Three Majority-Based Deterministic Dynamics for Three Opinions

Abstract

Phenomena from a variety of disciplines, including biology, computer science and sociology, can be modeled by graph dynamics in which nodes are associated with states and the node-state association changes in time. Although general k-state dynamics have been considered, most of the research in this area refers to binary dynamics especially as far as deterministic dynamics are regarded. In this paper 3-state deterministic dynamics are studied from the computational complexity perspective. A tractability result is proved when the third state is a state of neutrality, adopted by any node unable to establish a preference between the two remaining states. Subsequently, two hardness results are proved for two cases where each of the three states represents a semantically distinct state: the case in which a state change occurs in a node only if the most preferred state among the remaining two receives a suitable number of preferences, and the case in which a state change occurs in a node only if its current state lacks sufficient preferences and the most preferred state among the remaining two receives a suitable number of preferences. Finally, the relation of the last two results and a conjecture from the 1980s is discussed and it is shown that the conjecture is contradicted in both cases.

1. Introduction

Graph dynamics, in which nodes are assigned labels from a finite set and each node may change its label over time according to a predefined set of updating rules, generally depending on the environment the node belongs to, is a language that is well-suited to describe evolutionary phenomena across several application fields, ranging from automata networks [1,2,3] to general distributed computing models [4,5,6], from biological systems [7,8] to social networks.

Despite the wide applicability of the model, in order to focus the discussion, in what follows the topic will be mainly centered on the dynamical process underlying opinion formation in social networks like, for instance, the one occurring in political elections. Actually, understanding social influence, including opinion formation and evolution, has been a focus of research in sociology for a long time up until recent years [9,10,11,12,13,14]. Within this setting, a set of individuals are engaged with each other in stronger or weaker relations of friendship or enmity and such relations influence the opinion of each of them. Any individual’s tendency is to conform to his friends’ opinions and to go against his enemies ones by taking into consideration each relation’s strength. The rules according to which individuals change their opinion are usually referred to as dynamics and many questions of interest here concern the predictability of the social network behavior: will all the individuals ever reach consensus, or will they ever reach a qualified majority about the opinions on a topic, or will the network ever reach an equilibrium configuration (that is, an opinion assignment in which the opinion of every individual in the network will never change according to the specified dynamics), and similar.

The questions mentioned above have been largely studied in the literature, by introducing a wide variety of dynamics, both deterministic and randomized, mainly in the binary opinion setting; that is, when individuals have to choose between two alternatives. As far as randomization is concerned, it is proposed in different aspects: either in the graph generation process [15,16], or by assuming that individuals may decide not to change their opinion even if the conditions to do it are met [17,18,19,20], or, finally, in the definition of the opinion dynamics themselves. As a few examples, stochastic models for opinion dynamics when only friendship relations occur have been considered, starting with the French-DeGroot model [21,22] and continuing with the social impact model [23], the Voter model [24,25] and the majority rule model [26] among others. Some of these models have been subsequently adapted to the case of friendship/enmity relations [27,28,29,30].

Deterministic binary opinion dynamics models have also been proposed and analyzed [15,31,32,33,34,35,36]. In [37], a general framework for the definition of deterministic opinion dynamics is proposed. The dynamics model introduced in [34,35] encompasses both the deterministic majority rule and the underpopulation one [33]; this model has been subsequently extended to its maximum degree in [38]. Many of the cited papers consider the binary majority rule, one of the most intuitive and most widely studied deterministic dynamics models, in which an individual, at any step, adopts the opinion held by the majority of the individuals he is connected with.

However, binary opinion dynamics are only a simplification of real-world and practical settings, with examples ranging from everyday life issues (e.g., which product to buy, which set of political parties to vote for, and similar scenarios) to more technical issues. As an example, automata networks (both a computational model itself and a tool frequently used to model the behavior of biological networks) are graphs in which nodes are associated with states belonging to a (finite) set. The node-state association evolves at discrete time steps according to the transition maps each node is equipped with: at each time step, the node’s transition map specifies the node’s state at the next time step based on the states of its neighboring nodes.

Non-binary dynamics have been considered too, mainly with respect to randomized models and/or by simulation tools [39,40,41,42,43,44,45,46]. A short overview of some of the contents of these papers is now provided. In [47], the existence of non-binary asynchronous and non-deterministic dynamics allowing to maximize the diffusion of one target opinion starting with an assignment of the initial opinions of all individuals is studied, and the huge difference of such a property when considering binary and non-binary dynamics is highlighted. Indeed, while deciding if a dynamics leading to consensus exists is a problem in P in the binary opinions case, it is proved in [47] that the same problem is computationally intractable in the three opinions case. The consensus reachability of k-opinions probabilistic 3-majority dynamics (in which, at each time step, an individual and three of his friends are uniformly at random chosen and the individual takes the opinion held by the majority of the selected friends) is studied in [44]. In [46], a bound on the number of rounds necessary to reach plurality consensus with high probability with respect to the h-majority with k opinions problem is provided, where plurality consensus is the consensus on the opinion that is the most popular one at the beginning of the process. In [41,42,43], three-opinion dynamics are studied with the third opinion being a neutral one, adopted by an individual unable to choose one of the two, let us say, meaningful opinions according to the dynamics. In [48,49,50], a majority dynamic is proposed and studied, which will be briefly described later in this paper.

Paper Achievements

This paper studies the computational complexity of some prediction problems with respect to three majority-based three-opinions synchronous deterministic dynamics; simply speaking, a deterministic dynamics is synchronous when each individual checks for the opportunity of an opinion change at every time step. The primary motivation underlying this work is that of extending the study of binary deterministic dynamics—for which a considerable amount of effort has been spent—to the much less considered non-binary deterministic dynamics, with a special interest in the conjecture posed in [31]. Needless to say, this is a theoretical paper intended to be a first step to understanding the richness (in terms of complexity) of these, more general, models. In this respect, the specific dynamics here introduced play, first and foremost, the role of examples to serve as terms of comparison between tractable and intractable cases.

Regarding the prediction problems associated with synchronous deterministic dynamics, a couple of issues deserve attention. First, notice that, since the possible network opinion configurations are finite, then after a number of steps during which the network enters a set of distinct opinion configurations, a loop of opinion configurations starts. Hence, a natural solution to the above-mentioned prediction questions is to simulate the network evolution starting at a given opinion configuration until a configuration meeting the requests is met or a loop (of opinion configurations) is met.

In [2], it is shown that, from the computational complexity point of view, no better strategy than the simulation one can be devised for synchronous deterministic dynamics unless P = NC. Actually, it has been proved in a series of papers (since [31] and to [38]) that a wide variety of binary-opinion dynamics, including the majority rule one, exhibit evolution sets containing a polynomial number of configurations in the network size (so yielding polynomial-time prediction algorithms) and loops of size at most 2 when the relations between pair of individuals are symmetric. In [31], it has been conjectured that such a result (and, in fact, the same proof there proposed) could be generalized to general (non-binary) majority dynamics. The conjecture has been partially and positively solved since 1983 [48,49,50] by assuming that each individual exploits his own a priori ranking in order to break ties: this means that whenever the same amount of social pressure pushes an individual towards two or more opinions, he will take the one in the best position within his personal ranking.

The aim of this paper is to introduce a set of three-opinion majority-based dynamics assuming no a priori preferences of the individuals about the three opinions, and to study the above-mentioned topics with respect to them. The main difficulty in moving from binary-opinion majority dynamics to ternary-opinion majority dynamics when no a priori information is available lies in the definition of majority itself. Actually, the definition of binary-opinion majority dynamics is intuitive and straightforward, with only the possibility of choosing how to break ties (keeping or changing opinion in case of an equal number of relations pushing towards each of the alternatives, with both definitions leading to the same complexity behavior). On the other hand, there are plenty of majority definitions when more than two opinions are considered, among which the two natural and intuitive definitions of the relative and the conservative absolute majority dynamics deserve mention: the former definition makes an individual adopt the opinion having the maximum consensus among its connections, the latter one makes an individual change his opinion only if another alternative is held by more than half of his connections. Furthermore, while the absolute majority is not exposed to ties, when defining a relative majority, dynamics decisions have to be made about how to manage them. As an example, how should an individual having opinion a and two friends, one with opinion b and one with opinion c, behave?

In Section 3, we consider the same semantic difference between two opinions and the third one found in [41,42,43] (which, however, moved in a non-deterministic setting and/or by simulation tools); that is, two opinions are the meaningful ones and the third opinion stands for neutrality. The dynamics defined in this setting work as follows: if the same number of connections pushes an individual towards each of the two meaningful opinions, then the individual becomes neutral. This dynamics fits for modeling the inability of individuals to make a decision when neither (meaningful) alternative receives more support than the other among the set of individuals he is connected with. Using a coding plus simulation technique, it is proved here that the evolution set of a network evolving to the dynamics just outlined contains a polynomial number of configurations in the network size, even in the presence of positive/negative relations of different strengths. By the above discussion, it follows that the prediction problems are polynomial-time decidable in this setting.

In Section 4, a class of dynamics for three meaningful opinions is introduced that can be considered a generalization of the two-opinion underpopulation dynamics introduced in [33]. Specifically, in this case, the situation is modeled in which an individual changes his opinion only for a second opinion (if it exists) that receives among the set of individuals he is connected with at least as much support as a threshold value, and only if such second opinion is strictly more preferred than the third opinion. This corresponds to the behavior of somewhat conservative individuals whose tendency is to try to keep an opinion until a threshold amount of social pressure works in favor of a change. A simulation technique is used here to prove that (some of) the prediction problems are PSPACE-complete even in a network of equal-strength positive relations only. This implies that the number of configurations in the evolution set of a network evolving according to such dynamics is super-polynomial in the network size unless P = PSPACE. The same technique also allows us to prove that the length of the orbit loop is not upper bounded by 2, so disproving the conjecture in [31] in this case.

