Small groups occupy an important place in our lives: we live in a family, we work in a structural department of a company, and we can be members of a school class or a university group. SNA-methodology helps us to answer many life's vital questions, one of which is how to predict social group sustainability and its existence for a long time. In this paper it is argued that a structurally balanced group is likely to be a sustainable one. Some structural balance theories are under consideration in order to evaluate the structural balance level in a communication network. The triad census characteristic is used as an analytical basis. It was demonstrated that the triad census is a very useful and informative characteristic, and, while its consideration and comparison with the calculated census for a random Bernoulli graph, some additional phenomena were revealed. In this research work a new method of data collection and its processing was used; the advantages of this new approach are also discussed.
If we hear today “a social network”, we will imagine a global network, such as Facebook, Instagram, Twitter, and etc., and we almost forget that a small group is also a social network but of a smaller size. We are members of a number of small groups: we live in a family, we work in a structural department of a company, and we can be members of a school class or university group.
There are two clearly definable periods in research of small group networks: the
1950s and 1960s (the “early era”), and since1990s till our days (the “current era
”) [
Armed with information tools, SNA combines the graph theory, mathematical statistics, and sociological methods of data collection and its analysis and interpretation.
Barry Wellman [
Various type ties could be revealed between social group members, but we have focused on those of communication type. We tried to fix all the communications in which students took part while their education activities. These fixations were made by students themselves through their usage of the self-designed information system.
Many sociologists consider communications to be the main relations in a social group, and we paid the main attention to the communications which originated in the educational process.
N. Luhmann argued that social systems were constructed of communications.
“Their (social systems) elements are communications which are recursively produced
and reproduced by a network of communications and which cannot exist outside of
such a network” [
This idea was one of the main reasons to focus on communication ties in our research. Students from 4 different university groups took part in it during the educational year.
The main issue under consideration was a group sustainability. We were interested, why some social groups were keeping their integrity, maintaining internal relations, even when formal ties were destructed, while other groups failed to exist under the similar conditions. We try to find sustainability premises in the group structure. What is the group index which could propose the evidence of its sustainability, and what concepts are to deliver the theoretical foundation for this choice?
We argue that structurally balanced group is more likely to be a sustainable one. As the main index for balance evaluation a triad census characteristic is proposed. In the next paragraphs description of the balance theories, the triad census specification, the method of data collection and this research results are delivered. 2
There are several theories of a structural balance. They are described in details in
[
One of the first theories, which gave the birth to a set of balance models, was the
cognitive theory of Heider. Heider’s theory [
From the behavioral point of view, the important consequence of Heider’s theory is the assumption that any positive relation is a transitive one (e.g. “a friend of my friend is also my friend” is true), and, vice versa, any negative relation is an intransitive one (e.g. “an enemy of my enemy is my enemy” is false). If these conditions are satisfied, the structural balance exists.
This model, mathematically expressed by Harary and Cartwright [
Harary-Cartwright’s structure theorem confirms that “a s-graph (signed graph) is
the balanced one if and only if its points can be separated into two mutually exclusive
subsets such that each positive line joins two points of the same subset and each
negative line join points from different subsets” [
In another words, a group presented by a balanced graph can be partitioned into
two antagonistic subgroups (one of these subgroups can be empty) so, that internal
subgroup links must be positive, and external subgroup links must be negative.
According to E.C. Johnsen [
It is difficult to find such partition in reality, that is why, new empirical models
were proposed. Davis [
In all previously discussed models, only mutual (symmetric) and null ties are permitted. In reality, we often have the symmetric kind of relationships. Most of communication ties (such as who talks to whom, who gives information or advice to whom), affective ties (such as who likes whom, or who trusts whom) or formal ties (such as who reports to whom) are asymmetric in their nature.
The Ranked Clusters Model assumes that any group is a subgroup hierarchy where
every hierarchical level has at least one subgroup. This model extends clustering with
permission of directed (asymmetric) ties between subgroups, with orientation of the
directed ties consistent with hierarchical ordering in which ties are directed from
”lower” to “higher” levels (any lower level subgroup member prefers those who are
members of the higher level subgroups). Davis and Leinhardt are considered to be the
authors of this model [
The Transitivity Model is the most general model of all discussed and subsumes the other models as special cases. The main rule in this model is as follows: if there are 3 members named A, B, C in a group, and A has a tie with B, B has a tie with C, then .A must have a tie with C.
