Inferring monopartite projections of bipartite networks: an entropy-based approach

The first step of our algorithm prescribes to measure the degree of similarity of nodes r and ${r}^{\prime }$. A straightforward approach is counting the number of common neighbors ${V}_{{{rr}}^{\prime }}$ shared by nodes r and ${r}^{\prime }$. By adopting the formalism proposed in [16], our measure of similarity is provided by the number of bi-cliques ${K}_{\mathrm{1,2}}$ [17], also known as V-motifs [16]:

$$\begin{eqnarray}&&{V}_{{{rr}}^{\prime }}=\displaystyle \sum _{c=1}^{{N}_{C}}{m}_{{rc}}{m}_{{r}^{\prime }c}=\displaystyle \sum _{c=1}^{{N}_{C}}{V}_{{{rr}}^{\prime }}^{c},\end{eqnarray} \tag{ 2.1 }$$

where we have adopted the definition ${V}_{{{rr}}^{\prime }}^{c}\equiv \,{m}_{{rc}}{m}_{r^{\prime} c}$ for the single V-motif defined by nodes r and ${r}^{\prime }$ and node c belonging to the opposite layer (see figure 1 for a pictorial representation). From the definition, it is apparent that ${V}_{{rr}^{\prime} }^{c}=1$ if, and only if, both r and $r^{\prime}$ share the (common) neighbor c.

Notice that naïvely projecting a bipartite network corresponds to considering the monopartite matrix defined as ${{\bf{V}}}_{{rr}^{\prime} }^{{\rm{naive}}}={V}_{{rr}^{\prime} }$ whose densely connected structure, described by ${{\bf{R}}}_{{rr}^{\prime} }^{{\rm{naive}}}={\rm{\Theta }}[{V}_{{rr}^{\prime} }]$, is characterized by an almost trivial topology.

The second step of our algorithm prescribes to quantify the statistical significance of the similarity of our nodes r and $r^{\prime}$. To this aim, a benchmark is needed: a natural choice leads to adopt the ERG class of null-models [16, 18–22].

Within the ERG framework, the generic bipartite network ${\bf{M}}$ is assigned an exponential probability $P({\bf{M}})=\tfrac{{{\rm{e}}}^{-H(\vec{\theta },\vec{C}({\bf{M}}))}}{Z(\vec{\theta })}$, whose value is determined by the vector $\vec{C}({\bf{M}})$ of topological constraints [18]. In order to determine the unknown parameters $\vec{\theta }$, the likelihood-maximization recipe can be adopted: given an observed biadjacency matrix ${{\bf{M}}}^{* }$, it translates into solving the system of equations $\langle \vec{C}\rangle (\vec{\theta })={\sum }_{{\bf{M}}}P({\bf{M}})\vec{C}({\bf{M}})=\vec{C}({{\bf{M}}}^{* })$ which prescribes to equate the ensemble averages $\langle \vec{C}\rangle (\vec{\theta })$ to their observed counterparts, $\vec{C}({{\bf{M}}}^{* })$ [19].

Two of the null models we have considered in the present paper are known as the bipartite random graph (BiRG) model and the bipartite configuration model (BiCM) [16]; the other ones are the two 'partial' configuration models ${\mathrm{BiPCM}}_{r}$ and ${\mathrm{BiPCM}}_{c}$: the four null models are defined, respectively, by constraining the total number of links, the degrees of nodes belonging to both layers and the degrees of nodes belonging to one layer only (see appendix for the analytical definitions).

The use of linear constraints allows us to write $P({\bf{M}})$ in a factorized form, i.e. as the product of pair-specific probability coefficients

$$\begin{eqnarray}&&P({\bf{M}})=\displaystyle \prod _{r=1}^{{N}_{R}}\displaystyle \prod _{c=1}^{{N}_{C}}{p}_{{rc}}^{{m}_{{rc}}}{(1-{p}_{{rc}})}^{1-{m}_{{rc}}}\end{eqnarray} \tag{ 2.2 }$$

the numerical value of the generic coefficient p_rc being determined by the likelihood-maximization condition (see appendix). As an example, in the case of BiRG, ${p}_{{rc}}={p}_{\mathrm{BiRG}}=\tfrac{L}{{N}_{R}\cdot {N}_{C}},\forall r,c$ with L being the total number of links in the actual bipartite network.

Since ERG models with linear constraints treat links as independent random variables, the presence of each ${V}_{{rr}^{\prime} }^{c}$ can be regarded as the outcome of a Bernoulli trial:

$$\begin{eqnarray}&&{f}_{\mathrm{Ber}}({V}_{{rr}^{\prime} }^{c}=1)={p}_{{rc}}{p}_{r^{\prime} c},\end{eqnarray} \tag{ 2.3 }$$

$$\begin{eqnarray}&&{f}_{\mathrm{Ber}}({V}_{{rr}^{\prime} }^{c}=0)=1-{p}_{{rc}}{p}_{r^{\prime} c}.\end{eqnarray} \tag{ 2.4 }$$

It follows that, once r and $r^{\prime}$ are chosen, the events describing the presence of the N_C single ${V}_{{rr}^{\prime} }^{c}$ motifs are independent random experiments: this, in turn, implies that each ${V}_{{rr}^{\prime} }$ is nothing else than a sum of independent Bernoulli trials, each one described by a different probability coefficient.

