Heuristic methods for synthesizing realistic social networks based on personality compatibility

Abstract

Abstract Social structures and interpersonal relationships may be represented as social networks consisting of nodes corresponding to people and links between pairs of nodes corresponding to relationships between those people. Social networks can be constructed by examining actual groups of people and identifying the relationships of interest between them. However, there are circumstances where such empirical social networks are unavailable or their use would be undesirable. Consequently, methods to generate synthetic social networks that are not identical to real-world networks but have desired structural similarities to them have been developed. A process for generating synthetic social networks based on assigning human personality types to the nodes and then adding links between nodes based on the compatibility of the nodes’ personalities was developed. Two new algorithms, Probability Search and Compatibility-Degree Matching, for finding an effective assignment of personality types to the nodes were developed, implemented, and tested. The two algorithms were evaluated in terms of realism, i.e., the similarity of the generated synthetic social to exemplar real-world social networks, for 14 different real-world social networks using 20 standard quantitative network metrics. Both search algorithms produced networks that were, on average, more realistic than a standard network generation algorithm that does not use personality, the Configuration Model. The algorithms were also evaluated in terms of computational complexity.

Full text

R E S E A R C H Open Access
Heuristic methods for synthesizing realistic
social networks based on personality
compatibility
Daniel A. O’Neil
*
and Mikel D. Petty
* Correspondence: dao0030@uah.edu
University of Alabama in Huntsville,
Huntsville, AL 35899, USA
Abstract
Social structures and interpersonal relationships may be represented as social networks
consisting of nodes corresponding to people and links between pairs of nodes
corresponding to relationships between those people. Social networks can be
constructed by examining actual groups of people and identifying the relationships of
interest between them. However, there are circumstances where such empirical social
networks are unavailable or their use would be undesirable. Consequently, methods to
generate synthetic social networks that are not identical to real-world networks but
have desired structural similarities to them have been developed. A process for
generating synthetic social networks based on assigning human personality types to
the nodes and then adding links between nodes based on the compatibility of the
nodes’personalities was developed. Two new algorithms, Probability Search and
Compatibility-Degree Matching, for finding an effective assignment of personality types
to the nodes were developed, implemented, and tested. The two algorithms were
evaluated in terms of realism, i.e., the similarity of the generated synthetic social to
exemplar real-world social networks, for 14 different real-world social networks using 20
standard quantitative network metrics. Both search algorithms produced networks that
were, on average, more realistic than a standard network generation algorithm that
does not use personality, the Configuration Model. The algorithms were also evaluated
in terms of computational complexity.
Keywords: Social networks, Network generation, Network metrics, Personality
compatibility, Probability search, Compatibility-degree matching
Introduction and motivation
Social network analysis is the study of social structures and relationships. Built from
the theoretical foundation of graph theory, social networks are formal mathematical
structures, consisting in their simplest form of nodes corresponding to actors or
agents, where actors or agents may be individual people or identifiable groups of
people, and links between pairs of nodes corresponding to relations between them,
where relations may be any type of contact or connection between the actors or agents
the nodes represent (Knoke and Yang, 2008) (Scott 2000).
The study and use of social networks often begins from and depends on empirical
social networks. Empirical social networks are obtained directly from the real-world
group or organization they represent, by the process of investigators identifying the
A
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people in the group or organization of interest and determining if the relationships to
be represented in the network exist between them. Empirical social networks obtained
by observation are valuable, but there are issues with them. Empirical social networks
can be difficult and expensive to obtain, especially if the process for doing so is manual,
and consequently relatively few in number and less than comprehensive in covering the
range of possible social networks. They may not be available in the size, in terms of num-
ber of nodes or links, that an investigator needs. And while obtaining social networks from
social media or other digital sources is much easier today than in the past, such empirical
networks can be vulnerable to malicious recovery of private information from them using
de-anonymization methods (Narayanan et al, 2011) (Narayanan and Shmatikov, 2008).
Synthetic social networks, generated algorithmically rather than obtained empirically,
can mitigate these issues. Given effective social network synthesis methods, a user
could produce a set of synthetic social networks, individually non-identical but collect-
ively with specific desired structural characteristics, including size. A set of multiple so-
cial networks could be used to systematically test a network analysis or visualization
tool (Staudt et al., 2017), and would allow the deliberate introduction of deviations
from the defining characteristics of the class of social networks for testing purposes
(Tsvetovat and Carley, 2005). In addition, synthesizing social networks is an approach
to anonymization, which may protect the privacy of the individuals represented in an
empirical social network (Narayanan and Shmatikov, 2009). Researchers may use the
synthetic social networks without privacy concerns and freely share them with other re-
searchers to allow repeatable experiments (Zhou et al., 2008).
However, an arbitrary or random graph is unlikely to be suitable as a synthetic social
network for any particular application. To be useful a synthetic social network must “ap-
proximate certain qualities or parameters found in the empirical data”(Tsvetovat and
Carley, 2005). In other words, a useful synthetic social network must possess the structural
characteristics expected for the class of social networks it is intended to exemplify, without
being simply a copy of one of those networks. For brevity, a synthetic social network with
the structural characteristics of a desired class of social networks, perhaps as measured by
suitable quantitative network metrics, will hereinafter be described as realistic.
A number of synthetic social generation methods exist; several important ones will be
described later. Broadly speaking, the existing methods are based on replicating structural
characteristics of an exemplar network. Our goal in this work was to examine whether a
network generation method based instead on personality compatibility between nodes
(where the nodes are assumed to correspond to persons) could be effective. Social net-
works based on personality compatibility can be of significant interest to organizations that
must organize teams of persons to interact and work effectively, especially in challenging
circumstances. We sought to develop a capability to synthesize personality-based social
networks for future space exploration missions and colonies. In such missions, crew com-
patibility will be essential, so a capability to model social network formation and camarad-
erie within such circumstances could be very useful to mission planners and analysts.
Given the large number of people participating in online social networks, such as
Facebook and Twitter, it is unsurprising that much current social network research
tends to focus on large networks. Often, web based networks are scale free and the
thousands of links and nodes tend to result in similar metrics. The research presented
in this article is focused on relatively small networks with 10 to 100 nodes. The
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real-world networks used as exemplars are drawn from a wide range of organizations,
ranging from an accounting firm to a monastery.
Two algorithms able to automatically synthesize realistic social networks using per-
sonality compatibility are described and compared in this article. The algorithms are
given as input a set of nodes of the desired size. The algorithms then assign, using dis-
tinctly different methods, a personality type to each node that can be used as the basis
for stochastically generating links between the nodes. Link generation between a pair of
nodes depends of the relative compatibility of the personalities assigned to the two
nodes. Personality type compatibilities are encoded in a personality compatibility that is
an input to the generation process. Because link generation is stochastic given a per-
sonality type assignment to the nodes, multiple non-identical social networks can be
generated as needed from a single assignment once a suitable assignment has been
found. The algorithms have been shown to generate synthetic social networks that are
significantly more realistic, in terms of their structural properties as measured by a
range of standard graph metrics, than social networks generated using a standard net-
work generation algorithm that does not use personality, the Configuration Model. The
generation process has been demonstrated to work with multiple personality compati-
bility tables, and is thus adaptable to different personality type models.
The remainder of this article is structured as follows: Section 2provides background
information about social network analysis. Section 3 is a brief survey of important re-
lated work. Section 4 explains the social network synthesis algorithms developed in this
research. Section 5 describes the software implementation of the three algorithms and
discusses their execution. Section 6 reports the results of testing and comparing the al-
gorithms, including quantitative measures. Finally, Section 7 states the conclusions of
this work and suggests possible future work.
Background
This section provides background information on graph theory and social network ana-
lysis, and explains the metrics that were used to measure networks’structural similarity.
Social network analysis
The details vary by specific application, but in their simplest form, in a social network the
nodes may correspond to people in a group, organization, or population of interest. The pres-
ence of a link connecting two nodes represents some relationship, such as kinship, friendship,
collaboration, or information exchange, between the people corresponding to the nodes the
link connects. For example, social networks are used to represent social distance in (Li et al.,
2018) and information spreading in (Bouanan et al., 2018). The study of the structural prop-
erties of such social networks can provides insight into the group, organization, or population
it represents. As an example, Fig. 1shows a real world social network found to exist within a
corporate law firm in the northeastern United States (Lazega 2001).
Classes of social network
Not all social networks have the same structural characteristics and properties. Social
networks that represent communications in terrorist organizations might be expected
to differ in structure and activity from those that represent collaborations in a scientific
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community. A set of social networks that represent instances of some well-defined cat-
egory of group of organization will be termed a class. Some examples of classes of so-
cial networks are listed in Table 1; several of the examples in the table are based on
(Easley and Kleinberg, 2010). The examples in Table 1are all social networks, but intui-
tively they are not the same in terms of structure.
Note that in the last example in Table 1, the nodes of the social network correspond
to organizations, not individual people. That example is included in order to draw at-
tention to this distinction. This work focuses on social networks where the nodes cor-
respond to people. The potentially different structure of an organizational-node
network as compared to a people-node network will become of interest later.
A particular social network may be an element of one class, but not of another, by
virtue of its structural properties. Therefore, two operations are of interest: (1) Mem-
bership; given a social network, how can it be tested for membership in a particular
class of social networks? (2) Generation; given a description or example of a particular
class of social networks, how can a synthetic social network that is a member of that
class be generated? This work focuses on the second operation.
Fig. 1 Friendship within a law firm (Lazega 2001)
Table 1 Classes of social networks (Easley 2010)
Group or organization Nodes Possible link(s)
Terrorist organization People Communications
Recruitment
High school student body People Romantic relationship
Athletic teammates
Social club People Friendship
Sponsorship
Employees of a corporation People Exchange of email
Supervisory authority
Regional or national populace People Relatedness
Transmission of infection
Financial system Banks Interbank loans
Currency exchanges
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Data structures and attributes for social networks
In the implementations described later, social networks were stored internally using ad-
jacency matrices (Gersting 2014). More sophisticated data structures for social net-
works are available, but the networks used in this work were relatively small and
simple adjacency matrices were sufficient. As for the attributes of the networks, two
are important. First, networks may be weighted or unweighted. This work is concerned
solely with the absence or presence of links, and therefore only unweighted networks
were used. Second, networks may be symmetric or asymmetric. Links in symmetric net-
works typically represent mutual or two-way relationships, whereas links in asymmetric
networks represent one-way relationships. This work is concerned solely with mutual
relationships, and therefore symmetric networks were used.
Social network metrics
In this context, metrics are numerical measurements of a social network’s structure. A
wide range of different metrics are available. Graph theory provides a number of abstract
metrics, sometimes known as graph invariants, that quantify some aspect of a network’s
structure without attaching any specific semantic meaning to the metric’s values. Exam-
ples include maximal degree, girth, or vertex chromatic number (Bang-Jensen and Gutin,
2008). Social network analysis has defined additional metrics that are intended to measure
something about the network that has semantic meaning in the context of the social ap-
plication of the network. These metrics include centrality (Scott 2000), reciprocity (New-
man 2010) (Scott & Carrington, 2011), and clustering coefficient (Easley and Kleinberg,
2010). Finally, overarching empirically-derived structural properties common to categories
of networks, such scale-free and cellular, may apply to social networks (Tsvetovat and Car-
ley, 2005). All are intended to measure in an objective and quantitative way some aspect of
anetwork’s structure that may be useful for a particular application. The intent is that realis-
tic synthetic social networks would have metric values similar to those of the real-world so-
cial networks they were intended to mimic, without having identical structures.
Many network metrics have defined, and clearly not all could be used in this work.
From those available, 20 were carefully selected to assess the similarity of real-world
and synthetic social networks in this work. That selection was made in part based on
the motivation of studying the social networks of future space colonies. Thus metrics
that characterize information flow, integration of individuals into the network, level of
camaraderie indicated by clustering, and level of influence among the individuals are of
interest. Because this work used only undirected symmetric networks, only metrics
suitable for those networks were considered.
The metrics selected include both standard metrics of graphs’structural characteris-
tics (nodes, links, components, degree, radius, and eccentricity) and metrics considered
to be relevant to social network structure, per (Rapoport 1957) (Freeman 1978) and
(Bonacich 2007). In the former category, the number of nodes, links, and components,
the network’s radius and eccentricity, and the nodes’degrees fundamentally
characterize a network’s structure.
In the latter category, metrics found useful to study team structure and interaction
were of special interest. Global clustering coefficient, average clustering coefficient, Gini
coefficient, and number of communities provide some insight to the tight knit groups
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and the distribution of nodes among the communities. Average betweenness serves as
a basis of comparison for maximum betweenness to identify the information brokers or
potential bottle-necks in the network. Likewise, average closeness serves as a basis of
comparison for minimum closeness to identify the nodes that are at the heart of com-
munities. Mean path length, network radius, average eccentricity, and network diam-
eter are geodesic distances that can be used estimating the rate of information flow
across a network. Eigencentrality indicates the level of influence that a node may exert
on other nodes. In some similar applications, clustering, path length, betweenness,
closeness, and diameter were used in a study of information sharing and collaboration
in small groups (Manso and Manso, 2010), betweenness was used in a study of inter-
action in programming teams (Gloor et al., 2011), density and diameter were used in a
study of authorship collaboration (Gajewar and Das Sarma, 2012), eigencentrality was
used in a study of leadership in social groups (Bullington 2016), and Gini coefficients
have been used as a measure of inequality of participation in digital health social net-
works (van Mierlo et al., 2016). Table 2lists and defines the metrics used.
Personality models
In 1923, Jung described distinct human personality types based upon his clinical observa-
tions (Jung 1971). Using Jung’s ideas, in 1944 (Myers, 1962) developed a structured ap-
proach to identifying personality types and published a manual describing a personality
typing process that later became known as the (Myers & McCauley, 1985) Type Indicator
(MBTI) (Smathers 2003). In the MBTI typing scheme, each person is categorized on four
“dichotomies”or dimensions, held to correspond to different aspects of personality. Two
“preferences”or values are possible on each dichotomy, yielding a total of 16 different per-
sonality types. The four dimensions and their two preferences each are:
Attitude (inward or outward focus); Extraversion (E) or introversion (I).
Perceiving (information gathering) function; Sensing (S) or Intuition (N).
Judging (deciding) function; Feeling (F) or Thinking (T).
Lifestyle preference; Perceiving (P) or Judging (J).
Tabl e 3(a) shows the estimated proportion of the United States population who would be
categorized into each preference, with each dimension considered separately (Marioles et al.
1996;Mitchell,1996). Table 3(b) shows the result of calculating a proportion for each person-
ality type, based on the dimensions’proportions. A detailed description of the 16
Myers-Briggs types is beyond the scope of this article; for details see (Keirsey 1998). The im-
portant ideas here are that each person may be categorized as having one of the 16 types and
thatthelikelycompatibilityoftwopeoplemay be estimated from their personality types.
Critics of the MBTI personality model point to apparent problems. Metzner et al.
suggested that the “rigid”dichotomies of the Jungian personality types constitute a
“conceptual straight jacket”and proposed a reformulation of the dichotomies as pairs
of primary and inferior psychological functions (Metzner, Burney, and Mahlberg, 1981).
