The Logic of Homophily Dynamics in Heterogeneous Networks: Axiomatization, Model Checking and Validity Checking

: Social networks have received considerable attention from the modal logic community. In this article, we study and characterize one of the most important principles in the ﬁeld of social networks. Homophily, which means similarity breeds association, reveals the nature of social organization. In order to be able to express similarity and association together, we generalize the basic network and then deﬁne the heterogeneous network. The heterogeneous network is also deﬁned to provide a good foundation for the use of logical approaches. The Logic of Homophily LH G , M that we propose in this article is based on Computation Tree Logic and Formal Concept Analysis. LH G , M describes the homophily dynamics of the heterogeneous networks at a speciﬁed similarity coefﬁcient. Furthermore, we not only axiomatize the LH G , M and prove that the axiom system LH nG , M is sound and complete, but we also prove that the model checking and the validity checking for LH G , M are both PSPACE-complete.


Introduction
With the rapid development in recent years, social network analysis (SNA) has been considered as an effective tool for solving various problems in sociology, economics, and management. As a model that can intuitively represent knowledge, beliefs, and social problems, social networks also received a lot of attention from modal logicians (e.g., [1][2][3][4][5][6][7]). In this article, we will characterize one of the most influential principles of social networks, homophily, from the perspective of logic.
Homophily describes the basic social organization rule that similarity breeds association [8]. In other words, those who have more of the same attributes will be more likely to tend to associate with each other. For example, people that have the same interests, have the same academic background, and have the same hometown are more likely to become friends.
In short, a general network is a structure composed of an object set and a relation between objects, which is usually depicted by a graph, in which objects and relationships are represented by nodes and edges. For example, in Figure 1, objects 1, 3 and 4 are, respectively, associated with 2. Similarly, the affiliation network is a structure composed of an object set, an attribute set and a relation between object and attribute. For example, in Figure 2 (for all figures in this article, unshaded circles represent objects, shaded circles represent attributes, straight lines represent relationships between objects and attributes, and curved lines represent relationships between objects), the object 1 has the attribute a and b, both the object 3 and 4 have the attribute c. However, neither of the two most common types of networks can be directly used to represent homophily since they can only represent association and similarity, respectively. In order to study homophily on the basis of networks, sociologists often use additional data tables and some statistical methods,   In this article, the logic of homophily will be based on the HetNets to study and describe the dynamics of networks under the principle of homophily. In a HetNet, if a pair of objects has a certain number of the same attributes, it is considered that the two are similar in a certain degree, and thus they will be associated with each other. If in the HetNet, all similar objects have already been associated with each other, it is considered that the network is homogenized.
The most related work to this article is logic of allies and enemies (LAE) [5][6][7]9]. LAE is based on computation tree logic (CTL), and mainly characterizes another important structural theory in social networks-structure balance theory. Similarly, the logic of homophily in this paper will be also based on CTL to characterize the dynamics of the HetNet under homophily. In addition, logic of homophily will introduce the methods in formal concept analysis (FCA) to represent and extract the implicit information of the HetNet.
Previous logic study on social network structure is only about the structure balance theory and stays on the general object networks. In this article, we generalize the logic study of the structure of social networks to HetNets and the principle of homophily. We first combine general object networks with affiliation networks to propose HetNets that can be used to characterize the principle of homophily; secondly, we define similarity and homophily in HetNets and construct the homophily-evolution models; thirdly, we establish the logic of homophily, which is based not only on CTL but also on FCA, and is, therefore, more expressive, to describe and analyze the HetNet and its homophily dynamic; fourthly, we axiomatize the logic and prove that its axiom system is sound and complete; finally, we propose the model checking and validity checking problem and prove that they are both PSPACE-complete by improving the algorithms.
The structure of this article is as follows: In Section 2, we first introduce the definition of HetNet and define the similarity and homophily on it, and then establish the homophily-evolution model to describe the dynamics. In Section 3, we first introduce the implication between attributes on HetNet, then define the language and semantics of logic of homophily LH G,M , and give some valid formulas. In Section 4, we first introduce the description formula of the HetNet, and then define and explain the axiom system LH n G,M , and finally prove the soundness and the completeness of LH n G,M . In Section 5, we express what the model checking for LH G,M is and then prove it is PSPACE-complete. In Section 6, we express what the validity checking for LH G,M is and then prove it is PSPACE-complete. In Section 7, we summarize our work in this article and illustrate the directions for future work.

