# Percolation and Dissolution of Borromean Networks

$\hat{[}cowboy]{\text{D}}$onald G. Ferschweiler  Department of Physics and
Astronomy, California State University, Long Beach  Ryan Blair  Department of
Mathematics and Statistics, California State University, Long Beach  Alexander
R<EMAIL_ADDRESS>Department of Physics and Astronomy,
California State University, Long Beach

###### Abstract

Inspired by experiments on topologically linked DNA networks, we consider the
connectivity of Borromean networks, in which no two rings share a pairwise-
link, but groups of three rings form inseparable triplets. Specifically, we
focus on square lattices at which each node is embedded a loop which forms a
Borromean link with pairs of its nearest neighbors. By mapping the Borromean
link network onto a lattice representation, we investigate the percolation
threshold of these networks, (the fraction of occupied nodes required for a
giant component), as well as the dissolution properties: the spectrum of
topological links that would be released if the network were dissolved to
varying degrees. We find that the percolation threshold of the Borromean
square lattice occurs when approximately 60.75% of nodes are occupied,
slightly higher than the 59.27% typical of a square lattice. Compared to the
dissolution of Hopf-linked networks, a dissolved Borromean network will yield
more isolated loops, and fewer isolated triplets per single loop. Our
simulation results may be used to predict experiments from Borromean
structures produced by synthetic chemistry.

##  I Introduction

Borromean rings consist of three mutually entangled loops for which no two
loops share a binary entanglement. They were originally the heraldic symbol of
the House of Borromeo and feature prominently in centuries of art, can be
constructed from organometallic synthesis or DNA origami cantrill2005nanoscale
, and serve as an analogy for certain bound states in nuclear physics
bertulani2007geometry . The three-loop rings are not the only such topological
structure, and larger components with Borromean connectivity can be
constructed. Large, regular, planar structures are referred to as Borromean
networks, or Borromean lattices or chainmail, which are depicted with square
lattice symmetry in Figure 1. This extends the existing concept of the Olympic
gel, a network of Hopf-linked molecular rings igram2016resolving , into a
system with only Borromean connectivity. Borromean lattices are not Brunnian
in that the removal of a single component may remove several other but will
not dissolve the network entirely. Physical Borromean lattices have been
created by synthetic chemists thorp2015infinite ; hardie2016self ;
catalano2021open , and the crystallographic symmetry of Borromean lattices has
been discussed o2020crystallographic ; o2021borromean .

Figure 1: Left: Stylistic rendering of a Borromean chainmail square lattice by
Luc Devroye luc . No two rings share a direct topological link, but no ring
can be removed from the network. Right: Rendering of such a network in three
dimensions, visualized and annealed using KnotPlot.

Kinetoplasts are complex topologically linked networks of DNA molecules found
in the mitochondria of trypanosome parasites. Each kinetoplasts consists of
about 5000 linked DNA “minicircles” which decode the RNA produced by a few
dozen linked “maxicircles” shapiro1995structure . Each minicircle in a
kinetoplast is Hopf-linked (the simplest linking between two circles, as in
the Olympic emblem) to its spatial neighbors. The average valence, the number
of Hopf links that each minicircle shares with its neighbors, is estimated at
three, consistent with a honeycomb lattice topology chen1995topology ;
ibrahim2018estimating . These estimates are based on experiments that dissolve
the network using enzymes that break individual minicircles, and measure the
relative frequency of different topological structures by gel electrophoresis.
The network topology is determined by comparing the relative frequency of
different components to the predictions from graph theory models. The graph
theory models treat each minicircle as a node and each Hopf link between
minicircles as an edge between two nodes. This allows predictions, for
example, of the probability that a single minicircle is released from the
network as a function of the number of minicircles that are dissolved. It is
not expected that kinetoplasts have Borromean connectivity, and no such
experiments have been performed for Borromean materials. Simulations of dense
circle packing rodriguez2013percolation ; diao2012effects  or ring polymer
interlocking michieletto2015kinetoplast ; d2017linking , often applied to
percolation phenomena in kinetoplast DNA, typically compute only the Gauss
linking number which detects pairwise interactions, but cannot detect
Borromean connectivity. Here, we consider the outcome of comparable
dissolution experiments with Borromean networks.