How can we use these theories and reveal the tructural balance existence in a graph (and a social group also)? It is reasonable to analyze triadic census for this purpose. 3
The minimal structural element in a social network is a dyad – two actors who are
probably tied with each other. But most scholars argue that the minimal social group
is a triad. Arguments pro this concept were delivered by D. Krackhardt in [
Referring to Simmel, who was the first to state the fundamental difference between a dyad and a triad, D.Krachardt clearly expressed three grounds for this difference existence. The first ground is that a dyad preserves the individuality of both dyad members in comparison with a triad, where any member’s interests could be suppressed by the other group members for the interests of this group as a whole. The second ground is that in a triad every member has less bargaining power than in a dyad, because of the less threat of the group destruction, when one member leaves this triad. The third ground is that any conflict, which is inevitable in any relationship over time, is more readily managed and resolved in a triad.
Holland and Leinhardt’s [
The triad census for a network is summarized in a 16 element vector (): = (!!", !"#, !"#, !"#!, !"#!, !"#!, !!!!, !!!!, !"!!, !"!!, !"#, !"#!, !"#!, !"#!, !"#, !""), where ttype is the number (or frequency) of the “type” triads (of the appropriate isomorphism class) in a network.
This characteristic may be considered to be rather complex, but it definitely determines the network graph. It implicitly includes indexes of network reciprocity, clustering, density, and it reflects the structural balance level in a network graph. 003 021C 201 300
Interviews, questionnaires or observation are the most common survey methods for data collecting. Each of them has its own advantages and disadvantages and might be used under the certain conditions [16].
In this research we used the different approach: data was collected while “KOMPAS TQM” system implementation was processed. This information system was designed on the basis of ideas and concepts proposed by G. Vodyanov [17]. It was engineered by students and professionals of Ural Federal University and implemented in one of its economic educational departments. It is a quality management system and is geared towards customer feedback reception. But from sociometric point of view this collected data could be interpreted as communications evaluating data, as students tried to evaluate services delivered by different departments and enterprises, and their colleagues as well.
This system supports the process of regular communication result evaluation. System users enter positive or negative marks from the certain range. Every mark must be followed by a comment. Thus, the marks entered into the system reflect the real communications between system users and they are confirmed by comments. The mark sign (“+” or “-“) characterizes the “information receiver” attitude to the communication result in a whole (positive or negative) while numerical value (from 1 to 5 points) reflects the strength (weight) of this communication (i.e. the usefulness degree for receiver).
Data was collected for the period of two semesters of the 2013-2014 educational year. Students, having been registered in the system in advance, regularly input their marks, thereby fixing and evaluating their communications.
The engineered system has web interface, and every user can work with it from his/her home computer or even from his/her smartphone.
The kernel of the information system is a data base for collected data storing. Two data base sets were formed: the first one with data of winter semester and the second – for the summer semester. The “winter data set ” consists of 4846 records and the “summer” one has 4769 records (it means that 4846 marks were inputted into winter data base, and 4769 – into summer data base). If we take in account only marks related to personal communications (exclude marks which evaluate the results of interaction with different organizations), the data bases include 2805 and 3394 records accordingly.
This method of data collecting has real advantages over traditional methods. First, it delivers data which is more objective, because it is a sum result of several marks not the result of a momentary reaction, which can depend on the respondent’s mood or state of health.
Second, there is a time interval between a communication act and the mark input, a respondent has enough time for thinking and real evaluating of this communication.
Third, we can elicit different type data from data bases and form different sociomatrices for future research. One data set might include the sum marks put by every respondent to the other respondents, and the other set might be consisted of mark counts. The second set can be very useful for embedded ties analysis, one of the main issues under consideration in economical sociology [19]. 5
Four groups of students were in the focus of this research: two groups of the last year of education (identified as 34th and 35th) and two groups of freshmen (identified as 44th and 45th). For analysis binary data was received: every group matrix with mark counts was dichotomized. The rule was as follows: if a matrix element is greater than zero, it was converted into 1, and set equal to zero otherwise. Every social network of a student group is presented by a binary directed graph in this case. Graphs visualizing communication networks of the 35th and 44th groups are shown in Fig.2 (males are shown as blue squares and females as pink circles). It is distinct that the 35th group network is denser than the network of the 44th group. It is proved by the indices which are calculated.
a) b)
Several group characteristics are presented in Tab.2, some of them are rather clear and it is not necessary to discuss them.
Density is one of the main network characteristics (it is equal to the proportion of observed to possible edges). For directed graphs, the density is calculated as L/n(n −1), where L is the number of arcs in the observed graph, and n is the number of vertices [20]. The density values (∆)will be necessary for the further research. Grou Year of Number Males Fe- Net- Cluster- Reciprocip Id. Educa- of Stu- males work ing Co- ty Coeff tion dents Density eff. (∆) 34 4 13 6 7 0.66 0.715 0.515 35 4 17 9 8 0.809 0.822 0.719 44 1 17 10 7 0.250 0.616 0.528 45 1 30 15 15 0.201 0.371 0.336 As one can see from Tab.2 groups are comparable in their dimensions, the 45th group is larger than others, and the 34th group has the minimal number of members, but every group might be considered as a small group.
