This is a project I worked on and wrote in the second month of my very first semester as a 1st year PhD student in 2015. Originally, it was intended as a paper in the very first introductory course in computational social science. I’ve posted it here and left the original text as is – there may be typos or occasional incomplete logic, but surely this will be useful to someone. To be honest, I don’t actually know where the code is as I didn’t have very good code sharing and dissemination practices then.
So the analysis of negative ties in social networks is a growing field of study in academic circles and by social network theorists as well as agent-based modelers. Negative ties are basically a network relationship based on some negative or dislike or opposition attitude toward another actor in a network environment. Negative tie analysis is not clearly understood in the literature, and treatment of negative ties both structurally and mathematically is an area of early development in social network methodology and analysis.
But the space is going to become more important. Understanding group behavior under negative tie conditions is an increasingly important research space, especially as organizations are increasingly reliant on team-based work and innovation (See Felps et. al, 2006) and as society is tending towards social media tie formation based on who you don’t like – not just who it is you do like!
The literature on negative ties provides some insight into how negative tie formation affects group behaviors and group activities. Granovetter (1973) provides an overarching core theory for understanding how bridge nodes in networks possess greater access to novel information that allows them to gain value from the network through strategic dissemination of that information. But Granovetter does not necessarily describe attitudinal conflicts emanating from negative information. In fact, one might argue that the large majority of all social network theory research is based on the assumption that ties are either neutral or positive in nature.
To understand negative tie formation and how it relates to group behavior, one must understand the most fundamental building block of networks and how they behave under negative tie constraints—that is—what are the governing dynamics of negative tie dyadic and triadic behaviors in networks (See Labianca, 2014).
For this project I had hoped to begin the process of understanding how negative ties affects group behavior under two separate micro simulations. For the design of the model I combine personal anecdotal experiences with proven research findings that form some telling analysis of how perceptions of negative feelings, and reduced trust can affect group behavior from a social network perspective.
Model Design: Conceptual Foundations
Primarily I base the majority of the conceptual modeling on Felps et. al. (2006), which provides a detailed and generous review of how a single or multiple “bad apple[s]” can be catalysts in the formation of dysfunctional groups, and of the spreading of negative feelings and attitudes through organizational and personal networks.
I define “bad apple” in the context of the model as an actor who consistently and entirely forms negative relationships with any and all members of the network. In other words they exhibit effectively negative chronic behavior – almost a narcissist.
Both of my models focus on understanding the resulting centralities of consistent negative tie formations, and not necessarily the behaviors that must manifest in order to form the negative ties themselves. In other words my focus is on the structural dependencies of negative ties in network evolution, specifically longitudinal and dynamic networks.
Essentially, the two model variations amount to the simulation of flight or fight human responses, summarized as follows:
Conceptual Model: The Brave
“The Brave” model considers that network actors form positive defensive behavioral reactions to negative tie formation as a method to cope with and defend against threatening, negative ties. Conceptually, the brave model assumes that when a negative tie is formed with a positive-tie-forming actor, the positive-tie-forming actor will seek to protect oneself through increased strength and frequency of positive tie formations with other members of the network. Knight (2014) details a process by which the collective reaction of groups, once contextual idiosyncrasies are accounted for, attempt to protect one another through more cohesive positive relationship building.
Our aim in the brave model is to gain a fundamental understanding of how network actors change their relative connections and this their network positions with other network actors when faced with a negative-tie-forming actor. We call this model “The Brave” because nodes don’t shy away from a fight, instead they fight the negative-tie-forming actor by holding their ground in the network.
Conceptual model: The Coward
“The Coward” model considers that network actors break existing positive bonds with network actors as a defensive behavioral reaction to negative tie formation as a method to escape from threatening, negative ties. A random network member of the “Coward” variation of the model is someone who may see the whole network as more negative because one “bad apple” formed a negative tie with them. Thus they feel the safest option to avoid confrontation is to leave the network altogether, and thus over time reduce the number of existing ties they have with other network actors. This means that they continue to break existing relationships with random network members in order to get out until they no longer have any ties to break.
Just like “The Brave” model, our aim in simulating the Coward variation of our model is to get a sense of how the bad apple is able to create a sense of dysfunction in groups through gauging network positioning and the reduction of density of ties. We call the model “The Coward” as an exaggeration of the non-confrontational style of the nodes in the network.
For both variations of our model we designed a number of Python classes that perform various functions.
This class performs the function of creating a random network. In the current model we rely heavily on the use of am Erdos-Renyi random network for our testing and simulation. In future model permutations we wish to expand this class into a number of other more realistic network formations such as scale free and small world networks.
