# The Smallest Valid Extension-Based Efficient, Rare Graph Pattern Mining, Considering Length-Decreasing Support Constraints and Symmetry Characteristics of Graphs

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. Frequent Pattern Mining and Frequent Graph Pattern Mining

#### 2.2. Pattern Mining On Multiple Minimum Support Constraints

#### 2.3. Pattern Mining on Length-Decreasing Support Constraints

## 3. Smallest Valid Extension-Based Rare Graph Pattern Mining, Considering Length-Decreasing Support Constraints and Symmetry Characteristics of Graphs

#### 3.1. Preliminaries

**Definition 1.**

_{1}, v

_{2}, …, v

_{i}}, and a set of edges, E(P) = {e

_{1}, e

_{2}, …, e

_{j}}.

**Definition 2.**

_{1}, v

_{2}) and (v

_{2}, v

_{1}) are equal to each other.

**Definition 3.**

_{1}, v

_{n}, |V(P)| and |E(P)| are the first and last vertices and the number of vertices and edges comprising P, respectively. A free tree should have at least one vertex of which the degree is 3 or more. In addition, there is no cyclic relation in all of its edges. If P is a free-tree with k vertices, the following conditions are satisfied:

**Definition 4.**

_{G}= {Tr

_{1}, Tr

_{2},…, Tr

_{n}} be a given database storing n graph data records (also called graph transactions), where each graph transaction, Tr, is composed of multiple vertices and edges. Given a graph pattern, P, we can calculate the support of P, S(P), as follows:

_{G}. In other words, the result of S(P) signifies how many times P appears in DB

_{G}. If S(P) is not smaller than a user-given minimum support threshold, we regard P as a frequent sub-graph or a frequent graph pattern. Thus, the final goal of traditional frequent graph pattern mining is to extract all the possible graph patterns of which the support values are higher than or equal to this single minimum support threshold.

#### 3.2. Overall Architecture of the Proposed Method

#### 3.3. Mining SRGs from Graph Databases

#### 3.3.1. Length-Decreasing Support Constraints and Smallest Valid Extension on Graph Mining

**Definition 5.**

_{prev}be a sub-graph pattern just before S becomes a cyclic graph, l

_{prev}be a length of S

_{prev}, and k be the number of cyclic edges inserted into S. Then, l becomes an addition of l

_{prev}and k, where l can be denoted as l = L(S).

**Definition 6.**

**Definition 7.**

^{−1}(S(S)) and returns the minimum length that S must have in order to become a potentially frequent sub-graph pattern. Such a condition is also denoted as f

^{−1}(S(S)) = min(l|f(l) ≤ S(S)).

**Example 1.**

^{−1}(S(S)) returns 7 since the minimum value is 7 among the lengths corresponding to the supports lower than or equal to 4%. Therefore, S must have more than length of 7 to be frequent. However, it is eventually infrequent since its length is 5.

**Definition 8.**

^{−1}(S(S)) before it becomes a potentially frequent sub-graph pattern.

**Lemma 1.**

^{−1}(S(S)) such that S(S) < f(L(S)), then S’ is always an infrequent pattern.

**Proof.**

^{−1}(S(S)). Therefore, we can induce the inequality, f

^{−1}(S(S)) ≤ f

^{−1}(S(S’)). In order that S’ expanded from the infrequent sub-graph S becomes frequent, these two conditions, S(S) < f(L(S)) and S(S’) ≥ f(L(S’)) must be satisfied. After we multiply the inverse function by the conditions, the result can be denoted as follows: L(S) ≤ f

^{−1}(S(S)) ≤ f

^{−1}(S(S’)) ≤ L(S’). Therefore, S’ becomes infrequent if it does not satisfy these conditions. Because the current mining step performed up to S, S’ has not yet been expanded. Therefore, we can determine the values of L(S), L(S’), and f

^{−1}(S(S)) but cannot know the value of f

^{−1}(S(S’)) (L(S’) can be inferred from L(S)). Therefore, if L(S’) ≥ f

^{−1}(S(S)) is false, i.e., L(S’) < f

^{−1}(S(S)) is true, S’ becomes an infrequent graph pattern. For this reason, we can know whether or not S’ is valid in advance even though any actual expansion process for S’ is not performed. ■

#### 3.3.2. Pre-Pruning Infrequent Sub-Graphs by the SVE Property without Any Pattern Loss

**Lemma 2.**

_{S’}= {Tr

_{1}, Tr

_{2}, …, Tr

_{n}} be a set of graph transactions including S’. Then, if there is any element satisfying L(Tr

_{i}) < f

^{−1}(S(S)) among the elements of SET

_{S’}(1 ≤ I ≤ n), S’ can permanently be pruned.

