Univariate and multivariate models of positive and negative networks: Liking, disliking, and bully–victim relationships
Highlights
► General like, general dislike, and bullying relations were studied in 18 classrooms. ► Positive and negative network structures differ systematically in univariate models. ► General dislike and bullying are negative networks with different structures. ► Multiplex networks give insight into interdependence of positive-negative relations. ► Children with an equivalent position in the negative network are tied positively.
Introduction
Traditionally, the focus of social network analysis has mainly been on relations with a positive meaning, such as friendship, exchange, or cooperation. Although negative relations are important in classical theories such as balance theory (Heider, 1946) and its representation by signed graphs (Cartwright and Harary, 1956), overall they have been less frequently analyzed than positive relations. Our goal was to investigate the network of negative relations simultaneously with that of positive relations within the same group using a bivariate or multiplex approach. This is important because we expected to increase our understanding of positive and, especially, negative networks by investigating them simultaneously. For example, in bullying research, interventions are proposed based on the assumption that positive ties protect against bullying. Moreover, this is an alternative approach to analyzing signed digraphs, which are formed by combining a binary positive and binary negative digraph, following De Nooy (1999). The multivariate structure of negative and positive relations was analyzed in a classroom setting of social networks of Finnish elementary school students, using a bivariate network modeling (“ERGM”) approach with parameters for positive, negative, and mixed-tie configurations (Robins et al., 2009).
In this study, we aimed to gain insight into the typical structural patterns observable in both positive and negative relations. In performing a multivariate social network analysis of positive and negative ties, we investigated whether networks of positive and negative relations were meaningfully related, and whether the multivariate structures provided further insight into the interdependence between positive and negative networks. It was necessary first to study the network structures of the positive and negative ties on their own (i.e., univariately). Thus, we aimed to set a starting point for further empirical research about negative ties (univariately and multivariately).
To this end, we identified the network structure of positive and negative social networks to determine which structural parameters would be sufficient to model the network data of positive and negative networks. We presumed that this would be more revealing for the negative than for the positive relations, as less is known about the former, but it was important for both types of relations in preparation for the multivariate analysis. The findings of these univariate analyses would also enable us to establish differences – if any – between the structural network patterns that apply to positive and negative relations.
To address these research aims, we investigated classroom networks consisting of a positive relation of “general like” and two negative relations of “general dislike” and bullying. Several classrooms were considered, with the aim of obtaining results that would go beyond the network structure in just one particular classroom. The negative networks of general dislike and bullying were both included in this study to achieve greater generalizability in representing negative relations. We investigated the combination of general like and only one of the negative ties, because examination of two different networks simultaneously is currently the maximum for the available software.
The foundations of the simultaneous study of positive and negative tie networks were laid in the work on structural balance theory. Structural balance theory has a long and rich history; Hummon and Doreian (2003) provide a concise overview (see also Wasserman and Faust, 1994). If relations between actors create tension or “imbalance”, a social process is triggered by which actors change their relations in order to reach a “balanced” state. Structural balance theory can be used as a set of dynamic mechanisms to explain such tie formation in networks (at the micro level, see Heider, 1946) as well as the existence and evolution of group structures (at the macro level, see Newcomb, 1961). Hummon and Doreian (2003) argue that both micro- and macro-level processes should be incorporated to investigate balance theoretic processes. Cartwright and Harary (1956) formalized the ideas on balance theory, and they proposed use of signed graphs to represent the multiple structural relations of actors, making balance theory applicable to social networks.
Structural or social balance is regarded as a fundamental social process, and can account for the structure of affective relations (Heider, 1946, Newcomb, 1961). Usually, affective relations are measured using dichotomous sociometric data; relations between actors are present or absent. Such binary networks can be considered incomplete signed (directed) graphs, because they do not distinguish positive, negative, or neutral (absent) relations (De Nooy, 1999). For example, the absence of a relation between actors in a positive network does not inform us whether the null dyad is neutral or negative. More knowledge can be gained by including negative network relations.
