# Modelling Multilevel Interdependencies for Resilience in Complex Organisation

Authors: Justyna Tasic; Fredy Tantri; Sulfikar Amir
Journal: Complexity
 (2019)

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## Abstract

This paper aims to model multilevel interdependencies in complex organisational systems and proposes application for resilience analysis. Most of the existing research studied interdependencies only at the single-level and overlooked their multilevel character. In response to this gap, we propose a multilevel approach to better comprehend the complexity of interdependencies in organisational systems. More specifically, the study focuses on how interdependencies are shaped across multiple organisational levels. To understand the research problem, we use multilevel and social network theories to elaborate the concept at five organisational levels, namely, individual, intraunit, interunit, intraorganisational, and interorganisational. Further, we show the application of multilevel interdependencies into analysis of organisational resilience. To this end, we construct a multiplex model of a real world organisational system that comprises formal and informal relations of different social exchange strength. Using the agent-based simulations of the organisational system, we investigate the relations between organisational interdependencies and organisational performance in normal and disrupted conditions. With the results, we argue that managing multilevel interdependencies is crucial to reduce vulnerability of organisational systems. By introducing the multilevel conceptualisation of interdependencies and presenting their influence on organisational resilience, we hope to pave a path to managing the complexity of interdependencies and strategic resilience enhancement in organisational systems.


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## Body

## 1. Introduction

What are the consequences of interdependencies that occur at multiple levels on our capacity for responding to disruptions? This paper examines the emergence of multilevel interdependencies and how they affect organisational behaviour in mitigating crisis. Organisational behaviours have their origins and consequences at different levels [1, 2]. In analysing this correlation, most management researchers study individuals, work groups, organisations, and industry using approaches in which the structure of interactions of these entities is typically observed only at one level [3]. Research conducted at the micro-, meso-, and macro-levels often ignores the fact that organisational dynamics result from multilevel interactions. This particular aspect is crucial in understanding the impact of interdependencies on organisational performance. In management science, interdependencies are understood as exchange relationships [4, 5] within and between organisations [6]. In general, studies on interdependencies are divided between those looking at the micro- and mesolevel and those at the macro one. Studies on the micro- and mesointerdependencies are firmly grounded in organisational behaviour and social science theories, which have been used to explain individual or group behaviour in the context of task interdependence, workflows, and interpersonal or intergroup relations [7–11]. The macro approach has been usually adopted by economics, sociology, strategy, and supply chain management scholars. This approach provides the understandings of interorganisational relations between companies, supply chain partners, and regulatory and institutional entities that affect organisational performance [6, 12–17]. While single-level approaches have their own virtues, there is a gap of knowledge in how these levels interact with one another and what implications they bring on organisational resilience. Except for the divergent focuses, research on interdependencies at single levels often uses structural-functionalism and social exchange theory lenses [14, 18–24] to investigate interdependencies. However, due to disparate goals, organisational experts of each of the levels have seldom merged their perspectives to achieve a bigger picture of organisational dynamics. As a result, adopting either micro-, meso-, or macro-lens only yields an incomplete understanding of multilevel interactions and their consequences. It is this lack of knowledge in multilevel organisational interdependencies that this paper seeks to shed light on.Grounded in general systems theory [25, 26], a multilevel view on organisational analysis differs from analysing organisations through a single-level lens. It aims to shift the research field toward the view of organisations as complex and interconnected social systems [27, 28] where the phenomena are analysed at interlevel domains. Because interdependencies in organisational systems are too intricate phenomenon to be explained by only single-level terms, their comprehensive understanding requires integration of micro-, meso-, and macro-insights. Nowadays, due to economic and technological changes, the degree of interdependencies within and between organisations constantly grows [29, 30]. In this light, using the structural approach, we aim to introduce a preliminary conceptualisation of multilevel interdependencies in organisational systems and present its applications for resilience analysis. We elaborate the concept using themultilevel [31] andsocial network theories [32–34], which help us to explain in detail how interdependencies are formed and how they can be measured at each level. We argue that presenting interdependencies as a multilevel construct enables a more integrated and comprehensive understanding of the phenomenon’s complexity and dynamics that unfold across organisational levels, and affect organisational resilience. We prove it with agent-based simulations of a real organisational system under the crisis condition.To achieve this objective, this paper is structured as follows. The following two sections present the state-of-art in interdependencies and resilience studies of organisational systems, in which we highlight the key aspects of interdependencies and resilience studies and identify areas where multilevel analysis remains lacking. After reviewing the existing works on organisational interdependencies and resilience, we begin to introduce a multilevel conceptualisation of interdependencies in complex organisational systems. We move then to the empirical application of the concept in the area of resilience analysis. More specifically, we construct a multiplex model of a real world organisational system that comprises weighted formal and informal relations between and within two organisations. By giving the weights to multiplex relations we aim to reflect their different social exchange strength. We establish the weights based on the system of informal relationships introduced by Luo [35] and the hierarchical logic of formal employment relationships, which determine chains of command (authority) and formal information exchanges (reporting) [36]. We also describe in details qualitative and quantitative characteristics of existing edges. Further, we present the relations between multilevel interdependencies and organisational performance in normal and disrupted conditions. Next, we discuss the results and highlight that managing multilevel interdependencies is crucial to reduce vulnerability and enhances resilience of organisational systems. The conclusion summarises theoretical and empirical contributions of the paper.


## 2. Linking Interdependencies to Resilience

Interdependence began to be a subject of inquiry for organisational research in 1949 when Deutsch introduced a theory of social interdependence, which gained influences on management research and practices for several decades. Since then, organisation researchers have examined the nature and consequence of interdependence from various perspectives. A dichotomy between micro and macro levels appears to characterize the study of social interdependencies. In the micro studies of interdependencies, La Porte [4] defined the degree of organised system complexity as a function of number of components, differentiation of components, and interdependent links between the components. In similar direction, management science explored interdependencies with the theory firmly grounded in organisational psychology and social science to explain human behaviour in the context of task interdependence, interpersonal relationships and performance at micro- and mesolevels of analysis [7–11]. At the macro level, analysis of interdependencies in management studies drew theoretical underpinnings on disciplines such as economics, economic sociology, and supply chain management to assess interorganisational relations that shape organisational performance, for instance, flows of goods and services, trust-based relationships, and industry environment [6, 12–16]. In this area of research, only a few attempts looked into interdependencies across levels [37, 38].In further developments of organisational sciences, interdependencies are understood as social exchange relationships [4, 5] within and between organisations [6]. At micro level, Brass [21] underlined that an organisation is composed of interdependent networks of employees. At mesolevel, the interdependencies are defined as a relationship between two or more organisational units that are mutually affected [19]. Following this path, Gulati [39] and Tomkins [14] tracked down the source of interdependencies in the interorganisational context by focusing on the role of trust and information relations as determinants of the interdependent relationship. Conceptually, the notion of interdependence assumes at least one exchanged resource per interdependent dyad [4]. However, in real life, the relations become more complex, due to the multiplex types of independencies. It is extremely important to note that while organisational relations areformally maintained, through the existence of rules and regulations [6, 40, 41], they are alsoinformally shaped, through social structures, trust, and reciprocity [23, 40, 42]. Informal relations embody interpersonal trust, which lubricates social support and, thus, enables complex transactions and facilitates collective action. Therefore, structural position of actor in an organisation is formed by both formal and emergent informal interdependencies, which interact with each other. In this area, interdependencies were analysed as networks of formal work exchanges [e.g., [21, 43]], and informal relationships as acquaintance, friendship and familiarity networks [35, 44, 45]. While the formal relationships are easily identified, the informal interactions are hardly detectable and often overlooked; however, they do impact organisational performance [45–48].There are two commonly distinguished types of ties—instrumental and expressive [49]. Instrumental ties are information and cognition based exchanges of resources for instrumental purposes (e.g., reporting, acquaintance). Expressive ties include an affective factor and reflect common relationship identity as well as social support (e.g., friendship). In addition, researchers also recognised existence of mixed ties, which combine features of both types [35]. According to Luo [35], informal instrumental ties are based on rules of equity (acquaintance), informal expressive ties are governed by rules of need (friendship), and mixed ties by rules of favour (familiarity). Both strong instrumental and expressive ties have a positive effect on performance; in contrast weak instrumental ties are less likely to enhance performance [50].Overall, the research on organisational interdependencies has produced a growing body of knowledge to illuminate the dynamics of organisational systems as a result of complex interactions and ever expanding structures. However, as our review of this body of knowledge above indicates, most of the research is inclined to pay attention only at interactions and processes that occur at the same level. Characters and properties of interdependencies that emerge out of multilevel dynamics remain to be explored. In addition, the influence of informal interactions on organisational functioning requires in-depth analysis.It is posited in this paper that the multilevel structure of interdependencies has potential impact on organisational capacity for responding to crisis, especially large-scale ones. What remains unclear is how exactly structural interdependencies will affect organisational resilience. To have a clear explanation, we first need to have a basic understanding of what constitutes resilience and what renders organisation to be resilient. Up to date, the concept of resilience has been applied in a wide variety of academic fields, including among others, ecology [51–53], sociology [54], psychology [55], supply chain management [56], strategic management [57], disaster mitigation [58, 59], sociotechnical resilience [60, 61], resilience engineering [62, 63], organisational reliability [29, 64, 65], and resilience management [66–68]. In the following lines, we describe the concept of resilience with a primary reference to resilience engineering and management studies, which constitute the most relevant ground for this paper.Many scholars acknowledge that resilience offers a potential solution to overcome disruptions and uncertainties in organisational systems, and create the environment for organisational development. In 2006, a group of experts in industrial and system safety initiated an organisation-centred paradigm called “resilience engineering” [62]. Recently, the continuations of those developments led to the following definition of resilience: “Resilience is an expression of how people, alone or together, cope with everyday situations – large and small – by adjusting their performance to the conditions [69]. Resilience in the field of management is generally regarded as “an emergent property that relates to the inherent and adaptive qualities that enable an organisation to take a proactive approach to threat and risk mitigation” [70]. In general terms, an organisational system “can be seen as more resilient when it is more robust and less vulnerable to disruptions and recovers faster from disruptions when they occur” [71]. To sum up, resilient organisational systems are proactive in mobilizing their resources, use their abilities and adapt to effectively perform under a variety of conditions (both expected and unexpected).How do organisational structures affect resilience? As pointed out by a number of researchers, complex systems organisations are composed of the multiple levels of operation [72] in which interconnected agents form a network of nonlinear relationships that give rise to emergent behaviours [66]. The more complex organisation, the more complex response to disruptions will become [72]. Similar to safety, resilience is seen as a systemic property where individual, group, and organisational levels have a reciprocal influence on each other [73, 74]. From this vantage point, managing resilience is “as a matter of a balancing between individual resilience (individual responses to operational challenges) and system resilience (the large scale autocatalytic combination of individual behaviour)” [62]. Resilient employees and units do not guarantee resilient organisation. In addition, organisations are embedded in interorganisational relationship, which are crucial for sustaining their operations [13, 56, 75]. Organisations are more likely to be resilient if they can effectively mobilise internal and external resources and capabilities [74]. Therefore, interorganisational resilience is usually described as adaptability, flexibility and redundancy or diversification [56, 75]. Applying systems thinking approach, Leveson [63] argued that system safety is an emergent property and thus must be controlled at the system level. Therefore, building resilience should be seen as a collaborative process, which requires continuous learning and adaptation from all actors, i.e., across individuals, groups, organisations [68, 76], and interorganisational networks.The bottom line is that organisational resilience is strongly related to everyday work and organisational structure [60, 65, 74, 77]. Well-developed relationships (formal and informal) within and among organisations are fundamental for taking up a joint action during normal and crisis operations [29, 45, 47, 65]. The informal control instruments can substitute or complement formal control instruments in both normal [14, 23, 78–80] and crisis operations [47]. Thus, any resilient organisational system has a strong culture of awareness and is able to create improvised responsive networks to mitigate a crisis [47, 59, 65, 67].


