To demonstrate different supply
network architectures consisting of the four aforementioned
heterogeneous supply network ties (i.e., contractual, transactional,
professional, and personal ties), this study adopts social network
analysis (SNA), which has long been used in analyzing any social network
as a set of interrelated actors and ties. The field of SCM has stressed
the potential applicability of SNA in a supply network context. For
instance, Carter et al. proposed SNA as a valuable complement to
traditional methodologies which may be used to advance current knowledge
on various relationships existing within and beyond the supply chain.
This view was echoed by Borgatti and Li who pointed out that supply
chain settings are particularly suitable for SNA indices, which have
proven "highly portable" across other disciplines from economics to
physics. More recently, Galaskiewicz also noted that SCM theories
mostly captured at the local level (e.g., dyad or triad) can be tested
by using a supply network as the primary unit of analysis.
Despite
repeated calls for such approach, there are still very few SCM studies
that use SNA. Moreover, the vast majority of
existing studies on supply network are case-based research that uses SNA
measures defined for binary (i.e., "1" if a tie exists between two
supply network entities, "0" otherwise) and non-directional ties (i.e.,
if one supply network entity perceives a tie, its counterpart's
perception of the existence of the tie is automatically assumed). This
is commonly referred to as the binary network approach, and most of the
existing SNA indices have been devised solely based on this approach. The binary network approach specified by a symmetric adjacency
matrix is conceptually and computationally straightforward and
especially appropriate when a researcher focuses on cognitive ties
(e.g., who knows whom). An important limitation of this approach,
however, is that it involves an unrealistic premise - all ties are
completely homogeneous and symmetrical - which contradicts previous
findings in the literature. For instance, strong social ties strengthen
interpersonal obligations, facilitate change in the face of
uncertainty, and help to develop relationship-specific heuristics. Therefore, by using the binary network approach, network
researchers can inevitably overlook important information about network
properties embedded in network ties and consequently arrive at limited
or even misleading implications for network architecture.
We thus
adopted a directed valued network approach represented by an asymmetric
adjacency matrix to overcome the aforementioned shortcomings of the
binary network approach. This approach takes into account the
direction and strength (or magnitude) of each tie between different
network entities. In network terms, a directed valued network consists
of a set of actors (or nodes) \({n_1, n_2, ⋯, n_g}\), a set of arcs (i.e.,
directional ties or links) \({l_1, l_2, ⋯, l_L}\), and a set of values \({v_1, v_2,
⋯, v_L}\) attached to the arcs, subject to
\(l_k=<n_i,n_j>≠l_m=<n_j,n_i>\) where \(v_k\) is not necessarily equal to
\(v_m\). This is a more useful and realistic approach for exploring
supply network phenomena since it allows for the possibility that a
focal firm and its suppliers may view the strength (or even the
existence) of their ties differently. In this sense, there has been a
growing need for SNA indices that can be used in the directed valued
network setting when it is based on a different adjacency matrix.
More
specifically, this study focuses on four socio-centric network indices
(i.e., betweenness centralization, in-degree centralization, out-degree
centralization, and global clustering coefficient), which describe the
overall pattern of multiple actors within a single, bounded network.
While ego-centric indices, such as centralities, deal with a particular
actor's (i.e., ego's) position within the network, they provide a better
understanding of the directed valued network in that the network
architecture from one ego's viewpoint can be markedly different from
those of others linked directly or indirectly. They also fit
perfectly with the purpose of this study to explore the association
between an OEM's strategic orientation and the supply network
architectures it creates based on different types of supply network
ties. Table 2 proposes a new framework for the supply network
implications of the socio-centric SNA indices for the directed valued
networks used in this study for four types of supply network ties.
Table 2. Socio-centric indices, conceptual definitions, and interpretations by supply network tie.
| Socio-Centric SNA Index |
Conceptual Definition |
Tie Type |
Implications for Directed Valued Supply Network |
| Betweenness centralization (BTC) |
The extent to which particular network actors serve as hubs relative to the rest of the network |
Contractual |
The
extent to which there exist particular focal firms that have more or
less complete (or specific) contract terms than other supply network
members.
