¹¹institutetext: University of Groningen ²²institutetext: University of Amsterdam
²²email: {m.d.los,z.l.christoff,d.grossi}@rug.nl

On the Graph Theory of Majority Illusions

Maaike Venema-Los^(✉) 11 Zoé Christoff 11 Davide Grossi 1122

Abstract

The popularity of an opinion in one’s direct circles is not necessarily a good indicator of its popularity in one’s entire community. For instance, when confronted with a majority of opposing opinions in one’s circles, one might get the impression that one belongs to a minority. From this perspective, network structure makes local information about global properties of the group potentially inaccurate. However, the way a social network is wired also determines what kind of information distortion can actually occur. In this paper, we discuss which classes of networks allow for a majority of agents to have the wrong impression about what the majority opinion is, that is, to be in a ‘majority illusion’.

Keywords:

majority illusion social networks graph colorings.

1 Introduction

When making decisions, people often use information from the decisions of others in their circles and are influenced by those. For instance, if a lot of people around you buy a specific brand, vote for a specific political party, or have a specific opinion, you are more likely to buy, vote, or think the same (see e.g. [14, 17]).

However, one’s view of ‘the world around’ might be distorted by the social network one is in, and one might as a result misrepresent one’s situation with respect to the overall population. A well-known example of this is the so-called ‘friendship paradox’ [9]: agents in a network are likely to get the impression that their popularity is lower than average because their friends have more friends than they do. Similarly, on the basis of what they can see of others around them, agents might get the wrong impression with respect to how popular their opinions are in the entire population. Indeed, the proportions of opinions an agent observes in its neighborhood are not necessarily a representative sample of their overall distribution in the population.

From this local sampling, one could for instance get the impression that they disagree with the majority of the population on a particular opinion and get influenced by this impression when taking their decisions. As a consequence, one could in principle influence what people will decide by changing the network structure to tweak the distribution of opinions/behavior agents see locally. Figure 1 (a), (b), and (c) illustrate this: rewiring a few edges is sufficient to make all nodes observe a different majority in their neighborhood.

A paradigmatic example is polling bandwagon effects [19] in political decision making: if, for any reason, citizens prefer not to vote on a losing party, a party could increase the chances of actually winning by making a lot of voters think it is winning. The idea of manipulating the network towards this end, so-called ‘information gerrymandering’, is introduced in [4, 20] and shown to potentially lead to undemocratic decisions. Similar phenomena have been observed in a variety of social networks [15].

In this paper, we examine which networks allow for which types of distortion between local and global opinion representations. We focus on a specific type of network distortion, ‘majority illusion’ as introduced by [15], where an agent observes that more than half its neighbors are in a certain state, while in the total network, less than half of the agents are in this state. An example is shown in Figure 1(d): if all nodes were to believe that their neighborhood was representative of the entire network, the red nodes would believe that the majority of nodes in the network is blue, while in reality it is red. Whether it is possible for such illusions to exist, and if so, for how many agents, depends on the network structure. On the one hand, not all networks allow for most agents to be under this type of majority illusions. On the other hand, there exist network structures in which even all agents can be under a majority illusion. In this paper we focus on the possibility of a majority of agents being in a majority illusion. The network in Figure 1(d) is an example of such a ‘majority-majority illusion’.

Related work.

The concept of majority illusion was first introduced in [15], to show how network structures can distort individual observations. Computational simulations are used to study to which extent the majority illusion can occur in scale-free, Erdős-Rényi networks, and several real-world networks, and show that networks in which high-degree nodes tend to connect to low-degree nodes are most susceptible to this illusion.

In [20] a voter game is modeled with two competing parties to show that majority illusions can be used for the purposes of ‘information gerrymandering’, that is, influencing people’s votes by misrepresenting the opinion of the group to them. The authors predict this by a mathematical model and confirm their results with a social network experiment with human participants. They find that information gerrymandering can even take place when all agents have the same amount of influence (the same degree).

In [10], the computational complexity of majority illusions is studied. A $q$ -majority illusion is defined, where at least a $q$ fraction of agents are under majority illusion, and it is shown that the problem of deciding whether or not a given network can be colored into a $q$ -majority illusion is NP-complete for $q>\frac{1}{2}$ . Whether it also holds for $q\leq\frac{1}{2}$ is left as an open question. Since majority illusions can in some situations have detrimental effects and are generally regarded as undesirable, they also study the problem of eliminating an illusion from a network by adding or deleting edges. The problem to identify whether it is possible to change the network in such a way that the number of agents with a $q$ -majority illusion is below a given bound is also shown to be NP-complete for $q>\frac{1}{2}$ .

Contributions.

We study the possibility of majority illusion in its weak and strict versions, in different classes of graphs. Section 2 introduces our framework, definitions and terminology. In Section 3, we prove that a weak version of the majority illusion can occur on all network structures. In Section 4, we provide some stronger results on specific classes of networks: graphs with odd degrees, properly $2$ -colorable graphs, and regular graphs. Table 2 gives a summary of which graphs allow for which type of majority illusions.

2 Preliminaries

Binary opinion networks and 2-colored graphs.

A social network (a simple graph) $G=\langle V,E\rangle$ consists of a finite set $V$ agents (nodes/vertices), and a set $E$ of (undirected irreflexive) edges between agents. If two agents are connected by an edge, we call them neighbors. We assume that no agent is a neighbor of itself. We write $N_{i}$ for the set of neighbors of $i$ and $d_{i}$ for its degree $|N_{i}|$ . Each agent holds a binary opinion on a single issue. Since binary single-issue opinion distributions can be seen as graph $2$ -colorings, we will borrow the terminology of vertex colorings, and use the terms ‘color’ and ‘opinion’ interchangeably. We write $c_{i}$ to refer to agent $i$ ’s color and $c$ for the $2$ -coloring of the graph ( $c:V\to\{red,blue\}$ ). A 2-colored graph is a triple $C=\langle V,E,c\rangle$ . Thoughout the paper, the term ‘colored graph’ refers to such 2-colored graph.

Majority illusion and opposition: intuitions.

In such opinion networks (or 2-colored graphs), we are concerned with three types of information: individual opinions, local majority opinions, and global majority opinions. Any two of these three types of opinions can be in agreement or not. We systematize and illustrate all possible relations between the above types of opinions in Table 1.

Different fields have been studying disagreement between the different types of information mentioned above. On the one hand, in social network science and social choice theory, an agent is under majority illusion when its neighborhood majority disagrees with the global majority [10, 15]. On the other hand, graph theory has concerned itself with the disagreement between a node’s color and the color of its neighbors: a proper coloring requires that no two adjacent nodes are of the same color, that is, that everybody disagrees with all of their neighbors. A generalisation of that concept is that of majority-coloring [1, 5, 13], where no agent agrees with most of its neighbors. We call the local disagreement faced by an agent in a majority coloring majority opposition. In such a situation, one might have the impression that they belong to a global minority. For instance, in Figure 1(d) all nodes might have this impression, while it is only true for the blue ones. When all agents are under this impression, then some of them must be mistaken, it must be some sort of illusion. Clearly, the two concepts of majority illusion and opposition are related. In this paper, we explore this relation to get results about majority illusions.

Table 1: Possible combinations of local and global majority winners, and presence or absence of majority opposition and majority illusion. We assume w.l.o.g. that the color of the relevant individual (highlighted in the exemplary illustrations) is red, otherwise just swap ‘red’ and ‘blue’ everywhere. ✗ indicates absence of the opposition/illusion,

\checkmark

indicates presence of the opposition/illusion, ‘weak’ indicates the presence of a weak-majority opposition or a weak-majority illusion.

local majority	majority opposition	global majority	majority illusion
red	✗	red	✗
red	✗	tie	weak
red	✗	blue	$\checkmark$ [15, 10]
tie	weak [1, 5, 13]	red	weak
tie	weak [1, 5, 13]	tie	✗
tie	weak [1, 5, 13]	blue	weak
blue	$\checkmark$	red	$\checkmark$ [15, 10]
blue	$\checkmark$	tie	weak
blue	$\checkmark$	blue	✗

Formal definitions.

We start by introducing some notions to be able to talk about which opinion is prevalent in a network, be it locally or globally. Given a set $S$ of agents such that $|S|=n$ and a coloring $c$ , a color $x$ is a majority winner of $S$ (we write $M_{S}=x$ ) if $|\{i\in S\mid c_{i}=x\}|>\frac{n}{2}$ . When neither color is a majority winner among $S$ (there is a tie), we will write $M_{S}=tie$ . We say that an agent is under majority illusion if both the agent’s neighbourhood and the entire network have a majority winner (no tie) but they are different. This definition is equivalent to that in [10].

Definition 1 (majority illusion)

Given a colored graph $C=\langle V,E,c\rangle$ , an agent $i\in V$ is under majority illusion if $M_{N_{i}}\not=\text{tie}$ and $M_{V}\not=\text{tie}$ and $M_{N_{i}}\not=M_{V}$ . A graph is in a majority-majority illusion if more than half of agents are under majority illusion.