## 3. Multilevel Interdependencies in Complex Organisation

Multilevel theory in organisational studies has its conceptual underpinnings in general systems theory [25, 26], which presents organisations as complex and interconnected social systems [27, 28, 72, 81]. Complexity of organisational systems arises from interdependencies of the system components and emergent behaviour [82]. In this view, an organisation is conceptualized as a set of subsystems composed of more elemental components (Figure 1) that are arrayed in a hierarchical structure [25]. Organisational systems are seen as tightly coupled and nonlinear structures blended with bidirectional causal loops [83].Figure 1
Nested character of complex organisational systems.The main assumptions underlying the multilevel approach are that many phenomena emerge at multiple levels of analysis, and that organisational entities reside in nested arrangements [1–3]. The connection between levels of a system is determined by their couplings, which is the extent to which properties, dynamics, and processes at one level affect other levels [25]. Partial analysis of the system can be misleading, and thus to avoid wrong conclusions multilevel analysis is extremely crucial to represent all properties of the system [84]. The implication of the system approach is double-side. On one side, in line with general system theory, dynamics and relationships at lower levels emerge over time to higher levels of a system to yield structure [85]. This view coincides with theories of complexity, chaos, and self-organisation. On the other side, many organisational theories postulated that the contextual factors at higher levels may have direct or moderating effect on lower levels phenomena [31]. Although, we acknowledge that interdependencies, as many other phenomena in organisational systems, are a fuzzy concept that emerges bottom-up and is affected by top-down contextual processes, in this paper our focus is primarily on the emergent character of interdependencies in order to open a research discourse on multilevel character of interdependencies.In advancing multilevel interdependencies informed by the theoretical insights described above, we useconfigural compilation approach to analyse interdependencies as a multilevel construct [1, 31, 83, 86]. The compilational approach rests on the assumption of discontinuity and complex nonlinear processes of emergence. It describes phenomena that comprise a common domain but are distinctively different as they emerge across levels, i.e., simple aggregation of a construct at multiple levels is impossible [31]. In compilation processes, concept properties across all levels are discontinuous—qualitatively different— yet functionally equivalent—have the same role [1]. In this way, functional equivalence allows analysing a phenomenon by the elemental content of different types and amounts but possessing similar collective properties across levels [87]. Configural properties of a phenomenon are based on compilation models of emergence, i.e., phenomenon is nonlinear, not uniformly distributed, and not isomorphic across levels. They capture the differential patterns of relationship and variability of lower-level contributions to yield higher-level properties. This means unit-level interdependencies are a complex configuration of unique characteristics of unit members that emerge as a whole. In sum, with application of configural compilation approach, we argue that interdependencies across all levels are discontinuous phenomenon that does not express uniform pattern due to their specific nature.To further elaborate this concept, thesocial network theory[32, 33, 89] is employed to explain in detail how relational exchanges comprising interdependencies are formed in organisations, and to depict their pattern at various levels. The pattern includes both formal and informal structure of relations that form organisational systems. Formal relations are prescribed set of interdependencies between employees established by the organisation that determine its formal functioning, i.e., authority and reporting relationships. Equally important to underline are informal networks that comprise of trust-based relationships such as acquaintance, friendship and familiarity networks. These informal relations influence on individuals, group, organisation and interorganisational network performance [11, 13, 15, 43, 48, 90, 91]. In addition, informal organisational structure very often supplements the formal relations, especially in crisis situations [45, 47].Applying the system view above, we recognise interdependencies to be nested at five organisational levels:individual, intraunit, interunit, intraorganisational andinterorganisational. This span of levels represents a hierarchical structure in which each level represents unique characteristics of interdependencies between individuals, units, and organisations. The interdependencies at each level of representation are constituted by formal and informal relations. The interdependencies are coupled across levels and their content is meaningfully related in the whole network of relations. While lower levels such as individual and unit are composed of more elemental components, higher-level interdependencies, especially organisational and interorganisational, are relatively inclusive and encompass characteristics of lower-levels. In doing so, we assume that interpersonal interdependencies constitute individual interdependencies that contribute to intra- and interunit level interdependencies; intraunit and interunit interdependencies contribute to intraorganisational-level interdependencies (Figure 2).Figure 2
Overview of nested multilevel interdependencies.As elaborated above, each organisational system is a system of multiplex networks that comprises formal (reporting and authority) and informal (acquaintance, friendship, and familiarity) relations, which form interdependencies across all levels of analysis. In the following paragraphs, we use weighted degree centrality to decompose and describe the multilevel interdependencies in organisational systems. Determining the weights depends on the particular analysis needs and the organisations features such as industry and organisational culture. In line with the configural compilation approach, the applied measures vary across the functional equivalents at multiple levels due to different characteristics of interdependencies across the levels (individual, intraunit, interunit, intraorganisational, and interorganisational network).( 1 ) Individual interdependencies entail formal and informal relational exchanges that constitute weighted degree of an individual, which is the focal point of analysis. Formal relations (F) comprise relationships predetermined by the organisation, i.e., authority and reporting relationships between the individual and other directly connected individuals. Formal weighted individual degree (wi[F]) is the sum of weights associated to all formal edges attached to individuali, whereN is the total number of individuals in the interorganisational networkm(1)wiF=∑j=1j≠iNwijFInformal relations (I) capture emergent organisational features and are the by-product of formal daily interactions and interpersonal attachment. They consist of trust-based networks, which influence on the work behaviour of the individual, i.e., acquaintance, friendship, and familiarity networks [35, 44]. Informal individual degree (wi[I]) is the sum of weights associated to all informal edges attached to individuali(2)wiI=∑j=1j≠iNwijIFormal and informal relations interact with each other and affect individual’s behaviour. Therefore, the individual’s structural position is the result of particular combination of both formal and informal interdependencies. The individuals bonded in the direct neighbourhood will be highly interdependent and analysis of interdependencies at this level helps to assess individuals’ connectedness.( 2 ) Intraunit interdependencies emerge from the configuration of the unit members’ interdependencies that give a comprehensive picture of interdependent relations within a unit. The formal (F) and informal (I) exchange relationships remain the same features as at the individual level; however, the interdependencies are more complex due to the increased number of individuals and relationships. Formal intraunit degree (wu[F]) of unituis the sum of formal edges attached to individuals within unitu, where Ku is the number of individuals in unitu (3). Informal intraunit degree (wu[F]) of unitu is the sum of weights associated to informal edges attached to individuals within unitu (4).(3)wuF=∑i=1Ku∑j=1j≠iKuwijF(4)wuI=∑i=1Ku∑j=1j≠iKuwijITo allow the comparability of the unit structures between the units, we construct also the average formal (wuF) and informal (wuI) unit degree of a unitu ( (5) and (6)), which are calculated by normalization of intraunit degrees by the number of individuals in a unitu.(5)wuF=wuFKu(6)wuI=wuIKuAnalysis of interdependencies at this level helps to identify the crucial individuals on whom the unit performance is the most dependent both formally and informally. Formal structure is always well known by the individuals involved in the unit. The analysis of informal structure helps to identify hidden unit patterns and notice hidden needs and opportunities. Formal and informal relations should be well balanced to facilitate better the unit performance.( 3 ) Interunit interdependencies encompass formal and informal relations that emerge from the cross-unit relations between individuals. Formal interunit degree (wuint[F]) of unitu is the sum of weights of formal edges between individuals in unitu and individuals in other units (Ks) in organisationo,where Uo is the number of units in organisationo (7). Informal interunit degree (wuint[I]) of unitu is the sum of weights of informal edges between individuals in unitu and individuals in other units (Ks) in organisationo (8).(7)wuintF=∑i=1Ku∑s=1s≠uUo∑j=1KswijF(8)wuintI=∑i=1Ku∑s=1s≠uUo∑j=1KswijIFormal interunit relations determine how units operate and how much formally interdependent is their work. The units can be very independent (divisional structure), moderately interdependent (functional structure), or very interdependent (matrix). The informal relations, described in the social network theory as informal boundary spanning, emerge from informal interunit exchanges of information and support (acquaintance, friendship, and familiarity relations). Informal interunit relations crucial for a unit to raise unit effectiveness and gain access to external resources; however, they work the best when the unit is well connected externally and well as internally.We construct also the average formal (wuintF) and informal (wuintI) interunit degrees of a unitu ((9) and (10)), which are calculated by normalization of interunit degrees by the number of individuals in a unitu.(9)wuintF=wuintFKu(10)wuintI=wuintIKu(4)Intraorganisational interdependencies include formal and informal relationships within and between units of an organisation. Formal intraorganisational degree (wo[F]) of an organisationo is the sum of formal intra- and interunit degrees within this organisation (11). Informal intraorganisational degree (wo[I]) of an organisationo is the sum of informal intra- and interunit degrees within this organisation (12). The formal and informal intraorganisational degrees are crucial to assess internal connectedness of an organisation.(11)woF=∑u=1UowuintF+∑u=1UowuF(12)woI=∑u=1UowuintI+∑u=1UowuIThe average formal (woF) and informal (woI) intraorganisational degrees ( (13) and (14)) are calculated by normalization of the degrees by the total number of individuals (Lo) in an organisationo.(13)woF=woFLo(14)woI=woILo(5)Interorganisational interdependencies comprise formal and informal relations that emerge from the cross-organisation exchanges between individuals. Formal interorganisational degree (woint[F]) is the sum of weights of formal edges between individuals in organisationo and individuals in other organisations (Lr) in interorganisational networkm,whereOis the number of organisations in the interorganisational networkm (15). Informal interorganisational degree (woint[I]) is the sum of weights of informal edges between individuals in organisationo and individuals in other organisations (Lr) in interorganisational networkm (16). Interorganisational relations constitute an important part of interdependencies as they may facilitate interorganisational information exchange, knowledge sharing, innovation transfer, and support.(15)wointF=∑i=1Lo∑r=1r≠oO∑j=1LrwijF(16)wointI=∑i=1Lo∑r=1r≠oO∑j=1LrwijIThe average formal (wintF) and informal (wintI) interorganisational degrees ( (17) and (18)) are calculated by normalization of the degrees by a total of individuals (N) in interorganisational networkm.(17)wintF=∑o=1OwointFN(18)wintI=∑o=1OwointIN


## 4. Multilevel Interdependencies Model to Analyse Organisational Resilience

In this section, we bridge the multilevel conceptualisation of interdependencies with resilience analysis. More specifically, we present the application of conceptualisation to investigate the relation between interdependencies and organisational performance in normal and disrupted conditions. Based on the real world data, we construct an agent-based model of multilevel organisational interdependencies of two organisations. We demonstrate results of calculated interdependencies measures and performance simulations. The following research questions guided our analysis:(1)
What type of structure makes an organisational system more resilient?(2)
How does the degree of organisational interdependencies change at multiple levels?(3)
Does higher degree of organisational interdependencies contribute to better performance?(4)
Which of the interdependencies’ levels contribute the most to the organisational performance?