Finally, in Section 5, a different majority definition is introduced: the strongly anti-conservative three-opinion majority, which can be considered the counterpart of the conservative absolute majority. Within this new definition, for a parameter k, an individual keeps his opinion only if it is held by half plus k of his connections or if it is not the case, but the other two alternatives obtain the same number of preferences from his connections. This kind of dynamic lends itself to modeling volatile behaviors in which individuals tend to quickly change their current opinion and to enthusiastically embrace a different opinion as soon as they hear of it a suitable number of times. This phenomenon is typically amplified in social networks, where the huge amount of content and the speed at which it circulates make forming a solid and lasting opinion a hard task, so that individuals freely abandon opinions as soon as the media attention shifts. By suitably adjusting the proof in Section 4, the PSPACE-completeness of the prediction problems with respect to the strongly anti-conservative three-opinion majority dynamics and the existence of orbits having loop lengths greater than 2 are also proved in this case. The conjecture posed in [31] is then contradicted in this last case as well.

A summary of the results concerning the complexity of the prediction problems is shown in

Table 1. A summary of the state of the art. Graph topology and opinion dynamics names are described in bold text, while plain text is used for the achievements description. The term pseudo-polynomial is here used to mean that the orbit size is polynomial in the size of the graph and of its weight values (instead of its weight values size). We remark that a (pseudo-)polynomial orbit size implies the existence of a (pseudo-)polynomial-time algorithm deciding the prediction problems.

	Deterministic Synchronous Opinion Dynamics
	Binary Dynamics			Ternary Dynamics
	Threshold-based			Majority	Under-	Strongly
				with	population	anti-
				neutrality		conservative
						majority
	Local threshold-based
Graph	Under-	Majority
type	population
undirected	polynomial				Prediction problems
unsigned	orbit				are PSPACE-complete
unweighted	size [33]				⇒ existence of orbits
undirected		polynomial			of super-polynomial size
signed		orbit			and existence of orbit loops
unweighted		size [35]			with length $>2$
undirected	pseudo-polynomial orbit size			pseudo-	Section 4 and Section 5
signed	[38]			polynomial
weighted				orbits size
				Section 3
directed	Prediction problems are PSPACE-complete [34,35]
unsigned	(⇒ super-polynomial orbit size)
unweighted

.

2. Preliminary Definitions and Notations

Let V be a finite set of entities and let $\mathsf{\Sigma }$ be a finite set of labels or, equivalently, opinions. A $\mathsf{\Sigma }$-configuration of V is a labeling function $\omega :V\to \mathsf{\Sigma }$. A $\mathsf{\Sigma }$-configuration of V evolves into a different $\mathsf{\Sigma }$-configuration of G according to a set of rules defined by a dynamics. Formally, a $\mathsf{\Sigma }$-dynamics $\mathbf{d}$ is a functional $\mathbf{d}:{\mathsf{\Sigma }}^V\to {\mathsf{\Sigma }}^V$, which specifies for a given $\mathsf{\Sigma }$-configuration $\omega$ of V at some time step t, the opinion configuration $\mathbf{d}\left(\omega \right)$ of V at step $t+1$.

Generally speaking, elements of V influence each other based on the set E of (binary) relations among them, so that $\mathbf{d}$ actually depends on the graph $G=(V,E)$. Hence, $\omega$ and $\mathbf{d}$ are usually referred to as, respectively, a configuration of G and a dynamics for G (instead of a configuration of V and a dynamics for V); the evolution of a configuration $\omega$ of G according to a dynamics $\mathbf{d}$ for a graph $G=(V,E)$ is a configuration of G that will be denoted as $\mathbf{d}(G,\omega )$. A dynamics $\mathbf{d}$ is deterministic and synchronous if ${\mathbf{d}}_G\left(\omega \right)={\mathbf{d}}_G\left({\omega }^{\prime }\right)$ whenever $\omega ={\omega }^{\prime }$ and independently of the time the funcional is applied. In the remainder of this paper, only deterministic synchronous dynamics will be considered.

For any initial $\mathsf{\Sigma }$-configuration of a graph G, an opinion dynamics $\mathbf{d}$ for G implicitly describes the evolution set ${\mathcal{E}}_{\mathbf{d}}(G,\omega )$ of G starting at $\omega$; that is, an infinite discrete dynamic process during which, at each time step, each node might change the opinion it held at the previous step. Specifically, ${\mathcal{E}}_{\mathbf{d}}(G,\omega )=\{{\omega }_0=\omega ,{\omega }_1,{\omega }_2,\dots \}$, where for $t=1,2,\dots$, ${\omega }_t=\mathbf{d}(G,{\omega }_{t-1})={\mathbf{d}}^t(G,{\omega }_0)$.

Since $\mathbf{d}$ is deterministic and synchronous then ${\mathcal{E}}_{\mathbf{d}}(G,\omega )$ is made of an (eventually empty) initial sequence of distinct configurations that will never appear in ${\mathcal{E}}_{\mathbf{d}}(G,\omega )$ again, called the transient $\tau (G,\omega )$ of ${\mathcal{E}}_{\mathbf{d}}(G,\omega )$, followed by a (eventually unit size) sequence of distinct configurations that will repeat forever, called the loop $\ell (G,\omega )$ of ${\mathcal{E}}_{\mathbf{d}}(G,\omega )$.

The orbit of a configuration $\omega$ of a graph G according to a dynamics $\mathbf{d}$ (in short, the $\mathbf{d}$-orbit of G at $\omega$) is ${\mathcal{O}}_{\mathbf{d}}(G,\omega )=\tau (G,\omega )+\ell (G,\omega )$; that is, ${\mathcal{O}}_{\mathbf{d}}(G,\omega )=\langle {\omega }_0=\omega ,{\omega }_1,\dots ,{\omega }_T\rangle$ where $T\le {\left[2(k+1)\right]}^{\left|V\right|}$ and

• For $t=1,\dots ,T$, ${\omega }_t=\mathbf{d}(G,{\omega }_{t-1})$, and
• There exists $h\le T$ such that ${\omega }_{T+ki}={\omega }_{h+i-1}$ for $i=1,\dots ,T-h+1$ and $k>0$.

The opinion configuration ${\omega }_T$ is an equilibrium configuration if $h=T$.

In what follows, the sequence ${\mathcal{O}}_{\mathbf{d}}(G,\omega )$ will be considered a set when required.

Let $\mathsf{\Pi }:V^{\mathsf{\Sigma }}\to \{\mathbf{true},\mathbf{false}\}$ be a polynomial-time checkable in $\left|V\right|$ predicate over the set of $\mathsf{\Sigma }$-configuration of a graph $G=(V,E)$. The $\mathsf{\Pi }$-Reachability[$\mathbf{d}$] problem consists in deciding, given graph G in a $\mathsf{\Sigma }$-configuration $\omega$, if a configuration ${\omega }^{\prime }\in {\mathcal{O}}_{\mathbf{d}}(G,\omega )$ exists such that $\mathsf{\Pi }\left({\omega }^{\prime }\right)=\mathbf{true}$; this formalizes the informal idea of prediction problem used throughout the introduction.

In the next sections, directed and undirected graphs will be considered, and a major role in the specific dynamics defined there and in the analysis of the results presented will be played by the number of neighbors of each node in the graph to which it belongs. For the sake of completeness, some related definitions will now be provided. Let $G=(V,E)$ be an undirected graph; $N\left(u\right)=\{v\in V:(u,v)\in E\}$ is the set of neighbors in G of node $u\in V$ and the degree of u in G is $\delta \left(u\right)=|N_G\left(u\right)|$. Let $G=(V,A)$ be an undirected graph; the set of in-neighbors in G of a node $u\in V$ is the set $N^{in}\left(u\right)=\{v\in V:(v,u)\in A\}$ (taking into account the arcs entering u) and the in-degree of u in G is ${\delta }^{in}\left(u\right)=\left|N_G^{in}\left(u\right)\right|$; the set of out-neighbors in G of a node $u\in V$ is the set $N^{out}\left(u\right)=\{v\in V:(u,v)\in A\}$ (taking into account the arcs exiting u) and the out-degree of u in G is ${\delta }^{out}\left(u\right)=\left|N_G^{in}\left(u\right)\right|$. When several graphs are considered, subscripts will be used to denote which graph the notation is applied to: for instance if $G=(V,E)$ is an undirected graph a $H=(V,A)$ is a directed graph, both defined over the same set V of nodes, for $u\in V$ the notations ${\delta }_G\left(u\right)$ and ${\delta }_H^{in}\left(u\right)$ shall refer, respectively, to the degree of u in G and to the in-degree of u in H.

3. Majority or Neutrality in Weighted Signed Undirected Graphs

Let $G=(V,E,\lambda )$ be a weighted signed graph with $\lambda :E\to \mathbb{Z}$, and let $\omega :V\to \{-1,0,1\}$ be a $\{-1,0,1\}$-configuration of G. In this section, a dynamics ${\mathbf{d}}_{MN}$ is considered in which each node takes opinion 0 if and only if the number of its neighbors having opinion $-1$ is equal to the number of its neighbors having opinion 1. Formally, the configuration ${\omega }_1={\mathbf{d}}_{MN}(G,\omega )$ is defined as follows: for every $u\in V$

$${\omega }_1\left(u\right)=\left\{\begin{array}{cc}-1\hfill & \mathrm{if}\sum _{v\in N\left(u\right)}\lambda (u,v)\omega \left(v\right)<0,\hfill \\ 0\hfill & \mathrm{if}\sum _{v\in N\left(u\right)}\lambda (u,v)\omega \left(v\right)=0,\hfill \\ 1\hfill & \mathrm{if}\sum _{v\in N\left(u\right)}\lambda (u,v)\omega \left(v\right)>0.\hfill \end{array}\right.$$

We now show how to simulate the evolution of G according to ${\mathbf{d}}_{MN}$ by the evolution of a related weighted signed graph $\overline{G}=(\overline{V},\overline{E},\overline{\lambda })$ according to a two-opinions related dynamics $\overline{d}$. Intuitively, in order to describe three opinions by a set of just two opinions a (sort of) binary coding of the set $\{-1,0,1\}$ will be defined together with a new graph $\overline{G}$ in which each node of G is “doubled”; that is, each node in G is mapped to a pair of nodes in $\overline{G}$: the goal is accommodating things in order to have the opinion of each of the nodes constituting a pair in $\overline{G}$ being one (let us say) bit of the coding of the opinion of the corresponding node in G.

