Pluralized leadership in complex organizations: Exploring the cross network effects between formal and informal leadership relations

a b s t r a c t

Understanding the connection between leadership and informal social network structures is
important in advancing understanding of the enactment of pluralized leadership. In this article
we explore how the enactment of pluralized leadership is shaped by leadership influence and
informal (advice and support) networks and the interactions between the two. Building on
recent developments in Exponential Random Graph Modeling, we empirically model the
cross network effects across three leadership networks and explore different forms of cross
network effects and under what conditions they occur. Our findings suggest that patterns
of pluralized leadership have important endogenous qualities, as shaped through actors’
leadership and informal networks, and are important for understanding the required capability
for facing increasingly complex organizational situations.

Rather than the individualized heroic view of leadership, we consider leadership as an emergent network of relations, which is
a shared and distributed phenomenon, encompassing several leaders who may be both formally appointed and emerge more
informally (Balkundi & Kilduff, 2006; Carson, Tesluk, & Marrone, 2007; Mehra, Smith, Dixon, & Robertson, 2006). Scholars’
attempts to theorize the notion that leadership extends beyond the individual have spawned a range of different concepts
such as: distributed leadership (Currie, Lockett, & White, 2011; Fitzgerald, Ferlie, McGivern, & Buchanan, 2013; Gronn, 2002;
Mehra et al., 2006), collective leadership (Carter & DeChurch, 2012; Contractor, DeChurch, Carson, Carter, & Keegan, 2012;
Cullen, Palus, Chrobot-Mason, & Appaneal, 2012; Denis, Lamothe, & Langley, 2001; Friedrich, Vessey, Schuelke, Ruark, & Mumford,
2009; Mumford, Friedrich, Vessey, & Ruark, 2012; Yammarino, Salas, Serban, Shirreffs, & Shuffler, 2012), shared leadership
(Carson et al., 2007; Ensley, Hmieleski, & Pearce, 2006), and relational leadership (Uhl-Bien, 2006). In the face of a good deal
of inconsistency surrounding conceptual and definitional issues, a number of these scholars have provided prescriptions for a better
understanding of these labels (see: Denis, Langley, & Sergi, 2012; Yammarino et al., 2012). In particular, Denis et al. (2012)
present the idea of pluralized leadership, within which these other concepts of leadership, extending beyond the individual, are
encompassed. In doing so, they present an opportunity to better our understanding of how pluralized leadership arises.
Specifically, Denis et al. (2012) describe pluralized leadership as being characterized by the existence of multiple leaders
in organizations, whom exert influence through both formal and informal means, and is “naturally occurring” in complex organizations.
As such, leadership is continuously collectively enacted and becomes a consequence of actors’ relations; an effect which is
a product of their local interactions (Denis et al., 2012, p. 254). In fact, this is a view, shared by many scholars of pluralized
leadership (broadly defined) who see leadership as a collective product of actors’ interactions that emerges in social relations

(Balkundi & Kilduff, 2006; Carson et al., 2007; Kilduff & Tsai, 2003; Uhl-Bien, 2006; Yammarino et al., 2012). Here, scholars are
persistently pointing to a gap in our knowledge of pluralized leadership surrounding the influence of leadership on the network
relations that connect people, and vice versa (Friedrich et al., 2009; Mumford et al., 2012; Yammarino et al., 2012).
In this study we address the research gap above drawing on social network analysis theory and method, and so heed the calls
for more scholarly attention to be paid to the micro-dynamics through which pluralized leadership is enacted (Brass, 2001;
Carson et al., 2007; Carter & DeChurch, 2012; Contractor et al., 2012; Mehra et al., 2006). In doing so, our research addresses
the call for research that “may require new types of leadership operationalizations, methods, interventions, and assessments for
understanding and enhancing leadership science and practice” (Yammarino et al., 2012, p. 384).
Consistent with prior research (Carson et al., 2007), we focus on leadership influence networks as constituted in the influence
relationships that emerge among actors (Contractor et al., 2012). Unlike most leadership network studies, which rely on a single
network, we investigate patterns of pluralized leadership from a multi-network view (i.e. “multiplexity”), focusing on the
presence of multiple, and interdependent, types of relationships between the same set of actors (Shipilov, Gulati, Kilduff, Li, &
Tsai, 2014).
From a theoretical perspective we explore the multiplexity surrounding pluralized leadership by employing concepts of mutual
exchange and entrainment to explain two different forms of interdependency between networks of leadership influence and
informal social network relations. Mutual exchange involves a directed tie of one type being reciprocated with a tie of another
type between two actors (Hiller, Day, & Vance, 2006; Settoon, Bennett, & Liden, 1996; Sparrowe & Liden, 1997), and is driven
by the principle of direct reciprocity (Bearman, 1997; Yamagishi & Cook, 1993). Entrainment is a process through which behavioral
cycles related to different informal relations become co-occurring with one another (McGrath, Kelly, & Machatka, 1984;
Standifer & Bluedorn, 2006). To theoretically model the conditions under which the informal relations underpinning leadership
influence are characterized by mutual exchange or entrainment we focus on nature of the informal ties involved, drawing a
distinction between instrumental (i.e. goal oriented) and expressive ties (the tie is an end in itself) (Fombrun, 1982; Ibarra,
1993). In doing so, we are able to offer a unique micro-level perspective of the enactment of pluralized leadership.
From a methods perspective, we draw on recent developments in social network analysis (SNA) methods, specifically
Exponential Random Graph Models (ERGMs) (see: Lusher, Koskinen, & Robins, 2012; Pattison & Wasserman, 1999; Robins,
Pattison, Kalish, & Lusher, 2007) to examine how multiple relations are interrelated. ERGMs are superior to models that assume
independent observations as they take dependencies inherent in network relations into account (Robins et al., 2007). To date
there are few studies that have examined leadership using an ERGM approach (Box-Steffensmeier & Christenson, 2014; Mehra,
Marineau, Lopes, & Dass, 2009; White, Currie, & Lockett, 2014), and even less that takes a multiplexity view (see: Contractor
et al., 2012 for a discussion). ERGM is state of the art, and holds the potential to generate new insights into the structure of
leadership relations, as it enables us to specify and test the specific conditions under which informal relations and leadership
influence relations may be mutually exchanged and/or entrained.
We contribute to the literature on pluralized leadership by examining the type of complex and contentious organizational
situation that Denis et al. (2012) suggest is likely to prove illuminating in considering leadership influence and informal relations
within pluralized leadership. Our data is drawn from a complex organization of an inter-professional, inter-organizational network
delivering health and social care, specifically the safeguarding of children