We can generalise this strict definition to weaker cases. First, there exists a weaker type of disagreement between local and global majorities: the cases where exactly one of the two is a tie. Second, the majority of agents under an illusion can also be weak, when exactly half of the agents are under illusion. The corresponding generalisations of majority illusion includes both types of weakening:

Definition 2 (weak versions of majority illusion)

Given a colored graph $C=\langle V,E,c\rangle$ , agent $i\in V$ is under weak-majority illusion if $M_{N_{i}}\not=M_{V}$ . A graph is in a majority-weak-majority illusion if more than half of the agents are under weak-majority illusion. A graph is in a weak-majority-(weak-)majority illusion if at least half of the agents are under (weak-)majority illusion.

As indicated in Table 1, while it is the strict version of the majority illusion that is studied by [15]¹¹1They use the strict version throughout the paper, except for Figure 1 where the illusion can be weak., and by [10], it is the weak version of majority opposition that is studied by [1], [5], and [13]. As far as we know, the strict majority opposition and the weak majority illusions have not been studied before. Furthermore, note that, in the same network, different agents can be under a weak-majority illusion with respect to different opinions, since it is possible that exactly half of the nodes in the network are of one color and half of the nodes of the other color.

Before proceeding, we introduce some extra terminology. When an agent is under majority illusion and all its neighbors all have the same color, we say that that agent is in unanimity illusion. Similarly, when all agents are under a (weak)-majority illusion, we will call it a unanimity-(weak-)majority illusion. We say that an illusion is possible for a graph $G=\langle V,E\rangle$ if there exists a coloring $c$ such that $C=\langle V,E,c\rangle$ is in the respective illusion.

3 Illusions in Arbitrary Networks

Our overall goal is to discover which social networks allow for majority illusions to occur. Since this is equivalent to asking which graphs can be colored in some specific way, we build on existing results from vertex colorings to obtain results about majority illusions. Recall that a coloring is called proper when no two neighbors are assigned the same color. The weaker notion of majority coloring [1, 13] is immediately relevant to us. In a majority coloring, each vertex is in what we described as majority opposition: at least half of its neighbors are of a different color than its own. For coherence with the rest of the paper, we call this a weak majority coloring here:

Definition 3 (weak majority $2$ -coloring)

A weak majority $2$ -coloring of a graph $G=\langle V,E\rangle$ is a $2$ -coloring $c$ such that, for each $i\in V:M_{N_{i}}\neq c(i)$ .

A graph is called weak majority $2$ -colorable if there exists a weak majority $2$ -coloring of it. Given a colored graph, we call monochromatic the edges between nodes of the same color, and dichromatic the ones between nodes of different colors.

Remark 1

The main result involving majority colorings is credited to [16] in the literature [1, 5]: every graph is weak majority $2$ -colorable. The proof strategy for this result is commonly described as easy and relying on a simple ‘color swapping mechanism’ that can only reduce the total number of monochromatic edges in the network. However, [16] itself focuses on multigraphs and is of a much wider scope. Therefore, to make the paper self-contained, we provide both a proof of the general result in Appendix 6 and below a proof of the related lemma, crucial to our main result, Theorem 3.1.

Lemma 1

Let $G=\langle V,E\rangle$ be a graph, and let $c$ be a $2$ -coloring of $G$ that minimizes the number of monochromatic edges. Then, $c$ is a weak majority $2$ -coloring of $G$ .

Proof

Let $E_{M}$ be the set of monochromatic edges and $E_{D}=E\backslash E_{M}$ the set of dichromatic edges in graph $G$ colored by $c$ . Assume for contradiction that there is a node $i\in V$ that is an endpoint of strictly more monochromatic edges (we write $E_{M_{i}}$ for the set of such edges) than dichromatic edges ( $E_{D_{i}}$ ): $|E_{M_{i}}|>|E_{D_{i}}|$ . Consider now a second $2$ -coloring $c^{\prime}$ of $G$ that only differs from $c$ with respect to $i$ ’s color, i.e., $c^{\prime}$ assigns the same color as $c$ did to all nodes except for $i$ : $c_{i}\neq c^{\prime}_{i}$ . Let us write $E^{\prime}_{M}$ for the new set of monochromatic edges, and $E^{\prime}_{M_{i}}$ and $E^{\prime}_{D_{i}}$ for the new sets of monochromatic and dichromatic edges from $i$ . Given that $|E^{\prime}_{M_{i}}|=|E_{D_{i}}|$ and $|E^{\prime}_{D_{i}}|=|E_{M_{i}}|$ , we now have $|E^{\prime}_{D_{i}}|>|E^{\prime}_{M_{i}}|$ and $|E_{M_{i}}|>|E^{\prime}_{M_{i}}|$ . Given that no other edge of the graph is affected by this change, the total number of monochromatic edges is smaller with coloring $c^{\prime}$ than it was with $c$ : $|E_{M}|>|E^{\prime}_{M}|$ . But since we started by assuming that $c$ was such that $|E_{M}|$ was minimal, this is a contradiction. ∎

We now use the existence of such a majority coloring to prove the following general result:

Theorem 3.1

For any graph $G=\langle V,E\rangle$ , a majority-weak-majority illusion is possible.

Proof

Let $G=\langle V,E\rangle$ be a graph and let $c$ be a $2$ -coloring of $G$ that minimizes the total number of monochromatic edges. By Lemma 1, $c$ is a weak majority $2$ -coloring of $G$ . Two cases:

•

$M_{V}\neq tie$ . Assume w.l.o.g. that $M_{V}=red$ . Since $c$ is a weak majority coloring, for any red vertex $i$ , $M_{N_{i}}\in\{blue,tie\}$ , and therefore $M_{N_{i}}\neq M_{V}$ . Hence, a majority of the nodes (the red ones) is under (possibly weak) majority illusion: we have a majority-weak-majority illusion.
•
$M_{V}=tie$ . Two cases:
- –
  
  If $|\{i\in V:M_{N_{i}}\in\{blue,red\}\}|>\frac{|V|}{2}$ , we have a majority-weak-majority illusion.
- –
  
  Otherwise (if $|\{i\in V:M_{N_{i}}=tie\}|\geq\frac{|V|}{2}$ ) choose a node $j$ with $M_{N_{j}}=tie$ and define a new coloring $c^{\prime}$ that is equal to $c$ for all nodes except for $j$ : $c^{\prime}_{j}\neq c_{j}$ . Since $j$ has as many blue as red neighbors, this does not change the total number of monochromatic edges in the graph. Therefore, $c^{\prime}$ is also a coloring that minimizes this number. Hence, by Lemma 1, $c^{\prime}$ is also a weak majority $2$ -coloring of $G$ . Now, we have $M_{V}=c^{\prime}_{j}$ , and we can apply the logic of the first case: Assume w.l.o.g. that $c^{\prime}_{j}=red$ . Since $c^{\prime}$ is a weak majority coloring, for any red vertex $i$ , $M_{N_{i}}\in\{blue,tie\}$ . It follows that a majority of the nodes has $M_{N_{i}}\neq M_{V}$ : we have a majority-weak-majority illusion.

∎

One of the results in [10] is that checking whether or not a network allows for a majority-majority illusion is NP-complete²²2[10] does not actually speak about a majority-majority illusion, but about ‘at least a fraction of $q$ agents is under majority illusion’, with $q>\frac{1}{2}$ . The fact that it then also holds for majority-majority illusion follows from that ‘more than half’ is equivalent to ‘at least some fraction $q$ where $q$ is more than half’ since we only have to consider rational numbers.. Here, in stark contrast, we see that there is no need for checking whether a network allows for a majority-weak-majority illusion, since Theorem 3.1 shows that it is always the case.

4 Illusions in Specific Network Classes

While the above solves the question of the existence of weak majority illusions, we now aim to understand when the strict version of the illusion can occur. In order to obtain results in that direction, we turn to some classes of graphs with well-known global properties. Note that the results from this section are not intended to describe realistic classes of social networks but can instead be seen as a starting point for the systematic analysis of the types of graphs that allow for majority illusions. We focus on graphs with only odd-degree nodes, properly 2-colorable graphs, and regular graphs.

4.1 Graphs with Odd Degrees

Theorem 4.1

For any graph $G=\langle V,E\rangle$ such that for all $i\in V,d_{i}$ is odd, a majority-weak-majority illusion is either a unanimity-weak-majority illusion or a majority-majority illusion.

