netdiffuseR (version 1.22.6)

infection: Susceptibility and Infection

Description

Calculates infectiousness and susceptibility for each node in the graph

Usage

infection(
  graph,
  toa,
  t0 = NULL,
  normalize = TRUE,
  K = 1L,
  r = 0.5,
  expdiscount = FALSE,
  valued = getOption("diffnet.valued", FALSE),
  outgoing = getOption("diffnet.outgoing", TRUE)
)
susceptibility(
  graph,
  toa,
  t0 = NULL,
  normalize = TRUE,
  K = 1L,
  r = 0.5,
  expdiscount = FALSE,
  valued = getOption("diffnet.valued", FALSE),
  outgoing = getOption("diffnet.outgoing", TRUE)
)

Value

A numeric column vector (matrix) of size \(n\) with either infection/susceptibility rates.

Arguments

graph

A dynamic graph (see netdiffuseR-graphs).

toa

Integer vector of length \(n\) with the times of adoption.

t0

Integer scalar. See toa_mat.

normalize

Logical. Whether or not to normalize the outcome

K

Integer scalar. Number of time periods to consider

r

Numeric scalar. Discount rate used when expdiscount=TRUE

expdiscount

Logical scalar. When TRUE, exponential discount rate is used (see details).

valued

Logical scalar. When TRUE weights will be considered. Otherwise non-zero values will be replaced by ones.

outgoing

Logical scalar. When TRUE, computed using outgoing ties.

Discount rate

Discount rate, \(w_k\) in the formulas above, can be either exponential or linear. When expdiscount=TRUE, \(w_k = (1 + r)^{k-1}\), otherwise it will be \(w_k = k\).

Note that when \(K=1\), the above formulas are equal to the ones presented in Valente et al. (2015).

Author

George G. Vega Yon

Details

Normalization, normalize=TRUE, is applied by dividing the resulting number from the infectiousness/susceptibility stat by the number of individuals who adopted the innovation at time \(t\).

Given that node \(i\) adopted the innovation in time \(t\), its Susceptibility is calculated as follows

$$S_i = \frac{% \sum_{k=1}^K\sum_{j=1}^n x_{ij(t-k+1)}z_{j(t-k)}\times \frac{1}{w_k}}{% \sum_{k=1}^K\sum_{j=1}^n x_{ij(t-k+1)}z_{j(1\leq t \leq t-k)} \times \frac{1}{w_k} }\qquad \mbox{for }i,j=1,\dots,n\quad i\neq j$$

where \(x_{ij(t-k+1)}\) is 1 whenever there's a link from \(i\) to \(j\) at time \(t-k+1\), \(z_{j(t-k)}\) is 1 whenever individual \(j\) adopted the innovation at time \(t-k\), \(z_{j(1\leq t \leq t-k)}\) is 1 whenever \(j\) had adopted the innovation up to \(t-k\), and \(w_k\) is the discount rate used (see below).

Similarly, infectiousness is calculated as follows

$$I_i = \frac{% \sum_{k=1}^K \sum_{j=1}^n x_{ji(t+k-1)}z_{j(t+k)}\times \frac{1}{w_k}}{% \sum_{k=1}^K \sum_{j=1}^n x_{ji(t+k-1)}z_{j(t+k\leq t \leq T)}\times \frac{1}{w_k} }\qquad \mbox{for }i,j=1,\dots,n\quad i\neq j$$

It is worth noticing that, as we can see in the formulas, while susceptibility is from alter to ego, infection is from ego to alter.

When outgoing=FALSE the algorithms are based on incoming edges, this is the adjacency matrices are transposed swapping the indexes \((i,j)\) by \((j,i)\). This can be useful for some users.

Finally, by default both are normalized by the number of individuals who adopted the innovation in time \(t-k\). Thus, the resulting formulas, when normalize=TRUE, can be rewritten as

$$% S_i' = \frac{S_i}{\sum_{k=1}^K\sum_{j=1}^nz_{j(t-k)}\times \frac{1}{w_k}} % \qquad I_i' = \frac{I_i}{\sum_{k=1}^K\sum_{j=1}^nz_{j(t-k)} \times\frac{1}{w_k}}$$

For more details on these measurements, please refer to the vignette titled Time Discounted Infection and Susceptibility.

References

Thomas W. Valente, Stephanie R. Dyal, Kar-Hai Chu, Heather Wipfli, Kayo Fujimoto Diffusion of innovations theory applied to global tobacco control treaty ratification, Social Science & Medicine, Volume 145, November 2015, Pages 89-97, ISSN 0277-9536 tools:::Rd_expr_doi("10.1016/j.socscimed.2015.10.001")

Myers, D. J. (2000). The Diffusion of Collective Violence: Infectiousness, Susceptibility, and Mass Media Networks. American Journal of Sociology, 106(1), 173–208. tools:::Rd_expr_doi("10.1086/303110")

See Also

The user can visualize the distribution of both statistics by using the function plot_infectsuscep

Other statistics: bass, classify_adopters(), cumulative_adopt_count(), dgr(), ego_variance(), exposure(), hazard_rate(), moran(), struct_equiv(), threshold(), vertex_covariate_dist()

Examples

# Creating a random dynamic graph
set.seed(943)
graph <- rgraph_er(n=100, t=10)
toa <- sample.int(10, 100, TRUE)

# Computing infection and susceptibility (K=1)
infection(graph, toa)
susceptibility(graph, toa)

# Now with K=4
infection(graph, toa, K=4)
susceptibility(graph, toa, K=4)