entropy
Article
Towards Social Capital in a Network Organization:
A Conceptual Model and an Empirical Approach
Saad Alqithami 1, * , Rahmat Budiarto 1 , Musaad Alzahrani 1
1
2
*
and Henry Hexmoor 2
Department of Computer Science, Albaha University, Al Bahah 65527, Saudi Arabia;
Rahmat@bu.edu.sa (R.B.); Malzahr@bu.edu.sa (M.A.)
Department of Computer Science, Southern Illinois University, Carbondale, IL 62901, USA;
Hexmoor@cs.siu.edu
Correspondence: salqithami@bu.edu.sa
Received: 29 March 2020; Accepted: 28 April 2020; Published: 1 May 2020
Abstract: Due to the complexity of an open multi-agent system, agents’ interactions are instantiated
spontaneously, resulting in beneficent collaborations with one another for mutual actions that
are beyond one’s current capabilities. Repeated patterns of interactions shape a feature of their
organizational structure when those agents self-organize themselves for a long-term objective.
This paper, therefore, aims to provide an understanding of social capital in organizations that
are open membership multi-agent systems with an emphasis in our formulation on the dynamic
network of social interactions that, in part, elucidate evolving structures and impromptu topologies
of networks. We model an open source project as an organizational network and provide definitions
and formulations to correlate the proposed mechanism of social capital with the achievement of
an organizational charter, for example, optimized productivity. To empirically evaluate our model,
we conducted a case study of an open source software project to demonstrate how social capital can
be created and measured within this type of organization. The results indicate that the values of
social capital are positively proportional towards optimizing agents’ productivity into successful
completion of the project.
Keywords: social capital; open multi-agent systems; collaboration; interaction; complex networks
1. Introduction
There has been an increasing interest in service-oriented computing that aims to combine
computational resources dynamically across boundaries, e.g., semantic web and peer-to-peer networking.
A shared feature of all these systems is that different services from data and software can be invoked
remotely to achieve a common goal, i.e., organization oriented. This style of distributed services
may often allow a number of competing service providers to achieve their respective requirements
due to the applicability of shared resources. Nevertheless, the collaborative nature of these systems
means that they will invariably create uncertainty surrounding the incentives of agents offering these
services. Multi-agent systems (MAS), in this regard, have demonstrated their efficiency in modeling and
implementing distributed systems as it signifies dynamics of heterogeneous agents’ interactions.
An ad-hoc organization of networked agents may form to rally around a specific problem.
We explore the effects resultant from networking by addressing one type of network effects called
Social Capital (SC). Social capital in a cross-organizational network can be characterized as collocated
or virtual collaboration to produce successful outcomes and successful connections. There are two
major perspectives on SC in networks. In the macroscopic perspective, SC for the entire network is
considered. In this view individuals do not incrementally add to the system or withdraw units of SC.
Instead, the foci are on the system principles like norms and conventions that provide resources for
Entropy 2020, 22, 519; doi:10.3390/e22050519
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overall social welfare. In contrast, the microscopic perspective adopted here explores how individuals
can gain access to resources by their positions and connections in the network [1].
1.1. Understanding SC
The consideration of SC may transcend issues surrounding the heterogeneity of an agent’s
affiliations since it captures benefits resulting from the preferential treatment and collaboration among
agents [2]. The scale of SC should consider components of norm, trust, reciprocity, governance,
tolerance, friendships and acceptance, which might be bounded within the organizational networks.
Quantities of SC can be used to replace interpersonal trust among agents and that is due to when
an organization generates positive values of SC, constituent agents gain benevolence and behave
in a trusting manner [3]. Other benefits of SC are enhanced group communication, efficient use of
intellectual capital, better collective action and easy way of accessing resources [4].
There is still no theoretical or practical value in determining some quantifiable measures of SC,
and the main idea is of the driven value of SC for an agent or an organization. Han and Breiger [5]
are of the ones to propose a measurement of SC. There are a few main elements of the SC that have a
proportional relationship to one another. Topologically speaking, high bonding rates provide more
opportunities for interaction and growth of SC. However, network structure by itself is inadequate
for the determination of SC. We must examine the contents of interaction and dispositions that create
social forces that attract or repel individuals [6]. At the level of a single link, the nature of social flow,
i.e., information flow, in the link leads to accumulation of SC. Social flows can be benevolent and
positive or negative and lack benevolence. Whereas positive flow leads to network positive gain in
SC, negative flow leads to loss of SC. Apart from social flow, dyadic ties may harbor trust or promote
distrust [7]. Trust supports SC whereas distrust erodes it. If the topic of interactions between a pair
of agents is centered on the main problem for an organization, that link positively contributes to SC.
Thus, flow, trust, and topic are link attributes that are proportional determinants for SC.
Social capital in a link is the accumulation of positive values of social flow and trust plus abundance
of communication over a common topic. Since considering a topic of interaction is included in the
determination of SC over the link, we note that this formulation of social capital is relative only to links
in an organization. SC is generated in the links through dynamics of interaction on the links. Thus,
SC for a network linearly scales by summing SC for all links in the network. Increased values of links
are proportional to increase social capital, i.e., network bonding measure. The effects of organization
topology are overlooked in this network perspective but will be considered egocentrically. All bridged
communities contribute to accumulation of the overall SC, which is the instrumental purpose of SC [8].
From an egocentric perspective, bridging is said to contribute to social capital [3,9].
Network bonding leads to increased density and closure in the network, which increases
resource access [1]. The more interconnections a network has, the more opportunities it will have for
accumulation of SC. A network with the most links possible, i.e., a clique, supports the highest SC
because a clique structure is a complete graph that connects each agent with everyone else in the graph
which produces a high bonding rate [10]. Thus, SC is seen in this structure to arrive to a higher value
compared with other types of structures. Other organizational structures, by contradiction, yield lower
SC than a complete network graph due to their less of connections. Social capital for an agent, on a
differnt perspective, is the egocentric for an individual that deliberately mirrors the Bonacich Power
Index [11]. This coincidence helps us to exploit the topological position of nodes. An agent that is
well positioned by having a High Power Index, i.e., high Bonacich centrality value [11], will similarly
possess high SC [2].
The previous definitions and views of SC in a restrictive network open the discussion to consider
them in organizations. Organizations, in general, are bounded networks with purposeful interaction
between their agents. Organizations have multiple degrees of institutionalized culture, norms and
values that are essential in the development of SC. SC receives direct and indirect effects from formal
institutions due to formal relations that have been provided by the organization to create interpersonal
Entropy 2020, 22, 519
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relations that contribute positively to SC [12]. SC in an organization represents the resultant outcome
from SCs embedded in a social system or through a direct or indirect social relations of an agent,
including inherited norms and culture values. Values of SC are built within the social structure to
facilitate the agents’ actions and interaction [12].
1.2. SC in the Literature
Social capital has been studied by many previous researchers [13–20]; however, a unified
definition of it is a critical issue. Bourdieu [21] refers to SC as the actual or potential collective
resources in an institutionalized synergistic network of homogenous agents, which in some cases
may result into other forms of capitals. The point behind SC is to make use of the accumulation of
resources embedded in the social structure [22]. Other authors [23] have defined SC as an attribute of
individuals that enhances their abilities to solve collective action problems. Furthermore, Nahapiet and
Ghoshal [14] described SC through three different dimensions: (1) structure dimension to include the
properties of the whole network, (2) relational dimension to present the values of exchanges in agents’
connections, and (3) cognitive dimension to support the homogeneity by sharing interpretations and
mutual understanding between agents [14,16].
