Polarization and Fluctuations in Signed Social Networks

TL;DR

This paper introduces a new opinion dynamics model for signed social networks that captures polarization and fluctuations, incorporating the boomerang effect and analyzing behavior under structural balance conditions.

Contribution

It proposes a novel, intuitive model for opinion evolution in signed networks that accounts for psychological effects and analyzes opinion polarization and fluctuations beyond existing models.

Findings

Opinions polarize under structural balance conditions.

Opinions exhibit persistent fluctuations in unbalanced networks.

The model captures richer behaviors than previous models.

Abstract

Much recent research on social networks has focused on the modeling and analysis of how opinions evolve as a function of interpersonal relationships. It is also of great interest to model and understand the implications of friendly and antagonistic relationships. In this paper, we propose a new, simple and intuitive model that incorporates the socio-psychological phenomenon of the boomerang effect in opinion dynamics. We establish that, under certain conditions on the structure of the signed network that corresponds to the so-called structural balance property, the opinions in the network polarize. Compared to other models in the literature, our model displays a richer and perhaps more intuitive behavior of the opinions when the social network does not satisfy structural balance. In particular, we analyze signed networks in which the opinions show persistent fluctuations (including the case of the so-called clustering balance).

Equations

$$x_i(t+1)=\{\begin{array}{l}{a}_i{x}_i(t)+(1-a_i)x_j(t),\text{ if }\{i,j\}\in {\mathcal{E}}_{+},\\ a_i{x}_i(t)+(1-a_i)o_{\min },\\ \text{ if }\{i,j\}\in {\mathcal{E}}_{-}\text{ and }{x}_i(t)<x_j(t),\\ a_i{x}_i(t)+(1-a_i)o_{\max },\\ \text{ if }\{i,j\}\in {\mathcal{E}}_{-}\text{ and }{x}_i(t)\ge x_j(t),\end{array}$$

$$\begin{array}{rl}\mathbb{P}[& x(\tau )\in Z^{-1}(2)|x(0)=x_o]\\ & \ge \mathbb{P}[\omega (0)=s(0)|x(0)=x_o]\\ & \times \mathbb{P}[\omega (1)=s(1)|x(0)=x_o,\omega (0)=s(0)]\dots \\ & \times \mathbb{P}[\omega (\tau -1)=s(\tau -1)\}|\\ & x(0)=x_o,\omega (\ell )=s(\ell )\text{ for }\ell \in [0,\tau -2]]\\ & ={\mathbb{P}}^{\prime }[s(0)]{\mathbb{P}}^{\prime }[s(1)]\dots {\mathbb{P}}^{\prime }[s(\tau -1)]\\ & \ge ({\gamma }_o{)}^{\tau },\end{array}$$

$$\begin{array}{rl}\mathbb{P}[& {\mathcal{T}}_{1\to 2}(0)\ge (\tau +1)\Gamma ]\\ & =\mathbb{P}[x(t)\notin Z^{-1}(2),t\in [0,(\tau +1)\Gamma -1]|x(0)=x_o]\\ & =\mathbb{P}[{\cap }_{\ell =0}^{\Gamma -1}{A}_{\ell (\tau +1)}|x(0)=x_o]\\ & =\mathbb{P}[A_0|x(0)=x_o]\\ & \times \prod _{\ell =1}^{\Gamma -1}\mathbb{P}[A_{\ell (\tau +1)}|x(0)=x_o,{\cap }_{0\le {\ell }^{\prime }\le \ell }{A}_{{\ell }^{\prime }(\tau +1)}]\\ & \le (1-{\gamma }_o^{\tau }{)}^{\Gamma }=:{\gamma }^{\Gamma }.\end{array}$$

$$\mathbb{P}[x_i(t)\in (o_{\min },o_{\min }+ϵ)\cup (o_{\max }-ϵ,o_{\max })\text{ i.o.}]=1.$$

Full text

1 Introduction

††This work was supported by, or in part by, the Army Research Laboratory and the Army Research Office under grant number W911NF-15-1-0577, the Cooperative Agreement Number W911NF-18-2-0066, and the National Science Foundation under grant number DGE-1258507. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.††Pedro Cisneros-Velarde (e-mail: pacisne@gmail.com) and Francesco Bullo (e-mail: bullo@engineering.ucsb.edu) are with the Center for Control, Dynamical Systems and Computation, University of California, Santa Barbara.††Kevin S. Chan (e-mail: kevin.s.chan.civ@mail.mil) is with the U.S. Army Research Laboratory, 2800 Powder Mill Rd., Adelphi, MD 20783, USA.