Additionally, McCrae and Costa commented that the MBTI lacks a neuroticism factor,
perhaps because emotional instability was not part of Jung’s type definitions, and it ap-
pears that Myers and Briggs believed that each personality type was positive. The lack
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of a negative factor may make the interpretation of MBTI results easier to accept. How-
ever, it could also allow the omission of information that would be useful to employers,
coworkers, counselors, and individuals (McCrae and Costa, 1989).
Nonetheless, the MBTI model is used and accepted at the U. S. National Aeronautics
and Space Administration, the organization from which this work’s motivating applica-
tion is drawn, e.g., (Nelson and Bolton, 2008). It is also widely used in industry in the
United States, including 89 of the Fortune 100 companies (Grant 2013), for applica-
tions that include increasing self-awareness to support decision analysis (Malik and
Zamir, 2014) (Weiler 2017), improving team performance by explaining communica-
tion styles (Choo, Lou, Camburn, et al., 2014), identifying correlations between per-
formance and personalities (Felder 2002) (Felder 2005) (Kiss, Kun, Kapitány, and Erdei,
2014) (Furnham and Crump, 2015a) (Furnham and Crump, 2015b), and identifying
Table 2 Social network metrics used in this research
Metric Definition
Nodes Number of nodes in the network; here denoted n.
Links Number of links in the network; here denoted m.
Components Number of disjoint sets of connected nodes in a network.For a connected
network, the value of this metric is 1.
Network density Number of links in the network divided by the number of possible
links n·(n–1) / 2; here denoted p.
Average degree Average, or mean, of the nodes’degrees.
Standard deviation degree Standard deviation of the nodes’degrees.
Global clustering coefficient Ratio of closed nodes of vertices to connected triplets of nodes.
Average clustering coefficient Average of the nodes’local clustering coefficients;the latter is the ratio of actual
links to neighborsto possible links to neighbors for a given node.
Number of communities Number of clusters in the network
Cluster Gini coefficient Inequality of distribution of nodes among communities
Mean path length Mean of the number of links in the shortest path betweeneach pair of nodes.
Average betweenness Mean of the nodes’betweenness centrality values, which is the number
of shortest paths between pairs of node that pass through a node.
Maximum betweenness Maximum of the nodes’betweenness centrality values.
Average closeness Mean of the nodes’closeness centrality values, which is the sum of the path
lengths between the node and all other nodes.
Minimum closeness Minimum of the nodes’closeness centrality values.
Average eigencentrality Mean of the nodes’eigencentrality (also known as eigenvector centrality); the
latter is a measure of the number of links each of a nodes neighbors have.
Minimum eigencentrality Minimum of the nodes’eigencentrality.
Network radius Minimum of the nodes’eccentricities; the latter is the maximum length of the
shortest paths from a node to all other nodes.
Average eccentricity Mean of the nodes’eccentricities.
Network diameter Maximum of the nodes’eccentricities.
Table 3 Personality type frequencies in the U. S. population (Marioles et al. 1996)
(a) (b)
E 0.463 I 0.537 ENTJ 0.045 ESTJ 0.097 INTJ 0.053 ISTJ 0.112
N 0.319 S 0.681 ENTP 0.033 ESTP 0.070 INTP 0.038 ISTP 0.081
T 0.529 F 0.471 ENFJ 0.040 ESFJ 0.086 INFJ 0.047 ISFJ 0.100
J 0.581 P 0.419 ENFP 0.029 ESFP 0.062 INFP 0.034 ISFP 0.072
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correlations between professions and personalities (MH, 1977) (Freeman 2009) (Jafrani
et al., 2017) (Rosati, 1993) (Capretz, 2002) (Cohen et al, 2013) (Loffredo et al, 2008)
(Moutafi et al, 2007) (Emanuel, 2013).
Other personality models exist. Arguably among the best known is the Five Factor or
OCEAN model. After analyzing correlations among 35 personality traits, Tupes and Chris-
tal identified five personality factors: Surgency (Extraversion), Agreeableness, Dependability
(Conscientiousness), Emotional Stability (versus Neuroticism), and Culture (Openness)
(Tupes and Christal, 1992)(JohnandSrivastava,1999). Goldberg referred to these factors
as “The Big Five”(Goldberg 1990). McCrae and Costa interpreted the factor Culture as
Openness to experience (McCrae and Costa, 1987). Ruston and Irwing rearranged the first
letters of the factors to form the mnemonic OCEAN (Rushton and Irwing, 2008).
Personality compatibility
The National Aeronautics and Space Administration (NASA) defines Team Risk as the
risk associated with a decrease in performance and behavioral health due to inadequacy of
ateam’s cooperation, coordination, communication, and psychosocial adaption
(DeChurch et al., 2015). “Currently, NASA has no formalized process to compose mission
teams from a scientific perspective, but this is an identified need for future exploration
missions”(Landon 2015). Anania asserts that “crew compatibility on an interpersonal
level will need to be a major factor in order to ensure optimal communication and coord-
ination within the team”(Anania et al., 2017). Brandley and Herbert applied MBTI to
their study of Information Systems teams and found that a team’s personality type com-
position is partially related to performance (Bradley and Hebert, 1997).
Personality compatibility may play a significant role in link formation in real-world
social networks. Back asserted that “personality differences influence social relation-
ships”, but noted that social network research rarely considers the effects of individual
personalities (Back 2015). With that in mind, the algorithms described here both make
use of inferred personality types for the people represented by the network’s nodes and
base the probability of a link forming between two nodes on the compatibility of the
personality types associated with those nodes.
Table 4is such a personality compatibility table for the MBTI personality types. The
rows and columns are the 16 MBTI personality types. Each entry in the table is the
probability of a link forming in a social network between two nodes if the nodes’associ-
ated personality types are those of the entry’s row and column. Note that the table is
symmetric, i.e., the two entries for two personality types are the same regardless of
which type is on the row and the column. Table 4was constructed from the personality
type descriptions in (Keirsey 1998); the process for doing so is detailed in Appendix 1.
Homophily and heterophily can be modeled as likelihoods of link formation among
personality types. In Table 4, values on the diagonal of the table represent a level of
homophily because cells on the diagonal are the intersections of rows and columns
identifying the same personality type. Values in the cells other than the diagonal repre-
sent some level of heterophily because those cells are at the intersections of rows and
columns that identify different personality types.
MBTI was used in this work because of its wide application in practical settings.
However, the social network generation algorithms presented later do not depend on
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Table 4 Personality compatibility table for pairs of MBTI personality types
ESTP ISFP ISTP ESFP ESTJ ESFJ ISTJ ISFJ ENFJ INFJ ENFP INFP ENTJ INTJ ENTP INTP
ESTP 0.040 0.296 0.506 0.506 0.296 0.296 0.506 0.296 0.714 0.506 0.506 0.506 0.296 0.714 0.867 0.506
ISFP 0.296 0.110 0.506 0.139 0.296 0.296 0.139 0.296 0.296 0.867 0.867 0.506 0.714 0.714 0.506 0.506
ISTP 0.506 0.506 0.259 0.296 0.867 0.506 0.714 0.867 0.139 0.296 0.714 0.296 0.139 0.506 0.714 0.714
ESFP 0.506 0.139 0.296 0.460 0.506 0.867 0.714 0.506 0.506 0.296 0.296 0.714 0.506 0.139 0.296 0.296
ESTJ 0.296 0.296 0.867 0.506 0.680 0.714 0.867 0.952 0.296 0.139 0.506 0.139 0.296 0.296 0.506 0.506
ESFJ 0.296 0.296 0.506 0.867 0.714 0.840 0.867 0.714 0.296 0.506 0.506 0.506 0.714 0.051 0.139 0.139
ISTJ 0.506 0.139 0.714 0.714 0.867 0.867 0.940 0.867 0.506 0.296 0.296 0.296 0.506 0.139 0.296 0.296
ISFJ 0.296 0.296 0.867 0.506 0.952 0.714 0.867 0.940 0.296 0.139 0.506 0.139 0.296 0.296 0.506 0.506
ENFJ 0.714 0.296 0.139 0.506 0.296 0.296 0.506 0.296 0.840 0.506 0.139 0.506 0.714 0.714 0.506 0.506
INFJ 0.506 0.867 0.296 0.296 0.139 0.506 0.296 0.139 0.506 0.680 0.714 0.714 0.867 0.506 0.296 0.296
ENFP 0.506 0.867 0.714 0.296 0.506 0.506 0.296 0.506 0.139 0.714 0.460 0.296 0.506 0.506 0.714 0.296
INFP 0.506 0.506 0.296 0.714 0.139 0.506 0.296 0.139 0.506 0.714 0.296 0.250 0.506 0.506 0.296 0.714
ENTJ 0.296 0.714 0.139 0.506 0.296 0.714 0.506 0.296 0.714 0.867 0.506 0.506 0.110 0.296 0.139 0.139
INTJ 0.714 0.714 0.506 0.139 0.296 0.051 0.139 0.296 0.714 0.506 0.506 0.506 0.296 0.030 0.867 0.867
ENTP 0.867 0.506 0.714 0.296 0.506 0.139 0.296 0.506 0.506 0.296 0.714 0.296 0.139 0.867 0.110 0.714
INTP 0.506 0.506 0.714 0.296 0.506 0.139 0.296 0.506 0.506 0.296 0.296 0.714 0.139 0.867 0.714 0.250
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any particular personality compatibility table or even on a particular personality model. Any
personality model that satisfies the following two criteria could be used: (1) it has personal-
ity types that are discrete, or could be discretized; and (2) it provides, or enable the develop-
ment of, a quantitative measure of the relative compatibility of different personality types
that can be encoded as a personality compatibility table. In fact, a different personality table
was used in the early stages of this work, with similar results to those reported here.
Related work
This section briefly reviews selected prior work related to generating graphs and social
networks.
Real-world social networks
Social network analysis research requires real-world social networks to use as input data. First
developed in the early 1980s, UCINet is a social network analysis application that calculates a
variety of network metrics (Freeman 1988). UCINet includes functions for discovering cohe-
sive subgroups in a network (Borgatti et al., 2014). An associated archive of social networks,
represented as adjacency matrices, is maintained in the UCINet format (Freeman, 2009)
(Freeman, 2016).
Tabl e 5lists the real-world social networks used in this research as source data; they are
from the UCINet archive. In all but one of the networks, the nodes of the network corres-
pond to individual people and the links to a relationship of some kind between them.
(The exception is the Schwimmer Taro Exchange Network., where the nodes correspond
to Orokaiva households within the Papaun village Sivepe and the links represent the mu-
tual exchange of gifts, such as cooked taro (Schwimmer, 1979) (Schwimmer 1973).)
The real-world social networks used in this research include both symmetric and
asymmetric and both unweighted and weighted networks. The new network synthesis
algorithms to be described produce symmetric unweighted networks. Therefore the
Table 5 Real-world social network data sets used in this research
Real-world social network Source Nodes Symmetric Weighted
Robins Australian Bank (Pattison et al., 2000)11nono
Roethlisberger & Dickson Bank Wiring
Room
(Roethlisberger and Dickson,
1939)
14 yes no
Thurman Office (Thurman 1979) 15 yes no
Sampson Monastery (Sampson 1969) 18 no yes
Krackhardt Office CSS (Krackhardt 1987)21nono
Krackhardt High-Tech Managers (Krackhardt 1987) 21 yes no
Schwimmer Taro Exchange (Schwimmer 1973) 22 yes no
Webster Accounting Firm (Webster 1993) 24 yes yes
Zachary Karate Club (Zachary 1977)34nono
Bernard & Killworth Technical (Bernard et al., 1982) 34 yes yes
Bernard & Killworth Office (Bernard et al., 1982) 40 yes yes
Krebs Fortune 500 IT Department (Advice) (Chen 2007) 56 no yes
Krebs Fortune 500 IT Department (Business) (Chen 2007) 56 no yes
Lazega Law Firm (Lazega 2001)71nono
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real-world networks were converted to symmetric and unweighted if necessary before
being used as exemplar networks. The conversions were done in the obvious ways; if
an asymmetric network had directed link(s) in either or both directions between two
nodes, the converted network had an undirected link between those nodes, and if a
weighted network had a weighted link of any weight between two nodes, the converted
network had an unweighted link between the nodes.
Current trends in social network analysis include social networks developed from
massive data sets captured from online social media and communities, such as FaceBook,
Twitter, and Wikipedia; (Mislove et al., 2007), (Crandall et al., 2008), (Kwak et al., 2010),
(Catanese et al., 2011), (Yang and Leskovec, 2015), and (Grandjean 2016) are examples.
Common interests in careers, pastimes, politics, popular culture, and societal trends serve
as the motivation for joining groups within these online communities, so personality types
may be one of many factors determining how links form in real-world social networks.
However, according to Krebs social networks expressed as connections via Facebook and
LinkedIn can be misleading because site members may try to connect with as many
people as possible and others acquiesce to the creation of apparent links with no real con-
nection. “Two people might show to be connected but they really are not –one person
was too embarrassed to turn down a ‘friend request’from a total stranger. These ‘false
positives’tend to pollute the data of these social networking services”(Krebs 2008).
Existing models for generating synthetic social networks
Generating synthetic social networks that are more realistic than random graphs, such
as those generated by the classic Erdős-Rényi G(n,p) algorithm, also known as the ran-
dom graph model (Erdős1959) (Erdos and Rényi, 1960), requires attention to the prop-
erties of social networks that distinguish them from random graphs. Since 1960, several
social network generation models have been developed. A selection of existing social
network generation models that consider or exploit various structural characteristics of
networks includes the following; each will be described following the list:
Random graph model (Erdos and Rényi, 1960)
Configuration model (Bollobás 1980) (Milo et al., 2003) (Newman 2003) (Viger and
Latapy, 2005)
Exponential random graph model (Holland and Leinhardt, 1981) (Frank and
Strauss, 1986) (Wasserman and Pattison, 1996)
Stochastic block model (Holland et al., 1983) (Nowicki and Snijders, 2001)
Small world model (Watts and Strogatz, 1998)
Preferential attachment model (Barabási and Albert, 1999)
Popularity Similarity model (Papadopoulos et al., 2012)
Chung-Lu graph model (Chung and Lu, 2002)
Degree correlation dK series (Mahadevan et al., 2006)
Block two-level Erdős Rényi model (Seshadhri et al., 2012)
Replication of complex networks model (Staudt et al., 2017)
In random graphs, the nodes’degrees tend to follow a Poisson distribution
(Bollobás 1998). This can be unrealistic; real-world networks’node degree
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distributions are more often non-Poisson and heavy-tailed. The configuration
model extends the random graph model to address that inconsistency (Bender and
Canfield, 1978)(Bollobás1980) (Molloy and Reed, 1995) (Molloy and Reed, 1998)
(Newman et al., 2001) (Milo et al., 2003)(Newman2003)(VigerandLatapy,
2005). In the configuration model, network generation is initialized with both the
number of nodes nand a specific degree sequence K={k
1
,k
2
,…,k
n
}, where k
i
is
thedegreeofnodev
i
. The degree sequence Kmay be random variates drawn from
a suitable distribution (checked to ensure that Σk
i
is even), or more simply, the
actual degree sequence of a real-world network serving as an exemplar of the class
of networks to be generated. Given nnodes a degree sequence K, links are added
by randomly connecting each node v
i
to k
i
other nodes, with each link uniformly
possible. This produces networks with a realistic degree distribution, but if a single
exemplar is used for multiple synthetic networks, all the generated networks will
havethesamenodedegrees.