an organizational context prone to publicly visible
failures of leadership, which may result in child deaths (Laming, 2009).
Pluralized leadership
Beyond Denis et al. (2012), various commentators have examined pluralized leadership within complex settings (Currie &
Lockett, 2011; Gronn, 2002; Huxham & Vangen, 2000; Spillane, Halverson, & Diamond, 2004; White et al., 2014; Yammarino
et al., 2012). They agree pluralized leadership is always present in professionalized, complex organizations, but do not provide
an adequate theorization of the spread of leadership, which takes account of the interaction of informal relations and leadership
influence that underpin pluralized leadership to explain the extent to which it is more or less widespread. Many studies seeking
to explain the spread of pluralized leadership tend to highlight the effect of external context (cf. Currie & Lockett, 2011; Currie,
Lockett, & Suhomlinova, 2009; White et al., 2014) and argue that some sources of influence carry more weight than others,
and are anchored in different sets of resources; i.e. leadership influence derived from managerial accountability or professional
status. These studies tend to emphasize that pluralization of leadership is likely to be concentrated in an elite group of actors,
rather than widespread. Yet other studies empirically report that pluralized leadership is widespread in a way that cannot be
explained by exogenous factors, such as managerial accountability or professional status (Buchanan, Addicott, Fitzgerald,
Ferlie, & Baeza, 2007; Huxham & Vangen, 2000). Buchanan et al. (2007) focus upon the fluid, ambiguous, migratory dynamics
around social relations in making their claim that ‘nobody’s in charge’, but tend to ignore, or even eschew perceived leadership
influence. Meanwhile Huxham and Vangen (2000) take a ‘holistic’ view of leadership through which they consider how collaboration
is shaped and enacted. They take a more balanced view of the interaction of perceived leadership influence and social
relations, in considering the behavior of participants identified as leaders, but also what happens on the ground because of
structures and processes of collaboration. However, Huxham and Vangen (2000) focus upon a wide range of issues, as well as
the interaction of perceived leadership influence and social relations within their empirical study, as a consequence of which
we still lack sufficient in-depth understanding of the spread of leadership influence at a more micro-level of analysis, involving
local level interactions derived from social relations (Denis et al., 2012; Yammarino et al., 2012).

A conceptual perspective within the pluralized leadership literature, synthesized by Denis et al.’s (2012), of particular
relevance in addressing the research gap about the interaction of leadership influence with social relations at the micro-level is
collective leadership (e.g. Carter & DeChurch, 2012; Contractor et al., 2012; Cullen et al., 2012; Friedrich et al., 2009; Mumford
et al., 2012; Yammarino et al., 2012). This most offers a springboard for our theoretical concern in determining how leadership
might be more widespread. Friedrich et al. (2009) develop a framework for understanding “collective” leadership, which highlights
the utilization of leader and team expertise within networks that aligns with our concern for interaction of multiple
relations and leadership influence. Others have followed the direction set out by Friedrich et al. (2009) in researching collective
leadership (Kramer & Crespy, 2011). All argue that the sharing of leadership is much less a static condition in which role behaviors
are structured, and much more dynamic, engendering collective leadership through multiple network channels. Collective
leadership is not isolated to defined leaders, but leaders are embedded within wider team and network structures, with communication
central to the collective leadership phenomenon.
Friedrich et al. (2009) set out three core constructs that constitute collective leadership. First, networks are the channels
through which communication is enacted. Second, in addition to the leaders’ (plural) personal networks, the network among
team members is critical to collective leadership. Consequently, third, collective leadership is characterized by exchange behaviors
across formal and informal networks. Friedrich et al. (2009: 955) highlight that communication exchanges and relationships
across multiple social networks are lacking empirical analysis arguing: “(W)e must … evaluate the bases of social network
connections, how information flows through the social network, and how understanding your social network and the networks
of those around you can facilitate collective leadership efforts.” This last statement links to the call for research by Denis et al.
(2012), which represents the springboard for our own research concerns regarding our empirical and conceptual focus: How
do multiple relations interact in leadership networks and with what effect?
While we draw considerable insight from existing studies of pluralized leadership, it is important to note that a number of
these studies are conceptual (e.g. Friedrich et al., 2009), and of those that include empirics, some are more normative than
critically analytical (e.g. Carter & DeChurch, 2012; Cullen et al., 2012). Hence, we are left with an inadequate understanding of
the empirical basis of pluralized leadership.
Denis et al. (2012) highlight how pluralized leadership research is enacted in daily, often mundane activities inside organizations,
and that the direction of leadership is shaped by often subtle and complex dynamics of informal, as well as formal, interactions
between organizational members. Although enhancing our understanding of pluralized leadership, Denis et al. (2012)
critique the extant literature, which takes a more relational approach, on two main grounds. First, the relational stream does
not provide sufficient consideration of power (see Currie & Lockett, 2011, for how this might impact the pluralization of leadership
in complex organizations, which are significantly professionalized). Second, and more pertinent to our research concerns,
they question how the “mundane activity” that constitutes leadership influence might be distinguished from non-leadership
activity, such as decision-making, problem-solving or simply team working: “How can leadership be studied and what counts
as leadership in this case?” (Denis et al., 2012, p. 267). Denis et al.’s (2012) critique, therefore, highlights our need to understand
the relational mechanisms through which pluralized leadership is enacted and direction of leadership influence derived from
these. In order to do so, we argue that it is necessary to complement the existing qualitative approaches, which characterize
the relational stream of pluralized leadership research to date, with social network theory and associated quantitative methods.
Social network theory and pluralized leadership
While there is increasing scholarly attention on the application of network theory and methods to the study of leadership, few
explored how this may advance leadership research (Balkundi & Kilduff, 2006; Contractor et al., 2012). A number of scholars have
argued that the enactment of pluralized leadership should be viewed as a specific type of social network, comprising multiple
forms of relationships, and therefore amenable to investigation using social network theory and methods (see Carson et al.,
2007; Dansereau, 1995; Graen & Uhl-Bien, 1995; Mayo, Meindl, & Pastor, 2003). To date, however, scholarship applying social
network theory and method to the examination of the enactment of pluralized leadership has largely focused on the networks
surrounding individual leaders. For example, scholars have examined network properties and individual influence (Brass, 1984:
Brass, Galaskiewicz, Greve, & Tsai, 2004), and dyadic relations, particularly between a formally designated leader and a subordinate
(Graen & Scandura, 1987). Consequently, scant attention has been paid to the importance of informal network relationships
connecting individual actors, on dimensions such as support and advice (Carson et al., 2007). Indeed, few if any, have examined
the potential of (multi) network-level analysis for studying leadership (Yammarino et al., 2012), and even less so from a multiple
network view (Contractor et al., 2012). We contend that any examination of the enactment of pluralized leadership needs to embrace
how, and under conditions of uncertainty, multiple individuals interact through a variety of different forms of social relationship
(Contractor et al., 2012; Yammarino et al., 2012).
Leadership is enacted through different forms of formal and informal interactions and exchanges between individuals (Uhl-Bien,
Marion, & McKelvey, 2007). Informal networks can serve to support an organization and provide additional backstage support to formal
leadership relations; however, they can also undermine the authority of formal leaders if the two are disconnected. The propensity for
relations to co-occur in networks referred to here asmultiplexity (Lazega & Pattison, 1999; Shipilov et al., 2014), is a concept that allows
us to explore howthe structure of relations in one network influences the structure of relations in other networks (Lee & Monge, 2011).
Social network researchers viewmultiplexity as a coincidence of different types of relationships that havemultiple contents (Contractor
et al., 2012). Such multiplex relationships are expected to be stronger than uniplex relationships because they contain more than one
basis for interaction (Skvoretz & Agneessens, 2007). Multiplex relationships reflect not only the simultaneous presence of multiplex

