2.1 Definitions
In the following, we introduce a new model class for weighted and directed graphs consisting of a fixed number \(n \in {\mathbb {N}}\) of nodes. Furthermore, we assume that the edges are modelled as random variables. A network consisting of \(n \in {\mathbb {N}}\) nodes is given by a matrix \(L=(L_{ij})_{i, j \in \{1, 2, \ldots , n\}}\), where the \(L_{ij}\) are random variables modelling the weight of the directed edge from node i to node j. A weight of 0 indicates that the corresponding edge is not present. This definition of a network allows for at most one weighted directed edge between two nodes. In practice, these weights are often aggregates of several individual relationships between the nodes, which motivates our model choice.
We propose using a compound Poisson Gamma distribution for these weights, with parameters given by a regression model. A compound Poisson Gamma distribution can be defined via the random variable
$$\begin{aligned} X=\sum _{\nu =1}^N S_\nu , \end{aligned}$$
where
\(N\sim \text {Poisson}(\lambda )\) and
\(S_{\nu } \sim \text {Gamma}(\alpha ,\mu _S)\),
\(\nu =1,\dots ,N\), are independent, where
\(\text {Poisson}(\lambda )\) is the Poisson distribution with mean
\(\lambda \) and
\(\text {Gamma}(\alpha , \mu _{S})\) is the Gamma distribution with shape parameter
\(\alpha \) and mean
\(\mu _{S}\).
1 Then
\({\mathbb {V}}\text {ar}(S_{\nu }) = \mu _S^2/\alpha \) and
\({\mathbb {E}}[S_{\nu }^2] = \mu _S^2 \frac{1+\alpha }{\alpha }\).
It is well known that
\({\mathbb {E}}[X]= {\mathbb {E}}[N] {\mathbb {E}}[S_{\nu }]= \lambda \mu _S\) and
\({\mathbb {V}}\text {ar}[X]= {\mathbb {E}}[N] E[S_{\nu }^2] = \lambda \mu _S^2 \frac{1+\alpha }{\alpha }\). This can be seen as a special case of the so-called Tweedie distribution Jorgensen [
27] and Dunn and Smyth [
15], which is usually parametrised via its mean
\(\mu \) and parameters
\(\phi \),
p such that
\({\mathbb {E}}[X]=\mu \) and
\({\mathbb {V}}\text {ar}[X]=\phi \mu ^p\).
2
Our network
\(L=(L_{ij})_{1 \le i, j \le n}\) will be modelled as independent random variables having a compound Poisson Gamma distributions, with parameters defined via a regression.
3 We will propose two ways of doing this—the first (CPNet1) will model
\(\mu _{ij}:={\mathbb {E}}[L_{ij}]\) via regression and the second (CPNet2) will model both the mean of
N, i.e.
\(\lambda \) and the mean of
\(S_{\nu }\), i.e.
\(\mu _S\), via regression. The numbers 1 and 2 in the names of CPNet1 and CPNet2 indicate how many regressions are embedded in the model.
The parameters of CPNet1 are chosen as follows. The shape parameter of the Gamma distribution is a fixed constant \(\alpha \). As mentioned before, we would like to define the overall mean via regression—thus we want to achieve \({\mathbb {E}}[L_{ij}]=\mu _{ij}\) for given \(\mu _{ij}\). That leaves flexibility on how to define the means of the Poisson and Gamma part of the distribution. We resolve this by imposing a second moment condition, namely \({\mathbb {V}}\text {ar}[L_{ij}]=\phi \mu _{ij}^p\), where \(p=\frac{\alpha +2}{\alpha +1}\). This ensures that every element of L will follow a Tweedie distribution with parameters \(\mu _{ij},\phi ,p\), with \(p\in (1,2)\).
In the above, \(X_{ij}\), \(i \in \{1, \ldots , n\}\), \(j \in \{1, \ldots , p\}\), are the elements of the design matrix. The variable \(f_i\) can be interpreted as “fitness” of node i.
We refer to l in the definition above as a link function. Examples for link functions are \(l(x,y)=\exp (x+y)\), \(l(x,y)=\max (\exp (x),\exp (y))\) and \(l(x,y)=\exp (x)+\exp (y)\). We would usually choose link functions that are monotonically non-decreasing in each of their arguments. This then implies that higher values of the fitnesses imply higher means of the corresponding compound Poisson distributions.
Next we consider CPNet2, which is a model in which both the mean of the Poisson distribution and the mean of the Gamma distribution are modelled separately via regression. The shape parameter \(\alpha \) of the Gamma distribution is again a fixed constant.
In the above, \(X_{ij}^{k}\), \(i \in \{1, \ldots , n\}\), \(j \in \{1, \ldots , p_k\}\), \(k\in \{N,S\}\) are the elements of the design matrices. Examples for link functions are as above. The variables \(f_i^N\) and \(f_i^S\) can be interpreted as fitnesses of node i, one affecting the Poisson part of the model, the other the Gamma part of the model.