In the present study, we used a bivariate approach in which we investigated positive and negative networks simultaneously. The network structures to be investigated were composed of a network of positive ties and a network of negative ties on a common node set, and are directed signed graphs; a tie in the positive network expresses positive affect, a tie in the negative network expresses negative affect, and the absence of a tie in both networks expresses neutral affect. In such a bivariate network approach, it is possible to observe whether the non-present positive tie is null or negative. One might even observe both a positive and a negative tie from an actor i to another actor j; although such “love–hate” relationships were rare in the combined networks in our analysis, they do occur.
Before investigating positive and negative networks simultaneously, we needed to examine the network structure of single positive and negative networks. Below, we introduce the positive and negative networks investigated and describe possible structural characteristics. Typical network structures of negative relations are expected to differ from those of positive relations. An example of these differences was already given by Robins et al. (2009). In an ERGM analysis of a network of “work difficulties” in a business organization, they found that isolates and one-sided isolates (sinks and sources) are significant network configurations, which is not usually seen for positive relations.
Positive relations are usually characterized by persistent affective bonds between two individuals; therefore, it is natural to assume that reciprocity is one of the main components that drive the formation of such relations. Moreover, transitivity is frequently observed in networks of positive relations (e.g., Feld and Elmore, 1982, Veenstra and Steglich, 2012). Sharing a common friend makes the establishment of a friendship more likely (Davis, 1970, Holland and Leinhardt, 1971). The precondition for triadic closure is being linked at distance two, a two-path. In our study of the positive network of general like, we expected that reciprocity and transitive closure would be localized social processes characterizing the preferences of children (e.g., children prefer to reciprocate nominations of general like), and that these micro-preferences would combine to form the larger (global) network structure of such positive relations.
Negative relationships are persistent and repeated negative judgments, feelings, or behaviors toward another person (Labianca and Brass, 2006). Labianca and Brass (2006) distinguished four different characteristics of negative relations: strength (or behavioral intensity) of the relation, reciprocity (mutuality of the negative relation), cognition (whether ego knows how alter evaluates him/her), and social distance (whether the negative tie is direct or indirect, the latter meaning that someone is positively connected to a person who is involved in negative ties).
Social network analyses of empirical data on negative relations are relatively rare, but have recently received more attention, with topics such as work difficulties (Robins et al., 2009), negative gossip at the workplace (Ellwardt et al., 2012, Grosser et al., 2010), or bully–victim relations (Sijtsema et al., 2009, Veenstra et al., 2007, Zijlstra et al., 2008). Some examples of research about negative relations outside of the signed graph tradition are studies on the relations of aggressive students in schools (Cairns et al., 1988, Farmer and Xie, 2007) and research on relational problems in organizations (Labianca and Brass, 2006).
A reason for the scarcity of studies on negative tie networks might be that those networks are often relatively sparse (compared with positive tie networks) and, therefore, more difficult to model as there is less information on the network structure. Generally, networks with fewer ties will have less structure. Another possible explanation is that negative networks are more difficult to observe, because this involves asking sensitive questions, which is not always easy.
In this study, we investigated two different negative tie networks: networks of general dislike and bully–victim relations, which are both important to children in everyday peer interactions (e.g., Rubin et al., 2009). Regarding relations of general dislike in classrooms, it is not unusual for a child to dislike at least one classmate or to be disliked by one or more classmates. For example, it was found in a sample of 2000 sixth-grade students that about two-thirds of the students received at least one dislike nomination from a peer (Witkow et al., 2005). Moreover, general dislike is often reciprocal: in a meta-analysis of mutual antipathies in 26 studies, Card (2010) showed that about one third of children in classrooms had at least one mutual relationship of general dislike. Bully–victim relationships have a different nature and structure. They are typically defined as dyads in which there is an imbalance of power between bullies and victims, and where negative actions of the bully toward the victim are intended and repeated over time (Olweus, 1993). Bully–victim relations are much less common than general dislike: about 5–10% of the relations in a classroom are usually bully–victim relations (Sijtsema et al., 2009, Veenstra et al., 2007). The differences between these relations of general dislike and bullying are their prevalence, or average degree, the rare mutuality of bully–victim ties, and the larger behavioral intensity (strength) of the bullying relation.
