Stability in social networks

Dunbar’s number is the cognitive limit of human beings to maintain stable relationships with other individuals in their social networks, and it is found to be 150. It is based on the neocortex size of humans. Usually, Dunbar’s number and related phenomena are studied from the perspective of an individual. Dunbar’s number also plays a crucial role in evolutionary psychology and allied areas. However, no study done so far has considered a couple who are in a stable relationship as a system from the perspective of Dunbar’s number and its hierarchy layers. In this paper, we study the impact of Dunbar’s number and Dunbar’s hierarchy from the perspective of a couple by studying mathematically the conjoint Dunbar graphs for a couple. The cost of romance is the loss of almost two people from one’s support network when a human being enters into a new relationship. Thus, we obtain mathematically that there is no significant change in one’s friendship if human beings spend negligible time with their partners. Also, along with marriage and friendship development, we attempt to assess how a person’s social network structure holds up over the course of a romantic relationship. The stability of personal social networks is discussed through soft set theory and balance theoretic approach.


Introduction
The single most significant factor affecting our health, pleasure and well-being is friendship.Humans are social creatures, and the relationships they build with one another are essential to their growth and welfare in a society.However, the time required to establish and nurture friendships is quite expensive, as there are several cognitive processes that underlie them.In recent years, social networks have gained a lot of attention.The average size of a human's personal social network appears to be about 150 [1].These networks have a clearly defined layered structure with layers that are related fractally in a certain way.A human group size prediction is simple to make.In doing so, which is only a matter of extrapolating a value for group size from the primate equation using the human neocortex volume, a number in the range of 150 is obtained [2].This value is also found in face-to-face contacts [3], calling patterns in cellphone databases [4], or postings in online environments [5], hunter-gatherer communities [6], Facebook and Twitter networks [7,8], email networks [9], co-authorship networks in scientific collaboration [7], alliances in online gaming environments [10], etc.
The variation of a personal network mainly depends on how extroverted or introverted a person is [11][12][13][14].Compared to an introvert, an extrovert may have a few more friends, but the level of friendship depends on the size of the network of friends a person has, i.e. on average, people with a wider social network have weaker bonds [11][12][13][14].Similar results were also observed for introverts and extroverts on many social networking sites [15,16].A 20-30 year-old young person has a greater tendency to make friends than a 60 year-old person.This is due to the investment of time a person can afford for their friends.A 60 year-old person has less time to spend with his friends due to family obligations or other issues [3,17].The cognitive limits on the number of persons who can be known as individuals and the constraint that time imposes on the ability for interaction both contribute to the limitations on network size and structure.Burton et al. [1] mentioned that these impacts are dependent on age as well, especially impacting individuals under the age of 36.The support network is even more severely diminished for people who have children.The cost of romance is the loss of nearly two members when one accounts for the addition of a new member to the network when beginning a relationship [1].These social expenses are typically distributed evenly among network members who are related and unrelated.Dunbar [17] mentioned seven pillars of friendship, which are seven attributes that people hold in their legacies.Also, all the factors that affect the formation and decay of friendship can be considered as attributes.Since the theory of soft sets deals with attribute based approximations of objects or events [18][19][20], we find it the most suitable tool for discussing the whole scenario using soft sets.We have made a detailed analysis of these events, such as friendship, stability of friendship, marriage and divorce in the later sections.We focus mainly on the following questions: (i) How do the Dunbar's layers of friendship work when a person gets married, i.e. the study of the Dunbar graph for a couple?(ii) How can we measure the bonding of friendship between two people mathematically?(iii) How does time investment affect the relationship between friendship and romance?(iv) How do personal social networks develop, and how big can a network get before it starts to become unstable due to too many weak links, given the distribution of the seven pillar qualities in the population?(v) What is the likelihood of a married couple having extramarital affairs?

Soft sets and conjoint Dunbar graphs
We cannot effectively handle complex problems in social science, economics and the environment with classical methods due to a number of inherent uncertainties.However, a potentially useful general mathematical tool for handling ambiguous, fuzzily specified things is provided by soft set theory.In this paper, we will look at how numerous social elements affect people's connections with one another.The theory of soft sets, which deals with the parameters that influence the approximations of an attribute through mapping, offers a practical solution for this.Thus, we consider the concept of soft set theory to study the relationship between friendship and romantic relationships and the stability of one's personal social network with related events.Definition 2.1.( [18]) A pair (F, A) is called a soft set over the universal set X if and only if F is a mapping of A into the set of all subsets of the set X, i.e.F : A → 2 X .In other words, the soft set is a parametrized family of subsets of the set X.
Since parametrization is an auxiliary factor for the convenience of working with soft sets, it is natural to introduce the notion of equivalence of soft sets.If the soft set (F, A) is given, then the family τ(F, A) = {F(a) : a ∈ A} specifies those subsets that can be approximate descriptions, and the parameter set A is chosen for reasons of convenience by the person who introduces the definition of this soft set.
We procure some fundamental operations of soft sets below: royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 231500 It is easy to note that the equivalence of soft sets is an equivalence relation.

