# Citizen Science and Topology of Mind: Complexity, Computation and Criticality in Data-Driven Exploration of Open Complex Systems

## Abstract

**:**

## 1. Introduction

- Section 2: How can we formalize and treat the databases of varying quality from both machine and human observations, which range from subjective bias to objective fact? How can we set up scientific measures that should assure the compatibility with the principles of accuracy and reproducibility ?
- Section 3: How can we generalize the concept of complexity measures in application to the human–computer hybrid systems in citizen science?
- Section 4: What is the nature of computational complexities in actual data processing?
- Section 5: What is the general condition to yield guided self-organization for cost-effective citizen science?

## 2. Inter-Subjective Objectivity Model

#### 2.1. Formalization of Subjectivity, Inter-Subjectivity, Subjective–Objective Unity, and Objectivity

- Subjectivity is the quality of observation that is based on human perception without the substantial support of a machine.
- Inter-Subjectivity is the degree of commonality between the subjectivities of multiple subjects.
- Objectivity is the quality of observation that is based on a machine measurement whose consequence does not depend on the operator’s will.
- Subjective–Objective Unity is the degree of commonality between the subjectivity and objectivity.
- Inter-Subjective Objectivity is the quality of observation that satisfies the coincidence of both inter-subjectivity and subjective–objective unity.

#### 2.2. Representative Model: Buoy–Anchor–Raft Model

- Buoy refers to subjective data that fluctuates on the sea surface, representing subjectivity. Buoy can provide subjective estimates of an observation object lying on the objective sea floor, but the observation is biased by subjective fluctuations.
- Anchor refers to objective data that is fixed on the sea floor representing objectivity, without the influence from the subjective sea surface. Anchors can be connected to buoys, which provide the evaluation of subjective fluctuation with respect to objective machine measurements.
- Raft represents the relationship between buoys, and refers to inter-subjectivity of data without reference to anchors. A buoy can evaluate another buoy using relative difference of fluctuation on a subjective sea surface, and the overall commonality between buoys is represented as the raft. Nevertheless, it is based on an internal observation between buoys without an objective system of units, and is therefore susceptible to a global drift of collective standard.
- Buoy–Anchor connection rope defines the degree of subjective–objective unity. As a buoy’s movement is more controlled by its anchor, higher subjective–objective unity is assured.
- Raft–Anchor connection ropes define the degree of inter-subjective objectivity. In addition to the commonality between buoys represented as a raft, the effects of the global drift from subjective sea surface could be controlled with anchors within a plausible range of error with respect to the objective sea floor.

## 3. Complexity Measures

#### 3.1. Complexity Measure and Search Function

**Theorem**

**1.**

#### 3.2. Observation Commonality as Complexity

**Theorem**

**2.**

**Theorem**

**3.**

#### 3.3. Topological Structure of Complexity 1: Total Order of Observations

- For each pair of edges $\{{e}_{i},{e}_{j\ne i}\in E\}$, calculate the order relation ${e}_{i}\le {e}_{j}$ or ${e}_{i}\ge {e}_{j}$ with respect to the given complexity measure as an edge attribute such as length.
- Score each edge ${e}_{i}$ by mapping to integer $z:{e}_{i}\mapsto \mathbb{Z}$ by adding $+1$ if ${e}_{i}\ge {e}_{j\ne i}$ and by adding $-1$ if ${e}_{i}\le {e}_{j\ne i}$, with respect to all other edges ${e}_{j\ne i}$.
- The sorting with the score $\{z({e}_{i})\}$ provides the total order of E.