- -
-
The
lower the index, the more firms there are which have more equally
complete contract terms with their supply network counterparts.
- -
-
The
higher the index, the more firms there are which have more unequally
complete contract terms with their supply network counterparts.
|
| Transactional |
The
extent to which there exist particular focal firms that have a higher
or lower percentage of monetary exchanges than other supply network
members (i.e. distribution of sales and spending in the network).
- -
-
The
lower the index, the more firms there are which have equal percentages
of monetary exchanges with their supply network counterparts.
- -
-
The
higher the index, the more firms there are which have higher or lower
percentages of monetary exchange with their supply network counterparts.
|
| Professional |
The
extent to which there exist particular focal firms that have more or
less work-related interactions than other supply network members.
- -
-
The
lower the index, the more firms there are which have an equal amount of
work-related interactions with their supply network counterparts.
- -
-
The
higher the index, the more firms there are which have more or less
work-related interactions with their supply network counterparts.
|
| Personal |
The
extent to which there exist particular focal firms that have more or
less non-work-related interactions than other supply network members.
- -
-
The
lower the index, the more firms there are which have an equal amount of
non-work-related interactions with their supply network counterparts.
- -
-
The
higher the index, the more firms there are which have more or less
non-work-related interactions with their supply network counterparts.
|
In-degree centralization
(IDC) |
The extent to which network resources are converged on particular network actors |
Contractual |
The
extent to which particular focal firms obtain more complete (i.e. less
favorable) contract terms from the other supply network members.
- -
-
The lower the index, the more firms there are which have fair contract terms with their supply network counterparts.
- -
-
The
higher the index, the fewer particular focal firms possess less
favorable contract terms with their supply network counterparts.
|
| Transactional |
The
extent to which particular focal firms take up a greater percentage of
the monetary exchanges occurring inside the supply network than others.
- -
-
The lower the index, the more firms there are which have equal percentages of the monetary exchanges.
- -
-
The higher the index, the fewer particular focal firms account for higher percentages of the monetary exchanges than the others.
|
| Professional |
The
extent to which particular focal firms obtain more incoming
work-related interactions from the rest of the supply network members.
- -
-
The lower the index, the more equal the amount of work-related interactions between supply network members.
- -
-
The
higher the index, the more work-related interactions among supply
network members is focused on fewer particular focal firms.
|
| Personal |
The
extent to which particular focal firms obtain more incoming
non-work-related interactions from the rest of the supply network
members.
- -
-
The
lower the index, then each of the supply network members has a more
equal amount of non-work-related interactions with one another.
- -
-
The
higher the index, the more non-work-related interactions among supply
network members is focused on fewer particular focal firms.
|
Out-degree centralization
(ODC) |
The extent to which particular actors disseminate network resources to others |
Contractual |
The
extent to which particular focal firms provide more complete (i.e. less
favorable) contract terms for the rest of the supply network members.
- -
-
The lower the index, the more firms there are which have fair contract terms with their supply network counterparts.
- -
-
The
higher the index, the fewer particular focal firms yield less favorable
contract terms for their supply network counterparts.
|
| Transactional |
The
extent to which particular focal firms generate higher percentages of
the monetary exchanges occurring inside the supply network than others.
- -
-
The lower the index, the more firms there are which have equal percentages of the monetary exchanges.
- -
-
The
higher the index, the fewer particular focal firms send out higher
percentages of the monetary exchanges for the rest of the supply network
members.
|
| Professional |
The
extent to which particular focal firms have more outgoing work-related
interactions to the rest of the supply network members
- -
-
The
lower the index, the more equal the amount of work-related interactions
between each of the supply network members and the others.