Proof

Let $G=\langle V,E\rangle$ be such that for all $i\in V$ , $d_{i}$ is odd. By Theorem 3.1, there exists a coloring of $G$ that induces a majority-weak-majority illusion. Consider any such coloring $c$ . Two cases:

•

$M_{V}=tie$ . For all $i\in V$ , since $d_{i}$ is odd, $M_{N_{i}}\neq tie$ and therefore $M_{N_{i}}\neq M_{V}$ : we have unanimity-weak-majority illusion.
•

$M_{V}\neq tie$ . Assume w.l.o.g. that $M_{V}=red$ . Since $G$ is in a majority-weak-majority illusion, $|\{i\in V:M_{N_{i}}\in\{blue,tie\}\}|>\frac{|V|}{2}$ , but since for all $i$ , $d_{i}$ is odd, this implies that all those vertices cannot have a tie: we have $|\{i\in V:M_{N_{i}}=blue\}|>\frac{|V|}{2}$ , a majority-majority illusion.

∎

The intuition is that an agent with odd degree cannot see a tie in its neighborhood, which causes either all agents a to be in weak-majority illusion if there is a global tie, or, if there is no global tie, a majority of agents to be in a majority illusion.

Given a graph coloring we can define a ‘swappable node’ as a node whose neighbors all have at least 2 (so 3 for odd degree) more nodes of one color than nodes of the other color. Then, a corollary of Theorem 4.1 is the following:

Corollary 1

For a graph $G=\langle V,E\rangle$ such that for all $i\in V,d_{i}$ is odd, if the coloring $c$ witnessing that majority-weak-majority illusion is possible induces that $M_{v}=tie$ and that there is at least one $j\in V$ that is ‘swappable’, a weak-majority-majority illusion is possible.

Proof

W.l.o.g. assume $c_{j}=red$ and define $c^{\prime}$ which is equal to $c$ for all nodes except that $c^{\prime}_{j}=blue$ . Since $c$ was a majority 2-coloring, all red nodes had more than half blue neighbors. Since $j$ ’s neighbors all had a margin of at least 2 and nothing except $j$ ’s color changed, all red nodes except $j$ still have more than half blue neighbors in $c^{\prime}$ . Hence, half of the nodes are under majority illusion.∎

4.2 2-Colorable Graphs

In the same way as we used the existence of a majority coloring to obtain results about the existence of majority illusions we can also use the existence of a special type of weak majority colorings, the proper colorings, to obtain results about majority illusions in $2$ -colorable graphs.

Lemma 2

Any proper 2-coloring of a graph $G=\langle V,E\rangle$ is either a majority-majority illusion or a unanimity-weak-majority illusion.

The idea of the proof is similar to that of Theorem 4.1: no node can see a tie among its neighbors.

Proof

Let $c$ be a proper $2$ -coloring of $G$ . Two cases:

•

If $M_{V}\neq tie$ , then w.l.o.g. assume that $M_{V}=red$ . Since more than half the nodes are red and all red nodes have a majority of blue neighbors, we have a majority-majority illusion.
•

If $M_{V}=tie$ , then all the nodes are under weak-majority illusion, since for all nodes, all neighbors are the other color. We have a unanimity-weak-majority illusion.

∎

Both cases used in the above proof are cases of majority-weak-majority illusions (which were already guaranteed to exist by 3.1), but we can also show the existence of the strict majority illusion in two different cases. First, when the number of nodes is odd, there cannot be a global tie, so by using the first case in Lemma 2 we get the following proposition:

Proposition 1

For any properly $2$ -colorable graph $G=\langle V,E\rangle$ with $|V|$ odd, a majority-majority illusion is possible.

Proof

Let $c$ be a proper 2-coloring of $G$ . Since $|V|$ is odd, $M_{V}\in\{red,blue\}$ , we can use the first case of the proof of Lemma 2: W.l.o.g. assume $M_{V}=red$ . Since more than half of the nodes are red and all red nodes have only blue neighbors (since $c$ was a proper 2-coloring), we have a majority-majority-illusion. ∎

Second, when the color of a node can be swapped if needed, we can solve a tie:

Proposition 2

For any properly $2$ -colorable graph $G=\langle V,E\rangle$ with some $i\in V$ such that for all $j\in N_{i}$ $d_{j}>2$ , a weak-majority-majority illusion is possible.

Proof

Let $c$ be a proper coloring of $G$ . Two cases:

•

If $M_{V}\neq tie$ , then conformingly to Lemma 2, we have a majority-majority illusion.
•

If $M_{V}=tie$ , then swap the color of node $i$ : let $c^{\prime}$ assign the same colors as $c$ to all other nodes but $c^{\prime}_{i}\neq c_{i}$ . Now $M^{\prime}_{V}=c^{\prime}i$ . W.l.o.g. say $c_{i}=blue$ and $c^{\prime}_{i}=red$ . All of $i$ ’s neighbors are also red and have now exactly one red neighbor ( $i$ ), and more than one blue neighbor. Therefore, all red nodes except for $i$ have more than half of their neighbors blue. Therefore, exactly $\frac{|V|}{2}$ of the nodes are in a situation of majority illusion.

∎

In [10], the complexity of checking whether a network admits (what we call) a weak-majority-majority illusion was left as an open problem. Propositions 1 and 2 show that by checking whether a graph is properly 2-colorable (which can be done in polynomial time [6]) and whether there exists a node whose neighbors all have degree larger than $2$ or whether the number of nodes is odd, we can know that a graph admits a (weak-)majority-majority illusion. Still this does not give us the complexity of checking whether a network admits a weak-majority-majority-illusion: while this is a sufficient condition for a graph to allow for a (weak-)majority-majority illusion, it is not a necessary condition.

4.3 Regular Graphs

In [2], theoretical analysis and experiments where human subjects were asked to perform estimation tasks are used to study the influence of network structure on the wisdom of crowds. The authors find a remarkable difference between centralized networks, where the degree distribution varies a lot between nodes, and decentralized (regular) networks, in which all nodes have the same degree, regarding what social influence does to the accuracy of the estimates of individuals (when individual’s estimates are based on a weighted average of their own belief and the beliefs of their neighbors). They show that in decentralized networks, social influence significantly improves individual accuracy and the group mean estimate. Furthermore, an overview of research about collective intelligence by Centola [7] mentions several studies about decentralized networks in practical applications: in decentralized networks, political polarization and biases about climate change and immigration are reduced [3, 11], and social influence reduced biases about the risk of smoking [12], as well as (implicit) race and gender biases in clinical settings [8]. Since decentralized/regular networks seem to be beneficial for group accuracy and bias reduction, we wonder whether they also are ‘good networks’ in terms of the distortion we study: to what extend they allow for majority illusions. According to [15], differences between the degrees of nodes and their neighbors are one of the main factors enabling majority illusion. Therefore, one would expect that regular networks, where all nodes have the same degree, make majority illusions less likely. Nevertheless, we show that majority illusions (beyond the ones given by Theorem 3.1) are also possible in regular networks.

A $k$ -regular network is a network in which all nodes have degree $k$ . We start by considering the simplest cases of regular network: simple cycles, where $k=2$ , and complete networks, where $k=|V|-1$ .

Proposition 3 (simple cycles)

For any $2$ -regular graph $G=\langle V,E\rangle$ , a (weak-)majority-majority illusion is not possible.

Proof

Let $G=\langle V,E\rangle$ be any 2-regular graph. A node can only be under majority illusion if both of its neighbours are of the minority color. Every minority-colored node can serve as a neighbour for at most two nodes. Thus, to give at least half of the nodes a majority illusion, there must be at least $\frac{|V|}{2}$ nodes in the minority color, which is a contradiction with being a strict minority.∎

However, a majority-weak-majority illusion is possible, according to Theorem 3.1, and it is easy to find one (which we leave as an exercise to the reader).

Proposition 4 (weak-majority-majority illusion in complete graphs)

For any complete graph $G=\langle V,E\rangle$ (i.e. a $k$ -regular graph with $k=|V|-1$ ), a (weak-)majority-majority illusion is not possible.

Proof

This is a corollary of Proposition 8 in Appendix 7. If $|V|=n$ , according to Proposition 8, a weak- $p$ - $q$ illusion can occur iff there is an integer $x$ such that $q(n-1)<x<qn$ and $n-x\geq pn$ . This implies that a majority-weak-majority illusion can occur iff there is an integer $x$ such that $\frac{n-1}{2}<x<\frac{n}{2}$ and $n-x\geq\frac{n}{2}$ . Clearly, there is no $x$ that satisfies the first requirement.∎

We know (by Theorem 3.1) that a majority-weak-majority illusion is always possible on complete graphs too. We can go further and specify the types of colorings under which these graphs are in such an illusion.

Proposition 5 (majority-weak-majority illusion in complete graphs)

A complete $2$ -colored graph $G=\langle V,E,c\rangle$ is under majority-weak-majority illusion if and only if:

•

the difference in numbers of nodes of each color is one; or
•

the number of nodes of each color is equal, then it is under unanimity-weak-majority illusion.

