Towards a Model of Urban Evolution Part IV: Evolutionary (Formetic) Distance—An Interpretation of Yelp Review Data

Abstract

This paper is part IV of “towards a model of urban evolution”. It demonstrates how the Toronto Urban Evolution Model (TUEM) can be used to encode city data, illuminate key features, demonstrate how formetic distance can be used to discover how spatial areas change over time, and identify similar spatial areas within and between cities. The data used in this study are reviews from Yelp. Each review can be interpreted as a formeme where the category of the business is a form, the reviewer is a group and the review is an activity. Yelp data from neighbourhoods in both Toronto and Montreal are encoded. A method for aggregating reviewers into groups with multiple members is introduced. Longitudinal analysis is performed for all Toronto neighbourhoods. Transversal analysis is performed between neighbourhoods within Toronto and between Toronto and Montreal. Similar neighbourhoods are identified validating formetic distance.

1. Introduction

Central to the analysis of the evolution of cities is measuring how cities change over time. Using the Toronto Urban Evolution Model (TUEM) signatures of spatial areas (We refer to spatial areas instead of cities, as the model applies to parts of a city, such as neighbourhoods, as it does to a city as a whole. Each being a spatial area.), we can measure to what extent the forms, groups and activities differ between any pair of signatures. A transversal analysis compares two different spatial areas. A longitudinal analysis compares the same spatial area at different times. A longitudinal analysis allows us to study the trajectory of the city–how it evolves over time. By examining the formemes that comprise these signatures, we can determine what has changed over time, and more importantly the formemes that lead to success or failure. Transversal analysis tells us to what extent two different spatial areas share similar combinations of forms, groups and activities. The analysis allows us to understand how similar spatial areas are and why they are similar (or dissimilar).

This paper applies the Toronto Urban Evolution Model, defined in [1,2,3], to Yelp review data. It demonstrates how Yelp data can be mapped into our model, and how the model, in particular the use of similarity metrics (formetic distance), provides insight into the forms, groups and activities in the neighbourhoods where the review events take place.

Section 2 provides details on the Yelp data used in this study and how it is encoded in TUEM. It shows how a reviewer’s review of a business is mapped onto TUEM forms and groups. In particular, a business ID is mapped to a form, a user ID onto a group, and a neighbourhood onto a spatial area.

Section 3 provides a review of TUEM’s signature distance functions which are the basis for the formetic distance analysis. Two functions are described. The first is based on the degree to which forms and groups are shared across neighbourhoods. The second is based on a weighted version of the first where the number of times a shared form and group appear in reviews is included.

Section 4 focuses on measuring longitudinal and transversal signature distance within and between Toronto neighbourhoods, respectively. Longitudinal distance measures how a neighbourhood evolves over time with respect to the groups and forms captured in a spatial area’s signature as it changes over time. The greater the distance between a neighbourhood’s signatures over time, the more a neighbourhood has changed. In other words, the Yelp users and the categories of venues are a proxy for the types of people and types of businesses that occupy a neighbourhood. Transversal distance measures the degree of similarity between neighbourhoods, again using Yelp users and categories as a proxy for the types of people and types of businesses that occupy a neighbourhood.

Section 5 focuses on longitudinal and transversal signature distance analysis again but with aggregated forms. In the Yelp data each review is associated with a single business ID. If the business ID is retained as the form, then there could be no intersections across neighbourhoods. To address this problem we replace the business ID with the Yelp category of business associated with the business ID. Hence, multiple business IDs are aggregated by Yelp category.

Section 6 focuses on longitudinal and transversal signature distance analysis again but with aggregated groups. In the Yelp data, each review is associated with a single user (which we refer to in the TUEM as a singleton group). In this section, singleton groups are aggregated into groups containing multiple users. The method of aggregation is based on the Apriori algorithm where users are grouped according to their having reviewed the same subset of forms (i.e., business categories).

Section 7 applies transversal signature distance analysis to pairs of neighbourhoods in Toronto and Montreal. The goal is to determine whether the model can identify similar neighbourhoods between the two cities. It uses form and group aggregation to allow for common forms and groups to be found.

Section 8 discusses a number of questions regarding validity of aggregation, and evolutionary insights drawn from the model. We conclude with Section 9.

Appendix A summarizes the formal model defined in [1].

2. Encoding Yelp Data in TUEM

Yelp is a consumer review social media site, with more than 150 million unique users providing millions of reviews in several languages. It contains information about (i) points of interest (POI) and (ii) their users. POIs are mainly businesses such as restaurants and coffee shops, but also may include public spaces such as parks or hiking trails. Each POI is classified within a hierarchical list of categories and may contain reviews provided by users containing evaluative assessments. The reviews are date stamped and correspond to a GPS location, enabling geospatial and temporal analysis.

The Yelp data encoded in our model is from the Yelp.com academic dataset. They are reviews of restaurants that spanned 2189 days (6 years), covering 140 Toronto neighbourhoods (Figure 1), and containing 327,188 reviews. Details about the dataset may be found in [4]. For a review of research using Yelp as a data source, as well as of its strengths and limitations, see [5].

Figure 1. City of Toronto 144 Neighbourhoods.

The neighbourhoods discussed in this report (Note that our analysis was performed across all neighbourhoods and only a subset are discussed to illustrate the ideas.) are:

• Annex (95)—High income predominately residential with retail;
• Bathurst Manor (34)—Middle income bedroom community;
• Bay Street Corridor (76)—Middle to high income mixed commercial and residential;
• Church-Yonge Corridor (75)—Middle income with retail;
• Danforth (66)—Middle income with retail;
• Junction Area (90)—Middle and low income with retail;
• Palmerston–Little Italy (80)—Middle income residential;
• Waterfront Communities-The Island (77)—Middle income with retail and commercial;
• Wychwood (94)—Middle to high income residential.

The number following the neighbourhood name is the City of Toronto’s numbering. These neighbourhoods are highlighted in red in Figure 1.

Each review is composed of a:

• Time stamp.
• Business ID: a unique identifier for the business being reviewed.
• Neighbourhood ID: a unique identifier for the neighbourhood in which the business is located. There are 140 neighbourhoods in Toronto.
• User ID: a unique identifier for the reviewer. There are 73,504 unique reviewers in this set.
• Categories: one or more categories, from the Yelp taxonomy of business categories, that the business is classified as. There are 854 unique categories (excluding top level categories) in this Yelp dataset.

The Yelp review data was translated into the TUEM model as follows:

• Spatial Areas (C): a spatial area c corresponds to a Toronto neighbourhood.
• Forms (P): a form p corresponds to a unique Yelp business ID.
• Groups (G): a group g corresponds to a unique User ID, i.e., reviewer.
• Activities (A): an activity a corresponds to the implicit activity of reviewing.

The formemes were partitioned into spatial areas (i.e., neighbourhoods), and within each spatial area into years. Each year-neighbourhood partition corresponds to a signature; one signature for each neighbourhood for each of the six years, resulting in 840 signatures. The formemes associated with a particular signature were assigned to the signature’s Hunome H, as they reflect the users and uses in the neighbourhood.

3. Formeme and Signature Distances

As defined in Parts II [1] and III [3], an important concept in modelling urban evolution is the degree to which formemes are similar. By similarity, we mean the degree to which they share the same elements: forms, activities, and groups. The hypothesis is the more similar the population of formemes, the more similar the evolutionary path. However, if areas with similar initial populations of formemes diverge over time, what are the key differences in elements that lead to the change?

We reproduce from part II the function fdist that returns the formetic distance between two formemes.

fdist(f₁, f₂): measures the formetic distance between formemes f₁ and f₂.

The smaller the value, the more similar the formemes are. There can be many different distance metrics. We constrain the definition of fdist as follows:

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).

In order to measure how similar urban genomes or hunomes are, we defined the distance between two sets of formemes F₁ and F₂:

Fdist(F₁, F₂): measures the similarity between formeme sets F₁ and F₂.

The smaller the value, the more similar the formeme sets are. There can be many different distance metrics. We constrain the definition of Fdist() as follows:

Axiom 4.

Reflexivity Fdist (F₁, F₁) = 0.

Axiom 5.

Symmetry Fdist (F₁, F₂) = Fdist (F₂, F₁).

Axiom 6.

Subadditivity Fdist (F₁, F₂) + Fdist (F₂, F₃) ≥ Fdist (F₁, F₃).