The exponential random graph models (ERGM), also known as the p* model, as-
sembles a network from subgraph structures, such as stars, triangles, paths, and
cycle patterns (Wasserman and Pattison, 1996)(Snijders2002)(Robinsetal.,
2007). Holland and Leinhardt developed an exponential family of probability distri-
butions for directed graphs, which derived from empirical observations of stars
(nodes with multiple links), isolates (nodes without links), and their triad census
(the sixteen possible configurations of a directed triad) (Holland and Leinhardt,
1977)(HollandandLeinhardt,1981). Frank and Strauss developed a family of dis-
tributions for directed and undirected Markov graphs wherein there existed de-
pendence among the links (Frank and Strauss, 1986). Snijders applied Monte Carlo
Markov Chains to estimate network metrics such dyads, undirected and directed
two paths, and directed and undirected triangles (Snijders 2002). Hunter distin-
guished between ERGM and p* by associating the maximum pseudo-likelihood es-
timation (Wasserman and Pattison, 1996) with p* and maximum likelihood
estimation (Geyer and Thompson, 1992)withERGM(Hunter2007).
Among the existing methods, the stochastic block model (SBM) may have the
most similarity to the new methods developed in this work, and so we describe it
in a bit more detail. The SBM can be used to generate networks and to detect
communities within large scale networks (Holland et al., 1983) (Anderson et al.,
1992)(FaustandWasserman,1992) (Newman and Girvan, 2004) (Bickel and
Chen, 2009)(Fortunato2010) (Decelle et al., 2011) (Abbe 2017). The set of ac-
tors or agents involved is first partitioned into Bcommunities or clusters known
as blocks. This partitioning is often done by manual analysis, based on observa-
tion or data. Tightly interacting groups of actors are placed into the same group.
AB×Bpreference matrix Wspecifies the probabilities of link formation both
within and between the blocks (Nowicki and Snijders, 2001). The probabilities
may be provided manually or by automated analysis of the source data. The
on-diagonal entries in Wspecify the probabilities of links forming between nodes
in the same block, whereas the off-diagonal entries in Wspecify the probabilities
of links forming between nodes in different blocks. If the on-diagonal probabil-
ities are higher than the off-diagonal probabilities, then the intra-block link dens-
ity will be higher than the inter-block link density; such a network is known as
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assortative. Conversely, if the off-diagonal probabilities are higher than the
on-diagonal probabilities, then the resulting network will have a higher
inter-block link density; such as network is known as disassortative.InanSBM
implementation, the number of nodes in each block may be stored in an integer
vector with Bentries. If the blocks are assumed to be disjoint, the sum of the
vector’s entries is the total number of nodes in the network. To generate a syn-
thetic network, the probability of link formation in Wbetween each pair of nodes
is used to stochastically determine if a link is formed between those nodes.
The small world model starts with a one dimensional regular ring lattice where
each node has links to its knearest neighbors (Watts and Strogatz, 1998)(Strogatz
2001). Several iterations of random rewiring produce a network with a desired
density. For each node, rewiring involves stochastically determining whether an
existing link is deleted or a new link is formed between the current node and an-
other randomly selected node.
The preferential attachment model starts with a small set of nodes and then
adds nodes and links in an iterative process based upon the connectivity of the
nodes (Barabási and Albert, 1999)(Barabási,2003). The number of nodes in the
initial set determines the maximum degree for new nodes. In each iteration, or
“time step”, a new node is added to the network and then links from the new
node to the existing nodes are stochastically added, up to the maximum degree.
The process depends upon the existing nodes’current connectivity, which is cal-
culated as k=m·(t/t
i
)
1/2
where mis the node’s current degree, tis the current
iteration (or time step), and t
i
is the initial time step when the node was added.
The probability of link being added from the new node to existing node iis k
i
/
(Σk)wherek
i
is the connectivity of node iand (Σk) is the sum of the connect-
ivity of the other existing nodes. New nodes, and links from them to existing
nodes, are iteratively added until the network has the desired number of nodes.
This process produces a scale-free network.
The Popularity Similarity model bases the probability of link formation on
hyperbolic distances between nodes (Papadopoulos et al., 2012). In this model,
the network grows as nodes are added at successive time steps. Older (earlier
added) nodes tend to be popular because they have had more time to connect to
other nodes. To model similarity, new nodes are randomly placed on a circle; a
node’s birth time determines the radial coordinate r
t
=ln(t). Two nodes, with
polar coordinates (r
s
,θ
s
)and(r
t
,θ
t
), have an approximate hyperbolic distance
x
st
=r
s
+r
t
+ln(θ
st
/2)=ln(stθ
st
/2)wheresand tare the nodes’respective birth
times. This hyperbolic distance serves as a convenient metric that represents both
radial popularity and angular similarity.
TheChung-Lumodelusesanexemplardegreesequencetosettheprobabilityof
link formation between two nodes. For a pair of nodes, the link formation prob-
ability is proportional to the product of corresponding degrees in the sequence
(Chung and Lu, 2002).
The degree correlation dK series model uses probability distributions for node
degree correlations for subnetworks of size dto generate networks. A generated 0
K-graph reproduces the average node degree of an exemplar network. A 1K-graph
reproduces the degree distribution of an exemplar network. A 2K-graph
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reproduces the joint degree distribution and a 3K-graph reproduces similar inter-
connectivity among triangles as an exemplar network (Mahadevan et al., 2006).
The Block two-level Erdős-Rényi model introduces community structures by generat-
ing a set of independent networks and then randomly linking nodes among the com-
munities (Seshadhri et al., 2012). Typically, algorithms that implement this model
include input parameters for nodes and density and the algorithm returns a network
with the number of links based upon the density.
(Staudt et al., 2017) describes the replication of complex networks (ReCon)
model that generates scalable synthetic social networks based on an exemplar net-
work. An objective of ReCon is to generate networks of different sizes, up to 32
times larger than the exemplar. The ReCon algorithm first detects communities in
the exemplar network using the parallel Louvain method. It then generates a work-
ing graph as a disjoint union of xcopies of the exemplar, where xis a scaling fac-
tor. For each detected community in the working graph, the algorithm preserves
the degree distribution and rewires the intra-community links through random
edge switching. After rewiring the intra-community links, it rewires the
inter-community links and generates links among the copies of the network (Staudt
et al., 2017). In this work a realistic replica of an exemplar social network was de-
fined as a network that has similar metric values as the exemplar. The metrics that
were compared to the exemplar included sparsity, i.e. number of links versus num-
ber of nodes, the degree distribution’s Gini coefficient, maximum degree, average
clustering coefficient, diameter, number of connected components, and number of
communities. ReCon produces replicas that are realistic under this definition be-
cause it preserves the exemplar’s community structure and node degrees.
Comparison to the current work
In contrast to the algorithms reported later, with only one exception the existing
social network generation methods do not use any actual or inferred attributes of
the persons represented by the nodes to determine or influence the generation of
links between the nodes. The exception is the stochastic block model, which uses
a group attribute associated with each node to determine the probability of link
formation with other nodes within the same group. None of the prior methods
use personality type or compatibility, as is done in this work, to produce syn-
thetic social networks. This idea was hinted at in (Staudt et al., 2017), which de-
scribed a potential application of synthetic social networks as showing
interactions that are “determined by implicit psychological and social rules”,but
those “rules”were not used to generate networks.
The desirable features of a synthetic social network generation algorithm include
parsimony (i.e., few parameters), speed of execution, and network realism. Realism,
in particular, is a very important characteristic of synthesized social networks. Real-
ism in social networks has been defined in terms of network structural features,
dynamics, and evolution (Staudt 2017). The similarity, or lack thereof, of metric
values between a synthetic network and a real network is understood as a measure
of realism. (Chakrabarti et al., 2004) (Leskovec et al., 2010). A quantitative assess-
ment of realism is central to the current work.
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Synthesizing social networks based on personality compatibility
This section explains the new personality-based synthetic social network generation al-
gorithms developed in the current work. The section begins with placing the new algo-
rithms in the context of the overall process used for network synthesis; the details of
the individual algorithms in the process will follow the overview.
Synthesis process overview
Figure 2shows the algorithms and dataflow in the network synthesis process. That
process starts with a real-world social network T, which serves as an exemplar of
the class of social networks to be generated. (In this work, Tis any of the fourteen
real-world networks listed in Table 5). Network Tis input to three different algo-
rithms. The two algorithms developed in this work, Probability Search (PS) and
Configuration-Degree Matching (CDM), each construct an assignment Aof person-
ality types to the nodes of T. Both employ heuristic methods to find A,albeitin
completely different ways. The resulting personality type assignment Ais then in-
put to a network generator algorithm (GNAC), which generates a set of synthetic
social networks (denoted Pfor the PS algorithm or Mfor the CDM algorithm),
using the personalities in Aand the compatibility information in personality com-
patibility table C.
The challenge is to find a personality type assignment Awhich, when the GNAC al-
gorithm is used with personality compatibility table C, will produce realistic synthetic
social networks. A personality type assignment that produces realistic synthetic social
networks will be referred to in this context as effective.
Fig. 2 Algorithms and dataflow in the network synthesis process
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Real-world exemplar network Tis also be input to a standard network generation algo-
rithm, the Configuration Model (CM). CM also generates a set Fof synthetic social net-
works based only on the structure of Tand without using and personality compatibility
information.
The three sets of synthetic networks are then input to a process that calculates the net-
work metrics listed in Table 2and compares them to the exemplar T.
Generating networks from a personality type assignment
Synthetic social networks are generated by an algorithm that considers personality
compatibility by using a personality compatibility table C(e.g., Table 4) and a
personality assignment Ato the nodes of the network. The network generation algo-
rithm is denoted the G(n,A,C) (GNAC) algorithm, where nis the number of nodes, A
is an assignment of personality types to the nnodes, and Cis a personality compatibil-
ity table that includes the personality types in A. Given an assignment Aof personality
types to nodes and a compatibility table C, as many synthetic social networks as needed
can be generated using the GNAC algorithm. They will likely differ due to the random-
ness in the algorithm, but they will be related in that all were produced using the same
assignment Aand compatibility table C.
The GNAC algorithm first determines the degree sequence of an exemplar net-
work T. The degree sequence is used to initialize a link budget for each of the nodes
in the synthetic network. The algorithm then randomly selects two triads of nodes
in the synthetic network as candidates for triangles. The personality types assigned
to the triads’nodes by Aand the personality compatibility table Care used to find
the probability of link formation between each pair of nodes in the triads, and the
probabilities for each triad are summed. The triad with the larger sum is then con-
verted into a triangle by connecting all unlinked pairs in the triad and the link bud-
gets of any newly linked nodes are decremented. This procedure repeats until the
number of triangles in the synthetic network is the same as the number of triangles
in the exemplar network.
Producing the desired number of triangles typically does not completely deplete the
link budgets of all of the nodes. For the nodes with remaining link budgets, the algo-
rithm randomly selects pairs of those nodes. If the pair is not linked, then a link is
formed and the nodes’link budgets are decremented. When a pair of nodes with
remaining link budgets that are not already connected cannot be found, then the algo-
rithm randomly selects nodes that have no remaining link budget. If the randomly
selected node and a node needing a neighbor are not connected, the algorithm ran-
domly adds a link between the nodes with a probability determined by the nodes’
assigned personality types and the compatibility table C. The process repeats until the
sum of all nodes remaining link budgets is 0, at which point the synthetic social net-
work is returned.
In the following pseudocode, Tis an exemplar network, Ais a personality assign-
ment, Cis a compatibility table, S=(V,E) is a synthetic network and uand vare nodes
in the network. At three points in the gnac function links may be added to the network.
The addlink function, shown first, is called by the gnac function; it adds a link between
nodes uand vif they are not already connected.
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The overall computational complexity of the GNAC algorithm is O(n
4
). To see this,
consider first the function addlink; it does not loop over the nodes or edges and so is
O(1). The GNAC algorithm itself begins with some housekeeping that includes an
O(nlog n)sortofthenodes’degree sequence (line 3). The first main loop (lines
7–25)isoverthetrianglesofT.Anetworkwithnnodes may have as many as
C(n, 3) triangles; C(n,3)=n!/(3!(n–3)!) ∈O(n
3
). Within that loop, the do while
loop (lines 8–11) may execute an arbitrary number of times, but on average is
O(1). The set membership tests (line 19) are O(1) if the edge set is stored in a
suitable data structure, such as an adjacency matrix. All of the remaining computa-
tion in the first main loop is also O(1). Thus the second main loop is O(n
3
). Find-
ing all the potential dyads the first time (line 27) is O(n
2
). There are potentially as
many as C(n,2)suchdyads;C(n,2)=n!/(2!(n–2)!) ∈O(n
2
). The second main
loop (lines 28–33) iterates once for each of the O(n
2
) dyads, and in each iteration
it again finds all potential dyads O(n
2
), thus the second main loop is O(n
4
). The
third and final main loop iterates at most once for each node, i.e., O(n) iterations. Each
iteration scans O(n) nodes to find those with remaining link budgets, so the third main
loop is O(n
2
). Thus the complexity of the GNAC algorithm as a whole is O(n
4
).
Probability search algorithm
The Probability Search (PS) algorithm is based on the idea that the probability of a
given social network being generated algorithm from a given personality type as-
signment Aand personality compatibility table Ccan be calculated. That calcula-
tion can be done in either of two ways that differ in whether or not nodes are
assumed to be distinguishable. For this work, it is assumed that the nodes are
uniquely identified and are thus always distinguishable from each other. This as-
sumption is appropriate for many social network applications, where nodes corres-
pond to specific known persons. The implication of uniquely identified nodes is
that a different network, with the same connection structure (i.e., isomorphic in
graph theory terminology) but connecting different specific nodes, would not be
equivalent as a social network because different people would be connected.
The probability of the network will be calculated using a simple extension of the
Erdős-Rényi G(n,p)algorithm.IntheG(n,p) algorithm the probability of link for-
mation pis constant for the entire network. In the PS algorithm’s probability cal-
culation the constant pis instead replaced for each pair of nodes with the
probability of a link forming between those nodes, given a personality type assign-
ment Aand a personality compatibility table C.Letp(i,j)betheprobabilitygiven
in Cof a link being present between two nodes iand jfor the personality types
assigned to nodes iand jby A. The probability of a network G=(V,E)being
formed is therefore given by Eq. (1); we will call this the network probability.
PGðÞ¼
Yi;jϵV;i≠j
pi;jðÞ if i;j
fg
∈E
1−pi;jðÞif i;j
fg
not∈E
ð1Þ
GivenanexemplarnetworkTand a compatibility table C, the network probabil-
ity can be used to search for the personality type assignment Athat has the high-
est probability P(T) of producing the exemplar. Once found, that personality type
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assignment can be used by the GNAC algorithm to generate synthetic networks
that are likely to be similar to the exemplar.
In theory, the optimum personality type assignment, i.e., the assignment that has
the highest possible probability of producing the given exemplar network T,could
befoundbymethodicallygeneratingeverypossiblepersonalitytypeassignment
and calculating P(T) for each one. Unfortunately, this is not practical for any but
the smallest networks. If a personality type scheme has kdifferent personality types
and exemplar network Thas nnodes, there are k
n
different possible type assign-
ments. For the MBTI personality type scheme using in this work k= 16, thus for
even the smallest real-world exemplar network used in this research, the Robins
Australian Bank network with 11 nodes, there are 16
11
≈1.76 · 10
13
possible
personality type assignments. Calculating P(T) for that many assignments at the
rate of one per millisecond would require over 500 years. Thus an exhaustive
search is impractical.
