6.1 The study and the data
The data came from an internal study of a government organisation, comprising 60 senior managers. Respondents were asked a number of name generator questions. Here we concentrate on work frequency (daily work ties or not), work importance (critical or not), and the formal reporting structure. Figure 6.1 depicts these networks separately. A simple inspection of the three networks does not provide much clarity of the effects in this data. We decided to fit a stochastic blockmodel (Nowicki & Snijders, 2001), and an exponential random graph model, and then compare these two results in a post-hoc analysis. Here we provide a brief summary of the results.
6.2 A stochastic blockmodel
The a priori stochastic blockmodel procedure is implemented in the StOCNET suite of statistical network programs (Boer, Huisman, Snijders & Zeggelink, 2003). The procedure
computes diagnostic statistics to suggest an optimal number of classes of actors. Inspection of these statistics led us to select a 3-class model. This resulted in 29 managers being placed in the first block, 19 in the second and 12 in the third. The three networks with nodes rearranged to reflect the block structure are shown in Figure 6.2.
Table 6.1 depicts the reduced blockmodel for the three networks, with the densities within and between each block presented as a ratio of the overall density of the specific network. Each cell of the table, then, is based on the number of actual ties as a proportion of the total number of possible ties (density) within and between blocks. Blocks in rows direct ties towards blocks in columns. These densities are then divided by the overall density of the particular network. So a final number below 1 indicates there are fewer ties between the respective blocks than would be expected if they were distributed equally across all blocks; and a number above 1 indicates there are more ties than would be expected. Broadly, a number well above 1 indicates that this is where we see a large proportion of the ties in this network. These are bolded in the Figure.
Block 1 Block 2 Block 3
Block 1 Reporting = .08 Critical = .57 Daily = .62 Reporting = 1.53 Critical = .75 Daily = .69 Reporting = 2.46 Critical = .40 Daily = .54 Block 2 Reporting = .15 Critical = .58 Daily = .81 Reporting = 1.15 Critical = 1.75 Daily = 2.32 Reporting = 4.08 Critical = 4.64 Daily = 2.72 Block 3 Reporting = .00 Critical = .00 Daily = .09 Reporting = .00 Critical = .40 Daily = .59 Reporting =2.92 Critical = 3.48 Daily = 3.29 Table 6.1
Reduced blockmodel for the three networks,
with densities presented as a proportion of overall density of the respective network. (bolded values indicate regions of high density.)
In all three networks we see a strongly hierarchical pattern. People in block 1 tend to report to people in blocks 2 and 3, while some people in block 2 report to others in block 2 but more report
to block 3. People in block 3 tend to report to others in block 3. We see that critical and daily exchanges tend to be reported in the main by blocks 2 and 3. While people in block 2 have some critical and daily exchanges with others in block 2 as well as with block 3, those in block 3 tend to see their critical and daily exchanges as occurring mainly with other block 3 people. The block model neatly picks out a hierarchy with block 3 at the apex, block 1 at the base, and block 2 somewhere in between.
a. Reporting network
b. Critical network
c. Daily network Figure 6.1
Multiple networks in an organisation
6.3 An exponential random graph model
We also fitted an exponential random graph model using the reporting network as an exogenous predictor. This was intended only as an exploratory analysis, in order to identify
interesting joint network Markov configurations, to investigate in conjunction with the blockmodel (see below). Accordingly, we used approximate pseudo-likelihood estimation procedures (Strauss & Ikeda, 1990; see Robins et al, 2006, for a discussion.). We found evidence for the following joint network effects:
• entraining of daily and reporting ties • entraining of daily and critical ties
a. Reporting network
b. Critical network
c. Daily network Figure 6.2
Multiple networks in an organisation, with nodes reflecting the three blocks. (Yellow nodes = block 1; Green nodes = block 2; Red nodes = block 3).
6.4 Combining the blockmodel and exponential random graph model
We examined the configurations identified in the exponential random graph model in conjunction with the blockmodel. We looked for the frequency of each configuration within and between blocks and compared that to the frequency we would expect if the blocks had no effect on the configuration (more specifically, if ties within and between blocks were randomly distributed, based on the blockmodel densities). Some of the larger effects were as follows:
Entrainment of ties
• Daily and reporting ties were most strongly entrained from block 2 to block 3: 32% of simultaneous daily and reporting ties occurred from block 2 to block 3 whereas
we would expect to see only 9% based on density alone. In other words, daily interactions most closely follow the hierarchy from block 2 to block 3.
• Critical and daily ties were also most strongly entrained from block 2 to block 3: 24% of critical and daily ties were aligned (expect 9%). So we might infer that critical work passed from block 2 to block 3 is most likely to be urgent work. Exchange
• Reciprocation of daily and critical ties happens most frequently in block 2: 55% of these types of exchange took place within block 2, when we expect only 11%. In other words, within block 2 a person who interacts frequently is likely to be recognised as a critical partner.
Triadic dependency
• Across all three blocks, and in ways consistent with the hierarchy among blocks, daily ties were frequently extended to both ends of a reporting relationship. So if someone interacted frequently with a person, they were also likely to interact frequently with that person’s boss. This suggests some level of teamwork across the hierarchy of blocks.
• Again across all three blocks and consistent with the hierarchy, people who report to the same person tend to interact daily.
• Within and between blocks 2 and 3, when two people interact frequently, others are likely to be critical partners of both of them.
• Critical pathways tend to be spanned by reporting relations. In other words, a first person will send on critical work to a second, who will then send it back to that person’s boss. Note that this effect occurs even though critical and reporting ties tend not to be entrained, nor to be exchanged. In other words, critical pathways are longer than the formal reporting structure of the organisation. This suggests that coordination is occurring and in a way that deviates from – but not too far from – the reporting structure.
6.5 Some general conclusions from the analysis
We can summarise these results in broad terms.
• Most ties (including multiple ties) are within blocks and up the hierarchy. People at the top do not “look down” much.
• Reciprocity and exchange occurs mainly within blocks (particularly in block 2). • Daily ties occur around the sharing of reporting relations and critical tasks. • Transfer of critical tasks among the individuals does not mirror the reporting
network but reporting relations tend to span the movement of critical tasks. In general, then, we might infer that this is an organisation which has a principal focus towards the top of the hierarchy, yet there is evidence of teamwork, cooperation and coordination across the hierarchy. But the functioning of work seems to be organised through frequent
interactions, rather than the formal reporting structure, although frequent ties are indeed related to the reporting structure. Critical work tasks are organised around these frequent interactions, and frequent interactions play an important role in traversing the hierarchy of the blocks. The flow of critical tasks tends to wrap around the formal reporting structure rather than directly mirror it, suggesting attention to coordination but at the cost of slower work flow.
It is risky to make sweeping generalisations based on this data alone. But, in view of the structural signature that this analysis suggests, there are some questions that this organisation might consider:
(a) How effective is the lower end of the hierarchy operating? Given that so much attention seems focused on the critical and urgent work at the top, what does the organisation actually know about the functioning of the more junior members of the hierarchy?
(b) Given the importance of frequent exchanges in structuring work in this organisation, what would happen in a new crisis involving even higher demands of urgency and criticality? How much redundancy does the organisation have to cope with a sudden increase in demands?
(c) Is the balance between the positives of increased coordination and the costs of slower work flow right?