# Towards a Model of Urban Evolution: Part II: Formal Model

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## Abstract

**:**

## 1. Introduction

## 2. Basic Model

#### 2.1. Components

P: | the set of all possible types of physical Forms. |

A: | the set of all possible types of Activities (uses). |

G: | the set of all possible types of Groups (users). |

#### 2.2. Formeme

#### 2.3. Genome

u_{i}: | the i^{th} formeme in U |

u_{i} [p]: | the set of forms in the i^{th} formeme in U |

u_{i} [a]: | the set of activities in the i^{th} formeme in U |

u_{i} [g]: | the set of groups in the i^{th} formeme in U |

_{i}u

_{i}[p], set of all forms in U

_{i}u

_{i}[a], set of all activities in U

_{i}u

_{i}[g], set of all groups in U

#### 2.4. Users, Uses and Used

h_{i}: | the i^{th} formeme in H |

h_{i} [p]: | the set of physical forms in the i^{th} formeme in H |

h_{i} [a]: | the set of activities in the i^{th} formeme in H |

h_{i} [g]: | the set of groups in the i^{th} formeme in H |

_{i}h

_{i}[p], set of all forms in H

_{i}h

_{i}[a], set of all activities in H

_{i}h

_{i}[g], set of all groups in H

_{i}| Formeme(h

_{i}) } that exist for some space c at time t

#### 2.5. Signal

- formeme that communicates a fragment of a genome. This fragment may be assimilated by another spatial area, first as a change to hunome H, and if it survives, eventually as a change to U;
- the source of the signal a spatial area receives. Where a signal comes from affects how it is received;
- method of communication. A formeme may be communicated in more than one way, and depending on the method of communication, the signal may travel only within c (intra-spatial signal), or between c’s (inter-spatial signal), or both (bi-spatial signal);
- the capacity of a signal to alter the recoding costs in the area that receives the signal; and
- the number of times the signal has been received. A signal that is received with a high frequency may have a higher probability of assimilation in H.

_{i}| s

_{i}= <f, r, c, sf, cm, n> }

_{i}has been received from c during the time span of the signature

_{i}: the i

^{th}signal in S

_{i}[f]: is the formeme f of the i

^{th}signal

_{i}[r]: is the recoding cost transform

_{i}[c]: is the spatial area from which the signal originates

_{i}[sf]: is the formeme sf that is the source of the i

^{th}signal

_{i}[cm]: is the communication methods of the i

^{th}signal

_{i,}[n]: is the frequency of the i

^{th}signal from c during the time span of the signature

_{i}s

_{i}[f][p] set of all forms in S

_{i}s

_{i}[f][a] set of all activities in S

_{i}s

_{i}[f][g] set of all groups in S

_{i}s

_{i}[f] set of all formemes in S

_{i}| Signal(s

_{i}) } that exist for some space c at time t

_{i}satisfies the following requirements:

- Formeme Validity: Formeme (s
_{i}[f]) is true. - Source Validity: Given the source spatial area c and formeme s
_{i}[sf], then s_{i}[sf] ∈ UG(c,t) - Signal Method Validity: s
_{i}[cm] ⊆ CM (where CM is the set of all methods)

#### 2.6. Signature

## 3. Component Characteristics

Usize(c, t, e): | is the number of times e appears in formemes in the genome, |

i.e., e ∈ U. | |

Hsize(c, t, e): | is the number of times e appears in formemes in the hunome, |

i.e., e ∈ H. | |

Fsize(F, e): | is the number of times e appear in a set of formemes F. |

value(c, t, e): | is a function that returns the value of e at spatial location c |

at time t where e ∈ P ∪ A ∪ G. |

**Signal characteristics**

reach(s, c, t): | % of population within c at time t that receive and can process |

the signal. |

reachG(s, c, t, g): | % of a group g within spatial area c at time t that receive |

the signal. |

audienceG(s, c, t, g): | predicate that denotes the signal s has a target |

audience of group g at time t |

precisionG(s, c, t, g): | the percentage of group g in spatial area c at time t that |

is able to process the signal. |

clarityG(s, c, t, g): | probability that the recipient group g will correctly interpret |

the content of the signal s in spatial area c at time t. |

noise(s, c1, t1, c2, t2): | percentage of the signal s content that mutates in |

transmission from spatial area c1 at time t1 to spatial | |

area c2 at time t2. |

receivedSignals(c, t, t’) is the set of signals received by location c between t and t’ |

## 4. Similarity and Formetic Distance

Axiom 1: | Reflexivity fdist(f1, f1) = 0 |

Axiom 2: | Symmetry fdist(f1, f2) = fdist(f2, f1) |

Axiom 3: | Subadditivity fdist(f1, f2) + fdist(f2, f3) ≥ fdist(f1, f3) |

Axiom 1: | Reflexivity Fdist (F1, F1) = 0 |

Axiom 2: | Symmetry Fdist (F1, F2) = Fdist (F2, F1) |

Axiom 3: | Subadditivity Fdist (F1, F2) + Fdist (F2, F3) ≥ Fdist (F1, F3) |

_{1}, U

_{2}). Figure 6 illustrates urban change in terms of the reproduction of urban genomes over time. Since moving up on the Y-axis represents the passage of time, the evolutionary lineage of different c can be traced by following a particular trajectory upwards. Along the X-axis, the diagrams provide a simplified depiction of genomic similarity between different c’s. Since proximity on the X-axis indicates a high degree of similarity, the diagram can be interpreted as representing the increasing differentiation of a set of spatial areas over time. Within this overall differentiation, characteristics such as pace, stability/volatility, and convergence/divergence (discussed in Part 3 [5]) govern the development of each individual c.