2.3 Interpretation as fitness models
These new models were inspired by the classical fitness models (see Sect.
1.1) that assign fitnesses to every node which then determines the link existence probabilities for every edge. We, however, take a broader view by considering a general regression framework that enables us to characterise more general features of the random graph. In particular, our regression framework incorporates fitness models as special cases but with the additional feature that fitnesses are used to characterise properties of the weights of edges in addition to the existence of edges.
To see how CPNet1 can be interpreted as a classical fitness model (in which no regression is used to determine the fitness parameter), we can set \(X=I_n\), where \(I_n\) is the \(n\times n\) identity matrix in CPNet1. Then, \(f_i = \sum _{j=1}^p X_{ij} \beta _j = \beta _i\) for all \(i \in \{1, \ldots , n\}\). Hence, the overall mean of \(L_{ij}\) is given by \(\mu _{ij} = l(f_i, f_j) = l(\beta _i, \beta _j)\) which can be interpreted as a fitness model for the mean of the weighted edges where \(\beta _i\), \(i \in \{1, \ldots , n\}\) are the fitnesses.
Similarly, we can set \(X^N=X^S=I_n\) in CPNet2. Then, \(l^{N}(f^{N}_{i},f^{N}_{j}) = l^N(\beta _i^N, \beta _j^N)\) can be interpreted as a fitness model for the mean of the Poisson distribution where \(\beta _i^N\), \(i \in \{1, \ldots , n\}\) are the fitnesses and \(l^{S}(f^{S}_{i},f^{S}_{j}) = l^{S}(\beta ^{S}_i, \beta _j^{S})\) can be interpreted as is a fitness model for the mean of the Gamma distribution with fitnesses \(\beta ^{S}_i\), \(i \in \{1, \ldots , n\}\).
Both CPNet1 and CPNet2 could be extended to give every node an in-fitness and an out-fitness. For example, in CPNet2, we could have 4 instead of 2 design matrices, i.e. for
\(k\in \{N,S\}\) and
\(l\in \{\text {in},\text {out}\}\) we have
\(X^{k,l}\in {\mathbb {R}}^{n\times p_{k,l}}\) and corresponding fitnesses
that then define the model via
$$\begin{aligned} N_{ij}\sim \text {Poisson}(l^{N}(f_i^{N,\text {out}},f_{j}^{N,\text {in}})) \quad \text {and}\quad S^{\nu }_{ij}\sim \text {Gamma}(\alpha , l^{S}(f_{i}^{S,\text {out}},f_{j}^{S,\text {in}})). \end{aligned}$$
Similarly, one could define an extension of CPNet1 with in-fitness and out-fitness.
As discussed in our literature review fitness models have been studied before and it has been shown that they can also be used to construct degree distributions with heavy tails, see e.g. Gandy and Veraart [
19]. These results carry over to our class of compound Poisson models, since as one can see from the formulae for the link existence probabilities (
1) one can model a wide range of link behaviour with an appropriate choice of fitness parameters and link functions
l.
2.6 Possible applications of the models
Our modelling framework can be used to deal with missing information in network models. For example, situations in which a financial network is only partially observed and one would like to fill in the remaining parts. In contrast to the literature on network reconstruction, see e.g. Gandy and Veraart [
19,
20] we do not assume that the row and column sums of the network matrix
L are observed and the individual entries need to be estimated, but we have a situation in mind in which the row and column sums are not observable but some individual entries of the matrix are observable. In such a situation one could fit our new model class to the available data and predict the missing entries from the fitted model. We will demonstrate how this can be done in our empirical case study.
An alternative application would be that one observes a network in the past (on one or several occasions) and fits the new model class to these observations. One then uses these results to predict a network in the future.
Alternatively, one might be in a situation that one observes a network that is related to a network of interest, e.g., a derivative exposure network corresponding to Credit Default Swap exposures written on a given reference entity is observed (for example where the reference entity is a UK company) but one is interested in the same type of network written on a different reference entity (for example a non-UK company) and would like to make predictions about this network.
All these possible application areas could arise in the context of macro-prudential stress testing for systemic risk analysis in financial networks. To be able to conduct a macro-prudential stress test one needs to consider the financial system as a whole and analyse potential feedback and amplification mechanisms between the market participants. Often, the connections that give rise to such feedback mechanisms are not fully observable and therefore one will need to rely on statistical and simulation methods to deal with the missing information. This is where our compound Poisson model class can be used. In Gandy and Veraart [
19,
20] it was demonstrated how a network reconstruction method can be used in a macroprudential stress test if the network of interest is not fully observable. As mentioned before, in these papers the assumption was that the network matrix itself was not observable but its row and column sums were. Here we assume that a subset of the network is observable, and we use the subset to estimate a statistical model that will then be used to predict the missing edges in the original network that is not fully observable.