A lattice with regular geometry may be regarded as full if each geometric site
on the lattice is occupied by a node in a network or graph that shares edges
with its geometric neighbors. A full lattice has a single component that spans
the entire length scale of the network, known as the giant connected
component. As nodes or edges are removed from the network, smaller clusters
will be isolated when none of their nodes share edges with the giant
component. If a sufficient fraction of edges or nodes is removed, the largest
component in the system will no longer span the length scale of the system,
and the giant connected component will cease to exist. The minimum fraction of
filled nodes or edges in a graph that allows a giant connected component to
exist is known as the percolation threshold essam1980percolation  and depends
on the topology of the graph (for example, a square lattice requires 59.27% of
its sites occupied to reach percolation ziff1992spanning ). Systems at the
percolation threshold have universal properties (such as a cluster size
distribution exponent and a fractal dimension) that depends only on space
dimensionality but not graph topology mertens2017universal .

In this work, we explore the connectivity of Borromean lattices through a
computational graph theory model. We apply this model to square lattice
Borromean networks and analyze the cluster size distribution of Borromean
networks as individual loops are removed from the system. We compare these
results to the dissolution of a traditionally connected graph, which serves as
an analogy to Hopf-linked topological lattices. We investigate the percolation
threshold of Borromean lattices, comparing it to the standard square lattice
behavior expected from a Hopf-linked network and the universal properties of
two-dimensional percolation. We make predictions of the distribution of the
smallest clusters that would be isolated from the network: single loops and
three-component Borromean rings, and consider whether an experiment could
determine whether a topological network is Hopf-linked or Borromean based on a
dissolution reaction. Although it is not expected that kinetoplasts or HK97
viral capsids (another topologically linked biomolecular structure
wikoff2000topologically ) have Borromean connectivity, such analysis can
provide insight about whether a given network is Borromean or Hopf-linked.

##  II Mapping and Computation

We represent topologically linked networks as a square lattices with each node
representing an individual loop, similar to Figure 1. With “standard”
connectivity rules, two occupied sites on a lattice are considered adjacent
(sharing an edge in a graph) if they are orthogonally next to each other. This
can be taken as analogous to two loops sharing a Hopf link. Unlike a Hopf-
linked network with standard pairwise connectivity, Borromean lattice contains
no pairwise adjacency, at least three loops are required to establish
connectivity. Instead, we define Borromean connectivity, where groups of three
occupied sites are considered mutually adjacent if they form the smallest
possible right triangle on the lattice, such that the three loops represented
by the appropriate space curves at these sites would form Borromean rings. In
a graph representation of the lattice, each triangle forms a cluster with
three nodes and three edges, while nodes that are not part of a triangle share
no edges with other nodes. Connected nodes can be part of more than one
triangle, as clusters in the network can contain more than three rings.

Figure 2: Mapping between the Borromean lattice and our graph theory
representation. Left: A Borromean network with a large cluster of ten rings, a
triplet of traditional Borromean rings, and three isolated rings. Middle:
Representation of the Borromean network on a square lattice where black sites
are unoccupied and occupied sites are color-coded with the ring they
represent. Right: Graph representation in which each node corresponds to an
occupied lattice site, and three nodes share edges if their corresponding
lattice sites form a right triangle with unit-length sides, meaning their
corresponding rings are Borromeanly linked.

Figure 2 shows the mapping between a network of Borromeanly-linked rings, the
square lattice representation, and the graph representation. The ring network
contains a 10-ring component, a 3-ring Borromean component, and three unlinked
rings in spatial proximity. On the lattice, each occupied site in the large
component can form a right triangle with at least two of its neighbors, as can
each component in the Borromean triplet, but the three singlets cannot. In the
graph representation, edges are not shared between the unlinked components.

