Explosive percolation (https://science.sciencemag.org/content/323/5920/1453) is a recent concept where random networks *R*_{0} (basic reproductive number) is reached abruptly. When *R*_{0} crosses 1.0 there is complete connectivity (a network of nodes and links are completely connected; #nodes = #links makes *R*_{0} =1 because *R*_{0} = #nodes/#links). This 1912 paper https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3320609/ shows the first evidence of mathematical modeling of infections. Later Erdos & Reyni developed the math for random networks and further scholarship showed the link between R and networks.

Explosive percolation is an extension of random networks.

But first, let’s start with random networks. Your regular street network is a random network. That might seem strange at first but the definition of a random network: In a street network: each node (or intersection) has an average of 4 links (or streets, defined as a continuous uninterrupted segment that starts and ends in an intersection). Some nodes may have 3 links originating from it (a T- intersection), 4 links (regular 4-way stop), 5, 6, 7 or sometimes, rarely, more (sometimes called circle centers). However, it’s mean is 4 and varies between 1 through 8, but mostly it is 3, 4, or 5. This follows a statistical distribution called a bell curve (Gaussian) with mean 4 and spread approximately 1 (see left graph below from Barabasi (barabasi.com/f/226.pdf).

http://barabasi.com/f/226.pdf Barabasi showed that road networks are random whereas airline networks are scale-free.

Now one way to create a random network is to first have a fixed number of nodes, say *n*, then join any two randomly selected nodes with a link. When you join enough nodes something magical happens: there is complete connectivity in the system when the #links reaches #nodes. Suddenly no node is isolated! This a continuous process and the phase change (or total connectivity: *R*_{0} *= 1*) occurs smoothly.

In 2015, a paper by Dimitris Achlioptas, Raissa M. D’Souza & Joel Spencer:

**abruptly**. This abrupt change is called a phase change in physics and in this case it is

**not continuous**.That is: there is complete connectivity but it happens like water freezing to ice at a particular temperature of 32F. Raise one degree and all will melt. In the case of explosive percolation (proven to happen only with finite n), the links suddenly reach complete connectivity unlike with random random networks where you can see slowly the network is reaching complete connectivity.

*[One note on paired random links is that it relates to a*

**first**

**order markovian process**and can be thought of as random walks].*“Explosive Percolation in Random Networks.*

*Abstract:*

*Networks in which the formation of connections is governed by a random process often undergo a percolation transition, wherein around a critical point, the addition of a small number of connections causes a sizable fraction of the network to suddenly become linked together. Typically such transitions are continuous, so that the percentage of the network linked together tends to zero right above the transition point. Whether percolation transitions could be discontinuous has been an open question. Here, we show that incorporating a limited amount of choice in the classic Erdös-Rényi network formation model causes its percolation transition to become discontinuous.”*

I did some simulations to show that road networks can suddenly isolate using random networks when they need to access a central point in the network like connecting a house (a random node) to a hospital (a central node). Sometimes just a 20 percent drop in links cause a 100 percent drop in accessibility to central nodes (see below).

It is scale free (see above). It does not matter what spatial filter (e.g. zip code areas, blocks, voronoi service areas etc.). Some areas will lose all connectivity even with as low as 5 percent dropped links but some areas will be connected even with 90 percent loss of links. The above experiment can be further studied to show that dropping pairs of random links will create this scale free condition even faster and with abrupt phase change!

**What about paired random links? Explosive percolation**

The above graph shows explosive percolation (orange line) vs gaussian percolation (blue line). When you drop, say 20 percent of links we get gaussian percolation (blue line showing lack of access to e.g. central places like hospitals) where as explosive percolation (orange line showing access is “baaad”).

With gaussian percolation (dropping randomly say 20 percent of street segments) we see whole segments of the map becoming ‘marooned’ (‘marooned’ meaning 100 percent lack of access to, say, any hospital). With explosive percolation (dropping 20 percent but paired street segments randomly — i.e. dropping 10 percent + their neighboring segments jointly). Do we see **explosive ‘marooning’? **I am running experiments and seeing that no percolation is occuring. *Maybe 90 percent drop will do? Please wait for an update soon.*

Explosive marooning may have implications to contain, say explosive pandemics likes Covid19 (coronavirus). For example: by installing road barriers/medical screening at strategic points within optimally derived hospital coverage areas we can optimally isolated and treat marooned zones. For areas with resilient connectivity, isolation will not be possible so other strategies may be more appropriate.

**“Metastatic” growth**

The marooning in certain areas create what can be termed as “metastatic” growth (all at once) https://www.pnas.org/content/111/46/E4911 .

**Social Networks and Transportation networks: diffusion of disease implications **

The paper ‘Modelling disease outbreaks in realistic urban social networks (Eubanks et al 2004)’ http://www.uvm.edu/pdodds/research/papers/others/2004/eubank2004a.pdf provides an excellent exposition of social and transportation networks that models the intricate contact process by which diseases spread. They use bipartite graphs as a novel and enhanced way to match origin destination trips. Using a GIS, I model origin-destination trips the traditional way. I guess, using bipartite matching may improve the model.

**Explosive Percolation, Backward Bifurcation and Hysteresis: Is there a link?**

Above are two diagrams (Paul Gruenewald, PRC Berkeley) that show when population is mixed, R1 can be reached without reaching 1 (hysteresis)! But, the nature of population mixing is not explored much. With Achlioptas et. al. we may be able to identify this nature …