Proof

If the difference in numbers of nodes of each color is one, assume w.l.o.g. that $M_{V}=red$ . Then for all red nodes $r$ , $M_{r}=tie$ , so we have a majority-weak-majority illusion. If the number of nodes of each color is equal, we have $M_{V}=tie$ but every node will observe a majority of the other color: we have a unanimity-weak-majority illusion. ∎

We return to the analysis of general regular graphs. The number of minority-colored neighbours needed for an illusion gives a restriction on the possible values of $k$ depending on $|V|$ :

Proposition 6 ((weak-)majority-majority illusion in $k$ -regular graphs)

Let $G=\langle V,E\rangle$ be a $k$ -regular graph with $|V|=n$ . If a (weak-)majority-majority illusion is possible on $G$ , then $n$ and $k$ must satisfy:

•

$k\leq n-4$ if $n$ and $k$ are even;
•

$k\leq n-3$ if one of $n$ and $k$ is even and one is odd.

Proof

This is a direct corollary of Proposition 9 in Appendix 7. ∎

Example 1

Consider a $k$ -regular graph $G=\langle V,E\rangle$ with $|V|=6$ and $k=4$ . For any node to be in a majority illusion, at least $3$ of its neighbours have to have a different color than the global majority color. Assume that the global majority color is red. Then there are at least $4$ red nodes and therefore only 2 nodes can be blue. Therefore, no node can have 3 or more blue neighbours.

The number of available edges of the minority color brings another requirement on the relative values of $|V|$ and $k$ for the strictest version.

Proposition 7 (majority-majority illusion in $k$ -regular graphs)

Let $G=\langle V,E\rangle$ be a $k$ -regular graph with $|V|=n$ . If a majority-majority illusion is possible on $G$ , then $n$ and $k$ must satisfy:

•

$n\geq\frac{2(3k+2)}{k-2}$ (assuming $k\geq 3$ ) if $n$ and $k$ are even;
•

$n\geq\frac{2(3k+1)}{k-1}$ (assuming $k\geq 2$ ) if $n$ is even and $k$ is odd;
•

$n\geq\frac{3k+2}{k-2}$ (assuming $k\geq 3$ ) if $k$ is even and $n$ is odd.

Proof

If $G$ is in a majority-majority illusion, there are more than half of the nodes of one color. W.l.o.g., assume that this majority color is red, and that the minority color is blue.

•

When $n$ and $k$ both are even, in order for a majority-majority illusion to exist, at least $\frac{n}{2}+1$ nodes have to be red. Nodes with an illusion have to have at least $\frac{k+2}{2}$ blue neighbours. Then, there have to be at least $\frac{n}{2}+1$ such nodes with an illusion. Thus there have to be at least $\frac{k+2}{2}(\frac{n}{2}+1)=\frac{(k+2)(n+2)}{4}$ edges to a blue node. Hence, there must be at least $\frac{(k+2)(n+2)}{4k}$ blue nodes because every blue node can have at most $k$ edges. Since at least $\frac{n}{2}+1$ nodes were red, there are at most $\frac{n}{2}-1$ left over to be blue, so this means that $\frac{(k+2)(n+2)}{4k}$ must be at most $\frac{n}{2}-1$ . This is equivalent to $n\geq\frac{2(3k+2)}{k-2}$ assuming that $k>2$ ;
•

When $n$ is even and $k$ odd, in order for a majority-majority illusion to exist, at least $\frac{n}{2}+1$ nodes have to be red. Nodes with an illusion have to have at least $\frac{k+1}{2}$ blue neighbours. Then, there have to be at least $\frac{n}{2}+1$ such nodes with an illusion. Thus there have to be at least $\frac{k+1}{2}(\frac{n}{2}+1)=\frac{(k+1)(n+2)}{4}$ edges to a blue node. Hence, there must be at least $\frac{(k+1)(n+2)}{4k}$ blue nodes because every blue node can have at most $k$ edges. Since at least $\frac{n}{2}+1$ nodes were red, there are at most $\frac{n}{2}-1$ left over to be blue, so this means that $\frac{(k+1)(n+2)}{4k}$ must be at most $\frac{n}{2}-1$ . This is equivalent to $n+2\leq k(n-6)$ , which means $n\geq\frac{2(3k+1)}{k-1}$ assuming that $k>1$ ;
•

When $k$ is even and $n$ odd, in order for a majority-majority illusion to exist, at least $\frac{n+1}{2}$ nodes have to be red. Nodes with an illusion have to have at least $\frac{k+2}{2}$ blue neighbours. Then, there have to be at least $\frac{n+1}{2}$ such nodes with an illusion. Thus there have to be at least $\frac{k+2}{2}\cdot\frac{n+1}{2}$ edges to a blue node. Hence, there must be at least $\frac{(k+2)(n+1)}{4k}$ blue nodes. Since at least $\frac{n+1}{2}$ nodes were red, there are at most $\frac{n-1}{2}$ left over to be blue, so this means that $\frac{(k+2)(n+1)}{4k}$ must be at most $\frac{n-1}{2}$ . This is equivalent to $n\geq\frac{3k+2}{k-2}$ assuming that $k>2$ .

∎

Example 2

Consider a $k$ -regular network with $|V|=6$ and $k=3$ . Let us assume that red is the global majority color, so we have at least 4 red nodes and at most 2 blue nodes. Then any node with a majority illusion has at least 2 blue neighbours. Since for a majority-majority illusion there have to be at least 4 nodes with an illusion, there are at least $4\cdot 2=8$ edges to blue nodes necessary. However, since we have at most 2 blue nodes that each have only 3 edges, this is not possible.

For any $n$ and $k$ (with $k>2$ and $n$ or $k$ even) satisfying the above constraints, we can find a $k$ -regular graph of size $n$ that has a majority-majority illusion. Note that this does not mean that for any $k$ -regular graph of size $n$ we can find a coloring that gives a majority-majority illusion, because there exist many different regular graphs with the same $n$ and $k$ . We only show that, for all combinations of $n$ and $k$ not excluded by our previous results, there exists at least one such graph with the illusion, and that we know how to find it.

Theorem 4.2

Let $n$ and $k$ be any two integers such that $k>2$ and $k$ or $n$ is even. If the requirements of Propositions 6 and 7 are met, there exists a $k$ -regular graph $G=\langle V,E\rangle$ with $|V|=n$ on which a majority-majority illusion is possible.

Proof sketch. We prove this by construction: we give an algorithm that takes as input $n$ and $k$ and returns a regular graph with $n$ nodes of degree $k$ that has a majority-majority illusion. The algorithm and a proof that the algorithm outputs the desired graph are given in Appendix 8. See Figure 2 for an example with 12 nodes of degree 6.

Propositions 6 and 7 and Theorem 4.2 together give necessary and sufficient conditions for $n$ and $k$ for the existence of a $k$ -regular graph with $|V|=n$ nodes on which a majority-majority illusion is possible.

5 Conclusion and Outlook

We studied weak and strong versions of the majority illusion. Using results about majority-colorings, we proved that no network is immune to majority-weak-majority illusion, and that some classes of graphs are not immune to stronger types of illusions either. The results are summarized in Table 2. We also provided an algorithm to find a $k$ -regular graph of size $n$ with a majority-majority illusion, when it exists.

The most natural direction for further research is to broaden the scope of our study: first, as we have initiated with Appendix 7, by considering proportions other than majority; and second, by considering classes of graphs that are more realistic as social networks, including not necessarily irreflexive ones. Moreover, beyond verifying their sheer possibility, we keep measuring the likelihood of such illusions for future work.

A different direction is to investigate the relation between majority illusions and majority logic [18], which can ‘talk about’ local majorities, but not about global majority. We propose to enrich the logic with a global majority operator, to express the results from this paper. The idea is elaborated on in Appendix 9.

Last but not least, it would be interesting to measure the impact of proportional illusions on specific social phenomena. For instance, how do they affect opinion diffusion dynamics in a population? How do they interact with polling effects? And how do they relate to better known types of illusions, such as the above-mentioned ‘friendship paradox’ [9]?

Table 2: The (im)-possibility of majority illusions on different classes of graphs. ✓ indicates that the illusion is possible on all graphs of the class, ✗ indicates that the illusion is not possible on any graph in the class, ✓/ ✗ indicates that the illusion is possible on some but not all graphs of the class. For the majority-weak-majority illusion, some stronger results are shown. References to the relevant results are given.

Class of graphs	majority-weak-majority			weak-maj.-majority	majority-majority
All graphs	✓(Thm. 3.1)			✓/ ✗
Graph with only odd-degree nodes		majority- majority or unanimity- weak- majority	(Thm. 4.1)
2-colorable graphs			(Lem. 2)
2-colorable graphs with $\|V\|$ odd				✓ (Prop. 1)
2-colorable graphs with $i\in V:\forall j\in N_{i}:d_{j}>2$				✓ (Prop. 2)	✓/ ✗
2-regular graphs				✗ (Prop. 3)
Complete graphs with $\|V\|$ even		unanimity-weak-majority (Prop. 5)		✗ (Prop. 4)
Complete graphs with $\|V\|$ odd

Acknowledgments. We thank the anonymous reviewers of EUMAS 2023 for their helpful comments. Zoé Christoff acknowledges support from the project Social Networks and Democracy (VENI project number Vl.Veni.201F.032) financed by the Netherlands Organisation for Scientific Research (NWO). Davide Grossi acknowledges support by the Hybrid Intelligence Center, a 10-year program funded by the Dutch Ministry of Education, Culture and Science through the Netherlands Organisation for Scientific Research (NWO).