The distribution describing the behavior of each ${V}_{{rr}^{\prime} }$ turns out to be the so-called Poisson–Binomial [23, 24]. More explicitly, the probability of observing zero V-motifs between r and $r^{\prime}$ (or, equivalently, the probability for nodes r and $r^{\prime}$ of sharing zero neighbors) reads

$$\begin{eqnarray}&&{f}_{\mathrm{PB}}({V}_{{rr}^{\prime} }=0)=\displaystyle \prod _{c=1}^{{N}_{C}}(1-{p}_{{rc}}{p}_{r^{\prime} c}),\end{eqnarray} \tag{ 2.5 }$$

the probability of observing only one V-motif reads

$$\begin{eqnarray}&&{f}_{\mathrm{PB}}({V}_{{rr}^{\prime} }=1)=\displaystyle \sum _{c=1}^{{N}_{C}}\left[{p}_{{rc}}{p}_{r^{\prime} c}\displaystyle \prod _{\displaystyle \genfrac{}{}{0em}{}{c^{\prime} =1}{c^{\prime} \ne c}}^{{N}_{C}}(1-{p}_{{rc}^{\prime} }{p}_{r^{\prime} c^{\prime} })\right],\end{eqnarray} \tag{ 2.6 }$$

etc. In general, the probability of observing n V-motifs can be expressed as a sum of $\left(\genfrac{}{}{0em}{}{{N}_{C}}{n}\right)$ addenda, running on the n-tuples of considered nodes (in this particular case, the ones belonging to the bottom layer). Upon indicating with C_n all possible nodes n-tuples, this probability reads

$$\begin{eqnarray}&&{f}_{\mathrm{PB}}({V}_{{rr}^{\prime} }=n)=\displaystyle \sum _{{C}_{n}}\left[\displaystyle \prod _{{c}_{i}\in {C}_{n}}{p}_{{{rc}}_{i}}{p}_{r^{\prime} {c}_{i}}\displaystyle \prod _{{c}_{i}^{\prime }\notin {C}_{n}}(1-{p}_{{{rc}}_{i}^{\prime }}{p}_{r^{\prime} {c}_{i}^{\prime }})\right]\end{eqnarray} \tag{ 2.7 }$$

(notice that the second product runs over the complement set of C_n).

Measuring the statistical significance of the similarity of nodes r and $r^{\prime}$ thus translates into calculating a p-value on the aforementioned Poisson–Binomial distribution, i.e. the probability of observing a number of V-motifs greater than, or equal to, the observed one (which will be indicated as ${V}_{{rr}^{\prime} }^{* }$):

$$\begin{eqnarray}&&{p} \mbox{-} \mathrm{value}({V}_{{rr}^{\prime} }^{* })=\displaystyle \sum _{{V}_{{rr}^{\prime} }\geqslant {V}_{{rr}^{\prime} }^{* }}{f}_{\mathrm{PB}}({V}_{{rr}^{\prime} }).\end{eqnarray} \tag{ 2.8 }$$

Upon repeating such a procedure for each pair of nodes, we obtain an ${N}_{R}\times {N}_{R}$ matrix of p-values (see also the appendix). In order to speed up the numerical computation of p-values, a Python code has been made publicly available by the authors⁴ .

As a final remark, notice that this approach describes a one-tail statistical test, where nodes are considered as significantly similar if, and only if, the observed number of shared neighbors is 'sufficiently large'. In principle, our algorithm can be also used to carry out the reverse validation, linking any two nodes if the observed number of shared neighbors is 'sufficiently small': this second type of validation can be performed whenever interested in highlighting the 'dissimilarity' between nodes.

In order to understand which p-values are significant, it is necessary to adopt a statistical procedure accounting for testing multiple hypotheses at a time.

In the present paper we apply the so-called false discovery rate (FDR) procedure [25]. Whenever M different hypotheses, ${H}_{1}...{H}_{M}$, characterized by M different p-values, must be tested at a time, FDR prescribes to, first, sort the M p-values in increasing order, ${p} \mbox{-} {\mathrm{value}}_{1}\leqslant ...\,\leqslant \,{p} \mbox{-} {\mathrm{value}}_{M}$ and, then, to identify the largest integer $\hat{i}$ satisfying the condition

$$\begin{eqnarray}&&{p} \mbox{-} {\mathrm{value}}_{\hat{i}}\leqslant \displaystyle \frac{\hat{i}t}{M}\end{eqnarray} \tag{ 2.9 }$$

with t representing the usual single-test significance level (e.g. t = 0.05 or t = 0.01). The third step of the FDR procedure prescribes to reject all the hypotheses whose p-value is less than, or equal to, ${p} \mbox{-} {\mathrm{value}}_{\hat{i}}$, i.e. ${p} \mbox{-} {\mathrm{value}}_{1}\leqslant ...\,\leqslant \,{p} \mbox{-} {\mathrm{value}}_{\hat{i}}$. Notably, FDR allows one to control for the expected number of false 'discoveries' (i.e. incorrectly-rejected null hypotheses), irrespectively of the independence of the hypotheses tested (our hypotheses, for example, are not independent, since each observed link affects the similarity of several pairs of nodes).