One possible definition of Fdist(), which we refer to as the “basic distance metric” bFdist, counts the number of element types shared between the formeme sets divided by the total number of element types across both formeme sets. In particular, bFdist’s fraction numerator sums the size of the sets created by the intersection of formeme elements, namely p, a and g. The numerator is dived by the denominator which is the sum of the size of the sets created by the union of the same formeme elements. The more elements in common between F₁ and F₂, i.e., the more similar they are, the closer the value of the fraction is to one. We then subtract the fraction from zero to get the distance, i.e., a distance of zero implies the faction is one and the F₁ and F₂ share the same elements.

$$\mathrm{bFdist}\left(\mathrm{F}1,\mathrm{F}2\right)=1-\frac{\left|\mathrm{F}1\left[\mathrm{p}\right]{\cap }^{ }\mathrm{F}2\left[\mathrm{p}\right]\left|+\right|\mathrm{F}1\left[\mathrm{a}\right]{\cap }^{ }\mathrm{F}2\left[\mathrm{a}\right]\left|+\right|\mathrm{F}1\left[\mathrm{g}\right]{\cap }^{ }\mathrm{F}2\left[\mathrm{g}\right]\right|}{\left|\mathrm{F}1\left[\mathrm{p}\right]{\cap }^{ }\mathrm{F}2\left[\mathrm{p}\right]\left|+\right|\mathrm{F}1\left[\mathrm{a}\right]{\cap }^{ }\mathrm{F}2\left[\mathrm{a}\right]\left|+\right|\mathrm{F}1\left[\mathrm{g}\right]{\cap }^{ }\mathrm{F}2\left[\mathrm{g}\right]\right|}$$

We defined an alternative Fdist, which we refer to as the “weighted distance metric” wFdist. wFdist takes into account the frequency with which elements appear in a set of formemes. If two sets of formemes have the same elements, then bFdist will determine that they are the same, i.e., zero distance between them. However, if one set of formemes has a greater frequency of elements than the other, bFdist will still return a zero distance. For example, let’s assume that every neighbourhood has one of each category of venue and group, the bFdist will return a distance of zero between pairs of neighbourhoods. However, if one neighbourhood has a very large number of Chinese venues, we would refer to this neighbourhood as “Chinatown”. So, the frequency with which elements such as forms (venues in Yelp) occur in formemes is an important component of measuring distance. In order capture the possible imbalance in the number of occurrences of elements in one neighbourhood versus another, wFdist weights the results. Unlike bFdist where the number of shared elements are counted in the numerator, we substitute the sum of the minimum number of occurrences of each element measured by Fsize. Fsize(F, e) returns the number of formemes f in F that contain the element e. The denominator sums the maximum number of occurrences of each element measured by Fsize.

$$\mathrm{wFdist}\left(\mathrm{F}1, \mathrm{F}2\right)=1-\frac{{\sum }_{\mathrm{e}\in \mathrm{Elements}\left(\mathrm{F}1,\mathrm{F}2\right)}\min \left(\mathrm{Fsize}\left(\mathrm{F}1,\mathrm{e}\right),\mathrm{Fsize}\left(\mathrm{F}2,\mathrm{e}\right)\right)}{{\sum }_{\mathrm{e}\in \mathrm{Elements}\left(\mathrm{F}1,\mathrm{F}2\right)}\max \left(\mathrm{Fsize}\left(\mathrm{F}1,\mathrm{e}\right),\mathrm{Fsize}\left(\mathrm{F}2,\mathrm{e}\right)\right)}$$

where Elements(F1, F2) is the set of all elements contained in F1 and F2

Elements(F1, F2) = F1 [p] ⋃ F2 [p] ⋃ F1 [g] ⋃ F2 [g] ⋃ F1 [a] ⋃ F2 [a].

In this paper, we measure the distance between Hunomes H of different spatial areas, as we wish to measure how similar the conceptions of urban form carried by the users of urban space.

4. Signature Distances: Unaggregated Data

In this section we perform a longitudinal and transversal analysis on the “raw” signatures of two neighbourhoods in Toronto. Highlighting these two neighbourhoods helps to build intuition about our measures before moving to more general analysis. By “raw”, we mean neither the forms (venues) have their business ID replaced by their business category, nor are individual users/reviewers IDs aggregate into groups containing more than one unique user/reviewer.

4.1. Longitudinal Analysis

In this section the evolution of the Waterfront neighbourhood is analysed over a six year period. Waterfront Communities-The Island (77) is situated at the foot of Lake Ontario. It is both a popular tourist area with many event spaces, restaurants, and upper income condominiums. The choice of this neighbourhood is based on the changing demographics and activities. During the period of the Yelp data, it was initially a tourist destination, but became more residential with the continuing addition of residential condominium buildings. See Figure 2 for the boundaries of the neighbourhood, which is numbered 77.

Figure 2. Waterfront Communities-The Island (77).

Table 1 depicts the longitudinal distance of the Waterfront Neighbourhood for a six-year period, illustrating how to measure how a single neighbourhood changes over time. For example, row Yr 1 shows the weighted distance (wFdist) between Yr 1 and subsequent years. An analysis of longitudinal distance, the distance ranges from 0.536 from year 1 to 2, to 0.84 from year 1 to 6.

Table 1. Waterfront Longitudinal Distance-Unaggregated Data.

	Yr 1	Yr 2	Yr 3	Yr 4	Yr 5	Yr 6
Yr 1		0.536	0.649	0.743	0.81	0.84
Yr 2			0.582	0.698	0.781	0.816
Yr 3				0.597	0.713	0.76
Yr 4					0.592	0.669
Yr 5						0.549
Yr 6

It may seem puzzling that the distance from year 1 to 2 is 0.536 and then grows by 0.1 approximately to each subsequent year. An analysis of the data in Table 2 provides an explanation.

Table 2. Longitudinal Analysis of the Waterfront Neighbourhood.

Year (1096)	Dist. from Prior Year	Dist. from Year 1	Waterfront Communities-The Island (77) Exp # 10.0	P	G	A	Total
1			Number of non-unique elements in component	4031	4031	4031	12,093
			Number of unique elements in component	644	1723	1	2368
2	0.536	0.536	Number of non-unique elements in component	4462	4462	4462	13,386
			Number of unique elements in component	709	2275	1	2985
			Intersection of elements with prior year	511	399	1	911
			Union of elements with prior year	842	3599	1	4442
			Intersection %	60.7	11.1	100.0	20.5
			Cumulative intersection since year 1	511	399	1	911
			Cumulative union since year 1	842	3599	1	4442
			Cumulative Intersection %	60.7	11.1	100.0	20.5
3	0.582	0.649	Number of non-unique elements in component	5798	5798	5798	17,394
			Number of unique elements in component	809	3190	1	4000
			Intersection of elements with prior year	569	507	1	1077
			Union of elements with prior year	949	4958	1	5908
			Intersection %	60.0	10.2	100.0	18.2
			Cumulative intersection since year 1	430	196	1	627
			Cumulative union since year 1	1013	6170	1	7184
			Cumulative Intersection %	42.4	3.2	100.0	8.7
4	0.597	0.743	Number of non-unique elements in component	8054	8054	8054	24,162
			Number of unique elements in component	838	4692	1	5531
			Intersection of elements with prior year	636	640	1	1277
			Union of elements with prior year	1011	7242	1	8254
			Intersection %	62.9	8.8	100.0	15.5
			Cumulative intersection since year 1	373	117	1	491
			Cumulative union since year 1	1150	10,024	1	11,175
			Cumulative Intersection %	32.4	1.2	100.0	4.4
5	0.592	0.81	Number of non-unique elements in component	11,463	11,463	11,463	34,389
			Number of unique elements in component	938	6567	1	7506
			Intersection of elements with prior year	683	999	1	1683
			Union of elements with prior year	1093	10,260	1	11,354
			Intersection %	62.5	9.7	100.0	14.8
			Cumulative intersection since year 1	332	82	1	415
			Cumulative union since year 1	1306	15,221	1	16,528
			Cumulative Intersection %	25.4	0.5	100.0	2.5
6	0.549	0.84	Number of non-unique elements in component	12,699	12,699	12,699	38,097
			Number of unique elements in component	989	7596	1	8586
			Intersection of elements with prior year	757	1254	1	2012
			Union of elements with prior year	1170	12,909	1	14,080
			Intersection %	64.7	9.7	100.0	14.3
			Cumulative intersection since year 1	300	54	1	355
			Cumulative union since year 1	1438	21,085	1	22,524
			Cumulative Intersection %	20.9	0.3	100.0	1.6

Table 2 depicts the longitudinal metrics for the Waterfront neighbourhood.