Appendix 6 Constructive Argument for Majority-2-Colorability

Theorem Appendix 6.1 (attributed to [16])

Every graph is weak majority $2$ -colorable.

Proof

Consider a graph $G=\langle V,E\rangle$ and an arbitrary $2$ -coloring $c$ of $G$ . Then, search for a node with more monochromatic edges than dichromatic ones. If such a node does not exist, we are done, and our initial coloring $c$ is already a majority $2$ -coloring. If such a node does exist, swap its color. Even though this might make another node increase its number of monochromatic edges, the total number of monochromatic edges in the graph can only decrease by such a step. We can proceed with this until there is no node to swap color of anymore, i.e. all nodes have at least as many dichromatic edges as monochromatic ones. We can be sure that at some point this will be the case, since every step can only strictly decrease the total number of monochromatic edges in the graph, and this number has a lower bound.

input : graph

G

output : weak majority

2

-coloring of

G

c\leftarrow

random 2-coloring of

G

;

3 while $\exists$ node $i$ with $E_{M}(i)>E_{D}(i)$ do

c_{i}\leftarrow\bar{c_{i}}

;

6 end while

return $c$

Algorithm 1 Weak majority 2-coloring

∎

See Figure 3 for an illustration of the algorithm.

Appendix 7 More General Illusions

For the following results, we will further generalize Definitions 1 and 2 to arbitrary thresholds as follows:

Definition 4 ( $q$ illusion)

Given a colored graph $C=\langle V,E,c\rangle$ , an agent $i\in V$ is under $q$ illusion if, for some $x\in\{red,blue\}$ , $|\{j\in N_{i}\mid c_{j}=x\}|>q\cdot d_{i}$ , but $|\{j\in N\mid c_{j}=x\}|<q\cdot|V|$ .

Definition 5 (weak- $q$ illusion)

Given a colored graph $C=\langle V,E,c\rangle$ , agent $i\in V$ is under weak $q$ illusion if, for some $x\in\{red,blue\}$ , $|\{j\in N_{i}\mid c_{j}=x\}|\geq q\cdot d_{i}$ but $|\{j\in N\mid c_{j}=x\}|\leq q\cdot|V|$ , and it is not the case that $|\{j\in N_{i}\mid c_{j}=x\}|=q\cdot d_{i}$ and $|\{j\in N\mid c_{j}=x\}|=q\cdot|V|$ .

For $q=\frac{1}{2}$ , Definitions 4 and 5 reduce to Definitions 1 and 2.

Note that in the same network, different agents can be under a $q$ illusion with respect to different colors if $q>\frac{1}{2}$ (or a weak $q$ illusion if $q\geq\frac{1}{2}$ ), since then for both colors there can be less than (or equal to) $qn$ of the nodes in the network of that color, while in the different neighbourhoods there can be more than $q$ of different colors. We call a network illusion where all agents that are under illusion have an illusion of the same color (or all see a tie) a monochromatic illusion, and the general case where different agents can have illusions of different colors a polychromatic illusion or just an illusion (so the polychromatic illusion is more general and includes also the monochromatic one). Note that for $q\leq\frac{1}{2}$ (and $q<\frac{1}{2}$ for weak illusions), the illusion can only be monochromatic.

We can also make the first ‘majority’ in ‘majority- $q$ illusion’ an arbitrary fraction of agents $p$ instead of exactly $\frac{1}{2}$ , and we can define both the weak and strong version of this to study cases where either at least $p$ of the agents are under illusion or more than $p$ of the agents are under illusion.

Definition 6 ((weak)- $p$ -(weak)- $q$ illusion)

A graph $C=\langle V,E,c\rangle$ is in a $p$ -(weak)- $q$ illusion if more than a $p$ - proportion of the agents is under (weak-) $q$ illusion. A graph is in a weak- $p$ -(weak)- $q$ illusion if at least a proportion $p$ of the agents is under (weak-) $q$ illusion.

If we set either $p$ or $q$ to $\frac{1}{2}$ , we get the respective ‘majority’-versions. We examine the possibility of $p$ - $q$ illusions on complete graphs.

Proposition 8 ( $p$ - $q$ illusions on complete graphs)

If $G=\langle V,E\rangle$ is a complete graph with $|V|=n$ , a $p$ - $q$ illusion can occur iff there is an integer $x$ , representing the number of nodes of color $c$ , such that $q(n-1)<x<qn$ and $n-x>pn$ .
A weak- $p$ - $q$ illusion can occur iff there is an integer $x$ such that $q(n-1)<x<qn$ and $n-x\geq pn$ .
A $p$ -weak- $q$ illusion can occur iff there is an integer $x$ such that $q(n-1)\leq x\leq qn$ and $n-x>pn$ .
A weak- $p$ -weak- $q$ illusion can occur iff there is an integer $x$ such that $q(n-1)\leq x\leq qn$ and and $n-x\geq pn$ .

Proof

For a $p$ - $q$ illusion to be possible in $G$ , we need that for some nodes, more than $q\cdot k=q(n-1)$ of its neighbours has a color $c$ , but that at the same time less than $q\cdot n$ of the set of all its neighbours plus the node itself has color $c$ . Hence, if we call the total number of $c$ -colored nodes in the network $x$ , we need to have that $q(n-1)<x<qn$ , or $\frac{x}{n}<q<\frac{x}{n-1}$ . For some combinations of $n$ and $q$ , there is no such integer $x$ , for example when $q=\frac{1}{m}$ for some integer $m$ or when either $qn$ or $q(n-1)$ is an integer. If there does exist such an $x$ , at least some nodes can be under illusion, in fact, exactly all the nodes that do not have color $c$ are under $q$ illusion. Since $x$ was the number of nodes with color $c$ , there are $n-x$ nodes with the other color. Hence, there can be a $p$ - $q$ illusion iff there is an integer $x$ such that $q(n-1)<x<qn$ and $n-x>pn$ . For the illusions with weak $q$ , the first requirement becomes $q(n-1)\leq x\leq qn$ ; for the illusions with weak $p$ the second requirement becomes $n-x\geq pn$ . ∎

Proposition 6 can be written in a sligtly more general way:

Proposition 9

³³3We can make a similar theorem for general

q

instead of

q=\frac{1}{2}

, but then we only get that

k+2\leq n

for the possibility of an illusion, which we already knew because for

k+1=n

we have a complete network in which a (strict) illusion is not possible.

If a 2-colorored $k$ -regular graph $G=\langle V,E,c\rangle$ with $|V|=n$ is in a (weak-) $p$ -majority illusion (for any $p>0$ ), then $n$ and $k$ must satisfy

•

$k\leq n-4$ if $n$ and $k$ are even;
•

$k\leq n-3$ if one of $n$ and $k$ is even and one is odd.

Proof

Assume $G$ is in a (weak-) $p$ -majority illusion. Then there is at least one agent under majority illusion, so more than $\frac{k}{2}$ of its neighbours have color $c$ , but less than $\frac{n}{2}$ of the agents in the entire network have color $c$ . Hence, this agent has at least $\frac{(k+1)}{2}$ neighbours of color $c$ if $k$ is odd and at least $\frac{(k+2)}{2}$ if $k$ is even. But in total, there are less than $\frac{n}{2}$ nodes of color $c$ , so at most $\frac{n-2}{2}$ if $n$ is even and at most $\frac{n-1}{2}$ if $n$ is odd. Hence we must have:

•

if $n$ and $k$ are even: $\frac{(k+2)}{2}\leq\frac{n-2}{2}$ , so $k\leq n-4$ ;
•

if $n$ is even and $k$ is odd: $\frac{(k+1)}{2}\leq\frac{n-2}{2}$ , so $k\leq n-3$ ;
•

if $n$ is odd and $k$ is even: $\frac{(k+2)}{2}\leq\frac{n-1}{2}$ , so $k\leq n-3$ .

∎

Proposition 7 can be generalised for arbitrary $p$ and $q$ as follows:

Proposition 10

If a 2-colorored $k$ -regular graph $G=\langle V,E,c\rangle$ with $|V|=n$ is in a monochromatic $p$ - $q$ illusion (for $0\leq p\leq 1$ and $0\leq q\leq 1$ ), then for any $n>1$ , $k>1$ , $0<q\leq 1$ , $p$ must satisfy $0\leq p<\frac{kn}{(n+1)(k+1)}$ .

