Package 'SINT'

Aggregate manifest responses with a quota rule

Description

Returns $\Theta(\sum_i 1(P_i = 1) - q n)$, where $\Theta$ is the Heaviside step function (1 for positive arguments, 0 otherwise). Only responses equal to 1 count in favor; abstentions (0) and opposing responses (-1) count as not in favor. With quota = 0.5 this is the simple majority rule.

Usage

aggregate_quota(P, quota = 0.5)

Arguments

P	Vector of manifest responses, or a matrix with one row per time step as returned by sint_simulate().
quota	Quota $q$ in $[0, 1]$.

Value

An integer (0 or 1), or an integer vector with one outcome per row when P is a matrix.

Examples

aggregate_quota(c(1, 1, 0, -1, 1))
aggregate_quota(c(1, 1, 0, -1, 1), quota = 2/3)

---

Normative climate from manifest responses

Description

Computes $S = (n^+ - n^-) / (n^+ + n^- + \epsilon)$, where $n^+$ and $n^-$ count the responses equal to $+1$ and $-1$.

Usage

climate_balance(P, eps)

Arguments

P	Vector of manifest responses in $\{-1, 0, +1\}$.
eps	Positive regularization constant $\epsilon$.

Value

A single number in $(-1, 1)$.

Examples

climate_balance(c(1, 1, 1, 0, 0, 0, 0, 0, 0, 0), eps = 0.1)

# As a normative pressure in a response function
f <- response_threshold(delta = 0.5, theta = 0.4, gamma = 0.9,
                        S = function(P) climate_balance(P, eps = 0.1))

---

Check the inputs of a Friedkin-Johnsen model

Description

Validates an influence matrix and a susceptibility vector and reports the quantities that govern convergence of the Friedkin-Johnsen dynamics $y(t+1) = \Lambda W y(t) + (I - \Lambda) y(0)$.

Usage

fj_check(W, lambda, tol = sqrt(.Machine$double.eps))

Arguments

W	Square numeric influence matrix.
lambda	Numeric vector of susceptibilities in $[0, 1]$, of length one (recycled) or equal to nrow(W).
tol	Numerical tolerance used when checking row sums.

Details

A non-negative W must be row-stochastic. A signed W (at least one negative entry) must have absolute row sums not greater than one.

Value

A list with elements:

n

Number of agents.

signed

TRUE if W has negative entries.

spectral_radius

Spectral radius of $\Lambda W$.

norm_inf

Infinity norm of $\Lambda W$.

converges

TRUE if the spectral radius is below one, so that the dynamics converge to a unique fixed point.

Examples

W <- matrix(c(0.5, 0.5, 0,
              0.2, 0.6, 0.2,
              0,   0.3, 0.7), nrow = 3, byrow = TRUE)
fj_check(W, lambda = c(0.8, 0.5, 0.9))

---

Equilibrium opinions of a Friedkin-Johnsen model

Description

Computes the unique fixed point $y^* = (I - \Lambda W)^{-1} (I - \Lambda) y(0)$ of the dynamics $y(t+1) = \Lambda W y(t) + (I - \Lambda) y(0)$. An error is raised when the spectral radius of $\Lambda W$ is not below one.

Usage

fj_equilibrium(W, lambda, y0, tol = sqrt(.Machine$double.eps))

Arguments

W	Square numeric influence matrix.
lambda	Numeric vector of susceptibilities in $[0, 1]$, of length one (recycled) or equal to nrow(W).
y0	Numeric vector of initial opinions, of length nrow(W).
tol	Numerical tolerance used when checking row sums.

Value

A numeric vector of equilibrium opinions.

See Also

fj_influence(), fj_simulate()

Examples

W <- matrix(c(0.5, 0.5, 0,
              0.2, 0.6, 0.2,
              0,   0.3, 0.7), nrow = 3, byrow = TRUE)
fj_equilibrium(W, lambda = c(0.8, 0.5, 0.9), y0 = c(0, 0.5, 1))

---

Total influence matrix of a Friedkin-Johnsen model

Description

Computes $V = (I - \Lambda W)^{-1} (I - \Lambda)$, whose entry $v_{ij}$ is the total effect of the initial opinion of agent $j$ on the equilibrium opinion of agent $i$. When W is non-negative, the rows of V sum to one.

Usage

fj_influence(W, lambda, tol = sqrt(.Machine$double.eps))

Arguments

W	Square numeric influence matrix.
lambda	Numeric vector of susceptibilities in $[0, 1]$, of length one (recycled) or equal to nrow(W).
tol	Numerical tolerance used when checking row sums.