The graph $\overline{G}$ associated to G is defined as follows (see Figure 1)

• $\overline{V}=V_1\cup V_2$ with $V_i=\{v_i:v\in V\}$ for $i=1,2$; that is, for each $v\in V$, $\overline{V}$ contains the pair of nodes $v_1$ and $v_2$;
• For every $(u,v)\in E$, $\overline{E}$ contains the edges $(u_1,v_1)$, $(u_1,v_2)$, $(u_2,v_1)$ and $(u_2,v_2)$ with $\overline{\lambda }(u_1,v_1)=\overline{\lambda }(u_2,v_2)=\lambda (u,v)$ and $\overline{\lambda }(u_1,v_2)=\overline{\lambda }(u_2,v_1)=-\lambda (u,v)$.

Let $\overline{V_1\times V_2}=\{(x,y)\in V_1\times V_2:\exists u\in V[x=u_1\wedge y=u_2]\}$; the procedure deriving $\overline{G}$ from G naturally induces a bijection $\mathcal{B}:V\to \overline{V_1\times V_2}$. Let $\eta :\{-1,0,1\}\to \left\{\right(-1,1),(1,-1),(1,1\left)\right\}$ be the encoding of $\{-1,0,1\}$ such that

$$\begin{array}{ccc}\eta (-1)\hfill & =\hfill & (1,-1),\hfill \\ \eta \left(0\right)\hfill & =\hfill & (1,1),\hfill \\ \eta \left(1\right)\hfill & =\hfill & (-1,1).\hfill \end{array}$$

A $\{-1,1\}$-configuration $\overline{\omega }$ of $\overline{G}$ encodes a $\{-1,0,1\}$-configuration $\omega$ of G if, for every $v\in V$, it holds that $(\overline{\omega }\left(v_1\right),\overline{\omega }\left(v_2\right))=\eta \left({\mathcal{B}}^{-1}\left(v\right)\right)$. Less formally, $\overline{\omega }$ encodes $\omega$ if, for every $v\in V$,

• $\overline{\omega }\left(v_1\right)=1$ and $\overline{\omega }\left(v_2\right)=-1$ if and only if $\omega \left(v\right)=-1$;
• $\overline{\omega }\left(v_1\right)=-1$ and $\overline{\omega }\left(v_2\right)=1$ if and only if $\omega \left(v\right)=1$;
• $\overline{\omega }\left(v_1\right)=1$ and $\overline{\omega }\left(v_2\right)=1$ if and only if $\omega \left(v\right)=0$.

Notice that if $\overline{\omega }$ encodes $\omega$ then it never happens that $\overline{\omega }\left(v_1\right)=-1$ and $\overline{\omega }\left(v_2\right)=-1$ for any $v\in V$.

The dynamics $\overline{\mathbf{d}}$ for $\overline{G}$ is then defined by setting, for every $x\in \overline{V}$ and for every configuration $\omega$ of $\overline{G}$,

$${\omega }_1\left(x\right)=\overline{\mathbf{d}}(\overline{G},\omega )\left(x\right)=\left\{\begin{array}{cc}-1\hfill & \mathrm{if}\sum _{y\in \overline{N}\left(x\right)}\overline{\lambda }(x,y)\overline{\omega \left(y\right)}<0,\hfill \\ 1\hfill & \mathrm{if}\sum _{y\in \overline{N}\left(x\right)}\overline{\lambda }(u,v)\overline{\omega \left(y\right)}\ge 0\hfill \end{array}\right.$$

(1)

where $\overline{N}\left(x\right)$ is the set of neighbors of x in $\overline{G}$.

The next lemma proves that if G is in any $\{-1,0,1\}$-configuration $\omega$ and $\overline{G}$ is in the configuration $\overline{\omega }$ encoding $\omega$, then G ${\mathbf{d}}_{MN}$-evolves much the same way as $\overline{G}$ $\overline{\mathbf{d}}$-evolves (see Figure 2).

Lemma 1. 

Let $G=(V,E,\lambda )$ be an undirected signed weighted graph and let ω be a $\{-1,0,1\}$-configuration of G. If the $\{-1,1\}$-configuration $\overline{\omega }$ of $\overline{G}$ encodes ω then ${\overline{\omega }}_1=\overline{\mathbf{d}}(\overline{G},\overline{\omega })$ encodes ${\omega }_1={\mathbf{d}}_{MN}(G,\omega )$.

Proof. 

For $i=1,2$ and for any $(u,v)\in E$, the contribution of u to $v_i$ at ${\overline{\omega }}^{\prime }$ is defined as $\chi (v_i,u,\overline{\omega })=\overline{\lambda }(u_1,v_i)\overline{\omega }\left(u_1\right)+\overline{\lambda }(u_2,v_i)\overline{\omega }\left(u_2\right)$; hence, the contribution of u to $v_1$ at ${\overline{\omega }}^{\prime }$ is $\chi (v_1,u,\overline{\omega })=\lambda (u,v)[\overline{\omega }\left(u_1\right)-\overline{\omega }\left(u_2\right)]$ and the contribution of u to $v_2$ at ${\overline{\omega }}^{\prime }$ is $\chi (v_2,u,\overline{\omega })=\lambda (u,v)[\overline{\omega }\left(u_2\right)-\overline{\omega }\left(u_1\right)]$. Notice that, since $\overline{\omega }$ encodes $\omega$,

• If $\omega \left(u\right)=-1$ then $\chi (v_1,u,\overline{\omega })=2\lambda (u,v)$ and $\chi (v_2,u,\overline{\omega })=-2\lambda (u,v)$,
• If $\omega \left(u\right)=0$ then $\chi (v_1,u,\overline{\omega })=0$ and $\chi (v_2,u,\overline{\omega })=0$,
• If $\omega \left(u\right)=1$ then $\chi (v_1,u,\overline{\omega })=-2\lambda (u,v)$ and $\chi (v_2,u,\overline{\omega })=2\lambda (u,v)$;

that is, in any case $\chi (v_1,u,\overline{\omega })=-2\lambda (u,v)\omega \left(u\right)$ and $\chi (v_2,u,\overline{\omega })=2\lambda (u,v)\omega \left(u\right)$. As a consequence, for each $v\in V$,

$$\sum _{x\in \overline{N}\left(v_1\right)}\overline{\lambda }(x,v_1)\overline{\omega \left(u\right)}\begin{array}{cc}=& \sum _{u\in N\left(v\right)}[\overline{\lambda }(u_1,v_1)\overline{\omega }\left(u_1\right)+\overline{\lambda }(u_2,v_1)\overline{\omega }\left(u_2\right)]\hfill \\ =& \sum _{u\in N\left(v\right)}\chi (v_1,u,\overline{\omega })\hfill \\ =& -2\sum _{u\in N\left(v\right)}\lambda (u,v)\omega \left(u\right)\hfill \end{array}$$

and, similarly,

$$\sum _{x\in \overline{N}\left(v_2\right)}\overline{\lambda }(x,v_2)\overline{\omega \left(u\right)}=2\sum _{u\in N\left(v\right)}\lambda (u,v)\omega \left(u\right).$$

This implies that:

• If ${\omega }_1\left(v\right)=-1$ then ${\sum }_{u\in N\left(v\right)}\lambda (u,v)\omega \left(u\right)<0$ and, hence, ${\sum }_{x\in \overline{N}\left(v_1\right)}\overline{\lambda }(x,v_1)\overline{\omega \left(u\right)}>0$ and ${\sum }_{x\in \overline{N}\left(v_2\right)}\overline{\lambda }(x,v_1)\overline{\omega \left(u\right)}<0$ so that ${\overline{\omega }}_1\left(v_1\right)=1$ and ${\overline{\omega }}_1\left(v_2\right)=-1$;
• If ${\omega }_1\left(v\right)=0$ then ${\sum }_{u\in N\left(v\right)}\lambda (u,v)\omega \left(u\right)=0$ and, hence, ${\sum }_{x\in \overline{N}\left(v_1\right)}\overline{\lambda }(x,v_1)\overline{\omega \left(u\right)}=0$ and ${\sum }_{x\in \overline{N}\left(v_2\right)}\overline{\lambda }(x,v_1)\overline{\omega \left(u\right)}=0$ so that ${\overline{\omega }}_1\left(v_1\right)=1$ and ${\overline{\omega }}_1\left(v_2\right)=1$;
• If ${\omega }_1\left(v\right)=1$ then ${\sum }_{u\in N\left(v\right)}\lambda (u,v)\omega \left(u\right)>0$ and, hence, ${\sum }_{x\in \overline{N}\left(v_1\right)}\overline{\lambda }(x,v_1)\overline{\omega \left(u\right)}<0$ and ${\sum }_{x\in \overline{N}\left(v_2\right)}\overline{\lambda }(x,v_1)\overline{\omega \left(u\right)}>0$ so that ${\overline{\omega }}_1\left(v_1\right)=-1$ and ${\overline{\omega }}_1\left(v_2\right)=1$.