As for the densities, it appears that groups from the first year of education have less density values than those from the fourth year. It is a predictable fact: students from the first year groups have less time to set relations with each other and communicate rarely. The maximum density value is in the 35th group.
For every group triad census was engineered with the help of UCINET 6.0 for Windows, and the necessary percentage values were calculated with the help of Excel. All results are collected in Tab. 3.
In SNA-methodology comparison with random graph characteristics is a very popular and really useful method. This approach reveals how the social nature of a graph influences on its structure. We has followed this tradition and calculated the triad censuses for Bernoulli random digraphs of the same densities (they are presented in Table 3 as Theoretical data).
Mathematical formulas for triad type probability calculations are proposed in [21] and are shown in Table 3 (column 2). All values are presented in the percentage form.
Analyzing data from Table 3 one can make the following conclusions: 1. The less density value we have in a network, the more triads in this network are of the types from the beginning of a triad census, and, vice versa, the denser networks are constructed of triads which types are at the end of a triad census. For the 34th and 35th groups the most “popular” triads are of 210 and 300 types and for the groups with small density the most “popular” triad types are 003 and 012. In Fig. 3 one can see triad quantity distribution for real groups, and in Fig.4 we propose the same kind of distribution for Bernoulli graphs of various densities. 3 2 2 D U C D U 0T 0C 01 0D 0U 0C 01 00 0 1 0 1 1 1 1 1 0 0 1 2 2 2 1 1 3 3 2 2 2 2 2 3 0 0 0 1 1 0 0 1 1 1
Triad Census Fig.4. Triad Number Distribution for Bernouli Graphs of Various Densities 003 012 102 21D 21U 21C 11D 11U 30T 30C 201 20D 20U 20C 201 300 0 0 0 1 1 0 0 1 1 1
Triad Census
From Fig.3, Fig. 4 one can see that a triad census can be partitioned into three intervals: the first one includes three triad types {003, 012, 102}, frequency values greatly depend on the network density – significant values for a network of a low density (Δ≤ 0.5) and near zero for a dense network; the second interval includes the majority of triad types, and it is rather plain with the minimum values 60 I III
II Δ=0.1 Δ=0.2 Δ=0.4 Δ=0.5 Δ=0.7 Δ=0.9
1 2 3 4 for the cyclic triads (it will be discussed further); and the third interval for three triad types {120C, 210, 300} can be considered as the mirror reflection of the first interval – significant values for a dense network and near zero for a network of a low density.
The main purpose of this research was considering the issue of the structural
balance. The results are shown in Table 4. The 34th group network is “the best”
matched to the Ranked Clusters Model and the 45th group is better structurally
balanced in accordance with the Transitivity Model. It is worth to mention that
all groups have close results against the Transitivity Model, and these scores are
rather high. According to this model groups might be sustainable.
3. If we compare cyclic triad number in a real group with the corresponding
calculated value, we will see that cyclical structures are very seldom in the real life
(their percentages are less than the corresponding theoretical ones). One can
compare real and calculated values for “021C”, “030C”, “012C” triads and
assess this difference.
4. It was revealed that the proportion of triads of “300” type is greater than it was
calculated. This is a very special triad, as it is a clique. As Krackhardt stated,
these groups (cliques) restrict people, because “group norms develop rules by
which each member must play to stay a part of the group” [
In this research 4 small groups were under discussion. The main issue was the concept of a small group sustainability from the standpoint of the structural balance. The structural balance theories and the corresponding models form the theoretical basis of this research. It was revealed that all groups have close results against the Transitivity Model, the scores are rather high, and according to this model groups might be sustainable. If one consider the Ranked Clusters Model, one of the fourth course groups is better than the others matched to it.
During this research work the new method for social data collecting was used. In comparison with the common sociological methods of interviews, questionnaires, and observation, this method is based on information system usage. It proposes the higher degree of objectivity, but it takes more time for data collection.
It was demonstrated that a triad census is a very useful and informative characteristic, one can reveal very specific knowledge from it. Using this structural characteristic one can evaluate the structural balance in a communication network graph, and consequently predict group sustainability.
It was revealed that cyclic triads are very seldom in real communication networks, while cliques (triads with all mutual ties) are rather common.
The plans for the future research are as follows: to investigate embedded ties using triad census analysis, and to observe the first year groups in their development.
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