When networks of positive and negative ties are investigated simultaneously, their interrelatedness can occur in different network configurations. We considered dyadic, degree-level, and triadic/higher-order dependencies (see also Table 1).
At the dyadic level, actors can report having both positive and negative relations with the same peers. Mixed reciprocity can also occur; for instance, when i → j is positive (e.g., like) whereas j → i is negative (e.g., dislike).
At the degree-level, the number of nominations received (indegrees) by children for positive ties may be correlated with the number of nominations they received for negative ties. The outdegrees (number of nominations given) of children for positive ties may also be related to their outdegrees for negative ties; this may be interpreted as a general response tendency in nominating others. The number of times that children are nominated for a positive tie can also be related to their tendency to nominate others negatively; and conversely.
The third level represents several actors in triads or higher-order combinations. Various combined triadic patterns have been proposed (Lazega and Pattison, 1999, Robins et al., 2009); we elaborate on these below.
As noted, combined patterns of positive-negative relations are related to structural balance theory (see, for example, Doreian and Krackhardt, 2001, Heider, 1946, Newcomb, 1961). Doreian and Krackhardt (2001, p.48) elaborated for signed triples (i → k, k → j, i → j) that an even number of negative dyads is necessary to obtain a balanced subgraph (see also Wasserman and Faust, 1994, chapter 6). For networks of positive and negative relations, it appears that triads are balanced when all three dyads between three actors are positive or negative, or when two of the three dyads are negative and one positive. A group of three or more actors is considered to be structurally balanced if two people have a positive relation and they are consistent in their relations with other people (either positive or negative).
Doreian and Krackhardt (2001) found in the classical data of Newcomb (1961), however, that some of the triple types predicted by structural balance theory were not found in the observed data. Doreian and Krackhardt hypothesized that this inconsistency might be due to individual characteristics of actors, competing mechanisms for the attention of (popular) actors (i.e., children dislike each other but are friends with the same popular classmate, leading to an unbalanced structure), or patterns at the group level (such as peer group rejection). To investigate such mechanisms, we formulated hypotheses for triadic and higher-order dependence patterns in multiplex positive–negative relations; this is described in the next section.
The starting point for our elaboration is the configuration mentioned above: children (actors) agree about the (negative) evaluation of others. This is illustrated in the triangular configuration in Fig. 1a. When i and j agree about the (multiple) actors k they dislike, they are expected to be positively tied because they are balanced: they share the children whom they do not like. This pattern is also applicable to bully–victim relations, when using reports on the question: “By which classmates are you victimized?” When i and j perceive that they are being victimized by the same bullies k, they might seek each other for comfort and support against the bullies (Fox and Boulton, 2006). Such relational patterns can be considered examples of structural equivalence or structural homophily (e.g., Wasserman and Faust, 1994), where actors i and j have a similar network position. In this case, actors with structurally similar ties in the negative network have a positive tie. Thus, we expected that children would have positive relations when they shared negative out-ties.
