Erratum: Role of the plurality rule in multiple choices (J. Stat. Mech. (2016) 023405)

. People are often challenged to select one among several alternatives. This situation is present not only in decisions about complex issues, e.g. political or academic choices, but also about trivial ones, such as in daily purchases at a supermarket. We tackle this scenario by means of the tools of statistical mechanics. Following this approach, we introduce and analyse a model of opinion dynamics, using a Potts-like state variable to represent the multiple choices, including the ‘ undecided state ’ , which represents the individuals who do not make a choice. We investigate the dynamics over Erd ö s – R é nyi and Barab á si – Albert networks, two paradigmatic classes with the small-world property, and we show the impact of the type of network on the opinion dynamics. Depending on the number of available options q and on the degree distribution of the network of contacts, dierent ﬁnal steady states are accessible: from a wide distribution of choices to a state where a given option largely dominates. The abrupt transition between them is consistent with the sudden viral dominance of a given option over many similar ones. Moreover, the probability distributions produced by the model are validated by real data. Finally, we show that the model also contemplates the real situation of overchoice, where a large number of similar alternatives makes the choice process harder and indecision prevail.


I. Introduction
People frequently face diverse situations that oer a wide range of options, such as when looking for a restaurant, a hotel, a phone model or any basic goods in the supermarket. The number of goods increases every day. It is estimated that approximately 50 000 new products are introduced every year in the US 4 . Even within each category of items, there may be many brands and item variations without dierentiated attractiveness. This leads to the problem of facing too many choices, termed 'overchoice' or choice overload [1]. To make decisions in these situations can be costly, and this stressful process leads to poor decisions or no decisions at all [2][3][4]. Then, the advantages of multiple choices can be cancelled out by the disadvantages of a more complicated choice process. In fact, despite representing an apparently positive development, many options may hinder the process of choice. For example, people with many purchase options tend to have more diculty in choosing and may end up buying nothing [5]. Motivated by these observations, we wonder to what extent people's interactions, leaving aside their individual psychology, contribute to this scenario by introducing, for instance, conflict and frustration. Then, by means of a model of opinion dynamics, we investigate the distribution of adoptions made by a population facing a large number of choices.
In modelling people's interactions, one of the basic ingredients is imitation, or social contagion. In fact, imitation occurs in diverse social contexts, from the dynamics of language learning to decision making. Depending on the questions posed, diverse rules of contagion, from simple pairwise to group interactions, have been proposed and studied in recent years [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. However, very few works deal with many choices [16,17,26,27]. In the vast majority of opinion models, the opinion of an agent is represented by a binary variable, since many questions can be tackled through the J. Stat. Mech. (2016) 023405 assumption of two possible (opposite) attitudes, e.g. being either favourable or unfavourable to a given choice. This kind of binary variable was also inspired in the spin-1/2 Ising model, leading to transposing the known results from physical to social questions. For our present purpose of studying multiple-choice situations, it is natural to consider a Potts-like state variable that can take several (discrete) values.
We consider that changes from one state to another are governed, not by simple pairwise contagion, like in [17,26,27], but instead by a 'plurality' rule [16]. This is based on the idea that an individual makes the choice that is the most popular among its contacts. In fact, when we have to choose or buy something, especially when there are so many similar options that there is not a favourite one a priori and it is not feasible to examine them all, it is reasonable to take into account other people's preferences [28]. Naturally, the closer the person is in our network of contacts, the more importance we give to their opinion, since the nearest neighbours in the network typically have similar interests and tastes. One can use this strategy not only in trivial or daily life issues but also in major ones such as political elections, where many candidates are competing. For changing or adopting a new opinion, however, a minimum of consensus between the contacts is necessary. This is expressed in a plurality rule, according to which an individual is persuaded to adopt the opinion shared by the largest number of its nearest neighbours.
