Paul Ormerod and Rich Colbaugh (2006)
Cascades of Failure and Extinction in Evolving Complex Systems
Journal of Artificial Societies and Social Simulation
vol. 9, no. 4
<https://www.jasss.org/9/4/9.html>
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Received: 26-May-2006 Accepted: 28-Aug-2006 Published: 31-Oct-2006
Figure 1. Proportion of extinction events in the ranges 0 – 0.1, 0.1 – 0.2 , … , 0.9 – 1.0; 100 autonomous agents, 100 solutions each of 1000 steps. Figure 1(a) size and spatial impact of shock drawn from uniform distribution on [0,1]; Figure 1(b) size and spatial impact of shock drawn from normal distribution with mean = 0.5 s.d. =0.1; Figure 1(c) size of shock drawn from beta distribution with parameters 1 and 5 and spatial impact of shock drawn from uniform distribution on [0,1] |
f_{i,j} = f_{i,j-1} + v_{ij} - f_{i,j-1} · v_{ij} | (1) |
where f_{i,j } is the fitness of the i th agent in period j and f_{i,j-1} its fitness in the previous period, and the v_{ij} are drawn from a uniform distribution on [0,1]. The expression (1) ensures that the fitness of each agent is bounded in [0,1], and implies that the value of alliances is subject to diminishing returns. The value of each alliance is defined as (f_{i,j} - f_{i,j-1}).
Figure 2. Average degree of the system and the probability of forming an alliance in any given period. 100 agents, 100 solutions each of 1000 steps. Figure 2(a) size and spatial impact of shock drawn from uniform distribution on [0,1]; Figure 2(b) size and spatial impact of shock drawn from normal distribution with mean = 0.5 s.d. =0.1; Figure 2(c) size of shock drawn from beta distribution with parameters 1 and 5 and spatial impact of shock drawn from uniform distribution on [0,1] |
Figure 3. Average fitness and average degree. 100 agents, 100 solutions each of 1000 steps. Figure 3(a) size and spatial impact of shock drawn from uniform distribution on [0,1]; Figure 3(b) size and spatial impact of shock drawn from normal distribution with mean = 0.5 s.d. =0.1; Figure 3(c) size of shock drawn from beta distribution with parameters 1 and 5 and spatial impact of shock drawn from uniform distribution on [0,1] |
So the self-interested actions of agents lead to very distinct increases in the average fitness of all agents.
Figure 4. Size of principal component of graph in first 40 steps of a typical solution of the model. Beta refers to the size of the shock drawn from a beta distribution and its range from a normal; normal and uniform refer to both size and range drawn from a normal and a uniform respectively; pAlly is the probability of forming an alliance (p) |
With the beta distribution, the probability of drawing a large shock is low, so that large extinction events are infrequent. These can, however, occur and are seen in the size of the principal component, which rises rapidly to 100 but occasionally drops.
Figure 5. Proportion of extinction events in the ranges 0 – 0.1, 0.1 – 0.2 , … , 0.9 – 1.0; 100 agents, 100 solutions each of 1000 steps. Probability of an agent forming an alliance in any given period is 0.3. Figure 5(a) size and spatial impact of shock drawn from uniform distribution on [0,1]; Figure 5(b) size and spatial impact of shock drawn from normal distribution with mean = 0.5 s.d. =0.1; Figure 5(c) size of shock drawn from beta distribution with parameters 1 and 5 and spatial impact of shock drawn from uniform distribution on [0,1] |
Figure 6. Proportion of extinction events in the range 0.9 – 1.0; 100 agents, 100 solutions each of 1000 steps. Probability of an agent forming an alliance in any given period is 0.3. Figure 6(a) size and spatial impact of shock drawn from uniform distribution on [0,1]; Figure 6(b) size and spatial impact of shock drawn from normal distribution with mean = 0.5 s.d. =0.1; Figure 6(c) size of shock drawn from beta distribution with parameters 1 and 5 and spatial impact of shock drawn from uniform distribution on [0,1] |
^{2}The spatial impact of shocks is between 0 and 0.5 (since on a unit circle you can never be more than 0.5 away) so the size of the impact is drawn from the distributions as said, but its halved to bring it into this range.
^{3} The probability of observing large extinctions is extremely low when both the size and spatial impact of the shock are drawn from the beta distribution. So in the results below we draw the size from the beta and the spatial impact from the uniform. The qualitative nature of the results is the same when both are drawn from the beta.
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