### 4.1. Materials and Methods

We used the agent-based method and built a model of organisational interdependencies, which comprised formal and informal relations within and between two organisations. The model of interdependencies was constructed from the sociometric data gathered in August 2017. The data was collected at individual level (N=54) by roster questionnaires that were distributed in two collaborating organisations, which operate within the security services sector in Southeast Asia. Organisation A is a research, training, and operational support centre. Organisation B is an operational support centre to enhance well-being and operational effectiveness of another organisation’s employee. Each of the organisations comprised 5 work units. The data described six multiplex social networks: reporting relationships, authority, acquaintance, friendship, familiarity, and problem-solving (Table1).Table 1
Networks data used in the model.
Social Network Questions Relationships Application in model ( 1 ) With whom do you like to discuss your daily work? [35] Acquaintance(instrumental exchange) ( 2 ) With whom do you talk about your private affairs during your daily chats? [44] Friendship(expressive exchange) Informal relations (trust-based) ( 3 ) Suppose that your colleague asks you to help his/her friend. Whose friends would you help? (adapted from Luo [35]) Familiar(instrumental and expressive exchange) ( 4 ) To whom do you report about you work progress? Reporting(instrumental exchange) Formal relations ( 5 ) Who is your direct supervisor? Authority(instrumental exchange) ( 6 ) Whom do you ask for help when you encounter a work-related problem, for which you couldn’t find a solution yourself? Problem-solving(instrumental exchange) Task demandFollowing the work of Luo [35], Haythornthwaite [36], Luo and Cheng [44], Krackhardt and Hanson [45], and Soda and Zaheer [92], the formal relations in the model consisted from reporting and authority networks and the informal interactions comprised acquaintance, friendship and familiar ties. For informal relations only mutual links (links that were confirmed by both connected agents) were considered as reliable and included in the model. The multiplex model was specified by the vector of the symmetric adjacency matrices (undirected graph) of formal and informal relations: A=A[F],A[I]. Each of the matrices was constructed through the aggregation procedure that resulted in multiplex edge types, which allowed us to specify the simulation parameters (Appendix A). The pairs of agents are connected by either (1) both formal and informal edges, or (2) a formal edge, or (3) an informal edge (Figure 3).Figure 3
Multiplex model of organisational interdependencies. Nodes’ colours represent work units. Node size depicts node strength. Blue and green edges are formal and informal relations. Individual no. 4 works in both organisations.The strength of social exchanges was reflected in the weight values given to each type of formal and informal edges. Qualitative and quantitative characteristics of the edges are described in Table2. The formal and informal weight values range from 0 to 1. The weights of informal edges are based on the system of social relationships introduced by Luo [35] that governs complex social transactions dependently on proportions of instrumental and expressive exchange dimensions. According to Luo [35], Luo and Cheng [44] the strongest and most efficient resource exchanges are facilitated through familiarity ties (‘rules of favour’), succeeded by friendship (‘rules of need’), and acquaintance (‘rules of equity’). The weights of formal edges are aligned with the hierarchical logic of formal employment relationships reflected in an organisational chart, which determine chains of command (authority) and formal information exchanges (reporting) [36]. In this light, the authority relations, representing formal vertical relations, are defined as the strongest, and are followed by reporting relationships that include both formal horizontal and vertical relations. The weights of aggregated formal and informal links have been sums of single network weight values (e.g. an edge representing single reporting and authority has a weight equal to 0.8 which is a sum of 0.2 and 0.6).Table 2
Quantitative and qualitative characteristics of existing relations.
Structure type Edge types Weight Social exchange strength Qualitative characteristics Formal One way reporting 0.2 Weak (i) Weak instrumental exchange(ii) Limited reliability (iii) No expectation of reciprocation (iv) Occasional exchanges Mutual reporting 0.4 Moderate (i) Moderate instrumental exchange(ii) Moderate reliability (iii) Expectation of reciprocation (iv) Occasional exchanges Authority 0.6 Strong (i) Strong instrumental exchange(ii) High reliability (iii) No expectation of reciprocation (iv) Frequent exchanges Authority, One way reporting 0.8 Very strong (i) Strong instrumental exchange(ii) Very high reliability (iii) No expectation of reciprocation (iv) Very frequent exchanges Authority,Mutual reporting 1.0 Extremely strong (i) Strong instrumental exchange(ii) Extremely high reliability (iii) Expectation of reciprocation (iv) Very frequent exchanges  Informal Acquaintance 0.1 Weak (i) Weak instrumental and weak expressive exchange(ii) Moderate level of trust (iii) Rules of fair exchange (iv) Expectation of instant reciprocation (v) Limited reliability, often insufficient in obtaining valuable resources (vi) Occasional exchanges Friendship 0.3 Moderate (i) Weak instrumental and strong expressive exchange(ii) High level of trust (iii) Rules of need (iv) Expectation for reciprocation, but not instant (v) High reliability (vi) Long-term, occasional exchanges (ad hoc when needed) Friendship, Acquaintance 0.4 Strong (i) Moderate instrumental and strong expressive exchange(ii) High level of trust (iii) Expectation of reciprocation, but not instant (iv) High reliability (v) Long-term, occasional exchanges Familiarity 0.6 Very strong (i) Strong instrumental and moderate expressive exchange(ii) High level of trust (iii) Rules of favour exchange (iv) Expectation of reciprocation, but not instant (v) Very high reliability (vi) Long-term and frequent exchanges (vii) Strong enough to be a bridge to connect to other agents Familiarity, Acquaintance, Friendship >= 0.7 Extremely strong (i) Strong instrumental and medium or strong expressive exchange(ii) High level of trust (iii) Expectation of reciprocation, but not instant (iv) Extremely high reliability (v) Long-term and very frequent exchanges (vi) Strong enough to be a bridge to connect to other agents


#### 4.1.1. Simulations

To investigate how structure of organisational interdependencies affects organisational resilience, we used the model to conduct series of simulations both in normal and disrupted conditions. The simulations highlight two important aspects from the organisational systems functioning, i.e., task interdependence (agent must cooperate with other agents to complete a task) and adaptation through collective problem solving (the agent support each other directly and indirectly to solve a problem). In all simulations, more than 80% of all agents are given tasks to complete. All tasks are specified bytasks demand[T_D], which is a list of resources that agent needs to gather to complete the task. The task demand was generated from the directed problem-solving network (Table 1) through selecting resources that belonged to the agents connected by out-going edges (i.e., agents that an agent would contact if a work-related problem occurs). If the number of resources was more than five, only five resources were randomly selected, so the task demand lists ranged from one to maximum five resources. The task is completed when 100% of resources listed on the task demand are gathered (Algorithm 1). Each agent has a resource of his own (e.g., agent J has a resource J), and each network edge possesses a weight value that reflects strength of the relationship, as described in Table 2. The strength of relationship is reflected also in the parameter of speed and, thus, transfer time (Appendix A, Table 3). The transfer time needed to connect and pass the resource between the agents is depended on the edge type. For each agent’s step, the transfer time is calculated by subtracting the sum of formal and informal edges’ speed values from 100. In each simulation we calculate the total time based on all transfer times needed to accomplish tasks to measure performance of agents, units, organisations, and interorganisational network. In the resources search, the agent use both formal and informal structure in the maximum distance of two edges. Based on social exchange theory, the agent will choose always the fastest way. To ensure the realism of simulations, the speed priority is given to all formal types of links (the most time efficient), so the agent will choose and informal edge only if there is no formal connection. In line with work of Luo [35], the 2-step resource sharing is only possible when two conditions are fulfilled: (1) the first step edge is at least moderately strong informally (i.e., follows the rules of need or favour) or weak informally but the condition of instant reciprocation is fulfilled (rules of equity); and (2) the second step edge contains the familiarity component (rules of favour). In this case, the total transfer time is sum of all steps needed to share the resource (see more details in Algorithm 1 and examples in Appendix B).Table 3
Simulation parameters.
Relations Aggregated edge type Contained network edges No. of existing multiplex links Weight Speed Share of other agents resources in disruption condition (directional) Contribution to 2-step resource sharing Formal Rep_one_way One way reporting 55 0.2 40 0.4 . Rep_mutual Mutual reporting 3 0.4 45 0.4 . Superv Authority 6 0.6 50 . . Superv_rep_ one_way Authority,One way reporting 52 0.8 55 0.4 . Superv_rep_mut Authority,Mutual reporting 4 1.0 60 0.4 .  Informal Acq Acquaintance 15 0.1 5 0.2 Step1 + T_D Friend Friendship 12 0.3 10 . Step1 Acq_friend Friendship, Acquaintance 17 0.4 15 0.2 Step1 Famil Familiarity 28 0.6 20 . Steps1 and 2 Acq_famil Familiarity, Acquaintance 6 0.7 25 0.2 Steps1 and 2 Friend_famil Familiarity, Friendship 11 0.9 30 . Steps1 and 2 Acq_friend_famil Familiarity, Friendship, Acquaintance 8 1.0 35 0.2 Steps1 and 2Algorithm 1:Task completion and resource sharing.
Input: tasks, other_agents, networks, disrupted_agents Output: agent_task_completion_time t =0 ( 1 ) for task in  tasks  do ( 2 )  for  other_agent in  other_agents  do ( 3 )  if  task = other_agent[resource] then ( 4 )  if  other_agent not in  disrupted_list  then ( 5 )  task = 100% ( 6 )  t+=  transfer_time(other_agent,agent) ( 7 )  else ( 8 )  for  subs_agent in  other_agent[shared_resources]  do ( 9 )  if  subs_agent not in  disrupted_list  then ( 10 )  if  distance(subs_agent, agent)= 1  then▹1 step ( 11 )  agent[task]+ = subs_agent[subs_resources] ( 12 )  t+ = transfer_time(subs_agent, agent) ( 13 )  else if  distance(subs_agent, agent)= 2  then▹2 step ( 14 )  if  link(1) and link(2) fulfill the 2-step rule  then ( 15 )  agent[task]+ = subs_agent[subs_resources] ( 16 )  t+ = transfer_time(subs_agent, agent) ( 17 ) if  all tasks≥ 100  then ( 18 )  return t ( 19 )   else ( 20 )  return  nanOn the basis of the task demand lists, we create two types ofmajor work disruptions:(1) targeted disruption (unavailability) of agents the most needed (i.e. agents with highest in-degree centrality in each unit; for bigger units at least 20% of agents were disrupted) to complete the task demands (N=9, 20% of agents with tasks) and (2) random disruption of agents, which had a resource needed to complete at least one task demand (N=9, the same number of agents as in targeted disruptions). While in normal conditions, an agent, to complete his task, looks for a resource directly from the original resource (an agent who owns it); in disrupted conditions, the agents are allowed to use the substituting resources if the original resource is not available anymore (owning agent disrupted) (see Algorithm 1). The substituting resource is conditioned by existence of the edge type that comprises exchange of work-related information, i.e., reporting (formal relations) and acquaintance (informal relations). To create substituting resources we considered directed networks of reporting and acquaintance (only mutual links), in which incoming edges granted an agent 40% and 20% of substituting resource respectively. Thus, in case of simultaneous reporting and acquaintance incoming links, the maximum value of substituting resource in question is 60% (Appendix A, Table 3). When we disrupt the organisational system, to complete his task, an agent first checks his available substituting resources and next looks for other agents who can substitute the needed resource to reach the needed amount (100%).