Evolution rules based on a locally dominant opinion have been considered before, for instance, the majority rule for two states introduced by Galam [18,19]. It was later extended to multistate opinions [16], by considering all-to-all interactions where the individual and all its contacts adopt the same opinion as the majority at the same time. A variant where, instead of the local majority, the plurality opinion is considered, was also studied [16]. But in the situations we address here, the decisions are not taken in groups or simultaneously, rather the choices of the individuals are aected by their knowledge of the previous choices of their contacts. Therefore, we assume that one single individual opinion changes at a time. Moreover, we will include the possibility of undecided people, which is a realistic feature which has been taken into account in three-state models, as natural extensions of binary cases [21][22][23][29][30][31]. Furthermore, as another dierential, the dynamics of our model takes place in small-world networks that, even if they do not facilitate analytic treatment, are more realistic than regular or mean-field settings. As paradigms of small-world networks, we consider two classes with distinct degree distributions: Erdös-Rényi (ER) [32,33] and Barabási-Albert (BA) [34] networks. The details of the model will be defined in section II.
The model encompasses instances where the dierent alternatives have a similar initial attractiveness. Many products sold on the Internet with similar qualities and prices, e.g. music albums, shoes etc, are within the model's scope. We also make the simplification that individuals can dier only in the number of contacts. Other heterogeneities of the agents might also be introduced in further work. Now we ask a basic question: how does a plurality rule mold the decision spectrum in the simplest, most homogeneous case?
First, we address the classical issue in this kind of problem about whether a consensus state can be achieved or not, where all (or almost all) of the individuals share the same preferences. Then, in simulations of the model, we compute the fraction of the realisations reaching consensus. In non-consensus situations, we analyse the distribution of adoptions. We also compute other relevant quantities such as the fraction of undecided people and the fraction of the population adopting the most popular alternative. The results will be presented in section III.

II. Plurality modelling
A plurality rule governs the opinion dynamics of N agents interacting through their network of contacts. Each agent i, corresponding to a node in the network, has a Pottslike opinion state variable S i , that can take the values s s , , q 1 … representing q electable alternatives (options or choices, which we enumerate in an arbitrary order), as well as an 'undecided' state s 0 , assessed when the individual has not adopted a defined option. The addition of the undecided state reflects the fact that sometimes people do not have a favourite choice.
We focus on the dynamics developed in the ER and BA networks, as representative of small-world networks with homogeneous and heterogeneous degree distributions, respectively. However, for comparative purposes we will also consider random neighbours and nearest neighbours in a square lattice (with periodic boundary conditions).
We assume that most individuals do not have a formed opinion a priori, except for initiators representative of each oered choice. Then, we start the dynamics with all nodes in the S = s 0 state, except randomly chosen q nodes, to each of which we attribute a dierent opinion S s s , , q 1 = … . We consider the same number of initiators (one initiator) for each alternative, reflecting the equivalent attractiveness of all the alternatives. This kind of initial state has been used in opinion models for proportional elections [35].
At each Monte Carlo (MC) step of the dynamics, we visit all the nodes of the network in a random order and update them successively, in asynchronous mode. The state of the visited node i is updated according to the following steps: (i) We define the set of nodes, i A , formed by i and its nearest neighbours.
(ii) We determine the plurality state S s i p 0 ≠ , associated with node i, as the state shared by the largest number of nodes in the set i A .
(iii) The agent i will then adopt its corresponding plurality state.
Note that, when we measure the state S i p , we ignore the nodes in i A that have S = s 0 (see figure 1) but the current opinion of site i also counts to define S i p . The updates are repeated and the dynamics stops when an absorbing state is attained, i.e. if, at a MC step, none of the nodes changes its state.