For each year, except year 1, the following metrics are displayed:

• Number of non-unique elements in component: total number of elements across the three components, P, A and G, of H.
• Number of unique elements in component: total number of unique elements across the three components, P, A and G, of H.
• Intersection of elements with prior year: total number of unique elements across the three components, P, A and G, in the intersection of this year’s elements (metric #1) with the prior year.
• Union of elements with prior year: total number of unique elements across the three components, P, A and G, in the union of this year’s elements (metric #1) with the prior year.
• Intersection %: Percentage of unique elements that are common to this year’s metric #1 and the prior year.
• Cumulative intersection since year 1: total number of unique elements across the three components, P, A and G, in the intersection of this year’s elements (metric #1) with all prior years.
• Cumulative Intersection %: Percentage of unique elements that are common to this year’s metric #1 and all prior years.

The distance between any two adjacent years ranges from 0.536 to 0.597, and is relatively stable. The reason for this is that from year to year, about 60% of the venues reviewed do not change (Intersection %), while 90% of the reviewers change from one year to the next. However, the cumulative intersection of venues reviewed since Year 1 drops from 42% in year 3 down to 21% in year 6. In other words, the marginal increase in distance from year 1 to years 3 and beyond is due to the slow change in venues that are reviewed from year 1 to 6. Individual reviewers come and go, but the underlying types of venues remain more stable.

4.2. Transversal Distance

In this section we explore measuring the distance between pairs of neighbourhoods during Year 1.

Table 3 depicts the transversal distance between 8 neighbourhoods. The neighbourhoods were chosen for their variation in neighbourhood types, from inner suburb residential (Bathurst Manor), to midtown residential (Wychwood), to residential with vibrant nightlife (Annex, Palmerston-Little Italy), to commercial/residential (Church-Yonge), to downtown high tourist area (Waterfront), to a neighbourhood in transition from industrial to residential/nightlife (Junction). In most cases, the pair-wise distances are quite high with Annex, Church, and Waterfront being more similar to each other. All three have a mid/downtown residential component plus vibrant nightlife.

Table 3. Transversal Distance-Unaggregated Elements.

	Annex (95)	Bathurst Manor (34)	Church-Yonge Corridor (75)	Danforth (66)	Junction Area (90)	Palmerston-Little Italy (80)	Waterfront Communities-The Island (77)	Wychwood (94)
Annex (95)		0.997	0.736	0.96	0.948	0.824	0.81	0.936
Bathurst Manor (34)			0.998	0.975	0.982	0.993	0.998	0.983
Church-Yonge Corridor (75)				0.967	0.958	0.857	0.771	0.952
Danforth (66)					0.822	0.926	0.98	0.834
Junction Area (90)						0.899	0.974	0.796
Palmerston-Little Italy (80)							0.901	0.882
Waterfront Communities-The Island (77)								0.967
Wychwood (94)

Table 4 depicts the data underlying the Annex–Bathurst Manor transversal distance metric for year 1:

Table 4. Transversal Metrics: Annex-Bathurst.

Exp # 10.0	P	G	A	Total
Annex (95)	447	963	1	1411
Bathurst Manor (34)	4	11	1	16
Intersection of elements	0	5	1	6
Union of elements	451	969	1	1421
Intersection %	0.0	0.5	100.0	0.4

• The lines listing the neighourhoods specify the number of unique elements for each of the components P, G, and A.
• Intersection of elements specifies the number of unique elements in the intersection of the two neighbourhoods.
• The union of elements specifies the number of unique elements in the union.
• The intersection % specifies the number of unique elements in the intersection divided by the number in the union.

Since the venues are located in only one neighbourhood, the number of venues at the intersection of the neighbourhoods is zero. Consequently, the only information the distance measures is the extent to which reviewers have posted reviews for venues in both neighbourhoods in the same year. Being an inner suburb residential community, Bathurst Manor has few venue reviews and the percentage of reviewers posting reviews in both neighbourhoods is 0.5%.

The Church-Yonge Corridor neighbourhood (Table 5) is the closest to the Annex in terms of distance. 21.1% of the reviewers posted reviews in both neighbourhoods in the same year. The Palmerston-Little Italy and Waterfront neighbourhoods (Table 6) also share reviewers with the Annex, ranging between 16.9% and 21.1%.

Table 5. Transversal Metrics: Annex-Church.

Exp # 10.0	P	G	A	Total
Annex (95)	447	963	1	1411
Church-Yonge Corridor (75)	437	1176	1	1614
Intersection of elements	0	373	1	374
Union of elements	884	1766	1	2651
Intersection %	0.0	21.1	100.0	14.1

Table 6. Transversal Metrics: Annex-Waterfront.

Exp # 10.0	P	G	A	Total
Annex (95)	447	963	1	1411
Waterfront Communities-The Island (77)	644	1723	1	2368
Intersection of elements	0	426	1	427
Union of elements	1091	2260	1	3352
Intersection %	0.0	18.8	100.0	12.7

The transversal analysis indicates that relative to other neighbourhood pairings, the Annex, Palmerston-Little Italy and Waterfront neighbourhoods are more similar to each other in the types of people (reviewers) they attract. In subsequent sections, the analysis will focus on four neighbourhoods: Annex, Bathurst Manor, Church Corridor and Waterfront–The Island.

5. Signature Distances: Aggregated Forms

The previous experiment treated both groups (reviewers) and forms (venues) as unique. No attempt was made to aggregate either. The problem with the unaggregated data is that a venue’s business ID is unique to a neighbourhood and therefore the intersection of venues between neighbourhoods is zero, as we saw in the previous section. Hence, we cannot get a true picture of the distance between neighbourhoods. In this experiment we substitute venue category for a venue’s businessID. Each venue may have more than one category. For example, a restaurant may be categorized as both an Italian restaurant and a pizzeria. We transform a single review into multiple reviews, one for each assigned category. We also introduce a weight for a review where a single review with three categories has a weight of 1/3 assigned to each duplicate review. In terms of our model of evolution, each review is interpreted as a formeme composed of a reviewer (G), review (A) and venue category (P) plus a weight:

• Form f: single business category;
• Activity a: Review—same for all formemes;
• Group g: group ID—the reviewer’s unique User ID;
• Weight w.

If a single review assigned 3 categories to a business ID, then we would generate three formemes:

-

<category 1, Review, Group 28, 0.333>

-

<category 2, Review, Group 28, 0.333>

-

<category 3, Review, Group 28, 0.333>

In this measurement approach, formemes capture each category separately, but the weight of the formeme is reduced in order to reduce the impact of assigning multiple categories in TUEM’s distance metric.

The Yelp business categories are embedded in a taxonomy with the following top level categories (Table 7):

Table 7. Yelp Top Level Categories.

Active Life	Financial Services	Local Services	Real Estate
Arts & Entertainment	Food	Mass Media	Religious Organizations
Automotive	Health & Medical	Nightlife	Restaurants
Beauty & Spas	Home Services	Pets	Shopping
Education	Hotels & Travel	Professional Services
Event Planning & Services	Local Flavor	Public Services & Government

Venues choose their category and often choose multiple categories starting with the top of the taxonomy. This results in formemes dominated by top level categories which do not adequately differentiate neighbourhoods. Consequently, the distance between neighbourhoods is reduced because they contain many of these top level categories. To reduce the impact of the top level categories on the distance analysis, we remove any formemes whose form (category) is one of these top level categories.

5.1. Longitudinal Distance

5.1.1. Longitudinal Case: Waterfront Communities-The Island (77)

Table 8 returns to the Waterfront and depicts its longitudinal distance for a six-year period. For example, row Yr 1 shows the weighted distance between Yr 1 and subsequent years. The distance ranges from 0.471 from year 1 to 2, to 0.803 from year 1 to 6.

Table 8. Longitudinal Distance for Waterfront Neighbourhood.