**Proof.**

_{S}

_{’}= {Tr

_{1}, Tr

_{2}, …, Tr

_{n}}, each Tr is a graph transaction with S’ in DB

_{G}, and n becomes the support of S’. If there is any Tr

_{i}such that L(Tr

_{i}) < f

^{−1}(S(S)) (1 ≤ i≤ n), it means that lengths of all super patterns generated from S’ are also smaller than f

^{−1}(S(S)) because the super patterns cannot have more lengths than L(Tr

_{i}). Furthermore, since S’ and the super patterns of S’ do not satisfy the minimum length by the inverse function, neither of them naturally satisfies minimum support constraints. As a result, pruning S’ does not have any negative effect on maintenance of the anti-monotone property. That is, we can obtain intended mining results without any problem. ■

**Example 2.**

_{S’}includes 4 graph transactions (denoted as SET

_{S’}= {Tr

_{1}, Tr

_{2}, Tr

_{3}, Tr

_{4}}), where the length for each Tr is set to 7, 4, 10, and 5 respectively. Then, S’ becomes an invalid pattern according to the SVE property and Lemma 1. Furthermore, since L(Tr

_{2}) is smaller than f

^{−1}(S(S)), any super patterns of S’ also become useless ones and therefore, S’ can directly be pruned.

#### 3.3.3. Multiple Minimum Supports of Vertex and Edge Elements on Graph Mining

**Definition 9.**

_{G}= {Tr

_{1}, Tr

_{2}, …, Tr

_{n}}, a set of x vertices and y edges comprising DB

_{G}can be denoted as V(DB

_{G}) = {v

_{1}, v

_{2}, …, v

_{x}} and E(DB

_{G}) = {e

_{1}, e

_{2}, …, e

_{y}}, respectively. Then, each of minimum support threshold, δ, is set for each element as shown in Table 1, where they are assigned by a user, respectively.

**Definition 10.**

_{G}. Then, a set of vertices and edges can be denoted as V(P) = {v

_{1}, v

_{2}, …, v

_{i}} and E(P) = {e

_{1}, e

_{2}, …, e

_{j}}, respectively. According to Definition 9, we know that each element has its own minimum support threshold set by a user, and P is composed of multiple elements. Hence, the minimum support threshold for P, T(P), is computed as the minimum value among the threshold values of P’s elements.

**Definition 11.**

#### 3.3.4. Pre-Pruning Invalid Graph Patterns Based on Multiple Minimum Support Constraints

**Definition 12.**

_{G}has multiple graph transactions and the corresponding elements as mentioned in Definition 9. Then, the overestimated minimum support constraint for DB

_{G}, O(DB

_{G}), is computed as the smallest value among the valid minimum support constraints of all the elements comprising DB

_{G}(it is also called Least Minimum Support (LMS)). In other words, let SET

_{T(DBG)}= {δ

_{1}, δ

_{2}, …, δ

_{x+y}} (δ

_{1}≥δ

_{2}≥ … ≥ δ

_{x+y}) be a sorted set of minimum support constraints for all the elements in DB

_{G}(x and y are the numbers of vertices and edges, respectively). Then, we start comparing the smallest threshold δ

_{x+y}with the real support of the element corresponding to δ

_{x+y}. After that, δ

_{(x+y)−1}is compared to the corresponding element support. Such a comparison is performed until we find the first element of which the support is higher than or equal to the corresponding minimum support constraint, δ

_{k}(1 ≤ k ≤ x + y). Then, we consider δ

_{k}as O(DB

_{G}).

#### 3.4. Improving Efficiency of Graph Mining Performance Based on Symmetry Features of Graphs

_{1}, e

_{1}, v

_{2}, e

_{2}, …, e

_{k-1}, v

_{k}} be a given path and N = {v,e’} be a pair of one vertex and edge that are supposed to be attached to P. Then, when expanding P with N, we have two choices; the first one is to add N to the front of P and the second one is to add N to the rear of P because of the characteristics of paths. If we add N to P without any consideration, an enormous number of duplicated graph patterns can be generated as the graph pattern growth works are conducted during the mining process. Meanwhile, if we set a specific constraint for limiting expansion directions of paths, we can effectively prevent such a problem.