Proof

Assume $G$ is in a monochromatic $p$ - $q$ illusion, and assume that the color of illusion is $c$ . Then more than $np$ of the nodes are under $q$ illusion, so at least $(n+1)p$ nodes are under illusion⁴⁴4the exact value can differ depending on the divisibility of $n$ by $\frac{1}{p}$ , or maybe on the common divisors of $n$ and $p$ , but it always has the lower bound of $(n+1)p$ .. Of the nodes under illusion, more than $qk$ of the neighbours are colored $c$ , so at least $(k+1)q$ of the neighbours are colored $c$ . Hence, there have to be at least $(n+1)p(k+1)q$ edges to a $c$ -colored node, and since every node has $k$ edges, there have to be at least $\frac{(n+1)p(k+1)q}{k}$ nodes of color $c$ . However, in the total network there are less than $qn$ blue nodes, so $\frac{(n+1)p(k+1)q}{k}<qn$ , which, for $q>0$ , boils down to $p<\frac{kn}{(n+1)(k+1)}$ . ∎

We can also say something (trivial) about the size of the existing proportional illusion in all graphs with only odd degrees of at most $k$ . The point is that we know that the illusion exists, and that, when $k$ is smaller (and $n$ larger), the illusion is stronger (it will not just be a neglectable difference between local majority and global majority).

Corollary 2

For any graph $G=\langle V,E\rangle$ with all $v\in V,d(v)$ odd and $\leq k$ , a weak majority- $q$ illusion exists for $q=\frac{\frac{k+1}{2}}{k}$ .

This holds in particular for regular graphs (it doesn’t add anything):

Corollary 3

For any $k$ -regular graph $G=\langle V,E\rangle$ with $k$ odd (and therefore $n$ even), a weak majority- $q$ illusion exists for $q=\frac{\frac{k+1}{2}}{k}$ .

The generalization to $p$ - $q$ illusions gives us another open question: We know that any network allows for majority-weak-majority illusions. The problem of whether a $q$ -weak-majority illusion is possible on a network for greater $q$ remains.

Appendix 8 Algorithm for Finding a $k$ -Regular Graph with a Majority-Majority Illusion.

Appendix 8.1 Pseudocode

The pseudocode of the algorithm can be found in Algorithm 2.

input : number of nodes

n

, degree

k

output :

k

-regular graph of

n

nodes, with majority-majority illusion

n_{red}\leftarrow\lfloor{\frac{n}{2}+1}\rfloor

;

n_{blue}\leftarrow\lceil{\frac{n}{2}-1}\rceil

;

3 if $n$ is even then

4 if $k$ is even then

k_{blue}\leftarrow\frac{k}{2}-3

;

k_{red}\leftarrow\frac{k}{2}-1

;

8 else

k_{blue}\leftarrow\frac{k-5}{2}

;

k_{red}\leftarrow\frac{k-1}{2}

;

12 if $k_{blue}<0$ then

k_{blue}\leftarrow 0

14else

k_{blue}\leftarrow\frac{k}{2}-2

;

k_{red}\leftarrow\frac{k-2}{2}

;

18/* Now

n_{blue}

and

n_{red}

are the numbers of blue and red nodes, and

k_{blue}

and

k_{red}

are the degrees of the regular subgraphs built in the last step. */

N\leftarrow

set of

n

nodes;

E\leftarrow\emptyset

;

21 /* set of edges starts empty */

G\leftarrow\langle N,E\rangle

;

R\leftarrow

any subset of

n_{red}

nodes of

N

;

B\leftarrow N\backslash B

;

E\leftarrow

add_initial_edges

(E,B,R,k)

;

26 /* Algorithm 3 */

E\leftarrow

add_extra_blue_edges

(E,B,k,k_{blue})

;

28 /* Algorithm 4 */

29 if $k_{red}$ is even or $n_{red}$ is even then

E\leftarrow

add_regular_subgraph

(E,R,k_{red})

;

31 /* Algorithm 5 */

33if $k_{blue}$ is even or $n_{blue}$ is even then

E\leftarrow

add_regular_subgraph

(E,B,k_{blue})

;

36if $k_{red}$ is odd and $n_{red}$ is odd then /* Hence also

k_{blue}

and

n_{blue}

are odd */

k_{red_{temp}}\leftarrow k_{red}-1

;

38 /* Make regular graphs with one less edge per node */

E\leftarrow

add_regular_subgraph

(E,R,k_{red_{temp}})

;

40 if $k_{blue}>0$ then

k_{blue_{temp}}\leftarrow k_{blue}-1

;

E\leftarrow

add_regular_subgraph

(E,B,k_{blue_{temp}})

;

blue_{1}\leftarrow

blue node with least amount of edges;

red_{c}\leftarrow

a red node that

blue_{1}

is not yet connected to;

E\leftarrow E\cup\{(blue_{1},red_{c})\}

;

47 /* add an edge between (

blue_{1}

and

red_{c}

) */

48 if $k_{blue}>0$ then

49 for node in $B\backslash\{blue_{1}\}$ do

50 if number of neighbors of node $<k$ then

next\leftarrow

first blue node that is not yet connected to

node

and has less than

k

edges;

E\leftarrow E\cup\{(node,next)

};

54 for node in $R\backslash\{red_{c}\}$ do

55 if number of neighbors of node $<k$ then

next\leftarrow

first red node that is not yet connected to

node

and has less than

k

edges;

E\leftarrow E\cup\{(node,next)

};

G\leftarrow\langle N,E\rangle

;

return G

Algorithm 2 Finding a majority-majority illusion coloring

input : set of edges

E

, blue nodes

B

, red nodes

R

, degree

k

output :

E

with some edges added between blue and red nodes

x\leftarrow 0

;

2 /* value to avoid double edges */

3 if $k$ is even then

n_{edges}\leftarrow\lfloor|R|\cdot\frac{k+2}{2}\rfloor

;

5 /*

n_{edges}

is the nr of necessary edges */

7else

n_{edges}\leftarrow\lfloor|R|\cdot\frac{k+1}{2}\rfloor

9 end if

10for $0\leq i<n_{edges}$ do

index_{red}\leftarrow i\mod|R|

;

node_{red}\leftarrow R[index_{red}]

;

index_{blue}\leftarrow(x+i)\mod|B|

;

node_{blue}\leftarrow B[index_{blue}]

;

17 if ( $node_{red},node_{blue})\in E$ then

x\leftarrow 1

;

19 /* (edge already exists, we shift one position) */

index_{blue}=(x+i)\mod|B|

;

node_{blue}\leftarrow B[index_{blue}]

;

23 end if

E\leftarrow E\cup\{(node_{red},node_{blue})\}

;

26 end for

return $E$

Algorithm 3 add_initial_edges

input : set of edges

E

, blue nodes

B

, degree

k

, degree

k_{blue}

output :

E

with extra blue edges

1 sort

B

such that ones missing most edges come first;

2 for node in $B$ do

index_{next}\leftarrow(B.index(node)+1)\mod|B|

;

next\leftarrow B[index_{next}]

;

5 if (node, next_node) $\notin E\wedge|\text{neighbors(node)}|<k-k_{blue}\wedge|\text{neighbors}(next)|<k-k_{blue}$ then /* The node has more than

k_{blue}

open edges (which is the nr to be left over after this), so less than

k-k_{blue}

neighbors (the next-node check is only for the case where one node has to be left open (n odd, k even)) */

E\leftarrow E\cup\{(node,next)\}

;

8 end if

10 end for

return $E$

Algorithm 4 add_extra_blue_edges

input : set of edges

E

, colored nodes

C

, degree

k_{color}

output :

E

with added regular subgraph between nodes of

C

1 for node in $C$ do

2 if $|C|$ is even then

index_{start}\leftarrow C.index(node)+\lfloor\frac{|C|}{2}\rfloor\mod|C|

;

4 /* The index of the node opposite to the current node */

6 else

index_{start}\leftarrow C.index(node)+\frac{|C|}{2}

;

8 /* (this will be a half number) */

10 end if

11 if $|C|$ is even and $k_{color}$ is odd then

E\leftarrow E\cup(node,C[index_{start}])

;

13 /* (add an edge to the node opposite.) */

15 end if

r\leftarrow\frac{k_{color}}{2}

;

17 /*

r

is the range to both sides */

18 if $|C|$ is odd and $k_{color}$ is odd then

r\leftarrow\frac{k_{color}-1}{2}

;

21 end if

22 for $1\leq i\leq r$ do

index_{minus}\leftarrow\lceil index_{start}-i\rceil\mod|C|

;

index_{plus}\leftarrow\lfloor(index_{start}+i)\rfloor\mod|C|

;

E\leftarrow E\cup\{(node,C[index_{minus}]),(node,C[index_{plus}])\}

;

27 end for

29 end for

return $E$

Algorithm 5 add_regular_subgraph

Appendix 8.2 Proof of Correctness of the Algorithm

We go by cases over the combinations of the possible parities of $|V|=n$ and $k$ .