The assertion is so proved. □

The above lemma proves that there exists a one-to-one correspondence between the $\{-1,0,1\}$-configurations met by a graph G during its evolution according to ${\mathbf{d}}_{MN}$ starting at configuration $\omega$ and the $\{-1,1\}$-configurations met by $\overline{G}$ during its evolution according to $\overline{\mathbf{d}}$ starting at configuration $\overline{\omega }$ encoding $\omega$: that is, if $\overline{\omega }$ encodes $\omega$ then ${\mathcal{O}}_{{\mathbf{d}}_{MN}}(G,\omega )=\langle {\omega }_0=\omega ,{\omega }_1,\dots ,{\omega }_T\rangle$ and ${\mathcal{O}}_{\overline{\mathbf{d}}}(\overline{G},\overline{\omega })=\langle {\overline{\omega }}_0=\overline{\omega },{\overline{\omega }}_1,\dots ,{\overline{\omega }}_T\rangle$ with ${\overline{\omega }}_t$ encoding ${\omega }_t$ for $t=1,\dots ,T$. Hence, the following theorem follows.

Theorem 1. 

Let $G=(V,E,\lambda )$ be an undirected signed weighted graph and let ω be a $\{-1,0,1\}$-configuration of G. If the configuration $\overline{\omega }$ of $\overline{G}$ encodes ω then $|{\mathcal{O}}_{{\mathbf{d}}_{MN}}(G,\omega )|=|{\mathcal{O}}_{\overline{\mathbf{d}}}(\overline{G},\overline{\omega })|$.

The dynamics $\overline{\mathbf{d}}$ defined by (1) is a particular case of a threshold-based dynamics, introduced in [38] for signed weighted graphs. A threshold-based dynamics ${\mathbf{d}}_{{\theta }^{+},{\theta }^{-}}$ is ruled by a pair $({\theta }^{+},{\theta }^{-})$ of functions such that ${\theta }^{+}:V\to \mathbb{N}$ and ${\theta }^{-}:V\to \mathbb{N}$, and it is defined as follows: for any $\{0,1\}$-configuration $\omega :V\to \{0,1\}$ of G and for each $u\in V$,

$${\omega }_1\left(u\right)={\mathbf{d}}_{{\theta }^{+},{\theta }^{-}}(G,\omega )=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\omega \left(u\right)=1\mathrm{and}{P}_1(u,\omega )\ge {\theta }^{+}\left(u\right)\hfill \\ \hspace{1em}& \mathrm{or}\omega \left(u\right)=-1\mathrm{and}{P}_1(u,\omega )\ge {\theta }^{-}\left(u\right),\hfill \\ -1\hfill & \mathrm{if}\omega \left(u\right)=1\mathrm{and}{P}_1(u,\omega )<{\theta }^{+}\left(u\right)\hfill \\ \hspace{1em}& \mathrm{or}\omega \left(u\right)=-1\mathrm{and}{P}_1(u,\omega )<{\theta }^{-}\left(u\right),\hfill \end{array}\right.$$

where $P_1(u,\omega )={\sum }_{v\in N\left(u\right):\lambda (u,v)\omega \left(v\right)>0}\left|\lambda (u,v)\right|$. Notice that the dynamics $\overline{\mathbf{d}}$ is a threshold-based dynamics such that, for every $u\in V$, ${\theta }^{+}\left(u\right)={\theta }^{-}\left(u\right)={\displaystyle \frac{{\sum }_{v\in N\left(u\right)}\left|\lambda (u,v)\right|}{2}}$. In [38], it is proved that, for every pair of threshold functions ${\theta }^{+}$ and ${\theta }^{-}$, for every signed weighted graph $G=(V,E,\lambda )$, with ${\sum }_{(u,v)inE}\left|\lambda (u,v)\right|=\Lambda$, and for every $\{-1,1\}$-configuration $\omega$ of G, $|{\mathcal{O}}_{{\mathbf{d}}_{{\theta }^{+},{\theta }^{-}}}(G,\omega )|\le 2\left(4\Lambda +8\left|V\right|\Lambda +8\left|V\right|\left|E\right|+2\left|V\right|+2\right)$; that is, the orbit size is polynomial in the size of G and its edge weight values.

Hence, from the above discussion, as a consequence of Theorem 1, and by noticing that the number of nodes in $\overline{G}$ is twice the number of nodes in G and that the number of edges in $\overline{G}$ is twice the number of edges in G, the next theorem directly follows:

Theorem 2. 

For any signed weighted undirected graph G and for every $\{-1,0,1\}$-configuration ω for G, the size of ${\mathcal{O}}_{{\mathbf{d}}_{MN}}(G,\omega )$ is polynomially bounded in the size of G and in its edge weight values.

4. Generalizing the Underpopulation Rule

Let $G=(V,E)$ be an undirected, unsigned, unweighted graph. The underpopulation dynamics ${\mathbf{d}}_{p,q}$, with $p,q\in \mathbb{N}$, introduced in [33], is governed by the pair $(p,q)\in \mathbb{N}\times \mathbb{N}$ of parameters and is defined as follows: for any $\{0,1\}$-configuration $\omega :V\to \{0,1\}$ of G and for each $u\in V$,

$${\omega }_1\left(u\right)={\mathbf{d}}_{p,q}(G,\omega )=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\omega \left(u\right)=1\mathrm{and}{P}_1(u,\omega )\ge p\hfill \\ \hspace{1em}& \mathrm{or}\omega \left(u\right)=-1\mathrm{and}{P}_1(u,\omega )\ge q,\hfill \\ -1\hfill & \mathrm{if}\omega \left(u\right)=1\mathrm{and}{P}_1(u,\omega )<p\hfill \\ \hspace{1em}& \mathrm{or}\omega \left(u\right)=-1\mathrm{and}{P}_1(u,\omega )<q,\hfill \end{array}\right.$$

where $P_1(u,\omega )=\left|\{v\in N\left(u\right):\omega \left(v\right)=1\}\right|$. It is worthwhile to remark that an underpopulation dynamics ${\mathbf{d}}_{p,q}$ for a graph G is actually a threshold-based dynamics ${\mathbf{d}}_{{\theta }^{+},{\theta }^{-}}$ such that ${\theta }^{+}\left(u\right)=p$ and ${\theta }^{-}\left(u\right)=q$ for every node u in G; that is, an underpopulation dynamics is a threshold-based dynamics such that the threshold functions are constant functions.

Aiming to generalize underpopulation binary dynamics to the three opinions case, an underpopulation $\{0,1,2\}$-opinion dynamics ${\mathbf{d}}_{{\gamma }_0,{\gamma }_1,{\gamma }_2}$ ruled by the parameters ${\gamma }_0,{\gamma }_1,{\gamma }_2\in \mathbb{Q}$ is here defined as follows: for any (undirected) graph $G=(V,E)$ and for any $\{0,1,2\}$-configuration $\omega$ for G, the configuration ${\omega }_1={\mathbf{d}}_{{\gamma }_0,{\gamma }_1,{\gamma }_2}(G,\omega )$ is defined so that, for each $u\in V$,

$${\omega }_1\left(u\right)=\left\{\begin{array}{cc}i\in \{0,1,2\}-\left\{\omega \right(u\left)\right\}\hfill & \mathrm{if}{P}_i(u,\omega )>{\gamma }_i\hfill \\ \hfill & \mathrm{and}{P}_j(u,\omega )\le {\gamma }_i\hfill \\ \hfill & \mathrm{with}j\in \{0,1,2\}-\{i,\omega \left(u_1\right)\},\hfill \\ \omega \left(u\right)\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(2)

where, for $i=0,1,2$, $P_i(u,\omega )=\left|\{v\in N\left(u\right):\omega \left(v\right)=i\}\right|$.

In this section, the $\mathsf{\Pi }$-Reachability[${\mathbf{d}}_{{\gamma }_0,{\gamma }_1,{\gamma }_2}$] problem will be proved to be PSPACE-complete for some choices of $\mathsf{\Pi }$. Notice that membership to PSPACE directly follows by recalling the intuitive solution to any $\mathsf{\Pi }$-Reachability problem; that is, simulating the graph evolution starting at a given opinion configuration until a configuration meeting the requests or a loop of configurations is met. Hence, the remaining of the section is devoted to proving the hardness issue.

Let ${\mathbf{d}}_M$ be the binary deterministic majority dynamics that, for an unweighted unsigned directed graph $G=(V,A)$, is defined as follows: if ${\omega }_0:V\to \{0,1\}$ is an opinion configuration for G then ${\omega }_1={\mathbf{d}}_M(G,{\omega }_0)$ is such that, for every $u\in V$,

$${\omega }_1\left(u\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}{\omega }_0\left(u\right)=0\mathrm{and}{P}_0(u,{\omega }_0)\ge {\displaystyle \frac{{\delta }_G^{in}\left(u\right)}{2}}\hfill \\ \hspace{1em}& \mathrm{or} \mathrm{if}{\omega }_0\left(u\right)=1\mathrm{and}{P}_0(u,{\omega }_0)>{\displaystyle \frac{{\delta }_G^{in}\left(u\right)}{2}},\hfill \\ 1\hfill & \mathrm{if}{\omega }_0\left(u\right)=1\mathrm{and}{P}_1(u,{\omega }_0)\ge {\displaystyle \frac{{\delta }_G^{in}\left(u\right)}{2}}\hfill \\ \hspace{1em}& \mathrm{or} \mathrm{if}{\omega }_0\left(u\right)=0\mathrm{and}{P}_1(u,{\omega }_0)>{\displaystyle \frac{{\delta }_G^{in}\left(u\right)}{2}},\hfill \end{array}\right.$$

where ${\delta }_G^{in}\left(u\right)=\left|\{v\in V:(v,u)\in A\}\right|$ is the number of in-neighbors of u and, for $i=1,2$, $P_i(u,{\omega }_0)=\left|\{v\in V:(v,u)\in A\wedge {\omega }_0\left(v\right)=i\}\right|$ is the number of in-neighbors of u having opinion 0 in ${\omega }_0$.