Value

A numeric $n \times n$ matrix.

See Also

fj_equilibrium(), fj_check()

Examples

W <- matrix(c(0.5, 0.5, 0,
              0.2, 0.6, 0.2,
              0,   0.3, 0.7), nrow = 3, byrow = TRUE)
fj_influence(W, lambda = c(0.8, 0.5, 0.9))

---

Simulate the Friedkin-Johnsen dynamics

Description

Iterates $y(t+1) = \Lambda W y(t) + (I - \Lambda) y(0)$ until the largest absolute change between two steps falls below tol or max_steps is reached. Unlike fj_equilibrium(), the iteration is carried out even when the dynamics do not converge.

Usage

fj_simulate(W, lambda, y0, max_steps = 1000L, tol = 1e-10)

Arguments

W	Square numeric influence matrix.
lambda	Numeric vector of susceptibilities in $[0, 1]$, of length one (recycled) or equal to nrow(W).
y0	Numeric vector of initial opinions, of length nrow(W).
max_steps	Maximum number of iterations.
tol	Convergence tolerance on the largest absolute change between two consecutive steps.

Value

A list with elements:

trajectory

Matrix with one row per time step, from $t = 0$ to the last step, and one column per agent.

steps

Number of iterations performed.

converged

TRUE if the stopping criterion was met before max_steps.

See Also

fj_equilibrium()

Examples

W <- matrix(c(0.5, 0.5, 0,
              0.2, 0.6, 0.2,
              0,   0.3, 0.7), nrow = 3, byrow = TRUE)
sim <- fj_simulate(W, lambda = c(0.8, 0.5, 0.9), y0 = c(0, 0.5, 1))
tail(sim$trajectory, 1)

---

Build an influence matrix from a network

Description

Converts a matrix, an igraph graph or a network object into a row-normalized influence matrix suitable for the other functions of the package. The adjacency matrix, optionally valued with an edge attribute, is normalized with row_normalize().

Usage

influence_matrix(x, weights = NULL, direction = c("attention", "influence"))

Arguments

x	A square numeric matrix, an igraph graph or a network object.
weights	Optional name of a numeric edge attribute holding the tie weights; if NULL, all ties have weight one. Ignored when x is a matrix.
direction	Meaning of an edge from $i$ to $j$: "attention" ($i$ attends to $j$) or "influence" ($i$ influences $j$).

Details

With direction = "attention", an edge from $i$ to $j$ means that $i$ attends to $j$, so that $w_{ij} > 0$; this matches the row-wise reading of $W$. With direction = "influence", an edge from $i$ to $j$ means that $i$ influences $j$, and the adjacency matrix is transposed.

For network objects, self-ties are present only if the object was created with loops = TRUE.

Agents with no ties in their row cannot be normalized. They are given a unit self-weight, so that they attend only to themselves, and a warning is issued.

Value

A numeric matrix whose rows have unit absolute sums, with vertex names as dimnames when available.

See Also

row_normalize(), fj_check()

Examples

A <- matrix(c(0, 1, 1,
              1, 0, 0,
              0, 1, 1), nrow = 3, byrow = TRUE)
influence_matrix(A)
influence_matrix(A, direction = "influence")

if (requireNamespace("igraph", quietly = TRUE)) {
  g <- igraph::graph_from_literal(a -+ b, b -+ c, c -+ a, a -+ c)
  influence_matrix(g)
}

---

Logistic response function

Description

Builds a response function mapping latent opinions to binary manifest responses through $\Pr(P_i = 1) = \mathrm{logit}^{-1}(\beta_i (y_i - \delta) + \gamma_i S_i)$.

Usage

response_logistic(beta = 1, delta = 0, gamma = 0, S = 0, stochastic = TRUE)

Arguments

beta	Discrimination $\beta$, of length one or $n$.
delta	Adhesion threshold $\delta$.
gamma	Sensitivity $\gamma$ to normative pressure, of length one or $n$.
S	Normative pressure: a number, a vector of length $n$, or a function of the previous manifest state.
stochastic	If TRUE, responses are drawn from Bernoulli distributions; if FALSE, the response probabilities are returned.

Details

S may be a fixed number (or vector) or a function of the previous manifest state P, such as climate_balance(). When S is a function and no previous manifest state exists (P is NULL), S is taken as 0.

Value

A function with arguments ⁠(t, y, P)⁠ for use in sint_simulate().

