Planar Maximally Filtered Graph (PMFG)

The final network that we discuss in this section is the filtered graph proposed by [Tumminelloet al. (2005)] with particular focus on the planar filtered graph (PMFG – created when the graph is embedded into a surface with genus set equal to 0). The networks discussed so far are a severe form of data reduction, containing the minimum number of edges. The proposed filtered graphs allow us to choose how much information we filter from the complete network, so by increasing the genus of the surface we are able to construct a more complex network containing more edges. The PMFG is constructed in a similar way to the MST. For a graphG(V, E) with|V|=nand |E|=mall edges,e1, e2, . . . , em, from the upper triangular section of C are placed in descending order e(1), e(2), . . . , e(m). Select the first edge e(1) and

construct a graph with e(1) and the two vertices that it connects. Continue select-

ing the ordered edges and add them to the network structure only if the resulting network can be drawn on a planar surface without edges crossing. There are some tests for planarity based on Kuratowski’s theorem [Kuratowski (1930)] that a graph

G is planar if and only if it contains neither K5 nor K3,3 as a topological minor.

(For more detail on these and others see [Hopcroft & Tarjan (1974)]). The algorithm ends when all vertices v1, v2, . . . , vn are connected, using 3(n−2) edges (this is the maximum number of edges in a PMFG – for further details please see Subsection 4.1.2; Eqn. (8)).

The advantage of the PMFG is that it will always contain the corresponding MST and so shows some of the clusters between stocks, but also provides additional infor- mation. Unlike the MST, the PMFG does not have a unique path between each of the vertices. This means that we cannot identify the hierarchical clustering between

stocks using the subdominant ultrametric distances in the direct way that we can with the MST. However, as the construction algorithm allows the inclusion of cycles the PMFG contains cliques, as with the AG, so we can extract further information from the network by analysing these cliques.

Looking specifically at the PMFG we consider 3- and 4-cliques, as the maximum number of elements that can form a clique is four. By considering the topology of the PMFG we can see that the basic structure (or motif) of the PMFG is a series of 3-cliques. Consider a sphere, a surface with g = 0. The PMFG separates the sphere into a sequence of triangular faces, with each vertex of the network belonging to a 3-clique. We can say that the PMFG is the triangulation of a sphere as the network consists entirely of 3-cliques (triangulation of a surface is a partition of that surface by triangles into facets). With our dataset of 30 stocks, we have a total of 30₃

= 4060 possible combinations of 3-cliques from each complete graph. By constructing the maximally filtered graph we considerably reduce the connectivity of the network leaving the most relevant cliques. (The possible structures of 3-cliques are discussed further in Section 4.2). We analyse the 4-cliques by showing the sectors that the four stocks belong to as well as the average correlation coefficient inside the clique, the range between the highest and lowest correlation coefficient in the clique and the standard deviation. Note that [Tumminello et al. (2005)] states the maximum number of 4-cliques formed by a PMFG isn−3 and we also prove this in Section 4.3.

Let us consider some of the examples highlighted in the previous subsections. We have noted from the 2007 MSTs and AGs that BAS, BAYN and LIN often formed a cluster and they all belong to the chemical sector. For the PMFGs for 2007 the

three stocks are connected in 60% of the networks and actually form a 3-clique in 44% of the networks. We also considered the cluster of stocks in the automobile sector for the 2004 data. These clusters are also shown in the PMFGs, with the three stocks being connected in 72% of the networks and a 3-clique forming in 44% of the networks. Finally, the two stocks in the utilities sector, RWE and EOAN, were connected in a high proportion of the MSTs and AGs for 2009 and this was also the case with the PMFGs with a connection in 84% of the PMFGs for 2009.

Cliques also allow us to identify the most connected stocks so that they can not only be clustered but also separated into two sets: core and periphery. This can be done using the AG as, due to the construction algorithm, we often have clique components and unconnected vertices. However, the benefit of the PMFG is that, as it is a connected network, we have a better understanding of the relationships between the stocks that are not identified as being within the core.

Unlike the MST and AG, the PMFG does not necessarily favour the strong, pos- itive stocks. We highlighted VOW3 as an example of a stock that was not fairly represented in the 2008 networks due to its negative correlation. For the PMFG in 2008 we see that VOW3 is mainly connected to three other stocks (84% of the networks) and these are mostly other stocks from the automobile sector (BMW, CON and DAI). At most it is connected to 6 other stocks (this included BMW and DAI). It forms 3-cliques and in some networks a 4-clique, although this 4-clique has a lower average correlation compared to the others from the same PMFG due to the negatively correlated VOW3.