impact_susceptibility {NetworkRiskMeasures}

Impact Susceptibility, Fluidity and Diffusion

Description

The impact_susceptibility measures the feasible contagion paths that can reach a vertex in relation to its direct contagion paths. When the impact susceptibility is greater than 1, it means that the vertex is vulnerable to other vertices beyond its direct neighbors (remotely vulnerable).

The impact_fluidity is simply the average of the impact susceptibility in the network.

The impact_diffusion tries to capture the influence exercised by a node on the propagation of impacts in the network. The impact diffusion of a vertex is measured by the change it causes on the impact susceptibility of other vertices when its power to propagate contagion is removed from the network.

All these measures are based on the communicability of the vulnerability matrix (see vulnerability_matrix and communicability_matrix).

Usage

impact_susceptibility(
  exposures,
  buffer,
  weights = NULL,
  terms = Inf,
  sparse = TRUE,
  binary = TRUE,
  exposure_type = c("assets", "liabilities", "impact", "vulnerability")
)

impact_fluidity(
  exposures,
  buffer,
  weights = NULL,
  terms = Inf,
  sparse = TRUE,
  binary = TRUE,
  exposure_type = c("assets", "liabilities", "impact", "vulnerability")
)

impact_diffusion(
  exposures,
  buffer,
  weights = NULL,
  terms = Inf,
  sparse = TRUE,
  binary = TRUE,
  exposure_type = c("assets", "liabilities", "impact", "vulnerability")
)

Arguments

exposures

an adjacency matrix, (sparse) Matrix or an igraph object with the network of bilateral exposures between vertices. By default, the function expects the exposures in the form of an assets matrix in which A -> B means that A has an asset with B. However, you can change that with the parameter exposure_type. When using a matrix, preferably it should have rows and columns names.

buffer

a numeric vector with the capital buffer for each vertex. Values should be in the same row/column order as the network of bilateral exposures. The buffer is not needed if exposure_type = "vulnerability".

weights

default is NULL. You can use a numeric vector of weights to give some economic significance to the measures, like, for instance, the total assets of the nodes.

terms

truncates the communicability matrix evaluation up to a pre-defined number of terms. If terms = Inf the function computes the matrix exponential using expm.

sparse

should the function use sparse matrices when computing the communicability? However, if terms = Inf the function will use expm which uses dgeMatrix-class.

binary

if binary = TRUE the function computes a 'binary' impact or vulnerability matrix. It truncates all values less than 1 to 0 and all values greater than 1 to 1.

exposure_type

character vector indicating the type of the bilateral exposures. It can be an "assets" network (where A -> B means that A has an asset with B), a "liabilities" network (where A -> B means that A has a debt with B), a (binary) "impact" matrix (where A -> B indicates the relative impact of A in B's capital buffer), or a (binary) "vulnerability" matrix (where A -> B indicates the relative impact A suffers from B's default). The default is "assets".

Value

The impact_susceptibility function returns a vector with the (weighted) impact susceptibility

The impact_fluidity function returns a vector with the (weighted) impact fluidity of the network.

The impact_diffusion function returns a data.frame with the vertex name and the (weighted) start, intermediate and total impact diffusion.

References

Silva, T.C.; Souza, S.R.S.; Tabak, B.M. (2015) Monitoring vulnerability and impact diffusion in financial networks. Working Paper 392, Central Bank of Brazil.

Silva, T.C.; Souza, S.R.S.; Tabak, B.M. (2015) Network structure analysis of the Brazilian interbank market . Working Paper 391, Central Bank of Brazil.

Examples

# Creating example data
## Assets Matrix (bilateral exposures)
assets_matrix <- matrix(c(0, 10, 3, 1, 0, 2, 0, 3, 0), ncol = 3)
rownames(assets_matrix) <- colnames(assets_matrix) <- letters[1:3]

## Capital Buffer
buffer <- c(a = 2, b = 5, c = 2)

# Measures
impact_susceptibility(assets_matrix, buffer)
impact_fluidity(assets_matrix, buffer)
impact_diffusion(assets_matrix, buffer)