- -
-
The
higher the index, the fewer particular focal firms initiate most of the
work-related interactions with the rest of the supply network members.
|
| Personal |
The
extent to which particular focal firms generate more outgoing
non-work-related interactions for the rest of the supply network members
- -
-
The
lower the index, then each of the supply network members has more equal
amount of non-work-related interactions with one another.
- -
-
The
higher the index, the fewer particular focal firms make more
non-work-related interactions for the rest of the supply network
members.
|
Global clustering coefficient
(GCC) |
The
extent to which the network as a whole is cliquish (or tightly knit)
(i.e. the degree to which all the network actors tend to cluster
together) |
Contractual |
The extent to which members of the entire supply network are directly connected by contract relations
- -
-
The
lower the index, the lower the proportion of all supply network members
that are directly connected by contract relations (i.e. the supply
network has a more ‘hierarchical’ architecture as a whole).
- -
-
The
higher the index, the higher the proportion of supply network members
that are directly connected by contract relations (i.e. the supply
network has a more ‘lateral’ architecture as a whole).
|
| Transactional |
The extent to which the members of the entire supply network are directly connected by monetary exchanges
- -
-
The
lower the index, the more the supply network as a whole has a
“hierarchical” architecture in the monetary exchanges among supply
network members.
- -
-
The
higher the index, the more the supply network as a whole has a
“lateral” architecture in the monetary exchanges among supply network
members.
|
| Professional |
The extent to which all the supply network members freely communicate work-related subjects across firm boundaries
- -
-
The
lower the index, the more “hierarchical” the architecture of
non-work-related interactions among members in the supply network as a
whole.
- -
-
The
higher the index, the more the supply network as a whole has a
“lateral” architecture for work-related interactions among supply
network members.
|
| Personal |
The extent to which all the supply network members freely communicate non-work-related subjects across firm boundaries
- -
-
The
lower the index, the supply network as a whole has a more
‘hierarchical’ architecture of non-work-related interactions among
supply network members.
- -
-
The
higher the index, the more “lateral” the architecture of
non-work-related interactions among members in the supply network as a
whole.
|
First,
betweenness centralization (BTC) represents whether most network actors
are equally central, or some actors (i.e., hubs) are much more central
than others. This index can be calculated by dividing the variation in
the betweenness centrality by the maximum variation in betweenness
centrality scores possible in a network of the same size.
Betweenness centrality is an ego-centric index indicating how often an
actor lies on the shortest path between all combinations of pairs of
other actors. The higher an actor's betweenness centrality, the more its
immediate counterparts depend on this actor to reach out to the rest of
the network. This index focuses on the role of an actor as an
intermediary and posits that the dependence of others makes the actor
central in the network. BTC, a socio-centric version of betweenness
centrality, ranges from 0 where all network actors have the same
betweenness centrality, to 1, where there exists one single actor
connecting all the other actors. This study calculates the BTC of a
directed valued supply network by adopting the formula suggested by
Opsahl et al. for betweenness centrality (\(C^{wα}_{B}(n_i)\)) for network actor \(n_i\), defined as:
\(C^{wα}_{B}(n_i)=\dfrac{g^{wα}_{n_jn_k}(n_i)}{g^{wα}_{n_jn_k}}\)
where
\(g^{wα}_{n_jn_k}\) is the total number of geodesics between two actors (\(n_j\) and
\(n_k\)), \(g^{wα}_{n_jn_k}(n_i)\) is the number of geodesics passing through actor \(n_i\),and \(α\) is a positive tuning parameter that is set to the benchmark value
of 0.5 to equally value both the number of ties and their strengths (\(w\)).
Thus, BTC can be formally expressed as:
\(C_B=\dfrac{∑_{i∈G}{C^{wα}_{B}(n^∗)−C^{wα}_{B}(n_i)}}{max∑_{i∈G}{C^{wα}_{B}(n^∗)−C^{wα}_{B}(n_i)}}\)
where
\(C^wα_{B}(n^∗)\) is the largest value of the betweenness centrality that occurs
across the network \(G\); that is, \(C_{B}^{w \alpha}\left(n^{*}\right)=\max _{i} C_{B}^{w \alpha}\left(n_{i}\right)\).
