Robustness of Organizational Network

A network is a set of items (nodes) connected by links(edges) . In case of social networks, the nodes are representation of actors while the edges represent relations between actors. The settings of network analysis can be a befitting tools for analysis of organizations.Peter Sherian Dodds et al. applied this network analysis settings to gauge the robustness of an organizational network.

Construction of  organizational network

In a business organization setup, the head of the company and immediate subordinate and superior of an employee is predefined beforehand. For instance, a company may have a director, two managers and two analysts in one department. The director would likely report to the Chief Executive Officer, or CEO, and both managers would report to the director. In addition, each manager may have an analyst reporting to them. In this setup the director is the immediate subordinate of the CEO; similarly, the managers are immediate subordinate of the director and the analysts are immediate subordinates of the mangers. Now to model the organizational network we represent each employee as a node and the relationships between employees as edges. The point to note is that  the immediate superior and subordinates connection,which are always by default connected, forms the hierarchical backbone of the network with some predefined  number of levels and branching ratio for each node.
Keeping this in mind the backbone of the network is framed with a given set of nodes-N, and the branching level-b for all but leaf nodes and forming the edge between the immediate superior and subordinate.Now on top of this hierarchical backbone of the network the existence of edges between two nodes which are not connected in the backbone structure is a stochastic process. 

Fig 1. Schematic of Organizational Network¹

In this framework the distance between two nodes is defined as the 
Hence probability on the existence of edges between two nodes is to be assigned; the probability that two nodes- i and j being connected is denoted by P (i, j).

This way the problem of organizational structure representation is converted into the defining the functional form of P (i, j) for all nodes i,j within the network.

To achieve the functional form of P (i, j) the following assumptions are adopted.

1. The probabilistic function P(i,j) is symmetric with respect to i and j.
2. Being other things equal same two persons who are more close to their common ancestors are more likely to form organizational relationship than persons being far away from their common ancestors. This assumption is in line with the assumption of homophily. So the definition of organizational distance x_i,j between two nodes i and j has been introduced as follows:        for d_i+d_j ≥2
   where di, dj are are distance of the nodes i,j from their common ancestors.
3. As immediate subordinates and superiors are connected by default, the edge between i & j is not stochastic  for d_i + d_j=1 .
4. As P (i, j) is not stochastic for d_i + d_j=1 we are only interested in functional form of P (i, j) for d_i + d_j ≥2
5. The assumption is being other things equal for two persons having common ancestor close to the root i.e. close to the top of the company  are more likely to form organizational relationship than two persons whose common ancestor is bottom down the organizational tree.
6. Hence the probability P (i, j) is dependent on D_i,j , the depth  of their common lowest ancestor and their own  depths d_i, d_j beneath their common ancestors
7. Now the probabilistic distribution P (i, j) is assumed to be decrease monotonically with both x_i,j and D_i,j. But these two factors will contribute differently to P (i, j). Hence two parameters ζ, λ are brought in to tune the effect of D_i,j ,x_i,j  

Incorporating these assumptions, the probabilistic function P (i, j) is characterized in the following manner: 

As the organization structure will be dependent on the values of ζ, λ let us have a look at the four organizational structure corresponding to the four set of marginal values of the parameter set (ζ, λ).

Random: This set of organizational structure is associated with the parameter set (ζ, λ) → (∞, ∞). As both ξ and λ possess very high value; the effect of x_i,j and D_i,j wane down and effectively the probability of  building relation between any two  persons in the organization is totally random and same for any two persons.

Local team: This set of organizational structure is associated with the parameter set (ζ, λ) → (0, ∞). In this case D_i,j has no effect on forming  links between two persons. But x_i,j limits the possibility of growing up links severely. In fact in this case for x_i,j >0, the probability of forming links between two persons is zero. So links are formed only if x_i,j=0 between two persons. Essentially two persons having same superior can only form team in this kind of organization.

Random interdivisional: This set of organizational structure is associated with the parameter set (ζ, λ)→(∞,0). As a result x_i,j has no power on the probability function, rather it is totally governed by Di,j. In this case links between two persons is formed only if their common superior is the top of the organization. In other words, all the members of a team belong to different divisions. The interesting part is the probability of forming link is same for persons belonging to any two different divisions.

Core-periphery:    This set of organizational structure is associated with the parameter set (ζ, λ)→(0,0). Here the probability function is constrained by both x_i,j and D_i,j. In this organization setup links are formed between persons who are immediate subordinate of the head of the organization. No culture of group formation exists between other employees of the organization. The organization expects the employee to work alone and all the employees report their work only to their superiors and communication is held only at the highest level of the tree. This is an example of pure branching hierarchical structure.

Now we consider the organizational structure for intermediate values of ζ and λ[(ζ, λ) ≈(0.5,0.5)]The probability function is dependent on both x_i,j and D_i,j and it decreases with increase in either of them. Hence groups are more likely to form between subordinate of same superior. But that’s not the only feature. The density of linkage ebbs away with the depth of the organization. Although this set of structure shows links across different divisions and different ranks in term of density of links it is much closer to the networks of core-periphery. This set of organizational network is termed as multiscale network.