### 4.2. Results

To answer what type of structure makes an organisational system more resilient, we examined the importance of formal and informal relations in determining the system behaviour under normal and disrupted conditions (Figure4). We conduct simulations of systems including (1) only formal relations (blue line), (2) only informal relations (green line), and (3) both formal and informal relations, i.e., the overall network (red line). Figure 4 presents average values from simulations of normal conditions (n=1000), random disruptions (n=3000), and targeted disruptions (n=1000). We measure the system performance by number of completed tasks and time to complete the tasks. The simulations results showcase that organisational system with the overall structure (including formal and informal relations) performs better than the system, which has only formal structure or only informal structure. Importantly, the overall structure system performance is not a sum of formal and informal structures’ performance. In this way, the overall system performance is not directly related to the number of existing links but rather their strength (relations quality) and other structural properties. Furthermore, it is crucial to acknowledge that in all conditions the formal structure plays a significant role in system performance and the organisational system that possesses only informal structure cannot perform well. In normal condition, all tasks are completed in both systems, but the system with overall structure completes the tasks faster. In case of both disruptions, the performance of both systems drops, as some tasks cannot be completed. However, it is very important to highlight that during the disturbed conditions, informal structure complements the formal structure (by 8% and 15% in random and targeted disruptions respectively). More specifically, the informal structure supplements the unavailable formal connections, and this results in higher number of completed tasks as well as faster completion of the tasks. In sum, in general terms, the organisational system with both formal and informal relations performs better under normal and disrupted conditions, thus is more resilient.Figure 4
Performance of organisational structures in normal and disrupted conditions.In addition, to rule out the factor of different links number and ensure the correct interpretation of the results, we have sampled formal and informal structures to have the same number of edges (precisely 90). In this way, we constructed the new overall structure that comprised 90 formal and 90 informal edges. We revised the task demand lists in accordance with the agents existing in the new overall structure. The new structures were sampled five times and simulations for the three conditions were repeated (n=1000, n=3000, and n=1000). The average sampling results are presented in Figure5. In general, the sampled results are consistent with the non-sampled (Figure 4). However, due to the executed changes that resulted in less efficient topology of new structures (smaller density, smaller transitivity, and longer average path; see Table 4), we reported performance drop of sampled formal and informal structures in normal condition and random disruption. The performance drop was bigger in case of formal than informal structure, especially in normal condition. That was due to higher deteriorative discrepancy between the sampled and the nonsampled formal structure in comparison to the sampled and non-sampled informal structure. In particular, the biggest disadvantages sampled formal structure concerned smaller transitivity, longer average path, and longer diameter, which highly influenced structure dynamics (see Table 4).Table 4
Properties of nonsampled and sampled structures.
Structure No. nodes No. edges Density Transitivity Average path length Diameter Non-sampled overall 54 217 0.15 0.44 2.66 2.40 Non-sampled formal 54 120 0.08 0.33 2.94 3.00 Non-sampled informal 43 97 0.11 0.43 3.40 3.90 Average sampled overall 53 180 0.13 0.38 2.81 2.66 Average sampled formal 52 90 0.07 0.24 3.31 3.76 Average sampled informal 42 90 0.1 0.39 3.42 3.76Figure 5
Average performance of sampled organisational structures in normal and disrupted conditions.In relation to the formal and informal structure, we investigated which types of edges facilitated most of exchanges in normal and disrupted conditions structure contributes the most to the organisational performance (Figure6). The usage of all types was relatively stable in all conditions. In total, most of exchanges were facilitated by both formal and informal edges (on average 46%) or only by formal edges (on average 43%). Approximately 11% of exchanges were conducted by informal edges. Similar pattern was observed over the simulation time. These results highlight with more details and reiterate the important contribution of the informal structure to the organisational system performance in normal and disrupted conditions.Figure 6
Types of edges used in normal and disrupted conditions.
(a)
Used edges by structure type over simulation time (b)
Used edges by structure type in totalNext, we examined the values of the formal and informal interdependencies degrees at multiple levels (AppendixC, Tables 5–9). Table 10 presents the summarised results of the degree analysis. The higher levels of the analysis, the higher were the values of non-averaged degrees due to the increasing number of considered network elements and their complex nature. The nonaveraged values of individual, intraunit, interunit, intraorganisational, and interorganisational degrees ranged from 0 to 72.8, and the averaged degrees ranged from 0 to 5.5. While both formal and informal individual, interunit, and intraorganisational degree values had moderate dispersion, the informal intraunit (SD = 7.8, mean = 5.3) and average formal interunit values (SD = 1.6, mean = 1.4) were characterised by high dispersion; that is, the degree values were very widely distributed. As data concern the relations within one interorganisational network, the variation at the interorganisational-level interdependencies was not possible to assess.Table 5
Individual degree.
Agent Id Org. Unit Formal Individual Degree Informal Individual Degree 1 A A3 1.4 4.2  2 B B1 1.2 0.1  3 B B 4.6 0.0  4 A/B A/B 9.4 2.1  5 A A1 1.8 2.4  6 A A3 1.2 2.2  7 A A4 3.6 1.3  8 B B4 1.2 0.1  9 A A1 1.0 2.8  10 B B2 3.8 0.0  11 A A4 2.2 0.0  12 B B3 1.6 1.7  13 B B1 0.8 0.0  14 A A3 8.0 4.9  15 A A3 1.0 0.8  16 B B1 1.0 3.9  17 A A1 6.4 1.2  18 B B1 0.8 2.9  19 A A2 1.2 5.6  20 A A3 4.8 0.1  21 A A2 1.8 0.9  22 B B1 1.2 4.2  23 A A 5.6 0.0  24 B B 0.4 0.0  25 A A3 0.8 3.6  26 A A3 0.8 1.5  28 A A3 1.2 4.2  29 A A3 1.8 2.6  30 A A4 0.8 0.4  31 B B2 1.0 2.4  32 B B1 1.0 0.2  33 B B3 6.0 1.1  34 A A2 5.2 2.0  35 A A3 1.2 4.2  36 A A3 5.4 4.3  37 A A4 1.0 0.8  38 B B3 1.8 6.6  39 B B2 0.2 0.0  40 A A3 1.2 0.4  41 A A4 2.2 0.8  42 B B2 0.8 0.0  43 B B3 1.4 1.1  44 B B3 1.4 3.1  45 B B1 1.6 3.5  46 B B1 6.4 0.0  47 B B1 0.4 0.0  48 A A1 1.6 3.4  49 A A4 2.6 2.9  50 B B 0.4 0.0  51 B B2 1.6 2.3  52 B B2 1.4 3.0  54 A A2 1.2 2.7  55 B B4 2.2 0.5  57 B B3 2.2 2.6Table 6
Intraunit degree.
Unit Formal Intraunit Degree Informal Intraunit Degree Number of Individuals Average Formal Intraunit Degree Average Informal Intraunit Degree A 0.4 0.0 2 0.20 0.00  A1 6.8 5.0 4 1.70 1.25  A2 4.8 1.8 4 1.20 0.45  A3 25.2 27.4 12 2.10 2.28  A4 6.4 2.4 6 1.07 0.40  B 1.6 0.0 2 0.80 0.00  B1 12.4 8.2 9 1.38 0.91  B2 7.6 3.0 6 1.27 0.50  B3 10.8 7.6 6 1.80 1.27  B4 3.2 0.2 4 0.80 0.05Table 7
Interunit degree.
Unit Formal Interunit Degree Informal Interunit Degree Number of Individuals Average Formal Interunit Degree Average Informal Interunit Degree A 11.0 1.1 2 5.50 0.55  A1 4.0 4.8 4 1.00 1.20  A2 4.6 7.9 4 1.15 1.97  A3 3.6 4.4 12 0.30 0.37  A4 6.0 3.8 6 1.00 0.63  B 6.6 1.0 2 3.30 0.50  B1 2.0 6.6 9 0.22 0.73  B2 1.2 3.5 6 0.20 0.58  B3 3.6 7.1 6 0.60 1.18  B4 1.0 0.4 4 0.25 0.10Table 8
Intraorganisational degree.
Org. Formal Organisational Degree Informal Organisational Degree Number of Individuals Average Formal Organisational Degree Average Informal Organisational Degree A 72.8 58.6 28 2.60 2.09  B 50.0 37.6 27 1.85 1.39Table 9
Interorganisational degree.
Interorg. network Formal Interorg. Degree Informal Interorg. Degree Number of Individuals Average Formal Interorg. Degree Average Informal Interorg. Degree m 0.0 2.7 54 0.00 0.05Table 10
Summary of interdependence degrees values at multiple levels.
Level Interdependence Degree N SD Mean Max Min Individual Formal individual 54 2.3 2.4 9.4 0.2 Informal individual 54 1.7 1.9 6.6 0.0  Intraunit Formal intraunit 10 6.8 7.9 25.2 0.4 Informal intraunit 10 7.8 5.6 27.4 0.0 Average formal intraunit 10 0.5 1.2 2.1 0.2 Average informal intraunit 10 0.7 0.7 2.3 0.0  Interunit Formal interunit 10 2.8 4.4 11.0 1.0 Informal interunit 10 2.5 4.1 7.9 0.4 Average formal interunit 10 1.6 1.4 5.5 0.2 Average informal interunit 10 0.5 0.8 2.0 0.1  Intraorg. Formal intraorganisational 2 11.4 61.4 72.8 50.0 Informal intraorganisational 2 10.5 48.1 58.6 37.7 Average formal intraorganisational 2 0.4 2.2 2.6 1.9 Average informal intraorganisational 2 0.4 1.7 2.1 1.4  Interorg. Formal interorganisational 1 0.0 0.0 0.0 0.0 Informal interorganisational 1 0.0 2.7 2.7 2.7 Average formal organisational degree 1 0.0 0.0 0.0 0.0 Average informal organisational degree 1 0.0 0.05 0.05 0.05The multilevel interdependence degrees are network centrality measures, i.e., rankings, which can be investigated by the Spearman’s rank correlation coefficientρ and the Kendall’s rank correlation coefficient τ [93, 94]. Using the correlation coefficients we examined if higher degrees contribute to the better performance, that is, if there is a negative relation between formal and informal degrees and time to complete a task at multiple levels (Table 11 and Figure 7). We adjust the significance values applying the sequential Bonferroni method [88] to avoid inflated risk of Type 1 error related to multiple comparisons. Subplots of Figure 7 are attached in the Supplementary Material for comprehensive image analysis. As each agent had a task that required different number of resources, each value of time to complete the task was normalized by number of demanded resources. In normal and random disruption conditions, there was a significant negative correlation between informal individual degree and time to complete a task; i.e., the higher informal individual degrees (Figure 7(a)) the faster task completion (ρ = -.35, ρ = -.33). Also, agents with higher formal individual degree tended to complete tasks faster in targeted disruption (ρ = -.26).Table 11
Correlation between interdependence degrees and time to complete task.
Degree Time to complete task(s) Normal condition Random disruption Targeted disruption Formal individual -.024 (.000) -.098 (-.055) -.262∗ (-.193∗) Informal individual -.351 ∗ ∗ (-.252 ∗ ∗) -.325 ∗ (-.239∗) -.174 (-.134) Average formal intraunit .058 (.068) -.095 (-.023) -.442 (-.295) Average informal intraunit .073 (.091) -.085 (.000) -.427 (-.273) Average formal interunit -.537 (-.432∗) -.463 (-.341) -. 427 (-.250) Average informal interunit .450 (.270) .316 (.180) -. 377 (-.270)  Spearman’sρ (Kendall’s τ) Bold correlation values are significant atα level corrected by sequential Bonferroni method [88]. ∗ p < 0.001 level (1-tailed); ∗p < 0.05 level (1-tailed).Figure 7
Relation between average interunit level degrees and task performance in normal and disrupted conditions.
(a)
Individual level degrees (b)
Average intraunit level degrees (c)
Average interunit level degreesAt the intraunit level (Figure7(b)), while in normal condition and random disruption there was no correlation between the formal and the informal intraunit degrees and time, in the targeted disruption both average formal and informal intraunit degrees were moderately correlated with better performance (ρ = -.44, ρ = -.43). At the interunit level (Figure 7(c)), in all conditions, units with higher average formal interunit degree performed better, i.e., needed shorter time to complete tasks (ρ = -.54, ρ = -.46, and ρ = -.43) than units with lower degrees. At the intraorganisational level (Table 12), Organisation A with both higher average formal and informal intraorganisational degrees performed better than Organisation B, which had lower degrees. As the model concerns the only one interorganisational network, the relations between interorganisational degrees and performance were not possible to assess.Table 12
Relation between average intraorganisational level degrees and task performance in normal and disrupted conditions.
Org. Average formal intraorganisational degree Average informal intraorganisational degree Time to complete task(s) Normal condition Random disruption Targeted disruption A 2.60 2.09 126 118 122 B 1.85 1.36 145 134 157There are three edge levels, which contain unique information about the organisational interdependencies, i.e., intraunit, interunit, and interorganisational. We examined their usage to assess which of them contribute the most to the organisational performance (Figure8). In total, most of the organisational exchanges were facilitated through the intraunit (on average 62%) and interunit (on average 34%) edges. The interorganisational edges constituted only 4% of the total number of used edges. While the usage of inter- and intraunit edges was stable in the normal and disrupted conditions, the usage of interorganisational edges decreased in the targeted disruption. This was due to the small number of interorganisational edges and low redundancy, which made this level prone to the disturbances caused in the targeted disruption.Figure 8
Levels of edges used in normal and disrupted conditions.
(a)
Usage of edge levels over simulation time (b)
Usage of edge levels in total


