Anthony Dekker (2007)
Studying Organisational Topology with Simple Computational Models
Journal of Artificial Societies and Social Simulation
vol. 10, no. 4 6
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Received: 28-Feb-2007 Accepted: 25-Jun-2007 Published: 31-Oct-2007
|Figure 1. Characteristic of the three simple simulation models discussed in this paper, compared to real organisations|
|Figure 2. Illustration of the Kawachi process, starting with a 60-node antiprism (top right)|
|D = 2.04 + 1.07 p-0.32||(1)|
Mathematically, the Kawachi process defines a continuous mapping from the interval [0,5] to the set of probability distributions on networks, and hence induces continuous mappings from the interval [0,5] to various network metrics. Figure 2 can be viewed as illustrating the mappings from the interval [0,5] to the metrics C and D.
|Q = 12.5 + 85.8/D2||(2)|
This regression equation, illustrated in Figure 3, explained 57% of the variance in the data (a correlation of 0.76, statistically significant at better than the 10-100 level, by analysis of variance). Since the average network distances D ranged from 2.6 to 7.9, this regression equation corresponds to the average quality percentages ranging from 14% to 25%. Extrapolating, it suggests that if every agent were directly connected to every other (i.e. D = 1), then the quality percentage would average about 98%.
|Figure 3. Results for Assignment Problem simulation, showing fit of data to the regression equation|
|Figure 4. Six Kuramoto oscillators connected in a ring network, showing the position (phase) of each oscillator|
Here the summation is over all agents j directly connected to agent i, and the multiplier k is 0.001 (substantially greater values than this result in chaotic behaviour when discretized). This differential equation can be approximated as a difference equation, specifying the changing phase over one timestep. We iterated this for 100,000 time steps to see if self-synchronization would occur, with the average frequency fi being sufficiently low (around 0.02) to ensure that the timestep behaviour would be a good approximation to the continuous behaviour specified by the differential equation.
|Figure 5. Average values of the correlation r between oscillator phases, for different values of the Kawachi parameter used to generate networks and for different frequency distribution widths|
This equation explained 74% of the variance in the data (a correlation of 0.86, with all components statistically significant at better than the 10-100 level). Using a square root rather than a cube root explained only 73% of the variance, and other network characteristics (such as the clustering coefficient, or even polynomials in the Kawachi parameter) did not provide additional explanatory power.
|Figure 6. Mean adjusted completion time (T) for the Lagrange Model, for different values of the Kawachi parameter p and the irrationality factor I. Networks with p = 0.5 performed best|
|T = 22,200 I - 58.9 D3 - 15,700 K + 2,600 K D + 30,500||(6)|
This predicts 76% of the variance in the data (a correlation of 0.87). All components of this polynomial are statistically significant at the 10-30 level or better, with the choice of the cubic power for D significant at the 10-5 level, and no other cubic or quartic terms significant.
|Figure 7. Change in average network distance (D) and average connectivity (K) for Kawachi process, overlaid on contour lines for polynomial predictor of the adjusted completion time (T)|
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