# LOCAL DECOMPOSITION OF HEXAHEDRAL SINGULAR NODES INTO SINGULAR CURVES

Paul Zhang1 Judy (Hsin-Hui) Chiang2 Xinyi (Cynthia) Fan3 Klara Mundilova4

(1Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A.
<EMAIL_ADDRESS>
2Univeristy of Illinois at Urbana-Champaign, Urbana-Champaign, Illinois,
U.S.A<EMAIL_ADDRESS>
3 Davidson College, Davidson, North Carolina, U.S.A<EMAIL_ADDRESS>
4Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A.
<EMAIL_ADDRESS>
###### Abstract

Hexahedral (hex) meshing is a long studied topic in geometry processing with
many fascinating and challenging associated problems. Hex meshes vary in
complexity from structured to unstructured depending on application or domain
of interest. Fully structured meshes require that all interior mesh edges are
adjacent to exactly four hexes. Edges not satisfying this criteria are
considered singular and indicate an unstructured hex mesh. Singular edges join
together into singular curves that either form closed cycles, end on the mesh
boundary, or end at a singular node, a complex junction of more than two
singular curves. While all hex meshes with singularities are unstructured,
those with more complex singular nodes tend to have more distorted elements
and smaller scaled Jacobian values. In this work, we study the topology of
singular nodes. We show that all eight of the most common singular nodes are
decomposable into just singular curves. We further show that all singular
nodes, regardless of edge valence, are locally decomposable. Finally we
demonstrate these decompositions on hex meshes, thereby decreasing their
distortion and converting all singular nodes into singular curves. With this
decomposition, the enigmatic complexity of 3D singular nodes becomes
effectively 2D.

##  1 Introduction

Hexahedral meshes are commonly used to model complex geometries and to solve
numerical PDEs. The results they produce with tri-linear basis functions are
often superior to those produced with linear basis functions on tetrahedral
meshes [1, 2]. They can be preferable to other types of meshes for their
natural local coordinate systems and have been shown to perform better with
quadratic basis functions in the context of nonlinear elasto-plastic
simulation [3]. Due to the persistent demand for hex meshes, a variety of
methods have been developed to generate them.

Of particular relevance to our work are frame field based methods, where a
smooth boundary aligning coordinate system is computed over the domain,
followed by parameterization and hex extraction [4, 5, 6, 7, 8]. Frame field
based hex meshing has especially elucidated the significance of singularities
within a hex mesh since frame fields inevitably contain singular structures
that are reflected in the resulting mesh. The singularities of a frame field
and hex mesh typically consist of a set of singular curves that join up in
space at singular nodes. These nodes and curves form the singular graph of a
field or mesh as illustrated in Figure 1. Since hex meshable singularities are
a subset of frame field singularities, much attention has been devoted to the
restriction and correction of singularities in frame field computation [9,
10]. Other works have derived conditions and algorithms to compute frame
fields obeying singular constraints [11, 12].

Various works have also targeted the enumeration or simplification of hex mesh
singularities. [13, 14, 15] derive algorithms to simplify the singular
structures of hex meshes with collapse operations on a coarsened mesh. [11]
provide an enumeration algorithm for all hex mesh singular nodes, as well as
an exhaustive list of the most practically relevant singular node types. While
this list only contains eight singular nodes, they already form complex
junctions that are challenging to parse or manipulate.

In contrast to singular nodes, singular curves are simpler to understand,
since their local structure is only a 2D singularity extruded into 3D. In the
fully 2D setting, cross field and quadrilateral (quad) mesh singularities are
significantly easier to visualize, and in the case of parameterized cross
fields, singularities are governed completely by a few simple conditions [16].
In quad meshes, singular vertex pairs are shown to move almost fluidly within
a quad mesh [17]. None of these results obviously translates to hex meshes,
where 3D singular nodes exist as junctions of multiple 2D singularities.

In this paper, we investigate the structure of hex mesh singular nodes. We
uncover that singular nodes can be simplified by pulling their constituent
elements apart into singular curves. Our results show theoretically and
empirically that singular nodes can be removed from a hex mesh thus reducing
the complexity of any local neighborhood in a hex mesh. Our contributions are
as follows:

  * •

We show by construction that all eight of the most practically relevant
singular nodes are decomposable into just singular curves.

  * •

We show that all singular nodes, regardless of valence, are locally
decomposable.

  * •

We apply our decompositions to hex meshes demonstrating that entire singular
graphs can be separated into independent singular curves.

##  2 Preliminaries

Our work is motivated by the following question. What if a singular node is
formed when singular curves just barely skim past each other? If that were the
case, then we could separate the curves with a sheet and increase its
thickness to force the curves away from each other, thereby untangling the
singular node. To formalize this idea, we begin with the following
definitions.

###  2.1 Singular Vertices, Curves, and Nodes

We denote a hex mesh as $\\{\mathcal{V},\mathcal{H}\\}$ where $\mathcal{V}$ is
a list of vertices embedding the mesh and $\mathcal{H}$ is a list of hexes of
the mesh. Let $\mathcal{F}$ denote the quadrilateral faces of the mesh,
$\mathcal{E}$ denote the edges of the mesh, and $\deg(e\in\mathcal{E})$ denote
the number of hexes adjacent to edge $e$, i.e., its degree or valence. A
_singular edge_ is an interior edge $e$ satisfying $\deg(e)\neq 4$. We will
not treat cases where $\deg(e)\leq 2$ since these are not typically accepted
as valid hex meshes. We will also not consider singular boundary edges in this
paper but refer the interested reader to [11] for the corresponding
definition. For our purposes, one can ignore boundary singularities or push
them all to the interior by adding one layer of padding to the hex mesh
boundary. A _singular vertex_ is a vertex of the mesh that is adjacent to any
singular edge. A _singular node_ is a vertex of the mesh that is adjacent to
more than two singular edges. A _singular curve_ is an alternating sequence of
singular edges and singular vertices that either forms closed cycles or, ends
at a singular node or boundary vertex. Note that we have chosen to deviate
from the language of [11] by distinguishing singular nodes from singular
vertices. Singular nodes are reserved for the junctions of multiple singular
curves and will be the primary focus of this work. Figure 1 depicts a summary
of this terminology.

Singular nodeSingular curveSingular vertices

Figure 1: (Left) The singular graph of a hex mesh of a sphere is shown.
Singular edges are colored red, singular nodes are large black circles and
singular vertices are small blue vertices. (Right) A close-up view of a
singular node. Faces adjacent to the singular node are displayed in purple. A
yellow sphere is overlayed on top of the singular node. Its intersection with
the local hex mesh partitions the sphere into triangular regions.

For a singular node $v\in\mathcal{V}$, we denote $\mathcal{T}(v)$ as the
triangle mesh in bijection with that node according to [6]. This bijection is
formed by intersecting the singular node of the hex mesh with an
infinitesimally small sphere. Since the intersection of a corner of a hex with
a sphere forms a triangle, the hexes adjacent to the singular node partition
the sphere into triangular regions thus forming a sphere triangulation. This
is depicted in Figure 1. The sphere triangulation encodes the _singular node
type_. If two sphere triangulations are isomorphic, then their singular nodes
are of the same type. The _signature_ of a singular node is a list of numbers
indicating how many adjacent singular edges are of each degree. Since singular
edges have degree 3 or higher, the signature of a node starts with the number
of edges with degree 3. For singular nodes whose adjacent singular edges have
only valence 3, 4, or 5, the signature also uniquely encodes the singular node
type [11]. For this reason we will frequently identify singular nodes by their
signature e.g. (4,0,0) is the signature identifying the singular node type
generated by subdividing a single tetrahedron into four hexes as illustrated
in Figure 4.

###  2.2 Sheet Inflation

Let a _sheet_ $\mathcal{Q}\subset\mathcal{F}$ be a manifold quad mesh whose
boundary is a subset of the boundary of $\mathcal{H}$. A _sheet inflation_
based on sheet $\mathcal{Q}$ is a mesh modifying operation by which each
$q\in\mathcal{Q}$ is thickened from a quad face to a hex cell [18]. A sheet
inflation operation is depicted in Figure 2.

Figure 2: (Left) Red quads indicate a sheet inside of a hex mesh of an
ellipsoid. Red curves depict its singular graph. The sheet is manifold with
boundary on the boundary of the hex mesh. (Right) Blue quads indicate all
faces of the inflated sheet.

In the case that the inflated sheet passes through a singular node, the
singular type of that node can change. One can interpret this as the sheet
passing an infinitesimally small gap between singular curves and forcing them
apart, or as cutting a singular node into separate pieces. As singular nodes
are often more easily visualized as sphere triangulations we describe here how
sheet inflation through a singular node corresponds to a _splitting_ of the
sphere triangulation. This splitting is illustrated by Figure 3.

Given a singular node $v$, and a sheet $\mathcal{Q}\subset\mathcal{F}$ that
passes through $v$, the faces of $\mathcal{Q}$ map to edges of
$\mathcal{T}(v)$. These edges trace out a cycle in the graph of
$\mathcal{T}(v)$ effectively partitioning the sphere into two disk
triangulations $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$. When the sheet is
thickened into a layer of hexes, these two disk triangulations are cut apart.
Then the disks are patched into sphere triangulations again by adding one new
vertex to each disk and attaching triangles from the boundary of each disk to
their respective new vertex. This splitting operation is illustrated in Figure
3. The end result is two sphere triangulations one built from
$\mathcal{D}_{1}$ and one built from $\mathcal{D}_{2}$.

Figure 3: (Left) The yellow sphere triangulation indicates the structure of a
singular node. Red quads indicate a sheet intersecting the singular node. The
sheet intersects the sphere triangulation on a cycle of red curves that divide
the sphere triangulation into two disks. (Right) Two singular nodes are
visualized from inflation of the red sheet on the left. Sphere triangulations
of the resulting two nodes are shown. Blue quad faces indicate newly created
faces from the inflation. Blue edges indicate newly created edges in each
sphere triangulation.

##  3 Singular Node Decomposition (Valence 3,4,5)

We are now equipped with the mechanism by which all practically relevant
singular nodes can be decomposed into singular curves. These nodes are
enumerated in [11, Figure 6] and will also be shown at the beginning of each
respective decomposition. Various singular decompositions will be depicted
throughout the remainder of this paper. Valence 3 singular curves will be red,
valence 5 singular curves will be green and higher valence singular curves
will be blue. We omit drawing interior regular edges to minimize clutter.
Since these operations can be challenging to understand from static images, we
also include supplemental videos for many of the decompositions.

In the first column of Figure 4 we start with the (4,0,0) singular node, which
consists of four valence 3 singular curves joined at a junction. With a single
sheet inflation, this singular node is revealed to actually be two valence 3
singular curves that pass each other orthogonally. A similar theme follows for
(2,2,2) in the second column which is revealed to be a valence 3 and a valence
5 singular curve passing each other orthogonally. Finally in the third column,
the (0,4,4) node decomposes into two valence 5 singular curves passing each
other orthogonally.

