Cross-correlations of American baby names

We show in Fig. S1 the cumulative distribution of the occurrence of the different baby names, averaged over years. Although it is clear from Fig. S1 that the distribution of names has fat tails, it is also clear that it is more complex than a power law (Fig. S1). As discussed in ref. 28 this distribution can be fitted with a combination of a beta function and a power law. The vast majority of names appear and spread fast through the states. They stay popular for a few years and then disappear (or, more precisely, their frequency goes down to very low, endemic levels), without keeping a high popularity for a long time. Rise and fall in popularity is a general mechanism in social science, and various preferential attachment models with noise and/or limited attention have been proposed to capture these rise/fall curves (21–23). The popularity rise and fall of the most representative names can be observed in Fig. S2. Even if this process is not completely symmetric (17), the rate of ascent is very similar to the rate of descent. The main difference is the leftover tail at very large times (a name that has appeared does not completely disappear).

In all years of the time period under investigation the first eigenvalues of the matrix of correlations $C_{ij}$ are well separated from all of the other eigenvalues, as can be seen in Fig. S3. However, Fig. S4 shows that in the last part of the 20th century there is no clear separation between the first and the second eigenvalue. This is not an effect due to the random noise: This issue is rather subtle and we comment further in the following section. In this section we discuss how to deal with the noise and how to identify system-specific, nonrandom, correlations. One of the possible methods is to compare the spectrum of the correlation matrices with the one of random Wishart–Laguerre matrices (34). This method provides bounds for the random bulk of the spectrum and thus the eigenvalues (and the corresponding eigenvectors) outside this interval are thought to yield information on the genuine correlations of the underlying system. The largest eigenvalues have been shown to play an important role in many situations (35, 40) and we will see below that this is the case also in our problem. We also used a slightly different method to establish that noise is not affecting our findings. We compared the spectrum of correlation matrices with the one obtained from random correlation matrices: In each year, we made a random permutation of the occurrences of the $N_{\text{f}}$ names inside each state, and we used it to compute the correlation matrix $C_R$. Because $N_{\text{f}}\sim {10}^4$, the correlation matrices $C_R$ are similar to identity matrices, whose spectrum is obviously made by ones. The spectrum of $C_R$ is different from the one of C, even if the widely used unfolded nearest-neighbor spacing distribution (for example, ref. 34) is similar. To rule out a role of the random noise, we compared the inverse participation ratio (IPR) $I\left(\lambda \right)$ of our data to that of the null model. This test provides a direct evaluation of the nonrandom part of the spectrum, and it is defined as follows. Let $v_i^{\lambda }$ be the ith component of the (normalized) eigenvector $v^{\lambda }$ such that $C{v}^{\lambda }=\lambda v^{\lambda }$, andwhere M is the total number of components. This quantity is often used in localization problems, because its reciprocal $1/I\left(\lambda \right)$ equals the number of states over which a vector is localized. An easy way to understand this point is to use the normalization condition ${\displaystyle {\sum }_i{v}_i^2}=1$, from which we see that $v_i\sim 1/\sqrt{c}$ for a vector localized over c states, implying that $I\left(\lambda \right)\sim 1/c$. A localized vector is a vector for which $c\sim 1$, whereas for nonlocalized vectors $c\sim O\left(M\right)$. We compared the IPR of all of the eigenvectors of $C_{ij}$ and $C_{ij}^R$ and we found that although there is not a complete agreement for the respective $I\left(\lambda \right)$ in the right region of the spectrum (small eigenvalues), there is a clear separation in the left part of the spectrum, which can be interpreted as a deviation from the random matrices behavior, as can be seen in Fig. S5. The first eigenvalues carry relevant information for the detection of collective modes in the system: They are found to be nonlocalized and to give important information on the correlation between states. The fact that the distribution of names occurrences is fat tailed is crucial for the qualitative agreement we find in the right region of the spectrum, as it is known that, in cross-correlation matrices of signals sampled from a fat-tailed distribution, eigenvectors corresponding to small eigenvalues are localized (41), whereas this does not happen in the Gaussian case, where the IPR is flat $I\left(\lambda \right)\sim 3/M$ for all λ (34, 37, 40).

Let us introduce here principal component analysis (PCA) and hierarchical clustering (HC), which are the two methods we used to investigate the (genuine) correlations of the states through the years. A simple way to extract information from the correlation matrix C is PCA (42), which consists of a partial eigendecomposition of the correlation matrix. It uses only the largest eigenvalues and their corresponding eigenvectors of the matrix to reconstruct collective modes of the system. It can be easily described by means of an example. Suppose that the correlation matrix is such that the nondiagonal elements of the line i are all very small except for a single element $C_{ij}$, which is almost one, and that all of the other nondiagonal terms are also very small (but $C_{ji}=C_{ij}$). It is clear that this situation corresponds to some kind of connection between i and j. In this case the eigenvector of C that corresponds to the largest eigenvalue contains this information in its components; it can be shown that the only components of v significantly different from zero are $v_i$ and $v_j$. Now suppose that the correlation matrix is slightly more complicated and that many nondiagonal elements are large. Although in some situations this effect is purely due to the noise, it can also be related to the fact that complicated patterns (more complex than pairwise correlations) are hidden in the system. It is reasonable to assume that eigenvectors of C take into account this information and that eigenvectors corresponding to largest eigenvalues are localized on such patterns. Thus, PCA may lead to a reconstruction of the complex correlations among many states. To get rid of the noise contained in the correlation matrices and to be sure that eigenvectors corresponding to large eigenvalues contain genuinely nonrandom information, we proceed as explained in the previous section.