There are two types of social capital in an intra-organizational network: bonding and bridging [24].
Both types are generated from agents’ interaction, i.e., network homophily—the theory of homophily
helps in initiating attachments or interactions between agents with similar attributes, which allows
them to self-select based on their public profiles [25]. A major difference between those two types
is that bonding SC occurs between homogenous agents working on a common goal while bridging
involves interaction between heterogeneous agents who are not necessarily working for the same
goal [22]. Bonding SC increases through closure, which contributes positively to the values of relations.
Although bridging SC can be considered between agents within an organization, increase of its
value can, in some cases, be a resultant of interaction through an inter-organizational network and
bridging cross structural holes [26,27]. Social network analysis, presented in [28], studied a network of
wildlife tourism micro-entrepreneurs for the purpose of identifying forms of bonding and bridging
social capital. The results showed that interactions, e.g., customer exchange or referral, between the
micro-entrepreneurs fostered the formation of a bridging network structure that contains four ties
connecting potential sub-groups in the network. In addition, the results highlight the importance of
reciprocation between the micro-entrepreneurs for the success of the wildlife tourism business.
From an empirical perspective, Zou, et al. [29] conducted an experiment to investigate the
relationships between social capital, emotion experience and life satisfaction for sustainable community.
The results revealed that structural social capital and cognitive social capital of the community
positively influence the life satisfaction and joyous experience of the residents. However, they
have negative impact on painful experience of the residents. Moreover, Sung-Hoon et al. [30]
empirically explored the effects of social capital on Organizational Citizenship Behavior (OCB) in the
emotional labor context. The empirical study involved 330 participants from South Korea occupying
customer service-oriented positions. The results showed that there is a statistically significant positive
relationship between social capital and OCB that is sequentially mediated by deep acting and job
engagement. The impact of social capital and psychological capital on the entrepreneurial performance
of the new generation of migrant workers in China was quantitively analyzed in [31]. Quantitative data
were collected through a survey conducted on 525 rural households. The collected data was analyzed
using the structural equation modeling. The analysis of data showed that psychological capital and
social capital of the new generation of migrant workers have impact on their entrepreneurial outcomes.
The rich literature is a good addition to our view of SC, yet it falls short in differentiating SC from
social network analysis that is directly affected by the network topologies. Analyses on the structural
dimension, e.g., asymmetric emerging distribution of interrelations, of social capital considering the
impact of it on the success of Open Source Software (OSS) projects have been discussed in [4,32]. Open
source refers to any program in which the source code is made available for use or modification as
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users or other developers see fit. Open source software is usually developed as a public collaboration
and made freely available. The deployment of valuable parameters discussed in the literature into this
work in order to efficiently measure and exploit SC in an OSS results in several benefits. One of the
reasons for this is that traditional studies on SC consider only the total number of ties an individual or
organization has, ignoring the direction of the social flows. In our measurements, however, we signify
the inputs by considering the reciprocity exchange theory to measure it. In addition, we consider the
impact of SC on a real-world case study of an OSS project, which has set it apart from traditional prior
techniques. We take advantage of GitHub (https://github.com/) since it is the most popular platform
for open source collaboration. On GitHub, developers can join and contribute to projects by submitting
issues and contributing code. They submit issues when sending messages about errors in applications
and suggesting ways to fix them. The contribution of code involves sending pull requests with the
corrections and improvements. A project team is considered as an organizational network, which consists
of developers as nodes and each one may have relations with others through common tasks in modules.
To this end, we have introduced SC and briefly provided some related literature review, i.e.,
in Section 1. The rest of the article is organized as follows: In Section 2, we quantify social capital from
the ground-up starting from agents’ interactions to providing a measurement for the value of SC for an
agent and the organization. Section 3 contributes an extensive experiment on an open source project
development and details a discussion, while Section 4 draws conclusions and future possibilities.
2. Quantifying Social Capital
We consider SC to be a scalar value that can be accumulated as well as consumed either verbatim
or used as credit. In a network, SC might be used to trade for help or exchanges with others in the
form of delegation of tasks. Bartering with SC can be limited to a pair of agents through an immediate
link between them. Alternatively, an agent might enter bartering anonymously with another agent
with whom there may not be a directed relationship. Our measurements of SC on OSS Project is
based on a weighted task-based directed graph inherited from the general dependency-network graph.
An organization, i.e., the OSS Project in our case, is modeled as a directed graph of agents that are
contributors as vertices and their cumulative values of relations between the contributors as edges:
{N , Relation }, where N is the set of agents in an organization that is ≥ 2, and Relation ⊆ N × N is the
set of directed relations between agents. The organization has a common goal that is divided into a set
of tasks. Each task will be conducted by a subgroup of N ⊆ N . Each agent has a capacity extracted
from her public profiles, which include capability, willingness and previous relations.
2.1. Parameters of SC
We propose a measurement of relations from continual interaction and a quantification of an
agent’s capacity before attempting to measure the values of SC.
2.1.1. Relations Measure
In a dynamic organization, agents form subgroups when they tackle different problems for the
continuation of their organization. Even though their relations have a huge impact on the formation
as well as the coordination in this world, subgroup formation as well as task or problem allocation
is outside the scope of this work. We focus on measuring a network of relationships for subsequent
determinations of different values that an agent accumulates when interacting with others. The initial
values of relations are provided by every agent when she first joins an organization. Those relations
and their values are not static and agents are able to create, diminish, or improve each one of them
depending on current actions and interaction.
For every action an organization performs, there exists a goal Gj ∈ { G }. Each goal will be
distributed into a set of tasks, such that Gj → {Θ} = hθ1 , . . . , θn i, for possible assignments to agents.
The completion of one task θm ∈ {Θ} includes interaction between agents for a set of subtasks
1 , . . . , θ k i. The coordination as well as control of those tasks are determined by the
{θm } = hθm
m
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organization. We benefit from the dynamic interaction among agents while achieving multiple tasks
in order to update the current values of relations. Those values of relations depends on the nature of
interaction over every given task; therefore, we model relations in a task-based scenario to describe the
continual changes over time in inter-agent connections and to help with updating relations throughout
repeated task assignments. In the case of OSS Project, the goal is to develop the project and the
sub-goals are the releases of the project. The tasks are the software modules that need to be developed
in order to achieve a sub-goal, i.e., releasing the project. The subtasks of one module are the lines
of codes to be added or deleted in order to complete the module. The interaction between agents
(contributors) working in a task (module) occur through the completion of the subtasks.