In our case, the FDR prescription translates into adopting the threshold $\hat{i}t/\left(\genfrac{}{}{0em}{}{{N}_{R}}{2}\right)$ which corresponds to the largest ${p} \mbox{-} {\mathrm{value}}_{\hat{i}}$ satisfying the condition

$$\begin{eqnarray}&&{p} \mbox{-} {\mathrm{value}}_{\hat{i}}\leqslant \displaystyle \frac{\hat{i}t}{\left(\genfrac{}{}{0em}{}{{N}_{R}}{2}\right)}\end{eqnarray} \tag{ 2.10 }$$

(with i indexing the sorted $\left(\genfrac{}{}{0em}{}{{N}_{R}}{2}\right)$ p‐value $({V}_{{rr}^{\prime} })$ coefficients) and considering as significantly similar only those pairs of nodes r, $r^{\prime}$ whose ${p} \mbox{-} \mathrm{value}({V}_{{rr}^{\prime} }^{* })\leqslant {p} \mbox{-} {\mathrm{value}}_{\hat{i}}$. In other words, every couple of nodes whose corresponding p-value is validated by the FDR is joined by a binary, undirected link in our projection. In what follows, we have used a single-test significance level of t = 0.01.

Summing up, the recipe for obtaining a statistically-validated projection of the bipartite network ${\bf{M}}$ by running the FDR criterion requires that ${{\bf{R}}}_{{rr}^{\prime} }^{{nm}}=1$ if, and only if, ${p} \mbox{-} \mathrm{value}({V}_{{rr}^{\prime} })\leqslant \,{p} \mbox{-} {\mathrm{value}}_{\hat{i}}$, according to null model nm used. Notice that the validation process naturally circumvents the problem of spurious clustering (see also appendix).

The aforementioned approaches providing an algorithm to project a validated network differ in the way the issue of comparing multiple hypotheses is dealt with. While in some approaches this step is simply missing and each test is carried out independently from the other ones [3, 14], in others the Bonferroni correction is employed [12, 13]. Both solutions are affected by drawbacks.

The former algorithms, in fact, overestimate the number of incorrectly rejected null hypotheses (i.e. of incorrectly validated links). A simple argument can, indeed, be provided: the probability that, by chance, at least one, out of M hypotheses, is incorrectly rejected (i.e. that at least one link is incorrectly validated) is $\mathrm{FWER}=1-{(1-t)}^{M}$ which is $\mathrm{FWER}\simeq 1$ for just M = 100 tests conducted at the significance level of t = 0.05.

The latter algorithms, on the other hand, adopt a criterion deemed as severely overestimating the number of incorrectly retained null hypotheses (i.e. of incorrectly discarded links) [25]. Indeed, if the stricter condition $\mathrm{FWER}=0.05$ is now imposed, the threshold p-value can be derived as ${p} \mbox{-} {\mathrm{value}}_{{\rm{th}}}=t\simeq 0.05/M$ which rapidly vanishes as M grows. As a consequence, very sparse (if not empty) projections are often obtained.

Naturally, deciding which test is more suited for the problem at hand depends on the importance assigned to false positive and false negatives. As a rule of thumb, the Bonferroni correction can be deemed as appropriate when few tests, out of a small number of multiple comparisons, are expected to be significant (i.e. when even a single false positive would be problematic). On the contrary, when many tests, out of a large number of multiple comparisons, are expected to be significant (as in the case of socio-economic networks), using the Bonferroni correction may, in turn, produce a too large number of false negatives, an undesired consequence of which may be the impairment of, e.g. a recommendation system.

As a final remark, we stress that an a priori selection of the number of validated links is not necessarily compatible with the existence of a level t of statistical significance ensuring that the FDR procedure still holds. As an example, let us suppose we retain only the first k p-values; the FDR would then require the following inequalities to be satisfied: ${p} \mbox{-} {\mathrm{value}}_{k}\leqslant {kt}/M$ and ${p} \mbox{-} {\mathrm{value}}_{k+1}\gt (k+1)t/M$. This, in turn, would imply ${p} \mbox{-} {\mathrm{value}}_{k}/k\lt {p} \mbox{-} {\mathrm{value}}_{k+1}/(k+1)$. The aforementioned condition, however, can be easily violated by imaging a pair of subsequent p-values close enough to each other (e.g. ${p} \mbox{-} {\mathrm{value}}_{3}=0.039$ and ${p} \mbox{-} {\mathrm{value}}_{4}=0.040$).