Instead, the new Probability Search (PS) algorithm performs a heuristic search
through the space of possible personality type assignments. After generating an ini-
tial personality type assignment randomly, it iteratively changes the assignment,
one node at a time. To do so, it uses node probability, a quantity similar to
network probability, but calculated for a single node. Given a network G,a
personality compatibility table C, and a personality type assignment A,thenode
probability of a single node iin Gis given by Eq. (2).
Pi
ðÞ¼YjϵV;i≠j
pi;jðÞ if i;j
fg
∈E
1−pi;jðÞif i;j
fg
not∈E
ð2Þ
At each iteration, the PS algorithm selects a node i, either the node with the
smallest node probability P(i) under the current personality type assignment (with
probability 0.95), or a random node (with probability 0.05). It then calculates P(i)
for that node ifor each of the possible personality types, holding the network
structure and other nodes’personality types fixed. The personality type that gives
the highest node probability P(i) is assigned to node i. This process repeats until
the overall network probability improvement achieved in an iteration is less than a
threshold, subject to a required minimum number of iterations. Finally, to prevent
non-productive repetitive changes to the same node’spersonalitytype,whena
node’s personality type is changed it is added to a list of nodes excluded from
adjustment in the next iteration and remains in that list for a certain number of
iterations. The improvement threshold, the minimum number of iterations, and the
number of iterations a node remains on the excluded list are all parameters to the
algorithm. (For the results reported here, the values 0.0001, n·k· 1000, and ⌈n/
10⌉respectively were used for those parameters. Those values were found
empirically.)
In the following pseudocode for the PS algorithm, Vis a set of nodes, Eis a set
of links, Cis a personality compatibility table, Aisapersonalitytypeassignment,
nis the number of nodes, and kis the number of different personality types. In
the pseudocode, two subroutines (functions) precede the main logic of the PS
algorithm.
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The overall computational complexity of the PS algorithm is O(n
3
). To see this, con-
sider first the functions vprob and gprob;vprob loops once over the nelements of V
(lines 3–9), and so is O(n), whereas gprob has two nested loops (lines 3–11), each over
the nelements of V, and so is O(n
2
). The main body of PS begins with some O(1)
housekeeping (lines 2–9) and an O(n
2
)calltogprob. The main loop (lines 10–49) exe-
cutes O(n) times. Within the main loop, the search for the lowest probability vertex
(lines 15–23) begins with an O(n) call to vprob (line 16), then loops over the available
nodes O(n) times; within that loop is an O(n) call to vprob, thus this portion of the
main loop is O(n
2
). Next the search for the highest probability personality type (lines
25–35) calls gprob once, and then enters a while loop that iterates ktimes, each time
calling gprob, which is O(n
2
). Because kis a constant and not a function of n, this por-
tion of the loop is O(n
2
). The last part of the while loop includes two operations on the
excluded list (lines 37 and 39) which can be accomplished in amortized O(1) time if im-
plemented as a deque, and another O(n
2
) call to gprob. Thus the complexity of the
main loop, and PS algorithm as a whole, is O(n
3
).
Compatibility-degree matching algorithm
The Compatibility-Degree Matching (CDM) algorithm first determine the degree sequence
of a given exemplar network T. It then generates a personality type assignment Ain accord-
ance with an empirical distribution based the frequency of each personality type in the U. S.
population (Table 3). The columns of personality compatibility table Cprovides an overall
compatibility of each personality type. The CDM then orders the personality types by over-
all compatibility and the nodes of the exemplar network Tby decreasing order of degree.
Using those two orderings, the CDM personalitytypestothenodessothatthepersonality
types with the highest overall compatibility are assigned to the nodes with the highest de-
gree. In the pseudocode, personality type assignment Ais a vector of size n.
The overall computational complexity of the CDM algorithm is O(nlog n). The n
nodes are sorted (line 3), which is O(nlog n). The summing of the compatibility values
(lines 4–6) is O(k
2
), where kis the number of personality types, and the sort of the
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sums (line 7) is O(klog k), but for most networks k<< n. The assignment of personal-
ity types (lines 8–10) is O(n) and the sort of the assigned types (line 11) is O(nlog n).
The final loop (lines 12–14) is O(n). Thus the complexity of the CDM algorithm as a
whole is O(nlog n).
Configuration model algorithm
In order to assess the effectiveness of the personality-based algorithms (PS and CDM), they
were compared to an existing network generative model that was not personality-based. Two
were considered for the role of baseline. Because of its abstract representation of popularity,
the Popularity Similarity model (Papadopoulos et al., 2012), as implemented in the R package
NetHypGeom (Alanis-Lobato et al., 2016),wasexamined.However,perhapsbecauseofthat
model’s orientation to large scale-free networks, the implementation sets certain bounds on
its input parameters; in particular, the average degree must be ≥2 and the scaling exponent
must be ≥2and≤3. Of the fourteen real-world networks to be used as exemplars in this work
(see Table 5),onlyone(ZacharyKarateClub)hadvalues for these metrics that satisfied both
of these bounds; the other thirteen had an average degree < 2, a scaling exponent either < 2
or > 3, or both. Thus the exemplars to be used did not seem well suited to the capabilities of
the Popularity Similarity model, or its implementation.
On the other hand, the Configuration Model (CM), which was described earlier, produces
synthetic networks based upon the degree sequence of an exemplar network, and does not
consider personality. Because it is based on degree sequence, is usable with the exemplars.
Furthermore, it is considered by some to be a standard basis of comparison: “Following the
works of Barabási et al., the degree distribution has become accepted as the most fundamental
network characteristic…[I]t has become a standard to compare network quantities to a
null-model where the degrees of the network (the degree sequence) is fixed and everything else
random”(Barrenas et al., 2009).
Implementation and execution
This section describes the software implementation of the algorithms and supporting
functions. It also discusses their execution.
Implementation of the algorithms
The two new algorithms for finding effective personality type assignments (PS and CDM), as
well as the network generator GNAC algorithm, were implemented in the R language. R is an
open-source programming language and environment with powerful and extensive features
for data analysis, data visualization, and statistical computing (R Core Team 2016). R also in-
cludes a full range of general purpose programming language features, including control
structures, mathematical operations, and file input/output. It should be noted that for
medium and large networks, the network probability value P(G)computedbythePSalgo-
rithm can become quite small, as it is the product of n(n–1)/2 probabilities, all of which are
≤1. A computer implementation of P(G) meant to handle medium and large networks must
take care to avoid numeric underflow. In our implementation, we used the R gmp (GNU
Multiple Precision) package for arbitrary precision arithmetic.
As already mentioned, CM is an existing algorithm for generating synthetic social networks.
A prior implementation of CM in the R language is available in the R igraph package, which
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is a collection of R functions for network analysis and visualization (Csárdi and Nepusz,
2013). In that package function sample_degseq produces networks using CM. That function
was used for this work without modification.
Execution of the algorithms
Because R is an interpreted language, R programs often execute more slowly than comparable
programs written in a compiled language. In addition, the two algorithms to find effective per-
sonality type assignments (PS and CDM) both involve numerous iterations, especially the PS
algorithm. Consequently, the algorithms’run times during testing and analysis were some-
times quite lengthy. To keep the executions manageable, the programs were run on super-
computers provided and supported by the Alabama Supercomputer Authority. Typical run
times for the two algorithms were highly dependent on the number of nodes in the exemplar
graph; for the PS algorithm the run times ranged from a few minutes for the smallest
real-world network (Robins Australian Bank, 11 nodes) to several hours for the largest
real-world network (Lazega Law Firm, 71 nodes). Although the algorithms’implementation
code was not parallelized, scripts were used to initialize and initiate multiple instances of the
programs to execute concurrently.
Results
This section reports the results of testing and comparing the PS and CDM algorithms with
the Configuration Model. The comparison is in terms of quantitative measures of the gener-
ated social networks’realism.
Realism is measured by the absolute difference between the mean metrics of the synthetic
networks and the network metrics of the exemplar real-world social network. The metrics
used to measure realism are listed in Table 2. Smaller absolute difference is preferred. Abso-
lute differences between the metrics of the exemplar real-world social network and the mean
metrics of the synthetic networks were calculated for networks generated by the PS and
CDM algorithms and compared to networks generated by the CM algorithm.
As an example of the results, Table 6presents a comparison of the realism metrics for the
assignments found by the PS and CDM algorithms for only one of the real-world exemplar
networks, Bernard & Killworth Technical. (For brevity, this section presents the results for
only one of the exemplars in Table 6; the complete set of results are presented in Tables 9, 10,
11, 12, 13, 14, 15,16, 17, 18, 19, 20, 21, 22 in Appendix 2.) In the table, column 1 shows the
name of the metric and column 2 shows that metric’s value for the exemplar social network.
Columns 3–6 apply to the synthetic social networks generated by the CM algorithm, collect-
ively denoted F; column 3 shows the mean metric value for the networks generated by the
CM algorithm, column 4 show the absolute difference between that mean value and the ex-
emplar metric value, column 5 shows the L
1
norm for that metric, and column 6 shows the
L
2
norm for that metric. Columns 7–10 show the same for the synthetic networks generated
by the PS algorithm, collectively denoted P,andcolumns11–14showthesameforthesyn-
thetic social networks produced the CDM algorithm, collectively denoted M.Incolumns4–6,
8–10, and 12–14, the cells’content is set in bold type to show at a glance the PS- and
CDM-generated networks’realism compared to the CM-generated networks’realism. Bold
indicates that the PS or CDM networks’mean metric value was closer to the exemplar than
the CM networks’mean metric value.
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Table 6 Realism results for the Bernard & Killwork Technical network
Metrics TF|T-F| L1(F) L2(F)P|T-P| L1(P) L2(P)M|T-M| L1(M) L2(M)
Nodes 34.00 34.00 0.00 0.00 0.00 34.00 0.00 0.00 0.00 34.00 0.00 0.00 0.00
Links 175.00 143.63 31.37 941.00 173.03 175.00 0.00 0.00 0.00 175.00 0.00 0.00 0.00
Components 1.00 1.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00
Network density 0.31 0.26 0.06 1.68 0.31 0.31 0.00 0.00 0.00 0.31 0.00 0.00 0.00
Average degree 10.29 8.45 1.85 55.35 10.18 10.29 0.00 0.00 0.00 10.29 0.00 0.00 0.00
Standard deviation degree 4.63 3.55 1.08 32.41 5.97 5.03 0.40 12.12 2.45 5.00 0.37 11.02 2.31
Global cluster coefficient 0.48 0.30 0.17 5.17 0.95 0.44 0.03 0.98 0.20 0.45 0.03 0.77 0.16
Average cluster coefficient 0.47 0.32 0.16 4.76 0.88 0.53 0.05 1.61 0.32 0.53 0.06 1.69 0.33
Mean path length 1.81 1.90 0.09 2.70 0.52 1.77 0.04 1.08 0.21 1.78 0.03 0.95 0.19
Communities 4.00 6.37 2.37 75.00 16.82 7.37 3.37 103.00 21.10 6.50 2.50 77.00 16.70
Gini coefficient 0.49 0.49 0.01 1.42 0.31 0.50 0.02 1.18 0.27 0.51 0.02 1.49 0.32
Average betweenness 13.32 14.81 1.48 44.53 8.51 12.75 0.58 17.74 3.49 12.81 0.52 15.74 3.17
Maximum betweenness 63.29 53.03 10.26 368.03 75.25 104.94 41.65 1249.56 238.61 102.52 39.23 1176.86 229.79
Average closeness 0.02 0.02 0.00 0.03 0.01 0.02 0.00 0.01 0.00 0.02 0.00 0.01 0.00
Minimum closeness 0.01 0.01 0.00 0.03 0.01 0.01 0.00 0.02 0.00 0.01 0.00 0.02 0.00
Average eigencentrality 0.53 0.59 0.06 1.82 0.38 0.50 0.03 1.32 0.27 0.49 0.04 1.38 0.30
Minimum eigencentrality 0.06 0.06 0.00 0.44 0.10 0.07 0.01 0.46 0.10 0.07 0.01 0.45 0.10
Network radius 2.00 2.13 0.13 4.00 2.00 2.00 0.00 0.00 0.00 2.00 0.00 0.00 0.00
Average eccentricity 2.88 3.08 0.19 5.79 1.33 2.79 0.09 3.74 0.85 2.80 0.08 3.68 0.80
Network diameter 4.00 3.97 0.03 3.00 1.73 3.47 0.53 16.00 4.00 3.47 0.53 16.00 4.00
Boldfaced numbers indicate which algorithm performed better for a particular metric
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AscanbeseeninTable6, for the Bernard & Killworth. Technical exemplar network both
thePSandtheCDMalgorithmsproducedmorerealistic synthetic social networks than the
CM algorithm over the majority of the network metrics.
Tabl e 7summarizes the overall realism results. Two realism comparisons were made: PS
versus CM and CDM versus CM. Both are reported in the table. A total of 280 metric values
(14 real-world social networks · 20 metrics) were calculated for each of the comparisons. The
columns labeled with an algorithm’s abbreviation (PS, CDM, CM) show the number of met-
rics where that algorithm’s metric values were closer to the exemplar that the other algorithm
in the comparison, and a column labeled “=”shows the number where the two algorithms’
metric values were equally close. In the PS versus CM comparison, the values of 142 of the
280 metrics (~ 50.7%) for the PS networks were closer to the values of the exemplar network
than those of the CM algorithm, and another 31 values (~ 11.1%) were equally close; the CM
networks values were closer to the exemplar on only 107 (~ 28.2%) of the metrics. In the
CDM versus CM comparison, the values of 140 of the 280 metrics (50.0%) for the DCM net-
works were closer to the values of the exemplar network than those of the CM algorithm,
and another 35 values (~ 12.5%) were equally close; the CM networks values were closer to
the exemplar on only 105 (~ 37.5%) of the metrics.
A simple hypothesis test of proportion confirms that both PS and CDM come closer to
the exemplar than CM more often that can be expected from random chance. For PS versus
CM we treat each of the 280 metrics as a binomial trial. A closer metric value in a
PS-generated network is counted as a success, a closer metric value in a CM-generated net-
work is counted as a failure, and equal metric values are omitted from the sample. In a
right-tailed test the hypotheses are H
0
:p=0.50 and H
1
:p> 0.50, so the statistical assump-
tion is that PS is not better than CM. The level of significance is set to α= 0.05. The sample
data is r= 142 and n= 142 + 107 = 249. The results are test statistic p= 0.570281,
z= 2.218035, and p-value = 0.01326, which is < α, thus we reject the null
hypothesis and conclude that PS outperforms CM. The same test applied to
Table 7 Realism results summary
Exemplar Real-World Social Network PS vs. CM CDM vs. CM
PS CM = CDM CM =
Robins Australian Bank 15 4 1 14 5 1
Roethlisberger & Dickson Bank Wiring Room 9 10 1 9 9 2
Thurman Office 13 6 1 14 5 1
Sampson Monastery 10 8 2 7 9 4
Krackhardt Office CSS 9 10 1 10 9 1
Krackhardt High-Tech Managers 11 8 1 9 9 2
Schwimmer Taro Exchange 5 14 1 5 14 1
Webster Accounting Firm 9 9 2 9 9 2
Zachary Karate Club 9 8 3 10 8 2
Bernard & Killworth Technical 13 5 2 13 5 2
Bernard & Killworth Office 11 6 3 11 6 3
Krebs Fortune 500 IT Department (Advice) 9 8 3 10 7 3
Krebs Fortune 500 IT Department (Business) 7 9 4 8 7 5
Lazega Law Firm 12 2 6 11 3 6
Total 142 107 31 140 105 35
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CDM versus CM has r=140 and n= 140 + 105 = 245. The results are test statistic
p= 0.571429, z= 2.236068, and p-value = 0.01264, which is again < α,thuswe
again reject the null hypothesis and conclude that CDM outperforms CM.