HetNets and Homophily
Definition 1. HetNet is a quadruple N = (G, M, I, R), where G and M are non-empty finite sets, I ⊆ G × M, R ⊆ G × G, and R is reflexive and symmetric.
In the above definition, requiring the relation R to be reflexive is to reflect that "the object is always associated with itself". In order to avoid ambiguity, we specifically require that G ∩ M = ∅. Then, in order to simplify the problem, we will only consider the case that object relations always appear in pairs, that is, R is required to be symmetric. G and M denote the object set and attribute set, respectively, I and R denote the relationship between objects and attributes and the relationship between objects, respectively.
As mentioned in Section 1, the HetNet given in the Definition 1 is one of the simplest and most intuitive heterogeneous networks and is a direct composition of a general network and an affiliation network. In other words, any HetNet N = (G, M, I, R) can be regarded as composed of the affiliation network (G, M, I) and the general network (G, R). For example, the network in Figure 3 is combined by the network in Figure 1 and the network in Figure 2.

Similarity and Homophily
Consider sets X, Y. ↑: 2 X × 2 X×Y → 2 Y and ↓: 2 Y × 2 X×Y → 2 X , respectively, denote the following operators (where,A ⊆ X, B ⊆ Y, E ⊆ X × Y): For the sets X, Y, if the relation E is fixed, this pair of operators can be regarded as a pair of concept-forming operators in FCA.
FCA, first proposed by Rudolf Wille in [10], is a formal tool that has a solid theoretical foundation and is widely used in various areas, e.g., knowledge representation, information management, and data mining. Not only is the the basic structure in FCA, called formal context, isomorphic to the affiliation network, but the methods of FCA can also be used to represent and extract implicit information in networks [11,12]. Although our work is heavily inspired by FCA, in this article, we will only introduce FCA where necessary since this work has not yet covered the FCA methods in depth (See [13] for more details about FCA). Definition 2. Consider N = (G, M, I, R) and O ⊆ G, the similarity of the object set O is defined as Sim(O) = |O ↑ I |. Let 0 < n ≤| M | be a similarity coefficient. If Sim(O) ≥ n, the objects in set O are considered to be n-similar.
Intuitively, O ↑ I represents the common attributes in M of objects in O, and the more they have the same attributes, the more similar they are and the higher the similarity coefficient the set has. For example, in Figure 3, Sim({1, 2}) = |{a, b}| = 2, Sim({3, 4}) = |{d}| = 1. Therefore, objects 1 and 2 are 2-similar and also 1-similar, but objects 3 and 4 are only 1-similar. In other words, objects 1 and 2 are more similar than 3 and 4. Since ∅ ↑ I = M, the empty set always has similarities and has the largest similarity, For any g ∈ G, {g} ↑ I = {m ∈ M | gIm}, so if one object has attributes, it is always the most similar to itself.
From the perspective of FCA, · ↑ I is one of concept-forming operators of formal context K = (G, M, I) that is isomorphic to the component affiliation network (G, M, I) of HetNet (G, M, I, R), and any set of objects with similarity is a subset of the extension of one certain maximal non-empty formal concept in K. Therefore, as long as all the maximal non-empty formal concepts are obtained, all object sets with similarity in a HetNet can be also obtained. Existing polynomial-delay algorithms for computing formal concepts can do this efficiently (see [14] for details). In the above definition, · ↑ R can be understood as one of the concept-forming operators of formal context (G, G, R) that is isomorphic to the component general network (G, R) of HetNet (G, M, I, R), and O ↑ R represents the objects that are associated with all objects in O. Similarly, the empty set is always homogenized since ∅ ↑ R = G. And because R is reflexive, any single-point set with similarity is always homogenized. In particular, if a HetNet does not have any n-similar set, then it is vacuously homogenized at n. Take Figure 3 as an example. Since {3, 4} ↑ R = {3, 4} and {1, 2} ↑ R = ∅, {3, 4} is homogenized at 2 but {3, 4} is not even homogenized at 1.
The principle of homophily describes the process of individual aggregation and organization caused by similarity. In other words, homophily shows that the network has a tendency to be homogenized. (1) If N is homogenized, then N = N; (2) If N is not homogenized, there is only one n-similar object set {i, j} ⊆ G, where i, j are different objects, satisfying that the object set is not homogenized on N, but homogenized on N , i.e., Sim({i, j}) ≥ n, (i, j) ∈ R, (j, i) ∈ R, (i, j) ∈ R and (j, i) ∈ R .

