From Ecology to Finance (and Back?): A Review on Entropy-Based Null Models for the Analysis of Bipartite Networks

Abstract

Bipartite networks provide an insightful representation of many systems, ranging from mutualistic networks of species interactions to investment networks in finance. The analyses of their topological structures have revealed the ubiquitous presence of properties which seem to characterize many—apparently different—systems. Nestedness, for example, has been observed in biological plant-pollinator as well as in country-product exportation networks. Due to the interdisciplinary character of complex networks, tools developed in one field, for example ecology, can greatly enrich other areas of research, such as economy and finance, and vice versa. With this in mind, we briefly review several entropy-based bipartite null models that have been recently proposed and discuss their application to real-world systems. The focus on these models is motivated by the fact that they show three very desirable features: analytical character, general applicability, and versatility. In this respect, entropy-based methods have been proven to perform satisfactorily both in providing benchmarks for testing evidence-based null hypotheses and in reconstructing unknown network configurations from partial information. Furthermore, entropy-based models have been successfully employed to analyze ecological as well as economic systems. As an example, the application of entropy-based null models has detected early-warning signals, both in economic and financial systems, of the 2007–2008 world crisis. Moreover, they have revealed a statistically-significant export specialization phenomenon of country export baskets in international trade, a result that seems to reconcile Ricardo’s hypothesis in classical economics with recent findings on the (empirical) diversification industrial production at the national level. Finally, these null models have shown that the information contained in the nestedness is already accounted for by the degree sequence of the corresponding graphs.

Appendices

Appendix: Revealed Comparative Advantage

The revealed comparative advantage (RCA, also knows as Balassa index [11]), rescales the product export volumes in order to determine whether countries are relevant exporters of products. Be e(c, p) the export value of product p in country c’s export basket. The RCA is calculated by comparing the monetary importance of p in c’s export basket to the global average,

$$\begin{aligned} \text {RCA}_{c, p} = \frac{e(c, p)}{\sum _{p'} e(c, p')} \big / \frac{\sum _{c'}e(c', p)}{\sum _{c', p'} e(c', p')} . \end{aligned}$$

(17)

A country is a relevant exporter if RCA $\ge 1$. Using the RCA, the weighted country-product biadjacency matrix can be binarized by keeping only those matrix entries that identify relevant exports and setting them to 1.

Appendix: Bipartite Exponential Random Graph Model

We report some of the null models that have been obtained through entropy maximization and which have been applied to binary and weighted bipartite networks. In the following, all quantities marked with an asterisk refer to the real networks, expressed by their binary ($\mathbf{M}^*$) or weighted ($\mathbf{W}^*$) biadjacency matrix. The layer dimensions are $N_L$ and $N_\varGamma$.

1.1 Bipartite Random Graph

Constraining the expected number of links in the graph ensemble yields an extension of the Erdős-Rényi random graph to bipartite networks, the bipartite random graph (BiRG). The constraint $C \equiv E = \sum _{i,\alpha }m_{i\alpha }$, and thus the Lagrange multiplier $\theta$ as well, is scalar. The partition function can be calculated easily:

$$\begin{aligned} \mathcal {Z}_\text {BiRG}(\theta )= & {} \sum _{G_\text {B} \in \mathcal {G}_\text {B}}e^{-\theta E(G_\text {B})}\nonumber \\= & {} (1+e^{-\theta })^{N_L N_\varGamma }. \end{aligned}$$

(18)

The probability per graph reads

$$\begin{aligned} P(G_G|\theta )= & {} \dfrac{e^{-\theta E}}{(1+e^{-\theta })^{N_L N_\varGamma }}\nonumber \\= & {} \left( p_\text {BiRG}\right) ^E(1-p_\text {BiRG})^{N_L N_\varGamma -E}, \end{aligned}$$