For every self-selected task, agents define a task-based graph upon the initial relations and there
is at least one active edge that prescribes a plan. Agents are able to form an edge through successive
interaction. In other words, the network structure allows for the property of transitivity, which permits
interaction over that edge to improve giving it the chance to reach a threshold in order to be considered
active. Interactions are commonly observed of two types of affinities [3], where (a) explicit affinities become
evident through interaction over an existing relation, i.e., it is observed when two or more agents have
interaction with whom they have a previous experience over an existing edge in the graph, and (b) implicit
affinities allow for other possible interaction among agents without previously modeled relations. In the
case of OSS Project, explicit affinities between two agents exist when both agents contributed on a
common software module. Interactions emerge from the closure property of relations [33] and may help
in forming new edges when updating relations, i.e., previously un-modeled relations.
The current values of relations are updated every time interval ∆t and , in our case, the time
interval t is the time between releases. For the general assembly, we describe existing relations as
explicit links; otherwise, they will be considered as implicit. Values of links are proportional to the
frequency of interaction over them. The value on an explicit edge, ELink , between agent i, i′ ∈ N,
is computed accumulatively based on the frequency of interaction, i.e., I, between the two agents
s ,
throughout the time interval, i.e., ∆t = t2 − t1 . This is stated in Equation (1) at a specific subtask θm
where tr is the end of duration that spans from t1 toward t2 , ∀i 6= i′ ∈ N.
′
′
i,i
s
s
ELi,i
ink ( θm , t2 ) = ELink ( θm , t1 ) +
∑
r ∈∆t
′
s
Iii (θm
, tr )
(1)
Implicit links, i.e., ILink , are traditionally observed through triadic closure theory [34]. Triadic
closure, in short, asserts that for each three agents i, i′ and i′′ where two explicit affinities exist in
terms that link i ↔ i′ and i′ ↔ i′′ , there should exist an implicit affinity that links i ↔ i′′ . In a triadic
formation of two explicit affinities, there are different possibilities for the value ∈ R that the implicit
affinity should have. The possible value that an implicit affinity may obtain depends on the value of
′′
the current explicit edges. Thus, we can state that the initial value of the third implicit link, i.e., ILi,i
ink ,
in a triad can be approximated in Equation (2), which is ∀i 6= i′ 6= i′′ ∈ N.
′ ′′
′
′′
s
ILi,i
ink ( θm )
≡
i ,i
s
s
ELi,i
ink ( θm ) + ELink ( θm )
′
′ ′′
i ,i
Ri,i
elation ( θm ) + Relation ( θm )
2
(2)
We are considering the formation of implicit links through explicit links only. That means there
must be an explicit path from the source node to target node in order for an implicit link to exist.
The traversal in the path of unrepeated explicit links between i and i′ will consider the maximum
volume despite distances. An extension of the closure envisioned in Equation (2), where there existed
two disjoint, i.e., nonconsecutive, links with explicit affinities or possible undefined links in between,
is determined through Equation (3).
′
i,i′
s
(θm
)
ILink
≡
i,i
s )
∑i,i′ ELink (θm
′
i,i
∑i,i′ Relation (θm )
2
(3)
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Agents’ interaction are instrumental in forming new implicit links and updating the values of
existing explicit ones. During a task completion, it is possible for frequently used implicit relations
to gain a sense of actualization; thereby, the implicit relations will be treated the same as explicit
ones. Next, we model relations considering those measurements of explicit as well as implicit links.
As stated earlier, the initial values of relations are provided by the agents’ public profiles and are used
in forming a task-based socio-graph. We mapped those relations into explicit links in a task-based
graph in order to capture current interaction as well as to allow possible measures of implicit links.
By the time a new task is going to be assigned, an organization updates agents’ relations over all tasks
based on the new values of links. When a relation from an implicit link (ILink ) reaches a threshold
value of τ that has been specified previously by an organization, it will be treated as an explicit one
and an agent is able to explicitly form a relation over it. It is possible for those relations to have a value
of positive, negative, or mutual (i.e., equal) for non-existing or possible unprejudiced relations.
The relations in the graph are asymmetric relations, so we have to know the temporal direction of
those relations. That is, in a tuple hi, ELink , i′ i we have to know if the relation direction is from i to i′ ,
i.e., i → i′ , or from i′ to i, i.e., i → i′ . Equation (4) updates the initial value of relation between every
pair of agents by considering the most repeated value over an explicit or an implicit link at a given
s ∈ { Θ }.
subtask, that is ∀ i 6= i′ ∈ N and ∀ θm
′
→i
Rielation
(θm ) = mode
s
′
′
i,i
s
s
(θm
) + ILi,i
ELink
ink ( θm )
(4)
2.1.2. Capacity Measure
Agent’s capacity can be described as the absolute ability to accomplish tasks given the time
constrains and interests. A measurement of an agent’s capacity is a critical issue and should be
addressed once an agent joins an organization. This will eliminate the possibility of agent’s ineligibility
to accomplish tasks when allocated to it. The value of capacity is dynamic and rapidly changing from
one task to another. For simplicity, we consider capacity to be a combination of an agent’s innate (1)
capabilities for the ability to achieve different tasks, extemporaneous (2) willingness to perform certain
actions based on her preferences, and ad-lib (3) availability for her readiness to participate. Agents’
capabilities and willingnesses are provided in their public profiles while availabilities are ranging from
[0 → 1] based on the task they occupy. Willingness is the degree of commitment to which an agent is
ready to work hard to achieve the organizational objectives. The willingness of an agent is important
in determining her contributions for a task. Equation (5) shows a very direct measurement for agent i’s
capacity to achieve a certain task θm ∈ {Θ} and that ∀ i ∈ N.
h
i
i
Capacity
(θm ) = availabilityi (θm ) · capabilityi (θm ) + willingnessi (θm )
(5)
Due to the rapid changes in the agent’s capacity, an agent will not be able to preserve them for
future use. They must be updated instantaneously every time a new task is performed. We assume
that the capacity of an agent is independent ∀i ∈ N. Along with the presentation of a possible
capacity measure, we have proposed a measurement for an agent relations driven from their continual
interactions. Those two main parameters needed in determining the values of agents’ unconditional
contributions, which are going to be used in the following sections to help in defining measurements
of benevolence.
2.1.3. The Value of Benevolence
Agents entering an organization and interacting with those whom they have no previous
interaction are initializing their benevolent values with a constant of a Null; then, the benevolences are
derived from their relationships with others as well as their capacities to overcome certain problems.
Due to the fact that an organization is a formation that overlays a dynamic network, we model
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benevolence between agents based on a directed network’s graph of connected vertices and edges.
The resultant graph will be a task-based weighted graph of vertices as agents capacities and edges
as their relations. The weighted benevolent graph is connected, and there should at least be one
active relation between any pair of agents. We follow next with a formal definition of the weighted
benevolent graph while emphasizing on the parameters that contribute to its value.
Let N ⊆ N be a set of agents working on a goal Gi . There exists a set of tasks, i.e., {Θ} =
hθ1 , . . . , θm i, for each goal. Let w : 2 N → x, where w( N ) ∈ N is a world of N-agents working on θm ,
and x is a random variable with distribution that has not been determined yet. The parameters of the
i
:
w( N ) are attained from an organization and sampled over existing k-subtasks to all N. Let Benevolence
k
R → R be the benevolent function of real values that computes the benevolent value of w( N ) at θm
based on the distribution of k-sub-tasks. We are trying to find out the benevolent values resulting from
unilateral relationships between agents of N ⊆ N in the w( N ).