## 4.1. Materials and Methods

We used the agent-based method and built a model of organisational interdependencies, which comprised formal and informal relations within and between two organisations. The model of interdependencies was constructed from the sociometric data gathered in August 2017. The data was collected at individual level (N=54) by roster questionnaires that were distributed in two collaborating organisations, which operate within the security services sector in Southeast Asia. Organisation A is a research, training, and operational support centre. Organisation B is an operational support centre to enhance well-being and operational effectiveness of another organisation’s employee. Each of the organisations comprised 5 work units. The data described six multiplex social networks: reporting relationships, authority, acquaintance, friendship, familiarity, and problem-solving (Table1).Table 1
Networks data used in the model.
Social Network Questions Relationships Application in model ( 1 ) With whom do you like to discuss your daily work? [35] Acquaintance(instrumental exchange) ( 2 ) With whom do you talk about your private affairs during your daily chats? [44] Friendship(expressive exchange) Informal relations (trust-based) ( 3 ) Suppose that your colleague asks you to help his/her friend. Whose friends would you help? (adapted from Luo [35]) Familiar(instrumental and expressive exchange) ( 4 ) To whom do you report about you work progress? Reporting(instrumental exchange) Formal relations ( 5 ) Who is your direct supervisor? Authority(instrumental exchange) ( 6 ) Whom do you ask for help when you encounter a work-related problem, for which you couldn’t find a solution yourself? Problem-solving(instrumental exchange) Task demandFollowing the work of Luo [35], Haythornthwaite [36], Luo and Cheng [44], Krackhardt and Hanson [45], and Soda and Zaheer [92], the formal relations in the model consisted from reporting and authority networks and the informal interactions comprised acquaintance, friendship and familiar ties. For informal relations only mutual links (links that were confirmed by both connected agents) were considered as reliable and included in the model. The multiplex model was specified by the vector of the symmetric adjacency matrices (undirected graph) of formal and informal relations: A=A[F],A[I]. Each of the matrices was constructed through the aggregation procedure that resulted in multiplex edge types, which allowed us to specify the simulation parameters (Appendix A). The pairs of agents are connected by either (1) both formal and informal edges, or (2) a formal edge, or (3) an informal edge (Figure 3).Figure 3
Multiplex model of organisational interdependencies. Nodes’ colours represent work units. Node size depicts node strength. Blue and green edges are formal and informal relations. Individual no. 4 works in both organisations.The strength of social exchanges was reflected in the weight values given to each type of formal and informal edges. Qualitative and quantitative characteristics of the edges are described in Table2. The formal and informal weight values range from 0 to 1. The weights of informal edges are based on the system of social relationships introduced by Luo [35] that governs complex social transactions dependently on proportions of instrumental and expressive exchange dimensions. According to Luo [35], Luo and Cheng [44] the strongest and most efficient resource exchanges are facilitated through familiarity ties (‘rules of favour’), succeeded by friendship (‘rules of need’), and acquaintance (‘rules of equity’). The weights of formal edges are aligned with the hierarchical logic of formal employment relationships reflected in an organisational chart, which determine chains of command (authority) and formal information exchanges (reporting) [36]. In this light, the authority relations, representing formal vertical relations, are defined as the strongest, and are followed by reporting relationships that include both formal horizontal and vertical relations. The weights of aggregated formal and informal links have been sums of single network weight values (e.g. an edge representing single reporting and authority has a weight equal to 0.8 which is a sum of 0.2 and 0.6).Table 2
Quantitative and qualitative characteristics of existing relations.
Structure type Edge types Weight Social exchange strength Qualitative characteristics Formal One way reporting 0.2 Weak (i) Weak instrumental exchange(ii) Limited reliability (iii) No expectation of reciprocation (iv) Occasional exchanges Mutual reporting 0.4 Moderate (i) Moderate instrumental exchange(ii) Moderate reliability (iii) Expectation of reciprocation (iv) Occasional exchanges Authority 0.6 Strong (i) Strong instrumental exchange(ii) High reliability (iii) No expectation of reciprocation (iv) Frequent exchanges Authority, One way reporting 0.8 Very strong (i) Strong instrumental exchange(ii) Very high reliability (iii) No expectation of reciprocation (iv) Very frequent exchanges Authority,Mutual reporting 1.0 Extremely strong (i) Strong instrumental exchange(ii) Extremely high reliability (iii) Expectation of reciprocation (iv) Very frequent exchanges  Informal Acquaintance 0.1 Weak (i) Weak instrumental and weak expressive exchange(ii) Moderate level of trust (iii) Rules of fair exchange (iv) Expectation of instant reciprocation (v) Limited reliability, often insufficient in obtaining valuable resources (vi) Occasional exchanges Friendship 0.3 Moderate (i) Weak instrumental and strong expressive exchange(ii) High level of trust (iii) Rules of need (iv) Expectation for reciprocation, but not instant (v) High reliability (vi) Long-term, occasional exchanges (ad hoc when needed) Friendship, Acquaintance 0.4 Strong (i) Moderate instrumental and strong expressive exchange(ii) High level of trust (iii) Expectation of reciprocation, but not instant (iv) High reliability (v) Long-term, occasional exchanges Familiarity 0.6 Very strong (i) Strong instrumental and moderate expressive exchange(ii) High level of trust (iii) Rules of favour exchange (iv) Expectation of reciprocation, but not instant (v) Very high reliability (vi) Long-term and frequent exchanges (vii) Strong enough to be a bridge to connect to other agents Familiarity, Acquaintance, Friendship >= 0.7 Extremely strong (i) Strong instrumental and medium or strong expressive exchange(ii) High level of trust (iii) Expectation of reciprocation, but not instant (iv) Extremely high reliability (v) Long-term and very frequent exchanges (vi) Strong enough to be a bridge to connect to other agents