ties, they also contribute to the development of a local network structure that involves multiple types of ties, with interdependence
among ties within dyadic and triadic network structures (Koehly & Pattison, 2005; Lazega & Pattison, 1999). Thus, a central suggestion
for this study is that multiplex networks exhibit regularities that are underpinned by specific forms of interdependence among relational
ties. These types of interdependencies might be expected to reflect underlying social processes that guide the emergence of pluralized
leadership (Contractor et al., 2012). In the next section we examine the multiplexity of perceived leadership influence with informal
relationships to facilitate a better understanding of the micro-processes of the pluralized leadership.
Leadership influence and informal social networks
We focus on leadership influence, rather than formal leadership, because it represents a holistic view of leadership that is not
simply based on a formal organizational chart, allowing for more fluid encompassing ties created by “quasi-structures” including
committees, task forces, teams etc. (Ibarra, 1993; Schoonhoven & Jelinek, 1990). To understand the enactment of leadership
influence, however, it is important to recognize that leadership influence and informal ties are unlikely to occur independently
of each other (Monge & Contractor, 2003). What is unclear, however, is what type of interdependencies or cross network effect,
where leadership relationships are aligned with informal relations (Krackhardt & Kilduff, 1990; Riordan & Griffeth, 1995), will
occur between two (or more) actors. Examining interdependencies or cross network effects is important, in that exploring
multiple networks without these effects is equivalent to studying a number of independent single networks. Consequently, the
different relational ties in which an actor is embedded should be characterized as cross-network effects (Lazega & Pattison,
1999), as networks cannot be properly understood if such interdependencies are ignored (Rank, Robins, & Pattison, 2010).
These effects can be then thought of as the building blocks of the leadership network structure.
Our interest lies in exploring the co-structuration of informal interaction between members of a network, and their orientation
toward leadership influence, to provide us with clues as to how pluralized leadership is enacted. In addition, leadership influence
itself represents a focused activity that increases mutual awareness, and facilitates the development of informal relationships
(Feld, 1981, 1982). According to this view, the presence of informal network ties is not only an antecedent, but also in part an
outcome of leadership influence.
In conceptualizing the nature of the leadership influence and informal ties we distinguish between instrumental ties (i.e. goal
oriented) and expressive ties that primarily provide friendship and social support (i.e. the tie is considered to be an end in itself)
(Fombrun, 1982; Ibarra, 1993). The delineation between instrumental and expressive informal ties is one that has been employed
extensively in organizational research (e.g. Fombrun, 1982; Ibarra, 1993; Krackhardt & Hanson, 1993; Torenvlied & Velner, 1998;
Umphress, Labianca, Brass, Kass, & Scholten, 2003). We classify leadership influence ties as being instrumental in nature as they
are goal oriented, however, informal relations may be instrumental or expressive in nature.
Informal instrumental ties arise in the course of work role performance and involve the exchange of job-related resources
including expertise and advice (Fombrun, 1982; Ibarra, 1993; Krackhardt & Hanson, 1993; Lincoln & Miller, 1979). Actors seek
out others for advice whom they view as being high status in nature (Blau, 1964; Cook & Whitmeyer, 1992; Thye, 2000) and
as having good connections to other parts of the organization (Thye, 2000). For example, Sorrentino and Field (1986) found a
positive relation between advice giving and leadership emergence; and Carson et al. (2007) found that advice giving relates to
patterns of leadership influence. Informal instrumental networks based on advice enable individuals to identify others with
potential resources, and to be able to reach out to these others when seeking such resources (Ibarra, 1992).
Informal expressive ties involve the exchange of friendship and social support, and tend to be less bound to formal structure
and work role (Ibarra, 1993). In a complex public services organizational context, however, friendship will be difficult to perceive
as ties will not be exclusively personal in nature since they are work-based and span organizational boundaries (Umphress et al.,
2003). Consequently, we follow the lead of others to focus on informal expressive relationship ties as represented by those to
whom one goes for social support (e.g. Krackhardt & Hanson, 1993; Torenvlied & Velner, 1998). We suggest that the person
one might go to for social support may help in shaping one’s status in a group in relation to leadership. Here, individuals support
one another, and help to create an environment where other members are valued and appreciated. By actively providing support,
individuals are more likely to be recognized in relation to leadership status (Seers, Keller, & Wilkerson, 2003).
In the next section, we describe the types of interdependencies or cross network effects that are potentially relevant for the
underlying structural regularities of pluralized leadership networks. Drawing on social exchange theory, and the emerging
literature on entrainment (e.g. Kelley, Futoran, & McGrath, 1990; McGrath, 1991; Pérez-Nordtvedt, Payne, Short, & Kedia, 2008;
Shi & Prescott, 2012), we explain how the similarities and differences in the forms of leadership influence and informal network
ties between two actors may lead to different cross network effects in terms of mutual exchange and entrainment. Below, we
focus our arguments at the level of the dyad on the basis of parsimony; however, we note that our arguments do not preclude
the existence of cross network effects occurring beyond the dyadic level (Balkundi & Kilduff, 2006; Ekeh, 1974; Emerson, 1976;
Gillmore, 1987; Jones, Hesterly, & Borgatti, 1997; Sparrowe & Liden, 1997; Takahashi, 2000; Uzzi, 1996).
Cross network effects: mutual exchange and mutual entrainment
Social exchange theory focuses on the quality of social interactions of actors within their networks, with a focus on the
exchange of resources (Molm., 1994, 2001; Settoon et al., 1996), and is based on the calculus of mutual exchange when building
and maintaining ties with each other. Applying the principle of mutual exchange to the interrelationships between leadership influence
and informal ties, we suggest that mutual exchange occurs where there is a difference in the content of the leadership

influence tie and the informal tie involved; i.e. when one tie is instrumental and the other tie is expressive. For example, where
person i perceives person j to have leadership influence (instrumental), they are more likely to exchange their willingness to be
formally led for informal social support (expressive) from j. Where mutual exchange occurs there is neither complete overlap, nor
complete diversion between networks (Krackhardt, 1987). Rather, there exists a complex pattern of interdependences among
relational ties, where exchange occurs in the context of other exchanges (Lazega & Pattison, 1999), and different networks
influence and reshape each other. Hence, there is an interlocking of exchanges that go beyond any transfer of a single resource.
Based on the above we hypothesize that:
H1. Under conditions of uncertainty, actors will seek to balance their relationship with others by entertaining cross network
effects characterized by mutual exchange between leadership influence (instrumental) ties and informal social support (expressive).
The second type of structural regularity is the entrainment of leadership influence and informal ties. McGrath et al. (1984)
defined social entrainment as the process through which behavioral cycles become co-occurring with one another. The
concept of social entrainment has received increased attention in management research (e.g. Kelley et al., 1990; McGrath,
1991; Pérez-Nordtvedt et al., 2008; Shi & Prescott, 2012) because it enables researchers to examine the complex interdependencies
between different forms of behavior (Shi & Prescott, 2012). We suggest that a SNA application of social entrainment is
particularly relevant to the study of pluralized leadership, and the interactions between different forms of relational ties, for
three main reasons. First, actors’ activities need to be endogenous for social entrainment to occur, which is consistent with the
assumption of structural logic, in that social entrainment is one class of structural regularity that can explain the formation of
networks (Lazega & Pattison, 1999). Second, entrainment can facilitate complex and interdependent coordination across a
range of human activities (Ancona, Goodman, Lawrence, & Tushman, 2001). Third, scholars have theorized thatmutual entrainment
may lead to a positive effect on performance through coordination (Lazega & Pattison, 1999).
Entrainment, from a SNA perspective, focuses on the extent to which there is a shared cadence of different forms of ties
between actors (Rank et al., 2010). At the dyadic level, mutual entrainment is said to be present between two actors (i and j),
where the presence of one type of tie form(e.g. i to j) is interdependentwith the presence of another type of tie (e.g. for i to j); i.e. there
is the co-occurrence of two different forms of tie between two actors (i.e. both ties are directed from i to j). To date, however, SNA
scholars have treated entrainment as an empirical property of two networks (see Lazega & Pattison, 1999; Lomi & Pattison, 2006;
Rank et al., 2010; Robins et al., 2007), but have not explained the conditions under which entrainment is likely to occur. Drawing
on the logic of social entrainment (McGrath et al., 1984), we suggest that leadership influence and informal ties are more likely to
be aligned, and hence mutually entrained, when they are similar in terms of their function. For example, when i has (instrumental)
leadership influence over j, i will be more likely to also provide informal advice (instrumental). Simply stated, advice is likely to be
sought out in relation to the enactment of leadership influence (see Lazega & Pattison, 1999). Hence, in contrast to mutual exchange,
which is promoted by difference between formal and informal ties (as outlined above),mutual entrainment is promoted by similarity
between the function of leadership influence and informal advice network ties. Hence:
H2. Under conditions of uncertainty, actors adopt a more social orientation and are more likely to enter cross network effects
characterized by mutual entrainment between leadership influence ties (instrumental) and informal advice ties (instrumental).
Method and data
Our study focuses on a City Local Safeguarding Public Service Network (CLSPSN). CLSPSN, as an organizational entity, represents
a mandated public services network, comprised of several legally autonomous organizations that work together to achieve
not only their own goals, but also a collective goal; safeguarding children, for example, from domestic or sexual abuse. The
CLSPSN is situated within the children’s services department of the local level of government (in England, a host local authority),
which is ultimately accountable for safeguarding failures. Around half of safeguarding networks are formally led by an independent
chair, with the other half led by a senior manager from the host local authority, commonly the Director of Children’s Services
(France, Munro, Meredith, Manful, & Beckhelling, 2009). At the same time, the children’s services department alone does not hold
all the resources for service delivery or control, nor do they manage key staff delivering services, so they cannot, alone, ensure
high quality delivery of services. Hence, the CLSPSN brings together a multitude of different professionals and organizations
(i.e. health, social care, education, careers and youth work, police and voluntary organizations, as well other local level agencies)
deemed responsible for strategically overseeing the front-line handling of child abuse and related deaths (DES, Department for
Education and Skills, 2007). Unlike non-mandated networks, which develop organically, goal-directed public service networks
are established with a specific purpose, either by those who participate in the network or through mandate, and evolve largely
through conscious efforts to build co-ordination and encourage informal interaction (Agranoff & McGuire, 2001; Kilduff & Tsai,
2003; Provan & Kenis, 2007). The participants meet regularly at overview meetings, but also work together and interact outside
the formal network meetings.
Data
Similar tomany network studieswe focus on a whole network, the boundary ofwhich is recognized and defined by the members
themselves, and employ a model-based design because we were able to observe all members of the network (Sterba, 2009). Our

