On Complexity and the Prospects for Scientific Advancement

With the onset of areas such as complex systems, network science, and artificial intelligence, efforts have been invested in modeling science itself. In the present work, we report a related approach to modeling the influence of the complexity of knowledge on the respective prospects for scientific advancement. More specifically, we focus on the question of how much the topological complexity of the knowledge network can influence the prospects for scientific advancement. Once the knowledge has been represented as a complex network, we consider one of its subnetworks, the nucleus, as representing the currently known portion of that network. The relative number of nodes adjacent to the nucleus, and the ratio between this quantity and the quantity of edges interconnecting the nucleus with the remainder of the network, are taken as quantifications of the potential for scientific advancement and the efficiency with which these advances can take place. Subsequent nucleus sizes are considered in both a simpler network (Erdos-Renyi) and a more complex model (Barabasi-Albert). The results surprisingly tended to vary little between these two models, suggesting that the complexity of the knowledge network may have little effect on the prospects for scientific advancement as modeled in the present approach.

Keywords
Complexity; complex networks; scientific advancement; network science

Authorship

Luciano da Fontoura Costa ^** Correspondence email address: luciano@ifsc.usp.br

Universidade de São Paulo, Instituto de Física de São Carlos, São Carlos, SP, Brasil.Universidade de São PauloBrasilSão Carlos, SP, BrasilUniversidade de São Paulo, Instituto de Física de São Carlos, São Carlos, SP, Brasil.

https://orcid.org/0000-0001-5203-4366

* Correspondence email address: luciano@ifsc.usp.br

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Figures

Figures (5)

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Figure 1
A given overall knowledge network with N= 14 nodes and a possible nucleus (red nodes), corresponding to the portion of the overall network that is currently known. In the present work, we quantify the prospect for discovery established by such configurations in terms of the number of nodes n and links e not yet known that are adjacent to the nucleus (shown in yellow) and which can, therefore, be discovered while exploring from the nucleus (through links shown in orange). More specifically, we will give special attention to the ratio r = n/N as a relative indication of the prospect for scientific advancement and a ratio s = n/e taken as a measurement of the efficiency of the scientific advancement (observe that 0≤s≤1). In the case of the example in this figure, we have r = 3/14 and s = 3/5. How will the potential for scientific discovery defined by specific types of nucleus vary with respect to the topological structure and complexity of the nucleus and overall knowledge network?

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Figure 2
The adjacency matrix K of the network in Fig. 1. The columns corresponding to the nucleus nodes (i.e. 1, 3, 6, 7) are marked in red. The rows corresponding to still unexplored nodes covered by the red columns (i.e. 4, 5, 8) are marked in yellow. The number of connections between the nucleus and the nodes to be discovered can be obtained by adding the entries along the respective red rows intersected with the red columns.

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Figure 3
The two parameters r(c) and s(c) quantifying the current scientific prospects (left-hand side) and the efficiency (right-hand side) in terms of the relative size of the nucleus c. These results were obtained for an overall knowledge network with N=1000 and ⟨k⟩ = 6,20 and 100, for R=200 realizations. The nodes constituting the nucleus were drawn with uniformly random probability among the N=1000 nodes of the overall network. Interestingly, quite similar results have been obtained for both the ‘simpler’ (ER) and more ‘complex’ (BA) network, suggesting that the intrinsic complexity of the overall knowledge network had little impact on both r(c) and s(c). In addition, a peak of scientific opportunity is observed for r(c) in all cases, which is followed by a regime of linear decrease. This peak tends to be displaced to the left with increasing average degrees. The last point in each of the s(c) curves does not follow from the simulations and serves only as a reference. See text for a discussion of other interesting aspects implied by these results.

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Figure 4
Illustration of the basic result in which the number o nodes reachable (yellow) from the nodes in the nucleus (red) does not change substantially with different uniform choices of node to constitute the nucleus. The novel and non-reachable nodes are shown in green. Though this interesting property could be expected in the case of uniformly random network models such as ER, it is relatively surprising to be observed in a network presenting higher levels of degree heterogeneity, as is the case with the BA model. The network in this example is a realization of the BA network with N=30 nodes, C = 4 nodes, and ⟨k⟩ = 2. This property is believed to underly several of the obtained results.

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Figure 5
The values and positions of the peaks defined by the r(c) curves for average degree ⟨k⟩ varying from 0 to 100 and considering the ER and BA models with n= 1000 nodes, for R=50 realizations of each configuration. The peak position is shown in terms of C, i.e. the number of nodes in the respective nucleus where the peak of r(c) is observed.

Figure 1 A given overall knowledge network with N= 14 nodes and a possible nucleus (red nodes), corresponding to the portion of the overall network that is currently known. In the present work, we quantify the prospect for discovery established by such configurations in terms of the number of nodes n and links e not yet known that are adjacent to the nucleus (shown in yellow) and which can, therefore, be discovered while exploring from the nucleus (through links shown in orange). More specifically, we will give special attention to the ratio r = n/N as a relative indication of the prospect for scientific advancement and a ratio s = n/e taken as a measurement of the efficiency of the scientific advancement (observe that 0≤s≤1). In the case of the example in this figure, we have r = 3/14 and s = 3/5. How will the potential for scientific discovery defined by specific types of nucleus vary with respect to the topological structure and complexity of the nucleus and overall knowledge network?

Figure 2 The adjacency matrix K of the network in Fig. 1. The columns corresponding to the nucleus nodes (i.e. 1, 3, 6, 7) are marked in red. The rows corresponding to still unexplored nodes covered by the red columns (i.e. 4, 5, 8) are marked in yellow. The number of connections between the nucleus and the nodes to be discovered can be obtained by adding the entries along the respective red rows intersected with the red columns.

Figure 3 The two parameters r(c) and s(c) quantifying the current scientific prospects (left-hand side) and the efficiency (right-hand side) in terms of the relative size of the nucleus c. These results were obtained for an overall knowledge network with N=1000 and ⟨k⟩ = 6,20 and 100, for R=200 realizations. The nodes constituting the nucleus were drawn with uniformly random probability among the N=1000 nodes of the overall network. Interestingly, quite similar results have been obtained for both the ‘simpler’ (ER) and more ‘complex’ (BA) network, suggesting that the intrinsic complexity of the overall knowledge network had little impact on both r(c) and s(c). In addition, a peak of scientific opportunity is observed for r(c) in all cases, which is followed by a regime of linear decrease. This peak tends to be displaced to the left with increasing average degrees. The last point in each of the s(c) curves does not follow from the simulations and serves only as a reference. See text for a discussion of other interesting aspects implied by these results.

Figure 4 Illustration of the basic result in which the number o nodes reachable (yellow) from the nodes in the nucleus (red) does not change substantially with different uniform choices of node to constitute the nucleus. The novel and non-reachable nodes are shown in green. Though this interesting property could be expected in the case of uniformly random network models such as ER, it is relatively surprising to be observed in a network presenting higher levels of degree heterogeneity, as is the case with the BA model. The network in this example is a realization of the BA network with N=30 nodes, C = 4 nodes, and ⟨k⟩ = 2. This property is believed to underly several of the obtained results.

Figure 5 The values and positions of the peaks defined by the r(c) curves for average degree ⟨k⟩ varying from 0 to 100 and considering the ER and BA models with n= 1000 nodes, for R=50 realizations of each configuration. The peak position is shown in terms of C, i.e. the number of nodes in the respective nucleus where the peak of r(c) is observed.

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[1] * Correspondence email address: luciano@ifsc.usp.br