Zoom In Zoom Out Reset image size

2.4.1. Community detection

Let us now test our validation procedure on the first data set considered for the present analysis: the World Trade Web. In the present paper we consider the COMTRADE database (using the HS 2007 code revision), spanning the years 1995–2010⁶ . After a data-cleaning procedure operated by BACI [29] and a thresholding procedure induced by the RCA (for more details, see [30]), we end up with a bipartite network characterized by N_R = 146 countries and N_C = 1131 classes of products, whose generic entry ${m}_{{rc}}=1$ indicates that country r exports product c above the RCA threshold.

Countries layer. Figure 2 shows three different projections of the WTW. The first panel shows a pictorial representation of the WTW topology in the year 2000, upon naïvely projecting it (i.e. by joining any two nodes if at least one neighbor is shared, thus obtaining a matrix ${{\bf{R}}}_{{rr}^{\prime} }^{{\rm{naive}}}={\rm{\Theta }}[{V}_{{rr}^{\prime} }]$). The high density of links (which oscillates between 0.93 and 0.95 throughout the period covered by the data set) causes the network to be characterized by trivial values of structural quantities (e.g. all nodes have a clustering coefficient very close to 1).

Zoom In Zoom Out Reset image size

The second panel of figure 2 represents the projected adjacency matrix using the BiRG as a null model. In this case, the only parameter defining our reference model is ${p}_{\mathrm{BiRG}}=\tfrac{L}{{N}_{R}\,\cdot {N}_{C}}\simeq 0.13$. As a consequence, ${p}_{{rc}}={p}_{\mathrm{BiRG}}$ for every pair of nodes and formula (2.7) simplifies to the binomial

$$\begin{eqnarray}&&{f}_{\mathrm{Bin}}({V}_{{rr}^{\prime} }=n)=\left(\displaystyle \genfrac{}{}{0em}{}{{N}_{C}}{n}\right){({p}_{\mathrm{BiRG}}^{2})}^{n}{(1-{p}_{\mathrm{BiRG}}^{2})}^{{N}_{C}-n}.\end{eqnarray} \tag{ 3.1 }$$

The projection provided by the BiRG individuates a unique connected component of countries (notice that the two blocks at bottom-right and top-left of the panel are linked through off-diagonal connections) beside many disconnected vertices (the big white block in the center of the matrix). Interestingly, the latter represent countries whose economy heavily rests upon the presence of raw-materials (see also figure 3), in turn causing each export basket to be focused around the available country-specific natural resources. As a consequence, the similarity between these countries is not significant enough to allow the corresponding links to pass the validation procedure. In other words, the BiRG-induced projection is able to distinguish between two extreme levels of economic development, thus providing a meaningful, yet too rough, filter.

Zoom In Zoom Out Reset image size

On the other hand, the BiCM-induced projection (shown in the third panel of figure 2), allows for a definite structure of clusters to emerge. The economic meaning of the detected diagonal blocks can be made explicit by running the Louvain algorithm on the projected network. As figure 3 shows, our algorithm reveals a partition into communities enclosing countries characterized by similar economic development [31]. In particular, we recognize the 'advanced' economies (EU countries, USA and Japan—whose export basket is practically constituted by all products [8, 30, 32–36]), the 'developing' economies (as centro-american countries and south-eastern countries as China, India, Asian Tigers, etc, for which the textile manufacturing represents the most important sector) and countries whose export heavily rests upon raw-materials like oil (Russia, Saudi Arabia, Libya, Algeria, etc), tropical agricultural food (south-american and centro-african countries), etc. An additional group of countries whose export is based upon sea-food is constituted by Australia, New Zealand, Chile and Argentina, which happen to be detected as a community on its own in partitions with comparable values of modularity.

Our algorithm is also able to highlight the structural changes that have affected the WTW topology across the temporal period considered for the present analysis. Figure 4 shows two snapshots of the WTW, referring to the years 2000 and 2008. While in 2000 EU countries were split into two different modules, with the north-european countries (as Germany, UK, France) grouped together with USA and Japan and the south-eastern european countries constituting a separate cluster, this is no longer true in 2008. Furthermore, the structural role played by single nodes is also pointed out. As an example, Austria and Japan emerge as two of the countries with highest betweenness, indicating their role of bridges respectively between western and eastern european countries and western and eastern world countries. A second example is provided by Germany, whose star-like pattern of connections clearly indicates its prominent role in the global trade.

Zoom In Zoom Out Reset image size

The block diagonal structure of the BiCM-induced adjacency matrix reflects another interesting pattern of the world economy self-organization: the detected communities appear to be linked in a hierarchical fashion, with the 'developing' economies seemingly constituting an intermediate layer between the 'advanced' economies and those countries whose export heavily rests upon raw-materials. Interestingly, such a mesoscopic organization persists across all years of our data set, shedding new light on the WTW evolution.