Definition 2.5. ([21])
The unary operation complement of (F, A) over a universal set X, denoted by C(F, A) = (W, A), is defined as follows.The set of parameters remains the same, and the mapping is given by WðaÞ ¼ X n FðaÞ for any a ∈ A. Definition 2.6.( [21]) The binary operation union ðS, AÞ < ðF, DÞ ¼ ðH, A Â DÞ for a pair of soft sets (S, A) and (F, D) given over a universal set X is defined as follows: The parameter set is chosen equal to the direct product of the parameter sets, i.e. equal to A × D, and the corresponding mappings are given by the formula Hða, dÞ ¼ SðaÞ < FðdÞ, (a, d) ∈ A × D. Definition 2.7.( [21]) The binary operation intersection ðS, AÞ > ðF, DÞ ¼ ðW, A Â DÞ for a pair of soft sets (S, A) and (F, D) given over a universal set X is defined as follows: The parameter set is chosen equal to the direct product of the parameter sets, i.e. equal to A × D, and the corresponding mappings are given by the formula Wða, dÞ ¼ SðaÞ > FðdÞ, (a, d) ∈ A × D. Definition 2.8.( [22]) The binary operation product (S, A) × (F, D) = (X, A × D) for a pair of soft sets (S, A) and (F, D) given over a universal set X is defined as follows.The parameter set is chosen equal to the direct product of the parameter sets, i.e. equal to A × D and the corresponding mappings are given by the formula X(a, d) = S(a) × F(d ), (a, d) ∈ A × D. Now, we consider an illustrative example to discuss the above operations.
Example 2.9.Let us construct two soft sets first.Consider the mappings F : A À ! 2 X and G : B À ! 2 X for two soft sets (F, A) and (G, B) over a universal set X. Let X contain people from a particular place, and let A, B be two sets of attributes.Consider X = {a, b, c, d, e}, A ¼ ftall, shortg, and B ¼ fgirl, boyg.For more on soft set theory, one may refer to [18,21,22].The hierarchy of friendship, as shown in figure 1, depicts the friends of a person depending on emotional closeness.The innermost layer of an ego consists of 1.5 people, as obtained by regression royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 231500 analysis [23], within hierarchically inclusive friendship circles or layers that have a very specific scaling ratio, i.e. each layer is three times the size of the one inside it.Hence, we are considering the layers starting from L 1 during the formation of friendships.Similarly, later on, in the case of the analysis of social networks formation with respect to a couple, we take the couple as an ego.Layer L 1 is made up of the person's closest acquaintances and is also referred to as their support system.Compared to one's acquaintances from layer L 1 , those in layer L 2 are those with whom the person has less emotional intimacy.In a similar manner, we obtain the other layers L 3 , L 4 , L 5 and L 6 etc. of Dunbar's friendship hierarchy.They denote the degree of a person's friendships with their peers in each circle.The 150 layer, which is the typical size for personal social networks, is denoted by the thick line.The layer at 1500 appears to reflect the average number of faces we are able to assign names to.There are acquaintances (500 layer) after this layer.Each succeeding circle sees a drop in contact frequency, assessed emotional intimacy, and readiness to act benevolently towards a particular alter.Next to the innermost layer, there are five people who are very closely emotionally connected to the person.This circle generally consists of immediate close family, best friends, or a romantic partner as well [17].As shown in the figure, as we go outward from the centre of the circle, the number of people increases but the emotional closeness decreases gradually.In this section, we try to connect two persons' friendship circles so that if the two persons get married, then how many friends they may have in common within a specific layer of the hierarchy of friendship is determined through some notions of soft set theory.For this, we define the following definitions: We first consider the operations for two soft sets (F, A) and (G, B) defined on the universal set X.The relationship between the parameters a ∈ A and b ∈ B will be described as an ordered pair  Now, let us analyse the situations mentioned above.We define soft sets (F, A) and (G, B) for two individuals X and Y, respectively, before they get into a relationship with each other.We consider the universal set U consisting of four main layers of Dunbar's hierarchy, i.e.U = {L 1 , L 2 , L 3 , L 4 }.Also, we consider the sets of attributes A and B consisting of the friends of X and Y, respectively, in Dunbar's hierarchy layers not exceeding Dunbar's number 150.For the sake of ease in our current analysis, we consider F(a 1 ) and F(a 2 ) as approximations of a 1 and a 2 ∈ A for the soft set (F, A).From this example, along with the definitions of various types of unions and intersections between two soft sets as defined above, we conclude the following: In the cases of left narrow union and left narrow intersection, we calculate the expression f> b[pðCon,a,:Þ GðbÞg, if pðCon, a, :Þ = ;, which gives us the layers in Dunbar's hierarchy that consist of the people with whom Y has less emotional closeness than the others.After that, the left narrow union Fða 1 Þ < f> b[pðCon,a,:Þ GðbÞg indicates the layers of Dunbar's friendship hierarchy for the couple C in which either X or Y decides to keep a 1 and the persons that lie in the layers we get while taking the intersection f> b[pðCon,a,:Þ GðbÞg.On the other hand, when we take the left narrow intersection Fða 1 Þ > f> b[pðCon,a,:Þ GðbÞg, it indicates the layers of Dunbar's hierarchy for C in which both X and Y decide to keep the person a 1 and the persons that lie in the layers we get while taking the intersection f> b[pðCon,a,:Þ GðbÞg.
In the cases of left wide union and left wide intersection, we calculate the expression f< b[pðCon,a,:Þ GðbÞg, if pðCon, a, :Þ = ;.It gives us the layers in the Dunbar hierarchy that consist of the people with whom Y has better emotional closeness than the others.After that, the left wide union Fða 1 Þ < f< b[pðCon,a,:Þ GðbÞg indicates the layers of Dunbar's hierarchy for the couple C in which either X or Y decides to keep a 1 and the persons that lie in the layers we get while taking the union f< b[pðCon,a,:Þ GðbÞg.On the other hand, when we take the left wide intersection From the above definitions, we see that the basic difference between left narrow union/intersection and right narrow union/intersection is that, for left narrow union/intersection we consider all the attributes from the soft set (F, A) and only those attributes of the soft set (G, B) which are connected to a ∈ A through the set Con and for right narrow union/intersection, we consider all the attributes from the soft set (G, B) and only those attributes of the soft set (F, A) that are connected to b ∈ B through the set Con. Similarly, for left wide union/intersection we consider all the attributes from the soft set (F, A) and only those attributes of the soft set (G, B) that are connected to a ∈ A through the set Con and for right wide union/intersection, we consider all the attributes from the soft set (G, B) and only those attributes of the soft set (F, A) that are connected to b ∈ B through the set Con.These scenarios are quite natural in the case of couple formation.
Table 1.Position of the individuals a 1 and a 2 in four layers Dunbar's hierarchy of the couple C from the perspective of the individual X.
One of the most important properties of classical union and intersection is commutativity.So, for a narrow union we have to compare two soft sets ðF, AÞ < À Con ðG, BÞ and ðG, BÞ < À Con 0 ðF, AÞ, where Con Naturally, there is no point in talking about the equality of these soft sets.It only makes sense to find out whether they are equivalent.Unfortunately, for operations: narrow union, wide union, narrow intersection, wide intersection, this property, generally speaking, is not fair.
Theorem 2.13.Suppose that the following conditions are satisfied for two soft sets (F, A) and (G, B) over a universe X and the set Con⊆ A × B.
(i) For ðF, AÞ > Now, let us consider two soft sets (F, A) and (G, B), respectively, for two persons with the parameter sets equal to the seven pillars of friendship [17], i.e.A = B ={ language, place of origin, educational history, hobbies/interests, sense of humour, worldview, musical tastes} and the universal set X is equal to the attribute values of the elements of A or B, i.e. the languages all over the world, the places all over the world, and so on.Here, we shall take the set Con In this case, for every a i ∈ A or B, i = 1, 2, …, 7, the set π(Con, a i ,.) is a singleton set.Then, we find the soft sets ðF, AÞ > À LCon ðG, BÞ or ðF, AÞ > þ LCon ðG, BÞ to know how much is common between the two people under consideration.The higher cardinality of the sets tððF, AÞ > À LCon ðG, BÞÞ or tððF, AÞ > þ LCon ðG, BÞÞ will indicate a higher possibility of the two people to become friends.Let us take the example below: Example 2.15.We take minimum number of attribute values for the universal set X so that the calculations get easier.We consider five languages: L 1 , L 2 , . .., L 5 , five places around the world: P 1 , P 2 , . .., P 5 , five educational institutions: E 1 , E 2 , . .., E 5 , five hobbies: H 1 , H 2 , . .., H 5 , two senses of humour: S 1 and S 2 , two political views: V 1 , V 2 , five musical tastes: M 1 , M 2 , . .., M 5 for generating the universal set X. Now we define the soft set (F, A) as follows: . Now, we construct the set Con as, Con ={(language, language), ( place of origin, place of origin), (educational history, educational history), (hobby, hobby), (sense of humour, sense of humour), (moral view, moral view), (musical taste, musical taste)}.We calculate the sets tððF, AÞ > À LCon ðG, BÞÞ ¼ ffL 2 g, fP 3 g, fE 3 g, fH 1 , H 3 , H 5 g, fV 2 g, fM 1 , M 3 gg and tððF, Here, higher cardinalities of the sets tððF, AÞ > À LCon ðG, BÞÞ or tððF, AÞ > þ LCon ðG, BÞÞ will indicate higher possibilities of creating a relationship between the two people under consideration.