- For each triplet of observation ${V}_{i,j,k}:=\{{v}_{i},{v}_{j\ne i},{v}_{k\ne i,j}\in V\}$ and associated edges $\{{e}_{i}:=\{{v}_{i},{v}_{j}\},{e}_{j}:=\{{v}_{j},{v}_{k}\},{e}_{k}:=\{{v}_{k},{v}_{i}\}\}$, update score of each observation by mapping to integer ${z}^{\prime}:{V}_{i,j,k}\mapsto \mathbb{Z}$ with the following six rules:
- If ${e}_{i}\ge {e}_{j}\ge {e}_{k}$, then ${z}^{\prime}({v}_{i})={z}^{\prime}({v}_{i})-1$, ${z}^{\prime}({v}_{j})={z}^{\prime}({v}_{j})+1$, ${z}^{\prime}({v}_{k})={z}^{\prime}({v}_{k})+0$.
- If ${e}_{i}\ge {e}_{k}\ge {e}_{j}$, then ${z}^{\prime}({v}_{i})={z}^{\prime}({v}_{i})+1$, ${z}^{\prime}({v}_{j})={z}^{\prime}({v}_{j})-1$, ${z}^{\prime}({v}_{k})={z}^{\prime}({v}_{k})+0$.
- If ${e}_{j}\ge {e}_{i}\ge {e}_{k}$, then ${z}^{\prime}({v}_{i})={z}^{\prime}({v}_{i})+0$, ${z}^{\prime}({v}_{j})={z}^{\prime}({v}_{j})+1$, ${z}^{\prime}({v}_{k})={z}^{\prime}({v}_{k})-1$.
- If ${e}_{j}\ge {e}_{k}\ge {e}_{i}$, then ${z}^{\prime}({v}_{i})={z}^{\prime}({v}_{i})+0$, ${z}^{\prime}({v}_{j})={z}^{\prime}({v}_{j})-1$, ${z}^{\prime}({v}_{k})={z}^{\prime}({v}_{k})+1$.
- If ${e}_{k}\ge {e}_{i}\ge {e}_{j}$, then ${z}^{\prime}({v}_{i})={z}^{\prime}({v}_{i})+1$, ${z}^{\prime}({v}_{j})={z}^{\prime}({v}_{j})+0$, ${z}^{\prime}({v}_{k})={z}^{\prime}({v}_{k})-1$.
- If ${e}_{k}\ge {e}_{j}\ge {e}_{i}$, then ${z}^{\prime}({v}_{i})={z}^{\prime}({v}_{i})-1$, ${z}^{\prime}({v}_{j})={z}^{\prime}({v}_{j})+0$, ${z}^{\prime}({v}_{k})={z}^{\prime}({v}_{k})+1$.
- The sorting with the score $\{{z}^{\prime}({v}_{i})|i=1,\cdots ,N\}$ provides the total order of V.

#### 3.4. Topological Structure of Complexity 2: Permutation between Total Orders of Observations

**Theorem**

**4.**

- Two observers observing N objects: Commonality orders I and $I\phantom{\rule{-1.00006pt}{0ex}}I$ can correspond to either subjective (buoy) or objective (anchor) observation. The I–N resolution provides integrated commonality measure such as buoy–anchor connection and raft evaluation according to the nature of the observation. TDC provides connections between buoys and/or anchors.
- N observers observing two different objects: The commonality of N observers—whether it be subjective (buoy) or objective (anchor)—are ranked with respect to two different objects I and $I\phantom{\rule{-1.00006pt}{0ex}}I$. The I–N resolution provides a mean ranking of N observers’ commonality upon these observations. TDC provides the reproducibility of commonality among N observers.
- Application of two different complexity measures to N observations: For example, the case of raft–anchor connection where N subjective observers (buoys) are ranked with inter-subjective commonality (raft evaluation) and weighted with two different anchors. The I–N resolution provides mean ranking of N observers’ inter-subjective objectivity, integrating multiple criteria of inter-subjective and objective evaluation. TDC represents statistical dependencies between two complexity measures in response to a given inter-subjective objective measurement. While significant matching between two commonality orders assures the reproducibility based on the coincidence of observation with these measures, non-significance can also be used to quantify complementarity of different evaluations [32].