### 4.1.1. Simulations

To investigate how structure of organisational interdependencies affects organisational resilience, we used the model to conduct series of simulations both in normal and disrupted conditions. The simulations highlight two important aspects from the organisational systems functioning, i.e., task interdependence (agent must cooperate with other agents to complete a task) and adaptation through collective problem solving (the agent support each other directly and indirectly to solve a problem). In all simulations, more than 80% of all agents are given tasks to complete. All tasks are specified bytasks demand[T_D], which is a list of resources that agent needs to gather to complete the task. The task demand was generated from the directed problem-solving network (Table 1) through selecting resources that belonged to the agents connected by out-going edges (i.e., agents that an agent would contact if a work-related problem occurs). If the number of resources was more than five, only five resources were randomly selected, so the task demand lists ranged from one to maximum five resources. The task is completed when 100% of resources listed on the task demand are gathered (Algorithm 1). Each agent has a resource of his own (e.g., agent J has a resource J), and each network edge possesses a weight value that reflects strength of the relationship, as described in Table 2. The strength of relationship is reflected also in the parameter of speed and, thus, transfer time (Appendix A, Table 3). The transfer time needed to connect and pass the resource between the agents is depended on the edge type. For each agent’s step, the transfer time is calculated by subtracting the sum of formal and informal edges’ speed values from 100. In each simulation we calculate the total time based on all transfer times needed to accomplish tasks to measure performance of agents, units, organisations, and interorganisational network. In the resources search, the agent use both formal and informal structure in the maximum distance of two edges. Based on social exchange theory, the agent will choose always the fastest way. To ensure the realism of simulations, the speed priority is given to all formal types of links (the most time efficient), so the agent will choose and informal edge only if there is no formal connection. In line with work of Luo [35], the 2-step resource sharing is only possible when two conditions are fulfilled: (1) the first step edge is at least moderately strong informally (i.e., follows the rules of need or favour) or weak informally but the condition of instant reciprocation is fulfilled (rules of equity); and (2) the second step edge contains the familiarity component (rules of favour). In this case, the total transfer time is sum of all steps needed to share the resource (see more details in Algorithm 1 and examples in Appendix B).Table 3
Simulation parameters.
Relations Aggregated edge type Contained network edges No. of existing multiplex links Weight Speed Share of other agents resources in disruption condition (directional) Contribution to 2-step resource sharing Formal Rep_one_way One way reporting 55 0.2 40 0.4 . Rep_mutual Mutual reporting 3 0.4 45 0.4 . Superv Authority 6 0.6 50 . . Superv_rep_ one_way Authority,One way reporting 52 0.8 55 0.4 . Superv_rep_mut Authority,Mutual reporting 4 1.0 60 0.4 .  Informal Acq Acquaintance 15 0.1 5 0.2 Step1 + T_D Friend Friendship 12 0.3 10 . Step1 Acq_friend Friendship, Acquaintance 17 0.4 15 0.2 Step1 Famil Familiarity 28 0.6 20 . Steps1 and 2 Acq_famil Familiarity, Acquaintance 6 0.7 25 0.2 Steps1 and 2 Friend_famil Familiarity, Friendship 11 0.9 30 . Steps1 and 2 Acq_friend_famil Familiarity, Friendship, Acquaintance 8 1.0 35 0.2 Steps1 and 2Algorithm 1:Task completion and resource sharing.
Input: tasks, other_agents, networks, disrupted_agents Output: agent_task_completion_time t =0 ( 1 ) for task in  tasks  do ( 2 )  for  other_agent in  other_agents  do ( 3 )  if  task = other_agent[resource] then ( 4 )  if  other_agent not in  disrupted_list  then ( 5 )  task = 100% ( 6 )  t+=  transfer_time(other_agent,agent) ( 7 )  else ( 8 )  for  subs_agent in  other_agent[shared_resources]  do ( 9 )  if  subs_agent not in  disrupted_list  then ( 10 )  if  distance(subs_agent, agent)= 1  then▹1 step ( 11 )  agent[task]+ = subs_agent[subs_resources] ( 12 )  t+ = transfer_time(subs_agent, agent) ( 13 )  else if  distance(subs_agent, agent)= 2  then▹2 step ( 14 )  if  link(1) and link(2) fulfill the 2-step rule  then ( 15 )  agent[task]+ = subs_agent[subs_resources] ( 16 )  t+ = transfer_time(subs_agent, agent) ( 17 ) if  all tasks≥ 100  then ( 18 )  return t ( 19 )   else ( 20 )  return  nanOn the basis of the task demand lists, we create two types ofmajor work disruptions:(1) targeted disruption (unavailability) of agents the most needed (i.e. agents with highest in-degree centrality in each unit; for bigger units at least 20% of agents were disrupted) to complete the task demands (N=9, 20% of agents with tasks) and (2) random disruption of agents, which had a resource needed to complete at least one task demand (N=9, the same number of agents as in targeted disruptions). While in normal conditions, an agent, to complete his task, looks for a resource directly from the original resource (an agent who owns it); in disrupted conditions, the agents are allowed to use the substituting resources if the original resource is not available anymore (owning agent disrupted) (see Algorithm 1). The substituting resource is conditioned by existence of the edge type that comprises exchange of work-related information, i.e., reporting (formal relations) and acquaintance (informal relations). To create substituting resources we considered directed networks of reporting and acquaintance (only mutual links), in which incoming edges granted an agent 40% and 20% of substituting resource respectively. Thus, in case of simultaneous reporting and acquaintance incoming links, the maximum value of substituting resource in question is 60% (Appendix A, Table 3). When we disrupt the organisational system, to complete his task, an agent first checks his available substituting resources and next looks for other agents who can substitute the needed resource to reach the needed amount (100%).


## 4.1.1. Simulations

To investigate how structure of organisational interdependencies affects organisational resilience, we used the model to conduct series of simulations both in normal and disrupted conditions. The simulations highlight two important aspects from the organisational systems functioning, i.e., task interdependence (agent must cooperate with other agents to complete a task) and adaptation through collective problem solving (the agent support each other directly and indirectly to solve a problem). In all simulations, more than 80% of all agents are given tasks to complete. All tasks are specified bytasks demand[T_D], which is a list of resources that agent needs to gather to complete the task. The task demand was generated from the directed problem-solving network (Table 1) through selecting resources that belonged to the agents connected by out-going edges (i.e., agents that an agent would contact if a work-related problem occurs). If the number of resources was more than five, only five resources were randomly selected, so the task demand lists ranged from one to maximum five resources. The task is completed when 100% of resources listed on the task demand are gathered (Algorithm 1). Each agent has a resource of his own (e.g., agent J has a resource J), and each network edge possesses a weight value that reflects strength of the relationship, as described in Table 2. The strength of relationship is reflected also in the parameter of speed and, thus, transfer time (Appendix A, Table 3). The transfer time needed to connect and pass the resource between the agents is depended on the edge type. For each agent’s step, the transfer time is calculated by subtracting the sum of formal and informal edges’ speed values from 100. In each simulation we calculate the total time based on all transfer times needed to accomplish tasks to measure performance of agents, units, organisations, and interorganisational network. In the resources search, the agent use both formal and informal structure in the maximum distance of two edges. Based on social exchange theory, the agent will choose always the fastest way. To ensure the realism of simulations, the speed priority is given to all formal types of links (the most time efficient), so the agent will choose and informal edge only if there is no formal connection. In line with work of Luo [35], the 2-step resource sharing is only possible when two conditions are fulfilled: (1) the first step edge is at least moderately strong informally (i.e., follows the rules of need or favour) or weak informally but the condition of instant reciprocation is fulfilled (rules of equity); and (2) the second step edge contains the familiarity component (rules of favour). In this case, the total transfer time is sum of all steps needed to share the resource (see more details in Algorithm 1 and examples in Appendix B).Table 3
Simulation parameters.
Relations Aggregated edge type Contained network edges No. of existing multiplex links Weight Speed Share of other agents resources in disruption condition (directional) Contribution to 2-step resource sharing Formal Rep_one_way One way reporting 55 0.2 40 0.4 . Rep_mutual Mutual reporting 3 0.4 45 0.4 . Superv Authority 6 0.6 50 . . Superv_rep_ one_way Authority,One way reporting 52 0.8 55 0.4 . Superv_rep_mut Authority,Mutual reporting 4 1.0 60 0.4 .  Informal Acq Acquaintance 15 0.1 5 0.2 Step1 + T_D Friend Friendship 12 0.3 10 . Step1 Acq_friend Friendship, Acquaintance 17 0.4 15 0.2 Step1 Famil Familiarity 28 0.6 20 . Steps1 and 2 Acq_famil Familiarity, Acquaintance 6 0.7 25 0.2 Steps1 and 2 Friend_famil Familiarity, Friendship 11 0.9 30 . Steps1 and 2 Acq_friend_famil Familiarity, Friendship, Acquaintance 8 1.0 35 0.2 Steps1 and 2Algorithm 1:Task completion and resource sharing.
Input: tasks, other_agents, networks, disrupted_agents Output: agent_task_completion_time t =0 ( 1 ) for task in  tasks  do ( 2 )  for  other_agent in  other_agents  do ( 3 )  if  task = other_agent[resource] then ( 4 )  if  other_agent not in  disrupted_list  then ( 5 )  task = 100% ( 6 )  t+=  transfer_time(other_agent,agent) ( 7 )  else ( 8 )  for  subs_agent in  other_agent[shared_resources]  do ( 9 )  if  subs_agent not in  disrupted_list  then ( 10 )  if  distance(subs_agent, agent)= 1  then▹1 step ( 11 )  agent[task]+ = subs_agent[subs_resources] ( 12 )  t+ = transfer_time(subs_agent, agent) ( 13 )  else if  distance(subs_agent, agent)= 2  then▹2 step ( 14 )  if  link(1) and link(2) fulfill the 2-step rule  then ( 15 )  agent[task]+ = subs_agent[subs_resources] ( 16 )  t+ = transfer_time(subs_agent, agent) ( 17 ) if  all tasks≥ 100  then ( 18 )  return t ( 19 )   else ( 20 )  return  nanOn the basis of the task demand lists, we create two types ofmajor work disruptions:(1) targeted disruption (unavailability) of agents the most needed (i.e. agents with highest in-degree centrality in each unit; for bigger units at least 20% of agents were disrupted) to complete the task demands (N=9, 20% of agents with tasks) and (2) random disruption of agents, which had a resource needed to complete at least one task demand (N=9, the same number of agents as in targeted disruptions). While in normal conditions, an agent, to complete his task, looks for a resource directly from the original resource (an agent who owns it); in disrupted conditions, the agents are allowed to use the substituting resources if the original resource is not available anymore (owning agent disrupted) (see Algorithm 1). The substituting resource is conditioned by existence of the edge type that comprises exchange of work-related information, i.e., reporting (formal relations) and acquaintance (informal relations). To create substituting resources we considered directed networks of reporting and acquaintance (only mutual links), in which incoming edges granted an agent 40% and 20% of substituting resource respectively. Thus, in case of simultaneous reporting and acquaintance incoming links, the maximum value of substituting resource in question is 60% (Appendix A, Table 3). When we disrupt the organisational system, to complete his task, an agent first checks his available substituting resources and next looks for other agents who can substitute the needed resource to reach the needed amount (100%).