Figure 4: From left to right the (4,0,0), (2,2,2), and (0,4,4) singular nodes
are depicted. Top to bottom indicates steps to decompose each singular node.
Red quads indicate the sheet to be inflated. Blue quads indicate faces of the
newly inflated hexes. Red(Green) edges are valence 3(5) singularities. One
sheet inflation is sufficient to decompose each of these nodes.

From these three examples, it is tempting to think all singular nodes may
consist of singular curves glued together orthogonally. One might conclude as
well that the number of singular curves of a particular valence meeting at a
singular node from this construction must be even. The (1,3,3) singular node,
shown in Figure 5, presents a curious counterexample. Since it consists of one
valence 3 singular curve and three valence 5 singular curves, it is impossible
to decompose these into two singular curves passing each other orthogonally in
a valid hex mesh.

This conundrum is resolved by realizing that one of the regular edges adjacent
to this singular node is actually a pair of valence 3 and valence 5 singular
curves, glued together in parallel. Another way to understand this node is
that one valence 5 singular curve has split an otherwise parallel pair of
valence 3 and 5 curves. The decomposition of this node into one valence 3 and
two valence 5 singular curves is illustrated in Figure 5.

Figure 5: The (1,3,3) singular node is decomposed into two valence 5 and one
valence 3 curve via two sheet inflations.

In Figure 6 we decompose the (0,3,6) singular node into three valence 5
singular curves. It is also valuable to think of the (0,3,6) node as a
combination of two (0,4,4) singular nodes. In this way, one only needs to
decompose a singular node into constituent nodes that are already known to be
decomposable. The fourth image of Figure 6 shows exactly this decomposition
which we will notate as:

$(0,3,6)=(0,4,4)+_{5}(0,4,4)$

The subscript 5 on the plus symbol denotes that two singular nodes are joined
along a valence 5 edge, followed by an inverse sheet inflation (sheet
collapse). This notation serves only as a shorthand and does not uniquely
encode how to glue two singular nodes together. It serves more as a recipe
than an equation with any algebraic properties. For completion, we show the
rest of the decomposition of the constituent (0,4,4) nodes.

Figure 6: The (0,3,6) singular node is decomposed into three valence 5 curves
via three sheet inflations.

The (0,2,8) singular node is decomposed in Figure 7 into four valence 5
curves. In the fourth image, we see that

$(0,2,8)=(0,3,6)+_{5}(0,4,4)$

and both constituent singular nodes have already been shown to be
decomposable. For completion, the rest of the decomposition steps are also
shown.

Figure 7: The (0,2,8) singular node is decomposed into four valence 5 curves
via five sheet inflations.

The (2,0,6) singular node is decomposed in Figure 8 into four valence 5 and
two valence 3 curves. By the fourth image we see

$(2,0,6)=(1,3,3)+_{4}(1,3,3).$

The rest of this decomposition is still interesting however as it introduces a
valence 6 singularity in the fifth image and a singular node with signature
(1,3,3,1). This valence 6 singular curve is removed in image 9 by decomposing
it into two valence 5 curves.

Figure 8: The (2,0,6) singular node is decomposed into four valence 5 and two
valence 3 curves via seven sheet inflations.

Finally the (0,0,12) singular node decomposition is shown in Figure 9 to
become six valence 5 curves. This singular node is especially interesting as
we were unable to show a decomposition of the form

$(0,0,12)=(0,4,4)+_{n}(0,2,8).$

Even though the number of singular curves present is sufficient, we were not
able to perform an inverse sheet inflation between (0,4,4) and (0,2,8) to
obtain (0,0,12). This shows that the order in which singular curves are
combined matters. As this figure is especially complex to comprehend, we offer
the following roadmap of how the decomposition is performed.

$(0,0,12)=\underbrace{\left((0,3,6)+_{5}(0,3,6)\right)}_{(0,2,8,1)}\,+_{6}\,\underbrace{\left((0,4,4)+_{5}(0,4,4)\right)}_{(0,4,4,1)}$

Figure 9: The (0,0,12) singular node is decomposed into six valence 5 curves
via seven sheet inflations.

##  4 Decomposing General Singular Nodes

Given that the eight singular nodes of valence 3, 4, or 5 are decomposable
into singular curves, a natural next question is whether decomposition extends
to higher valence singular nodes. In fact, the decomposition of the (0,0,12)
and (2,0,6) both already required decomposing singular nodes with valence 6:
(0,4,4,1), (0,2,8,1) and (1,3,3,1). We will refer to previously known
decomposable singular nodes and their associated sphere triangulations as
_base cases_. To generalize decomposability of singular nodes we offer the
following result.

######  Proposition 1.

Given a sphere triangulation $\mathcal{T}$ with some vertex $u$ of degree
larger than 5, there exists a splitting such that either the number of
vertices in both resulting triangulations decreases or the resulting
triangulations are base cases.

###### Proof.

The local neighborhood of $u$ is an umbrella $\mathcal{U}$ of at least 6
triangles. The boundary of this umbrella is a cycle of at least 6 vertices
denoted by $\mathcal{C}$. To construct a splitting of $\mathcal{T}$ into
triangulations of fewer vertices, we need a pair of vertices $a$ and $b$
adjacent to $u$ that are at least 3 edges apart from each other in
$\mathcal{C}$ such that there is path $p$ from $a$ to $b$ through the interior
of $\mathcal{T}-\mathcal{U}$. This construction is illustrated in Figure 10.
The sequence of edges $[(ua),p,(bu)]$ partitions $\mathcal{T}$ into
$\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ where each disk triangulation has at
least 2 interior vertices. Since splitting a sphere triangulation replaces all
vertices on the interior of either side with just one new vertex each, both
resulting triangulations will have fewer vertices than $\mathcal{T}$. For
readability, we leave more detailed construction of the splitting to
supplementary materials. ∎

$b$$u$$a$$p$$\mathcal{D}_{1}$$\mathcal{D}_{2}$$Interior\;vertices$$Interior\;vertices$

Figure 10: Illustration of how to find a cycle such that splitting along that
cycle results in two sphere triangulations, each with fewer vertices. The only
requirement is that there is a vertex $u$ of degree $\geq 6$. The required
cycle is then $[(ua),p,(bu)]$.

Applying the splitting in Prop. 1 could result directly in base cases, where
the rest of the decomposition is already known. If the splitting does not
result in base cases, then it produces triangulations with fewer vertices.
This can be repeated until there are not enough vertices to have a degree 6
vertex. Since sheet inflation at a singular node corresponds to splitting of a
sphere triangulation, Prop. 1 allows us to find a sequence of sheets whose
inflation results in singular nodes that have lower than valence 6 singular
edges. We have already enumerated singular decompositions for all singular
nodes with valence lower than 6 and can therefore decompose any singular node
into singular curves.

A limitation of Prop. 1 is that it restricts attention to individual singular
nodes while ignoring the full singular graph of the mesh. It can be
challenging to extend a sheet known locally around a singular node to the rest
of the hex mesh while guaranteeing no self-intersection occurs. We present our
simplistic solution to extending sheets in subsection 5.1 and leave more
careful consideration of how to avoid self-intersection to future work.

##  5 Results

1:procedure Decompose-Singular-Graph($\mathcal{H}$)

2: do

3: $N\leftarrow$ GetRandomSingularNode($\mathcal{H}$)

4: if OnlyHasValence345($\mathcal{H}$,$N$) then

5: $C\leftarrow$ GetHardcodedCut($\mathcal{H}$, $N$)

6: else

7: $C\leftarrow$ GetGeneralCut($\mathcal{H}$, $N$)

8: end if

9: $S\leftarrow$ PropagateCut($\mathcal{H}$, $C$)

10: $H\leftarrow$ SheetInflation($\mathcal{H}$, $S$)

11: while $N\neq\emptyset$

12: return $\mathcal{H}$

13:end procedure

Algorithm 1 Decomposes all singular nodes of a hex mesh into singular curves.

###  5.1 Singular Graph Decomposition

We develop a procedure to perform singular mesh decompositions on general hex
meshes. Pseudocode for this procedure is given in Alg. 1 and Alg. 2 First, we
randomly select a singular node. For any singular node with valence restricted
to 3, 4, or 5, we hard code a subset of faces adjacent to the node to be
inflated. If the node has valence 6 or higher, we use Prop. 1 (denoted
GetGeneralCut in Alg. 1) to select these faces. These faces form a partial
sheet that decomposes the initially selected node, but need to be extended
through the rest of the mesh in order to be inflatable.

Next we propagate the partial sheet throughout the hex mesh following Alg. 2.
Let a face be _parallel_ to the partial sheet if they share a regular edge but
share no adjacent hexes. We greedily add parallel faces to the partial sheet
until no more parallel faces can be found. Next we look for any interior
singular vertices on the boundary of the partial sheet. If such a vertex is
found, then we compute the smallest number of new faces that need to be added
to the partial sheet so that its boundary excludes this singular vertex. This
is denoted by Put-v-In-S in Alg. 2 and is equivalent to a graph shortest path
computation on the triangulation representing this singular vertex.

These two steps are repeated until no more parallel faces can be found, and
the boundary of the partial sheet is entirely on the boundary of the hex mesh.
If at any stage of the algorithm, the partial sheet became non-manifold then
the sheet propagation algorithm has failed. If the sheet is manifold then we
inflate it resulting in the decomposition of at least one singular node. All
results shown were generated by Alg. 1.

1:procedure PropagateCut($\mathcal{H}$, $S$)

2: $\mathcal{Q}\leftarrow$GetFaces($\mathcal{H}$)

3: while $\exists f\in\mathcal{Q}:$Parallel($\mathcal{H}$, $S$, $f$) do

4: $S\leftarrow S\cup f$

5: end while

6: $\mathcal{V}\leftarrow$GetVertices($\mathcal{H}$)

7: $\mathcal{V}_{S}\leftarrow$GetInteriorSingularVertices($\mathcal{H}$)

8: if $\exists v\in(\partial S\cap\mathcal{V}_{S})$ then

9: $S\leftarrow$ Put-v-In-S($\mathcal{H}$, $S$, $v$)

10: else

11: if NonManifold($\mathcal{H}$, $S$) then

12: return ERROR

13: else

14: return $S$

15: end if

16: end if

17: return PropagateCut($\mathcal{H}$, $S$)

18:end procedure

Algorithm 2 Propagates a partial sheet into a full sheet recursively.