There have been various opinion dynamics models in the literature [3, 18]. Opinions can be modeled as real numbers taking values in the closed interval $[0,1]$, where [math] means an agent completely disagrees with a particular issue, and $1$ that she completely agrees. One important question to answer is how the evolution and final distribution of opinions in a social network depend on the underlying network’s topology and of the (positive) influence structure among the individuals. More recently, signed graphs were introduced into the opinion dynamics literature. Signed graphs represent a natural way to model positive and negative relationships among individuals. For example, a sociological relevant concept is structural balance, in which the members of a social network can either have only positive relationships or be divided in two factions in which members of the same faction have positive relationships but negative ones with members of the other faction. The seminal work by Altafini [4] proposed a continuous time model over a signed graph where the opinions can take any real value. It is shown that when the underlying graph satisfies structural balance, the opinions converge to bipartite consensus and polarize, i.e., all opinions have the same absolute value with their signs indicating which agents belong to the same faction (if there is one faction, all opinions have the same sign). A discrete-time signed opinion model which is a counterpart of the Altafini model has also been proposed [12, 17], in which bipartite consensus is also attained under structural balance. These two models have initiated a lot of research in the field of signed opinion dynamics, and are, arguably, the most popular models in the literature. Extensions of these models and further analysis have been done in the literature, as can be noted in the recent work [16] and the references therein. Note, however, that both Altafini models and their extensions present an unrealistic opinion vanishing behavior (i.e., the opinions converge to zero) whenever the property of structural balance is lost in the underlying social network.

Another class of models in opinions dynamics was proposed by Li et al. [14] and is based on an extension of the voter model to signed graphs. In this model, individuals initially take binary opinion values (e.g., [math] and $1$). Then, at each subsequent time step, an individual is selected according to some process and updates her opinions by copying the same or the opposite opinion of one of her neighbors according to the sign of their relationship. By design, opinions cannot vanish under generic signed networks; however, the opinion values are simply discrete. Whenever the graph satisfies structural balance, they showed that the opinions polarize: one faction takes one value, while the other faction takes the remaining one. Recently, Lin et al. [15] proposed a model which can be regarded as an extension to the one from Li et al. In this model, opinions can take $m$ different discrete values from a set $S$. Then, an individual will copy the same opinion from a positive neighbor, but when facing a negative one, will randomly select an opinion different from that neighbor from the set $S$.

In this paper, we propose a novel opinion model over signed graphs. We assume that the opinions are real numbers taking value in a closed interval and each edge of the graph indicates the friendly or antagonistic relationship between two individuals. Our model is inspired by the boomerang effect studied in social psychology [10, 7, 1], which aims to explain why in some situations where two individuals engage in communication, they may not end up being in a better agreement but rather their attitudes become more dissentive, i.e., their opinions do not go in the intended direction (e.g., consensus or agreement) but in the opposite direction (e.g., polarization). The early work [13] suggested that this phenomenon can be explained by “the relative distance between subjects’ attitudes and position of communication”. Our model is motivated by the empirical observations in the social sciences (e.g., from the study of interpersonal attraction [5]) that two friendly agents will be closer in their attitudes and perspectives than two unfriendly agents. Specifically, we make the following assumption: whenever two agents who have a positive relationship interact, they are more agreeable and their opinions will become closer or even be in consensus, i.e., the opinion changes in the intended direction. On the other hand, whenever two agents with a negative relationship interact, the differences in their opinions will be more polarized after the interaction because of their increasing disagreement, i.e., the opinion changes in the opposite direction. Our opinion model captures such behavior mathematically, and we call it the affine boomerang model. Mathematically, our proposed model is an affine model, which makes it remarkably simple, and its dynamics are self-explanatory. Besides a linear model like the discrete Altafini model, this is, arguably, the next simplest model structurally.

