Rebellions and Impeachments in a Neural Network Society
Basede on a study of the modern presidencial democracies in South America, we
present a statistical mechanics exploration of the collective, coordinated
action of political actors in the legislative chamber that may result on the
impeachment of the executive. By representing the legislative political actors
with neurla networks, we observed that the larger the effective number of
presidential-agenda items are treated, the smaller the chances for a
cross-party dialogue, which, if combined with a decrement in the president's
public approval rating, could trigger an impeachment process.
Juan Neirotti, Nestor Caticha
# Rebellions and Impeachments in a Neural Network Society

###### Abstract

Based on a study of the modern presidencial democracies in South America, we present a statistical mechanics exploration of the collective, coordinated action of political actors in the legislative chamber that may result on the impeachment of the executive. By representing the legislative political actors with neurla networks, we observed that the larger the effective number of presidential-agenda items are treated, the smaller the chances for a cross-party dialogue, which, if combined with a decrement in the president's public approval rating, could trigger an impeachment process.
Introduction

The mechanism for deposition of institutional power in several areas of the world has changed in the last half century. A new pattern of government overthrow took over the traditional military coup, specially in Latin America as extensively documented in [27]. These abrupt changes of collective behavior are typically seen in a society which exerts pressure on the parliament to promote changes outside the election model by parliamentary impeachment. External pressures which include elements such as the state of the economy, perception of corruption or their combination may be distal reasons, but at a closer look the correlated behavior of the parliament follows the emboldenment that derives from the collective perception that there is sufficient strength in the opposition camp to overthrow the executive. Technically still within the realm of constitutional order, despite being associated to affective rather than ideological affinity [14; 15], this transition mechanism seems to bring new theoretical challenges to comparative studies of presidentialism.

To illustrate the problem and to better understand the expected characteristics of the model, we will briefly discuss three different instances were the executive was either impeached or nearly impeached by the legislative. The first case corresponds to the presidency of Fernando Collor de Mello, from Brazil. Collor won the 1989 elections in the second round with 54% of the votes, but his party only had 8% of the sits in the Chamber of Deputies and 4% in the Senate. By March 1990, when Collor was sworn into office and his approval ratings were at +60%, the consumer price index rose by 84% (observe the evolution of these numbers in figure 1); at this point Collor lunched his first economic plan. But in spite of the extreme measures imposed, government control of inflation proved elusive and popular discontent increased. The application of a second (unsuccessful) economic plan (Collor II), and a number of corruption cases revealed by the press provoked a plummet on the approval ratings and triggered a number of streets demonstrations. With very few allays in the Legislative, impeachment procedures were triggered and by the end of September 1992 the Chamber of Deputies approved (by a vote of 441-38) the impeachment by the Senate.

The second case corresponds to the presidency of Carlos Andres Perez, from Venezuela. Perez won his second term in 1988 with 53% of the vote and, in contrast to Collor's case, he was the leader of the largest party in the country. In order to put under control the critical situation inherited from the previous president, Perez announced an economic package (the Great Turnaround) in February 1989. These measures triggered an abrupt rise in inflation, and an immediate popular response in the form of riots (Caracazo). Observe the evolution of the presidential approval rating and the number of popular marches per month in figure 2. The human rights violations occurring during the days of the Caracazo, the increasing number of protests, and media exposes revealing scandals involving members of the administration compromised the credibility of the government and its capacity to control the economy. By September 1991 the followers of president Perez within the ruling party lost their internal elections. On February and November

Figure 1: Time evolution of the consumer prise index (full-black line) [1] and the presidential approval rating (dashed-red line) from 1990 to September 1992 (adapted from [27]).
1992 there were attempts of coup d'etat. By May 1993 was suspended by the Senate.

The third case corresponds to the presidency of Ernesto Samper, from Colombia. He won the elections of June 1994 with 50.6% of the votes. In the weeks following the election, the press began to reveal a possible contribution from the Cali drug cartel to Samper's campaign. The Prosecutor General opened an investigation (Process 8000) and, preemptively, Samper asked the Congress (under his party's control) to also investigate the accusations. The investigations were closed and, although his presidential approval rating continue to decline, Samper kept control of the Congress and managed to finish his tenure (figure 3).

In these three cases we observe patterns also found in in other presidencies. These observations can be synthesize

Figure 3: Approval and disapproval presidential ratings during Ernesto Samper’s tenure, from August 1994 to August 1998 [27].

Figure 2: Time evolution of (dashed-red line) from 198
as follows: At the beginning of the presidential period the presidential approval ratings are high, and they fluctuate according to the information the electorate receives about the current issues (either policies or scandals). Meanwhile the members of the legislative chamber discus the president's proposals under the influence of their internal political alliances and the effective pressure exerted by the presidential approval ratings. The more items are discussed in the chamber the more is known about the presidential agenda, which rearranges the chamber's alliances and provides more information to the pollsters, feeding into the cycle of influences and interactions. Naively, we conclude that the actions of the agents in the legislative chamber are based on the opinions they form on the items propposed by the president to be discussed, and that such discussions are modified by the alliances they have with each other and by the pressure from the polls. The larger the difference in opinions between legislative agents and the president and the lower the presidential ratings the higher the chances of impeachment.

There is a need to devise mechanisms for opinion formation and for alliance formation in order to provide insight for the understanding of the mechanics of modern presidential democracies. Empirical evidence coming from research in psychology supports there is a cost of dissent, with humans trying to attain social conformity modulated by peer pressure [3; 9; 30; 32] and that conformity is learned from interactions within a social network e.g. [16]. The cost of dissension on some set of issues can be modeled using techniques of Statistical Mechanics of disordered systems and there is an extensive literature on polarization [4; 7; 18; 21; 28; 31; 33] and echo chambers [8; 12; 22; 35] in opinion dynamics models. Our aim in this paper is to address yet again collective behavior by studying Agent Based Models, but with the specific aim of addressing constitutional impeachments.

The model is introduced in Section II. The analytical approach derives from the use of Statistical Mechanics in the space of interactions in the Gardner style [11], here applied not only to a single perceptron agent, but to a population of such agents. The relevant time scales of the different changes that can occur are discussed and this leads to the methodology appropriate for the analysis. In Section III we present the structure of the agents, the relevant order parameters that characterize the macroscopic state of the society and the analytical framework. The two types of quenched disorder, from the issues under discussion and the graph of interactions, lead as shown in Section IV, to functional mean field equations that determine the thermodynamics of the model. In Section V we present analytical results obtained from the study of the saddle-point equations. Readers interested in the lessons that can be gleaned from this toy model should go to Section VI where the interpretation in terms of less mathematical terms can be found. A short version of the extensive calculations is shown in the Supplementary Material (SM) Section in VII.1.

## II The model

Our members-of-congress agents, are simple neural networks that discuss and emit for or against opinions about multidimensional issues. In addition there is a special agent, playing the role of the current executive leader, called the president, to be followed or deposed depending on the intensive parameters of the model.

The agenda consists of \(N\)-dimensional binary issues \(\mathbf{\xi}_{\mu}\), and a simple measure of the agenda's complexity is given by \(\alpha=P/N\), where \(P\) is the number of pertinent topics under discussion. The \(a^{\rm th}\) agent's for or against opinion is \(\sigma_{a\mu}\), arising from the issue and its internal state \(\mathbf{J}_{a}\), \(\sigma_{a\mu}=\varphi(\mathbf{J}_{a},\mathbf{\xi}_{\mu})\in\{-1,+1\}\), where \(a\) runs over the members of congress and the president is designed by a label \(B\), and its opinion on issue \(\mathbf{\xi}_{\mu}\) is \(\sigma_{B\mu}=\varphi(\mathbf{B},\mathbf{\xi}_{\mu})\), where \(\mathbf{B}\) is the internal state of \(B\). A specific choice for \(\varphi\) will be postponed for now, but its output is binary, i.e. \(\varphi\in\{-1,+1\}\).

The inner circle or click of a particular congress-agent is represented by an adjacency matrix \(\mathbf{G}\) with entries \(g_{ac}\neq 0\) if agents \(a\) cares about the opinion of agent \(c\) and zero if not. The weighed opinion on issue \(\mathbf{\xi}_{\mu}\) of \(a\)'s peers is \(\Sigma_{a\mu}=\sum_{c}g_{ac}\sigma_{c\mu}\).

We consider the cost for agent \(a\) to hold an opinion on the \(\mu\)th issue to arise from two contributions:

\[C_{a,\mu}=-\frac{1+\sigma_{B\mu}\sigma_{a\mu}}{2}-\frac{1-\sigma_{B\mu}\sigma _{a\mu}}{2}\sigma_{a\mu}\Sigma_{a\mu}. \tag{1}\]

Equation (1) implements a mechanism of _corroboration_ as follows. If agent \(a\) agrees with \(B\), i.e. \(\sigma_{B\mu}\sigma_{a\mu}>0\), only the first term contributes to the cost, which gets reduced in one unit. If \(a\) and \(B\) disagree, \(\sigma_{B\mu}\sigma_{a\mu}<0\) and the second term is different from zero. If the weighed opinion of \(a\)'s peers is in agreement with \(B\), i.e. \(\sigma_{B\mu}\Sigma_{a\mu}=-\sigma_{a\mu}\Sigma_{a\mu}>0\), the cost increases, and if \(\sigma_{a\mu}\Sigma_{a\mu}>0\) the cost decreases. If agreeing with its peers is less costly than agreeing with \(B\), \(a\) can form a local consensus against \(B\), through corroboration.

A simple rearrangement of terms allows us to write the cost as:

\[2C_{a,\mu}=-\sigma_{B\mu}\sigma_{a\mu}-\Sigma_{a\mu}\sigma_{a\mu}+\Sigma_{a \mu}\sigma_{B\mu}-1. \tag{2}\]

The first term describes the advantage of having the same opinion as the president. The second, of concurring with its peers. The third one can be attributed to the disadvantage that other members of its peer group are in alliance with the president. The last is just an additive constant.
The overall cost for the entire congress, defined by the microscopic states \(\{{\bf J}_{a}\}\) of the agents, the topological structure of alliances \({\mathbf{G}}\) and the complete presidential agenda \({\cal A}=\{{\mathbf{\xi}}_{\mu},\sigma_{B\mu}\}_{\mu=1,\ldots P}\), gives the full Hamiltonian cost of the system:

\[E(\{{\bf J}_{a}\},{\cal A},{\mathbf{G}})=\sum_{a,\mu}C_{a,\mu}. \tag{3}\]

The cost for agent \(a\), expression (2), depends on the \(g_{ac}\) in two places. In the first, which we leave as shown, it describes the interaction of the peers \(c\) with agent \(a\). The second describes the overall influence of the president on the group of peers of \(a\). To simplify matters we consider, in the second term, to have a mean \(\nu\eta_{0}\) and disregard its fluctuations, where \(\eta_{0}\) is the average intensity of the influence exerted by another agent and \(\nu\) the average size of the group. Then the overall cost, up to an additive constant, simplifies to:

\[E_{0}(\{{\bf J}_{a}\};{\cal A},{\mathbf{G}})=-\frac{1}{2}\left[(1-\nu\eta_{0})\sum _{a,\mu}\sigma_{B\mu}\sigma_{a\mu}+\sum_{\mu ac}g_{ac}\sigma_{a\mu}\sigma_{c \mu}\right]. \tag{4}\]

The choice of techniques used to analyze the system follow from a discussion of the relevant "physical" time scales of the problem. While there is no conservation law that applies to the global cost in any strict sense, there are several relevant time scales associated to this discussion. The agenda under discussion and the political alliances are supposed to remain valid on a long time scale \(\tau_{q}\) of around one year, certainly less than \(\tau_{P}\), the 4 or 5 years of the presidential cycle. For times of the order of \(\tau_{q}\) we expect the variables \(\nu\) (the co-conspirators clique-size), \(\eta_{0}\) (the co-conspirators interaction strength), and \(\alpha\) (the volume of the agenda covered) to remain constant. Agents interact and may change opinions about the issues on a faster scale \(\tau_{op}\) of a few days. \(\tau_{op}\) is the typical time elapsed between the treatment of subsequent agenda items \({\mathbf{\xi}}_{\mu}\) and \({\mathbf{\xi}}_{\mu+1}\). The expected value of the cost is sufficiently stable on an intermediate time scale \(\tau_{C}\), which is larger than the time scale associated to the dynamics of the agents but much shorter than changes of the issues of national interest, \(\tau_{op}\ll\tau_{C}\ll\tau_{q}<\tau_{P}\). \(\tau_{C}\) is of the order of weeks, similar to the time validity of presidential polls data. This separation of the time scales determines the methodology of analysis of the problem and leads to a description of the system with a Boltzmann distribution with a \(\beta\), conjugated to the expected value of the cost, that controls the size of the fluctuations above the ground state. It can be interpreted as the pressure that society at large exerts on congress. As an example, [2] choose \(\beta\) as a measure of the president's polling. Since the time scale in which the agenda and the political alliances changes is still larger, their random effect can be averaged over as quenched disorder. This is reasonable, since at least during \(\tau_{q}\) the prominent issues of the agenda are to some extent fixed, as are the intra-party alliances.