•

$n$ and $k$ are even For the case where $n$ and $k$ are both even: Let there be $n_{red}=\frac{n}{2}+1$ red nodes and $n_{blue}=\frac{n}{2}-1$ blue nodes. We assume that the requirements $n-4\geq k$ (and $n\geq\frac{2(3k+2)}{k-2}$ ) hold.
We start by just connecting every red node to $\frac{k+2}{2}$ blue nodes, and do this as evenly spread out over the blue nodes as possible by just going in rounds over the red nodes and over the blue nodes until we are done. After this, all red nodes have enough blue neighbours to be in majority illusion, and there are in total $\frac{n+2}{2}\frac{k+2}{2}$ blue edges used, so there are $k(\frac{n}{2}-1)-\frac{n+2}{2}\frac{k+2}{2}$ blue edge-ends left over, evenly spread over the blue nodes. We can write this as $k(\frac{n}{2}-1)-(\frac{n}{2}-1+2)\frac{k+2}{2}=(\frac{n}{2}-1)(k-\frac{k+2}{2})-(k+2)=(\frac{n}{2}-1)(\frac{k-2}{2})-(k+2)$ . Hence, every blue node has $\frac{k-2}{2}$ edge-ends left over, but $k+2$ of them have one less left over. But, $k+2$ may exceed the number of blue nodes $\frac{n}{2}-1$ , namely when $n<2k+6$ , and then there must be some nodes that have two less (because we cannot spread out the $k+2$ evenly over the $\frac{n}{2}-1$ . However, $k+2$ cannot be bigger than $2(\frac{n}{2}-1)$ (since then we would have $k>n-4$ , which is contradicted by our first assumption), so there are no blue nodes with more than 2 less open ends than $\frac{k-2}{2}$ . We can therefore approach it from the other side, if we start with stating that every blue node has $\frac{k-2}{2}-2=\frac{k-6}{2}$ edge-ends left open, then $(\frac{n}{2}-1)2-(k+2)=n-k-4$ have another extra left open (and if $(\frac{n}{2}-1)2-(k+2)>\frac{n}{2}-1$ some have two extra, that is, when $n>2k+6$ ). So, we have three kinds of open ends left over.
1. (a)
  
  $\frac{n}{2}+1$ red nodes that all have $k_{red}=\frac{k}{2}-1$ open ends;
2. (b)
  
  $\frac{n}{2}-1$ blue nodes that all have $k_{blue}=\frac{k-6}{2}=\frac{k}{2}-3$ open ends
3. (c)
  
  another $n-k-4$ open ends evenly divided over the blue nodes (where each blue node has at most two ends of this kind).
Now we can link those open edges to make a regular graph:
1. i
  
  we can tie the ends in (c) to each other (possible since $n-k-4$ is even, and no blue is yet connected to another blue);
2. ii
  if $k_{blue}=\frac{k}{2}-3$ (and hence also $k_{red}=\frac{k}{2}-1$ ) is even:
  1. (a)
    
    make a regular graph between the remaining blue edge ends ( $\frac{n}{2}-1$ nodes with all $\frac{k}{2}-3$ open ends);
  2. (b)
    
    make a regular graph between the remaining red edge ends ( $\frac{n}{2}+1$ nodes with all $\frac{k}{2}-1$ open ends);
  Where 2 (a) is possible because any blue node is connected to at most two other blue nodes as a result of 1, so there are at least $(\frac{n}{2}-1)-3$ other blue nodes left over (also subtracting 1 for the node itself). Since $n-2\geq k$ , we also have $(\frac{n}{2}-1)-3\geq\frac{k}{2}-3$ . If $k_{blue}<0$ , we just skip this step, because then all blue nodes already have enough edges due to step (i). Clearly, 2(b) is always possible, since there are no edges between any red nodes yet.
3. iii
  if $k_{blue}$ and hence also $k_{red}$ are odd,
  1. (a)
    
    make a regular graph between the remaining blue edge ends leaving one edge per node open ( $\frac{n}{2}-1$ nodes with all $\frac{k}{2}-4$ ends);
  2. (b)
    
    make a regular graph between the remaining red edge ends, leaving one edge per node open ( $\frac{n}{2}+1$ nodes with all $\frac{k}{2}-2$ ends);
  Then we are left with exactly one open end for all nodes, but since everything so far was spread out as evenly as possible over the network, it is possible to divide the nodes into pairs that are not yet connected to each other.
The cases where one of $k$ and $n$ is odd work very similar:
•
$n$ is even, $k$ is odd When $n$ is even and $k$ is odd, we still have $n_{red}=\frac{n}{2}+1$ red nodes and $n_{blue}=\frac{n}{2}-1$ blue. We assume that, as given in Proposition 6, $k\leq n-3$ , and, as given in Prop. 7, $n\geq\frac{2(3k+1)}{k-1}$ . Then we proceed as in Algorithm 2 above: we tie every red node to $\frac{k+1}{2}$ blue nodes, as evenly as possible. Then, there are still $k(\frac{n-2}{2})-\frac{n+2}{2}\frac{k+1}{2}=\frac{k-1}{2}\frac{n-2}{2}-(k+1)$ blue edge-ends left over. Since these are as evenly spread over the blue nodes as possible, every blue node has at least $\frac{k-1}{2}-2=\frac{k-5}{2}$ open ends, and then there are another $\frac{n-2}{2}\cdot 2-(k+1)=n-k-3$ blue edge-ends left over. (if $n-k-3>\frac{n}{2}-1$ , some blue nodes have two extra, otherwise those can be divided over $n-k-3$ blue nodes that all have one extra). Then, all red nodes have a majority illusion, and we still have the following open ends to fill the regular graph:
1. a)
  
  $\frac{n}{2}+1$ red nodes, all with $k_{red}=\frac{k-1}{2}$ open ends;
2. b)
  
  $\frac{n}{2}-1$ blue nodes, all with $k_{blue}=\frac{k-5}{2}$ open ends to start with;
3. c)
  
  another $n-k-3$ blue open ends, at most 2 per node.
Since $n-k-3$ is even, and no blue is yet connected to another blue, we can just connect the open ends in (c) to each other. Then, if $\frac{k-1}{2}$ and $\frac{k-5}{2}$ are even, just make two regular graphs between the two, otherwise leave one open and connect those, just as in the case with both $n$ and $k$ even.
•
$n$ is odd, $k$ is even When $n$ is odd and $k$ is even, we have $n_{red}=\frac{n+1}{2}$ red nodes, $n_{blue}=\frac{n-1}{2}$ blue nodes, and a node needs to be connected to $\frac{k}{2}+1$ blue nodes to have an illusion. We assume that $k\leq n-3$ conformingly to Proposition 6, and that $n\geq\frac{3k+2}{k-2}$ conformingly to Proposition 7. Just as in the other cases, we start by tying every red node to $\frac{k}{2}+1$ blue nodes, which leaves us with $k(\frac{n-1}{2})-\frac{n+1}{2}\frac{k+2}{2}=\frac{k-2}{2}\frac{n-1}{2}-\frac{k+2}{2}$ blue ends open. It is not possible that $\frac{k+2}{2}>\frac{n-1}{2}$ , since that would imply that $k>n-3$ , so the remaining $\frac{k+2}{2}$ can be divided over the blue nodes. What this means is that every blue node has $\frac{k-2}{2}-1=\frac{k}{2}-2$ open ends at least, and $\frac{n=-1}{2}-\frac{k+2}{2}=\frac{n-k-3}{2}$ blue nodes have one more open end (so those have $\frac{k-2}{2}$ ). This means that we still have the following open ends to fill the regular graph:
1. (a)
  
  $\frac{n+1}{2}$ red nodes, all with $k_{red}=\frac{k-2}{2}$ open ends;
2. (b)
  
  $\frac{n-1}{2}-\frac{n-k-3}{2}=\frac{k+2}{2}$ blue nodes that have $k_{blue}=\frac{k}{2}-2$ open ends; and
3. (c)
  
  $\frac{n-k-3}{2}$ blue nodes that have $\frac{k}{2}-1$ open end.
If the number of nodes mentioned in (c) is even, we can connect them to each other and make regular graphs between the remaining red resp. blue nodes. If the number mentioned in (c) is odd, then either both $\frac{n+1}{2}$ ( $n_{red}$ ) and $\frac{k-2}{2}$ ( $k_{red}$ ) are even or both $\frac{n-1}{2}$ ( $n_{blue}$ ) and $\frac{k}{2}-2$ ( $k_{blue}$ ) are even, but not both. That means that we can make a regular graph of the remaining edges of either the blue nodes or the red nodes, but not both. However, of the one that cannot form a regular graph, we tie one node to one of the nodes in $c$ , which makes the remaining number in $c$ even, and of the rest, we make an almost regular graph.

Faster algorithm (only for nice values of $n$ and $k$ ).

