Finding the most descriptive substructures in graphs with discrete and numeric labels

Abstract

Many graph datasets are labelled with discrete and numeric attributes. Most frequent substructure discovery algorithms ignore numeric attributes; in this paper we show how they can be used to improve search performance and discrimination. Our thesis is that the most descriptive substructures are those which are normative both in terms of their structure and in terms of their numeric values. We explore the relationship between graph structure and the distribution of attribute values and propose an outlier-detection step, which is used as a constraint during substructure discovery. By pruning anomalous vertices and edges, more weight is given to the most descriptive substructures. Our method is applicable to multi-dimensional numeric attributes; we outline how it can be extended for high-dimensional data. We support our findings with experiments on transaction graphs and single large graphs from the domains of physical building security and digital forensics, measuring the effect on runtime, memory requirements and coverage of discovered patterns, relative to the unconstrained approach.

Appendix

The substructure discovery algorithms referred to throughout the paper are included here for convenience. The original references are Yan and Han (2002) for gSpan and Cook and Holder (2000) for Subdue.

1.1 A gSpan

To apply our method to gSpan, we calculate numeric anomalies for all vertices and edges by Definition 8 as a pre-processing step. Then amend step 2 to Remove infrequent and anomalous vertices and edges.

Algorithm 1 GraphSet_Projection: search for frequent substructures

Require: Graph Transaction Database $\mathbb {D}$, minSup

1:  Sort the labels in $\mathbb {D}$ by their frequency

2:  Remove infrequent vertices and edges

3:  Relabel the remaining vertices and edges

4:  $\mathbb {S}^{1} \leftarrow$ all frequent 1-edge graphs in $\mathbb {D}$

5:  Sort $\mathbb {S}^{1}$ in DFS lexicographic order

6:  $\mathbb {S} \leftarrow \mathbb {S}^{1}$

7:  for all edge $e \in \mathbb {S}^{1}$ do

8:    Initialise s with e, set s. D to graphs which contain e

9:    Subgraph_Mining($\mathbb {D}$, $\mathbb {S}$, s)

10:    $\mathbb {D} \leftarrow \mathbb {D} - e$

11:    if $|\mathbb {D}| < minSup$ then

12:     break

13:  return Discovered Subgraphs $\mathbb {S}$

1.2 B Subdue

To apply our method to Subdue, calculate numeric anomalies for all vertices and edges by Definition 8 as a pre-processing step (as above). Prune all anomalous vertices and edges from the graph before step 5.

Algorithm 2 Subdue: search for frequent substructures

Require: Graph, BeamWidth, MaxBest, MaxSubSize, Limit

1:  let ParentList = {}

2:  let ChildList = {}

3:  let BestList = {}

4:  let ProcessedSubs = 0

5:  Create a substructure from each unique vertex label and its single-vertex instances; insert the resulting substructures in ParentList

6:  while ProcessedSubs ≤ Limit and ParentList is not empty do

7:     whileParentList is not empty do do

8:      let Parent = RemoveHead(ParentList)

9:       Extend each instance of Parent in all possible ways

10:       Group the extended instances into Child substructures

11:       for all Child do

12:        if SizeOf(Child) ≤ MaxSubSize then

13:         Evaluate the Child

14:         Insert Child in ChildList in order by value

15:         if Length(ChildList) > BeamWidth then

16:          Destroy the substructure at the end of ChildList

17:      let ProcessedSubs = ProcessedSubs + 1

18:       Insert Parent in BestList in order by value

19:       if Length(BestList) > MaxBest then

20:        Destroy the substructure at the end of BestList

21:     Switch ParentList and ChildList

22:  return BestList