Optimal majority dynamics for the diffusion of an opinion when multiple alternatives are available

Abstract

We consider opinion diffusion on social graphs where agents hold opinions and where social pressure leads them to conform to the opinion manifested by the majority of their neighbors. Within this setting, we look for dynamics that allows us to maximize the diffusion of a target opinion given the initial opinions of all agents. In particular, we focus on the setting where more than two opinions are available to the agents, and we show that the properties of this setting are entirely different from those characterizing the setting where agents hold binary opinions only. Indeed, while it is well-known that greedy dynamics are always optimal ones in the binary case, this is no longer true in our more general setting and—rather surprisingly—even if just three opinions are available. Moreover, while it is possible to decide in polynomial time if a dynamics leading to consensus exists when agents have two available opinions, the problem becomes computationally intractable with three opinions, regardless of the fraction of agents that have the target opinion as their initial opinion.

Introduction

Consider the following prototypical scenario. The members of a department are organizing a social dinner, and they have to decide which restaurant to go. Initially, each of them holds an opinion on her ideal choice. At a certain point, they will exchange their viewpoints and each of them will be affected by a social pressure leading to adapt her opinion to the one manifested by the majority of her friends. So, one may ask: Would they be capable to reach a consensus for some/all profiles of their initial opinions? Can a starting group of seeds orchestrate a campaign to influence all other members or at least a large amount of them?

In this work we pave the way to a thorough analysis of the above kinds of questions under the lens of algorithm design and computational complexity. Specifically, we focus on a setting where social relationships are encoded as the edges of a social graph $G=(N,E)$ whose nodes correspond to the agents. In particular, by starting from an initial configuration where agents hold some innate opinions, we consider a model of opinion diffusion over the underlying graph where each agent is stable if and only if her current opinion agrees with the opinion held by a (non-strict) plurality of her neighbors. Hence, at any time step of the dynamics, agents that are not stable can change their opinion asynchronously, thereby leading to a non-deterministic evolution where the final configuration may depend on the specific order in which updates have been performed. This behavior has been used as a reference model in a number of works related to opinion diffusion in social networks (see, e.g., [38], [21], [36], [27], [31], [22]).

The study of how much an opinion can spread according to this diffusion process received, very recently, a lot of attention [3], [16], [5], [6], [9], [17]. However, most of results focused on the case in which only two opinions are available. For instance, Bredereck and Elkind [16] show that in this setting, there is a greedy implementation of the above diffusion process that always maximizes the number of agents with a given target opinion in the final configuration: it assumes that at each time step, the agent(s) that are willing to adopt the target opinion are allowed to change before every remaining agent. This basic result turned out to be very useful for describing in more details the evolution of opinions in this setting. For instance, Auletta et al. [3], [5], [6] show that there always exists an initial configuration from which a minority of agents can become majority when opinions spread according to this greedy diffusion process (and some variant of it); moreover, an even stronger result has been proved by Auletta et al. [9], namely that there always exists an initial configuration from which any majority of agents can converge to consensus.

The question that we address in this paper is:

To what extent do these results for the case in which only two opinions are available extend to scenarios with more than two opinions?

The answer that we provide to this question is essentially negative. We first focus on the greedy diffusion process. We show that, when there are at least three available opinions, greedy dynamics may fail to maximize the number of agents adopting the target opinion at the end of the diffusion process.

Actually, this behavior does not emerge just in extreme cases. Indeed, we show that the graphs for which every greedy dynamics always maximizes the diffusion of the target opinion are only trivial ones, such as paths, cycles and stars. Moreover, even if we only require the weaker property that there exists at least one maximizing greedy dynamics (but we allow other greedy dynamics to be unable to reach this goal), the set of graphs enjoying this property is surprisingly very limited: indeed, these are networks with “core” nodes having degree only two and four. And, even in these networks, computing the best greedy dynamics is a hard problem. Hence, greedy dynamics do not help in analyzing the outcomes of the opinion diffusion process when multiple opinions are available.

In fact, one might still wonder whether other kinds of (computationally) simple dynamics are guaranteed to exist which lead to “optimal” outcomes. We show that this question has a negative answer too. Indeed, we show that it is NP-hard to decide if a dynamics exists leading to a final configuration in which the fraction of agents adopting the target opinion is above a given threshold. The result still holds if one simply focuses on reaching consensus, regardless of how many agents hold the target opinion at the beginning.

As we have already pointed out, most of the works on opinion diffusion have focused on the case where only two opinions are available. However, the case of multiple opinions has been considered rather recently by Chierichetti et al. [22]. But, this work did not address the question of how the dynamics of the opinions can be manipulated and their results are unrelated to ours.

On the other hand, some recent literature on opinion diffusion considered more complex behaviors in addition to majority-based ones—see, e.g., [4], [2]. Moreover, recent works considered opinion formation on evolving networks [14], [15], [28]. Understanding to which extent our results can be lifted to these more complex settings is an interesting avenue for further research.

More in general, note that the question we consider in our work is related with questions emerging in viral marketing. The study of this topic has been initiated by Domingos and Richardson [26], [39], with the main topic being known as target set selection and consisting in deciding which nodes in a network need to be seeded in order to maximize the diffusion of the target opinion/product. Specific algorithmic and computational properties of this problem have been found depending on the adopted model of diffusion [33]. For instance, it has been observed that, for many natural models, the presence of randomization leads to intractability [32]. Motivated by these bad news, target set selection has been also studied under the lens of approximation theory [21] and parametrized complexity theory (see [37], [12], [13]). Moreover, many heuristics that scale over large networks have been proposed (see [23] and the references therein).

Actually, our specific perspective is closer to a different kind of marketing issue: seeds are given and they are not under the control of the marketer, who may instead incentivize nodes to change their opinions in the desired order, by exhibiting a social proof of the quality of the product, that is, a list of “friends” or influential people that are already using that product. This kind of social proof marketing strategies are nowadays very common and greedy strategies are very appealing in this framework. Indeed, they mean that the marketer has only to nudge agents for adopting the target product, and they can leave other products to freely spread among remaining nodes. Non-greedy approaches turn out to be much less natural in this setting, since they indeed require that the marketer invites agents to adopt products different from the targeted one. Hence, our results can be seen as proving that natural social proof marketing strategies may fail to maximize the diffusion of the targeted product whenever there are at least other two competing products, and finding other strategies achieving this objective is a hard problem.

Our works is also related to the recent line of research about election manipulation on social networks [40], [25], [24], [1], [19], [20], [35]. These works however focuses on a different model in which the information spreads over the networks and cause the change of opinion by voters.

The rest of the paper is organized as follows. The model of opinion diffusion considered in our work is detailed and formalized in Section 2. The basic properties of greedy strategies for opinion diffusions are studied in Section 3, while a thorough complexity investigation is conducted in Section 4. A few final remarks and directions for further work are discussed in Section 5. Some longer proofs are omitted from the main text and moved to Appendix.