	Yr 1	Yr 2	Yr 3	Yr 4	Yr 5	Yr 6
Yr 1		0.471	0.591	0.701	0.778	0.803
Yr 2			0.53	0.661	0.754	0.782
Yr 3				0.563	0.684	0.722
Yr 4					0.559	0.622
Yr 5						0.494
Yr 6

Neighbourhood Venue (Form) Changes. Examination of the Waterfront data shows that greater than 80% of the reviews from one year to the next are for the same categories of venues, whereas over the six year period the cumulative intersection of venue categories reviewed drops to approximately 50%, possibly reflecting an evolution in the categories of venues in the neighbourhood over the six year period. Within the restaurant industry, the decrease in intersecting venues is consistent with the average life span of a restaurant—the majority do not last more than a year and those that do, 70% will close within five years (https://yourbusiness.azcentral.com/average-life-span-restaurant-6024.html, accessed on 17 January 2022). This is validated by the cumulative intersection of non-aggregated forms (venues) in year 6 of Table 9 of 20%, implying that many restaurants have probably closed, but still greater than the average. Nevertheless, the analysis also indicates that the neighbourhood retains a sizable share of its orientation over the period, with the churn in particular businesses matched by a greater stability in the types of businesses present. In other words, the genome U for the neighbourhood influences the persistence of venue categories which in turn influences a persistence in the activities associated with them. Although some change in the actual activities in Hunome H occur, the persisted impact of U is clearly felt.

Table 9. Longitudinal Analysis of Components for Waterfront Neighbourhood.

Year (1096)	Dist. from Prior Year	Dist. from Year 1	Waterfront Communities-The Island (77) Exp # 10.1	P	G	A	Total
1			Number of non-unique elements in component	11,891	11,891	11,891	35,673
			Number of unique elements in component	313	1720	1	2034
2	0.471	0.471	Number of non-unique elements in component	12,927	12,927	12,927	38,781
			Number of unique elements in component	334	2271	1	2606
			Intersection of elements with prior year	289	399	1	689
			Union of elements with prior year	358	3592	1	3951
			Intersection %	80.7	11.1	100.0	17.4
			Cumulative intersection since year 1	289	399	1	689
			Cumulative union since year 1	358	3592	1	3951
			Cumulative Intersection %	80.7	11.1	100.0	17.4
3	0.53	0.591	Number of non-unique elements in component	16,278	16,278	16,278	48,834
			Number of unique elements in component	376	3188	1	3565
			Intersection of elements with prior year	316	507	1	824
			Union of elements with prior year	394	4952	1	5347
			Intersection %	80.2	10.2	100.0	15.4
			Cumulative intersection since year 1	277	196	1	474
			Cumulative union since year 1	400	6161	1	6562
			Cumulative Intersection %	69.2	3.2	100.0	7.2
4	0.563	0.701	Number of non-unique elements in component	22,715	22,715	22,715	68,145
			Number of unique elements in component	377	4689	1	5067
			Intersection of elements with prior year	351	639	1	991
			Union of elements with prior year	402	7238	1	7641
			Intersection %	87.3	8.8	100.0	13.0
			Cumulative intersection since year 1	272	117	1	390
			Cumulative union since year 1	415	10,013	1	10,429
			Cumulative Intersection %	65.5	1.2	100.0	3.7
5	0.559	0.778	Number of non-unique elements in component	33,404	33,404	33,404	100,212
			Number of unique elements in component	399	6564	1	6964
			Intersection of elements with prior year	360	999	1	1360
			Union of elements with prior year	416	10,254	1	10,671
			Intersection %	86.5	9.7	100.0	12.7
			Cumulative intersection since year 1	270	82	1	353
			Cumulative union since year 1	438	15,207	1	15,646
			Cumulative Intersection %	61.6	0.5	100.0	2.3
6	0.494	0.803	Number of non-unique elements in component	37,211	37,211	37,211	111,633
			Number of unique elements in component	422	7592	1	8015
			Intersection of elements with prior year	373	1253	1	1627
			Union of elements with prior year	448	12,903	1	13,352
			Intersection %	83.3	9.7	100.0	12.2
			Cumulative intersection since year 1	264	54	1	319
			Cumulative union since year 1	470	21,068	1	21,539
			Cumulative Intersection %	56.2	0.3	100.0	1.5

Neighbourhood Reviewer (Group) Changes. From one year to the next there is about 10% repeat reviewers-the same person will generate at least one review each year over a two year period. However, over the six years the number of people who submit reviews for each of the years reduces to 0.3%. Hence, the reviewer population appears to be transient, which is consistent with the Waterfront being a tourist area. The Waterfront therefore illustrates a type of urban evolution where the population of local formemes is likely adapted to the expectations of a mobile tourist population of users.

5.1.2. Longitudinal Case: Annex (95)

Table 10 depicts the longitudinal distance of the Annex Neighbourhood for a six-year period, with aggregated forms (venues). The longitudinal distance ranges from 0.471 from year 1 to 2, to 0.727 from year 1 to 6, showing a slightly less cumulative change than the Waterfront.

Table 10. Longitudinal Distance for Annex Neighbourhood.

	Yr 1	Yr 2	Yr 3	Yr 4	Yr 5	Yr 6
Yr 1		0.47	0.571	0.663	0.709	0.727
Yr 2			0.531	0.642	0.695	0.715
Yr 3				0.559	0.631	0.658
Yr 4					0.531	0.57
Yr 5						0.496
Yr 6

Table 11 provides detailed data that underly the longitudinal metrics for the Annex neighbourhood. About 50% of the venue categories reviewed each year are the same throughout the sixth year period. In the early years, 7–11% of the reviewers are the same from year to year. Over the six-year period, the number of reviewers that posted reviews across all years was negligible.

Table 11. Longitudinal Analysis of Components for Annex Neighbourhood.

Year (1096)	Dist. from Prior Year	Dist. from Year 1	Annex (95) Exp # 10.1	P	G	A	Total
1			Number of non-unique elements in component	5060	5060	5060	15,180
			Number of unique elements in component	246	962	1	1209
2	0.47	0.47	Number of non-unique elements in component	5188	5188	5188	15,564
			Number of unique elements in component	250	1231	1	1482
			Intersection of elements with prior year	214	230	1	445
			Union of elements with prior year	282	1963	1	2246
			Intersection %	75.9	11.7	100.0	19.8
			Cumulative intersection since year 1	214	230	1	445
			Cumulative union since year 1	282	1963	1	2246
			Cumulative Intersection %	75.9	11.7	100.0	19.8
3	0.531	0.571	Number of non-unique elements in component	6637	6637	6637	19,911
			Number of unique elements in component	258	1573	1	1832
			Intersection of elements with prior year	226	228	1	455
			Union of elements with prior year	282	2576	1	2859
			Intersection %	80.1	8.9	100.0	15.9
			Cumulative intersection since year 1	206	88	1	295
			Cumulative union since year 1	308	3251	1	3560
			Cumulative Intersection %	66.9	2.7	100.0	8.3
4	0.559	0.663	Number of non-unique elements in component	8596	8596	8596	25,788
			Number of unique elements in component	279	2201	1	2481
			Intersection of elements with prior year	232	277	1	510
			Union of elements with prior year	305	3497	1	3803
			Intersection %	76.1	7.9	100.0	13.4
			Cumulative intersection since year 1	198	44	1	243
			Cumulative union since year 1	330	5076	1	5407
			Cumulative Intersection %	60.0	0.9	100.0	4.5
5	0.531	0.709	Number of non-unique elements in component	9973	9973	9973	29,919
			Number of unique elements in component	282	2722	1	3005
			Intersection of elements with prior year	241	357	1	599
			Union of elements with prior year	320	4566	1	4887
			Intersection %	75.3	7.8	100.0	12.3
			Cumulative intersection since year 1	188	21	1	210
			Cumulative union since year 1	354	7267	1	7622
			Cumulative Intersection %	53.1	0.3	100.0	2.8
6	0.496	0.727	Number of non-unique elements in component	11,027	11,027	11,027	33,081
			Number of unique elements in component	300	3054	1	3355
			Intersection of elements with prior year	254	416	1	671
			Union of elements with prior year	328	5360	1	5689
			Intersection %	77.4	7.8	100.0	11.8
			Cumulative intersection since year 1	184	13	1	198
			Cumulative union since year 1	367	9680	1	10,048
			Cumulative Intersection %	50.1	0.1	100.0	2.0

5.1.3. Observations

To the extent that Yelp reviews provide a window into neighbourhood “scene” [6], the categories of forms are ever changing, all be it at a slow rate. Anecdotally, the Annex scene has always been evolving. In 1960s and 70s, the restaurant scene was dominated by Hungarian restaurants. This evolved into one dominated by Japanese and then Korean restaurants. Rather than reflecting a change in the groups inhabiting the area, which has remained largely stable, i.e., students, faculty, retirees, it reflects their evolving culinary tastes.

5.2. Transversal Distance

In this section we explore transversal distance between pairs of neighbourhoods during Year 1, where forms (venues) have been aggregated by business category. The analysis focuses on 3 neighbourhoods, each very different from each other: Annex, Bathurst Manor and Waterfront.