The macroscopic state of the system is characterized by order parameters to be described below. Still within the presidential cycle, changes due to externalities may lead to changes in the intensive parameters. Phase transitions may occur for a congress divided into a situation and opposition parties, to a majoritarian congress that either supports a state of almost unanimous support for the president or it is in almost total opposition. These transitions are to a constitutional dictatorship regime or to a state where the conditions for impeachment are ripe. They signal presidential crises driven by the collective properties of congress and not by external or internal military forces that act by simple dissolution of congress.

## III Methods and order parameters

We have not yet made explicit the _a priori_ measure of the \(\sigma\) variables. If it were just a product of independent uniform measures, e.g. Ising like variables, several interesting features of the problem would remain untouched. Thus we decided for more structured agents, which we model by a neural network classifier with a binary for/against, \(\pm 1\) output. In order to keep it analytically manageable, we choose the simplest architecture, the single layer perceptron. Linearly separable models, in some manner similar to the Rescorla-Wagner model [29] from psychology have been shown to be useful in describing human performance in several cases. Therefore the dynamical variables of an agent are \(N\) dimensional vectors \({\bf J}_{a}\) and its opinion on an issue is \(\sigma_{a\mu}=\mathrm{sgn}({\bf J}_{a}\cdot{\mathbf{\xi}}_{\mu})\). The issues from the agenda are constructed by choosing independently \(P=\alpha N\) vertices of the \(N\) dimensional hyper-cube with coordinates of absolute value equal to one.

Under the assumption that the average value of \(E_{0}\) in equation 3, is approximately constant over a cycle of discussions of order \(\tau_{C}\) and the random agenda and alliances quenched on the \(\tau_{q}\) scale, standard arguments yield the probability distribution of the states of the congress-agents, given by:

\[{\cal P}({\bf J}_{a}|\beta,{\cal A},{\bf G})=\frac{1}{Z}{\cal P}_{0}({\bf J}_{ a})\exp\{-\beta E_{0}(\{{\bf J}_{a}\};{\cal A},{\bf G})\}, \tag{5}\]
where \({\cal P}_{0}({\bf J})=(2\pi{\rm e})^{-N/2}\,\delta\left({\bf J}\cdot{\bf J}-N\right)\) is the _a priori_ measure of the agents weights, taken to be independent and uniform over the spherical shell of radius \(\sqrt{N}\) in \(N\) dimensions. The discussion about the separation of time scales require the use of quenched disorder. The macroscopic properties of the system are obtained from the free energy \(f=-\beta^{-1}\overline{\ln Z}\), averaged of the possible agendas and alliances, taken to be fixed on the relevant time scale.

The interactions between agents, encoded in the matrix \(\mathbf{G}\), are assumed independent of each other and identically distributed. They are constructed in two steps. First, a Bernoulli with parameter \(p\), is used to decide if there is a connection present between two peers. Then, the strength of their interaction \(\eta\) is drawn from a Normal distribution centered at \(\eta_{0}\) with variance \(\Delta^{2}.\) This leads to \(\nu=2p(M-1)\) (see equation (66)) where the factor 2 accounts for the fact that the graph is directed, then:

\[{\cal P}(g,\eta,x|\nu,\eta_{0},\Delta^{2}) = {\cal P}(g|\eta,x){\cal P}(x|p){\cal P}(\eta|\eta_{0},\Delta^{2}) \tag{6}\] \[= \delta(g-x\eta)\left[(1-p)\delta(x)+p\delta(x-1)\right]{\cal N}( \eta|\eta_{0},\Delta^{2}).\]

The problem is complicated by the existence of two sources of quenched disorder, the agenda \({\cal A}\) and the alliances, encoded in the matrix \(\mathbf{G}\). An adaptation of ideas introduced in References [5; 24; 34] to treat this problems associated to coding theory are needed here. Due to the technical impossibility of computing the average of a logarithm we proceed by applying the replica formalism [23], i.e.:

\[f=-\beta^{-1}\lim_{n\to 0}\frac{Z^{n}-1}{n}, \tag{7}\]

where \(Z^{n}=\prod_{\gamma=1}^{n}Z^{\gamma}\) is the partition function of the replicated system, each of the \(n\) systems linked to an index \(\gamma\).

Taking expectations over the alliances brings forward the following population averages order parameters:

\[\varrho_{\mu_{1}\ldots\mu_{\ell}}^{\gamma_{1}\ldots\gamma_{\ell}}\equiv{\mathbb{ E}}\left[\frac{1}{M}\sum_{a}\left(\sigma_{a\mu_{1}}^{\gamma_{1}}\sigma_{B\mu_{1} }\ldots\sigma_{a\mu_{\ell}}^{\gamma_{\ell}}\sigma_{B\mu}\right)\right], \tag{8}\]

where \(\varrho_{\mu}^{\gamma}\) is the average agreement of the population with \(B\) on the \(\mu\)-th issue on the \(\gamma\) replica, and \(\varrho_{\mu_{1}\ldots\mu_{\ell}}^{\gamma_{1}\ldots\gamma_{\ell}}\) are population averages of the agreement of individual \(a\) with \(B\) across systems \(\gamma_{1}\) (on issue \({\bf S}_{\mu_{1}}\)) to \(\gamma_{\ell}\) (on issue \({\bf S}_{\mu_{\ell}}\)). Their expectation values are \(\ell\)-point correlation functions for the opinions. The introduction of these parameters also requires the introduction of conjugate parameters \(\tilde{\varrho}_{\mu_{1}\ldots\mu_{\ell}}^{\gamma_{1}\ldots\gamma_{\ell}}\). Observe that \(\varrho_{\mu_{1}\ldots\mu_{\ell}}^{\gamma_{1}\ldots\gamma_{\ell}}\) is the average of a local properties, thus the conjugate variable \(\tilde{\varrho}_{\mu_{1}\ldots\mu_{\ell}}^{\gamma_{1}\ldots\gamma_{\ell}}\) must represent the average effect of the local neighborhood on the local agent. By imposing the replica-symmetric ansatz [5; 24; 34], the order parameters should not present any dependency on either replica or agenda item indexes, they should only depend on their number \(\ell\). By observing that the definition of the order parameters equation (8) satisfy \(-1\leq\varrho_{\mu_{1}\ldots\mu_{\ell}}^{\gamma_{1}\ldots\gamma_{\ell}}\leq 1\), we suppose the existence of a field \(\tanh(\beta z)\) which is drawn from two normalized distributions \(\pi(z)\) and \(\hat{\pi}(z)\) such that:

\[\varrho_{\mu_{1}\ldots\mu_{\ell}}^{\gamma_{1}\ldots\gamma_{\ell}}=\int{\rm d}z \,\pi(z)\tanh^{\ell}(\beta z)\qquad\quad\tilde{\varrho}_{\mu_{1}\ldots\mu_{ \ell}}^{\gamma_{1}\ldots\gamma_{\ell}}=\nu\int{\rm d}z\,\hat{\pi}(z)\tanh^{\ell} (\beta z). \tag{9}\]

Graph disorder introduces these two probability densities \(\pi(z)\) and \(\hat{\pi}(s)\), that are functional order parameters that describe the level of consensus at the local and neighborhood levels, respectively. It is their behavior that signals the transitions from a two parties equilibrium to a consensus that can be either for or against the presidential agent.

Observe that the parameters defined in (9) have been introduced in (72) and (73).

The usual order parameters associated with the agents overlaps and with the president are also introduced:

\[R_{a}^{\gamma} = {\mathbb{E}}({\bf J}_{a}^{\gamma}\cdot{\bf B}/N),\quad q_{a}^{ \gamma\rho}={\mathbb{E}}({\bf J}_{a}^{\gamma}\cdot{\bf J}_{a}^{\rho}/N), \tag{10}\] \[W_{ab}^{\gamma} = {\mathbb{E}}({\bf J}_{a}^{\gamma}\cdot{\bf J}_{b}^{\gamma}/N), \quad t_{ab}^{\gamma\rho}={\mathbb{E}}({\bf J}_{a}^{\gamma}\cdot{\bf J}_{b}^{ \rho}/N). \tag{11}\]

Under the assumption of replica symmetric saddle points \(R_{a}^{\gamma}=R\), \(q_{a}^{\gamma\rho}=q\), and, by Reference [26], \(W_{ab}^{\gamma}=t_{ab}^{\gamma\rho}=W\), the properties of the system follow from the extrema of the free energy functional (see Section VII.1):

\[f[q,R,\pi,\hat{\pi}] = T\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!
where the averages are taken over the following distributed variables \(\eta\sim\mathcal{N}(\eta|\eta_{0},\Delta^{2})\), \(y\sim\mathsf{P}(y|\hat{\pi})\), and \(x\sim 2\mathcal{N}(x|0,1)\mathcal{H}\left(-Rx/\sqrt{q-R^{2}}\right)\), where \(\mathcal{H}(t)\) is the Gardner error function \(\mathcal{H}(t)=\int_{t}^{\infty}\mathrm{d}x\mathcal{N}(x|0,1)\). We used the short hand \(\epsilon=\epsilon(\beta,\nu\eta,y)=\left[\exp(2\beta(1-\nu\eta_{0}+y))-1\right] ^{-1}\). The free energy is a functional of the normalized distributions \(\pi\) and \(\hat{\pi}\). The new variable \(y\)'s distribution is:

\[\mathsf{P}(y|\hat{\pi})\equiv\int\frac{\mathrm{d}\hat{y}}{2\pi} \mathrm{e}^{-iy\hat{y}}\exp\left[\nu\left(\int\mathrm{d}s\hat{\pi}(s)\mathrm{e }^{i\hat{y}s}-1\right)\right]. \tag{13}\]

The characteristic function \(\phi_{s}(\hat{y})\) of \(\hat{\pi}(s)\), and the generator function of the cumulants of \(s\), \(K_{s}(\hat{y})\) are:

\[\phi_{s}(\hat{y}) = \int\mathrm{d}s\hat{\pi}(s)\mathrm{e}^{i\hat{y}s}, \tag{14}\] \[K_{s}(\hat{y}) = \log\phi_{s}(\hat{y}). \tag{15}\]

We observe that there exists a distributed variable \(u\) such its generator of cumulants function can be defined as: \(K_{u}(\hat{y})\equiv\phi_{s}(\hat{y})-1\). Add \(\nu\) independent copies of \(u\) to define \(y=\sum_{i=1}^{\nu}u_{i}\), since:

\[\mathsf{P}(y|\hat{\pi}) = \int\frac{\mathrm{d}\hat{y}}{2\pi}\mathrm{e}^{-iy\hat{y}}\exp \left[\nu K_{u}(\hat{y})\right], \tag{16}\] \[= \int\frac{\mathrm{d}\hat{y}}{2\pi}\mathrm{e}^{-iy\hat{y}}\left[ \phi_{u}(\hat{y})\right]^{\nu}, \tag{17}\]

where the \(u_{i}\) are random variables with the property that the \(r\)th cumulant of \(u\) is equal to \(\mathbb{E}(s^{r}|\hat{\pi})\), the \(r\)th moment of \(s\). Hence as \(\nu\) grows, \(y\) becomes normal. The \(r\)th order cumulant \(\kappa_{r}^{(y)}\) of \(y\) satisfies:

\[\kappa_{r}^{(y)}=\nu\kappa_{r}^{(u)}=\nu\mathbb{E}(s^{r}|\hat{\pi}), \tag{18}\]

so the cumulants of \(y\) are constructed by accumulation of the cumulants of \(u\) or of the moments of \(s\). It automatically follow that:

\[\mathbb{E}(y|\mathsf{P}) = \nu\mathbb{E}(s|\hat{\pi}) \tag{19}\] \[\mathbb{E}(y^{2}|\mathsf{P})-\mathbb{E}(y|\mathsf{P})^{2} = \nu\mathbb{E}(s^{2}|\hat{\pi}). \tag{20}\]

Since \(\mathbb{E}(s^{2}|\hat{\pi})\) turns out to be proportional to \(\eta_{0}^{2}\), \(y\)'s variance is proportional to \(1/\nu\) in the relevant region where \(\nu\eta_{0}\) is of order \(1\).

