Data

We utilized the following four data sets: (i) The Twitter followership network of the members of the 111th U.S. Senate; (ii) the roll call of the Senators on legislative bills; (iii) The Twitter followership network of the members of the 18th Korean National Assembly; and (iv) the roll call of the Assembly members between the years 2008 to 2010 (covering three quarters of the term). The Twitter data are current as of June 2011, and Twitter users who followed two or more legislators were included in the data. The data contain all legislators who owned an official Twitter account at the time of data collection and maintained their seat during the entire term considered. This resulted in 67 senators (out of 100), 698 roll calls, and 139 806 Twitter users for the U.S., and 194 assembly members (our of 299), 1 119 roll calls, and 124 341 Twitter users for Korea. Both countries have two-party systems, the Republican Party (GOP) and the Democratic Party (DEM) that account for 99% of the U.S. Senate, and the Grand National Party (GNP) and the Korean Democratic Party (DP) that account for 86% of the Korean National Assembly. [12]

The data were modeled as bipartite networks. See Fig 1 for the schematic illustrations of the data.

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Fig 1. Schematic illustrations for the data employed in our study.

(A) The legislators are included in two distinct bipartite networks. On the left is the legislator–Twitter user network, and on the right the legislator–legislative bill network. Of the two types of edges—a ‘nay’ vote and a ‘yea’ vote—we consider the ‘nay’, since they are believed to carry more information in determining the legislators’ political spectra (see the Method section for more detail). The structure of each bipartite network can reveal differences in political positions of the legislators, which is the origin of the online-offline discrepancy. Here, for example, the upper two legislators occupy similar positions that are different from that of the the lower one since their follower sets are disjoint. The two groups’ voting patterns may show less clear differences. In this study we use Multidimensional Scaling (MDS) and Kendall’s ranking correlation coefficient to quantify the spectra and their discrepancies. (B) A sample of the U.S. senator–Twitter followership network consisting of ten legislators and their Twitter followers. The red squares are Republican (GOP) senators, and the blue squares are the Democratic (DEM) senators. All other nodes are Twitter users.

https://doi.org/10.1371/journal.pone.0124722.g001

Online and offline political spectra

The data introduced above can be modeled as bipartite networks on which the political spectra are constructed using matrix decomposition methods [13, 14]. In this study we use Classical Multidimensional Scaling (CMDS) [15, 16]. The method requires a dissimilarity function that act as the distance between two legislators. We use the Kulczynski dissimilarity [17]. In essence, CMDS regards the dissimilarity values as the Euclidean distances between legislators in 1-dimensional space and determines the coordinates of the legislators that most closely matches the given distances. We then interpret the coordinates as the political positions of the legislators. Once the spectra of the legislators are determined this way using the Twitter network, we can construct the spectra of the Twitter users by averaging the positions of the legislators that each user follows (see the Methods section for more details).

We consider the offline, roll call-based spectra as revealing the true political positions of the legislators as they results from their actual actions of voting. Furthermore, we accept the roll call-based spectra as more accurately reflecting true political positions of the offline public and the society, as the legislators’ actions are likely to be heavily influenced by public interests.

The political spectra of the legislators thus constructed are shown in Fig 2A and 2B for the U.S., and in Fig 2C and 2D for Korea. The spectra of the Twitter users (the average of the positions of the legislators that they follow) are shown as the solid lines in Fig 2A and 2C.

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Fig 2. Political spectra determined from Twitter and roll call data.

(A) The Twitter-based spectrum of the legislators, and the spectrum of Twitter users in the U.S.; (B) The roll call-based spectrum of the legislators in the U.S.; (C) The Twitter-based spectrum of the legislators, and the spectrum of Twitter users in Korea; (D) The roll call-based spectrum of the legislators in South Korea. The legislators are indicated by the shapes (one shape corresponds to one legislator) and the spectra of the Twitter users are indicated by solid lines. The party-line splits of the legislators are evident in both the Twitter- and the roll call-based spectra for both countries. The general Twitter public (solid lines in Fig 2A and C) show a majority-minority reversal, more people aligning themselves with the minority parties.

https://doi.org/10.1371/journal.pone.0124722.g002

A close examination of these spectra can reveals the disparities between the online and the offline. First, we find a stark indication of the significance of the disparity between Twitter and the offline in the majority-minority reversal in both countries: The majority of Twitter users align themselves with the minority parties. The mean and the median positions, and the overall distribution of the spectra are shown in Fig 2A for U.S. and Fig 2C for Korea. We find that in the U.S., Republican Senators John McCain (the presidential candidate in 2008) and Jim DeMint (of the populist Tea Party movement) boasting a disproportionately large number of followers are primarily responsible for this behavior. In Korea, the DP (green) overwhelms the GNP (blue) in the Twitter sphere despite being outnumbered by a nearly two-to-one margin in the number of seats in the Assembly.