Table 12 depicts the transversal distances for eight neighbourhoods in Year 1. Focusing on The Annex, it is most similar to the Church-Yonge neighbourhood (distance of 0.53), and least similar to the Bathurst Manor neighbourhood (distance of 0.995). In the remainder of this section we dig deeper into the data.

Table 12. Transversal Distance for Year 1.

	Annex (95)	Bathurst Manor (34)	Church-Yonge Corridor (75)	Danforth (66)	Junction Area (90)	Palmerston-Little Italy (80)	Waterfront Communities-The Island (77)	Wychwood (94)
Annex (95)		0.995	0.53	0.935	0.908	0.7	0.668	0.899
Bathurst Manor (34)			0.996	0.973	0.98	0.99	0.998	0.976
Church-Yonge Corridor (75)				0.947	0.927	0.757	0.593	0.919
Danforth (66)					0.706	0.876	0.966	0.759
Junction Area (90)						0.835	0.955	0.704
Palmerston-Little Italy (80)							0.84	0.814
Waterfront Communities-The Island (77)								0.948
Wychwood (94)

5.2.1. Transversal Case: Annex (95)—Church-Yonge Corridor (75)

The Annex and Church-Yonge neighbourhoods are more similar than any other pairings with the Annex—0.53. An analysis of Table 13 shows that 52% of the venue categories are common to the neighbourhoods, but more interestingly, 21% of the reviewers are common to both—these evolution of these areas’ is therefore likely influenced to a relatively high degree by their shared users. This is especially notable when longitudinally, the commonality of reviewers between subsequent years in the Annex is between 7 and 11% (see Table 11).

Table 13. Year 1 Transversal Metrics Annex-Church.

	P	G	A	Total
Annex (95)	246	962	1	1209
Church-Yonge Corridor (75)	249	1176	1	1436
Intersection	173	373	1	547
Union	332	1765	1	2098
Intersection %	52.1%	21.1%	100%	26.1%

5.2.2. Transversal Case: Annex (95)—Bathurst Manor (34)

The Annex and Bathurst Manor neighbourhoods are very dissimilar, with a distance of 0.995. Table 14 shows that they neither share any significant numbers of venue categories (2.4%) nor reviewers (0.5%). Their evolution therefore proceeds largely in parallel to one another, though they are likely influenced by their common environment to some degree.

Table 14. Year 1 Transversal Metrics Annex-Bathurst.

	P	G	A	Total
Annex (95)	246	962	1	1209
Bathurst Manor (34)	11	11	1	23
Intersection	6	5	1	12
Union	251	968	1	1220
Intersection %	2.4%	0.5%	100%	1%

5.2.3. Transversal Case: Annex (95)—Waterfront Communities-The Island (77)

The Annex and Waterfront neighbourhoods (Table 15) are more similar, with a distance of 0.668, but not as similar as the Annex and Church-Yonge neighbourhoods. Both the venue categories and reviewers the Annex shares with Waterfront are much greater than shared with Bathurst Manor, but not as high as shared with Church-Yonge.

Table 15. Year 1 Transversal Metrics Annex-Waterfront.

	P	G	A	Total
Annex (95)	246	962	1	1209
Waterfront Communities-The Island (77)	313	1720	1	2034
Intersection	182	426	1	609
Union	377	2256	1	2634
Intersection %	48.3%	18.9%	100%	23.1%

6. Signature Distances: Aggregating Groups

The experiment conducted in Section 5 saw the aggregation of forms (venues) by business category, providing a more accurate picture of the distances between neighborhoods. However, the data remains unaggregated with respect to groups, i.e., each reviewer constitutes their own group in a formeme. This discrepancy between the data as it stands and the model of urban evolution results in an inaccurate measure of the distance between neighborhoods (because reviewer activity is likely localized, i.e., reviewers may be more inclined to visit certain venues near places of work or residence) as well as between years (because reviewers may change the frequency of their Yelp logging activity over time).

In this experiment, we use the Apriori algorithm [7] to aggregate reviewers by the types of venues they visit. Performing group aggregation using a level-wise search, this algorithm abstracts groups based on the forms they conducted reviewing activities for. In particular, singleton groups (i.e., individual reviewers) are aggregated into groups based on their reviewing a common set of forms (venue categories). The minimum group size was set to 5000 and the maximum number of shared forms (venue categories) was set to 10. This resulted in 29 groups that shared 2 forms, and 6 groups that shared 3 forms. Appendix B lists the groups and the forms (venue categories) they share. If a singleton group (reviewer) was assigned to more than one aggregate group, then when mapping a formeme’s singleton group into an aggregate group, the aggregate group chosen was determined by:

• The aggregate group whose defining forms contained the formeme’s form;
• the aggregate group with the largest number of forms used to define the aggregate group.

If there does not exist an aggregate group whose defining forms contain the formeme’s form, then no mapping is performed.

The formemes in all signatures were transformed by mapping unique forms (Business ID’s) to one or more aggregate forms (Yelp categories) as described in Section 5. Secondly, singleton groups (reviewers) that were a member of an aggregate group were mapped into the aggregate group as described earlier. The following describes the longitudinal and transversal distances based on the aggregated signatures.

6.1. Longitudinal Distance

6.1.1. Waterfront Communities-The Island (77)

Table 16 depicts the longitudinal distances within the Waterfront Neighbourhood over a six-year period. Again, a cell contains the weighted distance between corresponding years. For example, row Yr 1 shows the weighted distance between Yr 2 and subsequent years. With the aggregation of groups, we find that the difference ranges from 0.204 for years 1 and 2 to 0.703 for years 1 and 6 (with a distance growth of approximately 0.15 for subsequent years). Comparing these values to the range for unaggregated data (0.536 to 0.84) and for aggregation of forms only (0.471 to 0.803), it is clear that aggregation (of groups especially) plays a large role in regularizing the distances between longitudinal signatures and thus providing a more accurate depiction of the evolution of the Waterfront Neighbourhood over time.

Table 16. Longitudinal Distance for Waterfront Neighbourhood.

	Yr 1	Yr 2	Yr 3	Yr 4	Yr 5	Yr 6
Yr 1		0.204	0.364	0.539	0.67	0.703
Yr 2			0.308	0.497	0.64	0.676
Yr 3				355	0.537	0.58
Yr 4					366	0.433
Yr 5						0.238
Yr 6

Examining the Waterfront data in greater detail in Table 17, the evolution of forms (venue categories) in column P remains identical to the results in Table 9 (which has no group aggregation). This confirms that the differences in longitudinal distances are a direct result of the aggregation of groups (reviewers). With the aggregation of groups (reviewers), there are now approximately 75% fewer unique elements in the group component of each year. The intersection of group elements from year to year appears to be significantly lower than that which is observed inTable 9, but this is due to the fact that frequency of group elements is not taken into account with set operations.

Table 17. Longitudinal Analysis of Components for Waterfront Neighbourhood.

Year	Distance from Prior Year	Distance from Year 1	Waterfront Communities-The Island (77) Exp # 11	P	G	A	Total
1			Number of non-unique elements in component	2398	2398	2398	7194
			Number of unique elements in component	313	418	1	732
2	0.204	0.204	Number of non-unique elements in component	3016	3016	3016	9048
			Number of unique elements in component	334	579	1	914
			Intersection of elements with prior year	289	39	1	329
			Union of elements with prior year	358	958	1	1317
			Intersection %	80.7	4.1	100.0	25.0
			Cumulative intersection since year 1	289	39	1	329
			Cumulative union since year 1	358	958	1	1317
			Cumulative Intersection %	80.7	4.1	100.0	25.0
3	0.308	0.364	Number of non-unique elements in component	3922	3922	3922	11,766
			Number of unique elements in component	376	806	1	1183
			Intersection of elements with prior year	316	45	1	362
			Union of elements with prior year	394	1340	1	1735
			Intersection %	80.2	3.4	100.0	20.9
			Cumulative intersection since year 1	277	33	1	311
			Cumulative union since year 1	400	1713	1	2114
			Cumulative Intersection %	69.2	1.9	100.0	14.7
4	0.355	0.539	Number of non-unique elements in component	5308	5308	5308	15,924
			Number of unique elements in component	377	1265	1	1643
			Intersection of elements with prior year	351	48	1	400
			Union of elements with prior year	402	2023	1	2426
			Intersection %	87.3	2.4	100.0	16.5
			Cumulative intersection since year 1	272	33	1	306
			Cumulative union since year 1	415	2921	1	3337
			Cumulative Intersection %	65.5	1.1	100.0	9.2
5	0.366	0.67	Number of non-unique elements in component	7125	7125	7125	21,375
			Number of unique elements in component	399	1700	1	2100
			Intersection of elements with prior year	360	61	1	422
			Union of elements with prior year	416	2904	1	3321
			Intersection %	86.5	2.1	100.0	12.7
			Cumulative intersection since year 1	270	33	1	304
			Cumulative union since year 1	438	4541	1	4980
			Cumulative Intersection %	61.6	0.7	100.0	6.1
6	0.238	0.703	Number of non-unique elements in component	8703	8703	8703	26,109
			Number of unique elements in component	422	2091	1	2514
			Intersection of elements with prior year	373	67	1	441
			Union of elements with prior year	448	3724	1	4173
			Intersection %	83.3	1.8	100.0	10.6
			Cumulative intersection since year 1	264	33	1	298
			Cumulative union since year 1	470	6529	1	7000
			Cumulative Intersection %	56.2	0.5	100.0	4.3

