Diffusive Metrics Induced by Random Affinities on Graphs. An Application to the Transport Systems Related to the COVID-19 Setting for Buenos Aires (AMBA)

The aim of this paper is to apply the diffusive metric technique defined by the spectral analysis of graph Laplacians to the set of the 41 cities belonging to AMBA, the largest urban concentration in Argentina, based on public transport and neighborhood. It could be expected that the propagation of any epidemic desease would follow the paths determined by those metrics. Our result reflects that the isolation measures decided by the health administration helped at the atenuation of the actual spread of COVID-19 in AMBA.

Keywords:
weighted graphs; diffusion; graph Laplacian; metrization; COVID-19

Authorship

M. F. ACOSTA

Instituto de Matemática Aplicada del Litoral, CONICET, UNL, FICH, Santa Fe, Argentina - E-mail: mfacosta@santafe-conicet.gov.arCONICET, UNLArgentinaSanta Fe, ArgentinaInstituto de Matemática Aplicada del Litoral, CONICET, UNL, FICH, Santa Fe, Argentina - E-mail: mfacosta@santafe-conicet.gov.ar

H. AIMAR

Instituto de Matemática Aplicada del Litoral, CONICET, UNL, Santa Fe, Argentina - E-mail: haimar@santafe-conicet.gov.arCONICET, UNLArgentinaSanta Fe, ArgentinaInstituto de Matemática Aplicada del Litoral, CONICET, UNL, Santa Fe, Argentina - E-mail: haimar@santafe-conicet.gov.ar

https://orcid.org/0000-0003-3829-7738

I. GÓMEZ ^**Corresponding author: Ivana Gómez - E-mail: ivanagomez@santafe-conicet.gov.ar

Instituto de Matemática Aplicada del Litoral, CONICET, UNL, Santa Fe, Argentina - E-mail: ivanagomez@santafe-conicet.gov.arCONICET, UNLArgentinaSanta Fe, ArgentinaInstituto de Matemática Aplicada del Litoral, CONICET, UNL, Santa Fe, Argentina - E-mail: ivanagomez@santafe-conicet.gov.ar

https://orcid.org/0000-0003-2921-2392

F. MORANA

Instituto de Matemática Aplicada del Litoral, CONICET, UNL, Santa Fe, Argentina - E-mail: fmorana@santafe-conicet.gov.arCONICET, UNLArgentinaSanta Fe, ArgentinaInstituto de Matemática Aplicada del Litoral, CONICET, UNL, Santa Fe, Argentina - E-mail: fmorana@santafe-conicet.gov.ar

*Corresponding author: Ivana Gómez - E-mail: ivanagomez@santafe-conicet.gov.ar

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables | Formulas

Figures (12)
Tables (1)
Formulas (17)

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Figure 1:
A map of the 41 cities of AMBA. Buenos Aires city (CABA) has the label 30.

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Figure 2:
Unnormalized 20 × 41 submatrix of A ¹, the adjacency matrix provided by the neighborhood relation

A_{i j}^{1} = 1

when cities i and j share points of their boundaries.

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Figure 3:
The full unnormalized matrix A ⁰ built with the data of SUBE provided by the public transport system in AMBA.

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Figure 4:
Unnormalized 20 × 20 submatrix of A ², the adjacency weighted matrix provided by the length of the shared portions of the boundaries of the two cities.

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Figure 5:
Unnormalized 20 × 20 submatrix of A ³, the weighted matrix provided by the product of the lengths of the shared boundaries times the minimum of their population.

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Figure 6:
Unnormalized 20 × 20 submatrix of A ⁴, the matrix provided by the minimum of the population of any two neighboring cities.

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Figure 7:
Diffusion distances to CABA (30) for t = 0.25 with affinity matrix A ⁰ and node uniform weights given by

\vec{a_{1}}

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Figure 8:
Diffusion distances to CABA (30) for t = 0.25 in eigth different instances of affinities

(θ A^{0} + (1 - θ) A^{i}

with i = 1, 2, 3, 4 and

θ = 0, \frac{1}{2}

) and node uniform weights given by

\vec{a_{1}}

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Figure 9:
Days up to 0.1% of infections over the population (from 0.1% of CABA).

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Figure 10:
Days up to 0.3% of infections over the population (from 0.1% of CABA).

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Figure 11:
Percentage of total infections for the first and second halves for June, July and August 2020.

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Figure 12:
Evolution of cases in CABA (red) and La Matanza (blue).