Our second contribution is a formal analysis of our proposed model: under certain conditions on the sign structures of the network that corresponds to structural balance, our model expresses opinion polarization, i.e., the opinions of two groups converge to opposite extreme values of the closed interval.

Finally, it is important to compare our model and the aforementioned models in the literature. Our model has the property that opinions do not necessarily vanish whenever the graph is not balanced, but, for example, can continue fluctuating inside the closed interval. The vanishing behavior, which we mentioned happens in both types of Altafini models and their extensions, has been interpreted as if the agents in the network become neutral or indifferent towards a specific topic. In the case of three antagonistic groups in a connected network, this would mean that all groups will end up having a zero valued opinion, i.e., they will have consensus on not having an opinion. This might be difficult to interpret. Instead, our proposed model predicts that two groups will polarize their opinions and the third one will continue fluctuating its opinions since its members observe people they dislike having opposite opinions. Thus, this third group does not settle down to a definite opinion and its members are persistently disagreeing with each other. This is, arguably, more intuitive since individuals of a social network can always hold an opinion, independently of whether their network is balanced or not. Moreover if we have an unbalanced network that differs from a balanced one in just the sign of one edge, it is not clear why that would drive the whole social network towards an indifferent opinion. Instead, our model suggests that opinions may fluctuate around extreme values of opinion, which is more intuitive since the underlying social network is approximately balanced.

2 The model

A signed graph $G$ is an undirected graph with signed edges, i.e., with edge weights equal to either $+1$ or $-1$. Let $\mathcal{E}=\mathcal{E}_{+}\operatorname{\cup}\mathcal{E}_{-}$ be the edge set of $G$, where $\mathcal{E}_{+}$ is the set of positive edges and $\mathcal{E}_{-}$ the set of negative edges. $G$ is complete when there exists an edge between any pair of vertices. A path from vertex $i$ to $j$ in $G$ is a sequence of edges that connect a sequence of distinct vertices starting from $i$ and finishing at $j$. A connected component is any subgraph such that all of its vertices are connected to each other by paths, but they are not connected to any other vertex of $G$. $G$ is connected whenever it has only one connected component. The abbreviation $i.o.$ stands for infinitely often.

We model the structure of a social network composed by agents as a graph. Then, throughout the paper, we use the words graph and network interchangeably, as well as the terms vertex and agent. Each agent in the network holds an opinion about a particular statement of a discussion topic, and her opinion describes how much she agrees with it. An agent $i$ has an opinion $x_{i}\in[o_{\min},o_{\max}]$: $x_{i}=o_{\max}$ whenever $i$ completely agrees with the statement being discussed, and $x_{i}=o_{\min}$ whenever she completely disagrees with it. The opinion vector $x\in[o_{\min},o_{\max}]^{n}$ has in its $i$th entry the opinion $x_{i}$ of agent $i$.

Definition 2.1** (Sign arrangement property).**

Given a connected signed graph $G=(\{1,\dots,n\},\mathcal{E}_{+}\operatorname{\cup}\mathcal{E}_{-})$ with $n\geq 3$, let $G_{+}=(\{1,\dots,n\},\mathcal{E}_{+})$. For $k\in\mathbb{N}$, we say that $G$ satisfies the $k$-sign arrangement property if

1. (i)

$G_{+}$* has $k\geq 1$ connected components, and* 2. (ii)

each negative edge connects vertices belonging to different connected components of $G_{+}$.

If this property holds, then each connected component of $G_{+}$ is a faction.

Based on the works [8, 11] in the sociological literature, we definite the notion of structural and clustering balance for connected graphs.

Definition 2.2** (Structural and clustering balance).**

Consider a connected signed graph $G$ with $n\geq 3$. Assume the vertices of $G$ can be partitioned in $m$ groups such that each positive edge joins two vertices from the same group and each negative edge joins vertices from different groups. We say that $G$ satisfies

1. (i)

structural balance* if $m\leq 2$, and* 2. (ii)

clustering balance* if $m\geq 3$.*

The following result follows immediately from the previous definitions.