Homophily-Evolution Models
The Definition 4 means that each update of the non-homogenized HetNet can and can only make a pair of objects with n-similarity associated. Although it is more intuitive to associate objects with higher similarity first, the updates are "equal" to each pair of objects that will be homogenized since we do not intend to discuss this issue in depth in this article. Take Figure 3 as an example. The HetNet in Figures 3-5 are its 1-immediate successor, 2-immediate successor and 3-immediate successor, respectively. The update made in Figure 4 is to associate object 2 with 3 since they have one common attribute b. The update made in Figure 5 is to associate object 1 with 2 since they have two common attributes a and b. In other words, the HetNet in Figure 5 is also a 1-immediate successor. Because there are no 3-similar sets in Figure 3's HetNet, it is homogenized at 3 and thus its only 3-immediate successor is itself.  Definition 5. Consider HetNet N = (G, M, I, R) and 0 < n ≤| M |. If there is a sequence N 0 n N 1 n · · · n N m (m is a positive integer) such that N 0 = N, and either m = 0, or for any 0 ≤ i < m, N i is not homogenized, then if and only if N m is homogenized, we call this sequence the n-homophily-evolution line of N (n-timeline for short, denoted l n (N)) and call N i the n-successor of N. The n-homophily-evolution tree t n N = (N , S) of N (n-timetree, for short) is a tuple, where N = {N | N is the n-successor of N}, S = {(U, V) | U, V ∈ N and U n V}.
By the immediate successor relation given in Definition 4, a HetNet will gradually form an n-timetree, which is non-transitive, antisymmetric, and has self-loops only at the terminal nodes. Since the network will eventually reach homogenized, the depth of this timetree is finite. Let len(l n ) and dep(t n ) be the length of n-timeline and the depth of n-timetree, respectively. len(l n ) is the number of HetNets in an n-timeline and dep(t n ) is the length of the longest n-timeline in it.

Proposition 1. The depth of n-timetree is limited by
For a HetNet N = (G, M, I, R), only when any two of the objects are not associated but are all n-similar, the number of updates of N can reach the maximum.

Logic of Homophily LH G,M
In this section, we will introduce the implication between attributes (IbA) and generalize it on HetNet, and then establish the Logic of Homophily LH G,M .

Implication between Attributes and Index
IbA, or the IbA system is an important method in FCA, which is used to represent and extract information from formal contexts. As the "language" of formal contexts, the IbA system describes and characterizes a formal context by expressing the "implicative relationship" between attributes. In other words, "the attribute set A "implicates" the attribute set B" actually means" any object that has all the attributes in A also has all the attributes in B". Consider an affiliation network or a formal context (G, M, I) and attribute sets A, B ⊆ M. An IbA of the formal context is denoted by Since the HetNet and the homophily principle involve not only the relationship between objects and attributes, but also the association between objects, IbA in FCA cannot be directly used to describe the HetNet and its homophily. However, because the formal context is isomorphic to the affiliation network, and the general network (G, R) can also be expressed in the form of the affiliation network (G, G, R), i.e., the HetNet can also be similarly expressed, the IbA method can be naturally generalized to the HetNets.
In definition 6, IR is shorthand for I ∪R and This type of abbreviation in the following also means the same. In addition, for the convenience of writing and avoiding ambiguity, brackets around sets in IbA are usually omitted and the string will be arranged in the order of numbers, indexes, and letters, and the numbers and letters will be arranged in the order of natural numbers and alphabets, respectively. For example, we write 13a ⇒ bd as an IbA instead of {1, a,3} ⇒ {b, d}.
The generalization made by Definition 6 is essentially to use the symbols in the object set and index set to represent the attributes of being associated with the "object represented by this symbol" and attributes of only being associated with the "object represented by this symbol", respectively. For example, IbA 1 ⇒ a means "the objects associated with object 1 all have the attribute a", and "1" in this IbA means the attribute "has a relationship with the object 1", instead of "the object 1".1 ⇒ b means "the object only associated with 1 has the attribute b", and since all objects in the HetNet are associated with themselves,1 ⇒ b actually means "object 1 has attribute b".