## 4.2. Results

To answer what type of structure makes an organisational system more resilient, we examined the importance of formal and informal relations in determining the system behaviour under normal and disrupted conditions (Figure4). We conduct simulations of systems including (1) only formal relations (blue line), (2) only informal relations (green line), and (3) both formal and informal relations, i.e., the overall network (red line). Figure 4 presents average values from simulations of normal conditions (n=1000), random disruptions (n=3000), and targeted disruptions (n=1000). We measure the system performance by number of completed tasks and time to complete the tasks. The simulations results showcase that organisational system with the overall structure (including formal and informal relations) performs better than the system, which has only formal structure or only informal structure. Importantly, the overall structure system performance is not a sum of formal and informal structures’ performance. In this way, the overall system performance is not directly related to the number of existing links but rather their strength (relations quality) and other structural properties. Furthermore, it is crucial to acknowledge that in all conditions the formal structure plays a significant role in system performance and the organisational system that possesses only informal structure cannot perform well. In normal condition, all tasks are completed in both systems, but the system with overall structure completes the tasks faster. In case of both disruptions, the performance of both systems drops, as some tasks cannot be completed. However, it is very important to highlight that during the disturbed conditions, informal structure complements the formal structure (by 8% and 15% in random and targeted disruptions respectively). More specifically, the informal structure supplements the unavailable formal connections, and this results in higher number of completed tasks as well as faster completion of the tasks. In sum, in general terms, the organisational system with both formal and informal relations performs better under normal and disrupted conditions, thus is more resilient.Figure 4
Performance of organisational structures in normal and disrupted conditions.In addition, to rule out the factor of different links number and ensure the correct interpretation of the results, we have sampled formal and informal structures to have the same number of edges (precisely 90). In this way, we constructed the new overall structure that comprised 90 formal and 90 informal edges. We revised the task demand lists in accordance with the agents existing in the new overall structure. The new structures were sampled five times and simulations for the three conditions were repeated (n=1000, n=3000, and n=1000). The average sampling results are presented in Figure5. In general, the sampled results are consistent with the non-sampled (Figure 4). However, due to the executed changes that resulted in less efficient topology of new structures (smaller density, smaller transitivity, and longer average path; see Table 4), we reported performance drop of sampled formal and informal structures in normal condition and random disruption. The performance drop was bigger in case of formal than informal structure, especially in normal condition. That was due to higher deteriorative discrepancy between the sampled and the nonsampled formal structure in comparison to the sampled and non-sampled informal structure. In particular, the biggest disadvantages sampled formal structure concerned smaller transitivity, longer average path, and longer diameter, which highly influenced structure dynamics (see Table 4).Table 4
Properties of nonsampled and sampled structures.
Structure No. nodes No. edges Density Transitivity Average path length Diameter Non-sampled overall 54 217 0.15 0.44 2.66 2.40 Non-sampled formal 54 120 0.08 0.33 2.94 3.00 Non-sampled informal 43 97 0.11 0.43 3.40 3.90 Average sampled overall 53 180 0.13 0.38 2.81 2.66 Average sampled formal 52 90 0.07 0.24 3.31 3.76 Average sampled informal 42 90 0.1 0.39 3.42 3.76Figure 5
Average performance of sampled organisational structures in normal and disrupted conditions.In relation to the formal and informal structure, we investigated which types of edges facilitated most of exchanges in normal and disrupted conditions structure contributes the most to the organisational performance (Figure6). The usage of all types was relatively stable in all conditions. In total, most of exchanges were facilitated by both formal and informal edges (on average 46%) or only by formal edges (on average 43%). Approximately 11% of exchanges were conducted by informal edges. Similar pattern was observed over the simulation time. These results highlight with more details and reiterate the important contribution of the informal structure to the organisational system performance in normal and disrupted conditions.Figure 6
Types of edges used in normal and disrupted conditions.
(a)
Used edges by structure type over simulation time (b)
Used edges by structure type in totalNext, we examined the values of the formal and informal interdependencies degrees at multiple levels (AppendixC, Tables 5–9). Table 10 presents the summarised results of the degree analysis. The higher levels of the analysis, the higher were the values of non-averaged degrees due to the increasing number of considered network elements and their complex nature. The nonaveraged values of individual, intraunit, interunit, intraorganisational, and interorganisational degrees ranged from 0 to 72.8, and the averaged degrees ranged from 0 to 5.5. While both formal and informal individual, interunit, and intraorganisational degree values had moderate dispersion, the informal intraunit (SD = 7.8, mean = 5.3) and average formal interunit values (SD = 1.6, mean = 1.4) were characterised by high dispersion; that is, the degree values were very widely distributed. As data concern the relations within one interorganisational network, the variation at the interorganisational-level interdependencies was not possible to assess.Table 5
Individual degree.
Agent Id Org. Unit Formal Individual Degree Informal Individual Degree 1 A A3 1.4 4.2  2 B B1 1.2 0.1  3 B B 4.6 0.0  4 A/B A/B 9.4 2.1  5 A A1 1.8 2.4  6 A A3 1.2 2.2  7 A A4 3.6 1.3  8 B B4 1.2 0.1  9 A A1 1.0 2.8  10 B B2 3.8 0.0  11 A A4 2.2 0.0  12 B B3 1.6 1.7  13 B B1 0.8 0.0  14 A A3 8.0 4.9  15 A A3 1.0 0.8  16 B B1 1.0 3.9  17 A A1 6.4 1.2  18 B B1 0.8 2.9  19 A A2 1.2 5.6  20 A A3 4.8 0.1  21 A A2 1.8 0.9  22 B B1 1.2 4.2  23 A A 5.6 0.0  24 B B 0.4 0.0  25 A A3 0.8 3.6  26 A A3 0.8 1.5  28 A A3 1.2 4.2  29 A A3 1.8 2.6  30 A A4 0.8 0.4  31 B B2 1.0 2.4  32 B B1 1.0 0.2  33 B B3 6.0 1.1  34 A A2 5.2 2.0  35 A A3 1.2 4.2  36 A A3 5.4 4.3  37 A A4 1.0 0.8  38 B B3 1.8 6.6  39 B B2 0.2 0.0  40 A A3 1.2 0.4  41 A A4 2.2 0.8  42 B B2 0.8 0.0  43 B B3 1.4 1.1  44 B B3 1.4 3.1  45 B B1 1.6 3.5  46 B B1 6.4 0.0  47 B B1 0.4 0.0  48 A A1 1.6 3.4  49 A A4 2.6 2.9  50 B B 0.4 0.0  51 B B2 1.6 2.3  52 B B2 1.4 3.0  54 A A2 1.2 2.7  55 B B4 2.2 0.5  57 B B3 2.2 2.6Table 6
Intraunit degree.
Unit Formal Intraunit Degree Informal Intraunit Degree Number of Individuals Average Formal Intraunit Degree Average Informal Intraunit Degree A 0.4 0.0 2 0.20 0.00  A1 6.8 5.0 4 1.70 1.25  A2 4.8 1.8 4 1.20 0.45  A3 25.2 27.4 12 2.10 2.28  A4 6.4 2.4 6 1.07 0.40  B 1.6 0.0 2 0.80 0.00  B1 12.4 8.2 9 1.38 0.91  B2 7.6 3.0 6 1.27 0.50  B3 10.8 7.6 6 1.80 1.27  B4 3.2 0.2 4 0.80 0.05Table 7
Interunit degree.
Unit Formal Interunit Degree Informal Interunit Degree Number of Individuals Average Formal Interunit Degree Average Informal Interunit Degree A 11.0 1.1 2 5.50 0.55  A1 4.0 4.8 4 1.00 1.20  A2 4.6 7.9 4 1.15 1.97  A3 3.6 4.4 12 0.30 0.37  A4 6.0 3.8 6 1.00 0.63  B 6.6 1.0 2 3.30 0.50  B1 2.0 6.6 9 0.22 0.73  B2 1.2 3.5 6 0.20 0.58  B3 3.6 7.1 6 0.60 1.18  B4 1.0 0.4 4 0.25 0.10Table 8
Intraorganisational degree.
Org. Formal Organisational Degree Informal Organisational Degree Number of Individuals Average Formal Organisational Degree Average Informal Organisational Degree A 72.8 58.6 28 2.60 2.09  B 50.0 37.6 27 1.85 1.39Table 9
Interorganisational degree.
Interorg. network Formal Interorg. Degree Informal Interorg. Degree Number of Individuals Average Formal Interorg. Degree Average Informal Interorg. Degree m 0.0 2.7 54 0.00 0.05Table 10
Summary of interdependence degrees values at multiple levels.
Level Interdependence Degree N SD Mean Max Min Individual Formal individual 54 2.3 2.4 9.4 0.2 Informal individual 54 1.7 1.9 6.6 0.0  Intraunit Formal intraunit 10 6.8 7.9 25.2 0.4 Informal intraunit 10 7.8 5.6 27.4 0.0 Average formal intraunit 10 0.5 1.2 2.1 0.2 Average informal intraunit 10 0.7 0.7 2.3 0.0  Interunit Formal interunit 10 2.8 4.4 11.0 1.0 Informal interunit 10 2.5 4.1 7.9 0.4 Average formal interunit 10 1.6 1.4 5.5 0.2 Average informal interunit 10 0.5 0.8 2.0 0.1  Intraorg. Formal intraorganisational 2 11.4 61.4 72.8 50.0 Informal intraorganisational 2 10.5 48.1 58.6 37.7 Average formal intraorganisational 2 0.4 2.2 2.6 1.9 Average informal intraorganisational 2 0.4 1.7 2.1 1.4  Interorg. Formal interorganisational 1 0.0 0.0 0.0 0.0 Informal interorganisational 1 0.0 2.7 2.7 2.7 Average formal organisational degree 1 0.0 0.0 0.0 0.0 Average informal organisational degree 1 0.0 0.05 0.05 0.05The multilevel interdependence degrees are network centrality measures, i.e., rankings, which can be investigated by the Spearman’s rank correlation coefficientρ and the Kendall’s rank correlation coefficient τ [93, 94]. Using the correlation coefficients we examined if higher degrees contribute to the better performance, that is, if there is a negative relation between formal and informal degrees and time to complete a task at multiple levels (Table 11 and Figure 7). We adjust the significance values applying the sequential Bonferroni method [88] to avoid inflated risk of Type 1 error related to multiple comparisons. Subplots of Figure 7 are attached in the Supplementary Material for comprehensive image analysis. As each agent had a task that required different number of resources, each value of time to complete the task was normalized by number of demanded resources. In normal and random disruption conditions, there was a significant negative correlation between informal individual degree and time to complete a task; i.e., the higher informal individual degrees (Figure 7(a)) the faster task completion (ρ = -.35, ρ = -.33). Also, agents with higher formal individual degree tended to complete tasks faster in targeted disruption (ρ = -.26).Table 11
Correlation between interdependence degrees and time to complete task.
Degree Time to complete task(s) Normal condition Random disruption Targeted disruption Formal individual -.024 (.000) -.098 (-.055) -.262∗ (-.193∗) Informal individual -.351 ∗ ∗ (-.252 ∗ ∗) -.325 ∗ (-.239∗) -.174 (-.134) Average formal intraunit .058 (.068) -.095 (-.023) -.442 (-.295) Average informal intraunit .073 (.091) -.085 (.000) -.427 (-.273) Average formal interunit -.537 (-.432∗) -.463 (-.341) -. 427 (-.250) Average informal interunit .450 (.270) .316 (.180) -. 377 (-.270)  Spearman’sρ (Kendall’s τ) Bold correlation values are significant atα level corrected by sequential Bonferroni method [88]. ∗ p < 0.001 level (1-tailed); ∗p < 0.05 level (1-tailed).Figure 7
Relation between average interunit level degrees and task performance in normal and disrupted conditions.
(a)
Individual level degrees (b)
Average intraunit level degrees (c)
Average interunit level degreesAt the intraunit level (Figure7(b)), while in normal condition and random disruption there was no correlation between the formal and the informal intraunit degrees and time, in the targeted disruption both average formal and informal intraunit degrees were moderately correlated with better performance (ρ = -.44, ρ = -.43). At the interunit level (Figure 7(c)), in all conditions, units with higher average formal interunit degree performed better, i.e., needed shorter time to complete tasks (ρ = -.54, ρ = -.46, and ρ = -.43) than units with lower degrees. At the intraorganisational level (Table 12), Organisation A with both higher average formal and informal intraorganisational degrees performed better than Organisation B, which had lower degrees. As the model concerns the only one interorganisational network, the relations between interorganisational degrees and performance were not possible to assess.Table 12
Relation between average intraorganisational level degrees and task performance in normal and disrupted conditions.
Org. Average formal intraorganisational degree Average informal intraorganisational degree Time to complete task(s) Normal condition Random disruption Targeted disruption A 2.60 2.09 126 118 122 B 1.85 1.36 145 134 157There are three edge levels, which contain unique information about the organisational interdependencies, i.e., intraunit, interunit, and interorganisational. We examined their usage to assess which of them contribute the most to the organisational performance (Figure8). In total, most of the organisational exchanges were facilitated through the intraunit (on average 62%) and interunit (on average 34%) edges. The interorganisational edges constituted only 4% of the total number of used edges. While the usage of inter- and intraunit edges was stable in the normal and disrupted conditions, the usage of interorganisational edges decreased in the targeted disruption. This was due to the small number of interorganisational edges and low redundancy, which made this level prone to the disturbances caused in the targeted disruption.Figure 8
Levels of edges used in normal and disrupted conditions.
(a)
Usage of edge levels over simulation time (b)
Usage of edge levels in total