Let $L\in {\{0,1\}}^{*}$ be a PSPACE-complete language and let $\mathcal{T}$ be the deterministic Turing machine deciding L in space $O\left(n^k\right)$ for some constant $k\in \mathbb{N}$. In [35], for any $x\in {\{0,1\}}^{*}$, an unsigned directed graph $G_x=(V_x,A_x)$ and an opinion configuration ${\omega }_x$ have been designed such that $|V_x{|\in O(\left|x\right|}^k)$ and the opinion evolution according to the deterministic majority dynamics of $G_x$ starting at ${\omega }_x$ reaches a given final configuration ${\omega }_A$ if and only if the computation $\mathcal{T}\left(x\right)$ ends in $q_A$.

It turns out that the in-degree and out-degree of nodes in $G_x$ are all functions of $\mathcal{T}$ only and they are independent from x; that is, they are constant.

We now show how to simulate the $\{0,1\}$-opinion evolution of $G_x$ according to the deterministic majority dynamics by the $\{0,1,2\}$-opinion evolution of a related unsigned weighted undirected graph ${\overline{G}}_x=({\overline{V}}_x,{\overline{E}}_x)$ according to an underpopulation dynamics.

Denote as $N_x^{in}\left(u\right)=\{v\in V_x:(v,u)\in A_x\}$ the set of in-neighbors of u in $G_x$, as $N_x^{out}\left(u\right)=\{v\in V_x:(u,v)\in A_x\}$ the set of out-neighbors of u in $G_x$, as ${\delta }_x^{in}\left(u\right)=\left|N_x^{in}\left(u\right)\right|$ the in-degree of u in $G_x$, and as ${\delta }_x^{out}\left(u\right)=\left|N_x^{out}\left(u\right)\right|$ the out-degree of u in $G_x$.

The procedure to get ${\overline{G}}_x=({\overline{V}}_x,{\overline{E}}_x)$ from $G_x$ first transforms $G_x$ into a graph $G_x^{″}$ such that all nodes in $G_x^{″}$ have the same out-degree and odd in-degree.

(1)

The in-degree of all nodes in $G_x$ is made odd: this is accomplished by adding a self-loop $(u,u)$ to $A_x$ for every even in-degree node $u\in V_x$. Denote as $G_x^{\prime }=(V_x,A_x^{\prime })$ the so derived graph and as ${\delta }_{G^{\prime }}^{in}$ and ${\delta }_{G^{\prime }}^{out}$ the in-degree and out-degree functions in $G^{\prime }$. It can be easily verified that, for any $\omega :V_x\to \{0,1\}$, ${\mathbf{d}}_M(G_x^{\prime },\omega )={\mathbf{d}}_M(G_x,\omega )$.

(2)

The out-degree of all nodes in $G_x^{\prime }$ is made equal: by denoting as ${\delta }_{max}^{out}$ the maximum out-degree of nodes in $G_x^{\prime }$, for every $u\in V_x$, ${\delta }_{max}^{out}-{\delta }_x^{out}\left(u\right)$ dummy out-neighbors are added to u. Denote as D the set of dummy nodes, as $G_x^{″}=(V_x\cup D,A_x^{″})$ the resulting graph, as ${\delta }_{G_x^{″}}^{in}={\delta }_x^{in}$ its in-degree function; by construction ${\delta }_{max}^{out}$ is the out-degree of all its nodes.

(3)

Notice that, since ${\delta }_{max}^{out}$ has constant size (with respect to x), then a constant number of dummy out-neighbors is added to any node in $V_x$; hence, $\left|D\right|\in O\left(\right|V_x\left|\right)$ and, finally, $|V_x\cup D|\in O(|V_x\left|\right)$.

The graph $G_x^{″}$ is then used to derive ${\overline{G}}_x=({\overline{V}}_x,{\overline{E}}_x)$ (see Figure 3). Intuitively, each arc $(u,v)$ in $G_x^{″}$ must be transformed into a structure ensuring that, at each step, the opinion of u is communicated to v but the opinion of v is not communicated to u. This is accomplished by a) matching each node u of $G_x^{″}$ to a pair of nodes $u_1,u_2$ in ${\overline{G}}_x$ and b) matching each arc $(u,v)$ of $G_x^{″}$ to a pair of nodes $u{v}_1,u{v}_2$ in ${\overline{G}}_x$, both connected to both $u_1$ and $u_2$ and with $u{v}_1$ connected to $v_1$ and $u{v}_1$ connected to $v_1$. In this way, if $u_1$ and $u_2$ have the same opinion, then $u{v}_i$ is more influenced by it than by the opinion of $v_i$ (for $i=1,2$). Next, in order to forbid that $u_i$ is influenced by the opinion of $u{v}_1$ and $u{v}_2$: c) two sets, $B_u$ and $C_u$, of never changing opinion nodes are connected to $u_1$ and $u_2$, and d) two sets, $Z_{uv}$ and $R_{uv}$, of nodes whose opinions swing, respectively, between 0 and 2 and between 1 and 2 are connected to $u{v}_1$ and $u{v}_2$. Finally, $Z_{uv}$ and $R_{uv}$ are made part of two complete bipartite graphs, respectively, over the node set $Z_{uv}\times Y_{uv}$ and over the node set $R_{uv}\times S_{uv}$, so that their swinging behavior is ensured.

Formally, the procedure to derive ${\overline{G}}_x=({\overline{V}}_x,{\overline{E}}_x)$ from $G_x^{″}$ is described in the following.

• Define $k_{max}=max\{{\delta }_{G_x^{\prime \prime }}^{in}\left(u\right):u\in V_x\}+2{\delta }_{max}^{out}$, and, for each node $u\in V_x$, define $k_u={\displaystyle \frac{3k_{max}-[{\delta }_{G_x^{″}}^{in}\left(u\right)+2{\delta }_{max}^{out}]}{2}}$ (recall that both $k_{max}$ and ${\delta }_{G_x^{″}}^{in}\left(u\right)$ are odd).
• For each node $u\in V_x$, ${\overline{G}}_x$ contains the pair of nodes $u_1$ and $u_2$ and the pair of sets of nodes $B_u=\{b_u^1,\dots ,b_u^{2k_u}\}$ and $C_u=\{c_u^1,\dots ,c_u^{2k_u}\}$. For $i=1,\dots ,k_u$, $(u_1,b_u^{2i-1})\in {\overline{E}}_x$, $(u_2,b_u^{2i})\in {\overline{E}}_x$, $(u_1,c_u^{2i-1})\in {\overline{E}}_x$, $(u_2,b_c^{2i})\in {\overline{E}}_x$.
• For each dummy node $d\in D$, ${\overline{G}}_x$ contains the pair of nodes $d_1$ and $d_2$.
• For each arc $(u,v)\in A_x^{″}$, ${\overline{G}}_x$ contains the pair of nodes $u{v}_1$ and $u{v}_2$ and two complete bipartite graphs $Z_{uv}\times Y_{uv}$ and $R_{uv}\times S_{uv}$, with $Z_{uv}=\{z_{uv}^1,\dots ,z_{uv}^{4k_{max}}\}$, $Y_{uv}=\{y_{uv}^1,\dots ,y_{uv}^{4k_{max}}\}$, $R_{uv}=\{r_{uv}^1,\dots ,r_{uv}^{4k_{max}}\}$ and $S_{uv}=\{s_{uv}^1,\dots ,s_{uv}^{4k_{max}}\}$. Finally, for $i=1,\dots ,2k_{max}$, $(u{v}_1,x_{uv}^{2i-1})$, $(u{v}_1,r_{uv}^{2i-1})$, $(u{v}_2,x_{uv}^{2i})$, $(u{v}_2,r_{uv}^{2i})$ are edges in ${\overline{G}}_x$.
• For each $(u,v)\in A_x^{″}$, $(u_1,u{v}_1)$, $(u_2,u{v}_1)$, $(u{v}_1,v_1)$ and $(u_1,u{v}_2)$, $(u_2,u{v}_2)$, $(u{v}_2,v_2)$ are edges in ${\overline{G}}_x$.
• About node degrees.

  -

  All nodes $u_1,u_2\in {\overline{V}}_x$ such that $u\in V_x$ have degree $3k_{max}$ in ${\overline{G}}_x$.

  -

  Each dummy node $d\in D$ has degree 1 in ${\overline{G}}_x$ as well as each node in $B={\cup }_{u\in V_x}{B}_u$ and each node in $C={\cup }_{u\in V_x}{C}_u$.

• About the size of ${\overline{V}}_x$ and the space complexity of the construction.

  -

  For every $u\in V_x$, $|B_u|=|C_u|=2k_u\le k_{max}$.

  -

  For each arc $(u,v)\in A_x^{\prime \prime }$, $|Z_{uv}|=|Y_{uv}|=|R_{uv}|=|S_{uv}|=4k_{max}$.

  -

  Hence, since $k_{max}$ is constant (with respect to x) and since each node in $V_x\cup D$ is replaced by two nodes in ${\overline{V}}_x$, it follows that $|{\overline{V}}_x|\in O(|V_x\left|\right)$. This proves that deriving ${\overline{G}}_x$ from $G_x$ requires $O\left(\right|x{|}^k)$ space.

Denote as ${\overline{\delta }}_x$ the degree function in ${\overline{G}}_x$.

