Algunos estudios empíricos

As we investigate the results in more detail we see that Dublin is divided into 15 separate communities at the lowest level, which then merge into 4 communities at the highest level (Figure 6.14). There is a clear north-south and east-west divide visible which corresponds well with the known social divides. The north-south divide is perhaps more well known because it corresponds to the literal physical divide of the River Liffey running through Dublin city centre. As Corcoran (2004) notes, “[the] relations between the city’s northside

and southside are somewhat uneasy.”. The east-west divide is less evident in the general psyche of the people but is a real socio-economic divide with the coastal eastern parts of the city much more affluent than the western parts of the city and its suburbs. These results seem to suggest that both divides truly do exist at an interpersonal level.

We may note that the cities and counties of Cork & Kerry, Limerick and Galway all emerge as single homogeneous entities (communities 3, 8 and 11) at the lowest levels of the algorithm’s output, suggesting that they are indivisible. One could perhaps use these results to argue that these cities have retained their village like social structure and are much more socially cohesive than Dublin. Before we get carried away with these arguments however, we must remind ourselves of the issues raised in Section 4.2.3, particularly regarding the so-called resolution limit.

As Fortunato & Barthelemy (2007) warn, there is an inherent limit in the size of communities that modularity optimising methods are capable of de- tecting. We must therefore explore the results in more detail to understand if we really have found the complete hierarchy of community structure. We will do this by applying the same community detection method to the sub- graph of each of the communities found at the lowest level. We will take the community containing the city of Cork as an example. We apply the Louvain method to this subgraph and find a 2 level hierarchical structure with 24 and 12 communities respectively and modularity scores of 0.479 and 0.488. Once again we see in Figure 6.17 that the communities at each level are spatially contiguous.

This appears to contradict the previous finding that this was a single ho- mogeneous community with no substructure. As we explain in Section 4.2.3, this happens due to the fact that modularity is a global quality measure. Mod- ularity is calculated relative to the entirety of the current graph so edges that

Figure 6.15: Modularity optimisation of the tower network. Community as- signments in Dublin. Level 2

Figure 6.17: Modularity optimisation of the Cork/Kerry community subgraph. Level 1

Figure 6.18: Modularity optimisation of the Cork/Kerry community subgraph. Level 2

may appear to have unexpectedly high weights in the context of the larger graph may actually have unexpectedly low weights in the context of the sub- graph. This example clearly illustrates the fact that the communities found by modularity optimisation have no a priori definition, rather they only have a definition with respect to a given partition of a particular graph. However we may still test whether the communities we find in the subgraph could be classed as ‘real communities’ of the full graph. Fortunato & Barthelemy (2007) define a ‘real community’ within the modularity framework as a group of nodes which have a larger than expected number of internal connections. A subgraph s is a ‘real community’ if ls

L−

�_ds

2L

�2

> 0, where L is the total number of edges in the network, ls is the number of edges within the subgraph s, and ds is the total

degree of the nodes in s. We perform this check for each of the communities found at the lowest level of the hierarchy in the Cork/Kerry subgraph and find that each of them satisfies this condition, meaning they are ‘real communities’ in the context of the full graph also.

This result tells us that the hierarchy found by the Louvain method is not a true hierarchy as it fails to find some low level communities that are considered real communities within the modularity framework. The authors of the Louvain method claim that the agglomerative nature of the method circumvents the resolution limit problem (Blondel et al., 2008), but this is clearly not the case. The modularity measure itself is biased towards larger communities so it is unlikely that any modularity optimising method could circumvent the issue. Despite the appeal of the hierarchical unfolding of the Louvain method and other similar methods, it appears we must consider the intermediary levels as mere artefacts of the process of the algorithm rather than as significant outputs.