Let us remark that this dynamics diers from that of the Sznajd type [36,37], where two or more individuals sharing the same opinion impose it on all their neighbours. It also diers in several aspects from the plurality rule introduced in [16]: (i) while in our case only the central node is aected, in [16] the whole group i A changes its opinion to the plurality state S i p in a single update step; (ii) here the size of the interaction group i A is given by G = k i + 1, where k i is the connectivity of site i, instead of being constant (anyway, the parameter k 1 ⟨ ⟩ + plays the role of an eective G, and they coincide in the limit of a highly homogeneous, or regular, network); (iii) the possibility of indecision is not contemplated in [16]; (iv) in the event of a tie, the opinion of node i remains unchanged in our model, while in [16] one of the dominant options is randomly selected. With respect to this last item, the present dynamics is closer to the J. Stat. Mech. (2016) 023405 majority rule version of [16], where the dynamics becomes static because, when there is no local majority, the state of the group does not change; (v) in terms of the underlying network, here we consider small-world networks, while the dynamics in [16] was studied in the mean-field limit and over a square lattice; vi) finally, another important dierence is in the initial conditions: we consider that decided nodes are diluted in a sea of undecided nodes, instead of the equiprobability of definite opinions.

III.A. Plurality dynamics
We follow the evolution of each realisation of the dynamics until the final state is attained. Distinct distributions of opinions can emerge in the final state depending on the number of alternatives q, the average connectivity k ⟨ ⟩, the network topology and size N. Figure 2 illustrates the evolution of n s , the number of nodes that share the same opinion s, in representative realisations of the dynamics on ER networks of size N = 10 4 . A final state is reached in a few MC iterations. Figure 2 illustrates the distinct patterns that can arise, while the full phenomenology as a function of the model parameter q and network features will be shown in section III.B.  (d)-(f) When q is large, the number of undecided agents does not decrease monotonously when the connectivity increases. However, on increasing the connectivity, consensus is reached, although it can take a longer time than for small q.
In all cases, the number of undecided nodes decreases with time, because the undecided individuals are not produced by the dynamics in the present version of the model, but are only introduced in the initial condition.
The dynamics in ER networks can be qualitatively understood as follows: In a first regime, each opinion propagates invading the undecided neighbours. If the initiators are very diluted (q N ), and the connectivity is not too high, then each cluster of nodes with the same opinion can develop almost independently of each other, during several MC steps (non-competitive regime). In this case, the initial growth is nearly exponential, described approximately by n t k n d /d in ER networks. When two or more clusters collide, a competitive regime starts. Depending on the network, the competition can take place more or less evenly so that ties stagnate the and (e)). Otherwise, a sort of rich-get-richer or cumulative advantage mechanism can take place. In that case, the winner's opinion becomes noticeably larger than the other ones, convincing individuals from other opinion clusters (figure 2(b)) or even the whole network ( figure 2(c)). Similar patterns as those shown for ER networks are also observed for BA networks, although for dierent values of the parameters, as illustrated in the first column of figure 3. Note that the winner's opinion, as well as the number of decided people, for the same parameters, are favoured in BA networks, where the cumulative advantage eects are more accentuated.
Let us remark that in some extreme situations, given the initial conditions studied here (one initiator for each state), the system does not evolve. This occurs, for instance, in the limit case when q = N (hence, each individual has a defined opinion) or the connectivity of all sites is N (complete graph). In these extreme cases, ties forbid changes of state and the dynamics is frozen from the start. However, we will restrict the range of the parameters to the region q k N , ⟨ ⟩ . Aside from networks with the small-world property (ER and BA), we also analysed, for comparison, interactions with (i) first neighbours in SQs (with periodic boundary conditions), and (ii) K random neighbours (with K > 1). Representative examples of the evolution, in SQs and K random neighbours for the case K = 4, are shown in the second and third columns of figure 3.
In regular lattices, the evolution is essentially non-competitive. Clusters grow from their initiators and, when they collide, the dynamics freezes due to ties in the interfaces without entering a competitive phase, in contrast to ER and BA networks, where there are long-range links that break the ties. Since the initial growth occurs at the surface of the cluster, then n n d s s ∝ which, dierently from small-world networks, gives a quadratic increase of n s with t, as observed in figure 3 (second column). Alternatively, it is easy to show that, for the synchronised update, at short times, the number of adopters of each choice grows around its initiator, following, on average, the recursion relation that yields the predicted quadratic increase with t, valid for small t, until n N q / s holds. Despite the prediction is done for the synchronous update, it is in good agreement with the average value of the simulated curves, as can be seen in the second column of figure 3. As a consequence of the lack of competition, the final values of n s are less disperse in the square lattice than in random networks and, mainly, consensus, or even a wide dominance of an opinion, becomes unlikely for q > 1.