Lemma 2.3**.**

*Let $G$ be a complete signed graph. $G$ satisfies the $k$-sign arrangement if and only if it satisfies structural balance when $k\leq 2$ or clustering balance when $k\geq 3$. *

Note that a signed graph satisfying the $k$-sign arrangement property does not need to be complete.

Definition 2.4** (Affine boomerang model).**

Let $G=(\{1,\dots,n\},\mathcal{E}_{+}\operatorname{\cup}\mathcal{E}_{-})$ be a signed graph. Assume that each agent has an initial opinion $x_{i}(0)\in[o_{\min},o_{\max}]$, $o_{\min}<o_{\max}$, and a self-weight $a_{i}\in(0,1)$. At each time step $t\in\mathbb{Z}_{\geq 0}$, select randomly an edge of $G$; assume each edge $\{i,j\}$ has a time-invariant positive selection probability $p_{ij}$. Update the opinions of the two agents $i$ and $j$ according to:

[TABLE]

and similarly for agent $j$.

Note that the boomerang effect is captured by the last two cases of equation (1).

We remark that our model has asynchronous updating of the opinions since only two opinions are updated simultaneously and independently per time step instead of all opinions at once (which would be synchronous updating). This type of updating has been present in other previous opinion models, e.g., in the Deffuant-Weisbuch model [9]. An example of selecting edges for the opinion updating is to do it uniformly as follows: let $m$ be the number of edges in the graph (e.g., $m={{n}\choose{2}}$ for complete graphs), then we can assign to every pair of agents the same probability of being selected and have $p_{ij}=1/m$ for any pair $\{i,j\}$.

Remark 2.5**.**

In our model, opinions take values on an arbitrary closed interval $[o_{\min},o_{\max}]$. From a sociological (and intuitive) point of view, it is plausible to have bounded opinions since there is no clear interpretation of a diverging opinion. Indeed, bounded opinions are present throughout the literature on opinion dynamics. The case $o_{\min}=-\theta$ and $o_{\max}=\theta$, for $\theta>0$ is characteristic in the literature of bipartite consensus (e.g., [4, 19]), and the case $o_{\min}=0$ and $o_{\max}=1$ characterizes various works in the literature of opinion dynamics over graphs with positive weights (e.g., [3]) or bounded-confidence models (e.g., [9]).

3 Model analysis

3.1 Theoretical results

Theorem 3.1** (Consensus and polarization in signed graphs).**

Consider a network satisfying the $k$-sign arrangement property. Consider the evolution of the affine boomerang model (1) with initial opinion vector $x(0)\in[o_{\min},o_{\max}]^{n}$. Then

1. (i)

Consensus: if $k=1$, then, with probability one, $\lim_{t\to\infty}x(t)=c\mathbbold{1}_{n}$, where $c$ is a random convex combination of the entries of $x(0)$. 2. (ii)

Polarization: if $k=2$, then, with probability one, $\lim_{t\to\infty}x_{i}(t)=o_{\min}$ for each agent $i$ of one of the two factions and $\lim_{t\to\infty}x_{j}(t)=o_{\max}$ for each $j$ of the other faction.

Proof.

Formally, at any time step, the selected edge is a discrete random variable over some probability space $(\Omega^{\prime},\mathcal{F}^{\prime},{\mathbb{P}}^{\prime})$ with $\Omega^{\prime}$ being the set of all edges on the graph, $\mathcal{F}^{\prime}$ the power set, and ${\mathbb{P}}^{\prime}[\{i,j\}]=p_{ij}$. Let $\omega(t)$ be the random edge selected at time $t$, then, the collection of random variables $\{\omega(t)\;|\;t\in\mathbb{Z}_{\geq 0}\}$ forms a stochastic process of an independent sequence of random variables. Then, an adequate probability space $(\Omega,\mathcal{F},\mathbb{P})$ can be constructed with $\Omega=\prod_{t\in\mathbb{Z}_{\geq 0}}\Omega^{\prime}$, $\mathcal{F}$ being the product of $\sigma$-algebras $\mathcal{F}^{\prime}$ over $t\in\mathbb{Z}_{\geq 0}$, and $\mathbb{P}$ being the product probability measure $\prod_{t\in\mathbb{Z}_{\geq 0}}{\mathbb{P}}^{\prime}$. Therefore, given the sequence of edges $\{s(t)\}_{t\in S}$ with some finite set $S\subset\mathbb{Z}_{\geq 0}$, $\mathbb{P}[\{\omega\in\Omega\;|\;\omega(t)=s(t),t\in S\}]=\prod_{t\in S}{\mathbb{P}}^{\prime}[s(t)]$.