## IV Saddle point equations

The extreme of the free energy (12) is determined by the saddle point equations, which determine the order parameters in a self-consistent way. The distribution \(\hat{\pi}(s)\) satisfies:

\[\hat{\pi}(s)=\int\mathrm{d}z\int\mathrm{d}y\,\mathsf{P}(z,s,y|\hat{\pi})=\int \mathrm{d}z\int\mathrm{d}y\,\mathsf{P}(y|\hat{\pi})\mathsf{P}(z|y)\mathsf{P}( s|z), \tag{21}\]

where

\[\mathsf{P}(z|y) = \left\langle\delta\left[z-\beta^{-1}g(x;\epsilon,q)\right] \right\rangle_{x} \tag{22}\] \[\mathsf{P}(s|z) = \left\langle\delta\left(s-\beta^{-1}\mathrm{arctanh}\left[ \tanh(\beta\eta)\tanh(\beta z)\right]\right)\right\rangle_{\eta}, \tag{23}\]

and

\[g(x;\epsilon,q)=\frac{1}{2}\ln\frac{1+\epsilon}{\epsilon}+\frac{1}{2}\ln\frac{ 1-\mathcal{H}_{+}}{\mathcal{H}_{+}};\mathrm{with}\ \mathcal{H}_{+}=\mathcal{H}\left(\sqrt{\frac{q}{1-q}}x\right). \tag{24}\]

The equations for \(\pi(z)\) and \(\hat{\pi}(s)\) are:

\[\pi(z) = \int\mathrm{d}y\,\mathsf{P}(y|\hat{\pi})\left\langle\delta\left[z- \beta^{-1}g(x;\epsilon,q)\right]\right\rangle_{x}, \tag{25}\] \[\hat{\pi}(s) = \int\mathrm{d}z\,\pi(z)\left\langle\delta\left(s-\beta^{-1} \mathrm{arctanh}\left[\tanh(\beta\eta)\tanh(\beta z)\right]\right)\right\rangle_{ \eta}. \tag{26}\]
This shows that \(\pi(z)\) is the distribution of the local field \(z\) associated to the agent, that is constructed over the influence of its neighborhood through \(\mathsf{P}(y|\hat{\pi})\), the distribution of consensus in the neighborhood of the agent, and the influence of the agenda through the average over \(x\). These two contributions represent the sources the agent uses to form its opinion. The distribution of the neighborhood effective field \(\hat{\pi}(s)\) acting on the local agent is obtained by averaging over the distribution of the local agent field through \(\pi(z)\) and through the distribution of influences through the average over \(\eta\). Observe that if there is agreement between agent and president and if the influence between peers is strong (large \(\eta\)), the neighborhood field \(s\) becomes large and positive. If the agent does not give any importance to its peers (\(\eta=0\)), the distribution \(\hat{\pi}(s)\) becomes a delta function centered at zero, and the system decouples.

In addition, these functional saddle point equations also depend on the usual parameters \(q\) and \(R\), which satisfy

\[\frac{q-R^{2}}{1-q} = \frac{\alpha}{\pi}\int\mathrm{d}y\,\mathsf{P}(y|\hat{\pi})\int \mathcal{D}x\,\frac{\exp\left(-\frac{qx^{2}}{1-q}\right)\,\mathcal{H}\left(- \frac{Rx}{\sqrt{q-R^{2}}}\right)}{\left[\epsilon+\mathcal{H}\left(-\sqrt{ \frac{q}{1-q}x}\right)\right]^{2}} \tag{27}\] \[\frac{R}{\sqrt{1-q}} = \frac{\alpha}{\pi}\sqrt{\frac{q}{q-R^{2}}}\int\mathrm{d}y\, \mathsf{P}(y|\hat{\pi})\int\mathcal{D}x\frac{\exp\left\{-\left(\frac{q}{1-q} +\frac{R^{2}}{q-R^{2}}\right)\frac{x^{2}}{2}\right\}}{\epsilon+\mathcal{H} \left(-\sqrt{\frac{q}{1-q}x}\right)}, \tag{28}\]

The numerical solution of this set of equations is discussed in Section VII.2.

## V Macroscopic characterization of the model

In Section VII.2 we demonstrate that for sufficiently large neighborhoods (\(\nu>O(1)\)), for sufficiently high pressure \(\beta\), and for a very narrow distribution of social strengths, i.e. \(\Delta\ll\eta_{0}\) and \(\mathcal{P}(\eta)=\mathcal{N}(\eta|\eta_{0},\Delta^{2})\), there are three possible solutions for equations (25) and (26). Two of them are the pure _conservative_ state, obtained if \(\nu\eta_{0}<1\) and the other is the _polarized_ pure state if \(\nu\eta_{0}>1\). There is a possibility of a third solution which is a mixture of the two pure states, that appears in the region of the phase space where dialogue between opposite positions may exist. By defining the parameter \(\Lambda\) as:

\[\Lambda(R)\equiv\frac{\mathrm{sgn}(R)}{2\beta}\frac{q}{1-q}, \tag{29}\]

which allow us to plot a partial phase diagram, presented in figure 7. In the region \((|\Lambda|,\eta_{0})\in\mathbb{A}\) a convex combination of both pure states is found. For \((|\Lambda|,\eta_{0})\notin\mathbb{A}\) the distributions can be expressed as:

\[\hat{\pi}_{0}(z) \equiv \mathcal{N}\left(z\left|\mathcal{I}_{0}^{\star}(\Lambda,\eta_{0} ),\eta_{0}^{2}-[\mathcal{I}_{0}^{\star}(\Lambda,\eta_{0})]^{2}+\Delta^{2}\right.\right) \tag{30}\] \[\pi_{0}(s) \equiv \mathcal{N}\left(s\left|1+\nu[\mathcal{I}_{0}^{\star}(\Lambda, \eta_{0})-\eta_{0}]+\frac{1}{2}\Lambda,\nu(\eta_{0}^{2}+\Delta^{2})+\frac{3}{4 }\Lambda^{2}\right.\right)\] (31) \[\mathsf{P}_{0}(y|\hat{\pi}) \equiv \mathcal{N}\left(y\left|\nu\mathcal{I}_{0}^{\star}(\Lambda,\eta_ {0}),\nu(\eta_{0}^{2}+\Delta^{2})\right.\right), \tag{32}\]

where \(\mathcal{I}_{0}^{\star}\) is the only solution to

\[\mathcal{I}_{0}^{\star}=\eta_{0}\mathrm{erf}\left(\frac{1-\nu\eta_{0}+\nu \mathcal{I}_{0}^{\star}+\frac{1}{2}\Lambda(R)}{\sqrt{2\left[\nu(\eta_{0}^{2}+ \Delta^{2})+\frac{3}{4}\Lambda(R)^{2}\right]}}\right) \tag{33}\]

outside region \(\mathbb{A}\) (this equation is developed in Section VII.2, equation (94)). We have observed that in the region of interest \(\nu\eta_{0}\sim O(1)\), the variance of \(\mathsf{P}_{0}(y|\hat{\pi})\) is of order \(O(\nu^{-1})\), therefore we can approximate this distribution by:

\[\mathsf{P}_{0}(y|\hat{\pi})\approx\delta(y-\nu\mathcal{I}_{0}^{\star}). \tag{34}\]

Inside the region \(\mathbb{A}\) we have mixed states described by:

\[\hat{\pi}_{\mathrm{m}}(z) \equiv h_{+}\mathcal{N}\left(s\left|\mathcal{I}_{+}^{\star},\Delta^{2} \right.\right)+h_{-}\mathcal{N}\left(s\left|\mathcal{I}_{-}^{\star},\Delta^{2} \right.\right) \tag{35}\] \[\pi_{\mathrm{m}}(s) \equiv \mathcal{N}\left(s\left|1+\nu[\mathcal{I}^{\star}-\eta_{0}]+ \frac{1}{2}\Lambda,\nu(\eta_{0}^{2}+\Delta^{2})+\frac{3}{4}\Lambda^{2}\right.\right)\] (36) \[\mathsf{P}_{\mathrm{m}}(y|\hat{\pi}) \equiv \mathcal{N}\left(y\left|\nu\mathcal{I}^{\star},\nu\{\left(( \mathcal{I}^{\star})^{2}\right)+\Delta^{2}\}\right.\right), \tag{37}\]
where \({\cal I}_{\pm}^{\star}\) are the stable solutions to equation (33) in \(\mathbb{A}\), \(h_{\pm}\) are suitable weights (104) satisfying \(0\leq h_{\pm}\leq 1\) and \(h_{+}+h_{-}=1\), \({\cal I}^{\star}\) is the mixed solution:

\[{\cal I}^{\star} \coloneqq h_{+}{\cal I}_{+}^{\star}+h_{-}{\cal I}_{-}^{\star} \tag{38}\] \[\left\langle({\cal I}^{\star})^{2}\right\rangle \coloneqq h_{+}({\cal I}_{+}^{\star})^{2}+h_{-}({\cal I}_{-}^{\star})^{2}, \tag{39}\]

and, given that for all \((\lambda,\eta_{0})\in\mathbb{A}\), \(\nu\eta_{0}\sim O(1)\), we have that:

\[{\sf P}_{\rm m}(y|\hat{\pi})\approx\delta\left(y-\nu{\cal I}^{\star}\right). \tag{40}\]

The application of equations (34) and (40) into equations (27) and (28) produce the following expressions

\[\frac{q-R^{2}}{1-q} = \frac{\alpha}{\pi}\left({\rm e}^{\kappa}-1\right)^{2}\int{\cal D }x\,\frac{\exp\left(-\frac{qx^{2}}{1-q}\right)\,{\cal H}\left(-\frac{Rx}{ \sqrt{q-R^{2}}}\right)}{\left[1+\left({\rm e}^{\kappa}-1\right){\cal H}\left( -\sqrt{\frac{q}{1-q}x}\right)\right]^{2}} \tag{41}\] \[\frac{R}{\sqrt{1-q}} = \frac{\alpha}{\pi}\left({\rm e}^{\kappa}-1\right)\sqrt{\frac{q}{ q-R^{2}}}\int{\cal D}x\frac{\exp\left\{-\left(\frac{q}{1-q}+\frac{R^{2}}{q-R^{2}} \right)\frac{x^{2}}{2}\right\}}{1+\left({\rm e}^{\kappa}-1\right){\cal H}\left( -\sqrt{\frac{q}{1-q}x}\right)}, \tag{42}\]

where \(\kappa\equiv 2\beta(1-\nu\eta_{0}+\nu{\cal I}^{\star})\). Observe that these equations are invariant under the following transformation \((\kappa,q,R)\rightarrow(-\kappa,q,-R)\).

At very high pressure \(\beta\), the equations (27) and (28) can be expressed as:

\[\frac{q_{\pm}-R_{\pm}^{2}}{1-q_{\pm}} = \frac{\alpha}{\pi}\int{\cal D}x\,\frac{\exp\left(-\frac{q_{\pm}x ^{2}}{1-q_{\pm}}\right)\,{\cal H}\left(-\frac{R_{\pm}x}{\sqrt{q_{\pm}-R_{\pm} ^{2}}}\right)}{\left[{\cal H}\left(-\sqrt{\frac{q_{\pm}}{1-q_{\pm}}x}\right) \right]^{2}} \tag{43}\] \[\frac{R_{\pm}}{\sqrt{1-q_{\pm}}} = \pm\frac{\alpha}{\pi}\sqrt{\frac{q_{\pm}}{q_{\pm}-R_{\pm}^{2}}} \int{\cal D}x\frac{\exp\left\{-\left(\frac{q_{\pm}}{1-q_{\pm}}+\frac{R_{\pm}^ {2}}{q_{\pm}-R_{\pm}^{2}}\right)\frac{x^{2}}{2}\right\}}{{\cal H}\left(-\sqrt {\frac{q_{\pm}}{1-q_{\pm}}x}\right)}, \tag{44}\]

where the sub-index \(+(-)\) is valid for \(\nu\eta_{0}<(>)1\). The \(\beta\rightarrow\infty\) solutions satisfy \(q_{\pm}=\pm R_{\pm}\). These results justify naming the solution with sub-index \(+\) as conservative and the solution with sub-index \(-\) as polarized. Similar behavior is observed for finite but large values of the pressure.

For sufficiently large pressures, sufficiently large \(\nu\) and a volume of information \(\alpha\gg\beta\), we can also demonstrate that:

\[q = 1-\frac{Q_{0}^{2}}{\alpha^{2}}+o(\alpha^{-2}) \tag{45}\] \[Q_{0} = \frac{(2\pi)^{3/2}}{2+\sqrt{\pi}} \tag{46}\]

and

\[R=\begin{cases}q+2\pi\sqrt{3}Q_{0}^{3}\alpha^{-3}{\rm e}^{-2\beta}&\nu\eta_{0 }<1\\ -q-2\pi\sqrt{3}Q_{0}^{3}\alpha^{-3}{\rm e}^{-2\beta(\nu\eta_{0}-1)}&1<\nu\eta_ {0}\end{cases}. \tag{47}\]

Due to the odd parity of \(R(\kappa)\), we can conclude that the plane \((\nu\eta_{0},\beta^{-1})\) is divided into two phases, a conservative phase for which \(R>0\) and \(\nu\eta_{0}<1\) and a polarized phase with \(R<0\) and \(\nu\eta_{0}>1\).

Consider the set of points with coordinates \((\beta^{-1},\nu\eta=1,\alpha)\) such that the parameter defined in equation (29) becomes \(|\Lambda^{\star}|=0.411\) (see figure 7). In consequence, the solution of equation (33) over this line is \(\nu{\cal I}^{\star}=0.988\) and the correspondent index \(\kappa\) becomes a function of the overlap \(q\), i.e.

\[\kappa^{\star}(q)=\frac{\nu{\cal I}^{\star}}{|\Lambda^{\star}|}\frac{q}{1-q}. \tag{48}\]
By solving the equations (41) and (42) with \(\kappa\) given by (48), we obtain the curve presented in figure 4.

We solve for the properties of the equilibrium state valid in the \(\tau_{q}\) time scale in the intensive parameters of the system: \(\beta\) the pressure, \(\alpha\) a measure of the complexity of the agenda and \(\nu\eta_{0}\) a measure of the peer pressure by other agents in congress which arises from the mean number of interlocutors \(\nu\) and the mean intensity of their interaction \(\eta_{0}\). These results are presented as the phase diagrams shown in figure 5.