In the absence of any network structure (i.e. when the neighbours are chosen purely randomly), like in the examples of the last column of figure 3, one of the opinions dominates and attains consensus, even for very small K. This can be understood in terms of a mean-field approach, following the lines of [16]. In fact, the fraction of undecided sites f n N / where P K−1 is a polynomial of order K − 1 in f 0 , whose coecients depend on the fractions f n N / s S ≡ , with S s s , , q 1 = … and P K−1 (0) > 0. For all K and q, the factor f 0 arises from the central undecided node that is part of the group. Therefore, in the steady state it must be f 0 = 0. Moreover, since P K−1 (0) > 0, equation (2) is stable for f 0 = 0. The remaining equations for the other fractions f s have solutions of the type found in [16] for their majority and plurality versions. In particular, the stable solutions are those of consensus, where f s j = 1 for some j (hence, the remaining fractions vanish).
For instance, for K = 2 and q = 2 (let us call f f s j j ≡ , j q 0 ⩽ ⩽ ), we have  = = is unstable. The above equations, for small values of K and q, valid for f 0 = 0 in the present model, are the same obtained for the majority and plurality versions studied in [16], although the equations in three cases dier for large enough values of K and q.
Increasing K leads to equations of the form where P K−2 is a definite positive polynomial of order K − 2 in f 1 , whose coecients depend on f 2 . Therefore, consensus is always stable for q = 2.
For a large number of alternatives, f 0 must necessarily vanish as well, and consensus is also a stable solution. For instance, when q = 3 and K = 2, once f 0 = 0, we have Increasing q and K, the structure of fixed points becomes more complex and more routes to consensus emerge [16]. Nonetheless, consensus is always the final state, which was also verified through numerical simulations.
Conversely, when the structure of the interaction network is relevant, non-trivial behaviours occur, as those illustrated in figure 2. A population of undecided people can survive and consensus is not always attained.

III.B. Phase diagram
To summarise the nontrivial final configurations that emerge in ER and BA networks we built a phase diagram in the plane k q ⟨ ⟩ − . For each realisation, we monitored the fraction of decided people and also the fraction of nodes sharing the most adopted opinion, or winner's choice, These quantities were averaged at the final state of several realisations. Unless something dierent was said, at least 50 realisations were considered for each set of values of the parameters. For each realisation of the dynamics, a dierent network was generated. The averaged fractions will be denoted by f d and f w , respectively.
We also computed the fraction p c of the simulations that reach consensus (operationally meaning at least 99% of the population).

J. Stat. Mech. (2016) 023405
When performing computations over ER networks, only the main component of the graph was considered.
The phase diagram in the plane of parameters k q ⟨ ⟩ − for the ER and BA networks is depicted in figure 4. We restricted the analysis to the region q N /10 ⩽ and k N/100 ⟨ ⟩ ⩽ . The diagram shows the changeover from regions of consensus to regions of a fragmented final state, as indicated by the colours from red to blue. The filled symbols emphasise the points where consensus is certain (p c = 1, red) or uncertain (p c = 0, blue). Moreover, the solid lines depict the frontier where f w = 0.5 ( f w > 0.5 to the right of the curves), and the shadowed areas highlight the points for which undecided people are the majority ( f d < 0.5). It is clear that the consensus domain (p 1 c , red region) is larger for BA networks, indicating that heterogeneity favours consensus. In both the ER and BA networks, inserting a few links, for fixed q near the transition frontier, may trigger consensus.