## 4. Computational Complexity

#### 4.1. Topological Complexity of Commonality

**Theorem**

**5.**

#### 4.2. Algorithmic Complexity

**Theorem**

**6.**

#### 4.3. Big Data Integration

**Theorem**

**7.**

**Theorem**

**8.**

## 5. Conjectures on Guided Self-Organization

#### 5.1. Criticality by Limitation

- Limitation by principle: Deterministic chaos inherent in a natural system does not allow for long-term prediction, because the tiniest observation error of the present state will develop in exponential order [35]. Short-term validity of meteorological prediction is a typical example.
- Limitation by computational complexity: As explored in Section 4.2, extensive feedback based on exhaustive computing is often impossible with respect to available computing resources. The resolution of feedback may include time delay or incomplete optimization. Spatial-temporal scale of the forecast also sets the constraint as a general trade-off between prediction accuracy and computational resources. The coarser the forecast granularity is, the more costly the calculation becomes, but the more likely it is to realize an accurate long-term prediction.

#### 5.2. Criticality by Successful Learning

- When the actual prediction accuracy is high and human–machine interaction is high, this indicates the successful modelling of observing phenomenon with the use of computation.
- When actual prediction accuracy is high and human–machine interaction is low, it means the human has achieved a successful understanding of the phenomenon with less dependency on a machine.
- When actual prediction accuracy is low and human–machine interaction is high, it indicates the possibility that computational capacity is not sufficient to effectively treat the phenomenon. Otherwise, the observing phenomenon might be in dynamical transition that effective computational model needs to be changed.
- When actual prediction accuracy is low and human–machine interaction is low, more human effort needs to be engaged both on actual observation and the utilization of the machine interface.

#### 5.3. Criticality by Guided Optimization

**Theorem**

**9.**

- ${D}^{m}({P}_{a}:{P}_{o})$: Discrepancy between actual distribution and optimum portfolio strategy that orthogonally decomposes and attempts to achieve a balance between short-term management objective and long-term sustainability.
- ${D}^{m}({P}_{a}:{P}_{s})$: Target risk of short-term management objective.
- ${D}^{m}({P}_{o}:{P}_{s})={D}^{e}({P}_{s}:{P}_{o})$: Buffering element of robustness trade-off between short-term management objective and long-term sustainability.
- ${D}^{m}({P}_{l}:{P}_{o})$: Potential risk of optimum portfolio w.r.t. long-term sustainability.
- ${D}^{m}({P}_{l}:{P}_{s})$: Potential risk of short-term management objective w.r.t. long-term sustainability.
- ${D}^{m}({P}_{l}:{P}_{a})$, ${D}^{m}({P}_{a}:{P}_{l})$: Potential risk of actual distribution w.r.t. long-term sustainability.

## 6. Results from Biodiversity Management

## 7. Discussion

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Theorem 1.**

**Proof**

**of**

**Theorem 2.**

**Proof**

**of**

**Theorem 3.**

**Proof**

**of**

**Theorem 4.**

**Proof**

**of**

**Theorem 5.**

**Proof**

**of**

**Theorem 6.**

**Proof**

**of**

**Theorem 7.**

**Proof**

**of**

**Theorem 8.**

**Proof**

**of**

**Theorem 9.**

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**Figure 1.**Schematic representation of the inter-subjective objectivity model. (

**a**) Relations between two subjectivities A and B, objectivity, inter-subjectivity between A and B, subjective–objective unity for A and B, and inter-subjective objectivity are depicted as inclusion relations between each other set. (

**b**) Development of inter-subjective objectivity as effective measurements of citizen science. As the inter-subjectivity increases along with the training of subjective–objective unity and inter-subjective feedbacks, the accuracy and reproducibility of measurement based on subjectivity can be assured by the convergence to inter-subjective objectivity.

**Figure 2.**Schematic representation of buoy–anchor–raft model. Buoy, raft, anchor, and connection rope refer to subjectivity, inter-subjectivity, objectivity, and subjective–objective unity, respectively. Concrete real-world examples are given in Table 1.