Syntax and Semantics
The language L G,M of logic of homophily LH G,M is built on the given object set G and attribute set M (where, G and M is finite, non-empty and satisfies The formula A ⇒ B represents the IbA A ⇒ B, Hom represents the HetNet is homogenized, and the propositional connectives and temporal operators are consistent with the conventional definitions (more information about CTL temporal operators, please refer to [15,16]). Other propositional connectives and temporal operators are also defined as usual. For example, φ → ψ = ¬φ ∨ ψ, EXφ = ¬AX¬φ, AFφ = A( Uφ). In addition, also for the convenience of writing, in the following, the expression of "atomic formula" A ⇒ B is consistent with the specification of IbA in the Section 3.1.
Consider LH G,M . The formula φ ∈ L G,M is satisfied on a HetNet N = (G, M, I, R) and at a similarity coefficient n (denoted as N n φ) defined as follows (where A, B ⊆ G ∪ M∪G, ψ ∈ L G,M , 0 < n ≤| M |): Take Figure 3 as an example to briefly explain the semantics. Let N denote the network in it. First, N n 2a ⇒ b, N n ¬4 ⇒ b, N n ¬c ⇒ b represent "all objects associated with object 2 that have attribute a have attribute b", "object 4 does not have attribute b", and "there is an object that has attribute c and does not have attribute b". Then, N 1 EX(1 ⇒ 2) and N 1 ¬Hom represent, respectively, "there is a 1-immediate successor of N such that object 1 is associated with object 2" and "the HetNet N is not homogenized at 1". Next, N 1 A(¬1 ⇒ 3 ∨ ¬1 ⇒ 2UHom) means "On any timeline starting from N with the similarity coefficient 1 until the network homogenized, there is always that 'object 1 has no association with 3 or object 1 is no associated with 2'". Finally, N 2 ¬EX(1 ⇒ 2) and N 3 Hom shows that "there is not a 2-immediate successor of N such that object 1 is associated with object 2" and "the HetNet N is homogenized at 3". Given The three valid formulas in the Proposition 2 state the following: (1) The HetNet will eventually be homogenized; (2) The update will not change the existing relationship; (3) The update will not generate new relationships between objects and attributes.

Axiom System LH n G,M
The axiomatization of the logic of homophily LH G,M relies on the description method first introduced by [9]. In short, the description is a formula used to characterize the HetNet. The description formula in this article is based on the regular atomic formula. Consider is a conjunction defined as follows: For any g ∈ G, A g represents the subset of G ∪ M only determined and selected by g. The description, denoted as δ, is composed of the following series of regular formulas and their complements: For the conjunction ψ in which each branch is a regular atomic formula, let ∼ ψ represent the conjunction of the complement of all branch formulas in ψ. Let δ + be the conjunction of all regular atomic formulas in δ, and ∼ δ + be the conjunction of complements of all its regular atomic formulas, and obviously δ ↔ δ + ∧ ∼ δ + .
For the HetNet N = (G, M, I, R), the description of N, δ(N) is defined as follows (the order of branches of conjunction is determined and IR is shorthand for I ∪ R): Since in a HetNet, objects are always associated with themselves and the associations between objects are always symmetric, not all description formulas describe a HetNet. In this article, we specifically stipulate the order of branches of conjunction in the description formula, i.e., the atomic formula is always on the left side of its complement, arrange the atomic formulas in the order of natural numbers according to the object indexes in them, and arrange the branches of complements of atomic formulas in the order of natural numbers and alphabetical order according to the symbols on the right of ⇒. Therefore, it is not difficult to prove that if δ(N) is the description of HetNet N, δ(N) is the unique description of N. Take Figure 3 as an example. Let N denote the network in it. Its description formula Table 1 represents the axiom system LH n G,M of the logic of homophily LH G,M with the similarity coefficient n. In Table 1, φ, ψ, χ are arbitrary formulas, A, B, C, D ⊆ G ∪ M ∪G, a ∈ M, E n ⊆ M and | E n |≥ n, g, g ∈ G,g,g ∈G,O ⇒ A and ∼O ⇒ A, respectively, represent g∈Og ⇒ A and g∈O ¬g ⇒ A (O is not empty and O ⊆ G), σ (g,g ) means a∈M (g ⇒ a ↔ ¬g ⇒ a), δ represents the description of a HetNet, and homophily axiom is the following formula (where δ 1 , δ 2 , . . . , δ m enumerate the descriptions of all HetNets that are homogenized at n):
The axiom system LH n G,M consists of three parts: CTL axioms and rules, axioms about IbAs and axioms and rules about HetNet and update.
CTL axioms and rules are consistent with those in [16].
In the axioms about IbAs, (Ref), (Frag), (Comp), and (Trans) are the axioms of Armstrong's system (see [17] for more details). , (E-gen) and (Exp). However, for the convenience of the follow-ing proofs, these two formulas are still retained as axioms. (HF), (Tm 1 ) and (Tm 2 ) are consistent with the similar axioms and rules in [9].
Consider a formula set Φ and a formula φ. If φ can be derived by axioms and rules of LH n G,M , φ is a theorem of LH n G,M (denoted n φ). If φ can be derived from Φ by axioms and rules of LH n G,M , Φ n φ.