In the
case of a directed network, two additional degree indices are defined:
in-degree, or the number of links terminating at the actor \((k^{in}_{n_i})\),
and out-degree, or the number of ties originating from the actor
\((k^{out}_{n_i)}\). In-degree centralization (IDC) calculates the dispersion
of or variation in in-degree centrality, and the extent of an individual
actor's influence on other actors; thus, high IDC indicates the
incoming flows of different network resources are focused on a small
group of actors in the overall network. In the same sense, high
out-degree centralization (ODC) indicates that a small number of actors
send out most of the network resources to the rest of the network
actors. This study derives IDC and ODC of a supply network from
in-degree centrality (\(C^{wα}_{D-in}(n_i)\)) and out-degree centrality
(\(C^{wα}_{D-out}(n_i)\)) for actor ni of a directed valued network using the
following equations:
\( C^{wα}_{D-in}(n_i)=k^{in}_{n_i}×(\dfrac{s^{in}_{n_i}}{k^{in}n_i})^α \)
\( C^{wα}_{D-out}(n_i)=k^{out}_{n_i}×(\dfrac{s^{out}_{n_i}}{k^{out}_{n_i}})^α \)
where
\( s^{in}\) and \(s_{out}\) are the total strengths attached to the incoming and
outgoing ties, respectively. Therefore, the general IDC and ODC ranging
from 0 to 1 are respectively defined as:
\(C_{D-in}=\dfrac{∑_{i∈G}{C^{wα}_{D-i_n}(n^∗)−C^{wα}_{D-in}(n_i)}}{max∑_{i∈G}{C^{wα}_{D-in}(n^∗)−C^{wα}_{D-in}(n_i)}}\)
\(C_{D-out}=\dfrac{∑_{i∈G}{C^{wα}_{D-out}(n^∗)−C^{wα}_{D-out}(n_i)}}{max∑_{i∈G}{C^{wα}_{D-out}(n^∗)−C^{wα}_{D-out}(n_i)}}\)
where \(C^{wα}_{D-in}(n^∗)\) and \(C^{wα}_{D-out}(n^∗)\) are the largest in-degree and out-degree centrality values in the network \(G\).
Lastly,
this study uses a global clustering coefficient (GCC) varying from 0 to
1 to measure the overall level of cohesion among network actors. In social network terms, this indicates the probability that
network actors \(n_j\) and \(n_k\) are also connected to each other when \(n_i\) is
connected to both of them, collectively represented as \((n_i;n_j,n_k)\). In a
directed valued network setting, this socio-centric index is defined as
the total value of closed triplets (i.e., triples of network actors
where each actor is connected to the other two; \(τ_Δ\)) divided by the total
value of triplets (i.e., triples where at least one actor is connected
to the other two; \(τ\)). Triplet value (\(ω\)) calculation is based on the
geometric mean of the tie values for the nodes comprising the triplet in
that it: (1) Captures differences between tie strengths, and (2) is
robust to extreme tie strength. Thus, the general GCC (\(C_g\)) can be
formally stated as:
\( C_g=\dfrac{1}{N}∑_{i,j,k∈G}\dfrac{{∑_{(ni;nj,nk)∈{τ_Δ}}ω_{τ_Δ}(ni;nj,nk)}}{{∑(_{ni;nj,nk)∈{τ}}ω_τ(ni;nj,nk)}} \)
where \(N\) is
the number of possible triplets in network G. Readers can refer to the
recent study of Opsahl and Panzarasa for more details on this
technique.
Because SNA indices have been developed and used within a
sociological context, they cannot be directly applied and interpreted
within an interfirm supply network context. Table 2, consequently,
proposes a new framework for the supply network implications of the
socio-centric SNA indices for directed valued networks used in this
study for each of the four tie types previously defined in Table 1.