**Figure 3.**Schematic representation of complexity measures as non-linear feature space and search function as its inverse functions. (

**a**) Utility characteristics of a complex system, or complexity measure in general terms, is expressed with a complex configuration in parameter space. Parameters can also represent other complexity measures. (

**b**) Complexity measures transform parameter space into non-linear feature space, which provides easier interpretation by sorting the order of a given utility. The inverse functions of complexity measures therefore correspond to search functions with respect to the search condition on utility.

**Figure 4.**Numerical example of convolution ${\lambda}_{N}({r}_{N})$. For two kinds of probability measure ${\mu}_{1}$ (green distribution) and ${\mu}_{2}$ (blue distribution) on $r\subset \mathbb{R}$ (supported by black rug), the convolution ${\lambda}_{N}({r}_{N})$ with $N=2,4,8,16$ are shown with different colors based on random sampling of $600,000\times N$ points from $\frac{N}{2}$ pairs of ${\mu}_{1}$ and ${\mu}_{2}$. The case of $\Lambda =1$ is simulated, which shows the canonical convergence towards normal distribution following the central limit theorem with ${\sigma}_{N}\to \sqrt{N}\sigma $, where $\sigma =\frac{1}{2}({\beta}_{1}+{\beta}_{2})$ as defined in (33) and (A13). For simplicity, ${\nu}_{N}$ is adjusted to 0 by the symmetric selection of ${\mu}_{1}$, ${\mu}_{2}$, and r.

**Figure 5.**Schematic representation of the triplet order algorithm that calculates the total order of three observations with respect to the complexity defined on the pair-wise commonality between them. Three observations A, B, and C are expressed as vertices of triangle in a two-dimensional surface, whose edge lengths A–B, B–C, and A–C represent the commonality of each vertex pair. For simplicity, the triangles are projected as regular triangles, but the actual edge lengths generally differ, which provides the total order of edges. The six case statements of the algorithm are shown separately. Given the total order between the edges in blue magnitude relation, the corresponding total order of observations are depicted with orange axes at the side of each triangle. Orange axes superimposed with triangles signify that by orthogonally projecting the vertices onto them, the total order of vertices are obtained, whose generalization is developed in the Section 3.4. This holds for arbitrary three positive values of edge length without the constraint of triangular inequality, by considering appropriate projection of the triangles to a non-Euclidian surface.

**Figure 6.**Integration of two commonality orders. (

**a**) The correspondence between commonality orders I and $I\phantom{\rule{-1.00006pt}{0ex}}I$ (orange arrows) can be described as the permutation between N observations (black circles), providing the topology of I–I matching (green dotted line) and I–N compromise (blue dotted line); (

**b**) Affine space with respect to the commonality orders I and $I\phantom{\rule{-1.00006pt}{0ex}}I$ as coordinate system (orange arrows) for the resolution of I–N compromise. The I–N mean commonality order (red solid arrow) can be calculated from the pair-wise order algorithm (Section 3.3) applied on the commonality orders I and $I\phantom{\rule{-1.00006pt}{0ex}}I$, which makes the I–I matching identical to the I–I dimension (green arrow) and sets the mean order to I–N compromise. One I-N resolution dimension is required to resolve one I–N compromise (blue arrow). The implicit structure of the integrated commonality order with continuity assumption takes a complex form reflecting I–N compromises (red dotted arrow as an example), which corresponds to the complex utility configuration in Figure 3a; (

**c**) The general case with an arbitrary number of I–N compromises. Total commonality space of $N-1$ dimensions is divided between I–N resolution dimensions (blue arrows) and I–I dimensions (green arrow), between which I–N mean commonality order can be defined (red arrow). $k<N$ axes of I–N resolution dimensions are required to resolve k I–N compromises (blue arrows). Taking the I–I dimensions and I–N resolution dimensions as Affine coordinates, the integrated commonality order is projected onto the I–N mean commonality order as a simplest sorted order of utility, which corresponds to Figure 3b.