Analysis of the formation of friendships and their effect on romantic relationships
In [17], Dunbar noted that friendship is the most important factor for our happiness, mental well-being, and health.He defined friends as the people who share our lives in a way that is more than just the casual meeting of strangers.It may include friends, members of our extended family and our romantic partners.Emotional closeness is defined as how an individual feels about another and can be determined by a variety of psychological instruments.Here, we try to find the emotional closeness or deepness of friendship between two individuals through some mathematical tools.
Dunbar [17] demonstrated how people who are friends often have a lot in common.Personal social networks are frequently homophilous by gender; for instance, men's networks tend to contain a disproportionate number of men, and women's networks tend to contain a disproportionate number of women [12].McPherson et al. [24] also identified several factors in homophilous networks, viz., gender, ethnicity, age, religion, education, social values, etc.Similarly, Newmann [25,26] studied assortative mixing in networks from the perspective of factors like language and race.Assortative royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 231500 mixing is the tendency for vertices in networks to be connected to other vertices due to homophily.He also found strong variation with assortativity in the connectivity of the networks.Dunbar [17] identifies seven dimensions as the foundation of friendship: language, place of origin, career trajectory, hobbies/ interests, sense of humour, worldview, and musical tastes.He also mentioned that these seven pillars seem to be interchangeable, and a four-star relationship can involve any combination of these seven dimensions [17].From the above, at first glance, it may be concluded that all these seven pillars have equal weight in the formation of friendships.But this conclusion is not true in general.Bahns et al. [27] identified the shared importance of attitude as highly significant.On the other hand, Galupo & Gonzalez [28] compared general friendship values and cross-identity values.From [27,28], it can be seen that some factors weigh more in comparison to others for friendship between two individuals.Since there are many factors behind the friendship of two individuals, we are confined to only seven pillars, as mentioned by Dunbar [17] in this paper.
Let (F, A) denote a soft set, where the set of attributes A contains the people under consideration and the universal set on which it is defined contains seven dimensions for the foundation of friendship as mentioned above.Also, let (G, B) be a soft set where the set B contains one human being, for which we have to check whether the rest of the people belonging to A may or may not have a chance to become friends.Then In fact, for a more detailed analysis of the deepness of friendship between two people, we have to partition the set X as finely as possible.Let us now take an example to illustrate the above formula.
In the above example, a, b and c indicate that there are three individuals present.In this case, friends a and b have a below-average degree of closeness.Persons b and c are also friends, but their friendship is less close-knit than usual.Therefore, based on these two connections, it is impossible to accurately forecast the level of friendship between a and c.However, according to theorem 3.1, for a and c to be friends, they must share at least a few things in common if a and b and b and c have powerful, deep friendships.