The mapping between ring network, lattice, and graph allows an efficient
simulation of incomplete Borromean networks. Networks are initialized as
random binary matrices of width $L$ representing a square lattice with
unoccupied and occupied sites, and can be converted to the 3D representation
by substituting a space curve at each lattice site, or converted to the graph
representation by calculating which occupied sites form triangles.

On a square lattice, each node has eight neighbors and may form twelve
distinct nearest-neighbor triangles with them. Our simplest algorithm scanned
each node of the lattice and checked whether the neighboring nodes required to
form each of the twelve triangles were occupied. If they were, three bits were
flipped in an adjacency matrix, indicating that the three nodes are involved
in a Borromean connection, and an additional three were flipped to keep the
matrix symmetric under transposition.

A cluster size distribution for each lattice instance is determined from the
adjacency matrix using a reverse Cuthill-McKee algorithm liu1976comparative ;
reverse . To distinguish between unoccupied sites and occupied but isolated
sites (which would not register on an adjacency matrix), the rows and columns
corresponding to empty sites were removed from the adjacency matrix before
cluster size calculation.

For traditional square lattice connectivity, representing Hopf-link
connections, a graphics processing algorithm that uses a flood-fill technique
exists to quickly determine connected components (“bwconncomp” in MATLAB),
which is asymptotically faster than the adjacency matrix cluster calculation
used for the Borromean lattices. In practice, the Borromean calculations can
be sped up by first finding the components with the traditional Hopf
connectivity rule, then creating a sub-matrix of each square component and
applying the Borromean clustering algorithm to it. Typically a lattice with
$L^{2}$ sites requires an adjacency matrix with $L^{4}$ total sites. At
percolation, the largest square lattice cluster size grows as approximately
$0.45\times L^{1.9}$ meaning the Borromean cluster sizes can be computed with
an adjacency matrix with only $\approx 0.2\times L^{3.8}$ components. For
larger lattices, the Borromean clustering algorithm becomes computationally
unfeasible as the adjacency matrices cannot be accommodated in system memory.

Our second algorithm for large, dense Borromean lattices iterates through each
unvisited node, checking a given node for triangular connections with its
neighbors and marking the given node as visited. If neighboring nodes share a
Borromean connection, those nodes are idempotently added to a growing list of
components connected to the original node. The triangular search procedure is
repeated for each node that is added to the list, until no more new
connections are found. The algorithm then returns to the initial outer
iteration and repeats the connected component search for the next unvisited
node. This avoids the need for a large adjacency matrix, and is faster on a
laptop than our first algorithm for full lattices above $L\approx 50$ and
percolating lattices above $L\approx 200$.

To perform our computational measurements, we initialized partially occupied
square lattices as random binary matrices, varying the fraction of occupied
sites, $q$, also known as the site occupation probability. We calculated the
cluster size distribution for Borromean and Hopf connectivity rules from size
$L=12$ up to $L=1000$. Our algorithms were implemented in MATLAB, and periodic
boundary conditions were not used.

The knot-lattice-graph mapping and algorithm in which each node may be
Borromean linked in various ways with its eight lattice neighbors represents a
fully symmetric square lattice with Borromean connectivity, but does not
represent the system in Figure 1. In Figure 1, the blue and yellow loops do
not form Borromean triplets with each other, whereas the red and green loops
do. To describe the connectivity of this network, separate rules are needed
for green/red and blue/yellow loops. Green and red loops form Borromean
connections with two of their neighbors if one is a nearest neighbor and one
is a next-nearest neighbor along the hypotenuse (eight of the twelve possible
triangle connections), and yellow and blue loops form Borromean connections
with two of their nearest neighbors (four of the twelve possible triangle
connections). Results from this mapping are reported in the Appendix.