To support the quantitative realism results at an intuitive level, Fig. 3presents an example
visual comparison of a real world social network with a randomly generated network and two
networks that were generated using a personality compatibility table. Figure 3ashowsthe
Robins Australian Bank social network (Pattison et al., 2000). Figure 3b shows a network that
was generated using the random G(n,p) algorithm. That network has the same number of
nodes and network density as the exemplar real world social network. Figure 3cshowsasyn-
thetic social network generated using an assignment of personality types found by the PS algo-
rithm. Figure 3d shows a synthetic social network generated using an assignment of
personality types found by the CDM algorithm. In the figure, node communities found by the
walktrap.community function in the R igraph package are depicted with bounding boxes
around them. A visual inspection of the networks in the reveals what appear to be more real-
istic communities within Fig. 3candd.
Conclusions and future work
This section states the conclusions of this work and suggests possible future work.
Conclusions
The PS and CDM algorithms differ from most prior work on generating synthetic social
network in a significant way. Most prior algorithms do not consider the attributes of the
nodes, or of the people or entities the nodes represent, when adding links; instead they are
Fig. 3 Visual comparison of the real world and synthetic social networks
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based on retaining or replicating some of the structural characteristics of the exemplar net-
work in the synthetic networks. For example, CM is given a degree sequence, which may be
the actual degree sequence of the real-world network serving as an exemplar (Newman
2003). In contrast, the PS and CDM algorithms use the attributes of the nodes, in particular
the personality types assigned to them, as the primary driver of their calculations.
From the quantitative results, it is evident that both the PS and the CDM algorithms,
which use personality compatibility information, generate more realistic synthetic social
networks than the CM algorithm, which does not. The PS and CDM algorithms are
quite similar in terms of realism. However, the CDM algorithm is much more compu-
tationally efficient, requiring substantially shorter execution times for large networks.
Either PS or CDM could be used with small to medium exemplars; for exemplars with
more than ~ 40 nodes, PS becomes impractical, at least in its current implementation.
Close examination of the results in Table 6show that the PS and CDM both performed
worst on the Schwimer Taro Exchange exemplar. It is unlikely to be a coincidence that in
that network only among the fourteen exemplars the nodes correspond not to individual
people, but to households, which is intuitively not as good a fit with personality-based algo-
rithms. Thus PS and CDM, or future enhancements of them, should be considered when
the nodes correspond to individual people and personality compatibility is expected to have
a significant effect on whether two people have the relationship that a link represents.
Future work
Because the PS and CDM algorithms both produce personality assignments that are then
input to the GNAC algorithm to generate synthetic social networks, we make two conjec-
tures that motivate future work. First, we conjecture that any method to find an effective
personalitytypeassignmentAcould be combined with the GNAC algorithm to synthesize
realistic social networks. Second, we conjecture that the method does not depend on a sin-
gle personality type scheme, such as the MBTI scheme used in this work. Rather, we believe
that any personality type scheme from which a personality compatibility table is available or
can be inferred could be combined with the PS and CDM algorithms to generate realistic
synthetic social networks. For example, a similar table construction process could be applied
totheOCEANpersonalitytypemodel,withtheadditional preliminary step of discretizing
continuous scales for each personality factors into a finite number of discrete values or
intervals.
In this work all of the social networks were treated as symmetric and unweighted. As an
obvious generalization, applying these methods to asymmetric and/or weighted social net-
works is an opportunity for future work. Because multiple metrics generated in a single ex-
periment are analyzed, the multiple comparison problem may be present, and suitable
methods to compensate for it could be employed. Finally, the assumption in the PS algo-
rithm that the nodes can be distinguished could be changed to consider networks that con-
nect the same personality types in the same way, as opposed to connecting the same nodes
in the same way, as equivalent. (This is analogous to color isomorphism in graph theory
terms.) Changing the assumption would change the formula for calculating the network
probability P(G).
Finally, according to (Aiello et al., 2012), there has been considerable research aimed at
predicting the overall evolution of social networks, but very few attempts to predict future
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connections of individual people within such networks. Within an organization, managers
may wish to create a new project team or work group. The methods developed in this work
could be applied to simulating the potential formation of social networks within the team
or group, given a set of personality types and a compatibility table. We speculate that gener-
ating synthetic social networks using individuals’personality types has the potential to lead
to a predictive or semi-predictive capability to anticipate the future social network that
could emerge in a team or group. If such a capability was sufficiently reliable, managers
could use its predictions when considering personnel assignments. This idea requires of
careful validation, perhaps by comparing predicted social networks to actual social networks
for existing teams or groups.
Appendix 1
Constructing a personality compatibility table for the MBTI
This appendix details the process used to construct a personality compatibility table for the
16 MBTI personality types. The process had these steps:
1. Identify a set of environmental factors that are important in determining
personality compatibility; for this work eight such factors were identified.
2. Interpret the personality model to determine each personality type’s opinion
regarding each of the environmental factors.
3. Perform pair-wise comparisons of 16 MBTI personality types to determine the
number of shared or consistent opinions regarding the environmental factors be-
tween each pair of personality types.
4. Scale the counts of common opinions into probabilities of link formation for the
compatibility table.
In the first step, environmental factors important in determining personality compatibility
were identified by examining the sources describing the personality model. Within a work-
place environment, the factors that may determine compatibility of colleagues include:
Authority; a tendency to respect or work with the chain of command.
Communication; a tendency to value accurate and specific vernacular.
Consideration; a tendency to respect or incorporate other people’s opinions.
Empathy; a tendency to recognize or synchronize with other people’s feelings.
Harmony; a tendency to tolerate or relieve interpersonal tensions.
Loyalty; tendency to value relationships and defend alliances.
Productivity; a tendency to value efficient processes or creating something.
Rules; a tendency to follow and defend documented procedures.
The following quotations from (Keirsey 1998) illustrate the source content from which
the environmental factors could be identified and the various personality types’likely opin-
ions of them were determined. Environmental factors noted after each quotation indicate
that the associated MBTI may have positive or negative attitude about those factors.
Promoters (ESTP) “[havea]lowtoleranceforanxietyandareapttoleaverelationships
that are filled with interpersonal tensions.”(Harmony, Loyalty)
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Composers (ISFP) “will put up with a lot more interpersonal tensions than other
Artisans”(Harmony, Loyalty).
Crafters (ISTP) “can be fiercely insubordinate, seeing hierarchy and authority as
unnecessary and even irksome.”(Authority, Rules)
Performers (ESFP) “tolerance for anxiety is the lowest of all the types, and they will
avoid worries and troubles by ignoring the unhappiness of a situation as long as
possible.”(Harmony, Productivity)
Supervisors (ESTJ) “may not always be responsive to points of view and emotions of
others and have a tendency to jump to conclusions too quickly.”(Authority, Productivity)
Providers (ESFJ) “tend to listen to acknowledged authorities on abstract matters, and
often rely on officially sanctioned views as the source of their opinions and attitudes.”
(Authority, Rules)
Inspectors (ISTJ) “Because of [being adamant about rule compliance,] they are often
misjudged as having ice in their veins, for people fail to see their good intentions and
their vulnerability to criticism.”(Authority, Rules)
Protectors (ISFJ) “know the value of a dollar and abhor the squandering or misuse of
resources.”(Productivity)
Teachers (ENFJ) “When [they] find that their position or beliefs were not
comprehended or accepted, they are surprised, puzzled, and sometimes hurt.”
(Communications, Harmony, Consideration)
Counselors (INFJ) “value staff harmony and want an organization to run smoothly and
pleasantly, making every effort themselves to contribute to that end.”(Harmony,
Consideration, Productivity)
Champions (ENFP) “Sometimes [they] get impatient with their superiors; and they will
occasionally side with detractors of their organization, who find in them a sympathetic
ear and a natural rescuer.”(Authority, Communication, Empathy)
Healers (INFP) “have difficulty thinking in conditional ‘if-then’terms; they tend to see
things as either black or white, and can be impatient with contingency.”
(Communication, Empathy, Consideration)
Fieldmarshals (ENTJ) “For the [Fieldmarshall], there must always be a reason for doing
anything, and peoples’feelings usually are not sufficient reason.”(Authority, Rules,
Productivity)
Masterminds (INTJ) “Colleagues may describe [Masterminds] as unemotional and, at
times, cold and dispassionate, when in truth they are merely taking the goals of an
institution seriously, and continually striving to achieve those goals.”(Productivity, Rules)
Inventors (ENTP) “If an [Inventor’s] job becomes dull and repetitive, they tend to lose
interest and fail to follow through -- often to the discomfort of colleagues.”(Productivity)
Architect (INTP) “It is difficult for an [Architect] to listen to nonsense, even in a
casual conversation, without pointing out the speaker’s error, and this makes
communication with them an uncomfortable experience for many.”
(Communication, Consideration)
Based on these quotes and other similar descriptions of the personality types, their likely
opinions regarding the environmental factors were determined. Table 8shows the result.
The Keirsey temperaments scheme groups the 16 possible MBTI personality types into four
categories, referred to as Artisans, Guardians, Idealists, and Rationals (Keirsey, 1998); the
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table is organized by those categories. In the table, a 0 indicates that people of the personal-
ity type are likely to hold a low or negative opinion of the environmental factor, whereas a 1
indicates a relatively high or positive opinion.
For each pair of personality types Xand Y, the number of environmental factors on which
they agreed (both had 0 or both had 1 in the table) was calculated; let that value be denoted
as a(X,Y), with a(X,Y)∈{0, 1, 2, …, 6}. (The pairwise comparison considered six environ-
mental factors, hence six was the maximum number of possible agreements. The maximum
number of agreed upon factors by any pair of two distinct personality types was actually
five.) The probability of a link forming between personality types X and Y was calculated as
pX;YðÞ¼0:5∙1þerf x−μðÞ
σ∙ffiffiffi
2
p

! !
where erfðxÞ¼ 2
ffiffiffi
π
pRx
0e−t2dt is the Gauss error function, μ≈2.9747, and σ≈1.8185.
The values for μand σwere determined empirically. The result of this formula is that
0.05 ≤p(X,Y)≤0.95 for all personality types Xand Y, leaving a small but non-zero probability
(0.05) of a link forming and a small probability of link not forming (also 0.05) between any
two personality types. The p(X,Y) values were recorded in the personality compatibility table.
The resulting personality compatibility table produced by this process and used in this work
was shown earlier in Table 4.
Other methods of determining the compatibility table values are possible, of course. The
synthetic social network generation algorithm will operate with any reasonable and internally
consistent compatibility table.
Table 8 Inferred MBTI personality types’opinions of environmental factors
Category Personality type Environmental factor
Authority Communication Harmony Loyalty Productivity Rules
Artisans Promoter ESTP 0 1 0 0 0 0
Composer ISFP 0 0 1 1 1 0
Crafter ISTP 0 1 0 1 1 1
Performer ISFP 1 0 0 0 0 1
Guardians Supervisor ESTJ 1 1 0 1 1 1
Provider ESFJ 1 0 0 1 0 1
Inspector ISTJ 1 1 0 1 0 1
Protector ISFJ 1 1 0 1 1 1
Idealists Teacher ENFJ 1 1 1 0 0 0
Counselor INFJ 0 0 1 1 0 0
Champion ENFP 0 0 0 1 1 0
Healer INFP 0 0 1 0 0 1
Rationals Fieldmarshal ENTJ 1 0 1 1 0 0
Mastermind INTJ 0 1 1 0 1 0
Inventor ENTP 0 1 0 0 1 0
Architect INTP 0 1 1 0 1 1
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Table 9 Realism results for the Robins Australian Bank social network
Metrics TF|T-F| L1(F) L2(F)P|T-P| L1(P) L2(P)M|T-M| L1(M) L2(M)
Nodes 11.00 11.00 0.00 0.00 0.00 11.00 0.00 0.00 0.00 11.00 0.00 0.00 0.00
Links 16.00 12.83 3.17 95.00 18.41 16.00 0.00 0.00 0.00 16.00 0.00 0.00 0.00
Components 1.00 1.13 0.13 4.00 2.00 1.07 0.07 2.00 1.41 1.47 0.47 14.00 3.74
Network density 0.29 0.23 0.06 1.73 0.34 0.29 0.00 0.00 0.00 0.29 0.00 0.00 0.00
Average degree 2.91 2.33 0.58 17.27 3.35 2.91 0.00 0.00 0.00 2.91 0.00 0.00 0.00
Standard deviation degree 1.87 1.27 0.60 17.92 3.49 1.75 0.12 8.79 1.93 1.82 0.05 7.46 1.70
Global cluster coefficient 0.38 0.14 0.24 7.29 1.48 0.38 0.01 1.27 0.32 0.45 0.07 2.76 0.61
Average cluster coefficient 0.41 0.17 0.24 7.46 1.55 0.63 0.23 6.80 1.29 0.66 0.26 7.73 1.48
Mean path length 2.02 2.73 0.71 21.20 6.28 2.13 0.11 6.55 2.27 2.75 0.74 23.69 5.83
Communities 3.00 3.47 0.47 18.00 5.10 2.47 0.53 18.00 4.24 3.13 0.13 14.00 4.47
Gini coefficient 0.24 0.15 0.10 2.91 0.66 0.12 0.12 3.72 0.76 0.20 0.04 2.85 0.62
Average betweenness 5.09 6.44 1.35 53.64 11.35 5.06 0.04 21.09 4.75 4.52 0.57 29.00 6.08
Maximum betweenness 25.17 21.60 3.56 153.90 35.15 26.26 1.09 140.33 31.75 22.18 2.99 159.33 34.81
Average closeness 0.05 0.05 0.01 0.24 0.05 0.05 0.00 0.11 0.03 0.06 0.00 0.18 0.04
Minimum closeness 0.04 0.03 0.01 0.25 0.05 0.04 0.00 0.11 0.03 0.04 0.00 0.16 0.04
Average eigencentrality 0.49 0.54 0.05 1.91 0.46 0.55 0.06 1.96 0.43 0.56 0.07 2.11 0.44
Minimum eigencentrality 0.14 0.15 0.01 2.08 0.46 0.22 0.08 2.66 0.54 0.20 0.06 2.05 0.43
Network radius 2.00 2.67 0.67 20.00 4.69 2.10 0.10 3.00 1.73 2.00 0.00 2.00 1.41
Average eccentricity 3.09 3.78 0.69 23.73 5.26 3.01 0.09 10.73 2.32 2.86 0.23 11.55 2.56
Network diameter 4.00 4.73 0.73 26.00 6.48 3.57 0.43 17.00 4.12 3.57 0.43 13.00 3.87
Boldfaced numbers indicate which algorithm performed better for a particular metric
Appendix 2
Detailed realism results
The following tables report the detailed realism results for all fourteen of the real-world social networks used as exemplars.