The dynamics according to which opinions in ${\overline{G}}_x$ evolve is the underpopulation one ${\mathbf{d}}_{\overline{\gamma }}={\mathbf{d}}_{\overline{\gamma },\overline{\gamma },\overline{\gamma }}$ (that is, ${\gamma }_1={\gamma }_2={\gamma }_3=\overline{\gamma }$ in (2)) with

$$\overline{\gamma }={\displaystyle \frac{3}{2}}{k}_{max}-{\delta }_{max}^{out}.$$

Notice that $\overline{\gamma }>1$.

A $\{0,1,2\}$-configuration $\overline{\omega }:{\overline{V}}_x\to \{0,1,2\}$ of ${\overline{G}}_x$mirrors a $\{0,1\}$-configuration $\omega :V_x\to \{0,1\}$ of $G_x$ if

• For every $u\in V_x$: $\overline{\omega }\left(u_1\right)=\overline{\omega }\left(u_2\right)=\omega \left(u\right)$, and $\overline{\omega }\left(b\right)=0$ and $\overline{\omega }\left(c\right)=1$ for every $b\in B_u$ and $c\in C_u$;
• For every $(u,v)\in A_x$: $\overline{\omega }\left(u{v}_1\right)=\overline{\omega }\left(u{v}_2\right)=2$, and $\overline{\omega }\left(z\right)=0$, $\overline{\omega }\left(r\right)=1$, $\overline{\omega }\left(y\right)=\overline{\omega }\left(s\right)=2$ for every $z\in Z_{uv}$, $y\in Y_{uv}$, $r\in R_{uv}$, $s\in S_{uv}$;
• For every dummy node $d\in D$, $\overline{\omega }\left(d_1\right)\ne 2$ and $\overline{\omega }\left(d_2\right)\ne 2$.

The next lemma proves that, if ${\overline{G}}_x$ is in a $\{0,1,2\}$-configuration mirroring the $\{-1,1\}$-configuration G is in, then the $\{0,1,2\}$-configuration held by ${\overline{G}}_x$ after two ${\mathbf{d}}_{\overline{\gamma }}$-evolution steps mirrors the $\{-1,1\}$-configuration held by G after one ${\mathbf{d}}_M$-evolution step.

Lemma 2. 

Let ${\omega }_0$ be a $\{0,1\}$-configuration for $G_x$. If ${\overline{\omega }}_0$ is the $\{0,1,2\}$-configuration for ${\overline{G}}_x$ that mirrors ${\omega }_0$, then ${\overline{\omega }}_2={\mathbf{d}}_{\overline{\gamma }}^2({\overline{G}}_x,{\overline{\omega }}_0)$ mirrors ${\omega }_1={\mathbf{d}}_M(G_x,{\omega }_0)$.

Proof. 

Recall that ${\overline{\omega }}_1=\overline{\mathbf{d}}({\overline{G}}_x,{\overline{\omega }}_0)$ and ${\overline{\omega }}_2=\overline{\mathbf{d}}({\overline{G}}_x,{\overline{\omega }}_1)$.

First of all, it can be easily verified that, since the degree of any node in ${\cup }_{(u,v)\in A_x}{Y}_{uv}$ and in ${\cup }_{(u,v)\in A_x}{S}_{uv}$ is $4k_{max}>\overline{\gamma }$, since the degree of any node in ${\cup }_{(u,v)\in A_x}{Z}_{uv}$ and in ${\cup }_{(u,v)\in A_x}{R}_{uv}$ is $4k_{max}+1>\overline{\gamma }$, and since the degree of any dummy node in D, of every node in B and of every node in C is $1<\overline{\gamma }$, by the definition of ${\overline{\omega }}_0$ the following holds:

• For every $b\in {\cup }_{u\in V_x}{B}_u$, $c\in {\cup }_{u\in V_x}{C}_u$ and for every $t\ge 1$

  $${\overline{\omega }}_t\left(b\right)={\overline{\omega }}_0\left(b\right)=0{\overline{\omega }}_t\left(c\right)={\overline{\omega }}_0\left(c\right)=1;$$
• For every $z\in {\cup }_{(u,v)\in A_x}{Z}_{uv}$, $y\in {\cup }_{(u,v)\in A_x}{Y}_{uv}$, $r\in {\cup }_{(u,v)\in A_x}{R}_{uv}$, $s\in {\cup }_{(u,v)\in A_x}{S}_{uv}$, and for every $t\ge 1$

  $$\begin{array}{ccc}\hfill {\overline{\omega }}_{2t-1}\left(z\right)=2& {\overline{\omega }}_{2t-1}\left(y\right)=0,{\overline{\omega }}_{2t-1}\left(r\right)=2,& {\overline{\omega }}_{2t-1}\left(s\right)=1,\hfill \\ \hfill {\overline{\omega }}_{2t}\left(z\right)=0,& {\overline{\omega }}_{2t}\left(y\right)=2{\overline{\omega }}_{2t}\left(r\right)=1,& {\overline{\omega }}_{2t}\left(s\right)=2;\hfill \end{array}$$
• For every dummy node $d\in D$ and for every $t\ge 1$, ${\overline{\omega }}_t\left(u\right)={\overline{\omega }}_0\left(u\right)\ne 2$.

The assertion is still to be proved for nodes $u_1$, $u_2$, $u{v}_1$ and $u{v}_2$ for $u\in V_x$ and $(u,v)\in A_x$.

• Let $u\in V_x$; for $i=1,2$ the following holds (see Figure 4 for an example).

Figure 4. The opinion evolution in ${\overline{G}}_x$ of the nodes $u_1$ and $u_2$ (associated with node u in $G_x$), of the nodes $u{v}_1$ and $u{v}_2$ (associated with arc $(u,v)$ in $G_x^{″}$), and of the components $B_u$, $C_u$, $Z_{uv}$, $Y_{uv}$, $R_{uv}$ and $S_{uv}$ according to ${\mathbf{d}}_{\overline{\gamma }}$ when ${\omega }_0\left(u\right)\ne {\omega }_1\left(u\right)$. Blue colored nodes have opinion 0, red colored nodes have opinion 1, and green colored nodes have opinion 2.

Figure 4. The opinion evolution in ${\overline{G}}_x$ of the nodes $u_1$ and $u_2$ (associated with node u in $G_x$), of the nodes $u{v}_1$ and $u{v}_2$ (associated with arc $(u,v)$ in $G_x^{″}$), and of the components $B_u$, $C_u$, $Z_{uv}$, $Y_{uv}$, $R_{uv}$ and $S_{uv}$ according to ${\mathbf{d}}_{\overline{\gamma }}$ when ${\omega }_0\left(u\right)\ne {\omega }_1\left(u\right)$. Blue colored nodes have opinion 0, red colored nodes have opinion 1, and green colored nodes have opinion 2.

Opinions after one step. 

Since ${\overline{\delta }}_x\left(u_i\right)={\delta }_{G_x^{″}}^{in}\left(u\right)+2{\delta }_{max}^{out}\left(u\right)+2k_u$ for every $u\in V_x$ and $i\in \{1,2\}$, and since, by the definition of ${\overline{\omega }}_0$, $P_0(u_i,{\overline{\omega }}_0)={\displaystyle \frac{|B_u|}{2}}={\displaystyle \frac{|C_u|}{2}}=P_1(u_i,{\overline{\omega }}_0)$ then

$$P_0(u_i,{\overline{\omega }}_0)=P_1(u_i,{\overline{\omega }}_0)\begin{array}{cc}=& k_u={\displaystyle \frac{3k_{max}-[{\delta }_x^{in}\left(u\right)+2{\delta }_{max}^{out}]}{2}}\hfill \\ =& {\displaystyle \frac{3[{max}_{u\in V_x}\left\{{\delta }_{G_x^{\prime \prime }}^{in}\left(u\right)\right\}+2{\delta }_{max}^{out}]-[{\delta }_x^{in}\left(u\right)+2{\delta }_{max}^{out}]}{2}}\hfill \\ =& {\displaystyle \frac{3{max}_{u\in V_x}\left\{{\delta }_{G_x^{\prime \prime }}^{in}\left(u\right)\right\}-{\delta }_x^{in}\left(u\right)+6{\delta }_{max}^{out}]-2{\delta }_{max}^{out}]}{2}}\hfill \\ >& {\displaystyle \frac{2-{\delta }_x^{in}\left(u\right)+4{\delta }_{max}^{out}}{2}}\hfill \\ =& {\delta }_{G_x^{″}}^{in}\left(u\right)+2{\delta }_{max}^{out}\left(u\right).\hfill \end{array}$$

Since $P_2(u_i,{\overline{\omega }}_0)={\delta }_{G_x^{″}}^{in}\left(u\right)+2{\delta }_{max}^{out}\left(u\right)$, from the above derivation it follows that $P_0(u_i,{\overline{\omega }}_0)=P_1(u_i,{\overline{\omega }}_0)>P_2(u_i,{\overline{\omega }}_0)$. Hence, since $k_u={\displaystyle \frac{3k_{max}-[{\delta }_x^{in}\left(u\right)+2{\delta }_{max}^{out}]}{2}}<{\displaystyle \frac{3}{2}}{k}_{max}-{\delta }_{max}^{out}=\overline{\gamma }$ then ${\overline{\omega }}_1\left(u_i\right)={\overline{\omega }}_0\left(u_i\right)={\omega }_0\left(u\right)$.