Applying our decomposition to a hex mesh of a sphere reveals that it has the
same singular graph structure as that of a padded tetrahedron. Figure 11 shows
this correspondence where inflating one sheet that passes through seven
singular nodes, simultaneously decomposes three of them. The end result is a
singular graph composed of four (4,0,0) singular nodes. One of these nodes has
singular curves that all connect directly to the boundary. The other four of
these nodes connect to each other and the boundary via valence 3 singular
curves in a tetrahedral arrangement. This singular graph is exactly what one
obtains by padding a hex mesh of a regular tetrahedron i.e. padding a (4,0,0)
node.

Figure 11: (Left) Hex mesh of sphere with singular graph. (Mid-left) Blue
hexes are newly inflated hexes. (Mid-right) Hex mesh post-inflation. (Right)
Singular graph of a padded hex mesh of a tetrahedron. The last two images have
topologically equivalent singular graphs.

Changing how the sheet cuts through the singular graph produces different
intermediate and final singular graphs. In Figure 12, we decompose a padded
cube in two different sequences and show the their intermediate singular
graphs. To improve clarity, we provide schematics of a subset of the singular
graphs. The ending singular graphs from both sequences are also topologically
distinct i.e. no purely geometric deformation maps one singular graph into the
other. They do however appear to invariably contain a single singular cycle.

The first sheet inflation of the second sequence results in the same singular
graph as a padded hex mesh of a triangular prism: a padded (2,3,0). Since the
hex mesh of a sphere has the same singular graph as the padded cube, these
results indicate that singular graphs for a padded cube, padded tet, and
padded triangular prism are identical up to a series of sheet inflation and
collapses.

Figure 12: We show two sequences of singular graph decomposition starting from
the same hex mesh on the left to a fully decomposed singular graph on the
right. The first row indicates select singular graph schematics for the first
sequence. The last row indicates select singular graph schematics for the
second sequence. Even though both singular graphs start out identical, the
ending singular graphs are different due to different sheet inflations.

In Figure 13, we apply our decomposition to more complex singular graphs. The
first two rows depict the decomposition of the G1 hex mesh. The starting
singular graph consists of 12 nodes connected by 36 singular curves. This
graph is successfully decomposed into seven singular curves, one of which is a
closed cycle. The last two rows depict the decomposition of the G2 hex mesh.
The starting singular graph consists of 16 nodes connected by 40 singular
curves. This graph is successfully decomposed into 12 singular curves, two of
which are closed cycles. While the starting singular graphs are different,
both meshes are fully decomposed with the same number of sheet inflations.

Figure 13: We apply singular decomposition to the G1 and G2 hex meshes. The
first two rows correspond to the sequence of singular graphs from decomposing
G1. The last two rows correspond to the sequence of singular graphs from
decomposing G2. The number of singular nodes decreases each sheet inflation
ultimately resulting in a singular graph with no nodes at all.

Figure 14: We apply singular decomposition to the cactus mesh from [19]. The
number of singular nodes decreases with each sheet inflation ultimately
resulting in a singular graph with no nodes at all. While the original
singular graph consisted of only valence 3, 4 and 5 nodes, intermediate
singular graphs from this sequence contain singular nodes with signature
(2,3,0,2). The fully decomposed singular graph has valence 6 curves.

Finally, we apply our decomposition in Figure 14 to the cactus mesh from [19].
While the mesh starts with only singularities of valence 3, 4 and 5, the
decomposition results in intermediate singular graphs with nodes of signature
(2,3,0,2). The final configuration is seen to contain singular curves of
valence 6. Since our goal is only to remove singular nodes, we terminate with
valence 6 curves.

###  5.2 Scaled Jacobians

The minimum scaled Jacobian of a hex mesh is a common metric by which to
evaluate distortion of the mesh [20]. We maximize the minimum scaled Jacobian
before and after singular decomposition of each singular node with free
boundaries and present the resulting minimum scaled Jacobians in Table 1.
Unsurprisingly, singular nodes have lower scaled Jacobians than singular
curves.

By symmetry, the minimum scaled Jacobian of any hex mesh, regardless of
resolution, containing a (4,0,0) node is upper bounded by
$\frac{4}{3\sqrt{3}}=.7698$. The same bound for a hex mesh containing a
(0,0,12) node is $\frac{\sqrt{2(5+\sqrt{5})}}{5}=.761$. These bounds are
exactly attained in Table 1 for the (4,0,0) and (0,0,12) nodes. The same upper
bound computed for meshes containing valence 3 singular curves is
$\sin(\frac{2\pi}{3})=.866$ and for meshes containing valence 5 singular
curves is $\sin(\frac{2\pi}{3})=.951$.

By performing a sheet inflation to split singular nodes into singular curves,
the minimum scaled Jacobian of the (4,0,0) node is increased to $.86$, almost
the theoretic upper bound. For (0,4,4) as well, decomposing the singular node
into two valence 5 curves brings the minimum scaled Jacobian to almost the
theoretic upper bound. Decomposing (0,0,12) node into six valence 5 curves
brings significant improvement to the minimum scaled Jacobian, though it is
not as close to the theoretic upper bound due to interactions between singular
curves.

Moving towards full singular graphs, we perform the same scaled Jacobian
optimization for a sphere mesh. Maximization of its minimum scaled Jacobian
results in a value of .768, close to the upper bound for any mesh containing a
(4,0,0) node. We apply our decomposition to this mesh and re-optimize its
scaled Jacobian resulting in a significant improvement to .849. We run the
same optimizations on the padded tetrahedron, G1, and G2 resulting in similar
increases in the minimum scaled Jacobian. These results are summarized in
Table 1.

Mesh | Original | Decomposed | UpperBound
---|---|---|---
(4,0,0) | 0.769 | 0.86 | 0.866
(2,2,2) | 0.807 | 0.862 | 0.866
(0,4,4) | 0.896 | 0.943 | 0.951
(1,3,3) | 0.822 | 0.865 | 0.866
(0,3,6) | 0.863 | 0.939 | 0.951
(0,2,8) | 0.82 | 0.937 | 0.951
(2,0,6) | 0.745 | 0.856 | 0.866
(0,0,12) | 0.761 | 0.926 | 0.951
Sphere | 0.768 | 0.849 | 0.866
Padded Tet | 0.715 | 0.812 | 0.866
G1 | 0.769 | 0.811 | 0.866
G2 | 0.757 | 0.820 | 0.866
Ellipsoid | 0.767 | 0.825 | 0.866
Table 1: For various hex meshes, we indicate the maximized minimum scaled
Jacobian before and after singular decomposition. The first column indicates
the mesh, the second column indicates before singular decomposition, and the
third column indicates after. The fourth column indicates a theoretic upper
bound on the minimum scaled Jacobian for the decomposed mesh. It essentially
indicates the presence of a valence 3 or 5 singular curve. The maximized
minimum scaled Jacobian is invariably higher post singular decomposition.

##  6 Conclusions and Future Work

This paper presents singular nodes as the result of gluing singular curves
together at a point and shows that the reverse can be done via sheet inflation
to untangle singular nodes into simple singular curves. This removes the 3D
complexity of singular nodes leaving meshes with lower distortion. We
demonstrate this procedure on a variety of meshes showing in all cases that no
singular nodes are left behind.

The main limitation of our work is that the local sheets we prescribe for
decomposing a singular node are not guaranteed to propagate globally while
avoiding self-intersection. This can result in the inability to decompose a
singular graph by removing all of its singular nodes. We expect that a valid
sheet inflation can always be found and leave its efficient computation to
future work.

While our method decreases the number of singular nodes in a mesh, its _base
complex_[14] may increase in size. This tradeoff should be considered by the
user as they may have to choose between a larger scaled Jacobian or
maintaining a small number of base complex cells.

Our results can be extended to design new ways of modifying the singular graph
of a mesh. Instead of only decomposing nodes into curves, one can _rewire_
singular curves by merging them at a node with sheet collapse, and decomposing
them in a different way from how they were combined. For example, consider the
(4,0,0) node in Figure 4. It’s sphere triangulation is a tetrahedron which
contains three distinct cycles of length four. Therefore, it is possible to
bring two valence 3 singular curves together to form a (4,0,0) node and split
them apart again in three distinct ways. Each one results in a different
singular graph, none of which require introducing new singularities.

Many works aim to build minimal degree smooth parameterizations of quad meshes
with singularities [21, 22]. These methods do not clearly generalize to the
volumetric case where singular nodes may suffer decreased continuity from
methods designed for 2D singularities. A promising approach following our work
is then to decompose any given singular graph so that no singular nodes exist.
We expect that it is easier to adapt quad mesh singular parameterization
methods to singular curves that are just 2D singularities extruded into 3D
than it is to adapt parameterization methods for singular nodes. Even if one
derived a singular node parameterization method for a specific singular node
type, there is no guarantee that it extends to any other node type. This
problem is made easier by only needing to consider singular curves after
decomposition.

## Acknowledgements

This project was launched at the Summer Geometry Initiative (SGI) 2021,
supported by National Science Foundation grant DMS-2103933, Army Research
Office grant W911NF2110095, and generous donations from corporate partners.
Paul Zhang acknowledges the support of the Department of Energy Computer
Science Graduate Fellowship and the Mathworks Fellowship. The authors thank
Justin Solomon and David Bommes for many valuable discussions.