This class performs the creation of a bad apple network. At the moment, this network is primarily a single node named “bad_apple). Since this network is created only a by a single node we attach no probabilities or network structure to it. In the future we hope to be able to test the effects of a whole set of random networks formed only by negative ties.
This class performs the actual modeling and simulations, allowing us to designate the method to be used, the number of negative ties to be created, and the number of positive ties to be created or destroyed by recipients of a negative tie.
This class recalculates all centrality measures, as well as global network measures, such as density and path length in order to enable validation, testing, and analysis.
This class outputs a number of file used for visualizations in specialized network visualization packages and software.
The model is easily described in that other than the class functions delineated above, the heart of the model is a single (quadruple) nested loop which performs the following function:
If the method selected by the user is the brave method, then the bad apple node finds and connects the most central actor (by closeness centrality), and then forms a tie with that actor (a negative tie). The most central actor, having now felt threatened by the sudden negativity in the network, forms 2 random ties with 2 random nodes in the network, other than bad apple (we assume that the most central actor will not try to form a positive tie with bad apple because that would require reciprocity with bad apple. This is contrary to our initial definition of bad apple, which is that it is a network actor that only forms negative ties which is why we do not allow reciprocal formation of ties with bad apple. Additionally, negative ties are not required to be mutual while positive ties do from an attitudinal perspective.
If the model variation selected by the user is the coward method, then as bad apple creates a negative tie with the most central actor, the most central actor breaks existing positive ties from the rest of the network in an attempt to flee the threatening nature of the new group dynamics.
For both model variations we iterate this process until either the most central actor becomes the bad apple node or until a reasonable number of runs has been processed.
The agent-to-agent interactions are not uniform from the bad apple node to the rest of the network, as the most central actor changes over time. Thus, this is not a discrete event simulation, but an agent-based model with strict and clear agent-to-agent interactions.
No element of uniform randomness was introduced at this stage of model design and application, though in future designs of this model, and especially as a stage where model calibration is to be conducted, some element of randomness may be introduced. At this stage we found it to be potentially counter-productive to do so, and without theoretical merit or foundation.
Verification and Validation
I used a number of methods for model verification and internal validation. This included numerous print statements to test the model output at every stage, especially from the more complex nested loops used in the addition and removal of network ties. I also used graphing and visualization for a number of model runs utilizing a small number of iterations and observed how many ties were removed and added, as well as verified that ties are forming with the most central actor at any given moment in time/model run. We then compared with what we expected the outcome to be under those conditions.
I also utilized specialized graphing software such as Gephi in order to visualize the end state networks formed after a number of iterations, and the results matched our expectations of where specific nodes should be located structurally.
The model utilized interactions, adaptation, and stochasticity in tie formation, but did not utilize sensing, prediction, or learning in its assumptions or processes.
Finally, I allowed any user to choose the initial input variables which included, number of model runs, number of maximum iterations, size of bad apple network (kept at n=1 for this set of model runs,) size of original network with probability of edge formation (Erdos-Renyi), number of edges bad apple forms throughout each iteration, and number of positive or negative ties to be created or destroyed by the recipient of bad apple negative ties.
For each model variation we conducted 5 trial runs, and it is our hope to do more in the future. For the example we provide in this paper we let the number of maximum iterations be 50 (time steps), probability of an edge forming in the initial random network generation to be 0.06, 1 negative tie formation per iteration, and 2 negative or positive tie formation (based on method variation choice) to be created or destroyed.
The results provided interesting insights into the consequences of bad apple behavior in an Erdos-Renyi random network.
Since there were many measures we could analyze through this process, our focus was to understand the relative changes in closeness centrality of the participating nodes in each model variation. We also investigated the relative growth of density inconclusively.
The most important results of the simulation process was that when the coward method was applied, the bad apple actor became the most central actor within 10-12 iterations, consistently. This is because they became the focus of the group as group members broke existing positive relationships with each other in order to escape. We use the standard closeness centrality algorithm in order to calculate the centralities of the nodes.
Also I found increasing densities in the coward method application even while more edges were being removed than added. Though I did not investigate how many negative ties contributed to the overall density calculations. Interestingly, over time, when comparing the coward method to the brave method-less independent clusters formed. One cause of this is that as ties were removed from the coward method simulation there were less ties to allow for cliques and clusters to form to begin with, thus allowing the network to become more homogeneous.