Another form of structural equivalence applies to children who receive negative nominations, either for general dislike or for bullying. In line with the argument of peer group rejection (Doreian and Krackhardt, 2001), receiving negative ties is a form of being rejected. Receiving negative ties from multiple others may lead to a vicious cycle in which a lack of fit with the group and being rejected enhance each other, making it hard to return to the peer group once rejected (Juvonen and Gross, 2005, Mikami et al., 2010). Moreover, once children are rejected by a substantial number of peers, the pool of peers from which they can choose friends is limited. This leads to default selection (Sijtsema et al., 2010): children have difficulties realizing the friendships they want to have and are forced to choose friends they initially would not have chosen. The social distance (cf. Labianca and Brass, 2006) in the negative network is small: these children are rejected themselves and they are positively tied to other rejected peers. This is shown in Fig. 1b. When children i and j are rejected by the same (multiple) peers k, they are more likely to form a positive relation. Children i and j might have preferred other friends, but due to their rejected position, they end up befriending other rejected (and structurally equivalent) peers (Mikami et al., 2010). This may also apply to bullying, because some aggressive children lack the skills to provide emotional and practical support and are, therefore, unattractive to form friendships with (Sijtsema et al., 2010). Thus, it is possible that bullies i and j form a friendship when they bully many of the same peers k. In sum, we expected that children would have positive relations when they shared negative in-ties.
In addition to the patterns of structural equivalence, other triangular patterns may occur: children may have opinions about the enemies of the children with whom they have difficult relations. We explored these configurations in our empirical analysis, as they are interrelated, though not yet theoretically underpinned. This may further developments in explaining the interplay of positive and negative ties. In Fig. 1c, child i dislikes child k, who, in turn, dislikes child j. Because both i and j have a negative relation with peers k, i and j may have a positive relation (enemies of enemies are friends), in either direction (from i to j or from j to i). The first is a depiction of transitive closure (see Fig. 1c); the second can be seen as cyclic closure (see Fig. 1d). For bully–victim relations, the pattern might be different because if i is being bullied by k, who is being bullied by j, a social dominance hierarchy may ensue, with child j at the top. Because low social dominance typically leads to rejection (Hawley, 1999, Mikami et al., 2010), it may also be the case that child j bullies child i, who is the lowest in the hierarchy, such that a positive relation between i and j is unlikely.
We used exponential random graph models (ERGMs, also called p* models) to model the network structure of positive and negative relations. ERGMs are probability models for complete networks of a given set of actors that are used to estimate parameters of dyadic, triadic, and higher-order level effects (see, Robins et al., 2007a, for an introduction). We considered these models suitable to use in examining our research questions about the network structure, as the associated methodology provides ways for obtaining good representations of the network structure. Single tie networks of general like, general dislike, and bullying can be investigated simultaneously in multivariate ERGMs (Lazega and Pattison, 1999, Pattison and Wasserman, 1999).
We first investigated the structures of the networks for general like, general dislike, and bullying separately. To identify which structural parameters represented the relational structures of the single networks, we searched for a parsimonious model that could be applied to all classrooms; our aim was to select the lowest possible number of structural parameters, while achieving well-converged and reliable estimations for each classroom. Therefore, we focused on the structural patterns in the networks, and neglected possible actor-level (such as gender) differences in the structure of negative and positive networks. Although we acknowledge that gender might be an important factor in the three networks investigated for the question who nominates whom (Card et al., 2008, Dijkstra et al., 2007, Maccoby, 1998), it may be only one of many factors that drive nominations. For example, children are selective in whom they dislike, and one aspect that plays a role in that mechanism is gender, among a great deal of other factors. The univariate estimations of the networks of the 18 classrooms were summarized using a meta-analytic procedure, which describes the occurrence of (need for) the various structural parameters and their means and variability over the different networks. This enabled us to know if and how the general network patterns of different types of ties were different. Following the same procedures, we investigated the structures of the networks of positive and negative ties simultaneously, using multivariate exponential random graph models for positive (general like) and negative (general dislike or bullying) tie networks.
Section snippets
Participants
Data stem from the Finnish KiVa bullying intervention program, a representative sample of elementary schools in all five provinces of mainland Finland (for an extensive description, see Kärnä et al., 2011). The data used in the present study come from the third wave, collected in May 2008, involving classroom networks where children were allowed to nominate an unlimited number of their classmates for negative as well as positive relations. We selected three schools from the total sample that
General like
In total, we analyzed 18 classrooms for general like; see the descriptive statistics in Table 2. On average, children liked about four to five classmates, and 43% of the nominations were reciprocated. There were only eight isolated children, who were neither liked by classmates nor liked classmates themselves. Fifty-four children were liked but did not nominate peers themselves (actors with at least one indegree but with zero outdegree: the so-called ‘sinks’).