## References

  * [1] Weingarten V.I.  “The controversy over hex or tet meshing.”  _Machine design_ , vol. 66, no. 8, 74–76, 1994
  * [2] Cifuentes A., Kalbag A.  “A performance study of tetrahedral and hexahedral elements in 3-D finite element structural analysis.”  _Finite Elements in Analysis and Design_ , vol. 12, no. 3-4, 313–318, 1992
  * [3] Benzley S., Perry E., Merkley K., Clark B., Sjaardema G.  “A Comparison of All Hexagonal and All Tetrahedral Finite Element Meshes for Elastic and Elasto-Plastic Analysis.”  _International Meshing Roundtable_ , vol. 17, 01 1995
  * [4] Ray N., Sokolov D., Lévy B.  “Practical 3D frame field generation.”  _ACM Transactions on Graphics_ , vol. 35, no. 6, 1–9, Nov. 2016.  URL http://dl.acm.org/citation.cfm?doid=2980179.2982408
  * [5] Lyon M., Bommes D., Kobbelt L.  “HexEx: Robust Hexahedral Mesh Extraction.”  _ACM Trans. Graph._ , vol. 35, no. 4, 123:1–123:11, Jul. 2016
  * [6] Nieser M., Reitebuch U., Polthier K.  “CubeCover–Parameterization of 3D Volumes.”  _Computer Graphics Forum_ , vol. 30, no. 5, 1397–1406, Aug. 2011\.  URL http://doi.wiley.com/10.1111/j.1467-8659.2011.02014.x
  * [7] Huang J., Tong Y., Wei H., Bao H.  “Boundary aligned smooth 3D cross-frame field.”  _SIGGRAPH Asia_ , p. 1. ACM Press, Hong Kong, China, 2011.  URL http://dl.acm.org/citation.cfm?doid=2024156.2024177
  * [8] Solomon J., Vaxman A., Bommes D.  “Boundary Element Octahedral Fields in Volumes.”  _ACM Transactions on Graphics_ , vol. 36, no. 3, 1–16, May 2017.  URL http://dl.acm.org/citation.cfm?doid=3087678.3065254
  * [9] Li Y., Liu Y., Xu W., Wang W., Guo B.  “All-hex meshing using singularity-restricted field.”  _ACM Trans. Graph._ , vol. 31, no. 6, 177, 2012
  * [10] Jiang T., Huang J., Wang Y., Tong Y., Bao H.  “Frame Field Singularity Correction for Automatic Hexahedralization.”  _IEEE Trans. on Visualization & Computer Graphics_, vol. 20, no. 8, 1189–1199, Aug. 2014
  * [11] Liu H., Zhang P., Chien E., Solomon J., Bommes D.  “Singularity-constrained octahedral fields for hexahedral meshing.”  _ACM Transactions on Graphics_ , vol. 37, no. 4, 1–17, Jul. 2018\.  URL http://dl.acm.org/citation.cfm?doid=3197517.3201344
  * [12] Corman E., Crane K.  “Symmetric Moving Frames.”  _ACM Trans. Graph._ , vol. 38, no. 4, Jul. 2019.  URL https://doi.org/10.1145/3306346.3323029
  * [13] Gao X., Panozzo D., Wang W., Deng Z., Chen G.  “Robust structure simplification for hex re-meshing.”  _ACM Transactions on Graphics (TOG)_ , vol. 36, no. 6, 1–13, 2017
  * [14] Gao X., Deng Z., Chen G.  “Hexahedral mesh re-parameterization from aligned base-complex.”  _ACM Transactions on Graphics (TOG)_ , vol. 34, no. 4, 1–10, 2015
  * [15] Xu G., Ling R., Zhang Y.J., Xiao Z., Ji Z., Rabczuk T.  “Singularity Structure Simplification of Hexahedral Meshes via Weighted Ranking.”  _Computer-Aided Design_ , vol. 130, 102946, 2021
  * [16] Campen M., Shen H., Zhou J., Zorin D.  “Seamless parametrization with arbitrary cones for arbitrary genus.”  _ACM Transactions on Graphics (TOG)_ , vol. 39, no. 1, 1–19, 2019
  * [17] Peng C.H., Zhang E., Kobayashi Y., Wonka P.  “Connectivity editing for quadrilateral meshes.”  _Proceedings of the 2011 SIGGRAPH Asia conference_ , pp. 1–12. 2011
  * [18] Ledoux F., Shepherd J.  “Topological modifications of hexahedral meshes via sheet operations: a theoretical study.”  _Engineering with Computers_ , vol. 26, no. 4, 433–447, 2010
  * [19] Bracci M., Tarini M., Pietroni N., Livesu M., Cignoni P.  “HexaLab.net: An online viewer for hexahedral meshes.”  _Computer-Aided Design_ , vol. 110, 24–36, 2019.  URL https://www.sciencedirect.com/science/article/pii/S0010448518304238
  * [20] Quadros R.  “The CUBIT Geometry and Meshing Toolkit.”  https://cubit.sandia.gov/, 2021
  * [21] Karčiauskas K., Peters J.  “Minimal bi-6 G2 completion of bicubic spline surfaces.”  _Computer Aided Geometric Design_ , vol. 41, 10–22, 2016
  * [22] Karčiauskas K., Peters J.  “Refinable smooth surfaces for locally quad-dominant meshes with T-gons.”  _Computers & graphics_, vol. 82, 193–202, 2019


# Destructive social noise effects on homogeneous and heterogeneous networks:
Induced-phases in the majority-rule model

Didi Ahmad Mulya<EMAIL_ADDRESS>Roni Muslim
<EMAIL_ADDRESS>
###### Abstract

This paper explores the effects of destructive social noises, characterized by
independence and anticonformity, on the occurrence of order-disorder phase
transitions within the framework of the majority-rule model. The study
encompasses various network topologies, including the complete graph, two-
dimensional (2-D) square lattice, three-dimensional (3-D) square lattice, as
well as heterogeneous networks like Watts-Strogatz, Barabási-Albert, and
Erdős-Rényi networks. These social behaviors are quantified using the
parameter $p$, representing the probability of agents exhibiting independent
or anticonformist tendencies. Our results reveal that the occurrence and
nature of phase transitions depend on the fundamental characteristics of the
model and the underlying network structure. Specifically, the model exhibits
continuous phase transitions on the complete graph and the 2-D square lattice,
with critical points that vary based on the model’s attributes. However, on
the 3-D lattice, the independence model notably lacks a phase transition,
while the anticonformity model still exhibits a continuous phase transition.
By employing finite-size scaling analysis and evaluating critical exponents,
we confirm that the model falls within the Ising model universality class,
except in the 3-D model. This study provides insights into the dynamic
interplay between social dynamics and network topology, especially based on
the majority-rule model.

###### keywords:

Opinion dynamics model , majority-rule , networks , phase transition ,
universality

[brin] organization=Research Center for Quantum Physics, National Research and
Innovation Agency (BRIN), city=South Tangerang, postcode=15314,
country=Indonesia

[uty] organization=Department of Industrial Engineering, University of
Technology Yogyakarta, city=Yogyakarta, postcode=55285, country=Indonesia

##  1 Introduction

Scientists employ the principles and concepts of physics within the field of
social science to gain a deeper understanding of social phenomena occurring in
society [1, 2, 3, 4]. One of the most effective approaches utilized by
scientists involves opinion modeling, which is constructed based on various
social characteristics. These characteristics include individuals’ tendencies
to align with majority opinions [5, 6, 7], the presence of social pressures
that induce opinion change [8], and the influence of social validation, which
compels individuals to follow prevailing trends [9, 10]. In addition to the
discrete opinion models mentioned, there are also continuous opinion models
that imply two individuals can influence each other if there is trust between
them [11, 12]. In general, a common feature emerging from the aforementioned
opinion dynamics models is that the system eventually reaches a state of
homogeneity, with all agents holding the same opinion or state. However, in
real social systems, such uniformity does not always occur. Instead, there are
moments of coexistence between minority and majority opinions, even amidst
opinion turbulence. There are even instances where minority-majority opinions
vanish. Considering these factors, scientists have developed opinion dynamics
models with new features, such as taking into account destructive behavior
that contradicts societal norms. The existence of this destructive social
behavior can give rise to intriguing new phases that are worth exploring and
understanding from the perspective of physics.

Destructive social behavior refers to a state or condition in which an
individual or group deviates from prevailing societal norms, expectations, or
standards. It often involves departing from established conventions,
traditions, or cultural practices. Such behavior is more commonly known as
nonconformity [13]. Social scientists further classify nonconformity into two
types: independence and anticonformity [14, 15, 16, 17]. Independence refers
to the ability or disposition of an individual to think, act, or make
decisions autonomously, free from undue influence, coercion, or external
pressure. Conversely, anticonformity is a deliberate rejection or opposition
to conforming to established norms, rules, conventions, or societal
expectations. It involves consciously going against the grain, challenging the
status quo, or resisting pressures to conform.

The existence of nonconformity behavior in opinion dynamics models that
influences the emergence of new phases has been extensively studied with
various scenarios [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. The
emergence of these new phases is an intriguing subject from the perspective of
other sciences, such as Physics, because it shares similarities with the
phenomenon of ferromagnetic-antiferromagnetic phase transitions in spin
systems, which can be understood quite well. With a slight change in the noise
parameter, the complexity at the microscopic level manifests itself at the
macroscopic level, which can be well understood. Similarly, this occurs within
social systems. Phase transitions can be understood as a shift from consensus
to discord in social-political discourse. As the Ising model explains in
physics, this phase transition arises from a disruptive parameter known as
temperature, which opposes the alignment of interacting spins, leading to
anti-parallel orientations. At a specific critical temperature, the spin
states become completely random. Similarly, in opinion models, interactions
between individuals, such as discussions, lead to convergence and consensus
formation. Conversely, oppositional behavior results in a stalemate condition,
where individuals defend their opinions.

In the context of this destructive social behavior, apart from scenarios
involving interacting agents, network topology also plays a crucial role in
the occurrence and type of phase transitions. For example, the Ising model on
a 1D lattice does not exhibit phase transitions, while higher-dimensional
networks demonstrate this phenomenon, even though they have the same
microscopic interaction scenarios. Even in the majority rule dynamics of
opinion models on a 2-D lattice with an independence parameter, phase
transitions are absent [31]. In Ref. [31], we observe a continuous phase
transition only when four agents in a two-dimensional lattice adopt
anticonformity behavior, whereas it does not occur for independent agents.
Recent work has also shown continuous phase transitions in the majority-rule
model with independent behavior. Furthermore, this model falls within the same
universality class as the mean-field Ising model [32].

This paper is focused on investigating the impact of disruptive social
factors, specifically independence and anticonformity, on the occurrence of
the order-disorder phase transition in the majority-rule model. The model is
defined on various types of networks to provide a more comprehensive
understanding of the order-disorder phase transition phenomenon, as well as
the scaling behavior of the model. The examined networks include homogeneous
networks such as the complete graph, a two-dimensional lattice, and a three-
dimensional lattice, as well as heterogeneous networks such as the Watts-
Strogatz (W-S) network [33], Barabasi-Albert (A-B) network [34], and Erdos-
Renyi (E-R) network [35]. Our analytical and numerical findings indicate that
the complete graph model undergoes a continuous phase transition with critical
points of $p_{c}=1/3$ for the independence model and $p_{c}=1/4$ for the
nonconformity model. Both models exhibit identical critical exponents,
specifically $\beta\approx 0.5$, $\nu\approx 2.0$, and $\gamma\approx 1.0$,
placing them within the same universality class as the mean-field Ising model.
Moreover, numerical simulations reveal that the two-dimensional lattice model
experiences a continuous phase transition with critical points at
approximately $p_{c}\approx 0.106$ for the independence model and
$p_{c}\approx 0.062$ for the nonconformity model, sharing critical exponents
of $\beta\approx 0.125$, $\nu\approx 2.0$, and $\gamma\approx 1.75$. In other
words, both models belong to the same universality class as the two-
dimensional Ising model. The critical exponents satisfy the identity equation
$\nu d=2\beta+\gamma$ [36]. Furthermore, a phase transition is not observed
for the independence model on a three-dimensional lattice, but the
nonconformity model undergoes a continuous phase transition with a critical
point at approximately $p_{c}\approx 0.268$. The best-fitting critical
exponents that collapse all data for $N$ are approximately $\beta\approx
0.25$, $\nu\approx 1.40$, and $\gamma\approx 1.40$.