By fixing the value of \(\alpha\), we can study the behavior of the system for a given volume of information. We constructed figure 6 by solving equation (33) for different values of \(\beta^{-1}\) and \(\nu\eta_{0}\) at fixed values of \(\alpha\) with \(\nu=10\) and \(\Delta=0.01\). The

Figure 4: Critical pressure against volume of information. When the number of issues treated in the legislature is not sufficiently large, there is always a phase around the opinion boundary \(\nu\eta=1\) where polarized and conservatives states coexist. This area represents the collection of points \((\alpha,\beta^{-1})\) where a discussion between members of the chamber with different positions may occur. There is a critical volume of information \(\alpha^{*}=1.534(1)\) bellow which there is always room for discussion, no matter how high the pressure \(\beta^{-1}\) is. Above this threshold there is always a maximum pressure \(\beta^{-1}(\alpha)\) such that above it there is no more discussion and positions are definitely set.

Figure 5: Phase diagram of the system in terms of the parameters \(\nu\eta_{0}\), \(\beta^{-1}\) and \(\alpha\). There are two phases separated by the plane \(\nu\eta_{0}=1\). For \(\nu\eta_{0}<1\) we have that \(R>0\) and the average consensus is in favor of \(B\). For \(\nu\eta_{0}>1\), \(R<0\) and the average position of the agents is to form local alliances against the president \(B\). In all the points of the space above the surfaces \(\nu\eta_{+}\) and \(\nu\eta_{-}\), the distribution describing the position of the neighborhood, given by \(\hat{\pi}(z)\), is sharply picked at \(+\eta_{0}\), for the conservative position, i.e. \(\nu\eta_{0}<1\), or at \(-\eta_{0}\), for the polarized position, i.e. \(\nu\eta_{0}>1\). In the region bellow the surfaces \(\nu\eta_{+}\) and \(\nu\eta_{-}\) we have the same phase separation at \(\nu\eta_{0}=1\) but the contribution from the neighborhood is a mixture of a polarized component plus a conservative component. The circle at coordinates \(\nu\eta_{0}=1\), \(\beta^{-1}=0.824\) and \(\alpha=1.534\) is the critical point presented in figure 4. The phase diagram presented in figure 6 has been obtained by cutting sections at constant \(\alpha\) from this three-dimensional plot, and the the red sphere corresponds to the first value of \(\alpha\) (\(=\alpha^{*}\)) for which the behavior presented in figure 6 c) is observed.
full lines separate pure-state areas [in white, for \(R<0\) and in dark gray (orange on-line) for \(R>0\), given by equations (30), (31), and (32)] from mixed-state areas [in gray (yellow on-line), given by equations (35), (36), and (37)]. We also found that for values of \(\alpha<\alpha^{\star}=1.534(1)\) the mixed states are contained into a mixed-triangular-shaped area, with vertexes at \((\beta^{-1}=0,\nu\eta_{0}=1.651(1))\), \((\beta^{-1}=0,\nu\eta_{0}=0.717(1))\), and \((\beta=\beta(\alpha),\nu\eta_{0}=1).\) In particular we observe that \(\beta^{\star}\equiv\beta(\alpha^{\star})\approx 1.214(1)\) and for all \(\alpha^{\star}<\alpha^{\prime}<\alpha\), \(\beta(\alpha)>\beta(\alpha^{\prime})>\beta(\alpha^{\star}).\) The lightly shaded (yellow on-line) region close to the boundary (\(\nu\eta_{0}=1\)) is characterized by a mixture of states that represents a state of dialog, where the influence on the agents from their neighborhoods come from both sides of the argument. The larger the complexity of the agenda (\(\alpha\)) the smaller the size of this region.

To complete our analysis and for very low values of \(\beta\) we obtain the following values for the parameters \(R\) and \(q\):

\[q \approx \frac{2\alpha}{\pi}\beta^{2}(1-\nu\eta_{0})^{2}\left(\frac{2 \alpha}{\pi}+1\right) \tag{49}\] \[R \approx \frac{2\alpha}{\pi}\beta(1-\nu\eta_{0})[1-2\beta(1-\nu\eta_{0})]. \tag{50}\]
Discussion and conclusions

During the last decade [6] the application of statistical mechanics techniques to model social problems have produced a number of interesting results, not only providing new insight to the discussion of social phenomena but also showing predictive capabilities [10]. Inspired by these ideas, we have develop a model for the phenomenon of impeachment in presidential democracies. The political agents, represented by perceptrons, interact with an external meta-agent \(B\), which represents the executive, and with peers in the legislative chamber. The model has been tailored to balance the need to explain observed behavior [17], the complexity of the social interactions [19; 31], and the analytical tractability of the mathematical expressions constructed.

It is important to note that, for sufficiently large number of alliances \(\nu>O(1)\), the saddle point equations (25 to 28) can be solved in pairs. The first two (25) and (26) involving the distributions \(\pi\) and \(\hat{\pi}\) connected with the distribution of alliances and the pair (27) and (28), connected to the parameters associated with the discussion of the presidential agenda. The solution to (25) and (26) has been expressed using the parameter \(\Lambda\) defined in equation (29), which brings an input from the disorder-from-learning part of the problem into the disorder-from-graph part of the problem. In a similar manner, the parameter \(\kappa\) that helps to express the solution of equations (27) and (28), introduces effects from the disorder-from-graph part of the problem into the disorder-from-learning part of the problem. The constraints that emerged from expressing the solution in terms of these parameters have help constructing the phase diagram presented in figure 7.

We obtained a set of sensible results for a topology represented by a directed graph with an average of \(\nu\) links per vertex (the number of co-conspirators). In this setting and considering a steady president, i.e. \(B\) constant, we found that there exist two possible pure positions. One characterized by an overall average attitude in favor of the president characterized by a positive and increasing (with the volume of information) average agreement \(R\), that we dubbed the conservative state, and other with a negative and decreasing value of \(R\), the polarized state. These states are also characterized by a sharp picked distribution of neighbors' influences \(\hat{\pi}\), centered at \(\mathcal{I}_{+}^{\star}\) (\(\mathcal{I}_{-}^{\star}\)) for the conservative (polarized) state. From figure 6 we also showed that for volumes of information bellow a critical value \(\alpha^{\star}=1.534(1)\), there is a region in the plane \((\nu\eta_{0},\beta^{-1})\), in a form of a band around \(\nu\eta_{0}=1\), where mixed states, defined by the equations (35), (36), and (37), exist. The mixture is explicit in equation (35) that presents the influence on an agent by its neighborhood (\(\hat{\pi}\)) as a combination of the two sides of the argument. This band of mixed states collapses into a triangle with vertexes at \((\beta^{-1}=0,\nu\eta_{0}=1.651(1))\), \((\beta^{-}1=0,\nu\eta_{0}=0.71845(1))\), and \((\beta=\beta(\alpha),\nu\eta_{0}=1)\), for values of \(\alpha>\alpha^{\star}.\) We also observed that the larger the volume of information the smaller the triangle area, i.e. \(\alpha>\alpha^{\prime}\) implies that \(\beta(\alpha)>\beta(\alpha^{\prime}).\) The interpretation of this behavior is as follows: when information is limited (low \(\alpha\)) and for values of effective co-conspirators \(\nu\eta_{0}\simeq 1\) the influence from the neighborhood to the agent is formed by a combination of positions in favor and against the executive. In this region the overlap \(R,\) which represents the average agreement with \(B\) still has a well defined sign given by \(\mathrm{sgn}(R)=\mathrm{sgn}(1-\nu\eta_{0}),\) but is the result from two pure-state contributions. In this region coexist the two positions, in pro and against the executive \(B\). Definite positions are not set yet, thus propitiating a state of dialogue. The more information is fed to the system the smaller this region becomes. There is a critical value of information \(\alpha^{\star}=1.534(1),\) beyond which this behavior is only observed for pressures lower than \(\beta^{\star}=1.214(1).\) In other words, the more information is provided the purer the contribution to the agents opinion from their neighborhoods and the lower the chances for a dialog between opposed positions. For very large values of \(\alpha\) and \(\nu\eta_{0}\approx 1\) coexistence exists only if the pressure \(\beta\) is sufficiently high. Thus, only a president with high index of popularity can guarantee a discussion of the topics in the agenda between opposite positions of the legislative.

Under the light of the cases used as motivation for our model we observe that there are events, represented by particular items of the executive's agenda, that are so momentous in the formation of opinions that can be considered critical (Collor de Mello's economic plans, Perez's Grate Turn Around), to the point that, immediately after they occur, opposite positions in pro or against the executive's proposals become more consolidated, and the influence of the neighborhood on the agents becomes more polarized (on either position) and the dialogue-prone region gets reduced. If the public rejects the proposals, \(\beta\) diminishes and the executive may find itself in front of a polarized legislative chamber that either supports it (Samper's case) or not (Collor de Mello's case). If the negative information instances persist and neither the public or the chamber supports the president, the executive may find itself facing an impeachment procedure.

A natural extension to this work comes through the consideration of the case of a changing \(B\). In a previous work [25] we have study the evolution of opinions in the presence of an adaptive social rule that slowly changes following the average position of the population. As a consequence, the contribution from socially neutral issues (i.e. issues \(\boldsymbol{\xi}_{0}\) such that \(\mathbf{B}\cdot\boldsymbol{\xi}_{0}=0\)) becomes relevant [13], as it can be observed by the presence of the parameter \(W\), which represents the overlap between the representation of different agents (and it is a measure of the level of agreement between them). We expect that, if a similar setting is imposed in the present framework, the free energy functional so obtain should be dependent also on a parameter \(W\), revealing the contribution from the socially neutral issues to
the system.

**Acknowledgment:** This work received partial support from CNAIPS-NAP USP.

## VII Supplementary Material

### Calculation of the averages over \(\mathcal{A}\), \(\mathbf{B}\) and \(\boldsymbol{G}\)

By observing that \(\mathcal{P}(g_{ac})=\int\mathrm{d}\eta_{ac}\mathcal{N}(\eta_{ac}|\eta_{0}, \Delta^{2})\sum_{x_{ac}=0,1}[p\delta_{x_{ac},1}+(1-p)\delta_{x_{ac},0}]\, \delta(g_{ac}-x_{ac}\eta_{ac})\), where the Kronecker's delta is \(\delta_{X,Y}=1\) if \(X=Y\) and \(0\) otherwise and Dirac's delta is \(\int_{\Omega}\mathrm{d}x\delta(x-x_{0})=1\) if \(x_{0}\in\Omega\) and \(0\) otherwise, the replicated partition function is:

\[\overline{Z^{n}}(\beta) \equiv \int\mathrm{d}\mathbf{B}\mathcal{P}(\mathbf{B})\int\prod_{\mu} \mathrm{d}\boldsymbol{\xi}_{\mu}\mathcal{P}(\boldsymbol{\xi}_{\mu})\prod_{a} \prod_{c}\int\mathrm{d}g_{ac}\mathcal{P}(g_{ac})\int\prod_{\gamma=1}^{n}\prod _{a}\mathrm{d}\mathbf{J}_{a}^{\gamma}\mathcal{P}(\mathbf{J}_{a}^{\gamma}) \tag{51}\] \[\prod_{\gamma\mu a}\exp\left\{\beta\sum_{c\in\mathbb{N}_{a}}x_{ ac}\eta_{ac}\mathrm{sgn}\left(\frac{\mathbf{J}_{a}^{\gamma}\cdot\boldsymbol{\xi}_{ \mu}}{\sqrt{N}}\right)\mathrm{sgn}\left(\frac{\mathbf{J}_{c}^{\gamma}\cdot \boldsymbol{\xi}_{\mu}}{\sqrt{N}}\right)\right\}\] \[\prod_{\gamma\mu a}\exp\left\{(1-\nu\eta_{0})\beta\mathrm{sgn} \left(\frac{\mathbf{J}_{a}^{\gamma}\cdot\boldsymbol{\xi}_{\mu}}{\sqrt{N}} \right)\mathrm{sgn}\left(\frac{\mathbf{B}\cdot\boldsymbol{\xi}_{\mu}}{\sqrt{N} }\right)\right\},\]

and by defining the variables:

\[\lambda_{a,\mu}^{\gamma}\equiv\frac{\mathbf{J}_{a}^{\gamma}\cdot\boldsymbol{ \xi}_{\mu}}{\sqrt{N}},\qquad u_{\mu}\equiv\frac{\mathbf{B}\cdot\boldsymbol{ \xi}_{\mu}}{\sqrt{N}} \tag{52}\]

and, by defining the overlaps:

\[R_{a}^{\gamma}\equiv\frac{\mathbf{J}_{a}^{\gamma}\cdot\mathbf{B }}{N},\qquad\quad W_{ab}^{\gamma}=\frac{\mathbf{J}_{a}^{\gamma}\cdot\mathbf{J }_{b}^{\gamma}}{N},\] \[q_{a}^{\gamma\rho}\equiv\frac{\mathbf{J}_{a}^{\gamma}\cdot \mathbf{J}_{a}^{\rho}}{N},\qquad\quad t_{ab}^{\gamma\rho}\equiv\frac{\mathbf{J }_{a}^{\gamma}\cdot\mathbf{J}_{b}^{\rho}}{N}, \tag{53}\]