As shown in figure 5, the results obtained by running the ${\mathrm{BiPCM}}_{r}$ (defined by constraining only the degrees of countries) are, although less detailed, compatible with the ones obtained by running the BiCM. In this case, the ${\mathrm{BiPCM}}_{r}$ constitutes an approximation to the BiCM, providing a computationally faster, yet equally accurate, alternative to it. On the other hand, the ${\mathrm{BiPCM}}_{c}$ induces a projection which is close to the BiRG one, thus adding little information with respect to the latter.

Zoom In Zoom Out Reset image size

Products layer. While the BiCM provides an informative benchmark to infer the presence of significant connections between countries, this is not the case when focusing on products. For this reason, we consider the ${\mathrm{BiPCM}}_{c}$, i.e. the null model defined by constraining only the products degrees: figure 6 shows the ${\mathrm{BiPCM}}_{c}$-induced projection of the WTW on the layer of products⁷ . Several communities appear, the larger ones being machinery, transportation, chemicals, electronics, textiles and live animals (a partition that seems to be stable across time).

Zoom In Zoom Out Reset image size

The detected communities seem to be organized into two macro-groups: 'high-complexity' products (on the left of the figure), including machinery, chemicals, advanced electronics, etc and 'low-complexity' products (on the right of the figure), including live animals, wooden products, textiles, basic electronics, etc. This macroscopic separation reflects the level of economic development of the countries trading these products. As figure 7 clarifies, the 'advanced' economies focus their trading activity on products characterized by high complexity, while 'developing' economies are preferentially active on low-complexity products [30, 35, 36]. A simple topological index captures this tendency: ${I}_{{ \mathcal R }{ \mathcal C }}=\tfrac{{\sum }_{r\in { \mathcal R }}{\sum }_{c\in { \mathcal C }}{m}_{{rc}}}{| { \mathcal R }| | { \mathcal C }| }$, i.e. the link density between groups of nodes ${ \mathcal R }$ and ${ \mathcal C }$, indicating one of the aforementioned communities of countries and one of the aforementioned communities of products, respectively. For example, as evident upon inspecting figure 7, 'advanced' economies (left panel) and 'developing' economies (right panel) are active on different clusters of products: while the trading activity of the former is mainly constituted by, e.g. chemicals and machinery, the latter mainly trade textiles, wooden products, etc. A more in-depth analysis of the grand canonical projection of the World Trade Web can be found in [37].

Zoom In Zoom Out Reset image size

Let us now consider the second data set: MovieLens 100 k. MovieLens is a project by GroupLens [38], a research lab at the University of Minnesota. Data (collected from 19th September 1997 through 22nd April 1998) consist of 10⁵ ratings—from 1 to 5—given by N_C = 943 users to N_R = 1559 different movies⁸ ; information about the movies (date of release and genre) and about the users (age, gender, occupation and US zip code) is also provided. We binarize the dataset by setting ${m}_{{rc}}=1$ if user c rated movie r at least 3, providing a favorable recension.

In what follows we will be interested into projecting this network on the layer of movies. Figure 8 shows the three projections already discussed for the WTW. As for the latter, ${{\bf{R}}}_{{rr}^{\prime} }^{{\rm{naive}}}={\rm{\Theta }}[{V}_{{rr}^{\prime} }]$ is still a very dense network, whose connectance amounts to 0.58. Similarly, the projection induced by the BiRG provides a rather rough filter, producing a unique large connected component, to which only the most popular movies (i.e. the ones with a large degree in the original bipartite network) belong.

Zoom In Zoom Out Reset image size

While both the naïve and the BiRG-induced projections only allow for a trivially-partitioned structure to be observed, this is not the case for the BiCM. By running the Louvain algorithm, we found a very composite community structure (characterized by a modularity of $Q\simeq 0.58$), pictorially represented by the diagonal blocks visible in the third panel of figure 8. The BiCM further refines the results found by the BiRG, allowing for the internal structure of the blocks to emerge: in our dicussion, we will focus on the bottom-right block, which shows the richest internal organization.

Figure 9 shows the detected communities within the aforementioned block, beside the genres (provided together with the data)⁹ : action, Adventure, Animation, Children's, Comedy, Crime, Documentary, Drama, Fantasy, Horror, Musical, Mystery, Noir, Romance, Sci-Fi, Thriller, War, western¹⁰ . Since some genres are quite generic and, thus, appropriate for several movies (e.g. Adventure, Comedy and Drama), our clusters are often better described by 'combinations' of genres, capturing the users' tastes to a larger extent: the detected communities, in fact, partition the set of movies quite sharply, once appropriate combinations of genres are considered.