We start by considering the case $k=1$. In this case, the model is a linear system of the form $x(t+1)=W(t)x(t)$, where $W(t)$ is a random matrix that takes, at each time step, the value $W_{ij}=I_{n\times{n}}-(1-a_{i})e_{i}(e_{i}-e_{j})^{\top}-(1-a_{j})e_{j}(e_{j}-e_{i})^{\top}$ whenever the edge $\{i,j\}$ is selected to be updated with probability $p_{ij}$ (here, $e_{i}$ is the $i$th column of the identity matrix $I_{n\times{n}}$). With probability one, $W(t)$ is a row stochastic matrix with a strictly positive diagonal for any $t$; moreover $W(t)$ is independent and identically distributed for any $t$. Thus, $\mathbb{E}[W(t)]$ (with respect to ${\mathbb{P}}^{\prime}$) is a row stochastic matrix that, when interpreted as an adjacency matrix, corresponds to a connected undirected network. Under these assumptions, [6, Theorem 12.1] implies the first statement of the theorem.

Now we prove the case $k=2$. We present the following lemma (whose proof is in the appendix) which describes the finite-time proximity property:

Lemma 3.2** (Finite-time proximity property).**

Consider the same assumptions as in Theorem 3.1 with a network satisfying the $2$-sign arrangement property. There exists a finite sequence of edges such that, if they are selected sequentially by our affine boomerang model, then, inside the interval $[o_{\min},o_{\max}]$, the opinions of any two vertices become arbitrarily close if they belong to the same faction, or arbitrarily apart if they belong to different ones.

Now, for any opinion vector $x\in[o_{\min},o_{\max}]^{n}$, we define the variable $Z:[o_{\min},o_{\max}]^{n}\to\{1,2\}$ as

1. (C1)

$Z(x)=1$ when there is no value $\tau>0$ such that one faction has all of its opinions above $\tau$ and the other faction has them equal or below it; 2. (C2)

$Z(x)=2$ when there exists a value $\tau>0$ such that one faction has all of its opinions above $\tau$ and the other faction has them equal or below it.

Clearly, $Z$ exhausts all possible situations for the values of the opinion vector $x$, and, moreover, induces a partition over the set $[o_{\min},o_{\max}]^{n}$: $[o_{\min},o_{\max}]^{n}=\cup_{m=1}^{2}Z^{-1}(m)$ and $Z^{-1}(1)\cap Z^{-1}(2)=\emptyset$.

Now, let us remark that, from the random selection process of the edges, it immediately follows that $\{x(t)\}_{t>0}$ is a random process over the probability space $(\Omega,\mathcal{F},\mathbb{P}$); and, moreover, it is a Markov process, i.e., $\mathbb{P}[x(t)\in Z^{-1}(M)\,|\,x(t-1)=c_{t-1},\dots,x(0)=c_{o}]=\mathbb{P}[x(t)\in Z^{-1}(m)\,|\,x(t-1)=c_{t-1}]$ for any $m\in\{1,2\}$. Observe that, with probability one, $x(t)\in[o_{\min},o_{\max}]^{n}$ for any $t$ since $x(0)\in[o_{\min},o_{\max}]^{n}$.

