Metrics for Multiset-Theoretic Subgraphs

. We show how to define a plethora of metrics for graphs—either full graphs or subgraphs. The method mainly utilizes the minimal matching between any two multisets of positive real numbers by comparing the multiple edges with respect to their corresponding vertices. In the end of this article, we also demonstrate how to implement these defined metrics with the help of adjacency matrices. Thesemetricsareeasytobemanipulatedinrealapplicationsandcouldbeamendedaccordingtodifferentsituations.Byourmetrics, oneshouldbeabletocomparethedistancesbetweengraphs,trees,andnetworks,inparticularthosewithfuzzyproperties.


Introduction
In the real world, we face a lot of uncertain mathematical objects.Among them, some are easier to be formalized via graphs or tree structures or networks.Henceforth, the distances between such structures are vital as they provide deeper information between two different structures.In the article [1], we have shown how to define metrics for graphs with single-edged vertices.Single-edge graphs are graphs with at most one edge between any two vertices.Such graphical structures normally are deterministic.The basic idea is to, based on minimal matching concepts, define matched parts and mismatched parts of incoming or outgoing edges.However, due to the complexity of real applications, in particular some fuzzy or indecisive mathematical objects, the metrics we had defined in that paper are insufficient to cover the needs.Therefore, we put forward some novel metrics which could accommodate such complexity in this article.We would consider graphs with multiple edges between any two vertices.This mechanism could then be applied in modelling some indecisive objects.Our research is also partially motivated by some articles regarding fuzzy mathematical objects [2][3][4][5].Furthermore, if one is interested in other variants of metrics for graphs, he could consult either [6] or [7].

Multisets
Let us introduce some definitions and operations of multisets.Let R + denote the set of all positive real numbers.Let N 0 denote the set of all the natural numbers including 0. Let Γ denote the set of all the functions R + → N 0 .Let   be the domain of a function .Let  *  = { ∈ R + : () ̸ = 0} be the nonzero domain of .Define Γ < = { ∈ Γ : | *  | < ∞}.In this article, we name each element in Γ < a multiset.Let ,  ∈ Γ < be arbitrary multisets.We use the notation  ≤  (i.e.,  is a multisubset of ) to denote that for all  ∈ R + , () ≤ ().
Definition (empty multiset).We call the zero function in Γ < the empty multiset.
Definition (descending order).Define the -th element in  by function  as follows: Definition (ascending order).Define the -th element in  by function  as follows:

Metrics
Let  be a set of vertices and  be a set of directed edges.Let MP < (R + ) denote the finite multipower set of R + .In this section, we show how to define metrics for labelled graphs and unlabelled graphs.The distance is mainly defined based on weights of the corresponding edges between the graphs.
Definition .We call  = (, ,  :  → MP < (R + )) a multiset-theoretic graph if and only if (2) For all ,  ∈ , every element in (, ) is nonnegative and where  is a multiset-valued weight function.Let  denote the set of all the multiset-theoretic graphs.The main purpose of this article is to define some metrics for  which is divided into two categories: labelled vertices and unlabelled vertices.
Definition .Let ‖(, )‖ denote the sum of all the elements in the multiset (, ).
. .Metrics for Labelled Graphs and Subgraphs.In this section, we show how to define the distance between any two graphs (whose vertices are all named) that could be graphs with either compatible or incompatible vertices.In this subsection, we assume all the vertices are labelled.To begin with, we show how to define metrics for MP < (R + ).These metrics will serve as the foundations for further construction of metrics.By the representations of multisets in descending and ascending forms as shown in (1) and (2), we have the following definitions.Based on the minimal matchings between any two multisets of positive real numbers [8], we derive the following metrics.
In this article, we assume  =  = 1/2.Indeed  and  could be decided by some mechanisms.We omit this part.
Definition (inbound measure: halved).Proof.They all follow from the definitions, in particular the property of set difference and the facts that , , and ℎ are all metrics.For a full proof, one could also consult the results in [1,8].
. .Metrics for Unlabelled Graphs and Subgraphs.In this section, we show how to define the distance between any two graphs (whose vertices are all unnamed) that could be either compatible or incompatible graphs.Since all the vertices are unnamed, the distance could not be defined as the one in the labelled cases.Hence all the possibilities of the interactions between  1 and  2 (whose vertices are all unnamed) should be taken into consideration.However, we could fix one graph, say  1 , and permute  2 .Then one computes all the possible distances and chooses the optimal permutation of  2 , in the sense of minimal distance.Let   () denote the -th graph whose vertices are identical to  with -permutation of the names for the vertices. indeed is treated as a naming system.There are | 2 |! ways of assigning the names to the unnamed set  2 .Based on metrics   ,   ,  ℎ , we define the unlabelled distances as follows.
Proof.They follow from the definitions.For a full proof, one could also consult the results in [1,8].