(19)

where $p_\text {BiRG} \equiv \dfrac{e^{-\theta }}{1+e^{-\theta }}$ is the probability of observing a bipartite link between any node couple $i\in L$, $\alpha \in \varGamma$. Notice that $p_\text {BiRG}$ is uniform and independent of the links. Since Eq. (19) is a Binomial distribution, we see that the probability of observing a generic graph $G_\text {B}$ in the ensemble reduces to the problem of observing $E(G_\text {B})$ successful trials with the same probability $p_\text {BiRG}$. We can obtain an analytical expression for the Lagrange multiplier $\theta$ and thus for the link probability by maximizing the likelihood, which reads

$$\begin{aligned} \mathcal {L}=\ln P(G^*|\theta )=-\theta \, E^*- N_LN_\varGamma \ln (1+e^{-\theta }), \end{aligned}$$

(20)

and returns

$$\begin{aligned} p_\text {BiRG}=\frac{E^*}{N_L\,N_\varGamma }. \end{aligned}$$

(21)

1.2 Bipartite Partial Configuration Model

Without loss of generality, we constrain the degree sequence on the layer L such that $\langle k_i\rangle =k_i^*,\,\forall i \in L$. For each node degree $k_i$, we introduce one associated Lagrange multiplier, $\theta _i$. This gives us the bipartite partial configuration model (BiPCM, [82]). Following the same procedure as in Eq. (18), we can obtain

$$\begin{aligned} \mathcal {Z}_\text {BiPCM}({\varvec{\theta }}) =\prod _{i,\alpha }1+e^{-\theta _i}. \end{aligned}$$

(22)

The probability per graph reads

$$\begin{aligned} P(G_B|{\varvec{\theta }})= & {} \prod _{i,\alpha } (p_\text {BiPCM})_i^{m_{i\alpha }} \big (1-(p_\text {BiPCM})_i\big )^{1-m_{i\alpha }}\nonumber \\= & {} \prod _{i}(p_\text {BiPCM})_i^{k_i} \big (1-(p_\text {BiPCM})_i\big )^{N_\varGamma - k_i}, \end{aligned}$$

(23)

where $(p_\text {BiPCM})_i=\frac{e^{-\theta _i}}{1+e^{-\theta _i}}$ is the probability of connecting the node i with any of the node of the opposite layer $\varGamma$. The link probabilities are not uniform, but depend on the Lagrange multipliers of the nodes $i\in L$. The factors in the product in Eq. (23) express the probabilities of observing exactly the constrained node degrees: the probability of the degree $k_i$ of the node $i \in L$ is given by the probability of observing $k_i$ successes trials of a binomial distribution with probability $(p_\text {BiPCM})_i$. Maximizing the likelihood $\mathcal {L}$ returns the explicit expressions for the link probabilities:

$$\begin{aligned} (p_\text {BiPCM})_i=\dfrac{k_i^*}{N_\varGamma }. \end{aligned}$$

(24)

1.3 Bipartite Configuration Model

In the monopartite configuration model, the degrees of all the nodes are constrained. Analogously, in the bipartite configuration model (BiCM, [80]) the degrees of the two layer degree sequences are constrained, such that $\langle k_i\rangle = k_i^*,\,\forall i\in L$, and $\langle k_\alpha \rangle =k_\alpha ^*,\,\forall \alpha \in \varGamma$. If ${\varvec{\theta }}$ and ${\varvec{\rho }}$ are the corresponding Lagrange multipliers, the partition function reads [80]

$$\begin{aligned} \mathcal {Z}_\text {BiCM}({\varvec{\theta }}, {\varvec{\rho }})=\prod _{i,\alpha }1+e^{-(\theta _i + \rho _\alpha )}, \end{aligned}$$

(25)

following essentially the same strategy used in Eq. (18). Again, the probability per graph factorizes in a product of probabilities per link:

$$\begin{aligned} P(G_B|{\varvec{\theta }}, {\varvec{\rho }})= & {} \prod _{i, \alpha } \dfrac{e^{-(\theta _i + \rho _\alpha ) m_{i\alpha }}}{1+e^{-(\theta _i+\rho _\alpha )}}\nonumber \\= & {} \prod _{i, \alpha } (p_\text {BiCM})_{i\alpha }^{m_{i\alpha }} \big (1-(p_\text {BiCM})_{i\alpha }\big )^{1-m_{i\alpha }}, \end{aligned}$$

(26)

where the probability per link reads

$$\begin{aligned} (p_\text {BiCM})_{i\alpha }=\frac{e^{-(\theta _i + \rho _\alpha )}}{1+e^{-(\theta _i+\rho _\alpha )}}, \quad i\in L, \alpha \in \varGamma \end{aligned}$$