6.1.2. Annex (95)

Table 18 depicts the longitudinal distance of the Annex Neighbourhood for a six-year period with group aggregation. The difference ranges from 0.207 for years 1 and 2 to 0.562 for years 1 and 6. While the distance between subsequent years grows by around 0.1 at first, between years 5 and 6 the distance only increases by 0.027. Comparing these longitudinal distances to those for the Waterfront Neighbourhood (which have a range of 0.204 to 0.703 and a distance growth of approximately 0.15 for subsequent years), there is a larger persistence of form or group elements over time for the Annex than the Waterfront. This greater stability is a reflection of the greater stability of the formetic bases of the Annex in its recurrent users and forms.

Table 18. Longitudinal Distance for Annex Neighbourhood.

	Yr 1	Yr 2	Yr 3	Yr 4	Yr 5	Yr 6
Yr 1		0.207	0.302	0.456	0.535	0.562
Yr 2			0.286	0.449	0.53	0.556
Yr 3				0.334	0.429	0.459
Yr 4					0.284	0.342
Yr 5						0.243
Yr 6

Table 19 depicts the longitudinal metrics for the Annex. Again, the values for the form component (venue categories) in Table 19 are identical to those displayed in this table’s non-group-aggregated counterpart (Table 11). Unlike the longitudinal metrics for the Waterfront (Table 18), the cumulative intersection of the group components here is much higher than in the non-group-aggregated metrics (Table 11). This may be because the metrics have captured the activity of a diverse group of tourists visiting the Waterfront compared to the more consistent choices of the student and professional population visiting venues in the Annex Neighbourhood. Either way, such underlying evolutionary mechanisms are reflected in the smaller cumulative change over the six years depicted in Table 18.

Table 19. Longitudinal Analysis of Components for Annex Neighbourhood.

Year	Distance from Prior Year	Distance from Year 1	Annex (95) Exp # 11	P	G	A	Total
1			Number of non-unique elements in component	1369	1369	1369	4107
			Number of unique elements in component	246	220	1	467
2	0.207	0.207	Number of non-unique elements in component	1629	1629	1629	4887
			Number of unique elements in component	250	287	1	538
			Intersection of elements with prior year	214	36	1	251
			Union of elements with prior year	282	471	1	754
			Intersection %	75.9	7.6	100.0	33.3
			Cumulative intersection since year 1	214	36	1	251
			Cumulative union since year 1	282	471	1	754
			Cumulative Intersection %	75.9	7.6	100.0	33.3
3	0.286	0.302	Number of non-unique elements in component	2072	2072	2072	6216
			Number of unique elements in component	258	386	1	645
			Intersection of elements with prior year	226	39	1	266
			Union of elements with prior year	282	634	1	917
			Intersection %	80.1	6.2	100.0	29.0
			Cumulative intersection since year 1	206	30	1	237
			Cumulative union since year 1	308	816	1	1125
			Cumulative Intersection %	66.9	3.7	100.0	21.1
4	0.334	0.456	Number of non-unique elements in component	2758	2758	2758	8274
			Number of unique elements in component	279	607	1	887
			Intersection of elements with prior year	232	41	1	274
			Union of elements with prior year	305	952	1	1258
			Intersection %	76.1	4.3	100.0	21.8
			Cumulative intersection since year 1	198	30	1	229
			Cumulative union since year 1	330	1378	1	1709
			Cumulative Intersection %	60.0	2.2	100.0	13.4
5	0.284	0.535	Number of non-unique elements in component	3278	3278	3278	9834
			Number of unique elements in component	282	709	1	992
			Intersection of elements with prior year	241	45	1	287
			Union of elements with prior year	320	1271	1	1592
			Intersection %	75.3	3.5	100.0	18.0
			Cumulative intersection since year 1	188	30	1	219
			Cumulative union since year 1	354	2030	1	2385
			Cumulative Intersection %	53.1	1.5	100.0	9.2
6	0.243	0.562	Number of non-unique elements in component	4022	4022	4022	12,066
			Number of unique elements in component	300	943	1	1244
			Intersection of elements with prior year	254	47	1	302
			Union of elements with prior year	328	1605	1	1934
			Intersection %	77.4	2.9	100.0	15.6
			Cumulative intersection since year 1	184	30	1	215
			Cumulative union since year 1	367	2913	1	3281
			Cumulative Intersection %	50.1	1.0	100.0	6.6

6.2. Transversal Distance

In this section, we explore measuring the distance between pairs of neighbourhoods during Year 1, which we briefly touched upon when comparing the metrics for the Waterfront and Annex Neighbourhoods in the previous section. Table 20 depicts the transversal distances for eight neighbourhoods in Year 1. Focusing on the Annex Neighbourhood, it is most similar to the Church-Yonge Neighbourhood (distance of 0.37), and least similar to the Bathurst Manor Neighbourhood (distance of 0.994). Compared to the transversal distances calculated prior to group aggregation, distances of 0.53 and 0.995, respectively, the grouping of reviewers now provides a clearer picture of the distance between the Annex and Church-Yonge Neighbourhoods while the distance between the Annex and the Bathurst Manor Neighbourhoods groups was large enough to not be significantly affected.

Table 20. Transversal Distance for Year 1—Aggregated Forms and Groups.

	nnex (95)	Bathurst Manor (34)	Bay Street Corridor (76)	Church-Yonge Corridor (75)	Danforth (66)	Junction Area (90)	Palmerston-Little Italy (80)	Waterfront Communities-The Island (77)	Wychwood
x (95)		0.994	0.443	0.37	0.917	0.883	0.623	0.585	0.874
nor (34)			0.996	0.995	0.958	0.969	0.988	0.997	0.968
Bay Street Corridor (76)				0.338	0.942	0.921	0.739	0.425	0.914
Church-Yonge Corridor (75)					0.933	0.908	0.697	0.482	0.897
Danforth (66)						0.505	0.823	0.958	0.602
Junction Area (90)							0.774	0.943	0.535
Palmerston-Little Italy (80)								0.812	0.743
Waterfront Communities-The Island (77)									0.938
Wychwood (94)

6.2.1. Transversal Case: Annex (95)—Bathurst Manor (34)

The Annex and Bathurst Manor neighbourhoods are very dissimilar, with a distance of 0.994. Table 21 shows that even with group aggregation, they do not share any significant numbers of venue categories (2.4%) or reviewers (0.9%).

Table 21. Year 1 Transversal Metrics Annex-Bathurst Manor.

Exp # 11	P	G	A	Total
Annex (95)	246	220	1	467
Bathurst Manor (34)	11	4	1	16
Intersection of elements	6	2	1	9
Union of elements	251	222	1	474
Intersection %	2.4	0.9	100.0	1.9

6.2.2. Transversal Case: Annex (95)—Waterfront Communities-The Island (77)

The Annex and Waterfront neighbourhoods are more similar (Table 22), with a distance of 0.585, but not as similar as the Annex and Church-Yonge neighbourhoods. Both the venue categories and reviewers the Annex shares with Waterfront are much greater than shared with Bathurst Manor, but not as high as shared with Church-Yonge.

Table 22. Year 1 Transversal Metrics Annex-Waterfront.

Exp # 11-2	P	G	A	Total
Annex (95)	246	220	1	467
Waterfront Communities-The Island (77)	313	418	1	732
Intersection of elements	182	42	1	225
Union of elements	377	596	1	974
Intersection %	48.3	7.0	100.0	23.1

6.2.3. Transversal Case: Annex (95)—Hurch-Yonge Corridor (75)

The Annex and Church-Young Corridor are a much more similar than the Annex-Waterfront, with a distance of 0.37 (Table 23). Although the intersection of forms and groups are only marginally better (2–4%), it is the frequency with which these intersecting elements occur in both signatures that draws them closer.