Computations and Implementations
In this section, we show how to implement all the abovementioned metrics via adjacency matrices.Let  = {V 1 , V 2 , V 3 , V 4 , V 5 }.Suppose G() is the set of all the graphs whose vertices are comprised of part or all of .Let  1 ,  2 ,  3 ,  4 ∈ G() defined as follows.
1 = ( 1 ,  1 ,  1 ),  2 = ( 2 ,  2 ,  2 ), 3 = ( 3 ,  3 ,  3 ), and  4 = ( 4 ,  4 ,  4 ), where  1 =  2 = ,  3 = {V 1 , V 3 , V 4 }, and  4 = {V 2 , V 3 , V 4 };   = (   ); and  1 ,  2 ,  3 ,  4 are defined as follows: . .Computations for Labelled Full Graphs.We call  1 and  2 full graphs.Based on the definitions regarding their components of distances, we have the following computations (which correspond to the original definitions, in matrix ] and   ( 1 , [ ] and   ( 1 , . .Computations for Labelled Subgraphs.We call  3 and  4 subgraphs.To begin with, we show how to compute the distance between a full graph and a subgraph.Their interactive components of distances are illustrated in the following matrix form: ] and   ( 1 , ] and   ( 1 , Now we show how to implement the computation of the distance between any two subgraphs via a matrix form and its norm as follows: . .Computations for Unlabelled Full Graphs.Suppose the vertices in  1 ,  2 ,  3 ,  4 are all unnamed.To compute the distance between unnamed  1 and  2 , according to the definition, we need to pick the smallest distances between  1 and   2 () for 1 ≤  ≤ | 2 |!.To implement this, we fix the adjacency matrix of  1 and permutes the adjacency matrix of  2 and compute all the respective distances and then choose the least one and its resulting permutation.Through computation, we have the following result: By setting   2 () as follows: Similarly, by setting   2 () as follows: . .Computations for Unlabelled Subgraphs.As for the   ( 1 ,  3 ), after our computations, the optimal corresponding subgraph in  1 is the truncated one with vertex lying in {V 1 , V 3 , V 4 }; i.e., the optimal subgraph with respect to for some unique , one has   * ( 3 ,  4 ) =   ( 3 ,   4 ()) = 36.By setting   4 () as follows: To sum up all the results regarding different metrics and graphs, we have Table 1.
There are several observations worth mentioning: (1)   * ≥  ℎ * ≥   * ; (2) the distance between a full graph and a subgraph is higher than either the distances for full graphs or the subgraphs; (3) the distance between unlabelled graphs is less than or equal to the one between labelled ones.
All these results agree with our theoretical definitions and derivations.

Real World Application
In this section, we demonstrate how to make a decision via our derived metrics when facing some uncertain situation in the real.Suppose country B's strategic deployment of air planes depends on country A's attack force.The degree of attack force ranges from 0 to 100, in which 0 indicates that there is no loss in the combat while 100 indicates the opponent's airbase is completely wiped out.Suppose A and B both have fives airbases in other countries C1, C2, C3, C4, and C5.Suppose A has 6 types of air planes A1, A2, A3, A4, A5, and A6.The number of each type of planes installed across different countries is listed in Table 2. Their respective attack forces are 30, 45, 55, 76, 88, and 97.Suppose B's observation of A's planes and air flight of his planes from one base to other bases is recorded in Tables 3 and 4. The potential flights from The optimal decision for B to counterbalance A is argmin { V (, B) : B ∈ B} , where V ∈ {,,ℎ}.Their individual solutions could be obtained via integer programming.Here we omit the final execution.If there is inconsistency between the choices of V, one could resort to subjective judgement or assigning weights between   ,   and  ℎ to reach a final decision.

Conclusion
In this article, we have shown how to define distances between graphs over either a set of labelled or unlabelled vertices via a plethora of metrics for graphs.We also give computational approaches to implement the computation of these metrics via the operations on adjacency matrices.This implementation gives an efficient and fast computation of the distance between any two such graphs.We also demonstrate how to apply these metrics in uncertain decision-making.Indeed, these metrics could be further applied in measuring the distance between real networks or tree-like structures.

Table 2 :
Country A's installed air force.

Table 3 :
Country A's potential air flight.

Table 4 :
Country A's potential attack force.

Table 2 .
This table could be directly converted into an adjacency matrix (named ), in which "none" is replaced by 0 and contents in each   is rewritten in the forms of sets.Suppose B has 7 types of air planes: B1, B2, B3, B4, B5, B6, and B7.Their respective attack forces are 28, 41, 46, 52, 61, 70, 86.Suppose B owns  1 = 6, 2 = 4, 3 = 10, 4 = 7, 5 = 10, 6 = 22, and  7 = 11 planes for each corresponding type.Now the problem for B is how he should send his air planes to counterbalance his opponent.Based on the metrics in this article, we could make a decision toward such uncertain situation.Let    ,    denote the numbers of air flight of A's and B's air planes  from airbase   to   .Define