In our analysis of heterogeneous networks, we only ascertain whether the model
undergoes a continuous phase transition. Based on our numerical results, the
independence and nonconformity models exhibit continuous phase transitions for
all the networks under consideration. Finally, as we did for homogeneous
networks, we refrain from estimating critical points and critical exponents on
heterogeneous networks. Our study will likely be a foundation for further
investigations in this field.

##  2 Model and methods

In the majority-rule model, a randomly selected small group comprises several
agents. Within this group, all members interact to align their opinions or
states with the majority opinion or state of the group. In the realm of social
psychology, when agents conform to the majority opinion, they exhibit
conformity behavior, often referred to as “conformist agents”. Conformity, in
essence, involves aligning one’s attitudes, beliefs, and behavior with the
established norms of a group [37]. In contrast to conformity, another
noteworthy social behavior is nonconformity, which can be further subdivided
into two distinct categories: anticonformity and independence.

In this paper, we delve into these three types of social behaviors and
introduce a probability parameter denoted as $p$, which represents the
likelihood of agents adopting either independence or anticonformity
(nonconformist agents). In other words, with a probability of $p$, agents
choose to act independently or exhibit anticonformity. Conversely, with a
probability of $1-p$, agents conform by following the majority opinion. To
elucidate further, we outline the algorithm of the model as follows:

  1. 1.

To compute the macroscopic parameters of the system, the initial state of the
system is prepared in a disordered state, wherein the number of agents with
positive and negative opinions is equal.

  2. 2.

Microscopic Interaction within the model:

    1. (a)

Independence Model: Within the population, a group of agents is randomly
selected, and with a probability of $p$, all agents act independently.
Simultaneously, with a probability of $1/2$, all agents change their opinions
in the opposite direction, i.e., $\pm S_{i}(i+t)=\mp S_{i}(t)$.

    2. (b)

Anticonformity Model: In the population, a group of agents is randomly chosen,
and with a probability of $p$, all agents act in an anticonformist manner.
Subsequently, if all agents within the group share the same opinion, they will
reverse their opinions in the opposite direction, i.e., $\pm S_{i}(i+t)=\mp
S_{i}(t)$.

  3. 3.

Alternatively, with a probability of $1-p$, all agents opt to conform by
following the majority opinion.

The model is defined within three networks: a complete graph, a two-
dimensional square lattice, and a three-dimensional square lattice. Each agent
is associated with two potential opinions or states, represented by Ising
numbers $\pm 1$. All agent opinions are embedded within the network nodes, and
the links or edges between nodes symbolize social connections.

The complete graph represents a network structure in which every node is
connected to every other node. In other words, on the complete graph, all
agents are neighbors and can interact with each other with the same
probability. Furthermore, we have also defined this model on two and three-
dimensional square lattices and three other heterogeneous networks, namely,
the B-A, W-S, and E-R networks. Each agent has four nearest neighbors in the
two-dimensional square lattice, while in the three-dimensional square lattice,
each agent has six nearest neighbors. In the heterogeneous networks, we
examined networks in which the minimum degree of connectivity for each node is
2, and we selected three agents to follow the majority-rule model algorithm,
as mentioned above.

For the model on the complete graph, we can conveniently perform an analytical
treatment to compute the order (magnetization) of the model using the
following formula:

$m=\dfrac{N_{\uparrow}-N_{\downarrow}}{N_{\uparrow}+N_{\downarrow}}=2\,r-1,$
(1)

where $N_{\uparrow}$ represents the total number of agents with the “up”
opinion, $N_{\downarrow}$ represents the total number of agents with the
“down” opinion, and $r=N_{\uparrow}/(N_{\uparrow}+N_{\downarrow})$ denotes the
fraction of agents with the “up” opinion. In the numerical simulation
treatment, we utilize the expressions $\langle
m\rangle=\sum_{j}m_{j}/R,\langle\chi\rangle=\sum_{j}\chi_{j}/R,$ and $\langle
U\rangle=\sum_{j}U_{j}/R$, where $R$ represents the total number of
independent realizations for each data point.

We employ finite-size scaling analysis to compute the critical exponents
corresponding to the order parameter $m$, susceptibility $\chi$, and Binder
cumulant $U$. The finite-size scaling relations are given by:

$\displaystyle m$ $\displaystyle=\phi_{m}(x)N^{-\beta/\nu},$ (2)
$\displaystyle\chi$ $\displaystyle=\phi_{\chi}(x)N^{\gamma/\nu},$ (3)
$\displaystyle p-p_{c}$ $\displaystyle=c\,N^{-1/\nu},$ (4) $\displaystyle U$
$\displaystyle=\phi_{U}(x),$ (5)

where $\phi$ represents the dimensionless scaling function that fits the data
near the critical point $p_{c}$. The critical exponents $\beta$, $\gamma$, and
$\nu$ come into play near the critical point.

The fluctuation or susceptibility $\chi$ and Binder cumulant $U$ are defined
as follows [38]:

$\displaystyle\chi$ $\displaystyle=N\left(\langle m^{2}\rangle-\langle
m\rangle^{2}\right),$ (6) $\displaystyle U$
$\displaystyle=1-\dfrac{1}{3}\dfrac{\langle m^{4}\rangle}{\langle
m^{2}\rangle^{2}}.$ (7)

We can determine the critical point at which the system undergoes an order-
disorder phase transition by identifying the intersection point of the curves
of the Binder cumulant $U$ and the probability $p$.

##  3 Result and Discussion

###  3.1 Time evolution and stationary state

This section discusses the dynamic evolution of the fraction opinion denoted
as $r$ across various population sizes $N$. The density of spin-up, $r$ at a
given time $t$, may exhibit fluctuations across different realizations,
contingent upon the specific characteristics of the model under examination.
We will provide a concise overview of the formulation governing the time
evolution of the spin-up density $r$ within the probability distribution
framework.

The probability of finding an agent with an opinion “up” at time $t$, denoted
as $r(t)$, is given by:

$r(t)=\sum_{x}x\,\mathrm{P}(x,t),$ (8)

where $P(x,t)$ represents the probability distribution of the system in state
$x$ (associated with the “up” state) at time $t$. The probability distribution
$P(x,t)$ can be determined when the initial conditions are known, employing
the following recursive formula:

$\mathrm{P}(x,t+1)=\sum_{x^{\prime}}\rho(x^{\prime}\to
x)\mathrm{P}(x^{\prime},t),$ (9)

where $\rho(x^{\prime}\to x)$ denotes the transition probability of agent with
state $x$ to $x^{\prime}$ at time $t$, which depends on the specific model
under consideration. Eq. (9) is commonly known as the discrete-time Master
equation and describes the temporal evolution of the probability distribution.

To derive the recursive formula for the density opinion $r$, we can start by
combine Eq. (8) and Eq. (9). We express the fraction opinion $r$ at the
subsequent time step, denoted as $r(t^{\prime})$, as follows:

$r(t^{\prime})=\sum_{x^{\prime}}\mathrm{P}(x^{\prime},t)\sum_{x}\rho(x^{\prime}\to
x)x,$ (10)

and the next can be further simplified as:

$\displaystyle r(t^{\prime})$
$\displaystyle=\sum_{x^{\prime}}x^{\prime}\,\mathrm{P}(x^{\prime},t)+\dfrac{1}{N}\sum_{x^{\prime}}\mathrm{P}(x^{\prime},t)\left[\rho^{+}(x^{\prime})-\rho^{-}(x^{\prime})\right]$
$\displaystyle=r(t)+\dfrac{1}{N}\left[\rho^{+}(r)-\rho^{-}(r)\right].$ (11)

In this context, $r(t)$ represents the initial fraction opinion, and the total
probability $P(x^{\prime},t)=1$. It is important to emphasize that, given the
model’s foundation on a complete graph reminiscent of the mean-field character
in statistical physics, the parameters $x$ and time $t$ are deemed to be non-
fluctuating. Therefore, we can denote $x^{\prime}$ as the fraction opinion
$r$. Drawing from this equation, we track the temporal evolution of the
fraction opinion $r$ during the sampling event corresponding to a single Monte
Carlo sweep.

To express Eq. 3.1 in a differential form, we take into account that the
temporal progression of the fraction opinion $r$ is measured in one step
(Monte Carlo step) by adjusting $t$ using a factor of $1/N$. Consequently, the
time increment equals $\Delta t=1/N$, as a single Monte Carlo sweep
corresponds to $N\Delta t$. As a result, in the scenario where $N\to\infty$ or
$\Delta t\to 0$, Eq. (3.1) can be rephrased in differential form as follows:

$\dfrac{\mathrm{d}r}{\mathrm{d}t}=\rho^{+}(r)-\rho^{-}(r),$ (12)

which is recognized as the rate equation governing the fraction opinion $r$.

As mentioned previously, we can approximate the topology using a mean-field
approximation on the complete graph where all agents are neighbors.
Considering there are $N$ total opinions or individuals in the system, the
opinion “up” increases and decreases by $1/N$ during each time step of the
dynamic process. We can mathematically express the probabilities of spin-up
decreasing, increasing, and remaining constant as follows:

$\displaystyle\rho^{+}(r)$ $\displaystyle=\mathrm{prob.}\left(r\to
r+1/N\right)$ (13) $\displaystyle\rho^{-}(r)$
$\displaystyle=\mathrm{prob.}\left(r\to r-1/N\right)$ (14)

The explicit form of Eqs. (13) - (14) depends on the model. This paper
considers a simple case in which three agents are chosen randomly in the
population and interact based on the abovementioned algorithm. Because in the
complete graph, the analytical result is suitable for a large population
$N>>1$ when we compare to the numerical simulation, we will concern the model
for the large population. Thus, the explicit form of Eqs. (13) - (14) for the
model with independence can be written as:

$\displaystyle\rho^{+}(r)$
$\displaystyle=3\left(1-r\right)\left[\dfrac{p}{2}+r^{2}\left(1-p\right)\right],$
(15) $\displaystyle\rho^{-}(r)$
$\displaystyle=3r\left[\dfrac{p}{2}+\left(1-r\right)^{2}\left(1-p\right)\right],$
(16)

And for the anticonformity model, Eqs. (13) - (14) can be written as:

$\displaystyle\rho^{+}(r)$
$\displaystyle=3\left(1-r\right)\left[p\left(1-r\right)^{2}+r^{2}\left(1-p\right)\right],$
(17) $\displaystyle\rho^{-}(r)$
$\displaystyle=3r\left[pr^{2}+\left(1-r\right)^{2}\left(1-p\right)\right].$
(18)

Eqs. (15) - (18) represent critical formulations for analyzing the system’s
state on the complete graph, particularly in evaluating an order-disorder
phase transition occurrence.