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Table 10 Realism results for the Roethlisberger & Dickson Bank Wiring Room social network
Metrics TF|T-F| L1(F) L2(F)P|T-P| L1(P) L2(P)M|T-M| L1(M) L2(M)
Nodes 14.00 14.00 0.00 0.00 0.00 14.00 0.00 0.00 0.00 14.00 0.00 0.00 0.00
Links 13.00 10.43 2.57 77.00 14.93 13.00 0.00 0.00 0.00 13.00 0.00 0.00 0.00
Components 6.00 6.10 0.10 3.00 1.73 6.43 0.43 13.00 3.61 6.10 0.10 3.00 1.73
Network density 0.14 0.12 0.03 0.85 0.16 0.14 0.00 0.00 0.00 0.14 0.00 0.00 0.00
Average degree 1.86 1.49 0.37 11.00 2.13 1.86 0.00 0.00 0.00 1.86 0.00 0.00 0.00
Standard deviation degree 1.61 1.35 0.26 7.90 1.65 1.85 0.24 7.30 1.63 1.74 0.13 4.38 1.07
Global cluster coefficient 0.64 0.16 0.48 14.44 2.76 0.45 0.20 5.88 1.24 0.45 0.19 5.80 1.14
Average cluster coefficient 0.71 0.17 0.53 16.02 3.08 0.63 0.08 4.89 0.95 0.64 0.07 3.29 0.71
Mean path length 9.34 9.49 0.15 5.78 2.23 9.62 0.28 13.51 3.10 9.32 0.02 5.84 1.66
Communities 7.00 7.73 0.73 22.00 5.29 7.60 0.60 18.00 4.69 7.60 0.60 18.00 4.90
Gini coefficient 0.37 0.32 0.04 1.40 0.34 0.34 0.03 0.85 0.26 0.32 0.05 1.55 0.39
Average betweenness 3.14 3.16 0.01 18.57 4.12 1.73 1.42 42.50 8.28 2.28 0.86 25.93 5.22
Maximum betweenness 16.00 12.84 3.16 101.42 25.42 13.31 2.69 159.33 35.77 15.96 0.04 117.17 25.69
Average closeness 0.06 0.06 0.00 0.20 0.05 0.08 0.02 0.64 0.13 0.07 0.01 0.33 0.07
Minimum closeness 0.04 0.04 0.00 0.17 0.05 0.06 0.02 0.65 0.13 0.05 0.01 0.30 0.06
Average eigencentrality 0.59 0.65 0.05 1.98 0.43 0.68 0.08 2.67 0.69 0.65 0.06 2.44 0.52
Minimum eigencentrality 0.20 0.21 0.01 2.22 0.54 0.34 0.14 4.15 0.93 0.30 0.10 3.21 0.73
Network radius 3.00 2.67 0.33 12.00 3.46 1.83 1.17 35.00 6.71 2.00 1.00 30.00 5.48
Average eccentricity 2.57 2.38 0.19 10.50 2.37 1.55 1.02 30.71 5.87 1.91 0.67 20.00 3.85
Network diameter 5.00 4.57 0.43 19.00 4.58 3.00 2.00 60.00 11.40 3.57 1.43 43.00 8.31
Boldfaced numbers indicate which algorithm performed better for a particular metric
O’Neil and Petty Applied Network Science (2019) 4:19 Page 32 of 49
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Table 11 Realism results for the Thurman Office social network
Metrics TF|T-F| L1(F) L2(F)P|T-P| L1(P) L2(P)M|T-M| L1(M) L2(M)
Nodes 15.00 15.00 0.00 0.00 0.00 15.00 0.00 0.00 0.00 15.00 0.00 0.00 0.00
Links 33.00 25.53 7.47 224.00 42.07 33.00 0.00 0.00 0.00 33.00 0.00 0.00 0.00
Components 1.00 1.07 0.07 2.00 1.41 1.00 0.00 0.00 0.00 1.03 0.03 1.00 1.00
Network density 0.31 0.24 0.07 2.13 0.40 0.31 0.00 0.00 0.00 0.31 0.00 0.00 0.00
Average degree 4.40 3.40 1.00 29.87 5.61 4.40 0.00 0.00 0.00 4.40 0.00 0.00 0.00
Standard deviation degree 2.53 1.78 0.75 22.57 4.29 3.03 0.50 15.07 2.91 2.94 0.41 12.43 2.39
Global cluster coefficient 0.52 0.25 0.27 7.96 1.51 0.47 0.05 1.47 0.32 0.47 0.05 1.46 0.32
Average cluster coefficient 0.48 0.28 0.20 6.34 1.28 0.73 0.25 7.49 1.38 0.71 0.23 6.90 1.29
Mean path length 1.88 2.27 0.39 11.83 3.16 1.80 0.08 2.47 0.50 1.86 0.02 3.67 1.63
Communities 3.00 3.77 0.77 31.00 7.42 3.87 0.87 48.00 10.10 3.63 0.63 35.00 8.43
Gini coefficient 0.18 0.22 0.05 2.44 0.57 0.28 0.11 4.03 0.80 0.28 0.11 3.48 0.76
Average betweenness 6.13 8.02 1.89 56.67 11.24 5.61 0.53 17.27 3.53 5.59 0.54 16.73 3.67
Maximum betweenness 37.25 28.64 8.61 282.45 57.01 47.22 9.97 312.71 64.49 47.27 10.03 307.51 65.38
Average closeness 0.04 0.03 0.01 0.14 0.03 0.04 0.00 0.06 0.01 0.04 0.00 0.06 0.01
Minimum closeness 0.03 0.02 0.01 0.18 0.04 0.03 0.00 0.06 0.01 0.03 0.00 0.04 0.01
Average eigencentrality 0.53 0.54 0.01 1.08 0.24 0.49 0.04 1.21 0.24 0.49 0.04 1.21 0.24
Minimum eigencentrality 0.11 0.12 0.01 1.39 0.29 0.15 0.04 1.46 0.29 0.15 0.04 1.68 0.32
Network radius 2.00 2.67 0.67 20.00 4.47 2.00 0.00 0.00 0.00 2.00 0.00 0.00 0.00
Average eccentricity 2.80 3.48 0.68 20.33 4.09 2.67 0.13 7.13 1.45 2.65 0.15 5.93 1.31
Network diameter 3.00 4.30 1.30 39.00 7.68 3.27 0.27 8.00 2.83 3.10 0.10 3.00 1.73
Boldfaced numbers indicate which algorithm performed better for a particular metric
O’Neil and Petty Applied Network Science (2019) 4:19 Page 33 of 49
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Table 12 Realism results for the Sampson Monastery social network
Metrics TF|T-F| L1(F) L2(F)P|T-P| L1(P) L2(P)M|T-M| L1(M) L2(M)
Nodes 18.00 18.00 0.00 0.00 0.00 18.00 0.00 0.00 0.00 18.00 0.00 0.00 0.00
Links 41.00 34.07 6.93 208.00 39.19 41.00 0.00 0.00 0.00 41.00 0.00 0.00 0.00
Components 1.00 1.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00
Network density 0.27 0.22 0.05 1.36 0.26 0.27 0.00 0.00 0.00 0.27 0.00 0.00 0.00
Average degree 4.56 3.79 0.77 23.11 4.36 4.56 0.00 0.00 0.00 4.56 0.00 0.00 0.00
Standard deviation degree 2.09 1.55 0.55 16.35 3.25 2.84 0.75 22.36 4.35 2.99 0.90 26.91 5.12
Global cluster coefficient 0.26 0.20 0.07 2.31 0.50 0.36 0.10 2.93 0.56 0.37 0.11 3.20 0.60
Average cluster coefficient 0.29 0.21 0.07 2.75 0.59 0.63 0.34 10.22 1.92 0.66 0.38 11.36 2.10
Mean path length 1.97 2.15 0.18 5.33 1.06 1.83 0.14 4.07 0.77 1.82 0.15 4.54 0.87
Communities 3.00 3.67 0.67 28.00 7.07 3.60 0.60 22.00 5.10 3.83 0.83 25.00 5.75
Gini coefficient 0.07 0.19 0.12 3.52 0.82 0.18 0.10 3.21 0.66 0.19 0.11 3.46 0.72
Average betweenness 8.22 9.73 1.51 45.33 9.00 7.07 1.15 34.56 6.54 6.94 1.29 38.56 7.39
Maximum betweenness 37.62 36.81 0.81 170.51 39.48 78.27 40.65 1219.35 228.02 82.56 44.94 1348.16 250.19
Average closeness 0.03 0.03 0.00 0.08 0.02 0.03 0.00 0.07 0.01 0.03 0.00 0.08 0.02
Minimum closeness 0.02 0.02 0.00 0.07 0.02 0.03 0.00 0.09 0.02 0.03 0.00 0.10 0.02
Average eigencentrality 0.48 0.50 0.02 1.32 0.31 0.42 0.06 1.89 0.36 0.41 0.07 2.06 0.38
Minimum eigencentrality 0.17 0.16 0.01 1.78 0.40 0.23 0.06 1.79 0.37 0.22 0.05 1.92 0.40
Network radius 2.00 2.80 0.80 24.00 4.90 1.97 0.03 1.00 1.00 1.93 0.07 2.00 1.41
Average eccentricity 3.00 3.40 0.40 12.50 2.56 2.65 0.35 10.50 2.20 2.60 0.40 12.39 2.67
Network diameter 4.00 4.07 0.07 6.00 2.45 3.00 1.00 30.00 5.66 3.07 0.93 28.00 5.66
Boldfaced numbers indicate which algorithm performed better for a particular metric
O’Neil and Petty Applied Network Science (2019) 4:19 Page 34 of 49
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Table 13 Realism results for the Krackhardt Office CSS social network
Metrics TF|T-F| L1(F) L2(F)P|T-P| L1(P) L2(P)M|T-M| L1(M) L2(M)
Nodes 21.00 21.00 0.00 0.00 0.00 21.00 0.00 0.00 0.00 21.00 0.00 0.00 0.00
Links 14.00 12.50 1.50 45.00 9.75 14.00 0.00 0.00 0.00 14.00 0.00 0.00 0.00
Components 9.00 9.43 0.43 17.00 4.58 9.13 0.13 4.00 2.00 9.17 0.17 5.00 2.24
Network density 0.07 0.06 0.01 0.21 0.05 0.07 0.00 0.00 0.00 0.07 0.00 0.00 0.00
Average degree 1.33 1.19 0.14 4.29 0.93 1.33 0.00 0.00 0.00 1.33 0.00 0.00 0.00
Standard deviation degree 1.39 1.18 0.21 6.23 1.38 2.08 0.69 20.55 3.93 2.11 0.72 21.70 4.06
Global cluster coefficient 0.13 0.08 0.05 2.91 0.60 0.14 0.01 0.69 0.14 0.13 0.01 0.59 0.13
Average cluster coefficient 0.16 0.11 0.05 3.80 0.76 0.68 0.53 15.78 2.90 0.70 0.55 16.38 3.01
Mean path length 15.84 16.45 0.61 41.12 8.83 14.37 1.48 44.27 8.77 14.53 1.31 41.18 8.38
Communities 10.00 11.33 1.33 40.00 8.72 10.47 0.47 14.00 3.74 10.50 0.50 17.00 4.36
Gini coefficient 0.40 0.33 0.07 2.04 0.43 0.42 0.02 0.77 0.17 0.42 0.02 0.80 0.18
Average betweenness 3.67 3.86 0.20 53.33 12.11 3.90 0.23 27.43 5.73 3.51 0.16 22.48 5.23
Maximum betweenness 22.50 25.14 2.64 315.33 71.60 55.97 33.47 1004.0 186.61 54.70 32.20 966.0 182.25
Average closeness 0.04 0.06 0.01 0.66 0.24 0.04 0.00 0.17 0.04 0.05 0.00 0.17 0.05
Minimum closeness 0.03 0.04 0.01 0.52 0.19 0.04 0.00 0.18 0.05 0.04 0.01 0.21 0.06
Average eigencentrality 0.47 0.52 0.05 2.33 0.56 0.39 0.08 2.36 0.45 0.39 0.09 2.54 0.48
Minimum eigencentrality 0.11 0.17 0.07 2.62 0.64 0.25 0.14 4.30 0.82 0.25 0.14 4.19 0.81
Network radius 3.00 2.87 0.13 14.00 4.00 1.83 1.17 35.00 6.71 1.83 1.17 35.00 6.71
Average eccentricity 2.33 2.41 0.08 13.00 3.07 1.77 0.57 17.05 3.68 1.66 0.67 20.10 3.99
Network diameter 5.00 5.20 0.20 26.00 6.16 3.20 1.80 54.00 10.58 3.00 2.00 60.00 11.40
Boldfaced numbers indicate which algorithm performed better for a particular metric
O’Neil and Petty Applied Network Science (2019) 4:19 Page 35 of 49
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Table 14 Realism results for the Krackhardt High-Tech Managers social network
Metrics TF|T-F| L1(F) L2(F)P|T-P| L1(P) L2(P)M|T-M| L1(M) L2(M)
Nodes 21.00 21.00 0.00 0.00 0.00 21.00 0.00 0.00 0.00 21.00 0.00 0.00 0.00
Links 36.00 31.47 4.53 136.00 26.42 36.00 0.00 0.00 0.00 36.00 0.00 0.00 0.00
Components 5.00 5.00 0.00 0.00 0.00 5.10 0.10 3.00 1.73 5.00 0.00 0.00 0.00
Network density 0.17 0.15 0.02 0.65 0.13 0.17 0.00 0.00 0.00 0.17 0.00 0.00 0.00
Average degree 3.43 3.00 0.43 12.95 2.52 3.43 0.00 0.00 0.00 3.43 0.00 0.00 0.00
Standard deviation degree 2.14 1.88 0.26 7.82 1.56 2.70 0.56 16.83 3.18 2.72 0.59 17.60 3.37
Global cluster coefficient 0.50 0.19 0.31 9.23 1.70 0.37 0.13 3.83 0.72 0.39 0.10 3.12 0.60
Average cluster coefficient 0.59 0.20 0.39 11.63 2.14 0.60 0.02 1.44 0.32 0.63 0.05 1.79 0.41
Mean path length 8.98 8.79 0.18 5.71 1.10 8.81 0.17 11.42 2.45 8.67 0.30 9.13 1.68
Communities 7.00 7.73 0.73 24.00 6.48 7.63 0.63 23.00 5.75 7.53 0.53 20.00 4.69
Gini coefficient 0.44 0.41 0.03 1.48 0.34 0.42 0.02 1.09 0.27 0.41 0.03 1.17 0.26
Average betweenness 9.29 7.46 1.83 57.10 10.97 6.11 3.18 95.29 17.67 6.24 3.04 91.33 16.82
Maximum betweenness 44.67 27.47 17.20 544.5 104.29 57.55 12.88 394.28 87.09 65.20 20.54 636.21 127.4
Average closeness 0.03 0.03 0.00 0.10 0.02 0.03 0.01 0.20 0.04 0.03 0.01 0.19 0.04
Minimum closeness 0.02 0.02 0.00 0.14 0.03 0.03 0.01 0.18 0.04 0.02 0.01 0.16 0.03
Average eigencentrality 0.45 0.58 0.13 3.83 0.76 0.47 0.02 0.96 0.23 0.44 0.01 0.80 0.16
Minimum eigencentrality 0.04 0.19 0.15 4.56 0.91 0.15 0.11 3.21 0.63 0.14 0.10 2.91 0.58
Network radius 3.00 2.83 0.17 5.00 2.24 2.07 0.93 28.00 5.29 2.07 0.93 28.00 5.29
Average eccentricity 3.38 2.79 0.59 19.29 3.62 2.45 0.93 27.95 5.22 2.44 0.94 28.10 5.24
Network diameter 5.00 4.20 0.80 28.00 5.29 3.77 1.23 37.00 7.42 3.63 1.37 41.00 8.19
Boldfaced numbers indicate which algorithm performed better for a particular metric
O’Neil and Petty Applied Network Science (2019) 4:19 Page 36 of 49
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Table 15 Realism results for the Schwimmer Taro Exchange social network
Metrics TF|T-F| L1(F) L2(F)P|T-P| L1(P) L2(P)M|T-M| L1(M) L2(M)
Nodes 22.00 22.00 0.00 0.00 0.00 22.00 0.00 0.00 0.00 22.00 0.00 0.00 0.00
Links 39.00 35.47 3.53 106.0 20.98 39.00 0.00 0.00 0.00 39.00 0.00 0.00 0.00
Components 1.00 1.00 0.00 0.00 0.00 1.30 0.30 9.00 3.32 1.23 0.23 7.00 2.65
