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Network growth models: A behavioural basis for attachment proportional to fitness

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ABSTRACT

Several growth models have been proposed in the literature for scale-free complex networks, with a range of fitness-based attachment models gaining prominence recently. However, the processes by which such fitness-based attachment behaviour can arise are less well understood, making it difficult to compare the relative merits of such models. This paper analyses an evolutionary mechanism that would give rise to a fitness-based attachment process. In particular, it is proven by analytical and numerical methods that in homogeneous networks, the minimisation of maximum exposure to node unfitness leads to attachment probabilities that are proportional to node fitness. This result is then extended to heterogeneous networks, with supply chain networks being used as an example.

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Linear program solutions for the minimisation of maximum exposure to node unfitness, superimposed on attachment probabilities calculated as proportional to node fitness.Nodes are ordered according to node fitness. (a) homogeneous nodes: the underlying fitness distribution is log-normal with zero scale parameter and shape parameter σ = 1.0 (b) tiered nodes: the underlying fitness distributions are log-normal with zero scale parameter and shape parameters σ1 = 3, σ2 = 1, σ3 = 1, σ4 = 0.1 respectively for the four tiers. In both cases, the linear program solution matches exactly with the attachment probabilities derived from the proportional fitness rule.
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f1: Linear program solutions for the minimisation of maximum exposure to node unfitness, superimposed on attachment probabilities calculated as proportional to node fitness.Nodes are ordered according to node fitness. (a) homogeneous nodes: the underlying fitness distribution is log-normal with zero scale parameter and shape parameter σ = 1.0 (b) tiered nodes: the underlying fitness distributions are log-normal with zero scale parameter and shape parameters σ1 = 3, σ2 = 1, σ3 = 1, σ4 = 0.1 respectively for the four tiers. In both cases, the linear program solution matches exactly with the attachment probabilities derived from the proportional fitness rule.

Mentions: See Methods for an analytical proof. To demonstrate the validity of the above solution numerically, we begin first with smaller systems that can be solved by the Excel Solver. Thus, we generated a set of nodes (with 12 nodes in the set) and fitness values sampled from a log-normal fitness distribution with scale parameter μ = 0 and shape parameter σ = 1.0, since it has been argued in the literature that fitness distributions are typically log-normal3 (Log-normal distributions are characterised by scale and shape parameters; see ref. 3 for a discussion on their relationship with the mean and standard deviation of the distribution). For this set of nodes, we solved the above linear program using the Excel Solver. The results are presented in Fig. 1a, which also shows attachment probabilities calculated as being proportional to fitness. The results match exactly. This result does not depend on the nature or standard deviation of the underlying fitness distribution, and in fact we have verified that any fitness distribution will yield the same exact match.


Network growth models: A behavioural basis for attachment proportional to fitness
Linear program solutions for the minimisation of maximum exposure to node unfitness, superimposed on attachment probabilities calculated as proportional to node fitness.Nodes are ordered according to node fitness. (a) homogeneous nodes: the underlying fitness distribution is log-normal with zero scale parameter and shape parameter σ = 1.0 (b) tiered nodes: the underlying fitness distributions are log-normal with zero scale parameter and shape parameters σ1 = 3, σ2 = 1, σ3 = 1, σ4 = 0.1 respectively for the four tiers. In both cases, the linear program solution matches exactly with the attachment probabilities derived from the proportional fitness rule.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5304319&req=5

f1: Linear program solutions for the minimisation of maximum exposure to node unfitness, superimposed on attachment probabilities calculated as proportional to node fitness.Nodes are ordered according to node fitness. (a) homogeneous nodes: the underlying fitness distribution is log-normal with zero scale parameter and shape parameter σ = 1.0 (b) tiered nodes: the underlying fitness distributions are log-normal with zero scale parameter and shape parameters σ1 = 3, σ2 = 1, σ3 = 1, σ4 = 0.1 respectively for the four tiers. In both cases, the linear program solution matches exactly with the attachment probabilities derived from the proportional fitness rule.
Mentions: See Methods for an analytical proof. To demonstrate the validity of the above solution numerically, we begin first with smaller systems that can be solved by the Excel Solver. Thus, we generated a set of nodes (with 12 nodes in the set) and fitness values sampled from a log-normal fitness distribution with scale parameter μ = 0 and shape parameter σ = 1.0, since it has been argued in the literature that fitness distributions are typically log-normal3 (Log-normal distributions are characterised by scale and shape parameters; see ref. 3 for a discussion on their relationship with the mean and standard deviation of the distribution). For this set of nodes, we solved the above linear program using the Excel Solver. The results are presented in Fig. 1a, which also shows attachment probabilities calculated as being proportional to fitness. The results match exactly. This result does not depend on the nature or standard deviation of the underlying fitness distribution, and in fact we have verified that any fitness distribution will yield the same exact match.

View Article: PubMed Central - PubMed

ABSTRACT

Several growth models have been proposed in the literature for scale-free complex networks, with a range of fitness-based attachment models gaining prominence recently. However, the processes by which such fitness-based attachment behaviour can arise are less well understood, making it difficult to compare the relative merits of such models. This paper analyses an evolutionary mechanism that would give rise to a fitness-based attachment process. In particular, it is proven by analytical and numerical methods that in homogeneous networks, the minimisation of maximum exposure to node unfitness leads to attachment probabilities that are proportional to node fitness. This result is then extended to heterogeneous networks, with supply chain networks being used as an example.

No MeSH data available.


Related in: MedlinePlus