Zoom In Zoom Out Reset image size

As an example, the orange block on the left side of our matrix is composed by movies released in 1996 (i.e. the year before the survey). Remarkably, our projection algorithm is able to capture the peculiar 'similarity' of these movies, not trivially related to the genres to which they are ascribed to (that are quite heterogeneous: Action, Comedy, Fantasy, Thriller, Sci-Fi) but to the curiosity of users towards the yearly new releases.

Zoom In Zoom Out Reset image size

Proceeding clockwise, the violet block next to the orange one is composed by movies classified as Animation, Children's, Fantasy and Musical (e.g. 'Mrs. Doubtfire', 'The Addams Family', 'Free Willy', 'Cinderella', 'Snow White'). In other words, we are detecting the so-called 'family movies', a more comprehensive definition accounting for all elements described by the single genres above.

The next purple block is composed by genres Action, Adventure, Horror, Sci-Fi and Thriller: examples are provided by 'Stargate', 'Judge Dredd', 'Dracula', 'The Evil Dead'. This community encloses movies with marked horror traits, including titles far from 'mainstream' movies. This is the main difference with respect to the following blue block: although characterized by similar genres (but with Crime replacing Horror and Thriller) movies belonging to it are more popular: 'cult mass' movies, in fact, can be found here. Examples are provided by 'Braveheart', 'Blade Runner' and sagas as 'Star Wars' and 'Indiana Jones'.

The following two blocks represent niche movies for US users. The module in magenta is, in fact, composed by foreign movies (mostly European—French, German, Italian, English—which usually combine elements from Comedy and elements from Drama), as well as US independent films (as titles by Jim Jarmush); the yellow module, on the other hand, is composed by movies inspired by books or theatrical plays and documentaries.

The last, cyan block is composed by movies which are considered as 'classic' Hollywood movies (because of the presence of either iconic actors or master directors): examples are provided by 'Casablanca', 'Ben Hur', 'Taxi Driver', 'Vertigo' (and all movies directed by Hitchcock), 'Manhattan', 'Annie Hall'.

As in the WTW case, running the ${\mathrm{BiPCM}}_{r}$ (defined by constraining only the degrees of movies) leads us to obtain a coarse-grained (i.e. still informative, although less detailed) version of the aforementioned results. Only three macro-groups of movies are, in fact, detected: 'authorial' movies (as 'classic' Hollywood movies, Hitchcock's, Kubrick's, Spielberg's movies), recent mainstream 'blockbusters' (as 'Star Trek', 'Star Wars', 'Indiana Jones', 'Batman' sagas) and independent/niche movies (as Spike Lee's and European movies).

As a final remark, we point out that projecting on the users layer with the BiCM indeed allows several communities to be detected. However, interestingly enough, none of them seems to be accurately described by the provided indicators (age, gender, occupation and US zip code), thus suggesting that users tastes are correlated with hidden (sociometric) variables yet to be individuated.

The BiRG model is the random graph model solved for bipartite networks. This model is defined by a probability coefficient for any two nodes, belonging to different layers, to connect which is equal for all pairs of nodes. More specifically, ${p}_{\mathrm{BiRG}}=\tfrac{L}{{N}_{R}\,\cdot {N}_{C}}$, where $L={\sum }_{r=1}^{{N}_{R}}{\sum }_{c=1}^{{N}_{C}}{m}_{{rc}}$ is the observed number of links and N_R and N_C indicate, respectively, the number of rows and columns of our network. Since all probability coefficients are equal, the probability of a single V-motif (defined by the pair of nodes r and $r^{\prime}$ belonging to the same layer and node c belonging to the second one) reads

$$\begin{eqnarray}&&p({V}_{{rr}^{\prime} }^{c})={p}_{\mathrm{BiRG}}^{2}.\end{eqnarray} \tag{ C.1 }$$

Thus, the probability distribution of the number of V-motifs shared by nodes r and $r^{\prime}$ is simply a Binomial distribution defined by a probability coefficient equal to ${p}_{\mathrm{BiRG}}^{2}$:

$$\begin{eqnarray}&&{f}_{\mathrm{Bin}}({V}_{{rr}^{\prime} }=n)=\left(\displaystyle \genfrac{}{}{0em}{}{{N}_{C}}{n}\right){({p}_{\mathrm{BiRG}}^{2})}^{n}{(1-{p}_{\mathrm{BiRG}}^{2})}^{{N}_{C}-n}.\end{eqnarray} \tag{ C.2 }$$

The BiCM [16], represents the bipartite version of the configuration model [18–20]. The BiCM is defined by two degree sequences. Thus, our Hamiltonian is

$$\begin{eqnarray}&&H(\vec{\theta },\vec{C}({\bf{M}}))=\displaystyle \sum _{r=1}^{{N}_{R}}{\alpha }_{r}{k}_{r}+\displaystyle \sum _{c=1}^{{N}_{C}}{\beta }_{c}{h}_{c},\end{eqnarray} \tag{ C.3 }$$

where ${k}_{r}={\sum }_{c=1}^{{N}_{C}}{m}_{{rc}}$ and ${h}_{c}={\sum }_{r=1}^{{N}_{R}}{m}_{{rc}}$ are the degrees of nodes on the top and bottom layer, respectively; ${\alpha }_{r}$ and ${\beta }_{c}$, instead, are the Lagrangian multipliers associated with the constraints.