In BA networks, for suciently large q ( 200) the critical value k 7 c ⟨ ⟩ becomes independent of q. In contrast, in ER networks, the dependence on q is stronger. The nonconsensus domain for ER networks, (p 0 c , blue region), when q becomes suciently large, spreads over the region of large mean connectivities. This means that near the transition frontier, eliminating a few alternatives can trigger consensus, an eect which in BA networks only occurs for connectivities below k 7 c ⟨ ⟩ . Concomitantly, in the shadowed area in figure 4(a), the fraction of decided people becomes a minority ( f d < 0.5). Conversely, in the BA case, the majority of nodes is decided over the whole phase diagram. Moreover, even in the absence of consensus, the winner group can become the majority ( f w > 0.5) more easily in BA networks.
In the following sections III.C and III.D, we describe in more detail the dependency on q and k ⟨ ⟩, respectively.

III.C. Eect of the number of options q
In this section, we focus on the impact of the number of opinions q on the steady state, and we also discuss size eects. Figure 5 shows the three quantities of interest, f d , f w and p c , as a function of q, for ER and BA networks and SQ lattices of three dierent sizes. The plots of p c versus q, for the ER and BA networks, correspond to vertical cuts of the phase diagram in figure 4. All the fractions monotonically decrease with q in the studied range, indicating that independently of the lattice, increasing the number of options hinders the process of decision making, thus promoting indecision and also making more dicult the appearance of a popular choice.
Concerning size eects, in ER networks (panel (a)) the three quantities are almost independent of size N.
The results in the square lattices (panel (c)) are in agreement with the predictions made in section III.A. For instance, p c > 0 only for q = 1. The winner fraction f w is independent of L and decays subtly above 1/q, as expected. In fact, we have seen that the fractions of each opinion have a narrow distribution around the mean value 1/q. Meanwhile, the decided fraction f d depends on L. Since undecided sites are at the interfaces of each opinion cluster, then their quantity over all q clusters is n L q q / /2 s 2 0 ∝ × . Therefore, the decided fraction f n L 1 /  . The fraction f w , as well as the critical value at which p c vanishes, follow the same scaling, as shown in the inset of figure 5(b).
The average fraction of decided people f d typically decreases with q (for q N ), but a kind of saturation eect occurs for large enough q and a flat level appears in small-world networks, indicating that f d becomes insensitive to the introduction of new choices. However, the value of the flat level changes with the network connectivity (not shown). For a large number of options q, the values of f d are smaller in ER networks. Heterogeneity of degrees seems to be helpful in breaking ties. In the heterogeneous BA networks, many nodes have low connectivity, and can be easily convinced by a decided neighbour. On the other hand, the occurrence of a local plurality is less likely in a homogeneous ER network with a given connectivity. As a consequence, ties are more frequent, more nodes remain undecided and the dynamics freezes. However, in square lattices, despite the homogeneity, the undecided fraction is relatively small. This can be understood as follows. Ties occur when distinct opinion clusters collide, then the surviving undecided nodes are located at the 'interfaces'. In networks with long-range links the encounter between dierent clusters occurs early, and many nodes remain undecided; meanwhile, in regular lattices with nearest-neighbour interactions, undecided nodes are conquered until the collision, late in the dynamics, when the opinion groups have occupied most of the lattice and few undecided nodes remain at the interfaces.
The average fraction of the population adopting the winner option, f w (squares) is also a significant quantity. (Necessarily f f w d ⩽ . ) The fraction f w is greater in BA networks. That is, the winner choice conquers on average a large fraction of the population in BA networks, compared to ER networks and square lattices with equivalent k ⟨ ⟩. In fact, the cumulative advantage that drives the growth of an opinion group is facilitated in these heterogeneous networks due to the presence of hubs, and the winner conquers more adopters. Also note that, in ER networks, when the fraction f d attains the flat level, a dominant opinion is absent, as mirrored by the very small value of f w (see figure 5(a)). Meanwhile, in BA networks the winner can always conquer an important fraction of the population ( figure 5(b)), shown by the fact that f w remains finite (except in the limit q N → ). For all kinds of networks, the probability of occurrence of consensus, p c (circles), typically falls from 1 to 0 as q increases. This agrees with the intuition that when there are more options to choose from, it is more dicult to attain consensus. The probability of consensus decays rapidly with q and, above a critical value q c , the fraction p c becomes negligibly small. This eect is accentuated in the square lattice where for q c = 1.