Soundness
We will prove the validity of all other axioms and rules except the CTL axioms and rules and Armstrong's axioms which have been proven to be valid. Next, a lemma used to assist the following proofs will be first proved. Proof. By definition, left to right obviously holds. Only from right to left is shown here. Consider the description δ = g∈G (g ⇒ A g ∧ ∼ (g ⇒ A g )), the HetNet N = (G, M, I, R) and the similarity coefficient n. If δ does not describe N, then there exists g ∈ G such that A g = {g } ↓ IR , i.e., either A g ⊂ {g } ↑ IR , or {g } ↑ IR ⊂ A g . For the former, according to the definition of ∼ (g ⇒ A g ), there must exist a ∈ {g } ↑ IR so that ¬g ⇒ a is a branch of δ. Therefore, N n δ. Similarly, for the latter, there must exist a ∈ A g such that N ng ⇒ a (otherwise, a ∈ {g } ↑ IR ). Thus, N n δ.
Here, we only show the axioms and rules that should be proved, and the proof will be given in the Appendix A.

Completeness
In this section, we will first prove that the tomorrow formula is the theorem of LH n G,M . Tomorrow formula, denoted as (t(δ)), is the following formula: δ 1 , . . . , δ n is the description of the n-immediate successor of the HetNet described by δ.
Proof. Consider a description δ which describes a HetNet and a similarity coefficient n.
Proof. According to the definition of description, the description is obviously consistent, so n δ → φ and n δ → ¬φ cannot be held at the same time.
Applying induction on φ to prove that for any description δ and formula φ, either n δ → φ or n δ → ¬φ must hold.
(1) For any δ and an atomic formula in the form ofg ⇒ A (where g ∈ G, A ⊆ G ∪ M ∪G), either n δ →g ⇒ A or n δ → ¬g ⇒ A must hold. Let ψ be an atomic formula in the form ofg ⇒ A. If ψ is the conjunction branch of δ + , or can be derived from one of the conjunction branches by (Frag), then n δ → ψ. According to (Frag) and (Comp), n ψ ↔ a∈A (g ⇒ a). If ψ is not a conjunction of δ + and cannot be obtained by (Frag), then there is at least one conjunction branch in a∈A (g ⇒ a) that also satisfy this condition. Letg ⇒ a be the formula that satisfies the condition. Suppose thatg ⇒ a isg ⇒g. If there is no other formula that satisfies the condition, according to (Ref), n δ → ψ. If there are other formulas, it is obviously impossible to beg ⇒g. Ifg ⇒ a is of the formg ⇒g (where g ∈ G and g = g ), then according to (Frag) , (Loop) and (Loop ¬ ), n δ → ¬ψ. If it is not the above two cases, then according to the definition of δ, it is easy to know that ¬g ⇒ a must be the conjunction branch of ∼ δ + . Therefore, n δ → ¬g ⇒ a , i.e., n δ → ¬ψ. In summary, (1) holds.
(2) Consider δ and non-empty sets O ⊆ G and A ⊆ G ∪ M∪G.
Consider LH G,M . For any maximal consistent set Φ, the relation in the HetNet N Φ = (G, M, I, R) of Φ is defined as follows (where g, g ∈ G, m ∈ M andg ∈G): For any β m , we should only consider the m-step n-successors of the root HetNet and since ¬α i+1 should be satisfied first in every update, for any 1 ≤ j ≤ m, it is impossible for α 1 j and α 0 j to be satisfied at the same time in the above m-step n-successors. Therefore, N n m n β m if and only if ψ m is satisfied. For the HetNet N defined above, dep(t n (N n m )) = 2m + 1. And the length of β m is also linear-related to m. The above reduction is thus a linear-time reduction. The model checking problem for LH G,M is PSPACEhard.
Lemma 9. The model checking problem for LH G,M is in PSPACE.
Proof. First, determining whether an atomic formula is satisfied on a HetNet or not requires only a few basic calculations on sets. In addition, the model checking algorithms for basic propositional connectives are polynomial-time. Therefore, what we should consider carefully here are Hom and temporal operators.
Consider a HetNet N = (G, M, I, R) and a similarity coefficient n. The Algorithm 1 can determine whether N n Hom or not. Since Algorithm 1 performs at most |G|(|G|−1) 2 + 1 calculations, it is in PSPACE. The algorithms for temporal operators are based on the algorithms for CTL given in [19] since the temporal operators in LH G,M come from CTL. Here, we only give the algorithm of E(φUψ) (Algorithm 2) as an example and other cases are shown in Appendix B.