Network density 0.17 0.15 0.02 0.46 0.09 0.17 0.00 0.00 0.00 0.17 0.00 0.00 0.00
Average degree 3.55 3.22 0.32 9.64 1.91 3.55 0.00 0.00 0.00 3.55 0.00 0.00 0.00
Standard deviation degree 0.96 0.96 0.00 2.90 0.69 2.65 1.69 50.71 9.33 2.69 1.73 51.78 9.49
Global cluster coefficient 0.28 0.11 0.17 4.99 0.96 0.33 0.06 1.77 0.37 0.32 0.05 1.35 0.29
Average cluster coefficient 0.34 0.11 0.23 6.88 1.30 0.72 0.38 11.43 2.10 0.72 0.38 11.29 2.08
Mean path length 2.49 2.66 0.16 5.03 1.22 2.91 0.42 20.68 6.67 2.78 0.29 18.58 5.91
Communities 5.00 4.97 0.03 17.00 4.58 5.23 0.23 17.00 4.36 5.33 0.33 20.00 5.10
Gini coefficient 0.13 0.20 0.07 2.43 0.53 0.20 0.08 2.29 0.51 0.21 0.08 2.48 0.55
Average betweenness 15.68 17.41 1.73 52.77 12.79 12.95 2.73 81.91 17.33 12.86 2.82 84.64 17.09
Maximum betweenness 46.38 53.76 7.38 319.93 80.14 157.15 110.76 3322.93 613.21 157.83 111.44 3343.28 615.46
Average closeness 0.02 0.02 0.00 0.03 0.01 0.02 0.00 0.09 0.02 0.02 0.00 0.09 0.02
Minimum closeness 0.02 0.01 0.00 0.07 0.02 0.02 0.00 0.03 0.01 0.02 0.00 0.02 0.01
Average eigencentrality 0.62 0.51 0.10 3.08 0.64 0.33 0.29 8.69 1.59 0.33 0.29 8.67 1.58
Minimum eigencentrality 0.32 0.15 0.16 4.92 1.01 0.07 0.24 7.28 1.34 0.08 0.23 7.00 1.28
Network radius 3.00 3.40 0.40 12.00 3.46 2.17 0.83 25.00 5.00 2.20 0.80 24.00 4.90
Average eccentricity 4.09 4.40 0.31 11.14 2.86 3.32 0.77 23.32 4.59 3.32 0.77 23.00 4.45
Network diameter 5.00 5.40 0.40 16.00 4.47 4.17 0.83 25.00 5.00 4.20 0.80 24.00 4.90
Boldfaced numbers indicate which algorithm performed better for a particular metric
O’Neil and Petty Applied Network Science (2019) 4:19 Page 37 of 49
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Table 16 Realism results for the Webster Accounting Firm social network
Metrics TF|T-F| L1(F) L2(F)P|T-P| L1(P) L2(P)M|T-M| L1(M) L2(M)
Nodes 24.00 24.00 0.00 0.00 0.00 24.00 0.00 0.00 0.00 24.00 0.00 0.00 0.00
Links 150.0 104.7 45.30 1359.0 249.42 150.0 0.00 0.00 0.00 150.0 0.00 0.00 0.00
Components 2.00 2.00 0.00 0.00 0.00 2.00 0.00 0.00 0.00 2.00 0.00 0.00 0.00
Network density 0.54 0.38 0.16 4.92 0.90 0.54 0.00 0.00 0.00 0.54 0.00 0.00 0.00
Average degree 12.50 8.73 3.78 113.25 20.79 12.50 0.00 0.00 0.00 12.50 0.00 0.00 0.00
Standard deviation degree 5.51 3.48 2.03 60.90 11.18 5.20 0.31 9.19 1.88 5.22 0.29 8.81 1.81
Global cluster coefficient 0.81 0.46 0.36 10.67 1.95 0.71 0.10 2.96 0.55 0.72 0.09 2.84 0.52
Average cluster coefficient 0.78 0.47 0.31 9.32 1.72 0.75 0.03 0.96 0.20 0.74 0.04 1.06 0.21
Mean path length 3.48 3.48 0.00 0.54 0.13 3.30 0.19 5.62 1.03 3.29 0.19 5.63 1.03
Communities 5.00 4.70 0.30 37.00 8.78 4.97 0.03 27.00 7.14 5.07 0.07 28.00 7.07
Gini coefficient 0.52 0.41 0.11 3.36 0.72 0.49 0.03 1.29 0.30 0.48 0.04 1.71 0.39
Average betweenness 6.50 6.49 0.01 6.25 1.46 4.35 2.15 64.63 11.80 4.34 2.16 64.79 11.83
Maximum betweenness 26.80 19.42 7.38 228.43 45.64 17.36 9.45 286.41 55.33 18.18 8.63 260.14 51.33
Average closeness 0.03 0.03 0.00 0.02 0.01 0.03 0.00 0.11 0.02 0.03 0.00 0.11 0.02
Minimum closeness 0.02 0.02 0.01 0.15 0.03 0.02 0.01 0.21 0.04 0.02 0.01 0.22 0.04
Average eigencentrality 0.65 0.69 0.03 1.09 0.23 0.71 0.05 1.58 0.31 0.70 0.04 1.31 0.26
Minimum eigencentrality 0.02 0.21 0.19 5.69 1.07 0.18 0.16 4.80 0.90 0.18 0.17 4.98 0.92
Network radius 2.00 2.00 0.00 0.00 0.00 1.93 0.07 2.00 1.41 1.90 0.10 3.00 1.73
Average eccentricity 2.79 2.25 0.54 16.13 2.98 2.00 0.79 23.79 4.36 1.99 0.80 24.08 4.41
Network diameter 4.00 3.00 1.00 30.00 5.48 2.63 1.37 41.00 7.94 2.67 1.33 40.00 7.75
Boldfaced numbers indicate which algorithm performed better for a particular metric
O’Neil and Petty Applied Network Science (2019) 4:19 Page 38 of 49
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Table 17 Realism results for the Zachary Karate Club social network
Metrics TF|T-F| L1(F) L2(F)P|T-P| L1(P) L2(P)M|T-M| L1(M) L2(M)
Nodes 34.00 34.00 0.00 0.00 0.00 34.00 0.00 0.00 0.00 34.00 0.00 0.00 0.00
Links 78.00 65.63 12.37 371.00 68.61 78.00 0.00 0.00 0.00 78.00 0.00 0.00 0.00
Components 1.00 1.13 0.13 4.00 2.00 1.13 0.13 4.00 2.45 1.20 0.20 6.00 2.45
Network density 0.14 0.12 0.02 0.66 0.12 0.14 0.00 0.00 0.00 0.14 0.00 0.00 0.00
Average degree 4.59 3.86 0.73 21.82 4.04 4.59 0.00 0.00 0.00 4.59 0.00 0.00 0.00
Standard deviation degree 3.88 2.69 1.19 35.61 6.63 3.87 0.00 6.10 1.38 3.67 0.21 7.49 1.63
Global cluster coefficient 0.26 0.16 0.10 2.96 0.56 0.31 0.05 1.63 0.33 0.33 0.08 2.25 0.44
Average cluster coefficient 0.59 0.19 0.40 11.85 2.18 0.68 0.09 2.71 0.51 0.67 0.08 2.49 0.48
Mean path length 2.41 2.85 0.44 13.27 4.99 3.05 0.64 22.98 12.16 3.60 1.19 36.32 14.09
Communities 5.00 7.53 2.53 82.00 18.55 6.80 1.80 58.00 12.41 6.27 1.27 42.00 9.70
Gini coefficient 0.17 0.31 0.14 4.28 0.87 0.33 0.17 5.04 0.96 0.31 0.15 4.37 0.86
Average betweenness 23.24 25.26 2.03 68.62 14.83 22.13 1.11 72.03 18.05 23.14 0.10 72.18 17.73
Maximum betweenness 231.07 163.33 67.74 2032.19 409.13 234.37 3.30 1271.06 277.04 208.98 22.09 1321.97 293.17
Average closeness 0.01 0.01 0.00 0.02 0.00 0.01 0.00 0.03 0.01 0.01 0.00 0.02 0.01
Minimum closeness 0.01 0.01 0.00 0.02 0.01 0.01 0.00 0.03 0.01 0.01 0.00 0.03 0.01
Average eigencentrality 0.39 0.33 0.06 1.79 0.36 0.30 0.09 2.73 0.51 0.30 0.10 2.84 0.53
Minimum eigencentrality 0.06 0.04 0.03 0.99 0.20 0.03 0.03 1.01 0.20 0.03 0.04 1.07 0.21
Network radius 3.00 3.03 0.03 1.00 1.00 2.60 0.40 12.00 3.46 2.73 0.27 10.00 3.16
Average eccentricity 4.03 4.12 0.09 6.94 1.79 3.70 0.33 14.50 3.18 3.82 0.21 12.88 2.97
Network diameter 5.00 5.13 0.13 8.00 3.16 4.70 0.30 15.00 3.87 4.87 0.13 16.00 4.24
Boldfaced numbers indicate which algorithm performed better for a particular metric
O’Neil and Petty Applied Network Science (2019) 4:19 Page 39 of 49
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Table 18 Realism results for the Bernard & Killworth Technical social network
Metrics TF|T-F| L1(F) L2(F)P|T-P| L1(P) L2(P)M|T-M| L1(M) L2(M)
Nodes 34.00 34.00 0.00 0.00 0.00 34.00 0.00 0.00 0.00 34.00 0.00 0.00 0.00
Links 78.00 65.63 12.37 371.00 68.61 78.00 0.00 0.00 0.00 78.00 0.00 0.00 0.00
Components 1.00 1.13 0.13 4.00 2.00 1.13 0.13 4.00 2.45 1.20 0.20 6.00 2.45
Network density 0.14 0.12 0.02 0.66 0.12 0.14 0.00 0.00 0.00 0.14 0.00 0.00 0.00
Average degree 4.59 3.86 0.73 21.82 4.04 4.59 0.00 0.00 0.00 4.59 0.00 0.00 0.00
Standard deviation degree 3.88 2.69 1.19 35.61 6.63 3.87 0.00 6.10 1.38 3.67 0.21 7.49 1.63
Global cluster coefficient 0.26 0.16 0.10 2.96 0.56 0.31 0.05 1.63 0.33 0.33 0.08 2.25 0.44
Average cluster coefficient 0.59 0.19 0.40 11.85 2.18 0.68 0.09 2.71 0.51 0.67 0.08 2.49 0.48
Mean path length 2.41 2.85 0.44 13.27 4.99 3.05 0.64 22.98 12.16 3.60 1.19 36.32 14.09
Communities 5.00 7.53 2.53 82.00 18.55 6.80 1.80 58.00 12.41 6.27 1.27 42.00 9.70
Gini coefficient 0.17 0.31 0.14 4.28 0.87 0.33 0.17 5.04 0.96 0.31 0.15 4.37 0.86
Average betweenness 23.24 25.26 2.03 68.62 14.83 22.13 1.11 72.03 18.05 23.14 0.10 72.18 17.73
Maximum betweenness 231.07 163.33 67.74 2032.19 409.13 234.37 3.30 1271.06 277.04 208.98 22.09 1321.97 293.17
Average closeness 0.01 0.01 0.00 0.02 0.00 0.01 0.00 0.03 0.01 0.01 0.00 0.02 0.01
Minimum closeness 0.01 0.01 0.00 0.02 0.01 0.01 0.00 0.03 0.01 0.01 0.00 0.03 0.01
Average eigencentrality 0.39 0.33 0.06 1.79 0.36 0.30 0.09 2.73 0.51 0.30 0.10 2.84 0.53
Minimum eigencentrality 0.06 0.04 0.03 0.99 0.20 0.03 0.03 1.01 0.20 0.03 0.04 1.07 0.21
Network radius 3.00 3.03 0.03 1.00 1.00 2.60 0.40 12.00 3.46 2.73 0.27 10.00 3.16
Average eccentricity 4.03 4.12 0.09 6.94 1.79 3.70 0.33 14.50 3.18 3.82 0.21 12.88 2.97
Network diameter 5.00 5.13 0.13 8.00 3.16 4.70 0.30 15.00 3.87 4.87 0.13 16.00 4.24
Boldfaced numbers indicate which algorithm performed better for a particular metric
O’Neil and Petty Applied Network Science (2019) 4:19 Page 40 of 49
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

Table 19 Realism results for the Bernard & Killworth Office social network
Metrics TF|T-F| L1(F) L2(F)P|T-P| L1(P) L2(P)M|T-M| L1(M) L2(M)
Nodes 40.00 40.00 0.00 0.00 0.00 40.00 0.00 0.00 0.00 40.00 0.00 0.00 0.00
Links 238.00 197.60 40.40 1212.00 223.29 238.00 0.00 0.00 0.00 238.00 0.00 0.00 0.00
Components 1.00 1.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00
Network density 0.31 0.25 0.05 1.55 0.29 0.31 0.00 0.00 0.00 0.31 0.00 0.00 0.00
Average degree 11.90 9.88 2.02 60.60 11.16 11.90 0.00 0.00 0.00 11.90 0.00 0.00 0.00
Standard deviation degree 4.48 3.39 1.09 32.72 6.03 5.11 0.63 19.40 3.83 5.05 0.57 17.59 3.61
Global cluster coefficient 0.41 0.27 0.14 4.08 0.75 0.41 0.00 0.30 0.06 0.42 0.01 0.34 0.08
Average cluster coefficient 0.43 0.28 0.15 4.58 0.84 0.46 0.03 0.96 0.20 0.46 0.03 0.87 0.19
Mean path length 1.76 1.83 0.06 1.93 0.36 1.73 0.03 0.96 0.19 1.74 0.03 0.84 0.17
Communities 4.00 5.50 1.50 55.00 12.12 4.80 0.80 42.00 10.86 5.13 1.13 54.00 13.64
Gini coefficient 0.35 0.36 0.01 1.51 0.35 0.32 0.03 2.42 0.60 0.34 0.01 2.75 0.60
Average betweenness 14.90 16.15 1.25 37.63 7.10 14.28 0.62 18.70 3.65 14.41 0.49 16.45 3.26
Maximum betweenness 46.13 47.39 1.27 188.93 42.15 124.58 78.46 2353.65 456.14 118.82 72.69 2183.96 439.83
Average closeness 0.02 0.01 0.00 0.02 0.00 0.02 0.00 0.01 0.00 0.02 0.00 0.01 0.00
Minimum closeness 0.01 0.01 0.00 0.02 0.00 0.01 0.00 0.04 0.01 0.01 0.00 0.04 0.01
Average eigencentrality 0.58 0.60 0.02 0.87 0.20 0.45 0.13 3.97 0.76 0.45 0.13 3.93 0.76
Minimum eigencentrality 0.12 0.15 0.03 1.17 0.24 0.11 0.01 0.56 0.13 0.10 0.02 0.81 0.19
Network radius 2.00 2.00 0.00 0.00 0.00 2.00 0.00 0.00 0.00 2.00 0.00 0.00 0.00
Average eccentricity 2.83 2.84 0.02 1.50 0.39 2.53 0.29 8.88 1.76 2.59 0.24 7.10 1.48
Network diameter 4.00 3.07 0.93 28.00 5.29 3.03 0.97 29.00 5.39 3.00 1.00 30.00 5.48
Boldfaced numbers indicate which algorithm performed better for a particular metric
O’Neil and Petty Applied Network Science (2019) 4:19 Page 41 of 49
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Table 20 Realism results for the Krebs Fortune 500 IT Department (Advice) social network
Metrics TF|T-F| L1(F) L2(F)P|T-P| L1(P) L2(P)M|T-M| L1(M) L2(M)
Nodes 56.00 56.00 0.00 0.00 0.00 56.00 0.00 0.00 0.00 56.00 0.00 0.00 0.00
Links 203.00 182.53 20.47 614.00 113.61 203.00 0.00 0.00 0.00 203.00 0.00 0.00 0.00
Components 2.00 2.00 0.00 0.00 0.00 2.00 0.00 0.00 0.00 2.00 0.00 0.00 0.00
Network density 0.13 0.12 0.01 0.40 0.07 0.13 0.00 0.00 0.00 0.13 0.00 0.00 0.00
Average degree 7.25 6.52 0.73 21.93 4.06 7.25 0.00 0.00 0.00 7.25 0.00 0.00 0.00
Standard deviation degree 4.18 3.45 0.73 22.02 4.12 4.70 0.52 15.57 3.04 4.78 0.60 17.88 3.34
Global cluster coefficient 0.35 0.15 0.20 5.95 1.09 0.28 0.07 2.14 0.40 0.29 0.06 1.81 0.34
