Event chains

The other network features to be analysed for event networks are chains, which are larger in terms of size, but conceptually simpler, than motifs. In this context, a chain is a sequence of events in which each can be linked to the previous one as a close pair. It is therefore a set of events for which a conceptual link runs throughout, via the antecedence relationship, even though only sequential pairs are connected.

In terms of the specific network features to which it refers, the term ‘chain’ can be defined in various ways, and so it is necessary to specify the sense in which it is used here before discussing the concept further. The definition is made with a number of theoretical concerns in mind, which will be discussed subsequently.

In an event network, a chain is a directed path of maximal length (i.e. one which is not a subset of a longer path).

Several aspects of this definition are worthy of comment. Firstly, the fact that chains are directed means that they respect the temporal precedence of events; in any chain, the events in question must have happened one after the other, in a defi- nite ordering. It may be the case that larger groups could be found if directionality was disregarded (in network terms, these would be ‘weakly connected components’),

but these cannot be characterised easily as stylised series of events.

The ‘maximal’ aspect of the definition also has a number of implications. A path in an event network is only a chain if it cannot be extended further; in other words, the first vertex in the chain must have no in-links, and the final vertex must have no out-links. A chain of length 2, for example, does not comprise 2 chains of length 1, since this condition is violated in both cases. This also means that the number of chains of length 1 is not simply equal to the number of links in the network: only links which are isolated (i.e. not part of a longer chain) are counted as such.

This maximality has consequences in terms of what can be inferred from the identifi- cation of a chain. The existence of a chain of length l indicates not only the presence of a sequence of l events, but also the absence of any (l + 1)th event with which the sequence could be extended. Chain length therefore corresponds to an upper limit on the linkage of events, in this sense, and suggests an alternative perspective: disproportionate incidence of chains of length l could be interpreted as evidence that (l + 1)th events tend not to occur.

One final point concerns the uniqueness of chains, and the way in which they are counted during analysis. According to the definition, two chains are distinct if they differ in at least one link, and chain counts are calculated on the basis of unique chains. A given link may therefore feature in more than one unique chain, as long as they diverge at some stage. This is an unavoidable consequence when chains are defined in this way, and can be considered as both a positive and negative feature. In one sense, it leads to ‘double counting’ of certain paths; however, it could also be argued that paths which diverge at some point are genuinely distinct and should be treated as such.

Several of these issues are summarised by the hypothetical example shown in Figure 2.5. The diagram shows the 4 unique chains which are present in a hypothetical

6-vertex event network, together with their lengths. The effect of the directionality is manifested in the fact that the longest chains are of length 2, and the fact that those 2-chains are not broken into shorter fragments is due to the requirement of maximality.

G

Length 1 Length 1 Length 2 Length 2

Figure 2.5: The identification of chains. The 6-vertex event network denoted by GDT_dt contains four unique chains, each of which is shown below. In each case, the chain in question is indicated by red links and its length is stated.

The interpretation of chains, in terms of the real-world patterns which they rep- resent, is fairly straightforward. Assuming that the close pair relationship can be taken as evidence of conceptual linkage, analysis of chain length can be interpreted as an indication of the typical size of ‘bursts’ of activity. It represents a measure of the extent to which events tend to occur in ‘spates’ of sustained activity, and also provides a means of examining the extent to which the clustering of crime is fragmentary.

Extending this further, such results can also be used to make inferences about the behaviour of offenders. Sequences of close events are often hypothesised to be the work of the same offender, with arguments related to the concept of foraging fre- quently invoked (e.g. Johnson et al., 2009b). In such a context, the length of chains acts as a measure of the extent to which offenders engage in such bursts of behaviour before moving on. The decision to end such a sequence, and the stage at which it is taken, is a theoretically-important one, and relates to notions of ‘carrying capacity’ when crime is modelled by analogy with ecology.

Chains are not a feature for which spatial patterns can be characterised with any degree of certainty. The existence of a chain is consistent with ‘drifting’ behaviour (where locations move in some definite direction as the chain progresses) but also with repeated occurrences at a fixed location. Little can therefore be inferred about spatial trends, and the primary value is in the inferences which can be made about the ‘burstiness’ of offending patterns.