A benevolent socio-graph is basically a combination of agents and relations. The value of relations
can be different from one task to another; however, for the sake of simplicity, we will be evaluating
those relations in a task-based graph. We use the normal distribution to correspond to the average
values of agents benevolences with a peak and the variability with other agents in a symmetric spread,
i,1
i,k
i
= (Capacity
), where k is number of subtasks and Capacity is the agent’s capacity
i.e., Capacity
, . . . , Capacity
i,k
2 ). The benevolence between a pair of agents (i, i ′ ) can be presented in
∀ Capacity
∼ Capacity (µi,k , σi,k
Equation (6).
i →¬i
→¬i
i
Benevolence
(θm ) = Rielation
(θm ) · Capacity
(θm )
(6)
The values of relations are critical in this case, they are resulting from a weighted directed graph
of the network. The benevolence takes advantage of agents’ current relations and the rapid changes in
their values within the assignment of one task. We take into consideration an agent current interests
and readiness to contribute captured in the measurement of capacity. Although implicit links are not
considered when defining benevolence, current values of relations have already considered them, and
they will directly contribute to current values of benevolence once a specific threshold is reached.
2.1.4. The Value of Potential-Benevolence
Agents’ beliefs play an important role in the expected receipt of SC. When an agent believes that
another is able to provide resources to her, she will then try to obtain those resources. When resources
are obtained, trust is initiated. Agents providing resources are then of higher power and importance
than the agent acquiring them. Since the value of the SC that initializes the link from acquirer to
provider is proportional to the acquirer belief, we consider belief to be a function of the directed link to
the provider. The value obtained from this function is proportional to the value of the SC gained by
the provider. Given a graph of N-nodes and i is one of the nodes while ¬i are other member nodes
∈ N that are 6= i, the potential benevolence of agent i receiving a contribution from other agents within
N is obtained through Equation (7).
i
PBenevolence
(θm ) =
∑
∀¬i ∈ N
i
¬i →i
¬i
C
(
θ
)
·
R
(
θ
)
Belie
apacity m
elation m
f
(7)
Equation (7) states that the value of an agent’s capacity is a critical parameter for receiving a
benevolence. The value of a relation from i → ¬i is not the sum of all links an agent traverses through
to get to the provider. It can be calculated through an implicit link ILink if an explicit directed link, i.e.,
ELink , is not available.
2.2. Measurement of SC
The SC for an agent is based on her beliefs of receiving contribution from peers over the network.
The probability of an agent providing a continual benevolence to another is proportional to the
expected capacity that the acquirer may be interested in, as stated in Equation (8). We are considering,
Entropy 2020, 22, 519
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in our measurements of SC, a task-based graph for which the following equations are for a specific
task, e.g., θm .
¬i →i
¬i →i
f Benevolence
| PBenevolence
=
∑
∀¬i ∈ N
"
¬i →i
¬i →i
Benevolence
∩ PBenevolence
¬
i
→
i
PBenevolence
#
0
if PBenevolence > 0
(8)
Otherwise;
The intersection represents an expectation to receive benevolence considering the given
benevolence; otherwise, the value of "zero" is considered. In our case, we can eliminate the value of
potential benevolence after the intersection and assume that the value of benevolence is true if the
potential one exists. Equation (9) shows that directed SC is gained by the provider agent.
SCi =
∑
∀¬i ∈ N
i
Belie
f
¬i →i
¬i →i
f Benevolence
| PBenevolence
(9)
We consider belief to be a decay function that decreases the value of SC received when traversing
through multiple agents. It is exponential to how many explicit links the acquirer has to travel through
to obtain resources from the agent provider. We introduce the belief function Belie f : R+ → R+ ,
i →¬i ( Ri →¬i ) is the belief of the relation that returns the task based between agents i and ¬i.
where Belie
elation
f
Belief is a monotonically decreasing function so that a larger number of relations corresponds to a
−λ·( Rielation )
i
.
lower belief. The belief value is domain specific and an example of it can be: Belie
f = e
When an agent capitalizes on another, her current capacity is also accessible for that agent to take
advantage of, in-return. When both agents capitalize on each other, they form a cooperative behavior
that contributes positively to the organizational SC, feeding back to the organization member-agents.
3. The Case of an OSS Project
Social phenomena, such as a lack of engagement, cohesion, or even corruption, have not always
been observed instantly. Empirical researchers have traditionally been looking for pools to obtain
information on perceptions instead. This is because of the natural contextual dependency in various
abstract aspects of SC that only make sense in a unified context, which makes it difficult to come up
with standardized identical measures. We, therefore, consider OSS to define our bounded organization
and to ease the process of measuring different social aspects of SC that were challenging to compute
efficiently in traditional methods.
Open source software is a type of software projects with publicly released source code and the users,
in most cases, have the right to change the source code of the system. The development process of OSS
projects are different than industrial software projects. OSS developments are based on collaborations
between multiple independent developers, i.e., contributors, who aim to achieve a common goal.
The contributors are usually located in different geographical areas. Thus, OSS projects mostly have
online repositories, e.g., GitHub, that allow multiple developers to contribute independently to the
project [4]. Over the last two decades, OSS development has gained popularity, and we have witnessed
successful OSS projects such as Linux, MySQL, and Hadoop. However, the majority of OSS projects
have failed due to different reasons, e.g., [35,36]. In this paper, we try to understand the impact of SC on
OSS development and whether it has a relation with the success of OSS projects.
In order to find a relevant OSS project that is best fitted to the SC concept, we have taken different
aspects into account, for example, interactions, implicit/explicit links, relationships, and capacities.
Finally, we chose the Apache Software project as our research context, which has served as an example
for OSS in many previous studies, e.g., [37]. The Apache Hadoop project was initialized by Apache
Software Foundation in 2003 and has been one of the most active OSS projects since its beginning.
There are software releases and evolution of the software since then, which are possible because of the
Entropy 2020, 22, 519
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fast-moving development process and the broad foundation of contributors, ranging from hobbyists to
companies. Thus, it involves more people than other OSS projects.
3.1. Determining Parameters
We consider the OSS Project Hadoop as a case study to illustrate the SC value computation.
We focus on org.apache.hadoop.yarn.client.api package that has 10 classes and 31 contributors
involved in the package development. Apache Hadoop YARN is the resource management and job
scheduling technology in the open source Hadoop distributed processing framework. YARN is
responsible for allocating system resources to the various applications running in a Hadoop cluster and
scheduling tasks to be executed on different cluster nodes. The technology became an Apache Hadoop
subproject within the Apache Software Foundation (ASF) in 2012 and was one of the key features added
in Hadoop 2.0, which was released for testing that year and became generally available in October
2013. The data are taken from the GitHub portal from year 2013 to 2018, and we divide the data into
three time intervals (t1 : 2013–2014, t2 : 2015–2016, and t3 : 2017–2018). In this case study, the task is the
package org.apache.hadoop.yarn.client.api in the project, and the subtasks are the 10 classes in
{10}
the package, i.e., θm . We consider the number of line code as the value of interaction among the
contributors, i.e., adding and deleting lines.