The structure found in the networks
Discussion
The results of our study point toward essential configurations for modeling the network structure of positive and negative relations. Whereas positive networks have been modeled often, few researchers have empirically investigated the network structure of negative networks. In this study of 18 classrooms with 393 students, we have shown that negative tie networks of children's relations of general dislike and bully–victim relations can be meaningfully modeled. The findings of this study also
Acknowledgements
This research was partly supported by Toptalent Grant (021.002.022) from the Netherlands Organization for Scientific Research (NWO) to the first author, by grants from the Academy of Finland (121091, 134843) to the sixth author, and the Dutch Ministry of Education (Onderwijs Bewijs) to the last author.
References (59)
The sign of affection: balance-theoretic models and incomplete signed digraphs
Social Networks
(1999)- et al.
Who are the objects of positive and negative gossip at work? A social network perspective on workplace gossip
Social Networks
(2012) - et al.
Aggression and school social dynamics: the good, the bad, and the ordinary
Journal of School Psychology
(2007) The ontogenesis of social dominance: a strategy-based evolutionary perspective
Developmental Review
(1999)- et al.
Some dynamics of social balance processes: bringing Heider back into balance theory
Social Networks
(2003) - et al.
Multiplexity, generalized exchange and cooperation in organizations: a case study
Social Networks
(1999) - et al.
A comparison of various approaches to the exponential random graph model: a reanalysis of 102 student networks in school classes
Social Networks
(2007) - et al.
An introduction to exponential random graph (p*) models for social networks
Social Networks
(2007) - et al.
Recent developments in exponential random graph (p*) models for social networks
Social Networks
(2007) - et al.
Closure, connectivity and degree distributions: exponential random graph (p*) models for directed social networks
Social Networks
(2009)
Bullying and the peer group: a review
Aggression and Violent Behavior
Bullying victimization in youths and mental health problem: ‘much ado about nothing’?
Psychological Medicine
The behavioral basis of acceptance, rejection, and perceived popularity
Social networks and aggressive behavior: peer support or peer rejection
Developmental Psychology
Antipathetic relationships in child and adolescent development: a meta-analytic review and recommendations for an emerging area of study
Developmental Psychology
Shared targets for aggression by early adolescent friends
Developmental Psychology
Direct and indirect aggression during childhood and adolescence: a meta-analytic review of gender differences, intercorrelations, and relations to maladjustment
Child Development
Structural balance: a generalization of Heider's theory
Psychological Review
Clustering and hierarchy in interpersonal relations: testing two graph theoretical models on 742 sociomatrices
American Sociological Review
Same-gender and cross-gender peer acceptance and peer rejection and their relation to bullying and helping among preadolescents: comparing predictions from gender-homophily and goal-framing approaches
Developmental Psychology
Peer contagion in child and adolescent social and emotional development
Annual Review of Psychology
Pre-transitive balance mechanisms for signed networks
Journal of Mathematical Sociology
Statistical analysis of friendship patterns and bullying behaviors among youth
New Directions for Child and Adolescent Development
Patterns of sociometric choices: transitivity reconsidered
Social Psychology Quarterly
Friendship as a moderator of the relationship between social skills problems and peer victimisation
Aggressive Behavior
A social network analysis of positive and negative gossip in organizational life
Group & Organization Management
Twenty years’ research on peer victimization and psychosocial maladjustment: a meta-analytic review of cross-sectional studies
Journal of Child Psychology and Psychiatry
Attitudes and cognitive organization
Journal of Psychology
The power of friendship: protection against an escalating cycle of peer victimization
Developmental Psychology
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