(27)

Compared to the probability distributions of the BiRG and BiPCM, we can see that the BiCM distribution is more general and corresponds to the product of different Bernoulli events with link-specific success probabilities. Note that the distribution factorizes and link probabilities are independent. Maximizing the likelihood returns the equation system [80]

$$\begin{aligned} \left\{ \begin{array}{c} \sum _\alpha \dfrac{e^{-(\theta _i+\rho _\alpha )}}{1+e^{-(\theta _i + \rho _\alpha )}}=k_i^*,\quad \forall i\in L,\\ \\ \sum _i\dfrac{e^{-(\theta _i+\rho _\alpha )}}{1+e^{-(\theta _i+\rho _\alpha )}}=k_\alpha ^*,\quad \forall \alpha \in \varGamma . \end{array}\right. \end{aligned}$$

(28)

Solving this system allows us to evaluate the Lagrange multipliers and ultimately obtain the graph probabilities.

1.4 Bipartite Weighted Configuration Model

Constraining the node strengths as $\langle s_i \rangle = s^*_i, \forall i\in L$, and $\langle s_\alpha \rangle = s^*_\alpha , \forall \alpha \in \varGamma$, yields the bipartite weighted configuration model (BiWCM, [31]). Be ${\varvec{\theta }}$ and ${\varvec{\rho }}$ the corresponding Lagrange multipliers. As shown in [31], the partition function is

$$\begin{aligned} \mathcal {Z}_\text {BiCM}({\varvec{\theta }}, {\varvec{\rho }}) = \prod _{i,\alpha } \frac{1}{ 1 - e^{-(\theta _i +\rho _\alpha )}}. \end{aligned}$$

(29)

The graph probability yields

$$\begin{aligned} P(G_B|{\varvec{\theta }}, {\varvec{\rho }}) = \prod _{i, \alpha } \left( e^{-(\theta _i +\rho _\alpha )}\right) ^{w_{i\alpha }} (1 - e^{-(\theta _i +\rho _\alpha )}). \end{aligned}$$

(30)

Similar to the BiCM, the Lagrange multipliers can be obtained by solving an equation system, which reads [31]

$$\begin{aligned} \left\{ \begin{array}{c} \sum _\alpha \dfrac{e^{-(\theta _i + \rho _\alpha )}}{1 - e^{-(\theta _i +\rho _\alpha )}} = s_i^*,\quad \forall i\in L,\\ \\ \sum _i\dfrac{e^{-(\theta _i +\rho _\alpha )}}{1 - e^{-(\theta _i + \rho _\alpha )}} = s_\alpha ^*,\quad \forall \alpha \in \varGamma .\\ \end{array}\right. \end{aligned}$$

(31)

1.5 Bipartite Enhanced Configuration Model

The bipartite enhanced configuration model (BiECM, [31]) is a bipartite extension of the monopartite enhanced configuration model introduced in [62]. Both, degrees as well as strengths, are constrained.

Be $\theta _i$ and $\theta _\alpha$ the constraints associated to the degrees, and $\rho _i$ and $\rho _\alpha$ those associated to the strengths for the nodes $i\in \text {L}$ and $\alpha \in \varGamma$, respectively. Using the short-hand notation $\phi _i = e^{-\rho _i}, \xi _\alpha = e^{-\rho _\alpha }, \psi _i = e^{-\theta _i}$ and $\gamma _\alpha = e^{-\theta _\alpha }$, the partition function reads [31]

$$\begin{aligned} \mathcal {Z}_{BiECM}({\varvec{\theta }}, {\varvec{\rho }}) = \prod _{i, \alpha } \frac{1 - \phi _i \xi _\alpha ( 1 - \psi _i \gamma _\alpha )}{1 - \phi _i \xi _\alpha }. \end{aligned}$$

(32)

Consequently, the network probability is given by

$$\begin{aligned} P(G_B) = \prod _{i, \alpha } \frac{(1 - \phi _i \xi _\alpha ) (\phi _i\xi _\alpha )^{w_{i\alpha }} (\psi _i\gamma _\alpha )^{\varTheta (w_{i\alpha })}}{1- \phi _i \xi _\alpha (1 - \psi _i\gamma _\alpha )} \end{aligned}$$

(33)

and factorizes in single link probabilities. The values of the Lagrange multipliers can be obtained through a nonlinear system of equations, as shown in the Appendix of [31].