##  III Results and Discussion

###  III.1 Identifying the Percolation Threshold

Above the percolation threshold, almost all occupied lattice nodes are within
the largest cluster, while below the threshold almost none are. The sharpness
of this transition increases with the size of the lattice. The size of the
second-largest cluster is maximized at the percolation threshold
margolina1982size , which is our primary method of determining it from our
computations. Figure 3 shows the largest and second largest cluster sizes as a
function of the occupation probability for Borromean and Hopf-linked square
lattices of width $L=400$. The largest component size varies with occupation
probability as expected, with the jump associated with percolation occurring
at a higher occupation probability for the Borromean lattice. Examining the
second largest component provides more evidence that the Borromean lattice has
a higher percolation threshold. We observe that the standard square lattice
has a second-largest component that is maximized near the expected value of
0.5927 ziff1992spanning . Under Borromean connectivity rules, the percolation
threshold is slightly but significantly higher. It is difficult to precisely
locate the peak with finite numerical precision and a finite-sized lattice,
but we estimate the peak at approximately 0.606. A more precise analysis of a
regularly-connected square lattice at $L=400$ finds a second-largest component
peak at $q=0.5917$, 0.16% below the asymptotic value, suggesting that the
Borromean percolation threshold is slightly higher than 0.606. Analysis based
on the universality of the fractal dimension at percolation (below) puts our
estimate at 0.6075.

Figure 3: Left: Fraction of occupied sites in a 400$\times$400 square lattice
that are within the largest component, as a function of fraction of sites that
are occupied, for the Hopf and Borromean connectivity rules. Right: The second
largest component sizes from the same data. The peak of the second largest
component indicates the location of the percolation threshold, which is
slightly higher for Borromean lattices. In both charts, error bars represent
standard error over multiple randomized lattice instances and the established
square lattice percolation threshold of 0.5927 is shown by a dashed line.

The higher occupation ratio required for percolation with Borromean
connectivity is illustrated in Figure 4. The largest cluster at the square
lattice percolation threshold will span the entire system (in this case
defined as touching the four edges of the lattice), but consist of regions
where the cluster forms “bridges” only one site wide and several sites long.
These can be seen at the blue-teal boundaries in Figure 4. Under the Borromean
connectivity rule, the giant cluster does not span these bridges, and forms
additional smaller Borromean clusters. Extra occupied sites adjacent to these
bridges are required to keep them in the giant cluster, which is why a
slightly higher occupation is required for a Borromean network to percolate.
Another argument, put forth by Robert Ziff in a personal communication, is as
follows: at the square lattice percolation threshold of $q=0.5927$, the
probability that two adjacent sites are filled (and form a cluster) is
$q^{2}=0.3513$. However, for the sites to form a Borromean triplet, one of the
four sites adjacent to the doublet must be occupied, which occurs with
probability $q^{2}(1-(1-q)^{4})$. For that probability to equal 0.3513, $q$
must have a value of 0.6004, slightly higher than 0.5927 but slightly slower
than what we observe.

Results from the modified connectivity rules that describe the network in
Figure 1 are presented in the Appendix. We note that the difference between
Borromean and Hopf connectivity in that case is much stronger, and the
percolation threshold is located at approximately 0.66.

Figure 4: A percolating cluster touching all four edges of a 100$\times$100
square lattice at the percolation threshold of 0.5927. For clarity only the
largest cluster is shown, as the union of the teal and blue sites, which
touches all four sides of the lattice. The blue subset is the largest
component under the Borromean connectivity rule, and this cluster does not
span the entire system, only touching two sides.

###  III.2 Universal Behavior

Although the connectivity of the Borromean lattice is not pairwise, it is
still local, and at percolation is expected to obey the expected universal
behavior of percolating systems. In two dimensions, this includes a fractal
dimension for largest cluster size of $91/48\approx 1.89$ ziff2011correction ,
and a cluster size distribution that decays with a -187/91 $\approx 2.05$
power law macleod1998large . The largest component size at $q=0.6075$ is plot
against the system size in Figure 5, along with data for the Hopf lattice at
0.5927, which is essentially identical. The best fit value for the fractal
dimension is 1.893$\pm$0.002, or (90.9$\pm$0.1)/48. If the largest cluster
size from systems that are not at the percolation threshold is examined, it is
expected to slightly deviate from power-law behavior, and display a best-fit
exponent that is not the universal value. At $q=0.606$, we observed an
exponent of 1.82$\pm$0.02, and at 0.609 we observed 1.96$\pm$0.02. We
interpret this as additional evidence that the percolation threshold of the
Borromean square lattice is at approximately 0.6075.