Now, assume that $x(t)\in Z^{-1}(2)$ for some $t<\infty$. Let $F_{1}$ be the faction such that $x_{i}(t)\leq\tau$ for any $i\in F_{1}$; and $F_{2}$ the one such that $x_{i}(t)>\tau$ for any $i\in F_{2}$. Let $\theta_{F_{1}}(t)=\max_{i\in{F_{1}}}x_{i}(t)$ and $\theta_{F_{2}}(t)=\min_{i\in{F_{2}}}x_{i}(t)$. If at $t+1$ some $i\in F_{1}$ and $j\in F_{2}$ are selected, we have that $x_{i}(t+1)<x_{i}(t)$ and $x_{j}(t+1)>x_{j}(t)$. On the other hand, if at $t+1$ both $i$ and $j$ belong to the same faction with $x_{i}(t)\leq x_{j}(t)$, we have that $x_{i}(t)\leq x_{i}(t+1),x_{j}(t+1)\leq x_{j}(t)$, with equality if and only if $x_{i}(t)=x_{j}(t)$. From these two observations it is easy to show that $\theta_{F_{1}}(t+1)\leq\theta_{F_{1}}(t)$ with probability one; i.e., $\{\theta_{F_{1}}(s)\}_{s\geq t}$ is a non-decreasing sequence which is lower bounded by $o_{\min}$. This implies convergence of $\{\theta_{F_{1}}(s)\}_{s\geq t}$ to some lower bound $c_{\min}$ with probability one. Now, for any $\epsilon>0$ and $t^{*}\geq t$, there exists some finite $T>0$ such that if the sequence of edges $\{(\theta_{F_{1}}(s),k(s))\}_{s=t^{*}}^{t^{*}+T}$ with $k(s)\in F_{2}$ for $t^{*}\leq s\leq t^{*}+T$ is selected, then $|\theta_{F_{1}}(t^{*}+T)-o_{\min}|<\epsilon$. Such sequence has a positive probability of being selected sequentially by the affine boomerang model for any $t^{*}$, from which it follows that $c_{\min}=o_{\min}$. Therefore, there is polarization for any $i\in F_{1}$ towards $o_{\min}$. A similar reasoning leads to the proof that $\{\theta_{F_{2}}(s)\}_{s\geq t}$ has an analogous increasing monotonic behavior and thus that there is polarization for $i\in F_{2}$ towards $o_{\max}$ with probability one. In conclusion, if $x(t)\in Z^{-1}(2)$ for $t\geq 0$, then polarization occurs with probability one and we say that $Z^{-1}(2)$ is an absorbing set since the opinion vector cannot escape from it once it enters this set.

Therefore, to finish the proof of the theorem, we only need to prove that, given $x(t)\in Z^{-1}(1)$ at any time $t$, there always exists (with probability one) a finite sequence of edges such that eventually $x(t^{*})\in Z^{-1}(2)$ for some $t<t^{*}<\infty$. Then, since any finite sequence of edges has positive probability of being selected sequentially by the affine boomerang model and $Z^{-1}(3)$ is an absorbing set, it follows that $\mathbb{P}[x(t)\in Z^{-1}(1)\;i.o.\,|\,x(0)\in Z^{-1}(1)]=0$; and this finishes the proof for item (ii) of the theorem. Therefore, it suffices to prove that $\mathbb{P}[x(t+T)\in Z^{-1}(2)\text{ for some }T>0\,|\,x(t)=x_{o}]=1$ for any $x_{o}\in Z^{-1}(1)$. So, let us fix any $x_{o}\in Z^{-1}(1)$. Let $\mathcal{T}_{1\to 2}(t)=\inf\{t^{*}>t:x(t^{*})\in Z^{-1}(2)|x(t)=x_{o}\}$ be the first time, after starting in $x_{o}\in Z^{-1}(1)$ at time $t$, at which the opinion vector enters the set $Z^{-1}(2)$. If we show that $\mathbb{P}[\mathcal{T}_{1\to 2}(t)<\infty]=1$ for any $t$, then we have finished the proof.