Taking the first two eigenvectors of C, $v^{\left(1\right)}$ and $v^{\left(2\right)}$, a bidimensional representation of these eigenvectors can be obtained by plotting the $M=51$ points of coordinates $\left(v_i^{\left(1\right)},v_i^{\left(2\right)}\right)$, one for each state. This can be done year after year and the result is shown in Fig. S6. In these scattered plots, clusters of states that are similar, in a sense that is discussed below, naturally emerge.

The notion of similarity can be understood with the following qualitative description. Let us define $\widehat{C}=\left(1+C\right)/2$, which is a positive definite matrix, whose elements satisfy $0<{\widehat{C}}_{ij}<1$. Let us assume that the state i has large correlations with a group of states $\mathcal{N}\left(i\right)$, whereas it has small correlations with the rest of the states; i.e., ${\widehat{C}}_{ij}\simeq 0$ when $j\notin \mathcal{N}\left(i\right)$. Let us also assume that another state k has a large correlation with a group of states $\mathcal{N}\left(k\right)$ whose overlap with $\mathcal{N}\left(i\right)$ is big enough. In this case, i and k are said to be similar, even if the correlation coefficient ${\widehat{C}}_{ik}$ between them is small, and those two states belong to the same cluster.

These results have been compared with those obtained using an agglomerative hierarchical clusterization based on the matrix $\widehat{C}$. HC is a method to build a hierarchy of clusters starting from M clusters of one state each and such that clusters are merged as one moves up in the hierarchy. This can be done in many ways. First, one has to define a metric $d\left(i,j\right)$, for example the Euclidean one, between the columns of the correlation matrix. Then one has to minimize a distance $D\left(X,Y\right)$ between the clusters X and Y: At each step, the two clusters separated by the shortest distance are merged. There are several possible choices for $D\left(X,Y\right)$, each one defining a possible criterion to perform the clusterization. We used the complete criterion, which is defined bywhich tends to find compact clusters of approximately equal diameters. This procedure leads to the construction of a tree diagram called a dendrogram, which can be cut at the height one prefers. This introduces a sort of arbitrariness in the number of clusters that are considered. Looking at the distribution of the scatter plot in Fig. S6, we decided to fix the number of clusters to three.

It can be noted that states that are similar in the sense specified before are found to belong the same cluster. In each year, to measure the quality of the clusterization obtained, we computed the cophenetic coefficient. Given the Euclidean distance $E\left(i,j\right)$ between the points i and j and the corresponding distance $D\left(i,j\right)$ that these points have on the dendrogram, the cophenetic coefficient c is defined aswhere $\overline{E}$ and $\overline{D}$ are, respectively, the average value of the Euclidean and of the dendrogramatic distances. Values of c close to one indicate a good clusterization, because the correlation coefficient between the actual distances and the dendrogramatic one is large. In all years c is found to be close to 0.85 and never smaller than 0.7. The results from PCA and HC can be found in Figs. 3 and 4 of the main text.

As we already noted Fig. S4 shows that the first eigenvalues are always well separated during the first decades of the 20th century, showing that persisting patterns in the states’ interrelations survive during years. We also noted that the situation is less clear at the end of the 20th century, because in that period there is not a clear separation between the first two eigenvalues. Here we discuss this point further. In each year, we considered the first two eigenvectors of the correlation matrix $C_{ij}$ and we assigned a color to each state, corresponding to the position it had in the plane spanned by $v^{\left(1\right)}$ and $v^{\left(2\right)}$, i.e., the eigenvectors corresponding to the two largest eigenvalues. Although these persistent patterns are clear in the first decades, indicating a separation between northern states and southern states, the situation changes at the end of the 20th century, when states start to form two main groups, one formed by the coastal states and a second one by the central states (a better visualization of such a transition can be obtained from Movies S1–S4, which show data for all of the available years). To better understand what has happened in the transition years, we studied how eigenvectors changed during years. Fig. S4 suggests to choose, as a reference, the eigenvectors of the correlation matrix in 1950, because it is the year with the largest difference ${\lambda }_1-{\lambda }_2$. We then studied how the first three eigenvectors of the correlation matrix in the year y are related to those of 1950. For each year we evaluated the projections of the kth eigenvector on the jth 1950s eigenvector,The first eigenvector turns out to be rather stable until the 1990s, when it becomes mostly a superposition of 1950s second and third eigenvectors. Conversely, after this transition, the second eigenvector is basically substituted by the first 1950s eigenvector, as can be seen in Figs. S7 and S8. These figures provide a clear interpretation of the evolution of the correlations between states: For many decades of the 20th century there has been a stable configuration of groups (clusters) of states that eventually broke down at the end of the century. We do not further investigate the reasons behind this change. Looking at the distribution of baby names allows a sensible and realistic clusterization of states to emerge, and this clusterization is related to cultural influences. The deep motivations that lie behind this change deserve to be studied by sociologists, maybe helped by further quantitative testing of large datasets.