Table 1 shows the names of the 10 sub-tasks (classes), number of commits, and the number of
agents involved in each time interval. Class number 2 and Class number 8 involves more than 4 agents,
for each time interval, that constitute social networks, while Class number 5 involve more than 1 agent
in time interval t2 and t3 . The rest of the classes have only one agent involved, thus, they do not
constitute a social network, thus do not contribute any social capital value. We give an example of a
social capital computation of agents involved in Class number 2 for time interval t1 to illustrate the
overall social capital computation of this social organization network.
Table 1. Classes and contributors of the package: org.apache.hadoop.yarn.client.api.
No.
Class Name
1
2
3
4
5
6
7
8
9
10
AHSClient.java
AMRMClient.java
ContainerShellWebSocket.java
InvalidContainerRequestException.java
NMClient.java
NMTokenCache.java
SharedCacheClient.java
YarnClient.java
YarnClientApplication.java
Package-info.java
# of Commits
3
37
2
1
9
4
3
42
1
1
Involved Agents
t1
t2
t3
0
5
0
1
1
1
0
7
1
1
2
8
0
0
2
1
1
12
0
0
1
9
1
0
3
1
2
5
0
0
Total
3
16
1
1
6
3
3
21
1
1
Table 2 shows sample data that has been collected from the GitHub portal for AMRMClient.java
class of org.apache.hadoop.yarn.client.api package. The class involves 16 contributors
throughout the development period with the interval of: 2013–2018. The sample data illustrated in
Table 2 constitutes a network organization based on the social fabric of contributions. A smaller social
formation of the AMRMClient.java class at the time interval t1 : 2013–2014 has been constructed by
5 members with various contributions to the class. Figure 1 shows a graph-based agents contributions
to the class in focus—Class 2. Therefore, the aggregation of all social formations constructed to solve
a given problem, i.e., when the agents are working on specific classes, for the 10 classes within the
package forms the overall network of this package as an organization.
Entropy 2020, 22, 519
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Table 2. Class 2: AMRMClient.java, # of contributors = 16.
Commit Date
Contributor
Add #
Del#
Commit Date
Contributor
Add #
Del #
18 June 2013
10 June 2013
21 June 2013
15 July 2013
16 July 2013 (1)
16 July 2013 (2)
17 July 2013
19 July 2013
31 August 2013
31 October 2013
09 July 2014
09 August 2014
11 October 2014
06 February 2015
16 March 2015
18 March 2015
19 March 2015
12 November 2015
08 January 2016
Vinoduec (Vin)
Bikas Saha - (BS)
Vinoduec (Vin)
Bikas Saha (BS)
Bikas Saha (BS)
Bikas Saha (BS)
Bikas Saha (BS)
Acmurthy - (Acm)
Bikas Saha (BS)
Vinoduec (Vin)
jian-he - (JH)
zjshen14 - (ZjS)
Vinoduec (Vin)
jian-he (JH)
Oza - (Oz)
zjshen14 (ZjS)
JunpingDu - (JPD)
Wangdatan - (WDT)
Xslogic - (Xs)
4
1
0
10
85
1
6
5
11
32
15
63
34
2
2
18
3
23
40
3
1
12
1
16
1
41
9
0
0
8
0
2
1
2
0
3
2
3
26 April 2016
12 June 2016
10 July 2016
09 August 2016
27 August 2016
12 November 2016
15 November 2016
14 February 2017
14 February 2017
16 February 2017
30 August 2017
04 September 2017
09 September 2017
18 September 2017
31 October 2017
17 January 2018
16 February 2018
31 July 2018
Xslogic (Xs)
Xslogic (Xs)
Sjlee (SjL)
Rohithsharmaks (Roh)
Wangdatan (WDT)
Wangdatan (WDT)
Xslogic (Xs)
Xslogic (Xs)
Xslogic (Xs)
Sjlee (SjL)
jian-he (JH)
Varunsaxena (Var)
Haibchen (Hai)
Xslogic (Xs)
Aajisaka (Aaj)
wangdatan (WDT)
Sunilgovind (Sun)
Hungj (Hun)
2
35
5
1
169
88
116
31
31
30
3
20
4
7
4
37
12
11
1
8
5
1
31
21
1
2
2
10
2
2
4
2
3
1
65
0
A social network of contributors is constituted based on the data presented in Table 2. As an
illustrative example, the social network of AMRMClient.java class for time interval (t1 : 2013-2014) is
constructed by 5 members contribute to the class (Figure 1a) and create a social network as shown
in Figure 1b. Thus, the overall social network of this package is a union of social networks of the
10 classes in the package.
ZjS
36
ZjS
63
63
63
2
36
12
Vin
63
JH
7
4
11
2
23
32
63
Class 2:
AMRMClient.java
BS
23
80
169
2
68
Vin
11
14
36
23
11
101
BS
JH
63
Acm
Acm
51
14
(a)
(b)
Figure 1. Illustrative sample of the social network organization constituted by contributors in
class AMRMClient.java for time interval 2013–2018. (a) Mapping the contributions of every agent;
(b) Ascending order of contributions.
Table 3 shows the total involvement of every agent within the organization and their relations to
one another that have been calculated from all 10 classes of the package. Also, the number of classes,
in which each contributor was involved, has been counted in order to later help in the measurement of
capacity for every individual agent. Moreover, due to the openness nature of an organization such as
this OOS project, we based our assumption on the premise that the willingness of the contributors is at
a maximum level, i.e., the value of willingness = 1. Once the data input processing is complete, the SC
of every individual agent can be computed using the Algorithms 1 and 2 proposed.
Entropy 2020, 22, 519
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Table 3. Relations of contributors and their involvement.
No.
Contributor
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Vin
Oz
Roh
BS
Acm
JH
ZjS
JPD
WDT
Xs
SjL
Var
Hai
Aaj
Sun
Hun
# of Relations
t1
t2
t3
24
0
0
12
13
6
6
0
0
0
0
0
0
0
0
0
18
26
21
0
0
25
7
7
27
10
9
0
0
12
12
0
1
0
0
0
0
12
0
0
13
14
11
8
8
5
7
7
# Classes
No.
Contributor
6
5
3
3
3
3
1
1
4
2
2
1
1
3
2
1
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Bil
Mac
Ctr
Kam
Hit
Ale
Sry
Sub
Xgo
Jlo
Car
Min
Nag
Vas
Bib
# of Relations
t1
t2
t3
0
0
0
0
7
5
6
6
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
14
14
12
11
11
10
0
3
2
0
1
0
0
0
0
0
0
0
0
4
0
4
# Classes
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Algorithm 1: A Sociograph based on Relations.
Result: Values of relations ∀ n-agents involved in a class
initialization;
for t = 1 → 3 do
for Module = 1 → 10 do
for i = 1 → n do
Read capacity Agent[i ] ;
for j = 1 → n do
Read ELink [i, j]
// Agenti → Agent j ;
end
Relation [i ] = max {| ELink [i, j]|}
// Maximum frequency;
Read values of relations ;
end
end
end
Algorithm 2: Measurements of Social Capital.