approach enables us to explore the social processes andmechanisms in the network more generally (see: Frank, 2009), as the corresponding
standard errors provide an indication of how different these estimates might be if the study was repeated. Model-based
inference acknowledges that empirical random sampling would not always be feasible, particularly for observational studies in the
social sciences, andwhere it is plausible to consider that the observed network data could also have been different (i.e., the individuals
could have been different while the social and institutional context remained the same, or external influences could have been
different). The idea is that in such a population of different networks, the systematic patterns as expressed in the parameters of a
statistical model would be the same, while the particular outcome observed could be different (Snijders, 2011).
Informal social network data were collected via a questionnaire that was personally administered to all (23) members of the
CLSPSN (response rate 100%), which covered participants’ perceptions of leadership influence, advice seeking and support, and
personal attributes. In parallel to the SNA data collected by the sociometric survey, the program of research at the CLSPSN also
involved in-depth qualitative interviews with all members of the board. The qualitative interviews, although not formally utilized
in this article, enabled us ensure that we were confident that our SNA data responses were representative of the patterns of advice
and support among the individuals concerned. Based on field experience the research team also explored a list of concrete
circumstances that would help in rooting leadership influence relations more firmly in the specific safeguarding context and in
the understanding that participants have. We avoided selection bias in that we collected network data from all respondents to
construct the network measures. Podsakoff, MacKenzie, Lee, and Podsakoff (2003) noted that this is a powerful procedure such
that additional statistical remedies are unnecessary. Details of the population of respondents are presented in Table 1. The size
of the network is consistent with other studies, as evidenced by Provan, Fish, and Sydow’s (2007) review of the literature on
whole networks. In addition, as ERGMs are based on dyadic observations, the effective number of (non-independent) observations
for each network is N × (N − 1), where N is the number of nodes in the network. Thus, our total number of observations for all
three networks is 506.
Perceived leadership influence and informal relations measures
Perceived leadership influence
Our definition of a leader is someone who is perceived as such by others, which is reflected through a set of formal and
informal ties (Balkundi & Kilduff, 2006; Mehra et al., 2006; Zohar & Tenne-Gazit, 2008). The measure is similar to that used in
other studies to capture respondents’ personal and explicit theories of leadership (Mehra et al., 2006), and is consistent with
classic socio-metrical work on leadership (Calder, 1977). A leadership relationship is said to exist when one member perceives
another as exerting leadership influence across multiple individuals. Consistent with the work of Brass and Burkhardt (Brass,
1984; Brass and Burkhardt, 1993; Burkhardt & Brass, 1990), we focus on influence rather than power because of negative connotations
associated with the concept of power (Pfeffer, 1981). Furthermore, Brass and Burkhardt (1993) argue that although
scholars have made definitional distinctions between influence and power, such distinctions are not common in everyday
usage of the words. In addition, the research on leadership perceptions suggests that perception is a good way to assess leadership
influence, and is consistent with the attributional nature of power (Dahl, 1957; Wrong, 1968). That is, if actor i believes that actor
j has leadership influence, actor i then behaves toward actor j as if the latter had leadership influence. Consistent with Brass and
Burkhardt (1993) and Salk and Brannen (2000), who collected data in this way to assess perceived leadership influence, we
employed the roster method (as outlined above) to collect information from each dyad. Specifically, we provided a full list of
names of all members of the CLSPN and asked each individual to identify people on the list whom they perceived to have
leadership influence in the CLSPSN, and then rate the degree of leadership influence from 1 being low to 4 being high.
Informal instrumental relations
Employing Ibarra’s (1993) classification, and consistent with Balkundi and Kilduff’s (2006) theorization of a social network
approach to leadership, we focus on “advice seeking” as our example of informal instrumental ties. Consistent with our approach
to measuring leadership influence we employed the roster method and collected information for each dyad, drawing on the measure
employed by Lazega, Lemercier, and Mounier (2006). Specifically, each person was asked to identify members of the CLSPN
whom they had regularly asked advice from, or who they have had discussions with, outside of formal deliberations on CLSPN
matters, and then rate the degree of advice from 1 being low to 4 being high. In doing so we were careful to ensure that respondents
did not confuse informal advice seeking with formal advice seeking that may constitute a part of a person’s formal duties.