_{1}, e

_{1}, v

_{2}, e

_{2}, …, e

_{k-1}, v

_{k}}, we can extract two strings from P as follows: v

_{1}-e

_{1}-v

_{2}-e

_{2}-…-e

_{k-1}-v

_{k}(original string) and v

_{k}-e

_{k-1}-v

_{k-1}-e

_{k-2}-…-e

_{1}-v

_{1}(inverse string). Then, if they are equal to each other, we consider P as a symmetric path. In this case, we do not need to consider what direction we have to choose because any selection leads to the same result. If the first string is lower than the second one in terms of a lexicographical order, we expand P by attaching new elements to the front of P. Meanwhile, if the first string is higher than the second one, we add the new ones to the rear of P. From the above path expansion technique based on the symmetry features of paths, we can omit any path expansion causing duplicated path creation. In addition, once the symmetry result of P is calculated, we can easily determine the symmetry result of its expanded path in a few additional computations. Let Sym

_{total}(P), Sym

_{front}(P), and Sym

_{rear}(P) be symmetry functions for the entire part of P ({v

_{1}, e

_{1}, v

_{2}, e

_{2}, …, e

_{k-1}, v

_{k}}), the front part of P ({v

_{1}, e

_{1}, v

_{2}, e

_{2}, …, e

_{k-2}, v

_{k-1}}) and the rear part of P ({v

_{2}, e

_{2}, v

_{3}, e

_{3}, …, e

_{k-1}, v

_{k}}), where each function returns 0 when the corresponding string is symmetric, 1 when the corresponding original string is lower than the inverse one and −1 when the original one is higher than the inverse one. Using this method, we can easily know the symmetry result of super patterns of P. Let P’ be a longer path that adds a new vertex and edge to P. Then, if the new elements have been attached to the front of P’, we can determine that Sym

_{total}(P) = Sym

_{rear}(P’). Meanwhile, if the new ones have been added to the rear of P’, it is true that Sym

_{total}(P) = Sym

_{front}(P’). Therefore, based on these characteristics, we can efficiently determine the symmetry results of mined patterns. By restricting directions of graph expansion based on the symmetry features of paths, we can improve the mining efficiency of the proposed method.

#### 3.5. Algorithm Description: SVE-RGM

_{G}through the calculated minimum support and LMS value (lines 3–6). Then, for each frequent vertex, the algorithm extracts valid sub-graph patterns according to length-decreasing support constraints and multiple minimum support thresholds as it performs a series of graph pattern expansion works (lines 7–11). When function Expand_subgraphs is called, SVE-RGM determines whether G is frequent or not and then assigns a flag, true or false, into the isFrequent variable (lines 1–4), where G is entered to P if G is frequent (line 3). Thereafter, for each edge in E, appropriate pattern expansion works are selectively conducted according to the state of G such as a path, a free tree, and a cyclic graph (lines 6–8). After that, if the support of the expanded pattern, G’, is not smaller than LMS, the algorithm conducts the subsequent works (line 9). If isFrequent is false, then the algorithm decides whether to prune G’ (lines 10–13). If G’ is not pruned, SVE-RGM calls Expand_subgraphs recursively to perform the next pattern expanding operations (lines 14–16). After all of mining operations terminate, we can gain a complete set of SRGs considering the length-decreasing support constraints and the multiple minimum support constraints for rarity of graph patterns.

## 4. Performance Evaluation

#### 4.1. Experimental Environment

_{i}be an element within a given graph dataset. Then, the corresponding δ value for each e

_{i}is calculated as follows: δ

_{i}= MAXIMUM(β × S(e

_{i}), LS) (LS: the smallest δ value in all the δ values). Note that the threshold of Gaston and FGM-LDSC is set to the same value as LS for fair comparisons. In the equation, β (=1/α (0 < β ≤ 1, 1 ≤ α)) is a variable showing how closely the actual support value of each element is related to the corresponding δ value. In other words, each δ is more likely to have a value more similar to the real support of the corresponding element rather than LS, if β becomes closer to 1.

#### 4.2. Analysis of Runtime Performance

#### 4.3. Analysis of Memory Usage Performance

#### 4.4. Analysis of Pattern Generation Performance

#### 4.5. Analysis of Algorithm Performance on Changing Length-Decreasing Support Constraints

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Example of various graph pattern forms. (