Result: The SC values of agents
initialization;
for t = 1 → 3 do
for Module = 1 → 10 do
for i = 1 → n do
for j = 1 → n, and i 6= j do
1
Belie f [i ] = Relation
[i ] ,
e
Compute Benevolance [i ],
Compute PBenevolance [i ],
Compute Social Capital of Agent[i ],
end
end
end
end
// Belief of Agenti → Agent j ;
// Using Equation (6);
// Using Equation (7);
// Using Equation (9);
Entropy 2020, 22, 519
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3.2. An Illustrative Example: Class 2
To illustrate the overall computation of the social capital of this social formation, this section
presents a calculation of SC with a sample collection of data from a sub-task-2 (Class-2) throughout the
time interval t1 : 2013–2014. Table 4 depicts the data used in the social capital calculation of the class.
The relation values are the total number of contributions, i.e., the total code lines provided to include
addition as well as deletion.
Table 4. Class name: AMRMClient.java, (2013-2014), # of contributors: 5.
Commit Data
Contributor/Agent
18 June 2013
19 June 2013
21 June 2013
15 July 2013
16 July 2013 (1)
16 July 2013 (2)
17 July 2013
19 July 2013
31 August 2013
31 October 2013
09 July 2014
09 August 2014
11 October 2014
Vinoduec (Vin)
Bikas Saha (BS)
Vinoduec (Vin)
Bikas Saha (BS)
Bikas Saha (BS)
Bikas Saha(BS)
Bikas Saha (BS)
Acmurthy (Acm)
Bikas Saha (BS)
Vinoduec (Vin)
Jian-he (JH)
Zjshen14 (ZjS)
Vinoduec (Vin)
# Add
# Del
Relation
Capacity
4
1
0
10
85
1
6
5
11
32
15
63
34
3
1
12
1
16
1
41
9
0
0
8
0
2
7
2
12
11
101
2
42
14
11
32
23
63
36
6
3
0
0
0
0
0
3
0
0
3
1
0
3.2.1. Links
Based on the assumption that an agent benevolence to one class is equal to every individual
member within that class, the number of agents involved within this class have been sorted in
ascending order considering their time of contribution. Figure 2 illustrates the number of links in the
network to calculate the belief values.
5
2
Vin
1
ZjS
4
1
JH
1
1
Vin’
1
BS
Acm
1
BS’
1
Vin”
1
BS”
6
8
Figure 2. The value of relation of each node in the social network in order to measure the belief values.
Here, we consider the repetition of agents’ order in the network when showing up again with apostrophes.
We are now able to calculate the link for each contributor by implementing Algorithms 1
and 2. The links for contributor Vin, for instance, are: ELink (Vin, ZjS) = 1; ELink (Vin, JH ) = 2;
ELink (Vin, BS) = {4, 6, 8}; ELink (Vin, Acm) = 5. By the use of Equation (4), we are able to obtain the
value of relations for all contributors, as shown in Table 5.
→¬i ∀i 6 = ¬i ∈ N.
Table 5. Number of relations for all contributors in sub-task 2 (Class-2), i.e., Rielation
.
¬i
i
Vin
ZjS
JH
BS
Acm
Vin
ZjS
JH
BS
Acm
8
7
4
4
1
-
2
1
-
7
7
6
3
5
4
3
1
-
Entropy 2020, 22, 519
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3.2.2. Relation
Table 6 shows the values of relations among the contributors in the Class-2 network. The value
of an agent relation is the sum-up number of code lines to include addition and deletion from each
contributor (refer to Table 3).
Table 6. Values of relations among the contributors.
Contributor
Vin
ZjS
JH
BS
Acm
Total
Vin
ZjS
JH
BS
Acm
63
46
169
14
36
23
-
36
63
-
80
63
46
14
68
63
46
11
-
227
252
161
180
28
Total
299
59
99
203
177
-
Receiver
3.2.3. Belief
i
−λ·( Relation )
i
Belief: Belie
. Assume that λ = 1. Table 7 shows the Belief values from one
f = e
contributor relative to the others.
i →¬i .
Table 7. Belief values, Belie
f
i
¬i
Vin
ZjS
JH
BS
Acm
Vin
ZjS
JH
BS
Acm
0.0003
0.0009
0.018
0.018
0.367
-
0.135
0.367
-
0.049
0.0009
0.002
0.049
0.006
0.018
0.049
0.367
-
3.2.4. Benevolence
Benevolence is calculated as defined in Equation (6). Table 8 shows the calculation results.
i →¬i
Table 8. Benevolence for each contributor, i.e., Benevolence
.
i
Vin
ZjS
JH
BS
Acm
0
63 × 1 = 63
46 × 3 = 138
507 × 3 = 1521
42 × 3 = 126
216 × 6 = 1296
0
0
0
0
216 × 6 = 1296
63 × 1 = 63
0
0
0
480 × 6 = 2880
63 × 1 = 63
46 × 3 = 138
0
42 × 3 = 126
408 × 6 = 2880
63 × 1 = 63
46 × 3 = 138
33 × 3 = 99
0
¬i
Vin
ZjS
JH
BS
Acm
3.2.5. Potential Benevolence
The value of Potential Benevolences are calculated using Equation (7). The values of Relations are
obtained from Table 3 (sum up the addition and deletion), and the values of Belief are obtained from
Table 5. The results are as follows.