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Table 1:
Relative differences.

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Figure 1: A map of the 41 cities of AMBA. Buenos Aires city (CABA) has the label 30.

Figure 2: Unnormalized 20 × 41 submatrix of A ¹, the adjacency matrix provided by the neighborhood relation $A_{i j}^{1} = 1$ when cities i and j share points of their boundaries.

Figure 3: The full unnormalized matrix A ⁰ built with the data of SUBE provided by the public transport system in AMBA.

Figure 4: Unnormalized 20 × 20 submatrix of A ², the adjacency weighted matrix provided by the length of the shared portions of the boundaries of the two cities.

Figure 5: Unnormalized 20 × 20 submatrix of A ³, the weighted matrix provided by the product of the lengths of the shared boundaries times the minimum of their population.

Figure 6: Unnormalized 20 × 20 submatrix of A ⁴, the matrix provided by the minimum of the population of any two neighboring cities.

Figure 7: Diffusion distances to CABA (30) for t = 0.25 with affinity matrix A ⁰ and node uniform weights given by $\vec{a_{1}}$ .

Figure 8: Diffusion distances to CABA (30) for t = 0.25 in eigth different instances of affinities $(θ A^{0} + (1 - θ) A^{i}$ with i = 1, 2, 3, 4 and $θ = 0, \frac{1}{2}$ ) and node uniform weights given by $\vec{a_{1}}$ .

Figure 9: Days up to 0.1% of infections over the population (from 0.1% of CABA).

Figure 10: Days up to 0.3% of infections over the population (from 0.1% of CABA).

Figure 11: Percentage of total infections for the first and second halves for June, July and August 2020.

Figure 12: Evolution of cases in CABA (red) and La Matanza (blue).

Table 1: Relative differences.

$ϵ_{t}^{1, 0}$	0.12035607	$ϵ_{t}^{1, 0.5}$	0.0609088
$ϵ_{t}^{2, 0}$	0.17173178	$ϵ_{t}^{2, 0.5}$	0.091446
$ϵ_{t}^{3, 0}$	0.0644136	$ϵ_{t}^{3, 0.5}$	0.0306021
$ϵ_{t}^{4, 0}$	0.09062579	$ϵ_{t}^{3, 0.5}$	0.04661433

\sum_{i = 1}^{n} E a_{i} = E (\sum_{i = 1}^{n} a_{i}) = E (1) = 1, \sum_{i = 1}^{n} \sum_{j = 1}^{n} E A_{i j} = E (\sum_{i = 1}^{n} \sum_{j = 1}^{n} A_{i j}) = 1 .

\sum_{i = 1}^{n} a_{i} (ω) = 1, \ dis \sum_{i = 1}^{n} \sum_{j = 1}^{n} A_{i j} (ω) = 1

Ω = \underset{k \in ℤ}{\cup} Ω_{k}, Ω_{k} \cap Ω_{ℓ} = \emptyset .

E a_{i} = \int_{Ω} a_{i} (ω) d P = \sum_{k \in ℤ} \int_{Ω_{k}} a_{i} (ω) d P \geq \int_{Ω_{k_{0}}} a_{i} (ω) d P \geq 2^{- k_{0}} P (Ω_{k_{0}}) > 0 .

Δ_{Γ} f (i) = \frac{1}{a_{i}} \sum_{j = 1}^{n} A_{i j} (f (i) - f (j))

Δ_{Γ} = {\bar{\bar{a}}}^{- 1} (\bar{\bar{A}} - \bar{\bar{D}})

Δ_{E G} f (i) = \frac{1}{E a_{i}} \sum_{j = 1}^{n} E A_{i j} (f (i) - f (j))

Δ_{ω} f (i) = Δ_{G (ω)} f (i) = \frac{1}{a_{i} (ω)} \sum_{j = 1}^{n} A_{i j} (ω) (f (j) - f (i)),

Δ_{G (ω)} f (i) = \frac{1}{a_{i} (ω)} \sum_{j = 1} A_{i j} (ω) (f (j) - f (i)), ω \in Ω, i \in V,

E (Δ_{G (ω)} f (i)) = E (\frac{1}{a_{i}}) E (\sum_{j = 1}^{n} A_{i j} (f (j) - f (i))) = \frac{1}{{(E (\frac{1}{a_{i}}))}^{- 1}} \sum_{j = 1}^{n} E (A_{i j}) (f (j) - f (i)) = \frac{1}{b_{i}} \sum_{j = 1}^{n} E (A_{i j}) (f (j) - f (i)) = Δ_{\tilde{G}} f (i),