we have that the expectation over patterns is:

\[\left\langle\cdot\right\rangle_{\mathcal{A}} \equiv \int\prod_{\mu}\mathrm{d}\boldsymbol{\xi}_{\mu}\mathcal{P}( \boldsymbol{\xi}_{\mu})\exp\left(i\sum_{\gamma\mu a}\hat{\lambda}_{a\mu}^{ \gamma}\frac{\mathbf{J}_{a}^{\gamma}\cdot\boldsymbol{\xi}_{\mu}}{\sqrt{N}}+i \sum_{\mu}\hat{u}_{\mu}\frac{\mathbf{B}\cdot\boldsymbol{\xi}_{\mu}}{\sqrt{N}}\right) \tag{54}\] \[= \int\prod_{\gamma a}\frac{\mathrm{d}R_{a}^{\gamma}\mathrm{d}\hat{ R}_{a}^{\gamma}}{2\pi/N}\,\exp\left(i\sum_{\gamma a}\hat{R}_{a}^{\gamma}(NR_{a}^{ \gamma}-\mathbf{J}_{a}^{\gamma}\cdot\mathbf{B})\right)\] \[\int\prod_{\gamma}\prod_{a<b}\frac{\mathrm{d}W_{ab}^{\gamma} \mathrm{d}\hat{W}_{ab}^{\gamma}}{2\pi/N}\,\exp\left(i\sum_{\gamma}\sum_{a<b} \hat{W}_{ab}^{\gamma}(NW_{ab}^{\gamma}-\mathbf{J}_{a}^{\gamma}\cdot\mathbf{J}_ {b}^{\gamma})\right)\] \[\int\prod_{a}\prod_{\gamma<\rho}\frac{\mathrm{d}q_{a}^{\gamma\rho} \mathrm{d}\hat{q}_{a}^{\gamma\rho}}{2\pi/N}\,\exp\left(i\sum_{a}\sum_{\gamma< \rho}\hat{q}_{a}^{\gamma\rho}(Nq_{a}^{\gamma\rho}-\mathbf{J}_{a}^{\gamma}\cdot \mathbf{J}_{a}^{\rho})\right)\] \[\int\prod_{\gamma<\rho}\prod_{a<b}\frac{\mathrm{d}t_{ab}^{\gamma \rho}\mathrm{d}\hat{q}_{ab}^{\gamma\rho}}{2\pi/N}\,\exp\left(i\sum_{a<b}\sum_{ \gamma<\rho}\hat{t}_{ab}^{\gamma\rho}(Nt_{ab}^{\gamma\rho}-\mathbf{J}_{a}^{\gamma }\cdot\mathbf{J}_{b}^{\rho})\right)\] \[\exp\left\{-\frac{1}{2}\sum_{\mu}\left[\sum_{\gamma a}\left(\hat{ \lambda}_{a\mu}^{\gamma}\right)^{2}+2\sum_{\gamma a}\sum_{\gamma<\rho}\hat{ \lambda}_{a\mu}^{\gamma}\hat{\lambda}_{a\mu}^{\rho}q_{a}^{\gamma\rho}+2\sum_{ \gamma a}\sum_{a<b}\hat{\lambda}_{a\mu}^{\gamma}\hat{\lambda}_{b\mu}^{\gamma}W_{ ab}^{\gamma}+\right.\right.\] \[\left.\left.+2\sum_{\gamma a}\sum_{\gamma<\rho}\sum_{a<b}\hat{ \lambda}_{a\mu}^{\gamma}\hat{\lambda}_{b\mu}^{\rho}t_{ab}^{\gamma\rho}+2\sum_{ \gamma a}\hat{u}_{\mu}\hat{\lambda}_{a\mu}^{\gamma}R_{a}^{\gamma}+\hat{u}_{\mu} ^{2}\right]\right\}+O(N^{-1}).\]

By considering the distribution of the synaptic vector \(\mathbf{B}\) as \(\mathcal{P}(\boldsymbol{B})=\prod_{k}\delta(B_{k}-1)\) and by defining the matrices:

\[[\boldsymbol{\hat{Q}}]_{a,b}^{\gamma,\rho} \equiv i\left\{\delta^{\gamma,\rho}\left(\delta_{a,b}\hat{\ell}_{a}^{ \gamma}+(1-\delta_{a,b})\hat{W}_{a,b}^{\gamma}\right)+(1-\delta^{\gamma,\rho}) \left(\delta_{a,b}\hat{q}_{a}^{\gamma,\rho}+(1-\delta_{a,b})\hat{t}_{a,b}^{ \gamma,\rho}\right)\right\} \tag{55}\]
\[[{\mathbf{Q}}]^{\gamma,\rho}_{a,b} \equiv \delta^{\gamma,\rho}\left(\delta_{a,b}+(1-\delta_{a,b})W^{\gamma}_{a, b}\right)+(1-\delta^{\gamma,\rho})\left(\delta_{a,b}q^{\gamma,\rho}_{a}+(1- \delta_{a,b})t^{\gamma,\rho}_{a,b}\right) \tag{56}\]

we have that the average over synaptic vectors become:

\[\langle\cdot\rangle_{{\bf B},\{{\bf J}^{\gamma}_{a}\}} = \int\prod_{\gamma,a}\frac{{\rm d}\hat{\ell}^{\gamma}_{a}}{4\pi} \exp\left(i\frac{N}{2}\sum_{\gamma,a}\hat{\ell}^{\gamma}_{a}-N\ln|\hat{\mathbf{Q}} |-\frac{1}{2}\sum_{a,b}\sum_{\gamma,\rho}\hat{R}^{\gamma}_{a}[\hat{\mathbf{Q}}^{-1} ]^{\gamma,\rho}_{a,b}\hat{R}^{\rho}_{b}-\frac{nNM}{2}\right), \tag{57}\]

which renders the following expression for the partition function:

\[\overline{Z^{n}}(\beta) = \int\prod_{\gamma a}\frac{{\rm d}\hat{\ell}^{\gamma}_{a}}{4\pi} \int\prod_{\gamma a}\frac{{\rm d}R^{\gamma}_{a}{\rm d}\hat{R}^{\gamma}_{a}}{2 \pi/N}\int\prod_{\gamma}\prod_{a<b}\frac{{\rm d}W^{\gamma}_{ab}{\rm d}\hat{W}^ {\gamma}_{ab}}{2\pi/N}\int_{a}\prod_{\gamma<\rho}\frac{{\rm d}q^{\gamma\rho}_{ a}{\rm d}\hat{q}^{\gamma\rho}_{a}}{2\pi/N}\int\prod_{\gamma<\rho}\prod_{ab}\frac{{ \rm d}t^{\gamma\rho}_{ab}{\rm d}\hat{t}^{\gamma\rho}_{ab}}{2\pi/N} \tag{58}\] \[\exp\left(\frac{N}{2}\mbox{tr}{\mathbf{Q}}\hat{\mathbf{Q}}-\frac{N}{2} \ln|\hat{\mathbf{Q}}|-\frac{N}{2}\sum_{ab}\sum_{\gamma,\rho}\hat{R}^{\gamma}_{a} \left[\hat{\mathbf{Q}}^{-1}\right]^{\gamma\rho}_{ab}\hat{R}^{\rho}_{b}+iN\sum_{ \gamma a}\hat{R}^{\gamma}_{a}R^{\gamma}_{a}-\frac{nNM}{2}\right)\] \[\int\prod_{\gamma\mu a}\frac{{\rm d}\lambda^{\gamma}_{a\mu}{\rm d }\hat{\lambda}^{\gamma}_{a\mu}}{2\pi}\exp\left(-i\sum_{\gamma\mu a}\hat{ \lambda}^{\gamma}_{a\mu}\lambda^{\gamma}_{a\mu}\right)\] \[\int\prod_{\mu}{\cal D}u_{\mu}\exp\left(i\sum_{\gamma\mu a}\hat {\lambda}^{\gamma}_{a\mu}R^{\gamma}_{a}u_{\mu}+(1-\nu\eta_{0})\beta\sum_{ \gamma\mu a}\mbox{sgn}(\lambda^{\gamma}_{a\mu}u_{\mu})\right)\] \[\exp\left\{-\frac{1}{2}\sum_{\mu}\left[\sum_{\gamma a}\left[1-(R^ {\gamma}_{a})^{2}\right]\left(\hat{\lambda}^{\gamma}_{a\mu}\right)^{2}+2\sum_{ \gamma a}\sum_{\gamma<\rho}\left[q^{\gamma\rho}_{a}-R^{\gamma}_{a}R^{\rho}_{a} \right]\hat{\lambda}^{\gamma}_{a\mu}\hat{\lambda}^{\rho}_{a\mu}+\right.\right.\] \[\left.\left.+2\sum_{\gamma a}\sum_{a<b}\left[W^{\gamma}_{ab}-R^{ \gamma}_{a}R^{\gamma}_{b}\right]\hat{\lambda}^{\gamma}_{a\mu}\hat{\lambda}^{ \gamma}_{b\mu}+2\sum_{\gamma a}\sum_{\gamma<\rho}\sum_{b}\left[t^{\gamma\rho}_ {ab}-R^{\gamma}_{a}R^{\rho}_{b}\right]\hat{\lambda}^{\gamma}_{a\mu}\hat{ \lambda}^{\rho}_{b\mu}\right]\right\}\] \[\left\langle\exp\left\{\beta\sum_{\gamma\mu a}\sum_{a\neq c}x_{ac }\eta_{ac}\mbox{sgn}(\lambda^{\gamma}_{a\mu}\lambda^{\gamma}_{c\mu})\right\} \right\rangle_{\!\!\!\!G}+O(N^{-1})\]

in the limit of large \(N\) we find that

\[\hat{R}^{\gamma}_{a} = i\sum_{\rho,b}[\hat{\mathbf{Q}}]^{\gamma,\rho}_{a,b}R^{\rho}_{b} \tag{58}\] \[\left[\hat{\mathbf{Q}}^{-1}\right]^{\gamma,\rho}_{a,b} = [{\mathbf{K}}]^{\gamma,\rho}_{a,b}\equiv[{\mathbf{Q}}]^{\gamma,\rho}_{a,b}- R^{\gamma}_{a}R^{\rho}_{b} \tag{59}\]

and to express the extraction of the asymptotic behavior of integrals of the form \(I_{N}\equiv\int_{x_{1}}^{x_{2}}{\rm d}x\,{\rm e}^{-Ng(x)}\) in the limit \(N\to\infty\) through Laplace's method, we denote: \(\mbox{extr}_{x}\,I_{N}\equiv{\rm e}^{-Ng(x_{0})+O(\log N)}\) where \(x_{0}\) is such that \(g(x_{0})\leq g(x)\) for all \(x\in[x_{1},x_{2}]\), so we can write:

\[\overline{Z^{n}}(\beta) = \mbox{extr}_{\mathbf{K}}\left\{\exp\left(\frac{N}{2}\ln|{\mathbf{K}}| \right)\right. \tag{60}\] \[\int\prod_{\gamma\mu a}\frac{{\rm d}\lambda^{\gamma}_{a\mu}{\rm d }\hat{\lambda}^{\gamma}_{a\mu}}{2\pi}\exp\left(-i\sum_{\gamma\mu a}\hat{ \lambda}^{\gamma}_{a\mu}\lambda^{\gamma}_{a\mu}\right)\] \[\int\prod_{\mu}{\cal D}u_{\mu}\,\exp\left(i\sum_{\gamma a}\hat{ \lambda}^{\gamma}_{a\mu}R^{\gamma}_{a}u_{\mu}+(1-\nu\eta_{0})\beta\sum_{\gamma \mu a}\mbox{sgn}(\lambda^{\gamma}_{a\mu}u_{\mu})\right)\] \[\exp\left[-\frac{1}{2}\sum_{\gamma\mu a}\left[1-(R^{\gamma}_{a})^ {2}\right]\left(\hat{\lambda}^{\gamma}_{a\mu}\right)^{2}-\sum_{\gamma\mu a}\sum_ {\gamma<\rho}\left[q^{\gamma\rho}_{a}-R^{\gamma}_{a}R^{\rho}_{a}\right]\hat{ \lambda}^{\gamma}_{a\mu}\hat{\lambda}^{\rho}_{a\mu}-\right.\right.\] \[\left.\left.-\sum_{\gamma\mu a}\sum_{a<b}\left[W^{\gamma}_{ab}-R^{ \gamma}_{a}R^{\gamma}_{b}\right]\hat{\lambda}^{\gamma}_{a\mu}\hat{\lambda}^{ \gamma}_{b\mu}-\sum_{\gamma\mu a}\sum_{\gamma<\rho}\sum_{b}\left[t^{\gamma\rho}_ {ab}-R^{\gamma}_{a}R^{\rho}_{b}\right]\hat{\lambda}^{\gamma}_{a\mu}\hat{ \lambda}^{\rho}_{b\mu}\right]\] \[\left\langle\exp\left\{\beta\sum_{\gamma\mu a}\sum_{a\neq c}x_{ac }\eta_{ac}\mbox{sgn}(\lambda^{\gamma}_{a\mu})\mbox{sgn}(\lambda^{\gamma}_{c\mu })\right\}\right\rangle_{\!\!\!\!G}\Bigg{\}},\]
where \({\cal D}x\equiv(2\pi)^{-1/2}{\rm d}x\,\exp(-x^{2}/2)\). Also, by imposing the Replica Symmetric (RS) ansatz: \(R_{a}^{\gamma}=R,\,q_{a}^{\gamma,\rho}=q\), and by following [26] we can assume that \(t_{ab}^{\gamma\rho}=W_{ab}^{\gamma}=W,\) then by defining