Vin
=
PBenevolence
n
ZjS
ZjS→Vin
Belie f × Relation
o
JH →Vin
JH
Vin
Acm→Vin
BS
BS→Vin
Acm
× Capacity
+ Belie
+ Belie
+ Belie
f × Relation
f × Relation
f × Relation
o
JH → ZjS
BS→ ZjS
Acm→ ZjS
ZjS
JH
BS
Acm
+ Belie
+ Belie
+ Belie
× Capacity
f × Relation
f × Relation
f × Relation
= {(0.367 × 63) + (0.135 × 46) + (0.049 × 69) + (0.006 × 14)} × 6
= 22.176
ZjS
PBenevolence =
n
Vin→ ZjS
Vin
Belie
f × Relation
= {(0.0003 × 36) + (0.135 × 46) + (0.0009 × 0) + (0.018 × 0)} × 1 = 8.452
Entropy 2020, 22, 519
JH
=
PBenevolence
14 of 19
n
o
ZjS
ZjS→ JH
Acm→ JH
JH
Vin→ JH
BS→ JH
Vin
BS
Acm
Belie
+ Belie f × Relation
+ Belie
+ Belie
× Capacity
f × Relation
f × Relation
f × Relation
= {(0.0009 × 36) + (0 × 63) + (0.0009 × 0) + (0 × 0)} × 3 = 0.0972
BS
PBenevolence
=
n
o
JH → ZjS
JH → BS
JH
JH
Vin→ BS
Vin
Acm→ BS
BS
Acm
Belie
+ Belie
× Capacity
+ Belie
+ Belie
f × Relation
f × Relation
f × Relation
f × Relation
= {(0.018 × 169) + 0 + 0 + (0.367 × 14)} × 3 = 24.54
Acm
=
PBenevolence
n
o
JH → Acm
JH
BS
BS→ Acm
BS
BS→ Acm
Acm
Vin→ Acm
Vin
+ Belie
Belie
+ Belie
× Capacity
+ Belie
f × Relation
f × Relation
f × Relation
f × Relation
= {(0.018 × 14) + 0 + 0 + (0.049 × 11)} × 3 = 2.373
3.2.6. Social Capital
According to Equation (9), the value of SCs are calculated as follows:
SC
Vin
ZjS→Vin
ZjS→Vin
JH →Vin
JH →Vin
Benevolence ∩ PBenevolence
Vin
→
JH
+ B
×
×
elie
f
ZjS→Vin
JH
→
Vin
PBenevolence
PBenevolence
Acm→Vin ∩ PB Acm→Vin
BS→Vin ∩ PB BS→Vin
Benevolence
Benevolence
enevolence
enevolence
Vin→ BS
+ BVin→ Acm ×
×
+ Belie
elie
f
f
BS→Vin
Acm→Vin
PBenevolence
PBenevolence
n
o
o n
o n
o n
−8.452)
−1.944)
−24.54)
−2.373)
= 0.367 × (638.452
+ 0.135 × (1381.944
+ 0.049 × (152124.54
+ 0.006 × (1262.373
Vin→ ZjS
= Belie f
Benevolence ∩ PBenevolence
= 15.117
SC ZjS
Vin→ ZjS
Vin→ ZjS
JH → ZjS
JH → ZjS
B
∩
PB
enevolence
enevolence
ZjS→Vin
+ B ZjS→ JH ×
×
= Belie f
Vin→ ZjS
JH → ZjS
elie f
PBenevolence
PBenevolence
BS→ ZjS
Acm→ ZjS
BS→ ZjS
Acm→ ZjS
B
→
PB
B
∩
PB
enevolence
enevolence
enevolence
ZjS→ BS
+ B ZjS→ Acm × enevolence
×
+ Belie f
elie
f
BS
→
ZjS
Acm
→
ZjS
PBenevolence
PBenevolence
n
o
1.944)
24.54)
2.373)
−226.176)
+ 0.367 × (0−1.944
+ 0.0009 × (0−24.54
+ 0.018 × (0−2.373
= 0.0003 × (63226.176
Benevolence ∩ PBenevolence
= − 0.386
Vin→ JH
ZjS→ JH
ZjS→ JH
Vin→ JH
B
∩
PB
B
∩
PB
enevolence
enevolence
enevolence
JH → ZjS
+ B JH →Vin × enevolence
×
SC JH = Belie f
elie
f
ZjS
→
JH
Vin
→
JH
PBenevolence
PBenevolence
BS→ JH
Acm→ JH
Acm→ JH
BS→ JH
Benevolence ∩ PBenevolence
Benevolence ∩ PBenevolence
JH → Acm
JH → BS
B
×
+
+ Belie f
×
elie
f
BS→ JH
Acm→ JH
PBenevolence
PBenevolence
o
n
−226.176)
−8.452)
24.54)
2.373)
= 0.0009 × (1296226.176
+ 0.0 × (638.452
+ 0.0009 × (0−24.54
+ 0.018 × (0−2.373
= − 0.014
JH → BS
ZjS→ BS
ZjS→ BS
JH → BS
Benevolence ∩ PBenevolence
Benevolence ∩ PBenevolence
BS→ ZjS
BS
→
JH
BS
+ B
×
×
SC = Belie f
elie
f
JH
→
BS
ZjS
→
BS
PBenevolence
PBenevolence
(
"
#)
Vin→ BS ∩ PBVin→ BS
Acm→ BS ∩ PB Acm→ BS
Benevolence
Benevolence
enevolence
BS→Vin
BS→ Acm
enevolence
×
+ Belie f
×
+ Belie f
Acm→ BS
Vin→ BS
PBenevolence
PBenevolence
o
n
−8.452)
−1.944)
−2.373)
−226.176)
+ 0.0 × (688.452
+ 0.0 × (1381.944
+ 0.367 × (1262.373
= 0.018 × (2880226.176
= 19.33
Entropy 2020, 22, 519
15 of 19
ZjS→ Acm
ZjS→ Acm
JH → Acm
JH → Acm
Benevolence ∩ PBenevolence
+ B Acm→ JH ×
×
elie
f
JH → Acm
ZjS→ Acm
PBenevolence
PBenevolence
(
"
#)
Vin→ Acm ∩ PBVin→ Acm
BS→ Acm ∩ PB BS→ Acm
Benevolence
B
enevolence
Acm→ BS
Acm→Vin
enevolence
enevolence
B
×
+
×
+ Belie
f
BS→ Acm
Vin→ Acm
elie f
PBenevolence
PBenevolence
n
o
−8.452)
−1.944)
−24.54)
−226.176)
= 0.018 × (2880226.176
+ 0.0 × (638.452
+ 0.0 × (1381.944
+ 0.049 × (9924.54
Acm→ ZjS
SC Acm = Belie f
Benevolence ∩ PBenevolence
= 0.359
Table 9 shows the completed computation of SCs of Class number 2 for the three time intervals.
The SCs values are accumulated from previous time intervals.
Table 9. Social Capitals of the agents involved in Class-2.
SC
Agents
t1
t2
t3
Vin
ZjS
JH
BS
Acm
Oza
JPD
WDT
Xsl
SjL
Roh
Var
Hai
Aaj
Sun
Hun
15.117
−0.386
−0.014
19.33
0.359
0
0
0
0
0
0
0
0
0
0
0
15.117
3.17
4.773
19.33
0.359
0.766
−1.023
−0.986
4.781
14.433
16.551
0
0
0
0
0
15.117
3.17
10.171
19.33
0.359
0.766
−1.023
1.126
11.648
33.067
16.551
−1.237
2.459
7.889
1.252
0.746
Total
34.406
77.271
121.391
3.2.7. Measuring SC for Only One Agent in the Subtask
As shown in Table 1, in each time interval there are some sub-tasks (classes) that consist of only
one contributor. Since there is only one agent/contributor in the class, we assume the relation is 1
(self relation), then the Belief is 1e and PBenevolence = 0. Hence, PBenevolence = 0, then, according to
Equation (8), the value of f ( Benevolence | PBenevolence ) = 0, thus the value of SC is zero. The calculation of
the SC for this case is shown in Table 10.
Table 10. SC of a single agent in a class.
Interval
Class #
Agent
Capacity
Relation
Benevelence
PBenevelence
SC
t1
4
5
6
9
10
BS
Vin
Vin
Acm
Vin
3
6
6
3
6
38
100
143
21
52
114
600
858
63
312
0
0
0
0
0
0
0
0
0
0
t2
6
7
Oz
Ctr
5
1
58
108
290
108
0
0
0
0
t3
1
3
6
Vin
Bil
Aaj
6
2
3
10
163
2
60
326
6
0
0
0
0
0
0
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3.2.8. Overall Social Capital
As mentioned in Section 3.1, the overall value of social capital is computed using the Algorithms 1
and 2. Table 11 shows the overall SCs of each contributor for intervals t1 , t2 and t2 . The social capital
value of each agent/contributor is a sum up of contributions in the involved sub-tasks/classes.
Table 11. The SCs for contributing agents.