## 5. Discussion and Implications

Organisational interdependence is a multidimensional construct that can be conceptualised at multiple levels. On one hand, the interdependencies make an organisational system more complex [30, 95]. On the other hand, the existence of multiplex relationships that comprise interdependencies is a natural feature of modern organisations [96, 97]. Because system’s safety is an emergent property [63] and partial analysis of the system can be misleading [84], it is crucial to have an in-depth understanding of the interdependencies’ dynamics that happen at multiple levels. The analysis of structural properties of organisational interdependencies helps to identify patterns and assess needs and opportunities both in normal and disrupted conditions. By better understanding multilevel interdependencies we can reduce vulnerability and increase the ability to withstand dynamic changes and, thus, enhance organisational resilience.Resilience is embedded in people’s behaviour, and it is built by proactive approach to mobilizing resources, abilities to respond and perform under a variety of conditions. In this study, we view organisational resilience as a systemic property, which requires management of relational dynamics at multiple organisational levels. For that reason, the concept is highly relevant to resilience analysis in the organisational context. Our multilevel conceptualisation of interdependencies considered two dimensions of organisational relations—formal and informal—and proposed measures to investigate their structure across levels. With well-mapped organisational interdependencies, we can examine how an organisational system behaves under normal and disturbed circumstances. The empirical study proved the structure of interdependencies influences the efficiency of organisational performance. In the model, in line with the state-of-art literature [7–9], we highlighted that task interdependence affects the dynamics and outcomes of organisational relationships. The results showed that an organisational system with a rich structure that combined both formal and informal relations performed better both in normal and disrupted conditions and thus was more resilient than the system based only on the formal relations. The formal structure appeared to meaningfully contribute to organisational performance. However, our results also underlined the importance of informal structure that substantially complements and substitutes the formal structure, especially in the disrupted conditions. We showcased that the trust-based relationships strongly affect agent’s decision-making and flow of network resources. The stronger informal relations, the faster transfer time and more resources are shared, as agents are more willing to provide assistance to each other. In addition, in most of simulated exchanges, the actors were connected by both formal and informal edge at the same time. These results confirm that well-established relationships (both formal and informal) within and between organisations condition organisational performance both in normal and disrupted conditions.The analysis of introduced multilevel interdependence degrees can determine the organisation’s ability to both work as expected, in normal conditions, as well as to create emergent response networks to confront the unexpected, mitigate consequences and adapt to the ‘new normal’. The degree of organisational interdependence is not uniformed; rather it changes depending on the analysed relationship level. Facing the unexpected, task execution very often has to be changed and improvised due to unavailability of the needed resources. Individuals, units or organisations that normally are loosely coupled can be tightly coupled during a period of disruption. The interactions’ changes happen at multiple levels; therefore, it is crucial to not only acknowledge the importance of individual interdependencies, but also higher interdependencies levels, such as intraunit, interunit, intraorganisational and interorganisational. Even though shaped by individuals’ interactions, the higher-level degrees have unique features and provide new information that is crucial for resilience analysis of an organisational system. At multiple levels, the results showed there is a negative relation between the interdependence degrees, the time needed to complete a task. This was the most evident at the individual, interunit and organisational levels. Most of the simulated exchanges were facilitated through intra- or interunit edges, which at the same time contribute the most to the overall system performance. The analysis of the multilevel interdependence degrees indicated also that individuals, units, and organisations are more structurally embedded in the organisational system and, thus, potentially can benefit the most to organisational performance in normal and disrupted conditions and at the same time contribute to organisational resilience.The results of this research imply that well-managed interdependencies are crucial to ensure resilient organisational performance. In the following lines, we present a few implications of this research. From our results, we can conclude that a resilient organisation should aim to have the dense and strong of organisational interdependencies, especially at intra- and interunit levels, which are constituted by the individuals’ exchanges within the organisation. Some of the practices to reach this goal could include(1) decentralising (flattening) the organisational structure, (2) changing the organisational structure to the matrix model, which imposes higher number of formal interactions between units, and (3) nourishing organisational culture and providing environment that will help to activate formulation of new informal relations within and between units as well as strengthen the existing ones (organising social events, retreats, and common spaces). Furthermore, both formal and informal organisational structure contribute to organisational performance during normal and disrupted situations; thus both of them should be part of in-depth analysis while establishing crisis management plans, procedures and practices. A resilient organisation should have balanced amount of formal and informal interdependencies, which can facilitate complex relational exchanges when the performance conditions are both certain and uncertain. The formal structure is very efficient in enabling the collaborative work; however, it is prone to the disruptions, and often it is the informal structure that supplements it during the disruptions. It is important to stay aware of the differences and advantages of the two types of relations. The analysis of informal structure can help to unmask concealed organisational patterns, needs, and opportunities. This information can be used to eliminate hidden organisational vulnerabilities and use overlooked potentials. In practice, the strengths of informal structure could be recognised by acknowledgment of the role of informal leaders in emergency preparedness activities (e.g., drills and exercises), emergency response plans, contingency plans, and the business continuity plan.This article aimed to introduce a new way of conceptualising organisational interdependencies and present its usefulness for resilience analysis. Our analysis had limited focus on measuring the direct effects or causality between organisational interdependencies and organisational performance in expected and unexpected conditions. Future research should take up the challenge to broaden analysis of the impact of organisational interdependencies on system resilience. The described application of the multilevel perspective on organisational interdependencies gave new insights into the structure of organisational interdependencies embedded in the specific context, which shapes the perceptions and behaviours of the involved actors. Future studies should consider cross-disciplinary analyses of the organisational interdependencies in other organisational contexts and larger samples in order to build up the comprehensive of how the interdependencies are shaped in different organisational systems and how they affect system performance. These future analyses, along with the preliminary results described in this paper, should be the insightful base of next practices for resilient design and management of organisational systems.


## 6. Conclusion

Organisations are complex interdependent systems, which require management efforts to remain resilient. As noted above, we argue for correlation between resilience and organisational interdependencies. Interdependencies in organisations have been studied from various perspectives; however, most of the analyses were conducted only at the single level and overlooked the multilevel character of interdependencies. In response to this gap, we proposed a multilevel approach to better comprehend the complexity of interdependencies in organisational systems. With this paper we contributed to the study of interdependencies and resilience by introducing a multilevel conceptualisation of interdependencies in organisational systems and presenting its application for resilience analysis. Adapting the most plausible definition of interdependencies as exchange relationships, our paper sheds light on how those relationships are shaped across multiple organisational levels and suggests how they could be decoupled in order to handle their complexity. We used the system and multilevel theories to explore five organisational levels, including individual, intraunit, interunit, intraorganisational, and interorganisational. We argued that interdependencies are a discontinuous phenomenon across levels that does not express uniform pattern. Accordingly, we employed the configural compilation approach to describe interdependencies’ features and proposed their measures at multiple levels. In addition to formal relationships, our conceptualisation underlines the significance of informal trust-based relationships, a notion that provides a new insight on the origins of interdependence. Furthermore, we applied the multilevel interdependencies conceptualisation into the analysis of organisational resilience and presented the relations between interdependencies and organisational performance at multiple levels in normal and disrupted conditions. Finally, we discussed how managing multilevel interdependencies is crucial to reduce vulnerability and to build up, maintain, and enhance resilience of organisational systems. By introducing the multilevel conceptualisation, we hope to pave a preliminary path to managing the complexity of the interdependencies in organisational systems. At the same time, we advance the analysis of the multilevel relationships between interdependencies and resilience as a promising step towards improving organisational design and resilience management.



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