Fitness model for the Italian interbank money market

We use the theory of complex networks in order to quantitatively characterize the formation of communities in a particular financial market. The system is composed by different banks exchanging on a daily basis loans and debts of liquidity. Through topological analysis and by means of a model of network growth we can determine the formation of different group of banks characterized by different business strategy. The model based on Pareto’s law makes no use of growth or preferential attachment and it reproduces correctly all the various statistical properties of the system. We believe that this network modeling of the market could be an efficient way to evaluate the impact of different policies in the market of liquidity.

Figure 1

(Color online) A plot of the interbank network. The group of the vertices (banks) goes from light (group of small banks) to dark (large banks) (the color codes for the various groups are the following: $1=\text{yellow}$, $2=\text{red}$, $3=\text{blue}$, $4=\text{black}$). Note that the darkest vertices (bank of group 4) form the core of the system.

Figure 2

(Color online) A plot of the out-degree and in-degree (in the inset) distributions, respectively. As already noticed, the contribution to the tail of frequency distribution emerges from the banks of group 4. Using the division in four groups, we determine the average group of each bin of $F\left(k\right)$. Every bin is then represented with a particular color and/or texture according to the value of the group obtained. For noninteger value of this average we introduced intermediate colors.

Figure 3

Left-hand side: Probability distribution $P\left(k\right)$ for the degree $k$. With empty circles we have the experimental results to be compared with simulation of our model (filled circles). Right-hand side: Above comparison between experiment (empty circles) and results obtained with simulation of our model (filled circles) for the assortativity $⟨K_{nn}\left(k\right)⟩$ and below for the clustering coefficient $c\left(k\right)$.

Figure 4

The division on classes of vertices permits to represent in a very easy way the organizational principles of the network. Following results of Table we draw a link among two groups when the number of links between banks belonging to them is bigger than the average value. Using the net volumes as weight of links, we can represent the directed interactions among classes of nodes: class 4 appear to be clearly a borrower and class 1 is a lender.

Figure 5

Distribution of the total daily volume of transaction per bank. This quantity is used as fitness in our model.