**Figure 7.**Topological hierarchy of commonality between observations. For example, five observations A, B, C, D, E are depicted with correspondence to the commonality order of each topological subset. The Venn diagram on the left represents the commonality structure within observation probability database ${\mathbb{X}}_{5}$ on variable $\mathbb{X}$ ($N=5$ in Section 4.2), where coincident observation is superimposed. The maximum commonality order is the projection between these topological subsets to the natural number $\mathbb{N}$ in right axis, describing the number of matching observations. Venn diagram cited from [34].

**Figure 8.**Numerical observation of the proof of Theorem 8. (

**a**) Chebyshev’s inequality (A41) and asymptotic convergence to $\mathcal{O}(LlogL)$ (A44) with respect to $N,M\ge 1$ $(N+M=L)$, $L=10,{10}^{2},{10}^{3}$. Y-axis is plotted with log scale. The equality in (A41) is given at $N=M=\frac{L}{2}$; (

**b**) Behaviour of $f(N)\sqrt{L}$, $f(M)\sqrt{L}$, and $f(N)f(M)L$ with respect to $L=10,{10}^{2},{10}^{3}$. For visibility, the Y-axis scale is given as ${log}^{2}({Y}^{-1})$ that represents smaller Y value to the bottom, and Y-axis label shows the value of $-logY$. The surface below the solid line $f(N)f(M)L$ represents the convolution multiplied by L, $f*f(L)L$. The mean value of solid line $f(N)f(M)L$ therefore corresponds to the upper limit of u that satisfies the polynomial constraint (45) with respect to given L. $c=d=2$ were used for the simulation.

**Figure 9.**Information geometrical optimization of diversity strategy portfolio with respect to actual distribution ${P}_{a}$, short-term management objective ${P}_{s}$, and long-term sustainability ${P}_{l}$. On a dual-flat statistical manifold based on Fisher information metric, each distribution is represented as a point (black circles). The m-geodesic is depicted with a blue line, while the e-geodesic is shown with a red line, which orthogonally cross at the optimized strategy ${P}_{o}$. Topological correspondence between complexity measure ${H}^{\prime}$ (aligned on left orange arrow) and diversity strategy portfolio (${P}_{a},{P}_{s},{P}_{l}$ and ${P}_{o}$) is shown with dotted lines with respect to the magnitude relation.

**Figure 10.**Results of inter-subjective and subjective–objective commonality orders in citizen observation of biodiversity. Seven people represented with numerical ID are aligned with commonality orders (

**a**) based on inter-subjectivity; and (

**b**) based on subjective–objective unity, which showed a 3.92% residual error probability regarding the rejection of the random order distribution hypothesis with respect to the binomial test (38).

**Table 1.**Examples of buoy, raft, and anchor in various social systems and scientific domains. Examples are not comprehensive, but a partial list of typical data from the recently increasing public availability.

Economy | Judiciary | Biodiversity Record | Medical Treatment | |
---|---|---|---|---|

Buoy | Demand, satisfaction | Sense of justice, guilt | Visual identification of species | Pain, psychological state |

Raft | Price, exchange rate | Law, court decision | Identification with voting | Diagnosis, prescription |

Anchor | Goods abundance | Evidential matter | DNA sequences | Physiological markers |

Section Number | 2.2 | 3.1 | 3.2 | 3.3 | 3.4 | 4 | 5 |
---|---|---|---|---|---|---|---|

Buoy | $\mathbf{B}$ | $\mu (\xb7)$,$I(\xb7)$ | ${\mu}_{i}(\xb7)$,${q}_{i}(\xb7)$ | Data contained in vertices V | Com. order I and $I\phantom{\rule{-1.00006pt}{0ex}}I$between N objects | Observations A, B, C, D, E | $P(\xb7)$, ${P}_{a}(\xb7)$, ${P}_{s}(\xb7)$, ${P}_{l}(\xb7)$, ${P}_{o}(\xb7)$, ${H}^{\prime}$ |