The cluster size distribution at percolation is seen in Figure 5 for both
Borromean and Hopf connectivity rules. The distribution is slightly different
for the smallest clusters, notably due to the lack of Borromean dimers, but
they converge to a similar form for large clusters. We measure best-fit
exponents for these distributions as -1.883$\pm$0.001 for Borromean and
-1.906$\pm$0.001 for Hopf connectivity, inconsistent with the Fisher exponent
of -187/91. The cluster size distribution exponent is more sensitive to
finite-size effects: Ding et al. observed a power-law consistent with a -1.92
exponent on square lattices 1000 sites wide ding2014numerical , while in an
earlier study Hoshen et al. observed data consistent with a -1.78 exponent on
triangular lattices 4000 sites wide hoshen1979monte . Only on lattices 2
million sites wide have precise data consistent with the -2.05 exponent been
observed macleod1998large . Although finite-size effects persist, we find that
Borromean lattices at the percolation threshold display the same universal
properties as those with traditional connectivity.

Figure 5: Universal properties of the percolation threshold. Left: The largest
component size as a function of the lattice width for Borromean lattices at
$q=0.6075$, with a best-fit exponent of 1.893$\pm$0.002, consistent with a
fractal dimension of 91/48. Data for regular square connectivity at $q=0.5927$
is overlaid and effectively identical. Error bars are smaller than points.
Right: Cluster size probability distribution for 1000$\times$1000 lattices
with Borromean and Hopf connectivity at their respective percolation
thresholds. Both distributions are consistent with each other, although
presumably due to finite-size effects have a weaker decay exponent than the
expected -187/91. The Hopf data is more precise due to the faster algorithm
used to generate it.

###  III.3 Dissolution Spectrum

An experiment to determine the topology of a hypothetical Borromean material
may break down the network in a similar way that kinetoplasts are broken down
by enzymes, and examine the topology of the smaller clusters that are released
from the network (for example with atomic force microscopy). How might a
Borromean material be distinguished from a Hopf-linked catenated network?

Chen et al. chen1995topology  discuss the probability of isolated rings as
well as Hopf-linked doublets and triplets being released from kinetoplast
networks during enzyme digestion, assuming different lattice geometries. The
isolation probabilities depend on the linearization probability $p$ of a
minicircle being cut by the enzyme, equivalent to a node being removed from a
graph, as well as the co-probability $q=1-p$, which is equivalent to our
lattice occupation probability. We may calculate similar probabilities for
Borromean connectivity on the square lattice. To find the probability of
isolating a single ring from the network, we consider a 3x3 lattice with the
central node occupied, and find that 34 of the $2^{8}$ possibilities for the
surrounding nodes being occupied lead to the central node being isolated.
Summing the probability of each configuration yields:

$P_{1B}=p^{4}q^{5}+8p^{5}q^{4}+16p^{6}q^{3}+8p^{7}q^{2}+p^{8}q.$ (1)

To find the probability of a Borromean triplet being released from the
network, we consider three occupied sites in an L-shape in the center of a 4x4
grid, and find that 100 of the $2^{13}$ possibilities lead to the links being
released. The total probability is:

$P_{3B}=4\left(p^{7}q^{9}+8p^{8}q^{8}+24p^{9}q^{7}+24p^{10}q^{6}+24p^{11}q^{5}+8p^{12}q^{4}+p^{13}q^{3}\right),$
(2)

the factor of 4 admitting rotations of the triplet. For comparison, the
probabilities for finding an isolated component or a triplet with regular
square lattice Hopf connectivity are $P_{1H}=qp^{4}$ and
$P_{3H}=2q^{3}p^{7}(2+p)$.