III.D. Eect of the mean connectivity k ⟨ ⟩
The behaviour of the characteristic fractions f d , f w and p c as a function of the average connectivity of the network, k ⟨ ⟩, are shown in figure 6 for several values of q in ER and BA networks. The plots of p c versus k ⟨ ⟩ correspond to horizontal cuts of the phase diagram in figure 4.
Let us start with the case of BA networks (shown in figure 6(b)) that exhibit simple monotonic behaviour, for the range shown in the figure. The three fractions increase with the connectivity. As observed in the phase diagram in figures 4(b) and 6(b) shows in more detail how the jump to consensus becomes more abrupt as q increases and the critical value of the connectivity becomes nearly independent of the number of options ( k 7 c ⟨ ⟩ ), as can also be observed in the phase diagram. Dierently, in ER networks (see figure 6(a)), k c ⟨ ⟩ increases with q. Moreover, the average fraction of decided people f d first decreases with the connectivity down to a minimal value localised at k min ⟨ ⟩ . Up to that point, the fraction of simulations attaining consensus p c is negligibly small. But, at k min ⟨ ⟩ , a transition occurs and both p c and f d rapidly increase with k ⟨ ⟩, up to 1. One would expect to have more decided nodes when the connectivity is higher, like in the case of BA networks, since, in principle, more connections might facilitate information spreading. However, on a low connected network, opinion groups are typically isolated from each other. When links are added, and disconnected groups become connected, ties can occur. That is, on the one hand higher connectivity implies that groups of dierent opinions can be more connected between them and compete. On the other hand, a node will be aware of more opinions, making the decision dicult and keeping more undecided nodes. Therefore, not only overchoice (high q) may produce stagnation of the dynamics but also 'overlink', or excess of contacts due to high k ⟨ ⟩. This explains the initial decrease of f d with k ⟨ ⟩, which occurs up to a minimal value of f d . After that point, introducing more connections will allow a dominant group to impose its opinion, concomitantly p c increases until reaching its maximal value of 1. Also note that in ER networks, for each value of q, there is an interval of mean connectivity for which the fraction of decided people becomes a minority.
The existence of an abrupt transition from a situation where many opinions coexist to consensus indicates that, by adding just a few links or by removing a few choices, most people may come to adopt the same state. The transition to consensus is more abrupt in BA networks, and the jump width decreases with q. In these networks, as discussed J. Stat. Mech. (2016) 023405 above, there is a dominant winner opinion group, which represents an important fraction of the population (finite f w ). The largest group gains additional adopters more easily, with a cumulative advantage. Near the critical connectivity, when adding a few links at random, it would be more probable to connect the very large group to smaller ones, and, as a consequence, they would be conquered by the dominant opinion, rapidly leading to consensus. In homogeneous ER networks this transition is less abrupt, because a largely dominant group is less probable. The width of the transition region slightly increases with q. For the extreme case of nearest neighbour interactions in square lattices, cumulative eects are completely absent; therefore, a transition to consensus is unlikely.

III.E. Distribution of opinions and empirical data
In non-consensus steady states, a broad distribution of opinions across the population can emerge. In order to analyse the shape of the distributions, we built the normalised histograms of P(n s ), where n s is the number of nodes with a given opinion s. The histograms were computed by accumulating realisations ending in non-consensus states. Typical distributions are depicted in figure 7. One can identify exponential, log-normal and power-law behaviours.
In the ER case, far enough from the critical frontier of consensus, the preferences are almost uniformly distributed with an exponential cuto. When approaching consensus (for instance by increasing k ⟨ ⟩) the distribution adopts a log-normal shape. Note that this occurs in the region of the phase diagram where indecision prevails (the shadowed area in figure 4). In BA networks, the distribution can also resemble a log-normal, but when approaching consensus the tail rises due to the existence of dominant winners. Moreover, when the dynamics freezes early, P(n s ) tends to reflect the degree distribution with exponent −3.