Effect of romantic relationship on friendship
Burton et al. [1] mentioned that the cost of romance is the loss of nearly two members from an individual's support network when the individual begins a new relationship.The reason is due to the time investment for a friend or the romantic relationship.In [17], we get to know that the closest support network consists of five people on average.This situation can be analysed mathematically by defining a time function T. So, we have the following definition: Definition 3.4.Let A = {a 1 , a 2 , a 3 , a 4 , a 5 } be the set of five persons that forms the support network of X.Then, we define a function T such that T : fXg Â A À !½0, 1 by TðX, a i Þ ¼ t X,ai , where t X,ai [ ½0, 1 indicates the time spend by X with a i .Here, we assume Since the average number of people in one's closest support network is five, which we consider to be the cardinality of the set A, one can change the cardinality of A by giving it a slightly different value.For convenience, we are using the average.Now, let us define the weight of a friendship (W) as follows: From the above definition, we can see that the more time a person spends with another person, the greater their weight of friendship.Also, it is obvious that 0 ≤ W(a, b) ≤ 1.Now, we define some mathematical results related to weight-after and weight-before.
Let a person X have a supporting network of a set of five persons {a 1 , a 2 , a 3 , a 4 , a 5 }.Suppose X is getting into a new relationship with an individual p.Without loss of generality, we will study the change in weight of friendship between X and a i , i ∈ {1, 2, 3, 4, 5}.The weight of friendship before X gets into a relationship is given by The weight of friendship after X gets into a relationship is given by TðX, a i ÞÞ À TðX, pÞ 5 : Thus, i¼1 TðX, a i ÞÞ À TðX, pÞ royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 231500 Hence, From the above discussion, we conclude the following theorem: Theorem 3.6.If two persons X and a i are friends and X is getting into a new relationship with p, then the change in weight of friendship between X and a i can be determined by the following limits:

■
Let us conduct a visual analysis of the problem.We take into account three instances in figure 3. Before entering a relationship, a person has a weight of 0.8 with one of their friends.Gradually, when he begins to spend more time with the romantic partner after entering a relationship, the weight of friendship with the friend fades away.Similar circumstances arise if the person has two additional friends with friendship weights of 0.5 and 0.2, respectively.Additionally, it is clear from figure 3 that even though a person who has a high weight of friendship with friends begins to spend more time with his romantic partner, he will still have a higher weight of friendship with the friend than someone who has a low weight of friendship before a relationship.royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 231500 Burton et al. [1] mentioned that we lose two good friends after getting into a romantic relationship.That is, on average, almost 40% of the time we spend with our romantic partner.Thus, mathematically, if TðX, pÞ À !0:4 then W after ðX, a i Þ À !W before ðX, a i Þ Â 0:6.Definition 3.7.Let W Ã denote the total weight of friendship of five persons {a 1 , a 2 , a 3 , a 4 , a 5 } with another person X before entering into a romantic relationship.Then, W Ã is given by: W Ã ¼ P 5 i¼1 WðX, a i Þ.If W ÃÃ denotes the total weight of friendship of five persons {a 1 , a 2 , a 3 , a 4 , a 5 } with another person X after he enters into a romantic relationship with p, then W ÃÃ is given by: W ÃÃ ¼ P 5 i¼1 WðX, a i Þ À WðX, pÞ.