We can solve Eq. (12) to find the explicit expression of the fraction opinion
$r$ at time $t$ for both the model with independence and the model with
anticonformity. By inserting Eqs. (15) - (18) into Eq. (12) integrating it, we
obtain the following expressions for $r$ at time $t$ for the model with
independence as:

$\displaystyle
r(t,p,r_{0})=\dfrac{1}{2}\left[1\pm\left(\dfrac{1-3p}{1-p+2e^{-3\left(1-3p\right)\left(t+A\right)}}\right)^{1/2}\right],$
(19)

where $A$ is a parameter that satisfies the initial condition $r(t)$ at $t=0$.
Specifically, $A$ is given by the equation
$A=\ln[(1-2r_{0})^{2}/(2(1-p)(r_{0}^{2}+r_{0})+p]/[3(1-3p)]$. In the same way
for the model with anticonformity, we obtain

$r(t,p,r_{0})=\dfrac{1}{2}\left[1\pm\left(\dfrac{1-4p}{1-4\,e^{-3\left(1-4p\right)\left(t+A\right)}}\right)^{1/2}\right],$
(20)

where $A=\ln\left[(1-2r_{0})^{2}/(r_{0}^{2}-r_{0}+p)\right]/(1-4p)$. Eqs. and
provide an exact expression for the fraction opinion $r$ at time $t$ for both
models, with $p$ representing the probability of agents adopting either
independence or anticonformity, and $r_{0}$ denoting the initial fraction
opinion. For instance, when $p=0$ (indicating no independent or anticonformist
agents), both Eq. (19) and Eq. (20) reduce to the same form, namely,
$r(t)=1/2[1\pm(1-2r_{0})\exp(3t/2)/((r_{0}-1/2)^{2}\exp(3t)+r_{0}(1-r_{0}))^{1/2}]$.
This result suggests that the fraction opinion $r$ evolves towards complete
consensus states $r=1$ (all agents have an ‘up’ opinion) for $r_{0}<1/2$ and
$r=0$ (all agents have a ‘down’ opinion) for $r_{0}>1/2$. At $p=1/3$ for the
model with independence and $p=1/4$ for the model with anticonformity,
$r=1/2$, indicating a completely disordered state with an equal number of ‘up’
and ‘down’ opinions. Therefore, $p=1/3$ and $p=1/4$ are considered the
critical points for the model with independence and anticonformity,
respectively.

Fig. 1 demonstrates a comparison between Eq. (19) (red color) and Eq. (20)
against numerical simulations for a large population $N=10^{4}$ and various
values of $p$. The analytical treatment aligns closely with the numerical
results. Notably, at $p=0$, all initial fractions $r_{0}$ evolve towards
complete consensus with $r=1$ for $r_{0}<1/2$ and $r=0$ for $r_{0}>1/2$. This
behavior is expected, as when there are no nonconformist agents, only the
initial opinion’s size influences the system’s final state, with a larger
initial opinion prevailing. In general, fluctuations due to the system size
also impact the system’s final state for all networks or graphs. However, on a
complete graph with sufficiently large populations, the system’s evolution is
more stable toward the final state [39]. Additionally, at $0<p<p_{c}$, the
fraction opinion $r$ evolves toward two stable values $r_{st}$, while at
$p=p_{c}$, all initial fractions $r_{0}$ evolve to $r=1/2$, representing a
completely disordered state.

Figure 1: The comparison between analytical treatment and numerical simulation
for the model with independence (red) and anticonformity (blue) for various
values of probability $p$. As seen, at $p=0$, all data $r_{0}$ evolve to a
complete consensus with $r=1$ and $r=0$. At $0<p<p_{c}$, all data evolve to
two stable $r_{st}$, and at $p=p_{c}$ all data evolve to $r=1/2$ (disordered
state). The population size is $N=10^{4}$, and each data point averages over
$300$ independent realizations.

###  3.2 Phase diagram and critical exponents

The order-disorder phase transition of the model can be analyzed by
determining the stationary condition of Eq. (12), where $dr/dt=0$. For the
model with independence on the complete graph, the stationary results are as
follows:

$\displaystyle
r_{1}=1/2,\,r_{2,3}=\dfrac{1}{2}\left[1\pm\left(\dfrac{1-3p}{1-p}\right)^{1/2}\right]\Rightarrow
m_{2,3}=\pm\left(\dfrac{1-3p}{1-p}\right)^{1/2},$ (21)

where $r_{1}$ corresponds to $r=1/2$, while $r_{2,3}$ represents two
additional stationary values. The critical point occurs at $p_{c}=1/3$, where
$m_{2,3}=0$. Additionally, for the model with anticonformity, the stationary
condition for the fraction opinion $r$ yields:

$\displaystyle
r_{1}=1/2,\,r_{2,3}=\dfrac{1}{2}\left[1\pm\left(1-4p\right)^{1/2}\right]\Rightarrow
m_{2,3}=\pm\left(1-4p\right)^{1/2}.$ (22)

In this case, $r_{1}$ corresponds to $r=1/2$, and $r_{2,3}$ represents two
additional stationary values. The critical point is $p_{c}=1/4$. Both sets of
equations, Eqs. (21) and (22), can be expressed as power laws in terms of $p$,
where $m\sim(p-p_{c})^{\beta}$. In these equations, $\beta$ takes the value
$1/2$, typical of the critical exponent corresponding to the magnetization of
the mean-field Ising model [40].

As mentioned earlier, the topology of the complete graph can be approximated
using a mean-field formulation. Monte Carlo simulations were conducted with a
large population size $N=10^{6}$ to confirm the analytical results, as shown
in Fig. 2 (a). The analytical results closely match the Monte Carlo simulation
results, demonstrating that both models with independence and anticonformity
undergo a continuous phase transition. Furthermore, Fig. 2 (b) and (c)
illustrate the effective potential in Eqs. (23) and (24), respectively. These
plots reveal the following behaviors, namely; for $p<p_{c}$, the potential is
bistable, for $p>p_{c}$, the potential is monostable and at $p=p_{c}$, the
transition occurs between bistable and monostable, indicating the critical
point of the model. These results provide a comprehensive understanding of the
phase diagram and critical behavior of the model on the complete graph.

Figure 2: Phase diagram of the model on the complete graph with both
independence and anticonformity. Analytical results in Eqs. (21) and (22) are
compared with Monte Carlo simulation data. The models exhibit continuous phase
transitions with critical points at $p_{c}=1/3$ and $p_{c}=1/4$, respectively.
Panels (b) and (c) display the effective potential in Eqs. (23) and (24),
respectively, illustrating bistable and monostable behaviors for different $p$
values.

Another approach to analyzing the order-disorder phase transition of the model
is to consider the effective potential, which can be obtained through
integration. Traditionally, the effective potential is derived from the
effective force $V(r)_{\text{eff}}=-\int f(r)_{\text{eff}}\,\mathrm{d}r$,
where $f(r)_{\text{eff}}=\rho^{+}(r)-\rho^{-}(r)$ represents the force driving
the opinion change during the dynamics process. For the model with
independence on the complete graph, the effective potential is given by:

$\displaystyle
V(m,p)_{\text{indep.}}=3\left(1-p\right)^{-1}\left(1-3p-\left(1-p\right)m^{2}\right)^{2}/32.$
(23)

Moreover, for the model with anticonformity, the effective potential is
expressed as:

$\displaystyle V(m,p)_{\text{antic.}}=-3\left(1-4p-m^{2}\right)^{2}/32.$ (24)

Plot of Eqs. (23) and (24) are exhibited in panels (b) and (c) of Fig. 2. One
can see for both potentials, there are bistable states for $p<p_{c}$, the
transition bistable-monostable at $p=p_{c}$, and the system is in a monostable
for $p>p_{c}$, indicating the model undergoes a continuous phase transition at
$p_{c}$.

The critical point of the model can also be analyzed using Landau’s theory.
According to Landau’s theory, the potential can be expanded by the
magnetization $m$ as $V=\sum_{i}V_{i}m^{i}$, where $V_{i}$ generally depends
on thermodynamic parameters [41]. In this model, $V_{i}$ can depend on the
noise parameters of the model, such as probability independence and
anticonformity $p$. The Landau potential $V$ is symmetric under inversion of
the order parameter $m\to-m$. Therefore, only even terms of the potential are
considered. Thus, the simplified Landau potential takes the form:

$V=V_{0}+V_{2}m^{2}+V_{4}m^{4}+V_{6}m^{6}\cdots$ (25)

It is sufficient to know the terms $V_{2}$ and $V_{4}$ to analyze the model’s
phase transition using the potential V. The critical point can be determined
by setting $V_{2}=0$, while the nature of the phase transition is determined
by $V_{4}(p_{c})$, where $V_{4}(p_{c})\geq 0$ indicates a continuous phase
transition, and $V_{4}(p_{c})<0$ indicates a discontinuous phase transition.
Comparing Eq. (25) with Eqs. (23) and (24), we can determine $V_{2}$ and
$V_{4}$ for both the model with independence and the model with
anticonformity. For the model with independence, we obtain
$V_{2}(p)=3\left(1-3,p\right)/8$ and $V_{4}(p)=9\left(1-p\right)/4$. For the
model with anticonformity, we obtain $V_{2}(p)=-3\left(1-4p\right)/8$ and
$V_{4}(p)=9/4$. As a result, the critical points $p_{c}$ are consistent with
the values obtained from the equilibrium analysis: $p_{c}=1/3$ for the model
with independence and $p_{c}=1/4$ for the model with anticonformity.
Furthermore, $V_{4}(p_{c})\geq 0$ for both models confirms that they undergo a
continuous phase transition. This analysis provides an additional perspective
on the phase transition behavior of the model, showing agreement with the
previously obtained critical points and the nature of the transitions.

To numerically estimate the model’s critical point and critical exponents,
finite-size scaling relations in Eqs. (2)-(7) are employed. The population
size $N$ is varied in the range of $2000~{}\text{to}~{}10000$ to compute the
magnetization $m$, susceptibility $\chi$, and Binder cumulant $U$ as shown in
Fig. 3. We take for each data point is averaged over $10^{5}$ independent
realizations to obtain good results. In the inset graphs of Fig. 3, normal
plots are presented, while the main graphs show the scaling plots of the
model. The critical point of the model is determined using the Binder method,
which involves observing the crossing of lines between the Binder cumulant $U$
and the probability of anticonformity $p$. In this case, the critical point is
estimated to be $p_{c}\approx 0.25$ (inset panel a), which agrees with the
analytical result in Eq. (22).

The main graphs of Fig. 3 display the scaling plots of the model. The best
critical exponents that result in the best collapse of all data points are
$\beta\approx 0.5,\nu\approx 2.0,$ and $\gamma\approx 1.0.$. Here, $\nu$ is
the critical exponent corresponding to the critical dimension $d_{c}=4$, and
the effective exponent $\nu$ is $1/2$. Consequently, $\nu=d_{c}\nu^{\prime}=2$
is obtained numerically. These critical exponents satisfy the identity
relation $\nu d=2\beta+\gamma$’, and they indicate that the universality class
of the model belongs to the mean-field Ising universality class [40]. It is
important to note that the same critical exponents are obtained for the model
with independence, suggesting that both models with independence and
anticonformity are identical. These models are also identical to well-known
models in the field, such as the Sznajd and kinetics exchange models. This
finite-size scaling analysis provides robust numerical evidence supporting the
model’s critical point and exponents, corroborating the analytical findings
and classifying the model within the mean-field Ising universality class.