For every $(u,v)\in A_x$ and $i=1,2$, $P_2(u{v}_i,{\overline{\omega }}_0)=0$; since

$${\overline{\delta }}_x\left(u{v}_i\right)={\displaystyle \frac{|Z_{uv}|}{2}}+{\displaystyle \frac{|R_{uv}|}{2}}+\left|\{u_1,u_2,v_i\}\right|=4k_{max}+3$$

and since all nodes in $Z_{uv}$ have opinion 0 in ${\overline{\omega }}_0$, if ${\overline{\omega }}_0\left(u_1\right)={\overline{\omega }}_0\left(u_2\right)=0$ then $P_0(u{v}_i,{\overline{\omega }}_0)\ge 2k_{max}+2>\overline{\gamma }$ so that, since ${\overline{\omega }}_0\left(u{v}_i\right)=2$, ${\overline{\omega }}_1\left(u{v}_i\right)=0$; symmetrically if ${\overline{\omega }}_0\left(u_1\right)={\overline{\omega }}_0\left(u_2\right)=1$. Hence, ${\overline{\omega }}_1\left(u{v}_1\right)={\overline{\omega }}_1\left(u{v}_2\right)={\overline{\omega }}_0\left(u_1\right)={\overline{\omega }}_0\left(u_2\right)$.

Opinions after two steps. 

It follows from the above discussion that, for every $(u,v)\in A_x$, for every $u\in V_x$ and for every $i=1,2$, ${\overline{\omega }}_1\left(u{v}_i\right)\ne 2$ and ${\overline{\omega }}_1\left(u_i\right)\ne 2$; since all nodes in $Z_{uv}\cup R_{uv}$ have opinion 2 in ${\overline{\omega }}_1$ and since $|Z_{uv}|=|R_{uv}|=4k_{max}$,

then $P_2(u{v}_i,{\overline{\omega }}_1)={\displaystyle \frac{|Z_{uv}|}{2}}+{\displaystyle \frac{|R_{uv}|}{2}}=4k_{max}>\overline{\gamma }$ and, hence, $P_0(u{v}_i,{\overline{\omega }}_1)\le 3<4k_{max}$ and $P_1(u{v}_i,{\overline{\omega }}_1)\le 3<4k_{max}$, and ${\overline{\omega }}_2\left(u{v}_1\right)={\overline{\omega }}_2\left(u{v}_2\right)=2$.

• Without loss of generality, assume that ${\overline{\omega }}_0\left(u_i\right)=1$; that is,

$${\overline{\omega }}_0\left(u_1\right)={\overline{\omega }}_0\left(u_2\right)={\overline{\omega }}_1\left(u{v}_1\right)={\overline{\omega }}_1\left(u{v}_2\right)={\omega }_0\left(u\right)=1.$$

Since all of the $2k_u$ nodes in $C_u$ have opinion 1 in ${\overline{\omega }}_1$ and $k_u$ of them are adjacent to $u_i$, and since ${\overline{\omega }}_1\left(u{v}_i\right)={\overline{\omega }}_0\left(u_i\right)=1$ for every $(u,v)\in A_x$, then

$$\begin{array}{ccc}\hfill P_1(u_i,{\overline{\omega }}_1)& =& k_u+\left|\{z{u}_i:(z,u)\in A_x\wedge {\overline{\omega }}_1\left(z{u}_i\right)=1\}\right|+2{\delta }_x^{out}\left(u\right)\hfill \end{array}$$

Similarly, since all of the $2k_u$ nodes in $B_u$ have opinion 0 in ${\overline{\omega }}_1$ and $k_u$ of them are adjacent to $u_i$, then

$$\begin{array}{ccc}\hfill P_0(u_i,{\overline{\omega }}_1)& =& k_u+\left|\{z{u}_i:(z,u)\in A_x\wedge {\overline{\omega }}_1\left(z{u}_i\right)=0\}\right|\hfill \end{array}$$

Finally, $P_2(u_i,{\overline{\omega }}_1)=0$.

If ${\omega }_1\left(u\right)=0$ then $|\{z\in V_x:(z,u)\in A_x\wedge {\omega }_0\left(z\right)=0\}|>{\displaystyle \frac{{\delta }_{G_x^{″}}^{in}\left(u\right)}{2}}$ (by the definition of ${\mathbf{d}}_M$) so that $|\{z{u}_1:(z,u)\in A_x\wedge {\overline{\omega }}_1\left(z{u}_1\right)=0\}|=|\{z\in V_x:(z,u)\in A_x\wedge {\omega }_0\left(z\right)=0\}|>{\displaystyle \frac{{\delta }_{G_x^{″}}^{in}\left(u\right)}{2}}$ and

$$\begin{array}{ccc}\hfill P_0(u_i,{\overline{\omega }}_1)& =& |\{z{u}_1:(z,u)\in A_x\wedge {\overline{\omega }}_1\left(z{u}_1\right)=0\}|+{\displaystyle \frac{|C_u|}{2}}>{\displaystyle \frac{{\delta }_{G_x^{\prime \prime }}^{in}\left(u\right)}{2}}+k_u\hfill \\ & =& {\displaystyle \frac{{\delta }_{G_x^{\prime \prime }}^{in}\left(u\right)}{2}}+{\displaystyle \frac{3k_{max}-[{\delta }_x^{in}\left(u\right)+2{\delta }_{max}^{out}]}{2}}={\displaystyle \frac{3}{2}}{k}_{max}-{\delta }_{max}^{out}=\overline{\gamma }.\hfill \end{array}$$

Hence, since ${\overline{\omega }}_0\left(u_i\right)=1$ and $P_0(u_i,{\overline{\omega }}_1)>P_2(u_i,{\overline{\omega }}_1)$, then ${\overline{\omega }}_2\left(u_i\right)=0={\omega }_1\left(u\right)$.

If ${\omega }_1\left(u\right)=1$, then

$$|\{z{u}_1:(z,u)\in A_x\wedge {\overline{\omega }}_1\left(z{u}_1\right)=1\}|=|\{z\in V_x:(z,u)\in A_x\wedge {\omega }_0\left(z\right)=1\}|\ge {\displaystyle \frac{{\delta }_{G_x^{″}}^{in}\left(u\right)}{2}}$$

so that $P_0(u_i,{\overline{\omega }}_1)<{\displaystyle \frac{{\delta }_{G_x^{″}}^{in}\left(u\right)}{2}}+k_u=\overline{\gamma }$, and hence, since $P_2(u_i,{\overline{\omega }}_1)\le {\displaystyle \frac{{\delta }_{G_x^{″}}^{in}\left(u\right)}{2}}+k_u$ too, ${\overline{\omega }}_2\left(u_i\right)=1={\omega }_1\left(u\right)$. □

From the above lemma, the next theorem holds.

Theorem 3. 

For any pair of $\{0,1\}$-configurations ${\omega }_0:V_x\to \{0,1\}$ and ${\omega }_A:V_x\to \{0,1\}$ of $G_x$, ${\omega }_A\in {\mathcal{O}}_{{\mathbf{d}}_M}(G_x,{\omega }_0)$ if and only if ${\overline{\omega }}_A\in {\mathcal{O}}_{{\mathbf{d}}_{\overline{\gamma }}}({\overline{G}}_x,{\overline{\omega }}_0)$, where ${\overline{\omega }}_0$ and ${\overline{\omega }}_A$ are the $\{0,1,2\}$-configurations of ${\overline{G}}_x$ such that ${\overline{\omega }}_0$ mirrors ${\omega }_0$ and ${\overline{\omega }}_A$ mirrors ${\omega }_A$.

As a consequence of the above theorem and of the hardness proof in [35], it follows that

Corollary 1. 

Let ${\mathbf{d}}_{\overline{\gamma }}$ be the $\{0,1,2\}$-dynamics defined in this section. Deciding if a given graph in a given initial $\{0,1,2\}$-configuration will ever reach a given target $\{0,1,2\}$-configuration while evolving according to ${\mathbf{d}}_{\overline{\gamma }}$ is PSPACE-complete even when restricted to unsigned, unweighted, undirected graphs.

Finally, a second noticeable consequence of Lemma 2 has been drawn. Since one evolution step of $G_x$ (or, equivalently, of $G_x^{″}$) according to ${\mathbf{d}}_M$ is simulated by two evolution steps of ${\overline{G}}_x$ according to ${\mathbf{d}}_{\overline{\gamma }}$, then an equilibrium configuration for $G_x$ with respect to ${\mathbf{d}}_M$ corresponds to a loop of length 2 for ${\overline{G}}_x$ with respect to ${\mathbf{d}}_{\overline{\gamma }}$.

In [35], it has been proved that deciding if an unsigned, unweighted, undirected graph in a given initial binary configuration will ever reach an equilibrium configuration while evolving according to the deterministic majority dynamics is a PSPACE-complete problem still by describing the computation of a deterministic Turing machine by the ${\mathbf{d}}_M$-evolution of a directed graph $G_x\left[\mathcal{E}\right]$. It turns out that $G_x\left[\mathcal{E}\right]$ is obtained by adding to $G_x$ a constant in-degree and constant out-degree component, so that the same procedure to derive ${\overline{G}}_x$ from $G_x$ can be applied to $G_x\left[\mathcal{E}\right]$, resulting in an undirected (unsigned unweighted) graph ${\overline{G}}_x\left[\mathcal{E}\right]$ such that Lemma 2 holds for the pair of graphs $G_x\left[\mathcal{E}\right]$ and ${\overline{G}}_x\left[\mathcal{E}\right]$. Since deciding if $G_x\left[\mathcal{E}\right]$ in a given initial binary configuration will ever reach an equilibrium configuration while evolving according to ${\mathbf{d}}_M$ is a PSPACE-complete problem, then, starting from some binary configuration, the ${\mathbf{d}}_M$-evolution of $G_x\left[\mathcal{E}\right]$ reaches either an equilibrium configuration or a loop of length at least 2. Correspondingly, the ${\mathbf{d}}_{\overline{\gamma }}$-evolution of ${\overline{G}}_x\left[\mathcal{E}\right]$ in an initial ternary configuration mirroring a configuration of $G_x\left[\mathcal{E}\right]$ reaches either a loop of length 2 or a loop of (even) length $\ge 4$.

