infection function

Susceptibility and Infection

Calculates infectiousness and susceptibility for each node in the graph

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)
)

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.

Returns

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

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%S(i) = [\sum_k \sum_j (x(ij,t-k+1) * z(j,t-k))/w(k)]/[\sum_k \sum_j (x(ij,t-k+1) * z(j, 1<=t<=t-k))/w(k)] for j != i

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 $t-k$, $z(j, 1<=t<=t-k)$ 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%I(i) = [\sum_k \sum_j (x(ji,t) * z(j,t+1))/w(k)]/[\sum_k \sum_j (x(ji,t) * z(j, t+1<=t<=T))/w(k)] for j != i

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

$$$$

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

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).

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)

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()

Author(s)

George G. Vega Yon