1.6 Maximum Entropy Capital Asset Pricing Model

The elements of the weighted biadjacency matrix can be rescaled to yield the quantities of the capital asset pricing model (CAPM, [61, 66]). In the financial context, the vertex strengths are often described as the total asset size of a bank (or market value of their portfolio), $V_i = \sum _{\alpha } w_{i\alpha }$, and the market capitalization of an asset, $C_\alpha = \sum _{i} w_{i\alpha }$ [31, 87]. In the CAPM, banks choose their portfolio weights proportional to their market value and the asset’s capitalization:

$$\begin{aligned} w^{CAPM}_{i\alpha } = \frac{V_i C_\alpha }{w}, \end{aligned}$$

(34)

where we have used $w = \sum _{i', \alpha '} w_{i'\alpha '}$ The probability distribution for the MECAPM yields [31]

$$\begin{aligned} P(G_B) = \prod _{i, \alpha } \left[ 1 - \left( p_{CAPM}\right) _{i\alpha }\right] ^{w_{i\alpha }} \left( p_{CAPM}\right) _{i\alpha }, \end{aligned}$$

(35)

where the probability per link reads

$$\begin{aligned} \left( p_{CAPM}\right) _{i\alpha } = \frac{w^{CAPM}_{i\alpha }}{1 + w^{CAPM}_{i\alpha }}. \end{aligned}$$

(36)

Note that $P(G_B)$ is geometrically distributed for $w_{i\alpha } \in \mathbb {N}$ [31]. The link probabilities can be easily calculated using the identity in Eq. (34).

1.7 Enhanced Capital Asset Pricing Model

The so-called enhanced capital asset pricing model (ECAPM, [87]) reconstructs the link topology and subsequently the link weights. Their method makes only use of the strength sequence and is composed of two steps.

Firstly, the topology of the network is reconstructed by using the BiCM under the assumption that the exponential Lagrange multipliers $x_i \equiv \text {e}^{-\theta _i}$ and $y_\alpha \equiv \text {e}^{-\theta _\alpha }$ are proportional to node-specific fitness values, represented by their strengths:

$$\begin{aligned} x_i\equiv & {} \sqrt{z_\varGamma } s_i,\quad \forall i \in L\nonumber \\ y_\alpha\equiv & {} \sqrt{z_L} s_\alpha ,\quad \forall \alpha \in \varGamma . \end{aligned}$$

(37)

Constraining the network density with the total number of links $\langle E\rangle \equiv E^*$, the parameter $z = \sqrt{z_\varGamma z_L}$ can be estimated using [87]

$$\begin{aligned} \langle E \rangle = \sum _{i, \alpha } \frac{z V_i C_\alpha }{1 + z V_i C_\alpha },\quad \forall i \in L, \alpha \in \varGamma , \end{aligned}$$

(38)

Subsequently, the single link probabilities are simply given by the BiCM expression in Eq. (11), substituting the Lagrange multipliers with the expressions (37):

$$\begin{aligned} (p_{ECAPM})_{i\alpha } = \frac{z V_i C_\alpha }{1 + z V_i C_\alpha },\quad \forall i \in L, \alpha \in \varGamma , \end{aligned}$$

(39)

where z absorbs the proportionality constants.

Secondly, the link weights are reconstructed using the CAPM model while taking the network topology into consideration. Instead of setting $w_{i\alpha } = V_i C_\alpha / w$, a correction factor is applied [87]

$$\begin{aligned} w_{i\alpha }= & {} m_{i\alpha } \frac{V_i C_\alpha }{w\ (p_{ECAPM})_{i\alpha }} \nonumber \\= & {} (V_i C_\alpha + z^{-1}) \frac{m_{i\alpha }}{w}, \end{aligned}$$

(40)

where $m_{i\alpha }$ is 0 or 1,depending the link is present in the graph or not.