The probability of the generic matrix ${\bf{M}}$ thus reads

$$\begin{eqnarray}&&P({\bf{M}})=\displaystyle \frac{{{\rm{e}}}^{-H(\vec{\theta },\vec{C}({\bf{M}}))}}{Z(\vec{\theta })},\end{eqnarray} \tag{ C.4 }$$

where $Z(\vec{\theta })$ is the grand canonical partition function. It is possible to show that

$$\begin{eqnarray}&&P({\bf{M}})=\displaystyle \prod _{r=1}^{{N}_{R}}\displaystyle \prod _{c=1}^{{N}_{C}}{p}_{{rc}}^{{m}_{{rc}}}{(1-{p}_{{rc}})}^{(1-{m}_{{rc}})},\end{eqnarray} \tag{ C.5 }$$

where

$$\begin{eqnarray}&&{p}_{{rc}}={p}_{{rc}}({\alpha }_{r},{\beta }_{c})=\displaystyle \frac{{{\rm{e}}}^{-({\alpha }_{r}+{\beta }_{c})}}{1+{{\rm{e}}}^{-({\alpha }_{r}+{\beta }_{c})}}\equiv \displaystyle \frac{{x}_{r}{y}_{c}}{1+{x}_{r}{y}_{c}}\end{eqnarray} \tag{ C.6 }$$

is the probability for a link between nodes r and c to exist.

In order to estimate the values for x_r and y_c, let us maximize the probability of observing the given matrix ${{\bf{M}}}^{* }$, i.e. the likelihood function ${ \mathcal L }=\mathrm{ln}P({{\bf{M}}}^{* })$ [19]. It is thus possible to derive the Lagrangian multipliers ${\{{x}_{r}\}}_{r=1}^{{N}_{R}}$ and ${\{{y}_{c}\}}_{c=1}^{{N}_{C}}$ by solving

$$\begin{eqnarray}&&{k}_{r}^{* }=\langle {k}_{r}\rangle =\displaystyle \sum _{c=1}^{{N}_{C}}\,\displaystyle \frac{{x}_{r}{y}_{c}}{1+{x}_{r}{y}_{c}},{h}_{c}^{* }=\langle {h}_{c}\rangle =\displaystyle \sum _{r=1}^{{N}_{R}}\,\displaystyle \frac{{x}_{r}{y}_{c}}{1+{x}_{r}{y}_{c}},\end{eqnarray} \tag{ C.7 }$$

where $\{{k}_{r}^{* }\}\,{}_{r=1}^{{N}_{R}}$, $\{{h}_{c}^{* }\}\,{}_{c=1}^{{N}_{C}}$ are the observed degree sequences.

Dealing with bipartite networks allows us to explore two 'partial' versions of the BiCM (hereafter BiPCM), defined by constraining the degree sequences of, say, the top and bottom layer separately. Let us start with the null model ${\mathrm{BiPCM}}_{r}$, defined by the following Hamiltonian:

$$\begin{eqnarray}&&H(\vec{\theta },\vec{C}({\bf{M}}))=\displaystyle \sum _{r=1}^{{N}_{R}}{\alpha }_{r}{k}_{r},\end{eqnarray} \tag{ C.8 }$$

where ${k}_{r}={\sum }_{c=1}^{{N}_{C}}{m}_{{rc}},\forall r$ are the degrees of nodes on the top layer. Although the probability of the generic matrix ${\bf{M}}$ still reads

$$\begin{eqnarray}&&P({\bf{M}})=\displaystyle \prod _{r=1}^{{N}_{R}}\displaystyle \prod _{c=1}^{{N}_{C}}{p}_{{rc}}^{{m}_{{rc}}}{(1-{p}_{{rc}})}^{(1-{m}_{{rc}})},\end{eqnarray} \tag{ C.9 }$$

upon 'switching off' the multipliers ${\{{\beta }_{c}\}}_{c=1}^{{N}_{C}}$ the coefficient p_rc now assumes the form

$$\begin{eqnarray}&&{p}_{{rc}}={p}_{{rc}}({\alpha }_{r})=\displaystyle \frac{{{\rm{e}}}^{-{\alpha }_{r}}}{1+{{\rm{e}}}^{-{\alpha }_{r}}}\equiv \displaystyle \frac{{x}_{r}}{1+{x}_{r}},\forall c.\end{eqnarray} \tag{ C.10 }$$

Notice that the BiCM probability coefficients in (C.6) exactly reduce to the ones in (C.10) whenever the degrees of all nodes belonging to the bottom layer coincide (i.e. ${h}_{c}\equiv h,\forall c$). However, ${\mathrm{BiPCM}}_{r}$ provides an accurate approximation to the BiCM even when the values ${\{{h}_{c}\}}_{c=1}^{{N}_{C}}$ are characterized by a reduced degree of heterogeneity (e.g. as signaled by a coefficient of variation ${c}_{v}=s/m\lt 0.5$, with m and s being, respectively, the mean and the standard deviation of the bottom layer degrees).