Table 23. Year 1 Transversal Metrics Annex-Church-Yonge.

Exp # 11	P	G	A	Total
Annex (95)	246	220	1	467
Church-Yonge Corridor (75)	259	257	1	517
Intersection of elements	173	41	1	215
Union of elements	332	436	1	769
Intersection %	52.1	9.4	100.0	28.0

7. Transversal Distances between Toronto and Montreal Neighbourhoods

Next, we explore the transversal distances between pairs of neighbourhoods in Toronto and Montreal.

To do so, we conducted form and group aggregation for Yelp reviews collected in Toronto and Montreal and calculated the transversal distances between all neighbourhood pair combinations (The complete formetic distance matrix is too large to display in this paper, but can be found at: http://ontology.eil.utoronto.ca/urbangenome/yelpLogReviewsFull-cv-td-1096.html, accessed on 1 April 2022). In Table 24 we display some notable distances between three neighbourhoods in Toronto (Annex, Bathurst Manor, and Bay Street Corridor) and three neighbourhoods in Montreal (Lachine-Ouest, René-Lévesque and Sainte-Marie).

Table 24. Transversal Distances between Neighborhoods in Toronto and Montreal for Year 1–Aggregated Forms and Groups.

	Annex (95)	Bathurst Manor (34)	Bay Street Corridor (76)	Lachine-Ouest	René-Lévesque	Sainte-Marie
Annex (95)		0.994	0.488	1.0	0.565	0.962
Bathurst Manor (34)			0.996	0.974	0.994	0.931
Bay Street Corridor (76)				1.0	0.627	0.973
Lachine-Ouest					0.999	0.994
René-Lévesque						0.952
Sainte-Marie

Color coding Toronto neighbourhoods in blue and Montreal neighbourhoods in red, cells that represent the distance between pairs of neighbourhoods in different cities are indicated in purple. Most significantly, notice that the distances between the Annex in Toronto and René-Lévesque in Montreal as well as the Bay Street Corridor and René-Lévesque are lower than distances between most neighbourhoods in the same city. This finding indicates that formetic information is only loosely determined by common membership of areas in the same spatial environment, and that the relevant selection mechanisms often involve the tastes and expectations of users regarding what particular parts of different cities should offer. This means that much urban evolution operates at the level of neighbourhoods rather than the city or region as a whole.

To provide some context for our analyses, René-Lévesque is located in downtown Montreal and sits adjacent to both McGill and Concordia University, much in the same way that the Annex sits adjacent to the University of Toronto. (Interestingly, the Royal Ontario Museum, which is the largest museum in Canada, can be found in the Annex while the Montreal Museum of Fine Arts, which is the largest art museum in Canada, can be found in René-Lévesque). While the Annex also consists of a large number of residences, its business activity is defined by cafés, restaurants, and bars that draw students and university staff alike. Although René-Lévesque is also home to retail stores, its eateries are similarly frequented by the students and staff of the two nearby universities. Considering the overlap in forms and groups in the Annex and René-Lévesque, it is unsurprising that their transversal distance indicates a high degree of similarity between these two neighbourhoods. However, this unsurprising fact would not be easily detectable by analyzing each neighbourhood within the context of its own city alone, as their cross-city commonalities would not come into view.

The Bay Street Corridor runs along the eponymous Bay Street in Toronto and is the home to a variety of forms including eateries, retail chains, hospitals, and the Toronto Financial District. It also lies adjacent to the University of Toronto and Toronto Metropolitan (formerly Ryerson) University, suggesting the activity in this neighbourhood is also driven by students and staff in addition to the professionals that work in the vicinity. The similarities in forms (eateries and retail stores) and groups (university students and staff) that characterize the Bay Street Corridor and René-Lévesque again provides a reasonable explanation for the small transversal distance between these two neighbourhoods.

Taking a closer look at the distance between the Annex and René-Lévesque (Table 25), we can see that there is considerable overlap between the forms and groups found in both neighbourhoods. That is, comparing the intersection of form elements with the number of form elements found in each neighbourhood, the Annex and René-Lévesque each share over 60% of the business categories populating each area. Furthermore, comparing the intersection of group elements with the number of group elements found in each neighborhood, we find there is an overlap of over 20% of Annex groups with René-Lévesque groups and an overlap of nearly 16% of René-Lévesque groups with Annex groups. As these two neighbourhoods are in different cities, our model has clearly picked up on key visitor attributes through our use of aggregated groups instead of individual reviewers.

Table 25. Year 1 Transversal Metrics Annex–René-Lévesque.

Exp # 12	P	G	A	Total
Annex (T)	258	766	1	1025
René-Lévesque (M)	273	993	1	1267
Intersection of elements	174	155	1	330
Union of elements	357	1604	1	1962
Intersection %	48.7	9.7	100.0	16.8
Intersection % in Annex (T)	67.4	20.2	100.0	32.2
Intersection % in René-Lévesque (M)	63.7	15.6	100.0	26.0

From the Bay Street Corridor and René-Lévesque transversal metrics displayed in Table 26 we can see a similar intersection of forms and groups is found in this neighbourhood pairing. In fact, René-Lévesque shares over 73% of its forms and 22% of its groups with the Bay Street Corridor (which are the highest intersection percentages found in Table 26 and Table 27). Conversely, the Bay Street Corridor shares only 61% of its forms and 19% of its groups with René-Lévesque (which are among the lowest intersection percentages found in Table 25 and Table 26). This phenomenon can be explained by revisiting the neighbourhood characteristics that we previously identified. Namely, all of the major forms that define the René-Lévesque neighbourhood (eateries and retail stores) can be found in the Bay Street Corridor but not all of the forms of the Bay Street Corridor (eateries, retail stores, hospitals and the Toronto Financial Distract) can be found in René-Lévesque. Therefore, the intersection percentages in René-Lévesque and the Bay Street Corridor match our qualitative understanding of the area well. We call this phenomenon Signature Subsumption, where the signature of one spatial area subsumes another it implies that the formemes in the subsumed signature are an approximate subset of the subsuming signature. In this case, the Bay Street Corridor subsumes René-Lévesque.

Table 26. Year 1 Transversal Metrics Bay Street Corridor–René-Lévesque.

Exp # 12	P	G	A	Total
Bay Street Corridor (76)	327	1133	1	1461
René-Lévesque	273	993	1	1267
Intersection of elements	201	218	1	420
Union of elements	399	1908	1	2308
Intersection %	50.4	11.4	100.0	18.2
Intersection % in Annex (T)	61.4	19.2	100.0	28.7
Intersection % in René-Lévesque (M)	73.6	22.0	100.0	33.1

Table 27. Comparative Longitudinal Analysis.

Year	Form (P)		Group (G)
	Annex	Waterfront	Annex	Waterfront
2	75.9	80.7	7.6	4.1
3	66.9	69.2	3.7	1.9
4	60	65.5	2.2	0.7
5	53.1	61.6	1.5	0.7
6	50.1	56.2	1.0	0.5

Interestingly, although we see a greater intersection of elements for the Bay Street Corridor and René-Lévesque neighbourhood pair (Table 26) compared to the Annex and René-Lévesque neighbourhood pair (Table 25), the distance of the former pair is larger (0.627 compared to 0.565). This can be explained by our choice of weighted distance as the transversal distance heuristic and the fact that the numbers presented in Table 25 and Table 26 count the number of unique elements and do not take into account the frequency of elements. Thus, our chosen distance heuristic weighs the distance by the frequency with which elements appear in the reviews of each neighbourhood, which is not captured in Table 25 and Table 26.

All in all, our analysis of distances and the element intersection of pairs of neighbourhoods in markedly different cities is supported by the characteristics of the neighbourhoods themselves. Therefore, it seems that our model is correctly picking up on the forms and groups that are essential to defining the urban activity of a neighbourhood.

8. Discussion

Our formal model of representing and analysing urban evolution provides a means for determining the similarity of spatial areas based on formetic distance, i.e., the degree of sharing of formemes, composed of forms, groups and activities, weighted by the frequency of occurrence. In this section we discuss questions raised by this work.

Is Formetic Distance meaningful?

We use Yelp data as a source of forms and groups that populate the formemes. Its content is limited, relative to the representational capabilities of our model. Recognizing the limitation in using Yelp data as a proxy for more general types of forms, groups and activities, nevertheless it does provide an approach to validating the model.