Balance theoretic analysis of friendship through soft set approach
The quantity 'deepness of friendship' defined above has an important role in the formation of friendships, but it doesn't guarantee that having a high deepness will represent a closer friendship.It only indicates that if friendship happens between two people, then their bonding will be stable if they have a fair deepness of friendship.The reason is that the quantity D(a, b) is based on the seven pillars of friendship [17], and two individuals having all the elements in common in the seven pillars don't yet know each other, so there is no friendship between them.But in the case of the quantity 'weight of friendship', it also includes the time function, hence, we conclude that if two individuals having a fair deepness of friendship between them spend some time with each other, then there is a fair weight of friendship between them, and thus we say they are friends.Therefore, in this section, we will use the weight of friendship along with balance theory [29] to analyse the scenario of the formation of Dunbar's hierarchy layers for an individual and calculate the number of 'cracks' or 'weak bonding' a person can tolerate until the personal social network becomes unstable.For this, let us introduce some preliminaries regarding the balance theory [29].
Heider first presented the balance theory [29] as a justification for attitude shift, then Cartwright and Harary explicitly generalized it for graphs [30].Balance theory provides a tool for measuring how balanced or stable a relationship is.The original definition solely considered whether or not graphs were balanced; by 'balanced', it was understood that all cycles among all nodes contained exactly an even number of negative edges [30].Later research claimed that the majority of social science applications of balance theory only applied to triads of nodes, or connected triples [31].Here, we also consider the concept of balanced triples for constructing the social support network of an individual.Balanced configurations are still those with an even number of negative edges.In particular, we define the concepts of 'The friend of my friend is also my friend' [ + + + ] and 'The enemy of my friend is my enemy' [ + − − ] as balanced or stable.The two additional varieties of signed triad configurations, [ + + − ] and [ − − − ], are regarded as unstable and cause network fractiousness.As a further refinement, social scientists since [32,33] have observed that the two types of balanced triads are not equally balanced, and the two types of frustration are not equally frustrated.The state [ + + + ] is more stable than the state [ + − − ].In this case, we are not going to deal with unstable triads and the state [ + − − ] because our goal is to establish a stable friend network for an individual rather than identify their enemy network.Now, before constructing an algorithm for the formation of a stable support network for an individual we mention some of the interesting results about friendship.The amount of time devoted to a particular social relationship strongly affects the emotional quality of a friendship [34].According royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 231500 to one prospective study, it takes over 200 hours of face-to-face interaction spread out over a three-month period to make a good friend [35].However, time is naturally scarce; we only spend roughly 20% of each day engaging in direct social interaction (excluding those related to work), or 3.5 hours [36].Given that not all of our relationships are equally valuable to us, we divide up our precious time among our social network to make the most of the various advantages that friends of various calibres may offer [37].However, there are some patterns that are largely constant: we spend 40% of our time with our closest friends and family, and another 20% with the next 10 closest people.In other words, only 15 people receive 60% of the 3.5 hours per day that we spend interacting with others.Only 30 seconds per day, on average, are spent with social partners in the social network's outermost tiers.Our social networks, as a result, develop a very distinctive layering with levels that follow a specific fractal structure [36].From this data, we can conclude that time spent with someone is a very important factor for a stable relationship with that individual.Let us calculate the value of the friendship function between two individuals for which one would belong at least among the 150 people of the other person's Dunbar's hierarchy.Definition 3.9.Let S : fpg Â X À !½0, 86400 such that S( p, x) = t × D( p, x), where t ∈ [0, 86400], D( p, x) ∈ [0, 1] and X is the set of people that are known to the person p, and t denotes the time p spends with x in seconds in one day.Also, D( p, x) denotes the deepness of friendship between p and x.Here, we take the range of S as [0, 86400] because a day consists of 86400 seconds.The function S is called the friendship function for p and x ∈ X.Now, let us calculate the smallest value of the friendship function for which a person x resides at least in the outermost layer of Dunbar's hierarchy of the individual p.Since only 30 seconds per day, on average, are spent with social partners in the social network's outermost tiers of an individual, we get S( p, x) = 30 × D( p, x).The value of D( p, x) is subject to the individual p; hence, we have not assigned any particular value to it.For now, we consider this value a threshold for x to reside at least at the outermost layer of the social network structure of individual p.