By the Markov property, we only need to show that $\mathbb{P}[\mathcal{T}_{1\to 2}(0)<\infty]=1$. We start by noticing that, by the finite-time proximity property, there exists a sequence of edges $s(0),\dots,s(\tau-1)$ for some $\tau>0$ such that $x(\tau)\in Z^{-1}(2)$. Let $\gamma_{o}:=\min_{\{i,j\}\in\mathcal{E}}p_{ij}$. Then,

[TABLE]

where the first inequality comes from a repetitive application of the conditional probability and the following equality comes from the independence of the underlying stochastic process. Let $\Gamma>0$ be any integer and $A_{\ell}=\{x(t)\notin Z^{-1}(2),t\in[\ell,\ell+\tau]\}$, then $\mathbb{P}[A_{0}|x(0)=x_{o}]\leq 1-\gamma_{o}^{\tau}$. Likewise, in a way similar to how we obtained expression (2), we compute

[TABLE]

Now, we observe that $\sum_{t=1}^{\infty}\mathbb{P}[\mathcal{T}_{1\to 2}(0)\geq(\tau+1)t]\leq\sum_{t=1}^{\infty}\gamma^{t}=\frac{\gamma}{1-\gamma}<\infty$ because of geometric series since $0<\gamma<1$. Then, by the first Borel-Cantelli lemma, we conclude that $\mathbb{P}[\mathcal{T}_{1\to 2}(0)<\infty]=1$. This concludes the proof.∎

A consequence of Theorem 3.1 is that a complete social network that satisfies structural balance with two factions ends up having its agents with totally opposite opinions. This agrees with the intuitive result that antagonistic groups are expected to develop polarized opinions, as shown by other models in the literature [14, 16]. Also, as expected, if there are no negative relationships between the agents (i.e., there is only one faction), all agents reach consensus.

Lemma 3.3** (Fluctuations).**

Consider a network satisfying the $k$-sign arrangement property with $k\geq 3$ factions $\{F_{1},\dots,F_{k}\}$ and such that there exists at least one negative edge between any pair of factions. Consider the boomerang opinion dynamics model (1) with $x_{i}(0)=o_{\min}$ for any $i\in F_{1}$, $x_{i}(0)=o_{\max}$ for any $i\in F_{2}$, and $x_{i}(0)\in(o_{\min},o_{\max})$ for any $i\in F_{k}$, $k\geq 3$. Then, for any $0<\epsilon<({o_{\max}-o_{\min}})/{2}$ and any $i\in F_{k}$, $k\geq 3$,

[TABLE]

Proof.

Note that $x_{i}(t)\in(o_{\min},o_{\max})$ for any $t\geq 0$ and any $i\in F_{k}$, $k\geq 3$, with probability one. Pick a positive $\epsilon<({o_{\max}-o_{\min}})/{2}$ and define the intervals $A_{\epsilon}^{\ell}=(o_{\min},o_{\min}+\epsilon)$, $A_{\epsilon}^{u}=(o_{\max}-\epsilon,o_{\max})$ and $A_{\epsilon}^{c}=[o_{\min}+\epsilon,o_{\max}-\epsilon]$. Note that these three intervals are non-empty and form a partition of $(o_{\min},o_{\max})$.

Now, take any $i\in F_{k}$, $k\geq 3$. First, define the random stopping times $\tau_{c\to\ell}(t)=\inf\{t^{*}>t\,|\,x_{i}(t^{*})\in A_{\epsilon}^{\ell}|x_{i}(t)\in A_{\epsilon}^{c}\}$, $\tau_{\ell\to u}(t)=\inf\{t^{*}>t\,|\,x_{i}(t^{*})\in A_{\epsilon}^{u}|x_{i}(t)\in A_{\epsilon}^{\ell}\}$ and $\tau_{u\to\ell}(t)=\inf\{t^{*}>t\,|\,x(t^{*})\in A_{\epsilon}^{\ell}|x_{i}(t)\in A_{\epsilon}^{u}\}$. Note that, if the pair $\{i,j\}$ is chosen, then the opinion of $i$ is always pushed towards $o_{\max}$ if $j\in F_{1}$, and always pushed towards $o_{\min}$ if $j\in F_{2}$ (this follows from the fact that for any $k\in F_{1}\cup F_{2}$, $x_{k}(t)=x_{k}(0)$ for all $t\geq 0$ with probability one). Then, following a reasoning similar to the one adopted in the proof of Theorem 3.1, we conclude that $\mathbb{P}[\tau_{c\to\ell}(t)<\infty]=\mathbb{P}[\tau_{\ell\to u}(t)<\infty]=\mathbb{P}[\tau_{u\to\ell}(0)<\infty]=1$ for any $t\geq 0$, from which the result follows. ∎