• Hence, the following corollary follows.

Corollary 2. 

Let ${\mathbf{d}}_{\overline{\gamma }}$ be the $\{0,1,2\}$-dynamics defined in this section. Then:

• Deciding if a given graph in a given initial $\{0,1,2\}$-configuration will enter a length 2 loop while evolving according to ${\mathbf{d}}_{\overline{\gamma }}$ is PSPACE-complete even when restricted to unsigned, unweighted, undirected graphs;
• The evolution of an unsigned, unweighted and undirected graph in a given initial $\{0,1,2\}$-configuration and evolving according to ${\mathbf{d}}_{\overline{\gamma }}$ may enter a loop of length $>2$.

5. Strongly Anti-Conservative Majority Dynamics

A three-opinions anti-conservative majority dynamics is here defined. Formally, let $G=(V,E)$ be a graph and let $\sigma :V\to \mathbb{N}$ be a stability-requirement function; the ${\mathbf{d}}_{\sigma -AM}$ is defined as follows: for any $\{0,1,2\}$-configuration $\omega :V\to \{0,1,2\}$ of G, if ${\omega }_1={\mathbf{d}}_{\sigma -AM}(G,\omega )$, then for every $u\in V$

$${\omega }_1\left(u\right)=\left\{\begin{array}{cc}i\in \{0,1,2\}-\left\{\omega \right(u\left)\right\}\hfill & \mathrm{if}{P}_{\omega \left(u\right)}(u,\omega )\le {\displaystyle \frac{{\delta }_u}{2}}+\sigma \left(u\right)\hfill \\ \hspace{1em}& \mathrm{and}{P}_i(u,\omega )>{\displaystyle \frac{{\delta }_u}{2}}+\sigma \left(u\right)\hfill \\ \hspace{1em}& \mathrm{and}{P}_i(u,\omega )>P_j(u,\omega )\hfill \\ \hspace{1em}& \mathrm{with}j\in \{0,1,2\}-\{i,\omega \left(u_1\right)\},\hfill \\ \omega \left(u\right)\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

In this section, small changes will be applied to the construction described in Section 4 to derive the undircted graph ${\overline{G}}_x$ from the directed graph $G_x=(V_x,A_x)$ in order to prove the PSPACE-completeness of Reachability[${\mathbf{d}}_{\sigma -AM}$] with $\sigma$ arbitrarily close to half of the node degree function.

Let ${\overline{G}}_x=({\overline{V}}_x,{\overline{E}}_x)$ be the graph described in Section 4. In the construction deriving ${\overline{G}}_x$, for each $u\in V_x$, set

$${\widehat{k}}_u={\displaystyle \frac{3\alpha {\delta }_{max}^{out}{k}_{max}-[{\delta }_x^{in}\left(u\right)+2{\delta }_{max}^{out}]}{2}}$$

for some parameter $\alpha \in {\mathbb{N}}^{+}$ such that $\alpha \ge 1$ and ${\widehat{k}}_u$ is integer: indeed, since dummy out-neighbors may be added to any node in $G_x$ without affecting its evolution, without loss of generality it can be assumed that ${\delta }_{max}^{out}$ is odd and $\alpha$ can be chosen odd too, so that ${\widehat{k}}_u$ is in fact integer.

Finally, for every node $b_u^i\in {\cup }_{u\in V}{B}_u$, make it part of a 4-node clique; repeat the same procedure for every node $c_u^i\in {\cup }_{u\in V}{C}_u$ and for every node $d\in D$. This ensures the opinion stability of the involved nodes with respect to dynamics ${\mathbf{d}}_{\sigma -AM}$. Denote as ${\overline{G}}_{x,\alpha }$ the graph obtained by the same procedure to get $G_x$ with using ${\widehat{k}}_u$ instead of $k_u$ and by replacing nodes in ${\cup }_{u\in V}{B}_u{\cup }_{u\in V}{C}_u\cup D$ by cliques (see Figure 5).

Notice that ${\overline{G}}_x$ is actually a subgraph of ${\overline{G}}_{x,\alpha }$; hence, a $\{0,1,2\}$-configuration $\overline{\omega }$ of ${\overline{G}}_{x,\alpha }$mirrors a $\{0,1\}$-configuration $\omega :V_x\to \{0,1\}$ of $G_x$ if:

• Its restriction to ${\overline{G}}_x$ mirrors $\omega$;
• For every node z in the clique containing $b_u^i\in {\cup }_{u\in V}{B}_u$ it holds that $\overline{\omega }\left(z\right)=\overline{\omega }\left(b_u^i\right)=0$;
• For every node z in the clique containing $c_u^i\in {\cup }_{u\in V}{C}_u$ it holds that $\overline{\omega }\left(z\right)=\overline{\omega }\left(c_u^i\right)=1$;
• For every node z in the clique containing $d\in D$ it holds that $\overline{\omega }\left(z\right)=\overline{\omega }\left(d\right)\ne 2$.

Denote as

$${\gamma }_{\alpha }={\displaystyle \frac{3}{2}}\alpha {\delta }_{max}^{out}{k}_{max}-{\delta }_{max}^{out}$$

so that ${\mathbf{d}}_{{\gamma }_{\alpha }}$ is the underpopulation dynamics (2) with $\gamma ={\gamma }_{\alpha }$.

Since all nodes $u_i$, with $i=1,2$, such that $u\in V_x$ have degree ${\overline{\delta }}_{x,\alpha }\left(u_i\right)=3\alpha {\delta }_{max}^{out}{k}_{max}$ in the graph ${\overline{G}}_{x,\alpha }$, if ${\overline{\delta }}_{x,\alpha }$ denotes the node degree function in ${\overline{G}}_{x,\alpha }$, then for each $u\in V_x$ and $i=1,2$ it holds that

$${\gamma }_{\alpha }=\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{3\alpha k_{max}}}\right){\overline{\delta }}_{x,\alpha }\left(u_i\right).$$

By setting $\sigma \left(v\right)={\displaystyle \frac{\delta \left(v\right)}{3\alpha k_{max}}}$ (where $\delta \left(u\right)$ is the node degree function of a generic graph), and by noticing that, starting at a $\{0,1,2\}$-configuration in which all nodes in any 4-node clique in ${\overline{G}}_{x,\alpha }$ have the same opinion, such nodes never change their opinion when ${\overline{G}}_{x,\alpha }$ evolves according to ${\mathbf{d}}_{\sigma -AM}$, the same arguments in the proof of Lemma 2 allow to prove the following lemma

Lemma 3. 

For any $\{0,1\}$-opinion configuration ${\omega }_0$ for $G_x$, if ${\overline{\omega }}_0$ is the opinion configuration for ${\overline{G}}_{x,\alpha }$ that mirrors ${\omega }_0$, then ${\overline{\omega }}_2={\mathbf{d}}_{\sigma -AM}^2({\overline{G}}_{x,\alpha },{\overline{\omega }}_0)$ mirrors ${\omega }_1={\mathbf{d}}_M(G_x,{\omega }_0)$.

Since the above lemma holds for any $\alpha \ge 1$, the next theorem is a consequence of it.

Theorem 4. 

For any $ϵ>0$, deciding if a given graph in a given initial $\{0,1,2\}$-configuration will ever reach a given target $\{0,1,2\}$-configuration while evolving according to the ${\mathbf{d}}_{ϵ-AM}$ is PSPACE-complete even when restricted to unsigned, unweighted, undirected graphs, where ${\mathbf{d}}_{ϵ-AM}$ denotes the dynamics ${\mathbf{d}}_{\sigma -AM}$ such that σ is the constant function with $\sigma \left(u\right)=ϵ$ for every node v in the graph.

Finally, by the same considerations leading to Corollary 2, the following corollary holds.

Corollary 3. 

For any $ϵ>0$, let ${\mathbf{d}}_{ϵ-AM}$ be the dynamics ${\mathbf{d}}_{\sigma -AM}$ where σ is the constant function such that $\sigma \left(u\right)=ϵ$ for every node v in the graph. Then:

• Deciding if a given graph in a given initial $\{0,1,2\}$-configuration will enter a length 2 loop while evolving according to ${\mathbf{d}}_{ϵ-AM}$ is PSPACE-complete even when restricted to unsigned, unweighted, undirected graphs;
• tTe evolution of an unsigned, unweighted and undirected graph in a given initial $\{0,1,2\}$-configuration and evolving according to ${\mathbf{d}}_{ϵ-AM}$ may enter a loop of length $>2$.

6. Conclusions

This paper considers three majority-based, three-opinion dynamics, proving that:

• The “majority or neutrality” dynamics (Section 3) yields polynomial size orbits in the graph size (and, hence, computationally tractable $\mathsf{\Pi }$-Reachability problems);
• Under the hypothesis P ≠ PSPACE (a standard computational complexity conjecture) the underpopulation dynamics (Section 4) and the strongly anti-conservative majority dynamics (Section 5) yield intractable Reachability problems (and, consequently the orbits they induce may have super-polynomial size) and orbit loop length not upper bounded by 2, so contradicting the conjecture in [31].

Actually, the majority dynamics considered in this paper replace the a priori assumptions by the introduction of some new assumptions that are not statically fixed but are determined, let us say, at evolution time by the current opinion of nodes at each step. Such “evolution time” assumptions are closer to many natural generalizations of binary opinion dynamics: the relative majority dynamics (and the policy for breaking ties) and the conservative absolute majority dynamics introduced in Section 1, and many others. The issues considered in this paper are still open problems with respect to such dynamics.