In order to estimate the values for x_r, let us maximize the likelihood function ${ \mathcal L }=\mathrm{ln}P({{\bf{M}}}^{* })$ again [19]. It is thus possible to derive the Lagrangian multipliers ${\{{x}_{r}\}}_{r=1}^{{N}_{R}}$:

$$\begin{eqnarray}&&{k}_{r}^{* }=\langle {k}_{r}\rangle =\displaystyle \sum _{c=1}^{{N}_{C}}\displaystyle \frac{{x}_{r}}{1+{x}_{r}}\Longrightarrow {p}_{{rc}}=\displaystyle \frac{{k}_{r}^{* }}{{N}_{C}}.\end{eqnarray} \tag{ C.11 }$$

Notice that, in this case,

$$\begin{eqnarray}&&p({V}_{{rr}^{\prime} }^{c})=\displaystyle \frac{{k}_{r}^{* }{k}_{r^{\prime} }^{* }}{{N}_{C}^{2}}\end{eqnarray} \tag{ C.12 }$$

i.e. each V-motif defined by r and $r^{\prime}$ has the same probability, independently from c. This, in turn, implies that the probability distribution of the number of V-motifs shared by nodes r and $r^{\prime}$ is again a Binomial distribution defined as

$$\begin{eqnarray}&&{f}_{\mathrm{Bin}}({V}_{{rr}^{\prime} }=n)=\left(\displaystyle \genfrac{}{}{0em}{}{{N}_{C}}{n}\right){\left(\displaystyle \frac{{k}_{r}^{* }{k}_{r^{\prime} }^{* }}{{N}_{C}^{2}}\right)}^{n}{\left(1-\displaystyle \frac{{k}_{r}^{* }{k}_{r^{\prime} }^{* }}{{N}_{C}^{2}}\right)}^{{N}_{c}-n}.\end{eqnarray} \tag{ C.13 }$$

Let us now move to considering the second partial null model, ${\mathrm{BiPCM}}_{c}$, defined by the Hamiltonian

$$\begin{eqnarray}&&H(\vec{\theta },\vec{C}({\bf{M}}))=\displaystyle \sum _{c=1}^{{N}_{C}}{\beta }_{c}{h}_{c},\end{eqnarray} \tag{ C.14 }$$

where ${h}_{c}={\sum }_{r=1}^{{N}_{R}}{m}_{{rc}},\forall c$ are the degrees of nodes on the bottom layer. The probability of the generic matrix ${\bf{M}}$ still factorizes, with the coefficient p_rc assuming the form

$$\begin{eqnarray}&&{p}_{{rc}}={p}_{{rc}}({\beta }_{c})=\displaystyle \frac{{{\rm{e}}}^{-{\beta }_{c}}}{1+{{\rm{e}}}^{-{\beta }_{c}}}\equiv \displaystyle \frac{{y}_{c}}{1+{y}_{c}},\forall r;\end{eqnarray} \tag{ C.15 }$$

as for the previously-considered $\mathrm{BiPCM}$, the BiCM probability coefficients in (C.6) exactly reduce to the ones in (C.15) whenever the degrees of all nodes belonging to the top layer coincide (i.e. ${k}_{r}\equiv k,\forall r$). Again, when the values ${\{{k}_{r}\}}_{r=1}^{{N}_{R}}$ are characterized by a reduced degree of heterogeneity, ${\mathrm{BiPCM}}_{c}$ provides an accurate approximation to the BiCM.

The Lagrangian multipliers ${\{{y}_{c}\}}_{c=1}^{{N}_{C}}$ are again straightforwardly estimated as

$$\begin{eqnarray}&&{h}_{c}^{* }=\langle {h}_{c}\rangle =\displaystyle \sum _{r=1}^{{N}_{R}}\displaystyle \frac{{y}_{c}}{1+{y}_{c}}\Longrightarrow {p}_{{rc}}=\displaystyle \frac{{h}_{c}^{* }}{{N}_{R}}.\end{eqnarray} \tag{ C.16 }$$

In this case, each V-motif defined by r, $r^{\prime}$ and c has a probability which depends exclusively on c. As a consequence, the probability distribution of the number of V-motifs shared by any two nodes r and $r^{\prime}$ is the same one, i.e. a Poisson–Binomial whose single Bernoulli trial is defined by a probability reading

$$\begin{eqnarray}&&p({V}_{{rr}^{\prime} }^{c})={\left(\displaystyle \frac{{h}_{c}^{* }}{{N}_{R}}\right)}^{2}.\end{eqnarray} \tag{ C.17 }$$