In order to compare the distributions from simulations with those from real world, we considered products that are rated online. We analysed data about items whose alternatives are not significantly dierentiated (for instance, in price and/or quality), as assumed in our model. We identified q with the number of items within each category and we considered the number of positive reviews (those with 4 and 5 stars) received by each item as an indicator of its total number of adopters, which potentially might become spreaders of the product, in a situation like that described by the present model. We analysed all the music albums from Google Play music 5 whose prices are similar (U$ 9 3 ± ); see supplementary data (stacks.iop.org/JSTAT/2016/023405/mmedia). We also analysed male sneakers from  , of the number of (favourable) reviews n r , with 4 and 5 stars, for Netshoes and Google Play music. It is reasonable to identify the number of items with q, and the number of reviews giving 4 and 5 stars with the number of decided people n d . For Google Play music, the prices are similar (U$ ± 9 3), then all the data were used ( × n 1.6 10 d 6 ; q 7500). For Netshoes we split the data into two subsets: items with prices above ( Netshoes 6 , a Brazilian e-commerce for sport's goods; see supplementary data (stacks.iop. org/JSTAT/2016/023405/mmedia). Since in this case the prices are more diverse, we split the data into two subsets: items with prices below and above the median (about R$ 200). For each set of data we computed the histogram of the number of adoptions (i.e. the number of reviews awarding 4 and 5 stars), as shown in figure 8. Comparison of figures 7 and 8 evidenced a remarkable qualitative similarity between real and simulated distributions producing log-normal shapes. The shape for small values of s is also similar. Once service users have access to the reviews of any other user, the underlying network is expected to be similar to a random graph with relatively high connectivity. Since purely random (mean-field) interactions would lead to consensus, which is not observed in the empirical data, one concludes that the underlying network must have some structure. The absence of a fat tail, related to the presence of hubs, as in BA networks, indicates that the empirical cases are best modelled by ER networks, at least qualitatively. In fact, in a 'rating network', the reviews are equivalent and none of them are expected to act as a hub; then, it is reasonable that ER networks yield more realistic results in this case.

IV. Final remarks
We introduced a model based on a plurality rule that mimics decision making governed by the influence of the relative majority of the neighbours. The model also incorporates the possibility of undecided agents. This applies not only to situations involving the consumption of products or services, but also to other environments where there is a variety of options, as far as the options are homogeneous with similar attractiveness (similar quality, cost, etc) and people have no preferred choice a priori.
Dierent final steady states emerge from the dynamics, depending on the number of available options and on the degree distribution of the network of contacts: consensus, wide distribution of opinions or also situations where indecision dominates for a suciently large number of options. In fact, decision making governed by the plurality rule may yield ties (conflict and frustration), contributing to overchoice stagnation. This eect appears to be mitigated in BA networks. The model envisages that stagnation may result not only from overchoice but also from the excess of links (see figure 6(a)).
For both types of small-world networks, consensus is almost certain for suciently low number of options and suciently high connectivity. For ER networks, consensus can occur even if there are many options available. If the neighbours are random, consensus is the rule. But consensus is unlikely for a large number of options and low network connectivity, especially if the network is homogeneous. In the square lattice with periodic boundary conditions, consensus is not reached, but opinions tend to be equipartitioned. In small-world networks, there are nontrivial, non-consensus regimes, and a broad distribution of opinions can emerge, with a shape similar to that of real ones when the assumptions of homogeneity hold, like in the examples of Netshoes and GooglePlay music albums.

J. Stat. Mech. (2016) 023405
The model indicates that consensus can suddenly emerge simply by introducing a few connections or eliminating a few items. Furthermore, it also predicts that an item can become very popular (with relatively large f w ), even if the initial attractiveness of all the items is uniform. These observations furnish another possible explanation as to why there is so much amateur content viralising on the Internet, or why a service, good or cultural product can become a bestseller without having any apparent dierentiated attractiveness.