Average cluster coefficient 0.42 0.16 0.27 7.99 1.47 0.43 0.00 0.46 0.11 0.43 0.00 0.42 0.10
Mean path length 4.29 4.22 0.06 1.87 0.37 4.15 0.14 4.12 0.77 4.15 0.14 4.20 0.78
Communities 8.00 11.10 3.10 107.00 22.96 9.30 1.30 61.00 13.53 8.67 0.67 68.00 15.75
Gini coefficient 0.43 0.46 0.03 1.63 0.34 0.48 0.05 1.71 0.38 0.45 0.02 1.33 0.30
Average betweenness 36.32 34.61 1.71 51.38 10.04 32.55 3.78 113.30 21.17 32.47 3.85 115.46 21.49
Maximum betweenness 262.14 169.51 92.63 2778.80 524.32 457.03 194.89 5846.64 1154.39 500.81 238.67 7160.13 1355.01
Average closeness 0.01 0.01 0.00 0.01 0.00 0.01 0.00 0.02 0.00 0.01 0.00 0.02 0.00
Minimum closeness 0.01 0.01 0.00 0.01 0.00 0.01 0.00 0.01 0.00 0.01 0.00 0.01 0.00
Average eigencentrality 0.32 0.40 0.08 2.47 0.48 0.32 0.01 0.75 0.17 0.31 0.01 0.69 0.15
Minimum eigencentrality 0.02 0.05 0.03 0.80 0.17 0.03 0.01 0.28 0.06 0.03 0.01 0.32 0.07
Network radius 3.00 3.00 0.00 0.00 0.00 2.80 0.20 6.00 2.45 2.80 0.20 6.00 2.45
Average eccentricity 3.66 3.56 0.10 4.09 0.89 3.45 0.21 6.93 1.47 3.48 0.19 6.84 1.43
Network diameter 5.00 4.60 0.40 12.00 3.46 4.47 0.53 16.00 4.00 4.40 0.60 18.00 4.24
Boldfaced numbers indicate which algorithm performed better for a particular metric
O’Neil and Petty Applied Network Science (2019) 4:19 Page 42 of 49
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Table 21 Realism results for the Krebs Fortune 500 IT Department (Business) social network
Metrics TF|T-F| L1(F) L2(F)P|T-P| L1(P) L2(P)M|T-M| L1(M) L2(M)
Nodes 56.00 56.00 0.00 0.00 0.00 56.00 0.00 0.00 0.00 56.00 0.00 0.00 0.00
Links 387.00 331.40 55.60 1668.00 306.33 387.00 0.00 0.00 0.00 387.00 0.00 0.00 0.00
Components 1.00 1.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00
Network density 0.25 0.22 0.04 1.08 0.20 0.25 0.00 0.00 0.00 0.25 0.00 0.00 0.00
Average degree 13.82 11.84 1.99 59.57 10.94 13.82 0.00 0.00 0.00 13.82 0.00 0.00 0.00
Standard deviation degree 5.20 3.98 1.21 36.43 6.71 5.79 0.59 17.62 3.50 5.76 0.56 16.88 3.30
Global cluster coefficient 0.49 0.24 0.26 7.69 1.41 0.35 0.14 4.15 0.76 0.36 0.14 4.06 0.74
Average cluster coefficient 0.56 0.24 0.32 9.57 1.75 0.38 0.19 5.55 1.02 0.38 0.19 5.55 1.01
Mean path length 1.90 1.86 0.04 1.21 0.23 1.80 0.10 3.05 0.56 1.79 0.11 3.24 0.59
Communities 3.00 5.03 2.03 67.00 16.16 5.60 2.60 78.00 16.85 4.80 1.80 60.00 14.14
Gini coefficient 0.10 0.29 0.20 6.05 1.27 0.39 0.29 8.71 1.63 0.27 0.18 5.65 1.18
Average betweenness 24.77 23.66 1.11 33.36 6.42 21.98 2.79 83.77 15.37 21.80 2.97 89.00 16.31
Maximum betweenness 116.33 78.13 38.20 1145.89 215.07 195.97 79.65 2401.24 497.39 217.56 101.24 3037.07 594.40
Average closeness 0.01 0.01 0.00 0.01 0.00 0.01 0.00 0.02 0.00 0.01 0.00 0.02 0.00
Minimum closeness 0.01 0.01 0.00 0.01 0.00 0.01 0.00 0.01 0.00 0.01 0.00 0.01 0.00
Average eigencentrality 0.52 0.56 0.04 1.34 0.28 0.40 0.11 3.39 0.65 0.38 0.14 4.11 0.76
Minimum eigencentrality 0.14 0.17 0.03 1.08 0.23 0.09 0.05 1.54 0.30 0.09 0.05 1.44 0.28
Network radius 2.00 2.00 0.00 0.00 0.00 2.00 0.00 0.00 0.00 2.00 0.00 0.00 0.00
Average eccentricity 2.89 2.86 0.03 1.21 0.29 2.75 0.14 4.20 0.86 2.71 0.18 5.41 1.10
Network diameter 3.00 3.03 0.03 1.00 1.00 3.00 0.00 0.00 0.00 3.03 0.03 1.00 1.00
Boldfaced numbers indicate which algorithm performed better for a particular metric
O’Neil and Petty Applied Network Science (2019) 4:19 Page 43 of 49
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Table 22 Realism results for the Lazega Law Firm social network
Metrics TF|T-F| L1(F) L2(F)P|T-P| L1(P) L2(P)M|T-M| L1(M) L2(M)
Nodes 71.00 71.00 0.00 0.00 0.00 71.00 0.00 0.00 0.00 71.00 0.00 0.00 0.00
Links 726.0 603.0 123.0 3690.0 675.18 726.0 0.00 0.00 0.00 726.00 0.00 0.00 0.00
Components 1.00 1.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00
Network density 0.29 0.24 0.05 1.49 0.27 0.29 0.00 0.00 0.00 0.29 0.00 0.00 0.00
Average degree 20.45 16.99 3.47 103.94 19.02 20.45 0.00 0.00 0.00 20.45 0.00 0.00 0.00
Standard deviation degree 8.10 5.88 2.21 66.40 12.16 8.21 0.12 5.76 1.41 8.12 0.03 4.85 1.25
Global cluster coefficient 0.44 0.28 0.16 4.80 0.88 0.41 0.04 1.06 0.20 0.40 0.04 1.10 0.20
Average cluster coefficient 0.45 0.29 0.16 4.93 0.90 0.41 0.04 1.25 0.23 0.41 0.05 1.34 0.25
Mean path length 1.75 1.79 0.04 1.15 0.21 1.73 0.02 0.73 0.14 1.73 0.02 0.65 0.12
Communities 3.00 6.30 3.30 99.00 20.95 5.03 2.03 65.00 16.22 6.27 3.27 102.00 23.41
Gini coefficient 0.11 0.37 0.26 7.74 1.49 0.35 0.23 7.27 1.47 0.40 0.28 8.55 1.67
Average betweenness 26.30 27.64 1.34 40.28 7.45 25.45 0.85 25.37 4.73 25.54 0.76 22.65 4.22
Maximum betweenness 106.69 99.45 7.25 431.28 86.78 200.50 93.80 2814.09 577.96 193.7 87.00 2610.11 519.36
Average closeness 0.01 0.01 0.00 0.01 0.00 0.01 0.00 0.00 0.00 0.01 0.00 0.00 0.00
Minimum closeness 0.01 0.01 0.00 0.01 0.00 0.01 0.00 0.01 0.00 0.01 0.00 0.01 0.00
Average eigencentrality 0.45 0.54 0.10 2.86 0.54 0.42 0.03 0.94 0.20 0.42 0.03 0.84 0.18
Minimum eigencentrality 0.09 0.15 0.06 1.65 0.33 0.09 0.00 0.29 0.07 0.08 0.01 0.37 0.08
Network radius 2.00 2.00 0.00 0.00 0.00 2.00 0.00 0.00 0.00 2.00 0.00 0.00 0.00
Average eccentricity 2.75 2.66 0.09 2.56 0.56 2.47 0.27 8.18 1.57 2.54 0.21 6.30 1.21
Network diameter 3.00 3.00 0.00 0.00 0.00 3.00 0.00 0.00 0.00 3.00 0.00 0.00 0.00
Boldfaced numbers indicate which algorithm performed better for a particular metric
O’Neil and Petty Applied Network Science (2019) 4:19 Page 44 of 49
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Abbreviations
CDM: Configuration-degree matching; CM: Configuration model; E: Extraversion; ERGM: Exponential random graph
model; F: Feeling; GNAC: Generate network using assignment and compatibility; I: Introversion; IT: Information
technology; J: Judging; MBTI: Myers briggs type indicator; N: Intuition; NASA: National aeronautics and space
administration; OCEAN: Openness, conscientiousness, extraversion, agreeableness, neuroticism; P: Perceiving;
PS: Probability search; ReCon: Replication of complex networks; S: Sensing; SBM: Stochastic block model; T: Thinking
Acknowledgements
The Alabama Supercomputer Authority, which is funded by the State of Alabama, provided a generous grant of
supercomputer processing time to support this work. The general research topic, generating synthetic social networks, was
brought to our attention by Eric W. Weisel of Old Dominion University. The anonymous reviewers of earlier versions of this
article provided insightful and valuable comments that helped to substantially improve the final version. In particular, the
Probability Search algorithm is based on an idea provided by one of the anonymous reviewers.
Funding
O’Neil was partially funded by the 2014 RADM Fred Lewis Postgraduate I/ITSEC Scholarship, awarded in association
with the Interservice/Industry Training, Simulation and Education Conference and organized by the National Training
and Simulation Association. Petty received no specific funding.
Availability of data and materials
All data and program source code described in this article is available to any interested parties. The documentation,
source code, input data (the exemplar real-world social networks and compatibility table), as well as the results are
available at GitHub at the following URL, https://github.com/daoneil/NetworkMetricSearch, in a directory named
GenSynthNetMet.
Authors’contributions
DAO’N identified the network metrics, designed and implemented two of the algorithms (CDM and GNAC), executed
the computer runs, and wrote the initial version of this article. MDP created the initial project concept, designed and
implemented one of the algorithms (PS), defined the performance comparison methodology, and extensively revised
this article. Both authors read and approved the final manuscript.
Authors’information
Daniel A. O’Neil works in the Office of Strategy within the Office of Strategic Analysis and Communication at the
National Aeronautics and Space Administration’s Marshall Space Flight Center. He develops software prototypes to
demonstrate the applications of various technologies to strategic analysis, such as interactive text based scenarios,
social network analysis, and 3D orbital trajectory visualization web-apps. Additionally, he integrates Microsoft Share-
Point data lists via Nintex workflows. During his career spanning three decades, his employers included the Boeing
Company, the U.S. Army Strategic Defense Command, and NASA. His experience includes development of real-time
code for the B1-B flight training simulator, management of the development of one of the first web-based intranets,
management of the development of a system-of-systems life cycle technology portfolio analysis system, and author-
ship of tutorials and associates demonstration code for ontology driven orbital dynamics visualization web-apps. He re-
ceived a B.S. degree in Electrical and Computer Engineering in 1985 and an M.S. degree in Engineering Management
in 1997 from the University of Alabama in Huntsville. He is currently a Ph.D. candidate in Modeling and Simulation at
the University of Alabama in Huntsville.
Mikel D. Petty is currently Senior Scientist for Modeling and Simulation in the Information Technology and Systems
Center and an Associate Professor of Computer Science at the University of Alabama in Huntsville. Prior to joining
UAH, he was Chief Scientist at Old Dominion University’s Virginia Modeling, Analysis, and Simulation Center and
Assistant Director at the University of Central Florida’s Institute for Simulation and Training. He received a Ph.D. in
Computer Science from the University of Central Florida in 1997. Dr. Petty has worked in modeling and simulation
research and education since 1990 in areas that include verification and validation methods, simulation interoperability
and composability, and human behavior modeling. He has published over 215 research papers and has been awarded
over $16.5 million in research funding. He has served on both National Research Council and National Science
Foundation committees on modeling and simulation, is a Certified Modeling and Simulation Professional, and is
Editor-in-Chief of the journal SIMULATION: Transactions of the Society for Modeling and Simulation International. He has
been dissertation advisor to eight graduated Ph.D. students in four different academic disciplines (Computer Science,
Modeling and Simulation, Industrial and Systems Engineering, and Computer Engineering).
Competing interests
The authors declare that they have no competing interests.
Publisher’sNote
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 7 July 2018 Accepted: 20 February 2019
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