See Also

response_threshold(), sint_simulate()

Examples

f <- response_logistic(beta = 4, delta = 0.5, stochastic = FALSE)
f(1, y = c(0.2, 0.5, 0.9), P = NULL)

---

Double-threshold response function

Description

Builds a response function mapping latent opinions to ternary manifest responses. With $z_i = \beta_i (y_i - \delta) + \gamma_i S_i$, the response is $+1$ if $z_i > \theta$, $-1$ if $z_i < -\theta$ and 0 (silence) otherwise.

Usage

response_threshold(
  beta = 1,
  delta = 0,
  theta,
  gamma = 0,
  S = 0,
  stochastic = FALSE
)

Arguments

beta	Discrimination $\beta$, of length one or $n$.
delta	Adhesion threshold $\delta$.
theta	Positive half-width $\theta$ of the silence band.
gamma	Sensitivity $\gamma$ to normative pressure, of length one or $n$.
S	Normative pressure: a number, a vector of length $n$, or a function of the previous manifest state.
stochastic	If TRUE, a logistic error is added to $z_i$.

Details

In the stochastic version a standard logistic error is added to $z_i$, which yields an ordered logit model with thresholds $\pm\theta$. S is handled as in response_logistic().

Value

A function with arguments ⁠(t, y, P)⁠ for use in sint_simulate().

See Also

response_logistic(), climate_balance(), sint_simulate()

Examples

f <- response_threshold(delta = 0.5, theta = 0.4)
f(1, y = c(0.05, 0.5, 0.95), P = NULL)

---

Normalize the rows of an influence matrix

Description

Divides each row by the sum of the absolute values of its entries. For a non-negative matrix the result is row-stochastic; for a signed matrix the absolute row sums equal one, as required by fj_check().

Usage

row_normalize(V)

Arguments

V	Numeric matrix of unnormalized, possibly signed, weights.

Value

A numeric matrix with the same dimensions and names as V.

Examples

V <- matrix(c(2, 1, 1,
              0, 3, 1,
              1, 1, 2), nrow = 3, byrow = TRUE)
row_normalize(V)

---

Simulate latent and manifest opinion dynamics

Description

Simulates a Friedkin-Johnsen process whose influence matrix and susceptibilities may depend on time and on the current state, coupled with an optional response function that maps latent opinions to manifest responses. At each step from $t$ to $t + 1$:

$y(t+1) = \Lambda(t) W(t) y(t) + (I - \Lambda(t)) y(0)$

$P(t+1) = f(t + 1, y(t+1), P(t))$

where $f$ is response.

Usage

sint_simulate(W, lambda, y0, steps, response = NULL, P0 = NULL)

Arguments

W	Influence matrix, or a function ⁠(t, y, P)⁠ returning one.
lambda	Susceptibility vector, or a function ⁠(t, y, P)⁠ returning one.
y0	Numeric vector of initial opinions, of length nrow(W).
steps	Number of steps to simulate.
response	Optional response function; if NULL, only the latent dynamics are simulated.
P0	Optional initial manifest state.

Details

W and lambda may be fixed objects or functions with arguments ⁠(t, y, P)⁠, called with the current time $t$, the latent state $y(t)$ and the manifest state $P(t)$ (NULL when there is no response function). They must return inputs valid for fj_check().

response is a function with arguments ⁠(t, y, P)⁠, called with the new time $t + 1$, the new latent state $y(t+1)$ and the previous manifest state $P(t)$; see response_logistic() and response_threshold(). When P0 is NULL, the initial manifest state is computed as response(0, y0, NULL).

External sources with fixed opinions can be represented as agents with susceptibility 0 and a unit self-weight.

Value

A list with elements:

y

Matrix of latent opinions, one row per time step from $t = 0$ to steps, one column per agent.

P

Matrix of manifest responses with the same layout, or NULL when response is NULL.

See Also

fj_simulate(), response_threshold(), aggregate_quota()

Examples

W <- matrix(c(0.5, 0.5, 0,
              0.2, 0.6, 0.2,
              0,   0.3, 0.7), nrow = 3, byrow = TRUE)
f <- response_threshold(delta = 0.5, theta = 0.1, gamma = 0.2,
                        S = function(P) climate_balance(P, eps = 0.1))
sim <- sint_simulate(W, lambda = 0.8, y0 = c(0.1, 0.5, 0.9),
                     steps = 10, response = f)
sim$P
aggregate_quota(sim$P)

---