Anchor | $\mathbf{A}$ | ||||||

Raft | $\mathbf{R}$, $\mathbf{E}$ | ${\mu}^{\prime}(\xb7,\xb7)$, ${I}_{2}(\xb7,\xb7)$ | ${\lambda}_{N}(\xb7)$ | Edge attribute of E | Com. order I and $I\phantom{\rule{-1.00006pt}{0ex}}I$ b/w N observers, TDC, I-I and I-N res. dim. | $O(\xb7)$ | ${H}_{2}^{\prime}$, ${D}^{m}(\xb7:\xb7)$, ${D}^{e}(\xb7:\xb7)$ |

Buoy–Anchor | $\mathbf{C}$ | ||||||

Raft–Anchor | ${\mathbf{E}}^{\prime}$ |

**Table 3.**Algorithmic complexity for the calculation of commonality orders. With respect to the maximum commonality order in (39), an exhaustive number of combinations with the use of observation probability database ${\mathbb{X}}_{N}$ of size N and the time scale required for the sorting of the commonality measure is shown. Sorting time is based on the worst-case performance of canonical algorithms such as bubble sort and quick sort (polynomial degree $d=2$). $\mathcal{O}(\xb7)$ denotes asymptotic notation of Landau. $O(\mathbb{X})=\lfloor \frac{N}{2}\rfloor $ and $\lceil \frac{N}{2}\rceil $ require the maximum calculation and sorting time. Note that the total computation time is upper-bounded by the sorting process ($d=2$) than the combinatorics of commonality ($d=1$), though calculation time of each commonality such as convolution should be further considered in actual implementation.

Maximum Commonality Order $\mathit{O}(\mathbb{X})$ | Number of Combination | Sorting Time ($\mathit{d}=2$) |
---|---|---|

2 | ${}_{N}{C}_{2}$ | $\mathcal{O}({({}_{N}{C}_{2})}^{2})=\mathcal{O}({N}^{4})$ |

3 | ${}_{N}{C}_{3}$ | $\mathcal{O}({({}_{N}{C}_{3})}^{2})=\mathcal{O}({N}^{6})$ |

⋮ | ⋮ | ⋮ |

$\lfloor \frac{N}{2}\rfloor$ or $\lceil \frac{N}{2}\rceil$ | ${}_{N}{C}_{\lfloor \frac{N}{2}\rfloor}$=${}_{N}{C}_{\lceil \frac{N}{2}\rceil}$ | $\mathcal{O}\left({\left({}_{N}{C}_{\lfloor \frac{N}{2}\rfloor}\right)}^{2}\right)=\mathcal{O}\left({\left({}_{N}{C}_{\lceil \frac{N}{2}\rceil}\right)}^{2}\right)=\mathcal{O}\left({N}^{2\xb7\lfloor \frac{N}{2}\rfloor}\right)$ |

⋮ | ⋮ | ⋮ |

N | ${}_{N}{C}_{N}=1$ | $\mathcal{O}({({}_{N}{C}_{N})}^{2})=\mathcal{O}(1)$ |

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Funabashi, M. Citizen Science and Topology of Mind: Complexity, Computation and Criticality in Data-Driven Exploration of Open Complex Systems. *Entropy* **2017**, *19*, 181.
https://doi.org/10.3390/e19040181

**AMA Style**

Funabashi M. Citizen Science and Topology of Mind: Complexity, Computation and Criticality in Data-Driven Exploration of Open Complex Systems. *Entropy*. 2017; 19(4):181.
https://doi.org/10.3390/e19040181

**Chicago/Turabian Style**

Funabashi, Masatoshi. 2017. "Citizen Science and Topology of Mind: Complexity, Computation and Criticality in Data-Driven Exploration of Open Complex Systems" *Entropy* 19, no. 4: 181.
https://doi.org/10.3390/e19040181