Stability of Dunbar's hierarchy layers from the perspective of balance theory
In this section, we consider the balanced triad [ + + + ] only.As shown in figure 4, let the 'ego' reside in the middle, and their Dunbar's hierarchy consists of persons p 1 , p 2 , p 3 , and so on.We denote stable relations by the green lines and unstable relations by the red lines in the figure.Now we follow the following algorithm to develop the layers of Dunbar's hierarchy for the 'ego'.Before proceeding with the steps, we assume a hypothesis.We are taking this hypothesis because, according to a model developed by Conradt & Roper [38], in situations involving conflicting interests, decisions made by the majority of the group should be advantageous because they avoid extreme outcomes by averaging royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 231500 over individual preferences, keeping the consensus costs reasonable for each person.Also, in [39], Dyer et al. provided support for the model of Couzin et al. [40] showing that where differences in preference are large and there is an imbalance in the number of individuals with each directional preference, human groups tend to choose the direction preferred by the majority.
Hypothesis.If there are n persons in a network of an individual, where n is an odd number, then the (n + 1) th person can enter the network if at least dn=2e persons that already exist in the network have a good relationship with the (n + 1) th person, and if n is an even number, then the (n + 1) th person can enter the network if at least dðn=2Þ þ 1e persons that already exist in the network, have a good relationship with the (n + 1) th person to have a stable network.Here, dne denotes the standard ceiling function.
This hypothesis is totally subjective, i.e. it totally depends on the 'ego' that we consider.The least value of the persons ( i.e. dn=2e or dðn=2Þ þ 1e ) may differ for different 'ego'.Next, we move to the following steps: 1. Construct a set A of individuals in ascending order that have friendship functions with values above the threshold, where the 'ego' is concerned.Thus, let A = {p 1 , p 2 , …, p n }. 2. According to our hypothesis, p 1 will enter the social network of the 'ego' as shown in figure 4, because p 1 has a positive relationship with the ego, and similarly, p 2 will also enter the network because p 2 also has a positive relationship with both the ego and p 1 .However, p 3 will not join the network because p 3 has two negative relationships with the other existing members of the network.3. We continue the process until there are too many weak bonds between the members of Dunbar's hierarchy layers of the 'ego' so that no more tolerance can be imposed on her. 4. The people with fewer weak bonds will be placed in the inner layers compared to those with more weak bonds in Dunbar's hierarchy.
if n is even: Proof.We calculate the maximum number of cracks according to our hypothesis and algorithm when the first individual will start to come to the network of p from the set A. We consider the following two cases: Case 2: When n is even, then total number of cracks ■ Corollary 3.12.For Dunbar's number n = 150, the maximum number of cracks or weak bonds that can occur in the stable network of an individual is 5550.Hence, the minimum number of stable or strong bonds that is required in Dunbar's hierarchy of an individual is n C 2 − 5550 = 150 C 2 − 5550 = 5625.Thus, based on balance theory, at least 5625 interpersonal strong bonds are required among the individuals to have one's social networks with n = 150 persons stable or balanced.
According to our algorithm, an ego should have at least minimal interactions with the people that reside inside his Dunbar's hierarchy.But it is not at all possible to interact with all the people under consideration when time is limited.Hence, after a certain number of friends in our networks, we start to get people who are uncommitted and have little attachment to the ego.Since 60 percent of the time of an ego is spent with the first 15 people in their Dunbar hierarchy, after that, we are left with only 40 percent of 3.5 hours (i.e., 5040 seconds) for the remaining people in the network.For ease of calculations and because humans are rational [41], we assume that the ego divides the remaining time equally among their friends so that we can get an approximate number of friends who have a little royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 231500 attachment to the ego based on time spent.Since only 30 seconds, on average, are spent by an ego with the people in the outermost layers, we get approximately 168 people with whom they spent at least 30 seconds.Hence, the total number of people with at least a little personal interaction with the ego is 183.
Therefore, if the population of the ego's network consists of n people, then n > 183, in order to calculate the probability that we have discovered an uncommitted person with whom one does not spend time.Consequently, the probability of seeing an ambivalent person in one's Dunbar hierarchy is 1 − (183/n).