If $n\equiv 2\mod 4$ and $n\leq 2k-2$ , we can connect all red nodes to all blue nodes. Then all red nodes still have $k-\frac{n}{2}+1$ open ends, and all blue nodes still have $k-\frac{n}{2}-1$ open ends. Since $n\equiv 2\mod 4$ and $k$ is even, both $k-\frac{n}{2}+1$ and $k-\frac{n}{2}-1$ are even. Hence, we can make two regular graphs with the remaining open ends, one between the red nodes, and one between the blue nodes. Note that if $n\equiv 0\mod 4$ , both the numbers of open ends of each color and the number of nodes of each color is odd, so we cannot apply this algorithm.

Appendix 9 Global Majority Logic

Majority Logic, a modal logic that can express facts about a majority of accesible worlds is introduced in [18]. It is an extension of Graded Modal Logic with the following syntax:

\alpha:=p\mid\neg\alpha\mid\alpha\vee\alpha\mid\Diamond_{n}\alpha\mid W\alpha,

where $p\in\Phi$ , a set of atomic propositions, and $n\in\mathbb{N}$ .

The formula $\Diamond_{n}\alpha$ means that $\alpha$ is true in strictly more than $n$ accessible states. $W\alpha$ means that $\alpha$ is true in at least half of the accessible states, and its dual $M\alpha=\neg W\neg\alpha$ means that $\alpha$ is true in more than half of the accessible states. This logic, as it is, can already capture some of the central notions of our paper. For instance, a majority opposition can be formulated as $p\wedge M\neg p$ , a weak majority opposition as $p\wedge W\neg p$ .

To capture the other notions relevant to the majority illusion, we propose to enrich this logic with global modalities: a modality $E_{n}$ (there exists at least $n$ states) and a ‘global majority modality’ $GW$ (in at least half of all states). Then the syntax $\mathcal{L}_{\textbf{GMJL}}$ (Global Majority Logic) becomes:

\alpha:=p\mid\neg\alpha\mid\alpha\vee\alpha\mid\Diamond_{n}\alpha\mid W\alpha\mid E_{n}\alpha\mid GW\alpha.

We will write $GM$ for the dual of $GW$ (in more than half of all states).

A model $M=\langle G,v\rangle$ is such that $G=\langle V,E\rangle$ is a graph, as defined in the beginning of this paper, together with a valuation $v:V\to\mathcal{P}(\Phi)$ .

We use the notation that for any formula $\alpha$ , the set of neighbors of $i$ for which $\alpha$ is true is $N_{\alpha_{i}}=\{j\mid j\in N_{i}\text{ and }j\models\alpha\}$ , and for any formula $\alpha$ we denote by $V_{\alpha}=\{s\mid s\models\alpha\}$ the set of agents at which $\alpha$ is true. The truth of a formula $\alpha\in\mathcal{L}_{\textbf{GMJL}}$ at an agent $i$ in model $M$ is defined as follows:

•

$M,i\models p$ iff $p\in v(i)$ ;
•

$M,i\models\neg\alpha$ iff $M,i\not\models\alpha$ ;
•

$M,i\models\alpha\vee\beta$ iff $M,i\models\alpha$ or $M,i\models\beta$ ;
•

$M,i\models\Diamond_{n}\alpha$ iff $|N_{\alpha_{i}}|>n$ (for $n\in\mathbb{N}$ );
•

$M,i\models W\alpha$ iff $|N_{\alpha_{i}}|\geq\frac{|N_{{}_{i}}|}{2}$ ;
•

$M,i\models E_{n}\alpha$ iff $|V_{\alpha}|>n$ ;
•

$M,i\models GW\alpha$ iff $|V_{\alpha}|\geq\frac{|V|}{2}$ .

We give some examples of statements about majority illusions that can be expressed in $\mathcal{L}_{\textbf{GMJL}}$ .

First, we can talk about the situation a single agent might be in.

•

$i$ is under weak majority opposition: $M,i\models(p\land W\lnot p)\lor(\lnot p\land Wp)$
•

$i$ is under majority illusion: $M,i\models(GMp\land M\lnot p)\lor(GM\lnot p\land Mp)$
•

$i$ is under weak majority illusion: $M,i\models(GWp\land\lnot Wp)\lor(GW\lnot p\land\lnot W\lnot p)\lor((Wp\land W\lnot p)\land(\lnot GWp\lor\lnot GW\lnot p))$

Second, we can talk about whether an entire network is under a specific type of illusion:

•

$G$ is under majority majority illusion: $M,i\models GM((GMp\land M\lnot p)\lor(GM\lnot p\land Mp))$
•

$G$ is under weak majority majority illusion: $M,i\models GW((GMp\land M\lnot p)\lor(GM\lnot p\land Mp))$
•

$G$ is under majority weak majority illusion: $M,i\models GM((GWp\land\lnot Wp)\lor(GW\lnot p\land\lnot W\lnot p)\lor((Wp\land W\lnot p)\land(\lnot GWp\lor\lnot GW\lnot p)))$
•

$G$ is under weak majority weak majority illusion: $M,i\models GW((GWp\land\lnot Wp)\lor(GW\lnot p\land\lnot W\lnot p)\lor((Wp\land W\lnot p)\land(\lnot GWp\lor\lnot GW\lnot p)))$

Third, we can talk about the possibility of certain illusions on a network:

•

a majority majority illusion is possible on $G$ :

$G\not\models\lnot GM((GMp\land M\lnot p)\lor(GM\lnot p\land Mp))$
•

a weak majority majority illusion is possible on $G$ :

$G\not\models\lnot GW((GMp\land M\lnot p)\lor(GM\lnot p\land Mp))$

•

a majority weak majority illusion is possible on $G$ :

G\not\models\lnot GM((GWp\land\lnot Wp)\lor(GW\lnot p\land\lnot W\lnot p)\lor((Wp\land W\lnot p)\land(\lnot GWp\lor\lnot GW\lnot p)))

•

a weak majority weak majority illusion is possible on $G$ :

G\not\models\lnot GW((GWp\land\lnot Wp)\lor(GW\lnot p\land\lnot W\lnot p)\lor((Wp\land W\lnot p)\land(\lnot GWp\lor\lnot GW\lnot p)))

Note that, just as colorability is expressible in terms of satisfiability in basic modal logic extended only with the universal modality, the possibility of our majority illusions (in addition to colorability and majority-colorability) are expressible in terms of satisfiability in this globalized majority logic.

Axiomatization and further analysis of $\mathcal{L}_{\textbf{GMJL}}$ is left as a direction for further research.

References

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On the Graph Theory of Majority Illusions

Abstract

Keywords:

1 Introduction

Related work.

Contributions.

2 Preliminaries

Binary opinion networks and 2-colored graphs.

Majority illusion and opposition: intuitions.

Formal definitions.

Definition 1 (majority illusion)

Definition 2 (weak versions of majority illusion)

3 Illusions in Arbitrary Networks

Definition 3 (weak majority 22-coloring)

Remark 1

Lemma 1

Proof

Theorem 3.1

Proof

4 Illusions in Specific Network Classes

4.1 Graphs with Odd Degrees

Theorem 4.1

Proof

Corollary 1

Proof

4.2 2-Colorable Graphs

Lemma 2

Proof

Proposition 1

Proof

Proposition 2

Proof

4.3 Regular Graphs

Proposition 3 (simple cycles)

Proof

Proposition 4 (weak-majority-majority illusion in complete graphs)

Proof

Proposition 5 (majority-weak-majority illusion in complete graphs)

Proof

Proposition 6 ((weak-)majority-majority illusion in kk-regular graphs)

Proof

Example 1

Proposition 7 (majority-majority illusion in kk-regular graphs)

Proof

Example 2

Theorem 4.2

5 Conclusion and Outlook

Appendix 6 Constructive Argument for Majority-2-Colorability

Theorem Appendix 6.1 (attributed to [16])

Proof

Appendix 7 More General Illusions

Definition 4 (qq illusion)

Definition 5 (weak-qq illusion)

Definition 6 ((weak)-pp-(weak)-qq illusion)

Proposition 8 (pp-qq illusions on complete graphs)

Proof

Proposition 9

Proof

Proposition 10

Proof

Corollary 2

Corollary 3

Appendix 8 Algorithm for Finding a kk-Regular Graph with a Majority-Majority Illusion.

Appendix 8.1 Pseudocode

Appendix 8.2 Proof of Correctness of the Algorithm

Faster algorithm (only for nice values of nn and kk).

Appendix 9 Global Majority Logic

References

Definition 3 (weak majority $2$ -coloring)

Proposition 6 ((weak-)majority-majority illusion in $k$ -regular graphs)

Proposition 7 (majority-majority illusion in $k$ -regular graphs)

Definition 4 ( $q$ illusion)

Definition 5 (weak- $q$ illusion)

Definition 6 ((weak)- $p$ -(weak)- $q$ illusion)

Proposition 8 ( $p$ - $q$ illusions on complete graphs)

Appendix 8 Algorithm for Finding a $k$ -Regular Graph with a Majority-Majority Illusion.

Faster algorithm (only for nice values of $n$ and $k$ ).