Network decomposition: m-slice analysis

While the various network properties and measures discussed in the previous section characterize the overall typological structure of the network, different substructures may exist within the higher order structure. In general, substructures emerge when there are subsets of nodes and edges that share certain characteristics in a way that distinguishes them from the rest of the network. Therefore, in order to identify such subsets or substructures, a number of techniques for decomposing a network into smaller parts have been proposed based on different defining characteristics.

In this study, a technique called m-core (Scott, 2000) or m-slice (Nooy et al., 2005)17 was used for network decomposition. An m-slice is defined as “a maximal

17

Scott (2000) introduced this technique as a variation or an extension of k- core analysis. In fact, as Scott (2000) pointed out, k-core analysis and m-core analysis are based on essentially the same procedure, while each adopts a different definition or criterion of cohesion to draw substructures from a given network. While

sub-graph in which each line has a multiplicity greater than or equal to m” and shows “a chain of points connected by lines of the specified multiplicity” (Scott 2000, p. 112). When the two-mode affiliation network, consisting of information objects and users, is transformed into a one-mode social network of users, each information object shared by a pair of users produces a link between them. Therefore, if a pair of users shares multiple information objects, at some point during the transformation process, there are multiple lines (links) between them. This phenomenon is called line multiplicity. At the end of the transformation process, any multiple lines between two users are merged into a single edge, to which the number of lines (the line multiplicity) is then assigned as a weight. Since the weight on an edge reflects the number of shared information objects between the connected pair, it is in fact a useful indicator of the level of shared interests between the two users. The weight, however, was ignored in the first part of the analysis where network properties were measured, because all of them are defined assuming an unweighted network. The m- slice technique was chosen for the substructure analysis, among a number of alternatives, because it allows us to take different levels of shared interests (represented by the number of shared information objects) into account.

By definition, an m-slice consists of edges that have a value of m or higher and nodes that are incident on those edges. In order to obtain an m-slice from a network at a given m value, therefore, all the edges (connections) with a value less than m and any isolated nodes are removed. The basic procedure for m-slice analysis involves iterative removal of edges and nodes. Starting from the original network,

edges and nodes are progressively removed as the value of m increases, and the original network is iteratively broken down into smaller sub-networks. It is, in effect, filtering out the weakest ties at each step so that areas with stronger connections are brought forth. A result of the procedure is a nested structure where lower m-slices contain higher m-slices. It discloses the overall shape of the network created by different levels of tie strengths (different degrees of interest sharing in our network) like contours on a map.

In the network of delicious.com users, each m-slice represents the sub-network where each connected pair of users share m or more information objects. The nested structure of m-slices, therefore, depicts the internal structure of the network with varying degrees of shared interests. In order to see the effect of increasingly stricter conditions for making connections among users on the resulting structure of the network, the same set of network properties was measured in each m-slice as in the original network. In addition, the number and sizes of its components were examined to show how the overall composition of the network changes as m changes.

The m-slice analysis was also used to identify cohesive subgroups or communities within the network. In principle, detecting a cohesive region within the network relies on the assumption that naturally existing subgroups would be reflected in some non-random pattern of connections. The most basic and obvious subgroups that appear in the connective structure of a network are network components, since by definition each node in the network belongs to only one component and components are separated from one another. As mentioned above, network components, therefore, can be regarded as communities. In a large scale network,

however, it is often the case that a large component contains multiple subgroups, especially if there is a giant component covering the majority of the network. The m- slice analysis provides a way to detect subgroups or communities by repeating the removal of weak ties until a large component in a lower slice is broken down into separate components. Note that components in any given m-slice can be regarded as communities defined by the minimum strength of connections with the threshold value of m. The method in fact falls into the category of hierarchical clustering techniques18 which have been widely used in the tradition of social network analysis in an attempt to locate cohesive subgroups within a social network.