoms; logistics depends on fuel, roads, finance, and digital infrastructure. A small failure in network A causes failures in B, which feed back into A. Under these conditions the transition can become much sharper—almost an abrupt phase transition rather than a smooth deterioration.
Schematically:
A\downarrow \Rightarrow B\downarrow
\Rightarrow C\downarrow
\Rightarrow A\downarrow\downarrow
until the mutually sustaining giant component disappears.
This is why a society may appear oddly stable immediately before serious breakdown. Restaurants remain open, salaries are still paid, trains still run, bureaucratic offices still stamp documents. Yet the redundancy underneath them is being consumed. The visible nodes remain; the invisible links are disappearing.
I would therefore distinguish three stages:
- Robust phase — many alternative paths; local failures are absorbed.
- Critical phase — connectivity becomes sparse; correlation length becomes very large, so distant events begin influencing one another.
- Post-percolation fragmentation — there may still be many functioning pieces, but no longer a functioning whole.
This last point is crucial. Collapse does not necessarily mean that everything stops existing. A state, institution, hospital, company, or ecosystem can still contain almost all its old pieces while having ceased to operate as an integrated system.
And this connects beautifully with your earlier formulation of 「一尚且無，何來二」. In network language, one could almost reverse it:
there may be innumerable “ones,” but if the links among them no longer percolate, there is no “whole.”
A million isolated nodes are not a society.
There is, however, one limitation. Pure percolation theory normally treats nodes or bonds rather simply—occupied/unoccupied, functioning/failed. Political and social systems contain adaptation, agency, hierarchy, coercion, substitution, learning, and meaning. So percolation is best regarded as a structural skeleton rather than a complete theory of social collapse.
For a more realistic collapse model, I would combine:
\boxed{
\text{percolation}
+
\text{cascading failure}
+
\text{network interdependence}
+
\text{loss of redundancy}
+
\text{positive feedback}
}
And then one gets a rather powerful picture:
systems usually do not collapse because every component fails; they collapse because enough relations fail that the remaining components can no longer constitute a system.
In that sense, percolation theory is almost a mathematical theory of 「關係斷裂到某一點，整體突然不再成其為整體」.