Axiom 1: | Reflexivity gdist(g, g) = 0 |

Axiom 2: | Symmetry gdist(g1, g2) = gdist(g2, g1) |

Axiom 3: | Subadditivity gdist(g1, g2) + gdist(g2, g3) ≥ gdist(g1, g3) |

Axiom 1: | Reflexivity adist(a, a) = 0 |

Axiom 2: | Symmetry adist(a1, a2) = adist(a2, a1) |

Axiom 3: | Subadditivity adist(a1, a2) + adist(a2, a3) ≥ adist(a1, a3) |

## 5. Spatial Distance

Axiom 1: | Reflexivity distanceC(c,c,t) = 0 |

Axiom 2: | Symmetry distanceC(c1, c2, t) = distance(c2, c1, t) |

Axiom 3: | Subadditivity distanceC(c1, c2, t) + distance(c2, c3, t) ≥ |

distance(c1, c3, t) |

## 6. Evolutionary Trajectories

Axiom: | Reflexivity GPath(c,t,c,t) = True |

Axiom: | Symmetry GPath(c1, t, c2, t’) = GPath(c2, t’, c1, t) |

Axiom: | Transitivity If GPath(c1, t, c2, t’) ∧ GPath(c2, t’, c3, t’’) then |

GPath(c1, t, c3, t’’) |

Axiom: | Reflexivity fPathU(c, t, c, t, f) = True |

Axiom: | Symmetry fPathU(c1, t, c2, t’, f) = fPathU(c2, t’, c1, t, f) |

Axiom: | Transitivity If fPathU(c1, t, c2, t’, f) and fPathU(c2, t’, c3, t’’, f) |

Then, fPathU(c1, t, c3, t’’, f) |

Axiom: | Reflexivity DPath(c, t, c, t) = True |

Axiom: | Asymmetry DPath(c1, t, c2, t’) ≠ DPath(c2, t’, c1, t) for c1 ≠ c2 |

Axiom: | Transitivity If DPath(c1, t, c2, t’) and DPath(c2, t’, c3, t’’) |

Then, DPath(c1, t, c3, t’’) for t < t’ < t’’ |

Axiom: | Reflexivity fDPathU(c, t, c, t, f) = True |

Axiom: | Asymmetr fDPathU(c1, t, c2, t’, f) = fDPathU(c2, t’, c1, t, f) for c1 ≠ c2 |

Axiom: | Transitivity If fDPathU(c1, t, c2, t’, f) and fDPathU(c2, t’, c3, t’’, f) Then, |

fDPathU(c1, t, c3, t’’, f) for t < t’ < t’’ |

_{s}, if the Fdist of the two sets of Signals is less than k

_{s}. In other words, the signals received during the time period are similar to each other.

_{s}, else False

## 7. Formeme Survival

## 8. Activity Costs and Recoding

_{e∈F[a]}activityCost(F, e)

## 9. Discussion and Conclusions

- What is the appropriate size of a spatial area c?

- What are the possible elements of the components: P, G and A?

- Does the model scale?

- How does this relate to agent-based urban simulations?

- A Signature can represent the form, activities and groups at some space and time, but where does evolution appear?

- What about intangible elements that take part in the construction of urban spaces?

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Relationship | Description |
---|---|

distanceC(c_{i}, c_{j}, t) | distance between the centroids of c_{i} and c_{j} |

distanceB(c_{i}, c_{j}, t) | shortest distance between the borders of c_{i} and c_{j} |

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**MDPI and ACS Style**

Fox, M.S.; Silver, D.; Adler, P.
Towards a Model of Urban Evolution: Part II: Formal Model. *Urban Sci.* **2022**, *6*, 88.
https://doi.org/10.3390/urbansci6040088

**AMA Style**

Fox MS, Silver D, Adler P.
Towards a Model of Urban Evolution: Part II: Formal Model. *Urban Science*. 2022; 6(4):88.
https://doi.org/10.3390/urbansci6040088

**Chicago/Turabian Style**

Fox, Mark S., Daniel Silver, and Patrick Adler.
2022. "Towards a Model of Urban Evolution: Part II: Formal Model" *Urban Science* 6, no. 4: 88.
https://doi.org/10.3390/urbansci6040088