**a**) A simple, labeled and undirected graph without any self-edges; (

**b**) A multiple graph with multiple edges between vertices; (

**c**) A directed graph with a self-edge.

**Figure 4.**Example of length-decreasing support constraints. (

**a**) A table with length and support information; (

**b**) A graph corresponding to the length-decreasing support constraints in Figure 4a.

**Figure 6.**(

**a**) Length-decreasing support constraints of PTE; (

**b**) Length-decreasing support constraints of SYN100K; (

**c**) Length-decreasing support constraints of DTP.

**Figure 7.**(

**a**) Runtime results of PTE on changing minimum support threshold; (

**b**) Runtime results of SYN100K on changing minimum support threshold; (

**c**) Runtime results of DTP on changing minimum support threshold

**Figure 8.**(

**a**) Runtime results of PTE on changing α (minsup = 1.5%); (

**b**) Runtime results of PTE on changing α (minsup = 2%); (

**c**) Runtime results of SYN100K on changing α (minsup = 1.5%); (

**d**) Runtime results of SYN100K on changing α (minsup = 2%); (

**e**) Runtime results of DTP on changing α (minsup = 5.5%); (

**f**) Runtime results of DTP on changing α (minsup = 6%).

**Figure 9.**(

**a**) Memory usage results of PTE on changing minimum support threshold; (

**b**) Memory usage results of SYN100K on changing minimum support threshold; (

**c**) Memory usage results of DTP on changing minimum support threshold.

**Figure 10.**(

**a**) Memory usage results of PTE on changing α (minsup = 1.5%); (

**b**) Memory usage results of PTE on changing α (minsup = 2%); (

**c**) Memory usage results of SYN100K on changing α (minsup = 1.5%); (

**d**) Memory usage results of SYN100K on changing α (minsup = 2%); (

**e**) Memory usage results of DTP on changing α (minsup = 5.5%); (

**f**) Memory usage results of DTP on changing α (minsup = 6%).

**Figure 11.**(

**a**) Pattern generation results of PTE on changing minimum support threshold; (

**b**) Pattern generation results of SYN100K on changing minimum support threshold; (

**c**) Pattern generation results of DTP on changing minimum support threshold.

**Figure 12.**(

**a**) Pattern generation results of PTE on changing α (minsup = 1.5%); (

**b**) Pattern generation results of PTE on changing α (minsup = 2%); (

**c**) Pattern generation results of SYN100K on changing α (minsup = 0.1%); (

**d**) Pattern generation results of SYN100K on changing α (minsup = 0.2%); (

**e**) Pattern generation results of DTP on changing α (minsup = 5.5%); (

**f**) Pattern generation results of DTP on changing α (minsup = 6%).

**Figure 13.**(

**a**) Different settings of length-decreasing support constraints for PTE (α = 1); (

**b**) Runtime result for PTE (α = 1); (

**c**) Memory usage result for PTE (α = 1); (

**d**) Pattern generation result for PTE (α = 1).

**Figure 14.**(

**a**) Different settings of length-decreasing support constraints for DTP (α = 1); (

**b**) Runtime result for DTP (α = 1); (

**c**) Memory usage result for DTP (α = 1); (

**d**) Pattern generation result for DTP (α = 1).

**Figure 15.**(

**a**) Different settings of length-decreasing support constraints for SYN100K (α = 1); (

**b**) Runtime result for SYN100K (α = 1); (

**c**) Memory usage result for SYN100K (α = 1); (

**d**) Pattern generation result for SYN100K (α = 1).

Element | v_{1} | v_{2} | … | v_{x} | e_{1} | e_{2} | … | e_{y} |

Threshold | δ_{1} | δ_{2} | … | δ_{x} | δ_{x+1} | δ_{x}_{+2} | … | δ_{x+y} |

**Table 2.**Input format of the graph dataset transformed from Table 1.

Tr_{1} | Tr_{2} | Tr_{3} | Tr_{4} |
---|---|---|---|

v 0 A | v 0 A | v 0 D | v 0 B |

v 1 B | v 1 B | v 1 A | v 1 A |

v 2 A | v 2 A | v 2 C | v 2 A |

v 3 C | v 3 C | v 3 A | e 0 1 a |

e 0 1 a | e 0 1 a | e 0 1 a | e 0 2 a |

e 0 2 a | e 1 2 a | e 0 2 a | e 1 2 a |

e 1 2 a | e 2 3 a | e 1 3 a | - |

e 2 3 a | - | e 2 3 a | - |

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**MDPI and ACS Style**

Yun, U.; Lee, G.; Kim, C.-H.
The Smallest Valid Extension-Based Efficient, Rare Graph Pattern Mining, Considering Length-Decreasing Support Constraints and Symmetry Characteristics of Graphs. *Symmetry* **2016**, *8*, 32.
https://doi.org/10.3390/sym8050032

**AMA Style**

Yun U, Lee G, Kim C-H.
The Smallest Valid Extension-Based Efficient, Rare Graph Pattern Mining, Considering Length-Decreasing Support Constraints and Symmetry Characteristics of Graphs. *Symmetry*. 2016; 8(5):32.
https://doi.org/10.3390/sym8050032

**Chicago/Turabian Style**

Yun, Unil, Gangin Lee, and Chul-Hong Kim.
2016. "The Smallest Valid Extension-Based Efficient, Rare Graph Pattern Mining, Considering Length-Decreasing Support Constraints and Symmetry Characteristics of Graphs" *Symmetry* 8, no. 5: 32.
https://doi.org/10.3390/sym8050032