Algorithm 1: Model Checking Algorithm for Hom
According to Algorithm 2, it is not difficult to find that it is a recursive algorithm. Each run of the algorithm either shortens the input formula or makes input HetNet "jump to" one of its immediate successors. Therefore, since the formula is finite, and the HetNet will finally reach homogenized, the algorithm will eventually halt.
The model checking algorithms for CTL temporal operators are all in PSPACE with respect to the size of the input computation tree and the length of the input formula. However, the size of the computation tree of an input HetNet (G, M, I, R) is at most ( |G|(|G|−1) 2 + 1)!, i.e., the growth of the size of the computation tree with that of the HetNet can even be greater than exponential. To reduce storage space usage, we use the depth-first algorithm like Algorithm 2, and release the storage space of the non-overlapping part of the old branch before starting to calculate a new branch. In other words, we do not need the storage space that can store the whole time tree, but only the space that can store the longest timeline, i.e., the size of the storage space is linear to the depth of the tree.  Proof. Consider a QBF ψ m = Q 1 x 1 Q 2 x 2 · · · Q m x m φ(x 1 , x 2 , · · · , x m ), where m is any positive integer, Q i = ∃ or ∀, φ is a propositional formula, and the value range of x 1 , x 2 , · · · , x m is {0, 1}, and a positive integer n. Let ). It is obvious that δ n m is the description of the HetNet N n m in the Proof of Lemma 8. Here, we take the value of the similarity coefficient as n. Since | M |= 2 mn and hence 1 ≤ n ≤| M |, if N n δ m , N n β m if and only if ψ m is satisfied (β m is the same as that defined in the Proof of Lemma 8). Therefore, n δ n m → β m if and only if ψ m is satisfied. The validity checking for LH G,M is PSPACE-hard.

Lemma 11. The validity checking for LH G,M is in PSPACE.
Proof. When non-empty finite sets G and M and a similarity coefficient n are given, the number of HetNets composed of G and M is determined and apparently finite. Let N be the set of all those HetNets. We can check whether n φ by enumeration, i.e., we can use model checking algorithms to check whether N n φ for all N ∈ N . Since the model checking algorithm is in PSPACE and the storage only needs to keep one HetNet during each model checking process, the validity checking for LH G,M is in PSPACE.
Theorem 5. The validity checking for LH G,M is PSPACE-complete.