Informal expressive relations
Following Ibarra (1992), we focus on “social support” as our example of informal expressive ties because it has an important
influence on communication processes and the exchange of information (e.g.Brass, 1984, Ibarra, 1993, Ingram & Roberts, 2000).
Social support ties are often seen as more readily available than advice, as advice is exchanged at arm’s length (Uzzi, 1996).
Furthermore, social support entails a willingness to help in difficult situations by providing different types of resources like
emotional support and socialization (Lazega & Pattison, 1999). Consistent with our other measures of leadership influence, we
employ the roster method here and collected information for each dyad. Specifically, each person was asked to identify members
of the CLSPN whom you go to for support or to be able to talk freely about any personal matters outside of formal deliberations
on CLSPN matters, and then rate the degree of advice from 1 being low to 4 being high. Consistent with our measure for informal
instrumental network in terms of advice seeking, we ensured that respondents did not confuse informal support seeking with
formal duties.
As ERGMs require matrices to be binary, following Krackhardt (1987) we use transformed our three network measures into
binary variables. A value of one was given to a person that was nominated as having leadership influence with a rating of greater
or equal to three; those who were not nominated, and those nominated but who had low leadership influence ratings of two or
less, were given the value 0. In doing so, our aim was to focus on the more significant leadership relations, which is an approach
that is consistent with SNA method (see: Wasserman & Faust, 1994). While different cut-off thresholds could have been chosen,
our selection of the dichotomization criterion that we adopt follows other studies (e.g. Conaldi & Lomi, 2013). We tested alternative
dichotomization rules, as suggested by Wasserman and Faust (1994), and we examined the network statistics including the
degree distribution. We found that the dichotomized networks do not appear to be qualitatively different from the corresponding
degree distributions of the original valued network, and the results of the analysis that we report were robust to alternative
choice of dichotomization rule. It was also a theoretical decision to choose a higher threshold as our interest lies in understanding
the process of influence (or the existence of ties) between participants, not the intensity of influence in leadership activities.
Finally, we believe that the dichotomization criterion chosen allows us to represent the leadership process with a limited loss
of information.
Controls for actor attributes
In consideringmulti-relational ties, it is important that we control for actor attributes, which may shape participants’ inclinations
to give advice to each other because of their status or experience. Alternatively, two participantswho provide each otherwith support
may bemore likely to collaborate on an informal basis. Data relating to actor attributes enabled us to explore the influence of context
on the formation of network ties (Robins, Pattison, & Elliott, 2001). Actor attributes are individual-level measures on the nodes of the
network, andweremeasured in terms of a participant’s professional status, seniority andwhether or not they had a coordination role
in the network.
Professional status is measured employing a binary variable, distinguishing between high status professions (one if a doctor or
social worker) and low status professions (zero if other profession). We classified the status of actors, drawing on the sociology of
professions literature that indicates that doctors and social workers enjoy high status relative to nurses and others, within health
and social care networks (Nancarrow & Borthwick, 2005). Professional status matters in safeguarding networks as it may
influence whether a participant is perceived as exerting leadership influence (Currie, Grubnic, & Hodges, 2011).
Seniority (formal leadership) is measured employing a binary variable indicating whether the participants have a formal
managerial role on the safeguarding board (yes=1) or not (no=0). Researchers have highlighted the importance of managerially
accountable roles in public service networks (Denis et al., 2001; Ferlie, Ashburner, Fitzgerald, & Pettigrew, 1996; Harrison, Hunter,
Marnoch, & Pollitt, 1992). This is consistent with Morgeson, DeRue, and Karam (2009) who used the term “formal leaders” to refer
to individuals, such as managers or formal team leaders with legitimate authority over other organizational members.We take the
view that if an participant has a managerial role, then more participants will perceive them as exerting leadership influence.
Coordination role is measured employing a binary variable, which takes the value of one when a participant has a formal
coordination role and zero otherwise. We include coordination role as an actor attribute because we expect this role to influence
an individual’s embeddedness in the leadership and advice networks (Lazega & Pattison, 1999). Within CLSPSN, there is dedicated
administrative cadre of staff to coordinate the work of professionals.
Exponential Random Graph Models
Scholars are increasingly recognizing the importance of the statistical modeling of social networks in organizations, including
endogenous network parameters (Robins et al., 2007). Endogenous network parameters represent specific network dependencies
and important social processes through which network structures are built (Monge & Contractor, 2003). Accounting for the
endogenous tendencies of social networks to self-organize into a variety of local configurations can allow us to account for specific
ways in which network ties may generate other network ties. Accordingly, we can employ such models to explicitly state that
leadership relations can be characterized by endogenous components, whose effects on tie formation co-exist with the effects
of other networks and individual attributes.
To model howleadership networks have their own internal organizing principles, and also the extent towhich formal and informal
networks are exchanged and/or entrained, we employ ERGMs. ERGMs began with Frank and Strauss (1986), who advanced research
into stochastic social networks by proposing the notion of dependency between network ties, and has been extended in a series of

articles exploring how best to identify specific models for certain forms of network data (e.g. Handcock, Hunter, Butts, Goodreau, &
Morris, 2004; Pattison & Robins, 2002; Pattison & Wasserman, 1999; Robins, Pattison, & Wasserman, 1999; Snijders, Pattison,
Robins, & Handock, 2006; Wasserman & Pattison, 1996). To model the dependence between network ties, Frank and Strauss (1986)
proposed using the Markov dependence assumption and the adoption of Markov Random Graph Models (MRGMs) (Pattison &
Wasserman, 1999; Robins et al., 1999; Snijders et al., 2006;Wasserman & Pattison, 1996).
The Markov dependence assumption infers that two tie-variables are dependent if they share a node (i.e. any relational ties
involving the same actors (say i and j) can be defined in which a possible tie from i to j is assumed conditionally dependent
only on other possible ties involving i and/or j). Any proposed assumptions about potential conditional dependencies among
network tie variables can be inferred from the Hammersley–Clifford theorem (Besag, 1974), which informs us that MRGMs can
be completely characterized by the numbers of edges, stars and triangles (Pattison & Wasserman, 1999; Robins et al., 1999;
Snijders et al., 2006; Wasserman & Pattison, 1996). It is important to note that MRGM allows us to understand the intricate
arrangement of social processes by which network ties are formed. Network ties can organize themselves into patterns because
the presence of some ties encourage others to come into existence, which is an endogenous process in that the network patterns
arise solely from the internal processes of the system of network ties. The system of ties can be represented by a number of
different configurations. By incorporating a number of configurations (e.g., edges, stars and triangles) simultaneously, MRGMs
can test which processes contribute to the formation of a network structure (Monge & Contractor, 2003).
The conditions outlined above are endogenous or structural effects. MRGMs or the more general ERGMs enable us to include a
series of different types of network parameters, which provide important insights into the enactment of pluralized leadership in
the network (see Fig. 1). The general pluralized leadership properties of a network are represented by the density (edge) and
centralization (k-in-star) parameters, which are degree-based effects that can give rise to self-organization. Density (edge) configuration
is a baseline propensity for tie formation and corresponds to the amount of leadership interaction in a network, in terms
of the proportion of direct ties in a network relative to the total number possible. Centralization (K-in-star) network configurations
are equivalent to modeling the in-degree distribution (Snijders et al., 2006); and we found that modeling with a Markov
2-in-star (simple activity) was sufficient for convergence. High positive values of these parameters indicate network centralization.
For instance, a significant large positive parameter indicates that in-degrees are centralized on a few key actors; whereas
a small or even negative parameter indicates a relatively equal spread (de-centralization) of influence across actors (Robins,
Pattison, & Wang, 2009).
Dyadic effects through the reciprocity parameter represent restricted exchange. Reciprocity is defined at the level of the dyad, and
refers to the overall tendency of actors to reciprocate leadership influencewith similar others. Typically ERGMs provide an estimate of
reciprocity for all pairs of actors across the network and we consider the estimate as an average effect across the network.
Generalized forms of exchange are represented by the transitivity and cycle parameters, which we operationalize at the level
of the triad. Triadic effects reflect the human propensity to operate in a group structure. We adopt two main triadic configurations.
Cycle denotes the tendency for a relationship to be in the form of generalized reciprocity (i.e. if there is a tie from i to j,
and also from j to h, there is also a tie from h to i). Transitivity denotes that if actor i perceives actor j as a leader, and actor j
perceives actor h as a leader, then actor i will also perceive actor h as a leader. The inclusion of transitivity and cycle parameters
are strengths of ERGMs, which address the paucity of network models that incorporate these effects (Newman, 2003).
In modeling the three networks we ask questions about how different types of networks interact with each other, and how
these interactions affect the structure of each network. The model specification can be divided into two parts: within network
effects (as described above), and cross-network effects, which involve ties from all networks. For cross-network effects, we
focus on mutual exchange and mutual entrainment. Significant cross-network effects imply that there is an association between
the different networks. For directed networks there are two dyadic configurations. The mutual exchange parameter represents the
extent to which the dyad exchanges ties of different types, whereas the mutual entrainment parameter represents the extent to
which the different network ties align within the dyad (i.e. both ties are directed from i to j). This specific configuration is perhaps
the most commonly encountered mechanism in studies of inter-organizational networks (Uzzi, 1996). These are basic configurations
for network association and should always be considered in a multivariate model. The advantage of the ERGM approach is
that it allows us to investigate these cross network effects for multiple networks at the same time as taking into account any
relevant dependencies of each network that are present.
Finally, we include parameters to control for the influence of exogenous context using actor attributes. These are important in
that individual participants bring their own qualities, capabilities and predispositions to a network. These can be very important
to the formation of ties (Kilduff & Tsai, 2003). Actor covariate effects can be entered into an ERGM taking several forms including
sender effects, receiver effects, and homophily effects (Lusher et al., 2012). Homophily effects capture the increase in the
likelihood of a tie forming between two participants given that both participants share or are similar on a given attribute. Sender
effects indicate whether an individual with a particular trait is more or less likely to seek out a tie. Receiver effects capture
whether individualswith certain attributes are more or less likely to be recipients of ties.Wemodel our actor attributes as influencing
the presence of ties through homophily effects andwe also added a receiver effect for our seniority measure (i.e. formal leadership), in
that we expect that members from higher levels of organizational hierarchy to be relied on for leadership.
Model specification
Following Wasserman and Pattison (1996), ERGMs can be viewed in a standard form in which the response variable is the
log-odds of the probability that a relational tie is present. In modeling the network we consider each potential network tie