Note that the conditions for the underlying signed network in this lemma are immediately satisfied if the network is complete and satisfies clustering balance. This lemma is interpreted as follows. Assume there are multiple antagonistic groups of people such that for any two groups there exist two members that can communicate with each other. Additionally, assume that only two groups are already polarized in the opinion spectrum with the rest having opinions at intermediate values (i.e., mathematically, in the interval $(o_{\min},o_{\max})$). Then, these non-polarized groups will have their opinions always fluctuating at intermediate values, i.e., their opinions do not polarize or reach consensus at some specific value. Intuitively, since the boomerang effect is persistent on the agents with intermediate values, these agents cannot settle on a definite opinion since they continue to interact with antagonistic agents on both ends of the spectrum. This behavior of opinion fluctuation has been observed in other models in the presence of stubborn agents who forbid the consensus of opinions among the agents [2]. Our work is the first one to propose a persistent fluctuating behavior based on the structure of friendly and antagonistic relationships in a social network.

3.2 Numerical results

For a complete graph satisfying structural balance, which is a particular case satisfying the conditions of Theorem 3.1, Figure 1 shows some example evolutions for self-weights $a_{i}~{}=~{}a~{}\in~{}(0,1)$ for any agent $i$. We observed that, in general, the larger the self-weights, the more time the polarization process takes.

Figure 2 shows examples where the underlying signed network has three factions. Remarkably, under generic initial conditions (which are weaker initial conditions than the ones in Corollary 3.3), two factions tend to polarize and the opinions of the third one show persistent fluctuations.

Finally, we provide numerical evidence of the behavior under networks that are the result of perturbations on balanced networks. Consider the situation where a complete and balanced social network with two antagonistic factions is randomly perturbed by flipping the sign of some of its edges. Intuitively, for small perturbations, we would expect that opinions, though not being able to perfectly polarize, would still “attempt” to be in such a state and fluctuate near extreme values. Figure 3 shows some examples confirming this phenomenon.

4 Conclusion

We have proposed a novel simple model for opinion dynamics over signed graphs. This model provides intuitive behavior and results on the opinion evolution under sociologically relevant sign structures of the underlying social network. Future work may be the inclusion of directional updating (i.e., updating one opinion at a time) in the model, as well as its analysis under relevant directed network structures. Another open direction for research is an analytical understanding of the transient time and convergence analysis for the polarization of opinions of the factions in a balanced network.

Proof for Lemma 3.2.

Since the network satisfies the 2-sign arrangement, for any $i$ and $j$ that belong to the same faction, there exists a nonempty collection of paths $\mathcal{P}^{+}_{i\leftrightarrow j}$ between $i$ and $j$ in which each path contains only positive edges. Let $p\in\mathcal{P}^{+}_{i\leftrightarrow j}$, then, from statement (i) from Theorem 3.1, we observe that if we only update pair of vertices present along the path $p$, then they can become arbitrarily close. Then, we can construct a finite sequence of edges such that it includes only edges from one or more different paths in $\mathcal{P}^{+}_{i\leftrightarrow j}$ in a sufficient number so that $i$ and $j$ become arbitrarily close. This proves the first part of the lemma.

Now, we consider the case where $i$ and $j$ belong to different factions. Notice that equation (1) clearly shows that we can always make the opinions of two vertices joined by a negative edge arbitrarily apart by continuously sampling such edge. Let $\mathcal{P}^{-}_{i\leftrightarrow j}$ be the nonempty collection of paths between $i$ and $j$. Due to the structure of the network, any $p\in\mathcal{P}^{-}_{i\leftrightarrow j}$ must have an odd number of negative edges. Then, $p$ can be constructed by appropriately concatenating sequences of positive edges with sequences of negative edges. From our discussion above, we can make the opinions of the agents participating in any of these positive sequences (if any) arbitrarily close, and the opinions of the agents in any of the negative edges arbitrarily apart. Then, it is possible to come up with a finite sequence of edges such that $i$ and $j$ become arbitrarily apart. This finishes the proof of the lemma. ∎