Second, the comparison between the roll call-based spectra of the legislators as the proxy for the offline public at large and the spectra of the Twitter users also reveals a similar disparity. Although we cannot directly compare the online and offline spectra on the same plot because of the differences in scale, the comparison of relative biases in the distribution is still valid. For the U.S., the spectrum of the Twitter users is heavily skewed towards GOP in sharp contrast to the DEMs who are favored in the spectrum of the offline public (Fig 2A and 2B). The trend is similar in the Korean spectra: the offline public favors GNP in contrast to the Twitter user who favor DP (Fig 2C and 2D).

Third, there also exists a discrepancy between the real political positions of the legislators and their positions constructed from the Twitter data. The party-line split in the positions of the legislators (indicated as solid shapes in the figures) is immediately noticeable for both nations, and accordingly the roll call-based spectra and the Twitter-based spectra of the legislators exhibit some degree of overall agreement (similarity) as measured by Kendall’s τ-coefficient (1 means identity; 0 means independence), with 0.68 ± 0.02 for the U.S. and 0.48 ± 0.01 for Korea (see Methods section for more detail). When we consider each party separately, however, we find a weaker agreement—for the U.S. parties, the τ is 0.463 ± 0.003 for GOP and 0.291 ± 0.022 for DEM. For the Korean parties that is 0.029 ± 0.005 for GNP and 0.062 ± 0.051 for DP.

This indicates that Korean Twitter population is more noticeably impervious to the political positions of the legislators, resulting in the very low level of intra-party correlations between the Twitter based spectrum and the roll call based spectrum of the legislators. In U.S. the correlation is more sizable.

These disparities between the Twitter users and the offline public show the limits of Twitter as a source for fair and accurate opinion mining, especially pertaining to political or social issues. In fact, the majority-minority reversal shows us that Twitter may be functioning as a de facto “alternative media.”

Quantifying partisanship of twitter users

The spread structures of the Twitter users’ spectra in Fig 2A and 2C imply that Twitter users are biased (show partisanship) in choosing which legislators to follow. We can quantify the strength of a user’s bias by measuring the probability that a user follows certain parties, and comparing it with the random probability that a user becomes connected to a party if the connections were made at random. If a user is unbiased they must be identical, and deviations between the two probabilities signify the strength of the bias.

A widely-used measure for the discrepancy between two probability distributions is the Kullback-Leibler (KL) divergence D. It is non-negative, i.e. D ≥ 0, 0 when two distributions being compared are identical, and positive otherwise. To account for random fluctuations we calculate the Z-score of KL divergences (see Methods section for detail) Z_D as a function of the number of politicians (degree) k that a user follows, shown in Fig 3. Z_D ≳ 1 implies partisanship, while Z_D ≲ −1 implies the opposite. The radius of the disc around each plot point is proportional to the the log of the number of users with the given degree k. The figure shows the widespread nature of partisanship on Twitter [9], with the exception of high-degree Twitter users who are more balanced in following each party (see Methods section for more detail).

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Fig 3. Z-score of the Kullback-Leibler (KL) divergence as a function of the number of legislators k that a Twitter user follows in the U.S. (A), and in Korea (B).

The radius of each circle is proportional to the log of the number of Twitter users with given k. Z_D ≳ 1 implies a statistically significant level of partisanship (political bias). We see that partisanship is widespread in both countries, except for high-degree Twitter users for whom D → 0.

https://doi.org/10.1371/journal.pone.0124722.g003

Classical multi-Dimensional scaling

CMDS is a method to embed a set of objects in a pre-defined N-dimensional space such that the distance between objects match given dissimilarities as much as possible. Specifically, CMDS aims to produce the coordinates of the objects that minimize a loss function called STRAIN [15]. In this paper we set N = 1 to produce a linear political spectrum that agrees with the common way of thinking of political polarization (e.g. left wing versus right wing).