Figure 3: Monte Carlo simulation result of the model for the order parameter
$m$, susceptibility $\chi$, and Binder cumulant $U$ (inset graphs) for several
population sizes $N$. The critical point is obtained from the crossing lines
Binder cumulant $U$ versus probability $p$ that occurs at $p=p_{c}\approx
0.25$ (inset graph of the panel (a)). This critical point confirms the
analytical result in Eq. (22). The best critical exponents that make all data
collapse near the critical point $p_{c}$ are $\beta\approx 0.5,\nu\approx
2.0,$ and $\gamma\approx 1.0$ (see the main graph) that indicate the model
belongs to the mean-field Ising model [40].

###  3.3 The model on the 2-D lattice

We examined several population sizes, denoted as $N=L^{2}$, where $L$ assumes
values of $32,45,64,100,150$, and $200$, to investigate the model’s critical
point and critical exponents thoroughly. The numerical results, specifically
pertaining to the order parameter $m$, susceptibility $\chi$, and Binder
cumulant $U$, are presented in Fig. 4. The critical point, signifying the
point at which the model undergoes a continuous phase transition, is
identified as $p_{c}\approx 0.106$ (as seen in the inset panel of Fig. 4 (a)).
We determined the optimal critical exponents that result in the best data
collapse by utilizing the finite-size scaling relations detailed in Eqs.
(2)-(5). These critical exponents are approximately $\beta\approx
2.0,\gamma\approx 1.75$, and $\nu\approx 0.125$. Notably, these values
indicate that the model is akin to the Sznajd model [30, 42] and shares
characteristics with the universality class of the two-dimensional Ising model
[40].

Figure 4: (Main panel) Scaling plot of the Binder cumulant $U$ (panel (a)),
the order parameter $m$ (panel (b)), and the susceptibility (panel (c). These
results are derived from extensive numerical simulations. Notably, for a
critical point at approximately $p_{c}\approx 0.106$, we observe that all
data, with population sizes $N=L^{2}$, exhibit a remarkable collapse. The
associated critical exponents are estimated to be $\nu\approx 2.0$,
$\gamma\approx 1.75$, and $\beta\approx 0.125$. The critical point $p_{c}$ is
determined by identifying the intersection of the curves for the Binder
cumulant $U$ as a function of the independence probability $p$ (inset panel
(a)). These results align with the critical exponents of the two-dimensional
Ising model universality class. It is important to note that all inset panels
present data in a normal plot format, and each data point results from
averaging over $3\times 10^{6}$ independent realizations.

Similarly, we extended our analysis to the model with anticonformity, and our
findings reveal that this model also experiences a continuous phase
transition. The critical point for the model with anticonformity is
approximately $p_{c}\approx 0.062$, as shown in Fig. 5. Remarkably, our
investigations yielded identical critical exponents for this model,
specifically $\beta\approx 0.125,\nu\approx 2.0$ and $\gamma\approx 1.75$.
These shared critical exponents indicate that both the model with independence
and the model with anticonformity exhibit analogous behavior, suggesting that
they belong to the same universality class. Our results align these models
with the two-dimensional Ising universality class. Furthermore, the critical
exponents for both models adhere to the identity relation $\nu
d=2\beta+\gamma$.

Figure 5: (Main panel) Scaling plots for the model with anticonformity: the
Binder cumulant $U$ [panel (a)], the order parameter $m$ [panel (b)], and the
susceptibility $\chi$ [panel (c)]. These results have been obtained through
extensive numerical simulations. Notably, for a critical point located at
approximately $p_{c}\approx 0.062$, we observe a remarkable collapse of all
data sets with population sizes $N=L^{2}$. The associated critical exponents
are estimated to be $\nu\approx 2.0$, $\gamma\approx 1.75$, and $\beta\approx
0.125$. The critical point $p_{c}$ is determined through the intersection of
curves representing the Binder cumulant $U$ as a function of the independence
probability $p$ (inset panel (a)). These results are consistent with the
critical exponents characterizing the universality class of the two-
dimensional Ising model. It is important to note that all inset panels present
data in a normal plot format, and each data point results from averaging over
$3\times 10^{6}$ independent realizations.

###  3.4 The model on the 3-D lattice

We conducted investigations using various population sizes $N=L^{3}$, with
linear dimensions $L=15,20,25,30,35$. Each data point represents an average of
over $10^{6}$ independent realizations. The numerical results for the model
featuring independence are presented in Fig. 6. In panel (a), we observe that
at $p=0.0$, the system exists in an ordered state characterized by $m=1.0$. In
contrast, the magnetization decreases to zero at a small $p=0.01$, signifying
a disordered state. The inset graph visually illustrates the magnetization $m$
at equilibrium for $p=0.0$ and $p=0.01$. However, relying solely on the
magnetization data makes it challenging to determine whether the model
undergoes an order-disorder phase transition. Nonetheless, we gain more
insight into this by examining the Binder cumulant $U$, as depicted in panel
(b). Notably, there are no intersections between the curves of Binder cumulant
$U$ and the probability of independence $p$, which suggests the absence of an
order-disorder phase transition in this model. See the inset graph for further
clarity.

Figure 6: The numerical results for the order parameter $m$ [panel (a)] and
Binder cumulant $U$ [panel (b)] are presented for the model with independence
on the three-dimensional square lattice. Notably, the order parameter $m$
diminishes to zero at a low independence probability of $p=0.01$, indicating
the absence of an order-disorder phase transition in this model. Furthermore,
the data depicting Binder cumulant $U$ as a function of probability $p$
reveals a lack of intersections between the curves. For a clearer
illustration, please refer to the inset graph.

Our analysis of the model defined on the 3-D square lattice reveals that it
undergoes a second-order phase transition with a critical point at
$p_{c}\approx 0.268$, as illustrated in Fig. 7. By employing finite-size
scaling relations, detailed in Eqs. (2)-(5), we have determined the best-
fitting critical exponents, yielding values of $\beta\approx 0.25$,
$\nu\approx 2.0$, and $\gamma\approx 1.4$. The results suggest that this model
is not in the same universality class as the three-dimensional Ising model
[40]. These critical exponents are universal, as they remain consistent across
various different data sets for different system sizes $N$.

Figure 7: (Main graph) Scaling plots for the Binder cumulant $U$, order
parameter $m$, and susceptibility $\chi$ for the model with anticonformity on
the 3-D lattice. Based on the results, the model undergoes a continuous phase
transition with a critical point at $p_{c}\approx 0.268$ (see inset figure
(a)). Additionally, we determine the best critical exponents as $\beta\approx
0.25,\nu\approx 2.0$, and $\gamma\approx 1.40$, leading to the collapse of all
data points around the critical point, as indicated. Each data point averages
$10^{6}$ independent realizations.

###  3.5 The model on the heterogeneous networks

Compared to the aforementioned homogeneous networks, the W-S, A-B, and E-R
networks are more representative of real social networks [34, 43]. These three
networks have been extensively studied in various network research and applied
to a wide range of social phenomena, including applications in the field of
medicine [44]. In this section, we considered these three heterogeneous
networks, visually represented in Fig. 8.

Figure 8: Three heterogeneous networks, namely the Watts-Strogatz (W-S)
network, Barabasi-Albert (A-B) network, and Erdos-Renyi (E-R) network, were
generated using networkx [45]. The population size or number of nodes, denoted
as $N=200$, and all networks have a minimum degree of connectivity set to $2$.

Within these networks, we selected an agent possessing at least two randomly
chosen nearest neighbors, as illustrated in Fig. 9.

Figure 9: Illustration of the selection of three agents at random (initial and
updated state): agent $S_{0}$ is initially chosen, followed by the random
selection of its two neighboring agents, $S_{1}$ and $S_{2}$ (solid edges),
who interact based on the majority rule. The red and green colors represent
agents with opinions or states of “up” and “down,” respectively.

The three agents interacted with each other according to the model’s
algorithm. We assigned varying node degrees in all networks, with the minimum
node degree being $2$. In other words, each agent has at least two nearest
neighbors. The population size was $N=3000$, and each data point was an
average of $10^{5}$ independent realizations. In this part, we exclusively
focus on analyzing whether or not the model undergoes a continuous phase
transition. Therefore, we do not seek critical points and critical exponents
as in the previously discussed homogeneous networks. Our numerical results for
the order parameter $m$ are presented in Fig. 10. It can be observed that when
$p=0$, the system is in a state of complete order. This situation arises
because all nodes have at least two nearest neighbors, allowing them to
interact with each other following the majority rule. It is also shown that
the model undergoes a continuous phase transition in all three networks, each
with different critical points (not specified in this paper). Other types of
phase transitions, such as discontinuous phase transitions, can also occur for
groups with a larger number of the group of agents, for example, $7$, or under
different scenarios.

Figure 10: Phase diagrams illustrating the occurrence of continuous phase
transitions for the models with independence and nonconformity on Barabasi-
Albert (A-B), Erdos-Renyi (E-R), and Watts-Strogatz (W-S) networks. The
population size is $N=5000$, and each data point represents the average of
$10^{4}$ independent realizations.

To further observe the continuous phase transition in the model defined on
these heterogeneous networks, we analyzed the magnetization fluctuations m
versus Monte Carlo steps. The results are shown in Fig. 11. It can be seen
that a bistable state of the magnetization $m$ emerges for all models on all
networks, indicating that the model undergoes a continuous phase transition.

Figure 11: The Monte Carlo step for the magnetization $m$ of the model on
heterogeneous networks occurs at low values of $p$. It can be observed that
the magnetization exhibits fluctuations between two stable states, indicating
that the model undergoes a continuous phase transition.

##  4 Summary and outlook

This paper delves into the disruptive social phenomena’s impact, such as
independence and nonconformity, on the occurrence of phase transitions from
order to disorder based on majority rule. The model’s scope encompasses
homogeneous networks such as complete graphs, two-dimensional square lattices,
and three-dimensional square lattices, as well as heterogeneous networks like
Barabasi-Albert networks, Watts-Strogatz networks, and Erdos-Renyi networks. A
probability parameter measures the likelihood of agents adopting independent
or nonconformist behavior, denoted as $p$, while agents conform to the
majority opinion with a probability of $(1-p)$.