between the participants as a random variable. For each pair of individuals i and j, we define a random variable Yij so that Yij = 1
if a given relation exists between i and j, and Yij = 0 otherwise. As relations of leadership influence give rise to directed ties, Yij
may be different (in general) from Yji. The observed value is specified as yij and Y is the matrix of all such variables, with y the
matrix of observed ties. In addition, we employ the assumption of homogeneity; i.e. parameters do not depend on the identities
of the nodes in the configurations to which they correspond. Following Pattison and Wasserman (1999) for ERGM the basic model
has the following form:

where: (i) Y is the n × n array of network tie variables, with realizations y; (ii) ΖA(y) is the network statistic of for all
configurations A (hypothesized to affect the probability of this network forming) in the model (configurations might
include edges, stars, transitive triads and so on); (iii) λA is the corresponding parameter estimate (equal to one if a particular
configuration is observed or zero otherwise); and (iv) the value κ is the normalizing constant, included to ensure that Eq. (1)
is a proper probability distribution.
The summation in the model includes all network effects within the given model. Eq. (1) describes a probability distribution of
graphs on n nodes. The probability of observing any particular graph y is dependent both, on the statistics ΖA(y), and on the
corresponding parameter λA, for all effects in the model.
We perform a multiple network ERGM including cross-network effects. In order to extend the univariate ERGM to multivariate
relations, we adopt an assumption that simply allows us to state that the status of, say, a dyadic network tie in one network
(e.g. leadership influence) may be conditionally dependent on the status of ties in another type of network (e.g. informal advice
seeking). Invoking the Hammersley–Clifford theorem (Besag, 1974; Pattison & Wasserman, 1999), we constructed a probability
model for a multivariate random network. Eq. (1) describes a general probability distribution of graphs and is used to determine
the particular probability of observing a graph (or network). The specific probability of observing any graph P(Y=y) depends on
both networks statistics ΖA(y) and the non-zero parameters λA for all configurations A in the model.
For the multivariate case, ΖA(y) is a multigraph as presented in Eq. (2); where Ak is a collection of configurations A of
tie-variables.

For the social relation s we define a binary variable Yijs, which equals 1 if there is a relational tie of type s between actor i and
actor j, and is 0 if no such tie is present. Each of the s social relationships is intended to express a distinctive relational content. As
we are exploring cross-network effects, we are particularly interested in whether or not there may be a tendency for leadership
ties to be entrained and/or exchanged with our informal networks. For example, and taking the simple case of looking at our
leadership influence network and the advice network, on the basis that those perceived as influential are more likely to be sought
out as sources of advice than non-influential participants, we would expect to observe Yija = YijI = 1 more frequently than
expected from the baseline frequencies of YijA = 1 and YijI = 1. There may also be mutual exchange or reciprocation effects in
directed networks, such that we may observe more frequently than expected situations where YijA = YjiI = 1 (a person j chosen
by i for advice tends to perceive i as influential — advice is “exchanged” for influence) and where YijI = YjiI = 1 (perceptions of
influence tend to be reciprocated).
Model estimation
Estimation parameters for ERGM are complex, and only recently have statistical methods become available (see: Robins et al.,
2007). More recently, Monte Carlo Markov Chain Maximum Likelihood Estimation (MCMCMLE) methods have been developed to
obtain estimates of parameters and standard errors for Exponential Random Graph Models (see: Hunter & Handcock, 2006).
Software for the modeling and estimation of networks using ERGMs is widely available for a single network, and two networks
(see: Robins et al., 2007), but none exists presently for three or more networks.
Without the use of MCMCMLE, the estimation of the parameters in Eq. (2) (the multivariate case) is more complex. To estimate
such a model with complex dependency assumptions using maximum likelihood methods may not be viable, and therefore,
indirect methods need to be used to estimate model parameters. Pseudolikelihood techniques are an indirect method for estimating
the ERGM parameters which are good for estimating univariate models since the estimation of κ, the normalizing constant,
can be done directly for simple models. For our univariate case we used pseudolikelihood techniques for parameter estimates
the results as a starting point for building a univariate model amenable to estimation using MCMCMLE as suggested by Robins
et al. (2007). Pseudolikelihood techniques are also easily implemented to the case of multiple networks, and they have proven
useful in estimating model parameters (Lazega & Pattison, 1999; Rank et al., 2010). Hence, we employed pseudolikelihood
estimation for our multivariate ERGM (Pattison & Wasserman, 1999; Strauss & Ikeda, 1990). In order to do this, following
Koehly and Pattison (2005), we use the fact that the random variables Yijs is dichotomous in nature to re-specify Eq. (2) into a
generalized autologistic model. Maximization of the pseudolikelihood function is achieved by fitting a logistic regression model,
which builds on the logit form of the Exponential Random Graph Model (Strauss & Ikeda, 1990). Based on the empirical network
data, a vector of measurements of the response variable and a matrix of measurements on the explanatory variables are created.