\[{\cal C}_{a\mu}\equiv\frac{\sqrt{W-R^{2}}y_{\mu}+Ru_{\mu}+\sqrt{q-W}y_{a\mu}}{ \sqrt{1-q}} \tag{62}\]

and by observing that the logarithm of the matrix \(K\) in the RS approach is

\[\ln|\mathbf{K}|=nM\left(\ln(1-q)+\frac{q-R^{2}}{1-q}\right)+O(n^{2}), \tag{63}\]

we have that, after the integration over the variables \(\{\hat{\lambda}_{a\mu}^{\gamma}\}\), the partition function becomes:

\[\overline{Z^{n}}(\beta) = \mathop{\rm extr}_{RqW}\left\{\exp\left[\frac{nNM}{2}\left(\ln(1- q)+\frac{q-R^{2}}{1-q}\right)\right]\right. \tag{64}\] \[\left.\int\prod_{\mu}{\cal D}u_{\mu}\prod_{\mu}{\cal D}y_{\mu} \prod_{\mu a}{\cal D}y_{a\mu}\prod_{\gamma\mu a}\frac{{\rm d}\lambda_{a\mu}^{ \gamma}}{\sqrt{2\pi}}\right.\] \[\left.\exp\left[-\frac{1}{2}\sum_{\gamma\mu a}(\lambda_{a\mu}^{ \gamma}-{\cal C}_{a\mu})^{2}+(1-\nu\eta_{0})\beta\sum_{\gamma\mu a}{\rm sgn} \left(\lambda_{a\mu}^{\gamma}u_{\mu}\right)\right]\right.\] \[\left.\left.\left\langle\exp\left\{\beta\sum_{\gamma\mu a}\sum_{a \neq c}x_{ac}\eta_{ac}{\rm sgn}(\lambda_{a\mu}^{\gamma}){\rm sgn}(\lambda_{c \mu}^{\gamma})\right\}\right\rangle_{\mathbf{G}}\right\}.\]

The average over the graph variables is, by defining the temperature \(\beta^{\prime}\equiv\beta/(1+\nu\eta_{0})\):

\[\Upsilon \equiv \left\langle\exp\left\{\beta\sum_{\gamma\mu a}\sum_{a\neq c}x_{ ac}\eta_{ac}{\rm sgn}(\lambda_{a\mu}^{\gamma}){\rm sgn}(\lambda_{c\mu}^{\gamma}) \right\}\right\rangle_{\mathbf{G}} \tag{65}\] \[= \int\prod_{ac}\frac{{\rm d}\eta_{ac}}{\sqrt{2\pi\Delta^{2}}}\exp \left[-\frac{(\eta_{ac}-\eta_{0})^{2}}{2\Delta^{2}}\right]\prod_{ac}\left\{1-p +p\,\prod_{\gamma\mu}\exp\left[\beta{\rm sgn}(\lambda_{a\mu}^{\gamma}){\rm sgn }(\lambda_{c\mu}^{\gamma})\right]\right\}\] \[= (1-p)^{M(M-1)}\int\prod_{ac}\frac{{\rm d}\eta_{ac}}{\sqrt{2\pi \Delta^{2}}}\exp\left[-\frac{(\eta_{ac}-\eta_{0})^{2}}{2\Delta^{2}}\right]\] \[\prod_{ac}\left\{1+\frac{p}{1-p}\,\prod_{\gamma\mu}\left[\cosh( \beta\eta_{ac})+{\rm sgn}(\lambda_{a\mu}^{\gamma}\lambda_{c\mu}^{\gamma})\sinh (\beta\eta_{ac})\right]\right\}\] \[\prod_{ac}\left\{1+\frac{p}{1-p}\,\cosh(\beta\eta_{ac})^{nP}\prod _{\gamma\mu}\left[1+\tanh(\beta\eta_{ac}){\rm sgn}\left(\lambda_{a\mu}^{ \gamma}\lambda_{c\mu}^{\gamma}\right)\right]\right\}.\]

Observe that we are assuming that the number of neighbors, in average, must be \(\nu\ll M:\)

\[\sum_{a}x_{ac}{\cal P}(x_{ac}) = p(M-1)=\frac{\nu}{2} \tag{66}\]
\[\Upsilon = \left(1-\frac{\nu}{2(M-1)}\right)^{M(M-1)}\int\prod_{ac}\frac{ \mathrm{d}\eta_{ac}}{\sqrt{2\pi\Delta^{2}}}\exp\left[-\frac{(\eta_{ac}-\eta_{0}) ^{2}}{2\Delta^{2}}\right] \tag{67}\] \[\prod_{ac}\left\{1+\frac{p}{1-p}\,\cosh(\beta\eta_{ac})^{nP} \left[1+\tanh(\beta\eta_{ac})\sum_{\gamma\mu}\mathrm{sgn}\left(\lambda_{a\mu}^ {\gamma}\lambda_{\nu\rho}^{\gamma}\right)\right.\right.\] \[\left.\left.\qquad+\tanh(\beta\eta_{ac})^{2}\sum_{\langle\gamma_{ 1}\mu_{1};\gamma_{2}\mu_{2}\rangle}\mathrm{sgn}\left(\lambda_{a\mu_{1}}^{ \gamma_{1}}\lambda_{c\mu_{1}}^{\gamma_{1}}\right)\mathrm{sgn}\left(\lambda_{ a\mu_{2}}^{\gamma_{2}}\lambda_{c\mu_{2}}^{\gamma_{2}}\right)+\ldots\right]\right\}\] \[= \exp\left\{-\frac{\nu}{2}M\right.+\] \[+ \left.\frac{\nu}{2}M\sum_{\ell=0}^{nP}\left\langle\cosh(\beta \eta)^{nP}\tanh(\beta\eta)^{\ell}\right\rangle_{\eta}\sum_{\langle\gamma_{1} \mu_{1};\ldots;\gamma_{\ell}\mu_{\ell}\rangle}\left[\frac{1}{M}\sum_{a} \mathrm{sgn}\left(\lambda_{a\mu_{1}}^{\gamma_{1}}u_{\mu_{1}}\right)\ldots \mathrm{sgn}\left(\lambda_{a\mu_{\ell}}^{\gamma_{\ell}}u_{\mu_{\ell}}\right) \right]^{2}\right\}.\]

Observe that \(\Upsilon\) is the part of the replicated partition function that accounts for the interaction between peers and the interaction between peers and graph. If \(\eta_{0}=\Delta=0\) then \(\Upsilon=1.\) We define

\[\varrho_{\mu_{1}\ldots\mu_{\ell}}^{\gamma_{1}\ldots\gamma_{\ell}} \equiv \frac{1}{M}\sum_{a}\mathrm{sgn}\left(\lambda_{a\mu_{1}}^{\gamma_{ 1}}u_{\mu_{1}}\right)\ldots\mathrm{sgn}\left(\lambda_{a\mu_{\ell}}^{\gamma_{ \ell}}u_{\mu_{\ell}}\right) \tag{68}\] \[\varrho_{0} \equiv \frac{1}{M}\sum_{a}1, \tag{69}\]

where \(\varrho_{\mu_{1}\ldots\mu_{\ell}}^{\gamma_{1}\ldots\gamma_{\ell}}\) is the average level of agreement per individual, across issues and replicas and \(\varrho_{0}\), which is fancy way to write 1, will be left as a free parameter for the time being until we apply a variational technique (which will confirm its value, see below).

Applying Laplace's method to the integrals involving the parameters \(\varrho_{\mu_{1}\ldots\mu_{\ell}}^{\gamma_{1}\ldots\gamma_{\ell}}\) and their conjugates \(\tilde{\varrho}_{\mu_{1}\ldots\mu_{\ell}}^{\gamma_{1}\ldots\gamma_{\ell}}\), we can express the replicated partition function as:

\[\overline{Z^{n}}(\beta) = \mathop{\mathrm{extr}}_{q,W,R,\left\{\varrho_{\mu_{1}\ldots\mu_{ \ell}}^{\gamma_{1}\ldots\gamma_{\ell}},\tilde{\varrho}_{\mu_{1}\ldots\mu_{ \ell}}^{\gamma_{1}\ldots\gamma_{\ell}}\right\}}\left\{\exp\left[\frac{nNM}{2} \left(\ln(1-q)+\frac{q-R^{2}}{1-q}\right)\right]\right. \tag{70}\] \[\left.\exp\left[-M\sum_{\ell=0}\sum_{\langle\gamma_{1}\mu_{1}; \ldots;\gamma_{\ell}\mu_{\ell}\rangle}\varrho_{\mu_{1}\ldots\mu_{\ell}}^{ \gamma_{1}\ldots\gamma_{\ell}}\tilde{\varrho}_{\mu_{1}\ldots\mu_{\ell}}^{\gamma _{1}\ldots\gamma_{\ell}}-\frac{\nu}{2}M+\right.\right.\] \[\left.\left.+\frac{\nu}{2}M\sum_{\ell=0}\left\langle\cosh(\beta \eta)^{nP}\tanh(\beta\eta)^{\ell}\right\rangle_{\eta}\sum_{\langle\gamma_{1} \mu_{1};\ldots;\gamma_{\ell}\mu_{\ell}\rangle}\left(\varrho_{\mu_{1}\ldots\mu_ {\ell}}^{\gamma_{1}\ldots\gamma_{\ell}}\right)^{2}\right]\right.\] \[\left.\int\prod_{\mu}\mathcal{D}u_{\mu}\prod_{\mu}\mathcal{D}y_{ \mu}\left(\prod_{\mu}\mathcal{D}t_{\mu}\prod_{\gamma\mu a}\frac{\mathrm{d} \lambda_{\mu}^{\gamma}}{\sqrt{2\pi}}\exp\left[-\frac{1}{2}\sum_{\gamma\mu}( \lambda_{\mu}^{\gamma}-\mathcal{C}_{\mu})^{2}+\right.\right.\right.\] \[\left.\left.\left.+(1-\nu\eta_{0})\beta\sum_{\gamma\mu}\mathrm{ sgn}\left(\lambda_{\mu_{1}}^{\gamma_{1}}u_{\mu_{1}}\right)\right]\right)^{M}\right\}\]

where we have disregarded terms of \(O(n^{2})\), \(O(N^{-1})\) and \(O(M^{-1})\) in the argument of the exponential and now:

\[\mathcal{C}_{\mu}\equiv\frac{\sqrt{W-R^{2}}y_{\mu}+Ru_{\mu}+\sqrt{q-W}t_{\mu}}{ \sqrt{1-q}}. \tag{71}\]

Once more we consider the RS approach by introducing the distribution \(\pi(z)\) and its conjugate \(\hat{\pi}(z):\)

\[\hat{\varrho}_{\mu_{1}\ldots\mu_{\ell}}^{\gamma_{1}\ldots\gamma_{ \ell}}=\mathcal{C}_{\hat{\pi}}\int\mathrm{d}s\,\hat{\pi}(s)\tanh^{\ell}( \beta s) \varrho_{\mu_{1}\ldots\mu_{\ell}}^{\gamma_{1}\ldots\gamma_{\ell}}=\int \mathrm{d}z\,\pi(z)\tanh^{\ell}(\beta z) \tag{72}\] \[\hat{\varrho}_{0}=\mathcal{C}_{\hat{\pi}}\int\mathrm{d}s\,\hat{\pi }(s) \varrho_{0}=\int\mathrm{d}z\,\pi(z) \tag{73}\]
where equations (72) are the definitions of the fields \(\pi\) and \(\hat{\pi}\), and equations (73) are the normalization conditions they must satisfy. By using the symmetry of the RS supposition and by integrating over the variables \(\{\lambda_{\mu}^{\gamma}\}\), the Hubbard-Stratanovich variables \(u\), \(t\) and \(y\) and by applying the scaling condition \(P=\alpha N\), we have that the replicated partition function takes the form of:

\[\overline{Z^{n}}(\beta) = \mathop{\rm extr}\limits_{q,R,\varrho_{0},\hat{\varrho}_{0},\pi, \hat{\pi}}\left\{\exp\left[-M\left(\frac{\nu}{2}-\hat{\varrho}_{0}+\varrho_{0} \hat{\varrho}_{0}-\frac{\nu}{2}\varrho_{0}^{2}\right)\right]\right. \tag{74}\] \[\left.\left(1+nMN\frac{\nu}{2}\varrho_{0}^{2}\alpha\left\langle \ln\cosh(\beta\eta)\right\rangle_{\eta}+\frac{nMN}{2}\left(\ln(1-q)+\frac{q-R^ {2}}{1-q}\right)-\right.\right.\] \[\left.\left.-nMN\alpha\mathcal{C}_{\hat{\pi}}\int\mathrm{d}z \,\mathrm{d}s\,\pi(z)\hat{\pi}(s)\ln\left[1+\tanh(\beta s)\tanh(\beta z) \right]+\right.\] \[\left.+\frac{\nu}{2}nMN\alpha\int\mathrm{d}z_{1}\,\mathrm{d}z_{2 }\,\pi(z_{1})\pi(z_{2})\,\left\langle\ln\left(1+\tanh(\beta\eta)\tanh(\beta z _{1})\tanh(\beta z_{2})\right)\right\rangle_{\eta}+\right.\] \[\left.\left.+nNM\alpha\mathrm{e}^{-\hat{\varrho}_{0}}\sum_{C=0}^ {\infty}\frac{\mathcal{C}_{\hat{\pi}}^{\ell}}{C!}\int_{-\infty}^{\infty}\prod _{\ell=1}^{C}\mathrm{d}s_{\ell}\,\hat{\pi}(s_{\ell})\;2\int_{-\infty}^{\infty }\mathcal{D}x\mathcal{H}\left(-\frac{Rx}{\sqrt{q-R^{2}}}\right)\right.\right.\] \[\left.\left.\ln\left[\mathcal{H}\left(\sqrt{\frac{q}{1-q}}x \right)\mathrm{e}^{-\beta(1-\nu\eta_{0})}\prod_{\ell=1}^{C}[1-\tanh(\beta s_{ \ell})]+\right.\right.\] \[\left.\left.\left.+\mathcal{H}\left(-\sqrt{\frac{q}{1-q}}x\right) \mathrm{e}^{\beta(1-\nu\eta_{0})}\prod_{\ell=1}^{C}[1+\tanh(\beta s_{\ell})] \right]\right)\right\}.\]