CMDS is performed by the eigendecomposition of the matrix made from the dissimilarities. The quality of the resulting coordinates in reproducing the given dissimilarities is defined as the fraction of the first eigenvalue over the sum of all positive eigenvalues. The values are 0.297 for Korea and 0.872 for the U.S. with the roll call data, and 0.113 for Korea and 0.093 for the U.S. with the Twitter data. The noticeably high value for the U.S. roll call data may be due to the stable two-party system of the U.S.

While one could theoretically gain more information from higher-dimension CMDS, we could not observe meaningful patterns allowing straightforward political interpretation beyond the first. Our 1-dimensional solutions show clear separations of the legislators along party lines, thus justifying the political interpretation given.

We used the Kulczynski dissimilarity [17]. In Twitter data it is given between two legislators i and j as (1) where F_x denotes the set of the followers of legislator x. The Kulczynski dissimilarity is recommended as an alternative to the commonly used Jaccard measure when the size of F_x exhibits a wide range as in our Twitter data (Fig 4) [21].

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Fig 4. Degree distributions in the data.

(A) The distribution of the number of Twitter followers for each legislator. (B) The distribution of the number of legislators that Twitter users follow. All distribtuions are heavy-tailed.

https://doi.org/10.1371/journal.pone.0124722.g004

The dissimilarity between legislators i and j in the roll call data is, similarly, (2) where V_x denotes the set of nay (opposition to a bill’s passing) votes of legislator x. We ignore the yea votes here. The motivation for this is that most bills that pass are bipartisan, potentially limiting the discriminating power of the yea votes.

Notes on the spectrum of Twitter users

From the legislator-Twitter data we estimated the spectrum of the legislators first on their Twitter followership, and then used it to obtain the spectrum of Twitter users. A dual approach could have been employed where we perform CMDS on the Twitter users by defining the dissimilarities between the users based on the set of legislators that they follow, and determining the positions of the legislators based on the results. While there is no reason to object this approach in principle, its computational cost was prohibitive, requiring several tens of gigabytes of memory to store the dissimilarity matrix alone.

Kendall’s τ rank correlation coefficient

Kendall’s rank correlation τ quantifies the similarity between two orderings of objects. Letting (x₁, x₂, ⋯, x_n) and (y₁, y₂, ⋯, y_n) be the orderings we wish to compare, Kendall’s τ rank correlation coefficient is defined as (3)

In this definition C is the number of (i, j) pairs such that the (i, j) satisfies x_i > x_j and y_i > y_j or it satisfies x_i < x_j and y_i < y_j. D is the number of (i, j) pairs such that the (i, j) satisfies x_i > x_j and y_i < y_j or it satisfies x_i < x_j and y_i > y_j.

Errors in Kendall’s τ were estimated through the jackknife method [22].

Z-score of the Kullback-Leibler divergence

Kullback-Leibler divergence (KL divergence) measures deviation of a probability distribution from a reference probability distribution. Here the reference probability distribution is the fractions of seats occupied by each party, i.e. the probability that a random user would follow a legislator from a given party.

Let P_i be the fraction of the number of legislators from party i among the number of the legislators, and p_ij be the fraction of the number of legislators from party i followed by Twitter user j among all legislators followed by j. Then the KL divergence D_j of Twitter user j is (4) where N is the number of parties. KL divergence is non-negative. A large D_j means that user j’s followership pattern does not agree with the reference probability, i.e. favors one party disproportionately when in a two-party system.

While the average KL divergence ⟨D⟩_k of Twitter users who follow k legislators can be obtained easily from the empirical data, for statistical significance we use the Z-score (5) where μ_k and σ_k are the expectation and standard deviation of the KL divergence when a user choose to follow a party according to the reference probability P_i. If we assume a Twitter user follows k legislators randomly with uniform distribution, then the probability of $\vec{n}=(n_1,\cdots ,n_N)$, the vector of the number of legislators in each party that the user follows, is given by the multinomial distribution $f(\vec{n};\vec{P})$ where n₁+⋯+n_N = k and $\vec{P}=(P_1,\cdots ,P_N)$. Then we have (6) and (7)

Z_D(k) ≳ 1 indicates that Twitter users follow a biased set of legislators, and that the bias is larger than the typical random fluctuation. Similarly, Z_D(k) ≲ −1 means that Twitter users follow the parties in close agreement with the reference probability P_i. As k approaches the actual number of legislators p_{i, j} converges to P_i, Z_D(k) becomes more negative (since D → 0).