The brave method also provided interesting results namely in the form of a lack of change in the overall dynamics and structure of the network. In this simulation, the bad apple never became a central actor. In fact, it remained and continued to be on the fringes of the network. This can be attributed to two causes: The first is that more positive ties were being formed within the network and outpaced the number of negative ties being formed, thus by nature of the closeness algorithm, bad apple was always at a disadvantage. The second is that as more ties were formed the most central actor changed over time, allowing for dissipation and decreased centralization of negative tie formation. This, in turn, meant that bad apple would never be able to form enough ties with enough actors to become the central actor as they did in the coward method, especially that the rate of negative tie formation for our analysis is less than (by half) the rate of positive tie formation and that there is only one bad apple, whilst there are at least 75 regular network members.
Figure 1: Network visualization of final state of the network for coward method (sim=1, size=75, p =0.06, ER, 50, 2, 2). Note the increased size (closeness centrality) of bad apple (blue, top-left-center). For this particular simulation it only took 11 iterations for bad apple to become the most central actor.
Figure 2: Network visualization of final state of the network for brave method (sim=1, size=75, p =0.06, ER, 50, 2, 2). Note the increased size (closeness centrality) of origin_44 (purple, middle-right-center). For this particular simulation the model ran its assigned maximum 50 iterations.
Though there is much more analysis and data collection to be performed. One major insight from our model development and simulation is that when a bad apple enters a new network depending on the network’s pre-existing disposition towards dealing with chronic negative actors and negative ties, the results can be drastically different.
If the group is less akin to confrontation and the bad apple consistently attacks the most central actor (most popular), they are able to become the new focus of the group itself. By new focus, we mean that the majority of relationships in the networks become negative and become tied to bad apple. This ensures dominance of information dissemination and relationship building in the network by the negative node and most likely could contribute significantly to the dysfunction that occurs in small groups when faced with undesirable conditions.
When the group has the ability and motivation to react to negative tie formation by strengthening existing bonds as well as increasing the frequency of formation of those positive bonds, the bad apple is unable to gain the focus of the group by virtue of popularity and closeness centrality. In fact, one might argue the bad apple was ill-served by consistently attacking (forming negative ties) the most central actor in the brave method of our model. A new test for future model development and research would be to change the mechanism by which bad apple attempts to form negative ties with network actors. The most salient modification to the model, thus, would be to ask bad apple to form a negative tie with the least central actor by closeness centrality and to measure the relative popularity growth of bad apple when compared to the most central negative tie formation mechanism of the current method.
New and more in-depth analysis would also be necessary to understand the degree distribution of the network as well as analysis to understand cluster formations of the group. It is our hypothesis that due to increased clustering in the brave method versus the coward method that the true nature of negative ties would be more apparent when conducting this simulation experiment on scale free networks rather than random networks. This is because scale free networks are more structural and hierarchical in nature and thus have a higher probability of fragmentation than their random counterparts.
Additionally, it would be quite appropriate to conduct analysis on the individual centrality measures of each node in our generated networks, to determine the rate at which centrality measures increase or decline when compared to the rates of negative tie formation.
Furthermore, in future models we will consider adding edge and node attributes that allow us to better simulate the true nature of negative ties, specifically, the decreased likelihood of them occurring to begin with. In one article we studied for preparing our model (Everett et al, 2015), it was estimated that negative times form roughly 10 times less than their positive tie counterparts. In other words, calibration of the model to real world data and empirical analysis would yield further insight into the developments of network groups when faced with a chronic, systemic, negative-tie-forming actor.
In this model design we integrated the traditional view of the network perspective, with the format of modeling interacting, interchanging agents of a network. The model and process, though not entirely complete, yielded interesting results that can be used for further hypothesis and model development. More importantly, the model achieved broadness in scope and scale, allowing for continued experimentation with a number of different fundamental networks, sizes, probabilities, discrete timelines, and methods of analysis.
The results yielded from the model analysis also stood on their own merits. Being able to see and understand the relative movement of network actors as measured by centrality and other standard measures allowed us to understand better strategies for combating dysfunction in teams and organizations. Moreover, the analysis allowed us to develop some insight into the robustness of centrality measures for mixed network problems.
As mentioned in the introduction to this paper, negative and mixed tie standard measures for centrality as well as other measures are not yet fully developed in the literature. Thus some effort to understand how current methods of measurement and analysis can be used to understand the nature of negative ties is most appropriate.
Finally the take away conclusions of our analysis is simple yet powerful and can be summarized in one statement: If the group is made of brave network actors, protect the least central actor on the fringes of the network, and if the network is made of fearful network actors, protect the most central actor of the network against the bad apple.
 We use gexf, graphml, and png formats.
 We use the term iterations interchangeably with ticks or time-line progression.
 Gephi is a well-known network graphing and visualization software