Observe that during the integration process the dependency with respect to \(W\) disappears. By adding up the series, can be re-express as:

\[\overline{Z^{n}}(\beta) = \mathop{\rm extr}\limits_{q,R,\varrho_{0},\hat{\varrho}_{0},\pi, \hat{\pi}}\left\{\exp\left[-M\left(\frac{\nu}{2}-\hat{\varrho}_{0}+\varrho_{0 }\hat{\varrho}_{0}-\frac{\nu}{2}\varrho_{0}^{2}\right)\right]\right. \tag{75}\] \[\left.\left(1+nMN\frac{\nu}{2}\varrho_{0}^{2}\alpha\left\langle \ln\cosh(\beta\eta)\right\rangle_{\eta}+\frac{nMN}{2}\left(\ln(1-q)+\frac{q-R ^{2}}{1-q}\right)-\right.\right.\] \[\left.\left.-nMN\alpha\mathcal{C}_{\hat{\pi}}\int\mathrm{d}z\, \mathrm{d}s\,\pi(z)\hat{\pi}(s)\ln\left[1+\tanh(\beta s)\tanh(\beta z)\right]+\right.\] \[\left.+\frac{\nu}{2}nMN\alpha\int\mathrm{d}z_{1}\,\mathrm{d}z_{2 }\,\pi(z_{1})\pi(z_{2})\,\left\langle\ln\left(1+\tanh(\beta\eta)\tanh(\beta z _{1})\tanh(\beta z_{2})\right)\right\rangle_{\eta}+\right.\] \[\left.+nNM\alpha\left(-\beta(1-\nu\eta_{0})+\mathrm{e}^{-\hat{ \varrho}_{0}}+\mathcal{C}_{\hat{\pi}}\int\mathrm{d}s\,\hat{\pi}(s)\ln[1+ \tanh(-\beta s)]\right)\right.\] \[\left.\left.+2nNM\alpha\int_{-\infty}^{\infty}\mathcal{D}x \mathcal{H}\left(-\frac{Rx}{\sqrt{q-R^{2}}}\right)\right.\right.\] \[\left.\left.\int\frac{\mathrm{d}y}{2\pi}\int\mathrm{d}\hat{y} \mathrm{e}^{-iy\hat{y}}\exp\left[\mathcal{C}_{\hat{\pi}}\int\mathrm{d}s\,\hat{ \pi}(s)\mathrm{e}^{i\hat{y}s}-\hat{\varrho}_{0}\right]\ln\left[1+\left(\mathrm{ e}^{2\beta(1-\nu\eta_{0}+y)}-1\right)\mathcal{H}\left(-\sqrt{\frac{q}{1-q}}x \right)\right]\right)\right\}.\]

Observe that

\[\partial_{\varrho_{0}}\overline{Z^{n}} = \left(-M\hat{\varrho}_{0}+\nu M\varrho_{0}+O(n)\right)\overline{Z^{n}} \tag{76}\] \[\partial_{\hat{\varrho}_{0}}\overline{Z^{n}} = \left(M-M\varrho_{0}+O(n)\right)\overline{Z^{n}} \tag{77}\]

which implies in the extreme that \(\varrho_{0}=1\) (as it was expected from equation (69)) and \(\hat{\varrho}_{0}=\nu\). From this last equation we have that \(\mathcal{C}_{\hat{\pi}}=\nu\). By defining the weight function:

\[\mathsf{P}(y|\hat{\pi})\equiv\int\frac{\mathrm{d}\hat{y}}{2\pi}\mathrm{e}^{-iy \hat{y}}\exp\left[\nu\left(\int\mathrm{d}s\,\hat{\pi}(s)\mathrm{e}^{i\hat{y}s }-1\right)\right], \tag{78}\]

and the distribution

\[\mathcal{P}(x|q,R)\equiv 2\mathcal{N}(x)\mathcal{H}\left(-\frac{Rx}{\sqrt{q-R^{2}}} \right), \tag{79}\]
the free energy functional \(F\) can be defined as

\[F \equiv T\frac{1-\overline{Z^{n}}(\beta)}{nNM} \tag{80}\] \[= T\left\{-\alpha\left(-\beta(1-\nu\eta_{0})+\mathrm{e}^{-\nu}+ \frac{\nu}{2}\left\langle\ln\cosh(\beta\eta)\right\rangle_{\eta}\right)-\frac{1 }{2}\left(\ln(1-q)+\frac{q-R^{2}}{1-q}\right)+\right.\] \[\left.-\frac{\nu}{2}\alpha\int\mathrm{d}z_{1}\,\mathrm{d}z_{2}\, \pi(z_{1})\pi(z_{2})\,\left\langle\ln\left(1+\tanh(\beta\eta)\tanh(\beta z_{1} )\tanh(\beta z_{2})\right)\right\rangle_{\eta}+\right.\] \[\left.+\nu\alpha\int\mathrm{d}z\,\mathrm{d}s\,\pi(z)\hat{\pi}(s) \ln\left[\frac{1+\tanh(\beta s)\tanh(\beta z)}{1-\tanh(\beta s)}\right]- \alpha\beta\int\mathrm{d}y\mathsf{P}(y|\hat{\pi})(1-\nu\eta_{0}+y)\right.\] \[\left.-\alpha\int\mathrm{d}x\mathcal{P}_{Rq}(x)\,\int\mathrm{d} y\mathsf{P}(y|\hat{\pi})\ln\left[\mathrm{e}^{-\beta(1-\nu\eta_{0}+y)}\mathcal{H} \left(\sqrt{\frac{q}{1-q}}x\right)+\mathrm{e}^{\beta(1-\nu\eta_{0}+y)} \mathcal{H}\left(-\sqrt{\frac{q}{1-q}}x\right)\right]\right\}\]

density of the system can be expressed as \(f=\mathrm{ext}_{q,R,\pi,\hat{\pi}}\,F.\)

### Fourier Transform of the Distributions

By using the Fourier Transform in the saddle point equations (25) and (26) we define the functions:

\[\hat{\phi}(\omega) \equiv \int\mathrm{d}s\,\hat{\pi}(s)\mathrm{e}^{is\omega} \tag{81}\] \[= 1+i\mathcal{I}_{0}\omega-\frac{\mathcal{R}_{0}}{2}\omega^{2}+O( \omega^{3})\] \[\phi(\omega) \equiv\int\mathrm{d}s\,\pi(s)\mathrm{e}^{is\omega}. \tag{82}\]

Thus:

\[\mathsf{P}(y|\hat{\pi}) \equiv \int\frac{\mathrm{d}\omega}{2\pi}\,\mathrm{e}^{-ig\omega+\hat{ \phi}(\omega)-\nu} \tag{83}\] \[\approx \int\frac{\mathrm{d}\omega}{2\pi}\,\exp\left[-\frac{\nu\mathcal{ R}_{0}}{2}\omega^{2}+i(\nu\mathcal{I}_{0}-y)\omega\right]=\mathcal{N}\left(y\,|\nu \mathcal{I}_{0},\nu\mathcal{R}_{0}\right),\]

which is consistent with (19) and (20). Consider the definition (29), then the Fourier Transform of \(\pi(s)\) can be expressed as:

\[\phi(\omega) = \tag{84}\] \[\approx \int\mathrm{d}y\mathcal{N}\left(y\,|\nu\mathcal{I}_{0},\nu \mathcal{R}_{0}\,\right)\mathrm{e}^{is\omega}\left\langle\exp\left[i\omega \left(1-\nu\eta_{0}+\frac{\Lambda(x)}{2}x^{2}\right)\right]\right\rangle_{x},\]

where the expectation over \(x\) is approximated by:

\[\left\langle\exp\left\{i\omega\left[1-\nu\eta_{0}+\frac{\Lambda(x )}{2}\right]\right\}\right\rangle_{x} \approx 2\int_{-\infty}^{\infty}\mathcal{D}x\,\Theta(xR)\exp\left[i \omega\left(1-\nu\eta_{0}+\frac{\Lambda(x)}{2}x^{2}\right)\right] \tag{85}\] \[\approx 2\int_{0}^{\infty}\frac{\mathrm{d}x}{\sqrt{2\pi}}\exp\left[- \frac{x^{2}}{2}+i\omega\left(1-\nu\eta_{0}+\frac{\Lambda(R)}{2}x^{2}\right)\right]\] \[\approx \int_{-\infty}^{\infty}\frac{\mathrm{d}x}{\sqrt{2\pi}}\exp\left[ -[1-i\omega\Lambda(R)]\frac{x^{2}}{2}+is\left(1-\nu\eta_{0}\right)\right]\] \[\approx \frac{1}{\sqrt{1-i\omega\Lambda}}\exp\left[i(1-\nu\eta_{0})\omega\right]\] \[\approx \exp\left[-\frac{3}{8}\Lambda^{2}\omega^{2}+i\left(1-\nu\eta_{0}+ \frac{\Lambda(R)}{2}\right)\omega\right],\]
where (85) is reached by assuming that \(\Lambda\) sufficiently small. From this point onwards we will refer to the parameter \(\Lambda(R)\) as the amplified thermal noise. In such a case the function \(\phi\) is Gaussian:

\[\phi(s)\approx\exp\left[-\frac{\nu{\cal R}_{0}+\frac{3}{4}\Lambda(R)^{2}}{2}s^{2 }+i\left(1-\nu\eta_{0}+\nu{\cal I}_{0}+\frac{\Lambda(R)}{2}\right)s\right], \tag{86}\]

and so

\[\pi(z)\approx\frac{1}{\sqrt{2\pi\left(\nu{\cal R}_{0}+\frac{3}{4}\Lambda(R)^{2 }\right)}}\exp\left\{-\frac{\left(z-1+\nu\eta_{0}-\nu{\cal I}_{0}-\frac{1}{2} \Lambda(R)\right)^{2}}{2\left(\nu{\cal R}_{0}+\frac{3}{4}\Lambda(R)^{2}\right) }\right\}. \tag{87}\]

The argument in the Dirac's delta function in (26) is mildly sensitive to changes in temperature, therefore we can approximate it by:

\[\beta^{-1}{\rm arctanh}\left[\tanh(\beta\eta)\tanh(\beta z)\right]\ \approx\ \frac{|\eta+z|-|\eta-z|}{2}={\rm sgn}(z\eta)\min\{|\eta|,|z|\}, \tag{88}\]

which allows us to write the expression of the Fourier transform of the distribution \(\hat{\pi}\) as:

\[\hat{\phi}(\omega)\ =\ \int{\rm d}s\phi(s)\left\langle\int\frac{{\rm d}z}{2\pi} \,\exp\left(-isz+i\omega{\rm sgn}(z\eta)\min\{|\eta|,|z|\}\right)\right\rangle _{\eta}, \tag{89}\]

where the expectation over the social strengths can be demonstrated to be, disregarding terms of \(O(\eta_{0}\Delta^{2})\):

\[{\cal Q} \equiv\ \left\langle\int\frac{{\rm d}z}{2\pi}\,\exp\left(-isz+i \omega\beta^{-1}\,{\rm arctanh}\left[\tanh(\beta\eta)\tanh(\beta z)\right] \right)\right\rangle_{\eta} \tag{90}\] \[\approx\ \delta(s)+\frac{\omega}{\pi}\exp\left(-\frac{\Delta^{2}s^{2}}{2} \right)\frac{\eta_{0}}{s}-\omega^{2}\frac{\eta_{0}^{2}+\Delta^{2}}{2}\delta(s).\]

Expression (90), together with (89) and (81) produce the following expressions for the moments of \(\nu^{-1}\hat{\pi}\) :

\[1 = \phi(0) \tag{91}\] \[{\cal I}_{0} = -\frac{i\eta_{0}}{\pi}\int{\rm d}s\frac{\phi(s)}{s}{\rm e}^{- \frac{\Delta^{2}s^{2}}{2}}\] (92) \[{\cal R}_{0} = \eta_{0}^{2}+\Delta^{2}, \tag{93}\]

which implies that

\[{\cal I}_{0}^{\star}=\eta_{0}{\rm erf}\left(\frac{1-\nu\eta_{0}+\nu{\cal I}_{ 0}^{\star}+\frac{1}{2}\Lambda(R)}{\sqrt{2\left[\nu(\eta_{0}^{2}+\Delta^{2})+ \frac{3}{4}\Lambda(R)^{2}\right]}}\right). \tag{94}\]

The equation (94) has one, two or three solutions depending on the value of the parameters \(\nu\), \(\eta_{0}\), \(\Delta\) and the function \(\Lambda(R).\) It is easy to see that for the line:

\[\eta_{0}=\frac{2+\Lambda(R)}{2\nu} \tag{95}\]

\({\cal I}_{0}^{\star}=0\) is a solution. Thus we expect that for \(\nu\eta_{0}>1+\Lambda(R)/2\), \({\cal I}_{0}^{\star}\approx-(\eta_{0}-\epsilon)\) is a solution, and for \(\nu\eta_{0}<1+\Lambda(R)/2\), \({\cal I}_{0}\approx\eta_{0}-\epsilon\) is a solution (in both cases \(0<\epsilon<\eta_{0}\) is a suitable positive number).