Marriage, divorce and Dunbar's hierarchy
According to Afifi et al. [42], cultures influence the processes of marriages and divorces.Our society is what gives us our culture, and the members of our society are the ones we deal with on a daily basis.As a result, those who are higher up in Dunbar's hierarchy of an individual also have an impact on that person in these areas.More generally, if a society is used to divorce, then no one in that society would hesitate to divorce their partner, demonstrating that the process is at least in part influenced by the culture created by the individuals in one's society.[43].Our theory holds that when a male marries a female, the female will naturally reside in the core layer of the male in Dunbar's hierarchy.Because of emotional attachments, supporting nature to her husband and his family, intimacy, etc., the marriage may indicate an increase in the male's hierarchy with those who are coming from the bride's side and a decrease in the hierarchy with those who already existed in some outer layers of the male's hierarchy.However, the outcome of a divorce is the exact opposite.Additionally, the stability of any male's or female's network may be compromised by the loss of their former spouse if they find someone better or more comparable to them in the sense of the deepness and weight indicated above.
But the reason for divorce is very different from the seven pillars that we have discussed and the time factor that we introduced.Apostolou et al. [44] used qualitative research techniques to identify 62 probable causes of divorce.Through the use of quantitative research techniques, they categorized these causes into seven general components.The ability to have children and financial difficulties were found to be the least important factors.They discovered that being a harmful spouse-having extramarital relationships, being abusive, being addicted to gambling or drugs, etc-were the most important factors that could lead people to divorce, followed by incompatibility and in-law problems.Hawkins et al. [45] also discussed several reasons for divorce.As seen in figure 5, when the ego feels better with partner P 2 than with partner P 1 due to the aforementioned issues, there will be an imbalance triad in the network because, in this scenario, both P 1 and P 2 would be located in the ego's hierarchy and have both positive and negative relations with him.Therefore, they choose to make the triad balance as [ − − + ] in order to keep it stable, and in doing so, the ego may seek to divorce his previous partner P 1 .As a result, we may draw the conclusion that, in order to analyse the stability and instability of a personal network system during divorce, functions must be defined in terms of the variables that affect the process.
From the perspective of a divorce, we are attempting to determine the possibility that a person will have an extramarital affair.Extramarital affairs have a significant impact on divorce [44].Therefore, when a marriage dissolves due to extramarital activities, we assume that the person with whom the ego has an royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 231500 extramarital relationship is hated by his or her former spouse and that they have a weak or unfavourable bond while forming a triad with the ego.Instead of analysing the likely causes of divorce, we look at the likelihood of an extramarital affair that causes the divorce from the perspective of a personal network system based on the balance theory.Let there be m − 1 persons in the personal network of an individual p while he gets married to p 1 .Hence, according to our algorithm, there are some cracks that are already in the network, and while the spouse enters the network, the number of cracks increases or remains the same.After the succession of the marriage, we assume n people have entered the network, and we take the new partner p 2 , with whom the ego has an extramarital affair, as the nth person in the network.Now, we need to know how many weak bonds there are in the network between the time the marriage is over and the nth person enters it.It is due to the fact that one of these cracks will represent the weak relationship between p 1 and p 2 .Now, to find the possibility that p 2 is having an extramarital affair with p, we have the following four cases: Case 1: When m + n is odd and m is also odd, then the minimum likelihood that p has an extramarital affair with Case 2: When m + n is odd and m is even, then the minimum likelihood that p has an extramarital affair with Case 3: When m + n is even and m is odd, then the minimum likelihood that p has an extramarital affair with Case 4: When m + n is even and m is also even, then the minimum likelihood that p has an extramarital affair with If a male finds someone better than his former partner, then we assume that the person is coming to the Dunbar's hierarchy of the male after his meeting with the former partner.So, we should consider the individuals who enter Dunbar's hierarchy of a male either after he marries or after he first meets his former partner in order to determine the possibility that he will discover someone better than his former partner.Since the male always has a positive relationship with the people in his Dunbar's hierarchy, in this case, if the first partner of the male is the nth person in the hierarchy, and there are a total of m people in the network at a specific time while the male finds someone better than his former partner, then the probability that a person from the list of the m − n people that enters the hierarchy after the male meets his second partner is better than his former partner is 1/(m − n).But if the person with whom the male feels better after marriage existed in his network before the marriage, then the probability will be 1/(m − 1).The same result can be obtained for a female.