{<Δ_{Γ} f, g>}_{\bar{b}} = \sum_{i = 1}^{n} (Δ_{Γ} f) (i) g (i) b_{i} = \sum_{i = 1}^{n} (\frac{1}{b_{i}} \sum_{j = 1}^{n} B_{i j} (f (j) - f (i))) g (i) b_{i} = \sum_{j = 1}^{n} \sum_{i = 1}^{n} B_{i j} (f (j) - f (i)) g (i) = \sum_{j = 1}^{n} (\sum_{i = 1}^{n} B_{i j} f (j) g (i) - \sum_{i = 1}^{n} B_{i j} f (i) g (i)) = \sum_{j = 1}^{n} \sum_{i = 1}^{n} B_{i j} f (j) g (i) - \sum_{j = 1}^{n} \sum_{i = 1}^{n} B_{i j} f (i) g (i) = \sum_{i = 1}^{n} \sum_{j = 1}^{n} B_{i j} f (j) g (i) - \sum_{i = 1}^{n} \sum_{j = 1}^{n} B_{i j} f (i) g (i) = \sum_{i = 1}^{n} (\frac{1}{b_{i}} \sum_{j = 1}^{n} B_{i j} (g (j) - g (i))) f (i) b_{i} {<f, Δ_{Γ} g>}_{\bar{b}} .

{<- Δ_{Γ} f, f>}_{\bar{b}} = = \sum_{i = 1}^{n} (- Δ_{Γ} f) (i) f (i) b_{i} = \sum_{i = 1}^{n} \sum_{j = 1}^{n} B_{i j} (f (i) - f (j)) f (i) = \sum_{i = 1}^{n} \sum_{j = 1}^{n} B_{i j} f^{2} (i) - \sum_{i = 1}^{n} \sum_{j = 1}^{n} B_{i j} f (i) f (j) = \sum_{i = 1}^{n} \sum_{j = 1}^{n} B_{i j} (f^{2} (i) - f (i) f (j)) = \frac{1}{2} [\sum_{i = 1}^{n} \sum_{j = 1}^{n} B_{i j} (f^{2} (i) - f (i) f (j)) + \sum_{i = 1}^{n} \sum_{j = 1}^{n} B_{i j} (f^{2} (i) - f (i) f (j))] = \frac{1}{2} \sum_{i = 1}^{n} \sum_{j = 1}^{n} B_{i j} (f^{2} (i) + f^{2} (j) - 2 f (i) f (j)) = \frac{1}{2} \sum_{i = 1}^{n} \sum_{j = 1}^{n} B_{i j} (f (i) - f (j))^{2} \geq 0 .

{\vec{a}}_{2} = (0.0023, 0.0009, 0.0004, 0.0014, 0.0015, 0.0009, 0.0012, 0.0030, 0.0007, 0.0009, 0.0011, 0.0015, 0.0008, 0.0016, 0.0049, 0.0005, 0.0006, 0.0018, 0.0015, 0.0031, 0.0013, 0.0008, 0.0012, 0.0010, 0.0019, 0.0022, 0.0014, 0.0006, 0.0019, 0.0095, 0.0011, 0.0004, 0.0015, 0.0018, 0.0018, 0.0026, 0.0013, 0.0018, 0.0029, 0.0018, 0.0034)

B_{t}^{i, θ; j} (k, r) = {ℓ \in V : d_{t}^{i, θ; j} (k, ℓ) < r}

ϵ_{t}^{i, θ} = \frac{||d_{t}^{0, 0; 1} - d_{t}^{i, θ; 1}||}{||d_{t}^{0, 0; 1}||},

{||d_{t}^{0, 0; 1} - d_{t}^{i, θ; 1}||}^{2} = \sum_{k, ℓ = 1}^{n} {|d_{t}^{0, 0; 1} (k, ℓ) - d_{t}^{i, θ; 1} (k, ℓ)|}^{2}

{||d_{t}^{0, 0; 1}||}^{2} = \sum_{k, ℓ = 1}^{n} {(d_{t}^{0, 0; 1} (k, ℓ))}^{2} .

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Diffusive Metrics Induced by Random Affinities on Graphs. An Application to the Transport Systems Related to the COVID-19 Setting for Buenos Aires (AMBA)

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[1] *Corresponding author: Ivana Gómez - E-mail: ivanagomez@santafe-conicet.gov.ar