Our results, analytically (for the complete graph) and through numerical
simulations, indicate that the model in homogeneous networks undergoes a
continuous phase transition with different critical points, as summarized in
Table 1. However, we found no phase transition for the 3-D lattice model with
independence. The critical exponents for the complete graph and the two-
dimensional lattice model, obtained through finite-size scaling analysis,
suggest that the model belongs to the same class as mean-field and 2-D Ising
models. These critical exponents satisfy the scaling identity relation $\nu
d=2\beta+\gamma$. Furthermore, for the model with nonconformity on the 3-D
lattice, the obtained critical exponents are $\beta\approx 0.25,\nu\approx
1.40,$ and $\nu\approx 2.0$, indicating that the model does not belong to the
same class as the 3-D Ising model.

Table 1: Critical exponents of the majority-rule model on regular networks. Networks |  S. Noises | Critical Point | $\beta$ | $\gamma$ | $\nu$
---|---|---|---|---|---
Com.-graph |  Independence | $1/3$ | 0.5 | 1.0 | 2.0
Anticonformity | $1/4$ | 0.5 | 1.0 | 2.0
2-D lattice |  Independence | 0.106 | 0.125 | 1.75 | 2.0
Anticonformity | 0.062 | 0.125 | 1.75 | 2.0
3-D lattice |  Independence | - | - | - | -
Anticonformity | 0.268 | 0.25 | 1.40 | 2.0

The model defined on heterogeneous networks such as B-A, W-S, and E-R networks
undergoes a continuous phase transition with distinct critical points, as
demonstrated by the magnetization data versus probability $p$. This conclusion
is further supported by the fluctuations in the two stable magnetic states,
denoted as $m$, indicating that the model undergoes a continuous phase
transition. We did not estimate the critical points of the model any further;
however, based on numerical data for magnetization $m$ versus probability $p$,
it is evident that the critical point of the model with nonconformity is
smaller than that of the model with independence. This difference also holds
consistently for models defined on complete graphs and 2-D lattices. From
these data, we can assert that models with nonconformity have a greater
tendency to undergo an order-disorder phase transition.

From a social systems perspective, we can assert that complete consensus is
achieved when no independent or nonconformist agents are present. Below the
critical point is a coexistence of majority and minority opinions. However, at
$p\geq p_{c}$, the system reaches an equilibrium state akin to a deadlock.
After comparing the critical points of the independence and nonconformity
models, it becomes evident that the model with nonconformity exhibits a lower
critical point than its independence counterpart. These results imply that
systems containing nonconformist agents tend to reach a stalemate state more
than systems containing independent agents. These findings are consistent with
what we previously reported in our prior study [31]. Of course, differences in
such phenomena are highly dependent on the applied model or scenario.

## CRediT authorship contribution statement

D. A. Mulya and R. Muslim: Conceptualization, Methodology, Writing, Software,
Formal analysis, Validation, Writing, Visualization, Review & editing. R.
Muslim: Main Contributor & Supervision. All authors read and reviewed the
paper.

## Declaration of Interests

The contributors declare that they have no apparent competing business or
personal connections that might have appeared to have influenced the reported
work.

## Acknowledgments

The authors thank the BRIN Research Center for Quantum Physics for providing
the mini HPC (Quantum Simulation Computer) for conducting numerical
simulations. D. A. Mulya expresses gratitude for the support received from the
Research Assistant program of BRIN talent management, as evidenced by Decree
Number 60/II/HK/2023.

## References

  * [1] S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, Springer-Verlag, New York, 2012.
  * [2] C. Castellano, S. Fortunato, V. Loreto, Statistical physics of social dynamics, Rev. Mod. Phys. 81 (2009) 591.
  * [3] S. Galam, Sociophysics: a personal testimony, Physica A: Statistical Mechanics and its Applications 336 (1-2) (2004) 49–55.
  * [4] P. Sen, B. K. Chakrabarti, Sociophysics: an introduction, Oxford, Oxford University Press, 2014.
  * [5] S. Galam, Majority rule, hierarchical structures, and democratic totalitarianism: A statistical approach, Journal of Mathematical Psychology 30 (4) (1986) 426–434.
  * [6] M. Mobilia, S. Redner, Majority versus minority dynamics: Phase transition in an interacting two-state spin system, Phys. Rev. E 68 (2003) 046106.
  * [7] P. L. Krapivsky, S. Redner, Dynamics of majority rule in two-state interacting spin systems, Phys. Rev. Lett. 90 (2003) 238701.
  * [8] S. Biswas, P. Sen, Model of binary opinion dynamics: Coarsening and effect of disorder, Physical Review E 80 (2) (2009) 027101.
  * [9] K. Sznajd-Weron, J. Sznajd, Opinion evolution in closed community, Int. J. Mod. Phys. C 11 (2000) 1157–1165.
  * [10] K. Sznajd-Weron, J. Sznajd, T. Weron, A review on the sznajd model—20 years after, Physica A: Statistical Mechanics and its Applications 565 (2021) 125537.
  * [11] G. Deffuant, D. Neau, F. Amblard, G. Weisbuch, Mixing beliefs among interacting agents, Advances in Complex Systems 3 (3) (2000) 87–98.
  * [12] G. Weisbuch, G. Deffuant, F. Amblard, J.-P. Nadal, Meet, discuss, and segregate!, Complexity 7 (3) (2002) 55–63.
  * [13] S. E. Asch, Studies of independence and conformity: I. a minority of one against a unanimous majority., Psychological monographs: General and applied 70 (9) (1956) 1.
  * [14] R. H. Willis, Two dimensions of conformity-nonconformity, Sociometry (1963) 499–513.
  * [15] R. H. Willis, Conformity, independence, and anticonformity, Hum. Relat. 18 (1965) 373–388.
  * [16] G. MacDonald, P. R. Nail, D. A. Levy, Expanding the scope of the social response context model, Basic Appl. Soc. Psych. 26 (2004) 77–92.
  * [17] P. R. Nail, G. MacDonald, On the development of the social response context model, in: The science of social influence: Advances and future progress, New York, Psychology Press, 2007, pp. 193–221.
  * [18] K. Sznajd-Weron, M. Tabiszewski, A. M. Timpanaro, Phase transition in the sznajd model with independence, Europhys. Lett. 96 (2011) 48002.
  * [19] P. Nyczka, K. Sznajd-Weron, Anticonformity or independence?—insights from statistical physics, J. Stat. Phys. 151 (2013) 174–202.
  * [20] M. A. Javarone, Social influences in opinion dynamics: the role of conformity, Physica A: Statistical Mechanics and its Applications 414 (2014) 19–30.
  * [21] N. Crokidakis, P. M. C. de Oliveira, Inflexibility and independence: Phase transitions in the majority-rule model, Phys. Rev. E 92 (2015) 062122.
  * [22] A. Chmiel, K. Sznajd-Weron, Phase transitions in the q-voter model with noise on a duplex clique, Physical Review E 92 (5) (2015) 052812.
  * [23] A. Abramiuk, J. Pawłowski, K. Sznajd-Weron, Is independence necessary for a discontinuous phase transition within the q-voter model?, Entropy 21 (5) (2019) 521.
  * [24] R. Muslim, R. Anugraha, S. Sholihun, M. F. Rosyid, Phase transition of the sznajd model with anticonformity for two different agent configurations, Int. J. Mod. Phys. C 31 (2020) 2050052.
  * [25] B. Nowak, B. Stoń, K. Sznajd-Weron, Discontinuous phase transitions in the multi-state noisy q-voter model: quenched vs. annealed disorder, Scientific Reports 11 (1) (2021) 1–13.
  * [26] J. Civitarese, External fields, independence, and disorder in q-voter models, Physical Review E 103 (1) (2021) 012303.
  * [27] R. Muslim, R. Anugraha, S. Sholihun, M. F. Rosyid, Phase transition and universality of the three-one spin interaction based on the majority-rule model, International Journal of Modern Physics C 32 (09) (2021) 2150115.
  * [28] R. Muslim, M. J. Kholili, A. R. Nugraha, Opinion dynamics involving contrarian and independence behaviors based on the sznajd model with two-two and three-one agent interactions, Physica D: Nonlinear Phenomena 439 (2022) 133379.
  * [29] R. Muslim, H. Lugo H, Effect of social behaviors in the opinion dynamics $q$-voter model, arXiv preprint arXiv:2307.14548 (2023).
  * [30] A. Azhari, R. Muslim, D. A. Mulya, H. Indrayani, C. A. Wicaksana, A. Rizki, Independence role in the generalized sznajd model, arXiv preprint arXiv:2309.13309 (2023).
  * [31] R. Muslim, S. A. Wella, A. R. Nugraha, Phase transition in the majority rule model with the nonconformist agents, Physica A: Statistical Mechanics and its Applications 608 (2022) 128307.
  * [32] A. L. Oestereich, M. A. Pires, S. M. Duarte Queirós, N. Crokidakis, Phase transition in the galam’s majority-rule model with information-mediated independence, Physics 5 (3) (2023) 911–922.
  * [33] D. J. Watts, S. H. Strogatz, Collective dynamics of ‘small-world’networks, nature 393 (6684) (1998) 440–442.
  * [34] R. Albert, A.-L. Barabási, Statistical mechanics of complex networks, Reviews of modern physics 74 (1) (2002) 47.
  * [35] P. ERDdS, A. R&wi, On random graphs i, Publ. math. debrecen 6 (290-297) (1959) 18.
  * [36] M. E. Fisher, The theory of equilibrium critical phenomena, Reports on progress in physics 30 (2) (1967) 615.
  * [37] R. B. Cialdini, N. J. Goldstein, Social influence: Compliance and conformity, Annu. Rev. Psychol. 55 (2004) 591–621.
  * [38] K. Binder, Finite size scaling analysis of ising model block distribution functions, Z. Phys. B: Condens. Matter 43 (1981) 119–140.
  * [39] R. Muslim, R. A. NQZ, M. A. Khalif, Mass media and its impact on opinion dynamics of the nonlinear $q$-voter model, arXiv preprint arXiv:2305.06053 (2023).
  * [40] H. E. Stanley, Phase transitions and critical phenomena, Vol. 7, Clarendon Press, Oxford, 1971.
  * [41] L. D. Landau, On the theory of phase transitions, Zh. Eksp. Teor. Fiz. 7 (1937) 19–32.
  * [42] M. Calvelli, N. Crokidakis, T. J. Penna, Phase transitions and universality in the sznajd model with anticonformity, Physica A: Statistical Mechanics and its Applications 513 (2019) 518–523.
  * [43] M. Newman, Networks, Oxford university press, 2018.
  * [44] D. L. Barabási, G. Bianconi, E. Bullmore, M. Burgess, S. Chung, T. Eliassi-Rad, D. George, I. A. Kovács, H. Makse, T. E. Nichols, et al., Neuroscience needs network science, Journal of Neuroscience 43 (34) (2023) 5989–5995.
  * [45] A. Hagberg, P. Swart, D. S Chult, Exploring network structure, dynamics, and function using networkx, Tech. rep., Los Alamos National Lab.(LANL), Los Alamos, NM (United States) (2008).