If the derivative with respect to \({\cal I}_{0}\) of the right-hand-side of (94) evaluated at \({\cal I}_{0}^{\star}\) is equal to 1, and \({\cal I}_{0}^{\star}\) is also a solution of (94), then (94) has two solutions, \({\cal I}_{0}^{\star}\) and \(\eta_{0}-\epsilon\) or \(-(\eta_{0}-\epsilon)\) depending on whether \(\nu\eta_{0}<1+\Lambda(R)/2\) or \(\nu\eta_{0}>1+\Lambda(R)/2\), respectively.

If \(\eta_{-}(\Lambda)<\eta_{0}<\eta_{+}(\Lambda)\), where:

\[\eta_{\pm} = \frac{2\mp|\Lambda|}{2\nu}\mp\left[\frac{1}{\nu}\sqrt{\left[\nu( \eta_{\pm}^{2}+\Delta^{2})+\frac{3}{4}\Lambda(R)^{2}\right]\log\left(\frac{2} {\pi}\frac{\nu^{2}\eta_{\pm}^{2}}{\nu(\eta_{\pm}^{2}+\Delta^{2})+\frac{3}{4} \Lambda(R)^{2}}\right)}-\right. \tag{96}\] \[\left.-\eta_{\pm}{\rm erf}\left(\sqrt{\log\left(\sqrt{\frac{2}{ \pi}\frac{\nu^{2}\eta_{\pm}^{2}}{\nu(\eta_{\pm}^{2}+\Delta^{2})+\frac{3}{4} \Lambda(R)^{2}}}\right)}\right)\right]\]
are the superior (\(\eta_{+}\)) and inferior (\(\eta_{-}\)) limits to the area in the plane (\(|\Lambda|,\eta_{0}\)) where three solutions to the equation (94) can be found. Let as define the set \(\mathbb{A}:=\{(|\Lambda|,\eta_{0}):\eta_{-}(\Lambda)<\eta_{0}<\eta_{+}(\Lambda)\}\). (The use of \(|\Lambda|\) instead of \(\Lambda\) is due to the fact that realizable solutions must satisfy \(\mathrm{sgn}(R)=\mathrm{sgn}(1-\nu\eta_{0})\). This point will be clarified when the equations involving \(R\) and \(q\) are contemplated. See bellow.)

It is important to note that for very high values of the pressure \(\beta\) we have that:

\[\nu\eta_{\pm}\approx\frac{1}{1\pm\sqrt{\frac{1}{\nu}\log\left(\frac{2\nu}{\pi }\right)}\mp\mathrm{erf}\left(\sqrt{\frac{1}{2}\log\left(\frac{2\nu}{\pi} \right)}\right)}, \tag{97}\]

which implies that the segment (at very high pressure) of the coexistence of states grows with \(\sqrt{\nu}\). From equation (94) we conclude that, to satisfy the saddle-point equations (25,26), there must be a set of values of \(\eta\) such that, for a very high pressure \(\beta\) there coexist states with different attitudes.

For all the points \((\Lambda,\eta_{0})\notin\mathbb{A}\), we have that both conditions (92) and (93) are satisfied for a distribution:

\[\hat{\pi}(s)\ =\ \mathcal{N}\left(s\left|\mathcal{I}_{0}^{\star},\eta_{0}^{2}-( \mathcal{I}_{0}^{\star})^{2}+\Delta^{2}\right.\right) \tag{98}\]

where \(\mathcal{I}_{0}^{\star}\) is the only solution of (94). In this case we also have that:

\[\mathsf{P}(y|\hat{\pi})\approx\mathcal{N}\left(y\left|\nu\mathcal{I}_{0}^{ \star},\nu(\eta_{0}^{2}+\Delta^{2})\right.\right). \tag{99}\]

For the points \((\Lambda,\eta_{0})\in\mathbb{A}\) we propose the following form for \(\hat{\pi}(s)=\mathcal{Z}^{-1}\exp[-\Phi(s)]\), where \(\mathcal{Z}\) is a normalization constant and the function \(\Phi(s)\) is defined as:

\[\Phi(s) \coloneqq \frac{s^{2}}{2\Delta^{2}}-\frac{\eta_{0}}{\nu}\frac{1-\nu\eta_{ 0}+\nu s+\frac{1}{2}\Lambda(R)}{\Delta^{2}}\mathrm{erf}\left(\frac{1-\nu\eta_ {0}+\nu s+\frac{1}{2}\Lambda(R)}{\sqrt{2\left[\nu(\eta_{0}^{2}+\Delta^{2})+ \frac{3}{4}\Lambda(R)^{2}\right]}}\right)- \tag{100}\] \[-2\frac{\eta_{0}}{\nu}\frac{\nu(\eta_{0}^{2}+\Delta^{2})+\frac{3 }{4}\Lambda(R)^{2}}{\Delta^{2}}\mathcal{N}\left(\nu s\left|\nu\eta_{0}-1- \frac{1}{2}\Lambda(R),\nu(\eta_{0}^{2}+\Delta^{2})+\frac{3}{4}\Lambda(R)^{2} \right.\right).\]

Figure 7: Phase diagram of the system in terms of the parameters \(\nu\eta_{0}\) and \(|\Lambda|\). There are two phases separated by the line \(\nu\eta_{0}=1\). For \(\nu\eta_{0}<1\) we have that \(R>0\) and the average consensus is in favor of \(B\). For \(\nu\eta_{0}>1\)\(R<0\) and the average position of the agents is to form local alliances against the president \(B\). In all the points of the plain outside the region \(\mathbb{A}\), the distribution describing the position of the neighborhood, given by \(\hat{\pi}(z)\), is sharply picked at \(+\eta_{0}\), for the conservative position, i.e. \(\nu\eta_{0}<1\), or at \(-\eta_{0}\), for the polarized position, i.e. \(\nu\eta_{0}>1\). In region \(\mathbb{A}\) we have the same phase separation at \(\nu\eta_{0}=1\) but the contribution from the neighborhood is a mixture of a polarized component plus a conservative component. We also observe that the vertexes of \(\mathbb{A}\) are (0,1.651), (0.411,1) and (0,0.717).
Observe that \(\Phi^{\prime}(\mathcal{I}_{0}^{*})=0\) is identical to (94) and, for all points in \(\mathbb{A}\) this equations have three roots \(\mathcal{I}_{-}^{*}<\mathcal{I}_{0}^{*}<\mathcal{I}_{+}^{*}\). The asymptotic behavior of \(\lim_{s\rightarrow\pm\infty}s^{-2}\Phi(s)>0\) indicates that \(\mathcal{I}_{\pm}^{*}\) are minima with \(\mathcal{I}_{0}^{*}\) an intermediate maximum. Let us compute the second derivative of \(\Phi\) at the minima:

\[\Phi^{\prime\prime}(\mathcal{I}_{\pm}^{*}) = \left.\frac{1}{\Delta^{2}}\left(1-\eta_{0}\ \frac{\mathrm{d}}{\mathrm{d}s} \mathrm{erf}\left(\frac{1-\nu\eta_{0}+\nu s+\frac{1}{2}\Lambda(R)}{\sqrt{2 \left[\nu(\eta_{0}^{2}+\Delta^{2})+\frac{3}{4}\Lambda(R)^{2}\right]}}\right) \right|_{s=\mathcal{I}_{\pm}^{*}}\right), \tag{101}\]

therefore \(\Phi^{\prime\prime}(\mathcal{I}_{\pm}^{*})=\mathcal{L}_{\pm}^{2}\Delta^{-2}\), with:

\[\mathcal{L}_{\pm}\coloneqq\sqrt{1-2\nu\eta_{0}\mathcal{N}\left(\nu\mathcal{I} _{\pm}^{*}\left|\nu\eta_{0}-1-\frac{1}{2}\Lambda(R),\nu(\eta_{0}^{2}+\Delta^{ 2})+\frac{3}{4}\Lambda(R)^{2}\right.\right)}, \tag{102}\]

which is larger than zero for all \((\Lambda,\eta_{0})\in\mathbb{A}\). Thus:

\[\hat{\pi}(s) \approx h_{+}\mathcal{N}\left(s\left|\mathcal{I}_{+}^{*},\Delta^{2} \right.\right)+h_{-}\mathcal{N}\left(s\left|\mathcal{I}_{-}^{*},\Delta^{2}\right.\right) \tag{103}\] \[h_{\pm} \equiv \frac{1}{2}\pm\frac{1}{2}\tanh\left(\frac{\Phi(\mathcal{I}_{+}^{* })-\Phi(\mathcal{I}_{-}^{*})}{4}-\frac{1}{2}\ln\frac{\mathcal{L}_{+}}{ \mathcal{L}_{-}}\right)\] (104) \[\int\mathrm{d}s\hat{\pi}(s)s \approx h_{+}\mathcal{I}_{+}^{*}+h_{-}\mathcal{I}_{-}^{*}\] (105) \[\int\mathrm{d}s\hat{\pi}(s)s^{2} \approx h_{+}(\mathcal{I}_{+}^{*})^{2}+h_{-}(\mathcal{I}_{-}^{*})^{2}+ \Delta^{2}, \tag{106}\]

observe that the expectation (104) represents a convex combination between the solutions to the equation (94), and \(\hat{\pi}^{\prime}(\mathcal{I}_{\pm}^{*})\approx 0\). In order to compute the distribution \(\mathsf{P}(y|\hat{\pi})\) we first need to compute the Fourier transform of \(\hat{\pi}\):

\[\hat{\phi}(\omega) \approx h_{+}\exp\left[-\frac{\Delta^{2}}{2}\omega^{2}+i\omega\mathcal{I }_{+}^{*}\right]+h_{-}\exp\left[-\frac{\Delta^{2}}{2}\omega^{2}+i\omega \mathcal{I}_{-}^{*}\right], \tag{107}\]

and if we define

\[\Upsilon(\omega,y) \coloneqq -iy\omega-\nu+\nu\int\mathrm{d}s\frac{\exp[-\Phi(s)+is\omega]}{ \int\mathrm{d}s^{\prime}\,\exp[-\Phi(s^{\prime})]} \tag{108}\] \[\approx -iy\omega-\nu+\nu\left\{h_{+}\exp\left[-\frac{\Delta^{2}}{2} \omega^{2}+i\omega\mathcal{I}_{+}^{*}\right]+h_{-}\exp\left[-\frac{\Delta^{2} }{2}\omega^{2}+i\omega\mathcal{I}_{-}^{*}\right]\right\}\] \[\approx -iy\omega+\nu\left\{i\omega(h_{+}\mathcal{I}_{+}^{*}+h_{-} \mathcal{I}_{-}^{*})-\frac{h_{+}(\mathcal{I}_{+}^{*})^{2}+h_{-}(\mathcal{I}_{ -}^{*})^{2}+\Delta^{2}}{2}\omega^{2}\right\}+O(\omega^{3})\]

\[\mathsf{P}(y|\hat{\pi}) = \int\frac{\mathrm{d}\omega}{2\pi}\,\mathrm{e}^{\Upsilon(\omega,y)} \tag{109}\] \[\approx \mathcal{N}\left(y\left|\nu(h_{+}\mathcal{I}_{+}^{*}+h_{-} \mathcal{I}_{-}^{*}),\nu[h_{+}(\mathcal{I}_{+}^{*})^{2}+h_{-}(\mathcal{I}_{-}^ {*})^{2}+\Delta^{2}]\right.\right)\]

These distributions should be applied depending on the value of \(\eta_{0}\) and \(\Lambda(R)\), following the diagram presented in figure 7.