The statistical importance of a particular variable is assessed by fitting two models, one with the variable and the other without it,
with the difference in the pseudolikelihood ratio statistics the indicator of variable importance.
Pseudolikelihood estimation techniques, however, are only approximate, and so assessment of the model is based on heuristics
that compare the observed values with the fitted values. As such, the approximate standard errors that accompany the
pseudolikelihood estimates are given only for guidance as to likely order of magnitude. All comparisons among models are
based on two indices of model fit, namely −2 times the log of the maximized pseudolikelihood, and the mean absolute residual
(MAR) for each possible network (Pattison & Wasserman, 1999). The MAR is the mean of the absolute value of the difference
between the observed values yijs and the fitted values, yijs. It is an index of model fit (Koehly & Pattison, 2005).
Empirical findings
Network descriptive sstatistics are presented in Table 2, and indicate that the leadership influence and informal networks have
a similar density, with just over 16% of all ties realized in all the cases. The centralization statistics are scaled to percentages, with
0% indicating that no participant in the network plays a more central role than any other participant, and 100% indicating that all
ties are through only one star participant. In the leadership influence network, relatively few leaders are recipients of incoming
ties (at a little over 73%), which stands in sharp contrast with the informal networks, in which the interaction is much more
distributed (at a little over 36%). Further insights can be gained by examining the co-occurrence of the relational ties in the
networks, which we achieved by providing the associations between the networks using QAP correlation (Krackhardt, 1988).
The results suggest that the informal (instrumental) leadership and informal (instrumental) advice networks are significantly
correlated, whereas, the informal (instrumental) leadership and informal (expressive) support networks are not correlated. The
QAP results give some indication of entrainment but not exchange, and that QAP is insufficient to look at the effects simultaneously.
We can, however, explore several effects simultaneously with ERGM.
We now present more details of the leadership influence and informal (advice seeking and support) networks, before
progressing to the multivariate ERGM, encompassing informal networks and informal leadership influence network. In presenting
the results, some clearly non-significant effects were dropped from the model. Several non-significant effects were retained in the
results presented below because they were of primary interest (cross-network effects). Dropping further non-significant effects
did not lead to important changes in the remaining results.
The results for the univariate ERGMs are presented in Table 3. Evidence of the enactment of pluralized leadership is represented by
density and centralization parameters. For each of our univariate networkswe find strongly negative parameters for the density effect
(single directed ties) from some actor i to another actor j. The negative parameters indicate that ties occurring at random are rare,
suggesting that ties aremore likely to appear in regular combinationswith other ties.Wesuggest that because building and maintaining
relational ties are costly, exchange relationships in the form of one-sided alignment to arbitrary others are unlikely, unless there
are additional desirable properties to the ties. These properties constitute the basis for tie interdependence,whichwe explore below.
In terms of a centralization parameter (2 in star), the results (which were positive and significant) suggest a tendency for some level
of centralization of incoming ties for the all three networks.We also find significant and negative values for the 3-in-star parameters
for both informal networks (advice and support), which reveal that informal ties are exhibiting more de-centralization than the
leadership influence network. In practical terms, there are a number of participants sought for informal advice and support by
many of their colleagues, as compared to the informal leadership influence, which is more concentrated.
Our findings suggest that both leadership influence and informal advice relations are likely to be directly exchanged, as
reflected in positive values of the reciprocity parameters. However, we did not find a significant parameter estimate for informal
support. In terms of generalized exchange, we found no significant parameter estimates on cycle triads for our leadership
influence network.We did, however, find a significant parameter estimate on 3 cycle for our informal support network (an indication
that support tie formation responds to a logic of generalized exchange), and a significant parameter estimate on transitivity for our
informal advice network.
Our findings for the leadership influence network, however, suggest that the emergent structuring of ties beyond dyads occurs
throughmeans other than leadership influence. In contrast, the informal networks (advice and support) show structural regularities
that are of considerable importance in the context of pluralized leadership, both at the dyadic level aswell as at the triadic level. In our

support network, the ties may be typically based on shared interests among participants. If so, itmay be that support can contribute to
influence in the network rather than influence being only an outcome of ongoing formal relationships.
Finally, in terms of exogenous controls, we found significant actor attribute parameters (seniority, professional status and
being in a coordination role) for the leadership influence network, suggesting that they are driving the concentration of ties in
the network. Actor attributes were modeled as homophily effects, and we find that similarity in status in our leadership influence
network provides a strong signal of hierarchy and may be an important mechanism in reducing uncertainty in the selection of
leadership partners. The lack of structuring leadership influence of the formal organization is particularly interesting because
the pluralized leadership literature tends to promote the view that formal leadership structures matter for pluralization, requiring
intensive cooperation among the participants in terms of shared responsibilities and tasks (Gronn, 2002). In contrast, only one of
our actor attribute parameters was found to be significant for our advice network (professional status) and none for the support
network. The findings suggest that actor attributes in terms of similarity of status matter little for the formation of informal
network ties. In terms of the receiver effects for our actor covariates, there is no indication of a receiver effect in our univariate
models. This suggests that members with higher status are not any more likely to be sought out for influence or informal ties.
Multivariate ERGM
We now explore cross-network effects. We focus our discussion mainly on the multivariate model and refer to the univariate
models only to highlight interesting changes. The multivariate ERGM presented in Table 4 indicates that, in terms of structural
effects, the density influence, density advice and density support parameters are all negative and significant. The findings suggest
that there is significant interaction between the participants. Our 2 star measure was also significant, which suggests that our
multivariate network exhibits centralization. In terms of restricted exchange, the parameter estimates for reciprocity influence
and reciprocity advice are positive and significant, indicating that each network interaction is through direct reciprocity and rarely
occurs in isolation. In both cases this indicates that the participants tend to reciprocate leadership nominations as well as advice.
None of the parameters capturing generalized exchange (3-cycle and transitivity) were significant. Additionally, it should be noted
that actor attributes effects that are present in the univariate models drop out when considering the multivariate level, suggesting
that actor attributes do not matter in our multivariate model. The inclusion of these effects did not lead to an improvement of the
pseudolikelihood ratio statistics, and were thus eliminated from the model.
Exchange
In terms of the cross-network effects of mutual exchange, we argue that the exchange of ties is more likely to occur where
there is a difference in the content of the ties involved; i.e. when one tie is instrumental and the other tie is expressive.
From Table 4 it can be seen that the only significant parameter was that for informal instrumental leadership ties and informal
expressive (support) ties being characterized by mutual exchange, which supports hypothesis 1. The finding can be interpreted
as indicating the following: those that receive support tend to nominate the supporter as having leadership influence. Informal

(expressive) support, therefore, seems to be important in the context of reciprocity, as conferring informal support on another
participant may encourage reciprocation in the form of leadership influence.
Entrainment
Drawing on the logic of social entrainment McGrath et al. (1984), we suggest that two ties are more likely to be entrained
when they are similar in terms of their function. Regarding mutual (dyadic) entrainment, the parameter estimates were positive
and significant for informal instrumental leadership and informal instrumental (advice) networks, indicating that if an informal
instrumental leadership tie connects two participants, they are more likely to be also linked by an informal instrumental advice
tie. The finding is also re-enforced through the QAP correlation results in Table 2, which indicated a strong association between
informal leadership influence and informal instrumental (advice) networks. Hence we find support for hypothesis 2. In contrast
we found no evidence for the mutual entrainment between informal leadership influence and informal expressive (support)
networks, which were more likely to be mutually exchanged as outline above.
Discussion
Our study addresses calls for more scholarly attention to be paid to the micro-dynamics through which pluralized leadership is
enacted (Brass, 2001; Carson et al., 2007; Carter & DeChurch, 2012; Contractor et al., 2012; Mehra et al., 2006). In particular we
extend understanding of the spread of leadership influence at a more micro-level of analysis, involving local level interactions
derived from social relations (Denis et al., 2012; Friedrich et al., 2009; Yammarino et al., 2012). We built on the insights of
Sparrowe and Liden (1997) and Balkundi and Kilduff (2006), to examine how the enactment of pluralized leadership is
shaped by local interactions derived from social relations. Drawing on recent developments in the area of SNA, employing
ERGMs (see: Pattison & Wasserman, 1999; Robins et al., 2007), we have been able to show a complex pattern of interdependencies
between informal leadership influence and informal networks, where patterns of interaction occur in the context of
other interactions (Lazega & Pattison, 1999). We contribute to the existing literature in the following ways.