Agent
Vin
BS
Acm
JH
ZjS
Oz
JPD
WDT
Xs
SjL
Roh
Sun
Hun
Xgo
Jlo
Car
SC
t1
t2
t3
35.558
22.21
0.161
−0.014
−0.386
0
0
0
0
0
0
0
0
0
0
0
79.461
22.21
0.161
26.016
3.17
297.831
−1.023
18.909
4.781
14.433
16.649
0.067
0
1.347
1.002
0.92
79.461
22.21
0.161
31.414
3.17
297.831
−1.023
22.386
13.192
33.067
16.649
1.319
0.746
1.347
1.002
0.92
Agent
SC
t1
t2
t3
Aaj
Min
Nag
Var
Bib
Bil
Hit
Ale
Sry
Sub
Vas
Ctr
Kam
Hai
Mac
0
0
0
0
0
0
1.112
0.887
1.086
−0.245
0
0
0
0
0
0.788
1.001
0.742
0.689
0
0
1.112
0.887
1.086
−0.245
0
0
0
0
0
8.238
1.001
2.188
−0.548
0.994
−0.897
1.112
0.887
1.086
−0.245
0
0
−1.779
2.459
1.266
Total
60.369
552.363
1091.977
3.3. Discussion
Here, we discuss social capital from four different perspectives as observed in the previous stated
case study.
1.
2.
3.
From the point of Relations, as mentioned in Section 2.1.1, the existing relations among the agents
are explicit links among them and the calculation of relations is illustrated in Figure 2 and Table 5.
As we can see from Figure 2, relations/links are constructed based on the agent’s commit dates.
The more agents involved, the more commit dates, thus, the more links created. Then, a class with
only two contributors/agents implies that the agents have only 1 link and gives a high belief value.
A sole agent in a class does not develop any social capital since it has no relations. Her contribution
will be taken into consideration when we consider the relation among classes, which is based on
inherited methods from each others classes reflecting inter-organizational perspective is, however,
out of the scope of the experiment.
From the Beliefs perspective, we recall from Section 2.2 that Belief is a monotonically decreasing
function so that a larger number of relations corresponds to a lower belief. In the case of the OSS
organization, each contributor does not know each other; thus, we merely measure the belief
based on the number of links corresponding to the agents. So, it makes sense that in a class with
only two agents/contributors, the agent trusts each other.
From the view of Benevolences, in the observed Package, there are three classes that involve more
than 2 contributors: Class 2 (AMRMClient.java) with 16 contributors, Class 5 (NMClient.java)
with 6 contributors, and Class 8 (YarnClient.java) with 21 contributors. Due to the nature of
the OSS organization that contributors/agents are not bonding (volunteer based involvement),
we have observed that only one agent (JH) consistently contributes in the three time intervals.
The results have shown that some individual agents, at the end of each time interval or even
the complete project, ended up with negative values of SC, which is because of the higher
expectations to receive benevolences from their peers. This means current connections and
Entropy 2020, 22, 519
4.
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the promising values they may entail may not always be available to share, which is, when
considering the benevolent behavior, what this paper is about. Nevertheless, the overall values
of SC for each time interval are positive. The SC value increases 814.97% and 102.36% for time
interval t2 and time interval t3 , respectively. The value of SC during time interval t2 has peaked
due to the high benevolence a single agent has contributed to the network. During this time
interval, two agents—including agent Oz—are involved in class number 5. Since only two agents
are involved, the relation among the agents is 1, giving the value of Belie f to be close to 1; therefore,
agent Oz contributes a higher benevolence value (Table 11). This, in part, indicates that an agent
productivity contributes positively to the organizational outcome.
Form the Organizational goal, the publicly available data of the observed package of Apache
Hadoop can be summarized as the number of commit dates of agents to complete designated
tasks for each time interval, t1 , t2 , and t3 , are 37, 37, and 28, respectively. We conclude, when we
consider the number of commits, that the agents in this organizational network have successfully
delivered the project and, accordingly, they have positive SC values that have led to a positive
value of their organization.
4. Conclusions and Future Work
The idea of this paper is to reflect the effects of existing social fabric among individual agents
by quantifying “Social Capital” in an ad-hoc organization using benevolence as a measure of their
collaborations. We illustrated that SC is achievable in the context of a dynamic network organization
with the help of a popular open source software project through the presentation of heuristics to
compute numeric values for measuring SC, which, in turn, can translate to degree and effectiveness
of collaboration in open ad-hoc agent organizations. This article defined SC on three different levels,
i.e., network, link and agent, and proposed a measurement for it based on the benevolences between
autonomous agents operating in a large-scale open service-oriented organization. Incorporating
benevolence, in measuring the social capital for individual agent and for the organization as whole,
gives more tangible values. Those values contribute positively towards the cooperative nature of an
organization. We showed an empirical evaluation of the proposed approach using a real-world case
study of an open-source project development, and we assessed the validity of each measure of SC
in different settings within a network organization. Furthermore, the empirical evaluation showed
that the social capital of an individual agent may result in a negative value as the agent expected
contributions from other agents; however, the contribution received may not be as expected. This
finding is a result of considering the benevolence in a social capital measurement. Another finding is
that the belief and trust also contributes towards the measurement of social capital. As we observe, the
social capital of a group or an organization increases when it involves more agents; on the other hand,
few numbers of agents involved in a subtask with significant line code contributions provides a higher
social capital, due to a higher value of belief. The authors believe that the amount of data used in the
empirical evaluation represents the behavior of the OSS social network, as the computations are to be
easily scaled up.
The concept of SC has shown its usefulness, fruitfulness, and efficiency in genuine empirical
research, such as the available empirical approach of SC presented here. The heterogeneity of the
proposed conceptualization has been less reflected in the empirical heterogeneity as to what has
been expected. Future work should consider the use of multi-method and multi-level strategies
to improve the current role of empirical evidence in the debate on SC. It should also consider the
proposition of a detailed social capital assessment model that is required to estimate the future behavior
of agents and agents’ peers in order to simplify the interaction process with those peers. Another
further direction is to benefit from a systematic analysis and recommendations on how agents ought to
behave for better performance to include comparisons with baselines and other metrics that measure
ad-hoc organizations.
Entropy 2020, 22, 519
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Author Contributions: Conceptualization, S.A. and H.H.; methodology, S.A.; software, R.B.; validation, S.A., R.B.
and M.A.; formal analysis, S.A.; investigation, S.A.; resources, R.B.; data curation, R.B. and M.A.; writing—original
draft preparation, S.A.; writing—review and editing, S.A.; visualization, S.A.; supervision, H.H.; project
administration, S.A. All authors have read and agreed to the published version of the manuscript.
Acknowledgments: We acknowledge the inputs and directions we have received from anonymous reviewers for
all the stages and submissions this work has been gone through in the past couple of years.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this manuscript:
OOS
MAS
SC
′
Iii
′
ELi,i
ink
′
ILi,i
ink′
→i
Rielation
i
Capacity
′
i →i
Benevolence
i →i ′
PBenevolence
Open Source Software.
MultiAgent System.
Social Capital and SCi is the social capital value of agent i.
Interaction between agent i and i′ .
Explicit link between agent i to i′ .
Implicit link between agent i and i′ .
Direct relationship from agent i to agent i′ .
The capacity that agent i has.
The directed benevolence from agent i and agent i′ .
Directed potential benevolence agent i′ expect to receive from agent i.
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