Figure 6 shows the expected mass fraction of different sized components as a
function of the undissolved fraction $q$, for Borromean and Hopf networks. A
greater proportion of the mass is found in single loops under Borromean
connectivity, in part due to the absence of 2-component links which occupy
part of the mass in dissolved Hopf systems. There are subtle differences in
the relative mass found in 3- and 4-component links between the two
connectivity rules. The ratio of triplets to singlets, a parameter that could
in principle be measured experimentally (and has been, for kinetoplast DNA),
is shown on the right, with the ratio for Borromean networks peaking close to
20% near $q=0.4$, but only 10% for Hopf networks. The exact expressions
describe the computational data well, with some finite-size edge discrepancies
at large $q$.

Figure 6: Left. Fraction of total mass found in isolated loops, three-
component loops, and four-component loops after a dissolution reaction of
Borromean and Hopf-linked topological networks, simulated for 200x200
lattices. The exact predictions for the singlet distribution are overlaid,
with slight discrepancies due to edge effects. Right. The ratio of triplets to
singlets after a dissolution under Borromean and Hopf connectivity.

##  IV Conclusion

We have simulated the dissolution of Borromean square lattices to calculate
their percolation threshold, which we find to be 2.4% higher than that of the
traditional square lattice. We have also made predictions about the population
of simple links released from a network during dissolution, finding marked
differences from Hopf lattices due in part to the lack of Borromean dimers. In
addition to being the first such analysis, it also makes predictions for
potential experiments on Borromean materials thorp2015infinite .

This analysis is possible because the square lattice structure allows us to
map a knot theory problem, the Borromean connection of three loops, to a graph
theory problem, the continuity of connected triangles. More general linked-
ring systems cannot necessarily rely on geometry to impose topology, but
invariants have been derived which can detect triplet-wise Borromean
connectivity when pairwise linking connectivity is absent deturck2009triple .
A number of studies have used the Gauss linking number to calculate the graph
structure of densely packed circles diao2012effects ; rodriguez2013percolation
or ring polymers michieletto2015kinetoplast ; d2017linking , but the linking
number cannot detect Borromean connectivity. Building upon these results,
future work can examine the Borromean networks that form within dense packings
once Hopf connections have been accounted for. This analysis need not end at
square lattices, nor at Borromean connectivity: lattices may be constructed of
any N-connected Brunnian link, which may present a range of percolation
phenomena.

##  V Acknowledgements

The authors wish to thank Robert Ziff for his helpful discussions about
percolation theory. They are also grateful to Kenny Umenthum for suggesting a
recursive algorithm, Raphael Candelier for his helpful blog, and Luc Devroye
whose artwork inspired this project. DF is supported by the HSI-STEM Bridge to
the Beach Program. This work is supported by the National Science Foundation,
grant number 2105113.

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##  VI Appendix

For completeness, we present results for the second-largest component of
Borromean lattices that correspond to Figure 1, in which red and green loops
share Borromean links with each other mediated by yellow or blue loops, but
blue and yellow loops do not share Borromean links mediated by red or green
loops. Compared to the generalized lattice mapping we describe in the main
text, this version breaks translational symmetry. To count connectivity
between between sites, green and red sites must form a triangle with one
nearest neighbor and one next-nearest neighbor, while blue and yellow sites
must form a triangle with two nearest neighbors. As can be seen from Figure 7,
the deviation from the square lattice is stronger, with the percolation
threshold occurring at approximately $q=0.66$.

Figure 7: The second largest component size for 200$\times$200 square lattices
corresponding to the network in Figure 1. The vertical lines are at 0.5927 and
0.6075, the percolation thresholds for traditional square lattice
connectivity, and fully symmetric Borromean square lattice connectivity. The
peak is located close to 0.66.