Conclusions
In this article, we explore the homophily dynamics of HetNets at a determined similarity coefficient from the perspective of logic. More precisely, we establish the Logic of Homophily LH G,M , which describes the homophily dynamics, construct a sound and complete axiom system LH n G,M , and propose the PSPACE-complete model checking and validity checking algorithms for LH G,M . IbAs in FCA are actually some first-order formulas constrained by universal quantifiers. Index expansion enables these first-order formulas to represent "exists and only exists", and to represent the edges in a HetNet. The method of description formulas based on index expanded IbAs is the key to proving the completeness of axiom systems and solving the validity checking problem. The axiom system LH n G,M not only provides the axioms and rules that directly describe the update, but can also be able to prove the tomorrow theorems, which can only be axioms in previous work [9]. Finally, the model checking algorithms provide the basis for implementation.
There are two main directions for future work: 1.
The Logic of Homophily LH G,M is based on CTL and FCA. However, in this article, we only discussed the temporal logic aspect of LH G,M , but did not conduct an in-depth study of the aspect of FCA. Therefore, we will explore the properties of LH G,M from the perspective of IbA systems (e.g., Attribute Exploration) in the following work.

2.
Cognitive balance theory, as another important dynamic theory in SNA, is not only closely related to the theory of homophily, but we have also conducted some work on this theory (e.g., [9]). Thus, we will consider further exploring the principle of sentiment homophily by introducing cognitive balance theory. In other words, for sentiment homophily, only when two objects like or dislike a certain attribute at the same time, they can be associated with. Since sentiment homophily can better and more accurately describe the formation of object relationships, it is very important and meaningful to generalize the logic of homophily to the logic of sentiment homophily.
Funding: This research received no external funding.

Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Data Availability Statement: Not applicable.
Acknowledgments: Thanks in advance to the anonymous reviewers for their help with the article.

Conflicts of Interest:
The author declare no conflicts of interest.

Abbreviations
The following abbreviations are used in this manuscript: Proof. Consider a HetNet N = (G, M, I, R) and a similarity coefficient n.
(Frag ¬ ) Suppose N n ¬A ∪ B ⇒ C. There is a g ∈ G s.t. g ∈ A ∪ B ↓ IR and g ∈ C ↓ IR , i.e., g ∈ A ↓ IR and g ∈ C ↓ IR . Therefore, N n ¬A ⇒ C.

(Comp ¬ )
Suppose N n ¬A ⇒ B ∧ ¬A ⇒ C. There is a g ∈ G s.t. g ∈ A ↓ IR , g ∈ B ↓ IR and g ∈ C ↓ IR i.e., g ∈ A ↓ IR and g ∈ B ∪ C ↓ IR . Therefore, N n ¬A ⇒ B ∪ C. (Incl) Suppose N n A ⇒ B ∧ ¬A ⇒ C. There is a g ∈ G s.t. g ∈ A ↓ IR , g ∈ B ↓ IR and g ∈ C ↓ IR Therefore, N n ¬B ⇒ C.

(Abs)
Suppose N nO ⇒ A ∧O ⇒ B∧ ∼(G \ O) ⇒ A. By the definition ofO, for any o ∈ O, N no ⇒ A ∧o ⇒ B and for any o ∈ G \ O, N n ¬o ⇒ A. Therefore, A ↓ IR = O and O ⊆ B ↓ IR , i.e., A ↓ IR ⊆ B ↓ IR . Thus, N n A ⇒ B.

(Symm)
Since R is symmetric, N ng ⇒ g ↔g ⇒ g. (A-gen) Suppose N n ¬AXg ⇒ A. N n EX¬g ⇒ A. There is an n-immediate successor of N, N = (G, M, I, R ), s.t. N n ¬g ⇒ A. Therefore, g ∈ A ↓ IR . Since R ⊆ R , i.e., A ↓ IR ⊆ A ↓ IR , g ∈ A ↓ IR . Thus, N n ¬g ⇒ A. (A-gen ¬ ) Suppose N n ¬g ⇒ a. Since for any n-immediate successor of N, I keeps the same. Therefore, N n AX¬g ⇒ a.

(E-gen)
Suppose N ng ⇒ E n ∧g ⇒ E n . By the definition of satisfication, {g, g } is n-similar on N. If (g, g ) ∈ R, N n EXg ⇒ g by (A-gen). If (g, g ) ∈ R, there is an nimmediate successor N = (G, M, I, R ) of N satisfying (g, g ) ∈ R , i.e., N ng ⇒ g and N n EXg ⇒ g .

(HF)
Since the n-immediate successor of a HetNet that is homogenized at n is itself, the homophily axiom is valid.