Pairwise analysis of neighbourhoods both within Toronto and between Toronto and Montreal demonstrates that neighbourhoods that are descriptively similar (By descriptively similar, we mean the informal characterization of a neighbourhood by those knowledgeable of the area.), such as the Annex and Bay St. Corridor in Toronto, are also similar using formetic distance. More interestingly, neighbourhoods in Toronto and Montreal, such as the Annex and René-Lévesque that were found to be similar using formetic distance, were also found to be descriptively similar upon subsequent examination.

For example, the Montreal neighbourhood found to be most similar (0.7) to Bathurst Manor in Toronto is Sault-au-Récollet. The latter is described as:

“is a small, semi-suburban and scenic neighbourhood, … there’s some scenic spots and great parks, … a bit out of the way from the busier parts of the city, … isn’t a place where you go to party, … one of the parts of the city where the middle-aged and elderly might outnumber the younger folks. A lot of families move out here to settle and buy homes for the long run. As such, many of the residents of this area are long-term inhabitants who want to enjoy the feeling of suburbia but still be close to downtown, … bit out of place, especially since there are some industrial zones nearby, … find restaurants, drugstores and groceries. Though there aren’t too many choices for amenities overall …” (https://nexthome.ca/neighbourhoods/montreal-sault-au-recollet/142959/, accessed on 27 January 2021)

Bathurst Manor is also an inner suburb and primarily a bedroom community bordering both a commercial/industrial area and two major parks. While Bathurst Manor may not be characterized as scenic, the major parks are. Similarly, it has two shopping plazas and several small strip malls that provide restaurants, drugstores, groceries, etc., but with limited choices.

Neighbourhoods that are descriptively dissimilar, such as the Annex and Bathurst Manor, and Bathurst Manor and René-Lévesque are also dissimilar using formetic distance.

The coupling of form, group and activity into formemes and using them to measure formetic distance between neighbourhoods demonstrates that our distance metric is consistent with the descriptive characterization of neighbourhoods. Neighbourhoods that are similar, have similar forms and to some extent groups—to the extent that Yelp data allows.

Is aggregating individuals from different cities into the same group valid?

The Yelp review data pairs an individual reviewer with an individual venue. It also provides one or more categories for each venue reviewed. Our group aggregation algorithm generates groups composed of individuals whose reviewed venue categories have maximum overlap. As venue categories are common across all cities, it is possible to group individuals regardless of the neighbourhood the venue is located in. Individuals within the same group have reviewed (a subset of) venues within the same categories. The implication is that a group denotes a set of individuals who are interested in similar types of food or services. That should imply that where there is an overlap in venue categories between neighbourhoods, there should be an overlap in groups visiting those venues.

By plotting and comparing the weighted intersection of forms to the weighted intersection of groups for every pair of neighbourhoods (This corresponds to a weighted version of the “intersection %” row in Table 19.) in Toronto, we can determine if our intuition is correct. Figure 3 depicts the plot of all Toronto neighbourhood pairs where the x-axis represents the percentage intersection of a pair of neighbourhoods’ forms, and the y-axis the percentage intersection of the two neighbourhoods’ groups. It appears that the weighted intersection of groups is dependent on the weighted intersection of forms above a 20% threshold. That is, the higher the weighted intersection of forms is for a pair of neighbourhoods, the higher the weighted intersection of groups is likely to be as well. However, this relationship is asymmetric in that form intersection does not seem to be dependent on group intersection. Overall, these observed patterns strongly indicate that where there is a critical mass of similar venues, similar groups of reviewers will be attracted. In other words, the genome of a neighbourhood influences the groups in the Hunome more strongly than vice versa, at this time scale. Additionally, our calculations of R² indicate that between 31 and 55% of the variation in the weighted intersection of groups observed in the Yelp data can be explained by the weighted intersection of forms. Therefore, our exploration of the correlation between the intersection of forms and groups seems to confirm that our aggregation of individuals from different cities based on their reviewing activity is valid.

Figure 3. Weighted intersection of forms against weighted intersection of groups.

Additionally, we observe (Figure 4) an interesting temporal pattern in the distribution of form and group intersections, wherein the weighted intersection of groups appears to decrease over time, resulting in the nested structure observed in our scatterplot. In fact, when only comparing the intersections of forms and groups for neighbourhoods similar to the Waterfront, this temporal trend is even more apparent. In Figure 4 the blue dots represent year 1 and the brown dots year 6. There is a clear downward progression each year.

Figure 4. Subset of weighted intersection of forms against weighted intersection of groups.

Looking at the most prolific group of reviewers (by far) from all 4 neighbourhoods, we found that while the number of reviews they made over time increased (Figure 5), the proportion of their reviews to the total number of reviews made in each neighbourhood in each year decreased. This pattern also held true for other groups common to all 4 neighbourhoods.

Figure 5. Review Counts and Proportions of the Most Active Group.

Because we found this trend present in several groups, this suggested that the number of groups making reviews has been increasing over time, resulting in the smaller proportions of the total number of reviews per neighbourhood per year. We determined the exact number of groups active as well, finding a clear increase in the number of groups making reviews in each neighbourhood over time (Figure 6).

Figure 6. Number of Groups in a Subset of Neighbourhoods.

Therefore, as the number of groups making reviews in each neighbourhood increases from a few hundred to a few thousand over the course of six years, it is unsurprising that we observe a decrease in the intersection of groups between neighbourhoods over time (it is more difficult to share thousands of common groups than hundreds). The correlation between the weighted intersection of forms and groups would have likely been stronger without this dampening effect on the weighted intersection of groups.

What does Yelp data tell us about the Model?

In [1], several propositions regarding density and its role in the selection process are defined. In our aggregation discussion above, a different type of density proposition emerges. The 20% form intersection threshold can be interpreted as a “density threshold” where higher levels of form intersection leads to higher levels of group intersection. On the other hand, Figure 5 tells us that the most active groups continue to remain active, but represent a smaller percentage of reviews over time due to the attraction of new groups interested in the forms. This demonstrates the attractive force of neighbourhoods that specialize in forms and their associated activities. Secondly, it implies that signalling is occurring from the neighbourhood to other groups interested in the forms and activities they support, leading to an increase in groups in the neighbourhood over time, providing a rough indication of the notion of signal reach. An open question is whether the change in groups over time will in turn alter the forms.

What does the model tell us about Yelp data?

Table 27 compares the longitudinal cumulative intersection of both forms and groups with respect to the first year of the Annex and Waterfront. With respect to forms, both neighbourhoods show decrease over the six year period from 76–81% commonality of venues reviewed to 50–56% by the sixth year. Whether this represents a change in the categories of venues found in each neighbourhood over time, or reviewer fatigue is unclear. On the other hand, the cumulative intersection of groups indicates a less transient reviewer population in the Annex than the Waterfront—which is understandable given the Waterfront is more of a tourist area. Nevertheless, over the six year time period the cumulative intersection of both forms and groups consistently decreases, signifying change in both venues and reviewers.

As we observed in Figure 6 this decrease in the cumulative intersection of groups is driven by the increasing number of groups making reviews per neighbourhood over time. The increase in groups could be due to a larger, more varied Yelp user base as the platform establishes itself. However, it may also be due to the signalling effect of Yelp.

9. Conclusions

The Toronto Urban Evolution Model provides an interesting framework for thinking about how urban areas may evolve overtime. The introduction of formemes as both the representation of a spatial area’s “genes” and as what is signalled among spatial areas provides an intriguing way of how to think about urban evolution. The model has been applied descriptively to several examples of urban evolution [4,8,9,10,11]. The goal of this research is to explore how large amounts of data can be encoded in the model, and to see what insights the model enables us to draw. Yelp data was chosen both for its availability and that it reduces model complexity by having a single activity. However, Yelp data presents many challenges, including the lack of precision in its category taxonomy, business choice of categories, lack of groupings of individual reviewers, and the extent which reviews are consistent with actual activity in a neighbourhood.

Despite the limitations of the Yelp data, the application of TUEM has led to a number of insights, such as the persistence of forms in certain neighbourhoods over time (amidst broader social change), and the ability of these persistent forms to draw new groups to the neighbourhood. However, the most interesting insights emerged from the formetic distance between pairs of neighbourhoods in Toronto and between Toronto and Montreal. Neighourhoods found to be formetically close to each were confirmed by their descriptive characterizations.

The analysis performed here is only the tip of the iceberg. With higher quality data and more diverse types of data, we can only imagine the insights TUEM will afford us.