Conclusion
The most crucial elements that affect a person's mental health are friendship and relationships [46][47][48].In §2, we can see how getting married changes the conjoint Dunbar graph and a person's relationship with another person through some concepts of soft narrow unions and soft narrow intersections.On the other side, we use soft set theory to gauge the strength of friendship between two people, which is dependent on the seven factors [17] that Dunbar listed.Additionally, we determined the importance of friendships between two people, which is based on both the amount of time spent with friends and the seven criteria.The similarities between these concepts and actual world circumstances have been graphically illustrated.Based on Dunbar's approximation [17], we have also found the supremum value of a person's overall weight of friendship under the influence of relationship.We next procure the balance theory to get the findings about the stability of a person's social network.We developed an algorithm to determine how large a person's social network can be so that it stays stable.We also looked at the marriage and divorce scenario and determined the likelihood of extramarital affairs.
We found that the seven pillars of friendship [17] do not weigh the same.There are several other factors responsible for friendship between two people [24][25][26].Some factors weigh more in comparison to others.In this paper, we confined ourselves only to the seven pillars of friendship [17].From [17], we know that the closest support network consists of five people on average.Although uniform numbers of friends can be found in this case, this is a skewed distribution.This skewed distribution does not affect our results in any way.For example, if we assume three members in one's support network, then the equality W after (X, a i ) = (1 − T(X, p))W before (X, a i ) is still consistent, except for the alternation of the upper limit of the summation for i as 3 in lieu of 5.In this case, W before (X, a i ) for i = 1, 2, 3 will be more because the person will get more time for friends in comparison to royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 231500 W before (X, a i ) for i = 1, 2, 3, 4, 5. Similarly, the person will also spend more time with a romantic partner.It yields a higher value of T(X, p).But the figure in this case, after simulation, will also be similar, as shown in figure 3.In short, we will get a family of straight lines altering the values of W before (X, a i ) as shown in figure 3.
The uses of soft set theory help to provide better models of relationships in comparison to others because soft set theory connects attributes and related approximations [18,19,21].We found that both friendship and relationship are connected to several attributes; hence, we defined the deepness of friendship D(a, b) between two persons a and b using soft set theory.The definition 3.2 uses D(a, b) to define the weight of friendship W(a, b) between two persons a and b.In general, a person spends more time with a romantic partner compared to a new friend entering the support network [49][50][51].Moreover, the person becomes more emotionally or physically attached to the romantic partner or trustworthy of the romantic partner as time passes in comparison to other friends [49][50][51].There are several other factors that differentiate a romantic partner from other friends [52].It may be possible that a new friend will become a romantic partner.In that case, the results of §3.1 will be valid.For one's extramarital affair or new relationship in the presence of a previous romantic partner, §3.4 provides a suitable description.Moreover, theorem 3.4 helps us study marriage, divorce, and Dunbar's hierarchy.It tells us about the stability of one's supporting network depending on the position of the romantic partner by using balanced triads.We found that a minimum of 5625 stable or strong bonds are required for the stability of one's social network of 150 people.It means that a minimum of 5625 C 3 = 29 647 267 500 balanced triads of [ + + + ] are required for the stability of one's social network of 150 people.Thus, the above results help balance theory to find the stability of one's social network.Since one has the scope to associate different weights with factors of friendship, one may explore new theories regarding this, as motivated by this paper.In the near future, we anticipate that these mathematical concepts will be useful in analysing human behaviour regarding friendship, relationship, etc. from the perspective of human-computer interactions and artificial intelligence.
Similarly, G (b 1 ), G(b 2 ) and G(b 3 ) are approximations of b 1 , b 2 and b 3 ∈ B for the soft set (G, B).Now, if a 1 lies inside layer L 3 of Dunbar's hierarchy layers, then obviously a 1 will be inside layer L 4 .So, we assume that F(a 1 ) = {L 3 , L 4 }, and F(a 2 ) = {L 2 , L 3 , L 4 }.Similarly, we consider G(b 1 ) = {L 1 , L 2 , L 3 , L 4 }, G(b 2 ) = {L 2 , L 3 , L 4 }, and G(b 3 ) = {L 4 }.Now, imagine the situation when X marries Y and they become a couple C. Thus, we have to analyse the scenario of Dunbar's hierarchy layers for this couple.We define the set Con = {(a, b) : a is interested in being a friend of b}⊆ A × B. If a 1 is interested in being a friend of b 1 , b 2 and b 3 , and a 2 is interested in being a friend of only b 1 and b 2 then Con = {(a 1 , b 1 ), (a 1 , b 2 ), (a 1 , b 3 ), (a 2 , b 1 ), (a 2 , b 2 )}.Thus, we get table 1.
, we can calculate the possible deepness of friendship D(a, b) between two persons a ∈ A and b ∈ B by the following formula: Dða, bÞ ¼ jFðaÞ > GðbÞj maxfjFðaÞj, jGðbÞjg , where |F(a)| and |G(b)| denote the cardinalities of the sets F(a) and G(b), respectively.If D(a, b) = 1, then there is a full chance that a and b are good friends with each other and if D(a, b) = 0, then there is no chance that a and b are friends with each other.Here, |F(a)| and |G(b)| are at least 7 (because a person maps to at least one value of each kind of attribute in the universal set X through the mappings F and G).

FDHFigure 2 .
Figure 2. (a) Friendship between two persons a and c exists if the deepness of friendship between persons a and b is greater than 0.5 and the deepness of friendship between persons b and c is greater than 0.5; (b) possibility of non-existence of friendship between two persons a and c if the deepness of friendship between persons a and b is less than 0.5 and the deepness of friendship between persons b and c is less than 0.5; (c) possibility of existence of friendship between two persons a and c exists if the deepness of friendship between persons a and b is less than 0.5 and the deepness of friendship between persons b and c is less than 0.5.

Definition 3 . 5 .
The weight of friendship between two persons a and b, denoted by W(a, b), is defined by W(a, b) = D(a, b) × T(a, b).

WFigure 3 .
Figure 3. Relation between W after and W before with time T(X, p).

2 Figure 4 .
Figure 4. Graphical representation of the algorithm to construct a stable support network for a person P 0 .

Definition 3 . 10 .
Let there be n people in the support network of an individual p.If a new person enters the network of p and has bad relations with m people that already exist in the network of p, then we call the number m as m cracks or m weak bonds in the network.Theorem 3.11.If there are n people in the social network of an individual p, then the maximum number of cracks or weak bonds in the network that can be tolerated by p

Figure 5 .
Figure 5. Representation of divorce in an imbalanced triad.
It is denoted by ðH, AÞ ¼ ðF, AÞ < þ LCon ðG, BÞ. (iii) A left narrow intersection of soft sets (F, A) and (G, B) with the set Con⊆ A × B is the soft set (H, A), where For every a ∈ A; the set π(Con, a,.) is a singleton set, 2. For every b ∈ B; the set π(Con,., b) is a singleton set.