Next two types of robustness of the organizational network is considered:

1. Congestion robustness: The robustness of the network against congestion of messages.

2.  Connectivity robustness: The robustness in connectivity of the network against failure of nodes.

Measures of Robustness:

Congestion robustness is associated with the probability that any given message will be processed by a given node. For node i, let us call this probability as ρ_i. The higher this probability the node the more likely to fail to process messages within a given time-frame. As different node in the organizational network will have different ρ_i,so the maximum of them i.e. ρ_max  over the entire organizational network is considered as  a measure

Connectivity robustness is measured by the metric C= S/(N-N_r) where S is the size of the largest connected component after removal of N_r nodes.

The robustness will not only be a function of network topology, but also a function of task environment. The task environment in case of information processing is measured by two factors:

1. Rate(μ): The average number of messages initiated by each node in unit time.
2. Distribution of messages(ξ): This parameter quantifies the distribution of the recipient of a message which initiated from a source node. Starting from a source node say, s, the desired recipient is assumed to be distributed in proportion to exp (-x/ ξ), where x is the organizational distance. ξ =0 implies the target node is within the unit organizational distance of the source node and for ξ →∞ the target is randomly and uniformly spread all over the network. This parameter ξ provides a good measure of the kind of task performed in the organization.when for most of the messages ξ →0 ,which corresponds to to the target node being within the same team ,implies the task are mostly locally dependent and highly decomposable.
   On the other hand, messages having high values of ξ signifies most of the tasks require involvement of different divisions  of  the organization meaning the tasks are not decomposable.

Analysis of Robustness

In this part the results of the author's findings is discussed.

Congestion Robustness:

Dependence on Network Topology
The author finds out that maximum congestion centrality  ρ_max is minimum for multiscale network.Hence multiscale network can be thought as most efficient in terms of congestional robustness.

Fig.2:ρ_max as a function of ζ & λ¹

Lighter regions correspond to lower values of 

Description of network:

random: ∇

local team:♢

inter-divisional:Δ

core-periphery:○

multi scale: □

Dependence on Network Density:

The author tested the congestional robustness for different networks as a function of m-the number of edges in the network

Fig.3: ρ_max as a function of network density¹

Description of network:

random: ∇

local team:♢

inter-divisional:Δ

core-periphery:○

multi scale: □

As the number edges increases  ρ_max eventually decreases for all types of networks. But the point to note is that for multiscale network the drop in occurs for relatively small value of m.And for core periphery network congestion centrality does not monotonically decrease, rather it exhibits oscillatory pattern with increase in m.

Dependency on Distribution of messages:

Now the congestion centrality is plotted against  ξ. 

Fig.4: ρ_max as a function of distribution of messages¹

Description of network:

random: ∇

local team:♢

inter-divisional:Δ

core-periphery:○

multi scale: □

For ξ →0 i.e. where the messages are distributed mostly locally, congestional centrality is low for all types of networks, signifying stronger congestion robustness.Although with increase in  ξ congestional centrality increases for each type of network, multiscale and coreperiphry network reveals better congestion centrality compared to other types  of networks.   

This result states that for organizations governed by local dependencies congestion centrality is low meaning better congestion robustness. On the other hand congestion robustness falls with increase in global dependencies and multiscale and core-periphery performs best in such environment.

Dependency on Network size:

Fig.5: ρ_maxas function of Network size¹

Description of network:

random: ∇

local team:♢

inter-divisional:Δ

core-periphery:○

multi scale: □

As the number of nodes increases congestion centrality decreases before achieving a constant value. The point to note multi-scale and core periphery network the centrality decreases a lot before attaining the saturated value,but other types of networks attain the saturated level far too quickly.

Core periphery and multiscale network are found to be most efficient in term of congestion robustness. But core periphery networks exhibit instability as shown in fig.3 and they are very sensitive to the choice of parameters making them unreliable in tackling congestion attack. 

Connectivity Robustness:

To measure the connectivity robustness certain nodes are removed and then the size of the largest component normalized by the size of the remaining network is considered as a measure. But the choice of order of removal of nodes is crucial. The authors found out the top-down approach of removal of nodes blows the most damaging impact on the connectivity robustness. Hence the performance under this top-down elimination of each type of network is measured. 

Fig.6: connectivity robustness as function of number of nodes removed¹

Description of network:

random: ∇

local team:♢

inter-divisional:Δ

core-periphery:○

multi scale: □

 As the result reveals, the random and inter-divisional networks are most robust in terms of connectivity robustness. Multiscale networks produces similar result as that of random network until the bottom ranks are removed.Local team network are the most vulnerable because in such organizational network the links between two employees are formed  only if they have common superior hence one person at the top makes the way for the the link between two division.So the  removal of one person at the top of the hierarchy causes reduces the size of the connected component significantly. connectivity of the network, 

Although the performance can be affected by the choice of removal strategy, all types of network performs better than the top-down removal. In all cases random network produces the superior result and ordering of the performance of different types of networks remains more or less same. 

The final conclusion is that multiscale networks are most robust in term of congestion robustness, and also produces satisfactory result in terms of connectivity robustness.

References:

1. "Information exchange and the robustness of organizational networks",Peter Sheridan Dodds, Duncan J. Watts, and Charles F. Sabel,Oct 14,2003,PNAS, vol.100, no. 21