First, we contribute to the emerging stream of research that emphasizes the need for an understanding of how leadership
influence interacts with local interactions derived from social relations (Friedrich et al., 2009). The findings from our univariate
ERGMs reveal that extra-dyadic regularities tend to be found in informal leadership ties, whereby transitive relations tend to appear
with advice ties and cyclical relations with informal support, but appear absent in relation to formal leadership ties. Then,
building on the work of Lazega and Pattison (1999) and Rank et al. (2010), we examined the patterns of interactions across
multiple forms of leadership ties using multivariate ERGMs, focusing on the interactions between informal instrumental leadership
networks and informal instrumental (advice) and expressive (support) networks. Our findings suggest that pluralized
leadership is not limited to singular types of network tie, but as suggested by Sparrowe and Liden (1997), may encompass several
different forms of network ties simultaneously. The cross network effects (via exchange and entrainment) seen in these relationships,
however, depend on the types of ties involved as detailed below. Incorporating both leadership influence and informal
social relations enables us to explore a more widespread enactment of pluralized leadership than evident in some empirical
studies (Buchanan et al., 2007; Huxham & Vangen, 2000).
Second, with regard to the cross-network effects of exchange, we drew on social exchange theory to argue that participants
use the calculus of mutual exchange when building and maintaining ties with each other. Based on the principle of direct
reciprocity (see: Bearman, 1997; Yamagishi & Cook, 1993) we argue that mutual exchange will be more likely to occur when
there is a difference in the content of the ties involved; i.e. when one tie is informal instrumental and the other tie is informal
expressive. Our findings support this argument, and offers important insight into the conditions under which mutual exchange
is more likely to occur between different ties.
Third, extending the work of Lazega and Pattison (1999) and Rank et al. (2010), we modeled the cross network effects of
entrainment across informal networks and informal leadership networks. To develop our understanding of the conditions
under which informal networks and informal leadership ties will be entrained we drew on the logic of social entrainment
(McGrath et al., 1984). As expected, we found entrainment between instrumental leadership influence and informal instrumental
advice network ties. In addition, we also were able to model for, and find, the presence of collective entrainment, which was
enacted in a pattern consistent with mutual entrainment.
Fourth, our work attests to the complex patterns of interdependencies that exist between perceived leadership influence
and informal social relations, and how leadership resources can flow in the opposite or the same direction. We suggest, therefore,
that it is important to be able to examine the different ways in which informal networks may interact, as only focusing on
one or two types of relations will provide a limited and partial insight into the extent to which leadership is really pluralized.
Furthermore, we highlight that the significance of actor attributes in shaping networks, fall away when we move from our
univariate analysis of leadership influence to our multivariate analysis modeling the cross network effects of leadership influence
and informal networks. We suggest that such a finding may indicate the importance of modeling multiple informal networks, as
they may have a greater effect on the shaping of differences in status and/or network role.
Finally, the vast bulk of extant research has tended to focus on a general conception of leadership, and as a consequence, our
knowledge of pluralized leadership remains under-developed (Denis et al., 2012). We suggest that by focusing on the relational
mechanisms through which leadership is enacted our study offers insight to how activities that constitute leadership influence
might be distinguished from other organizational activities. In particular, we draw on a broader concept of leadership by
examining the informal support and advice networks that constitute an important component of pluralized leadership
(Friedrich et al., 2009). By focusing on informal support and advice networks, as well as leadership influence, we are able to
explore aspects of pluralized leadership that extant research may not recognize as “leadership influence”, but are reflected in
the patterns of social exchange and entrainment across the different networks. Hence, our approach enables us to develop a
broad concept of leadership, moving beyond a narrow conceptualization of formal leadership influence, yet at the same time
differentiating between leadership and non-leadership activities.
Limitations and future research
In terms of future research, we suggest that it is important that scholars of PL broaden their focus to acknowledge that
leadership may be embedded in multiple forms of network relations. The importance of multiplex relationships between two
(or more) actors has had a long history in social network research, being viewed as a defining feature relational pluralism
(Shipilov et al., 2014). There exists, however, very little research on effects of multiplexity or the effects of heterogeneous
relations on leadership (and specifically pluralized leadership). Furthermore, although interest in the mechanisms behind multiple
network ties in organization has increased (Tortoriello, Reagans, & McEvily, 2012), the focus has been on dyadic relations, with
more collective forms of tie interdependencies remaining an important research gap (Shipilov et al., 2014). Hence, we argue
that scholars should build on our work, drawing on recent developments in SNA, to examine and better understand the complexity
of cross-network relationships from which more widespread pluralized leadership might derive.
In addressing the point above, we suggest that our work can help in contributing toward the construct validity via providing
some insight into a potential “nomological network” of leadership endeavor (Mumford et al., 2012). In modeling the relationships
between perceptions of leadership influence, and the cross network effects with informal relations, we have attempted to enhance
our understanding of pluralized leadership by considering under what conditions we will find multiplexity. As our study contains
multiple variables that are proposed as being differentially related to pluralized leadership, we regard that our work can inform a
nomological network approach to the validation of this concept. However, our research is only a first attempt to model the

multiplexity of relations encompassed within pluralized leadership, and much work would be required to refine and validate any
such approach.
As with all research our work is not without limitations, some of which provide additional potential avenues for future
research. First, our data is drawn from a single case study of an English public services network form of organization. Clearly,
any single case study limits the generalizability of findings, therefore, there is a need for others to conduct similar studies across
a range of empirical settings.
Second, our data is based on a cross-sectional research design, which only provides a snap-shot of the network structures. We
suggest that in the future researchers should try and employ more longitudinal research designs to explore the dynamics and
enactment of pluralized leadership over time. Recently, there have been developments around longitudinal versions of ERGMs
to investigate network dynamics (e.g. Hanneke, Fu, & Xing, 2010; Krivitsky & Handcock, 2014). Several scholars have reported
early work along these lines, including the evolution of multiple networks (e.g. Cranmer, Desmarais, & Kirkland, 2012; Snijders,
Van de Bunt, & Steglich, 2010). However, multiple network variants of existing models for longitudinal networks are still
under-developed. Over the next few years, attention to dynamics will be at the cutting edge of social network research, both
empirically and methodologically.
Third, a final limitation concerns the validity of our network measures. Our data collection method may have inflated the observed
correlations among our variables, via common method bias, so obscuring their true relationship (Avolio, Yammarino, &
Bass, 1991). In socio-metric research, other researchers have routinely acknowledged this problem (see: Podolny & Baron,
1997). In this study, the informal social networks measures were constructed differently to the influence construct, and the
(QAP) correlations were below the threshold suggested by Podsakoff et al. (2003). Thus, common methods bias can neither be
completely ruled out nor does it clearly present as a problem; however, the evidence seems to suggest that it is unlikely to
have been a serious problem in our work.
Conclusion
In this article we have addressed the gap in our knowledge of pluralized leadership surrounding the interplay between the
leadership influence and the informal network of relations that connect people, and vice versa (see Friedrich et al., 2009;
Mumford et al., 2012; Yammarino et al., 2012). In theoretical terms, we explored the conditions under which leadership influence
relations may exhibit cross-network effects with informal social relations and how, focusing on mutual exchange and mutual
entrainment. We did so through utilizing recent advances in SNA methods, through which we were able to demonstrate the
value of the ERGM approach for understanding how the micro-level enactment of pluralized leadership is shaped by local
interactions derived from social relations. We hope that our work will serve to focus future research on understanding the
complex social processes that underpin the enactment of pluralized leadership, and how they are shaped over time.
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