lolog function

Fits a LOLOG model via Monte Carlo Generalized Method of Moments

lolog is used to fit Latent Order Logistic Graph (LOLOG) models. LOLOG models are motivated by the idea of network growth where the network begins empty, and edge variables are sequentially 'added' to the network with an either unobserved, or partially observed order $s$. Conditional upon the inclusion order, the probability of an edge has a logistic relationship with the change in network statistics.

lolog(
  formula,
  auxFormula = NULL,
  theta = NULL,
  nsamp = 1000,
  includeOrderIndependent = TRUE,
  targetStats = NULL,
  weights = "full",
  tol = 0.1,
  nHalfSteps = 10,
  maxIter = 100,
  minIter = 2,
  startingStepSize = 0.1,
  maxStepSize = 0.5,
  cluster = NULL,
  verbose = TRUE
)

Arguments

• formula: A lolog formula for the sufficient statistics (see details).
• auxFormula: A lolog formula of statistics to use for moment matching.
• theta: Initial parameters values. Estimated via lologVariational if NULL.
• nsamp: The number of sample networks to draw at each iteration.
• includeOrderIndependent: If TRUE, all order independent terms in formula are used for moment matching.
• targetStats: A vector of network statistics to use as the target for the moment equations. If NULL, the observed statistics for the network are used.
• weights: The type of weights to use in the GMM objective. Either 'full' for the inverse of the full covariance matrix or 'diagonal' for the inverse of the diagonal of the covariance matrix.
• tol: The Hotelling's T^2 p-value tolerance for convergence for the transformed moment conditions.
• nHalfSteps: The maximum number of half steps to take when the objective is not improved in an iteration.
• maxIter: The maximum number of iterations.
• minIter: The minimum number of iterations.
• startingStepSize: The starting dampening of the parameter update.
• maxStepSize: The largest allowed value for dampening.
• cluster: A parallel cluster to use for graph simulation.
• verbose: Level of verbosity 0-3.

Returns

An object of class 'lolog'. If the model is dyad independent, the returned object will also be of class "lologVariational" (see lologVariational, otherwise it will also be a "lologGmm" object.

lologGmm objects contain:

• method: "Method of Moments" for order independent models, otherwise "Generalized Method of Moments"

• formula: The model formula

• auxFormula: The formula containing additional moment conditions

• theta: The parameter estimates

• stats: The statistics for each network in the last iteration

• estats: The expected stats (G(y,s)) for each network in the last iteration

• obsStats: The observed h(y) network statistics

• targetStats: The target network statistics

• obsModelStats: The observed g(y,s) network statistics

• net: A network simulated from the fit model

• grad: The gradient of the moment conditions (D)

• vcov: The asymptotic covariance matrix of the parameter estimates

• likelihoodModel: An object of class *LatentOrderLikelihood at the fit parameters

Details

LOLOG represents the probability of a tie, given the network grown up to a time point as

$$\textrm{logit}\big(p(y_{s_t}=1 | \eta, y^{t-1}, s_{ \leq t})\big) = \theta \cdot c(y_{s_t}=1 | y^{t-1}, s_{ \leq t})$$

where $s_{\leq t}$ is the growth order of the network up to time $t$, $y^{t-1}$ is the state of the graph at time $t-1$. $c(y_{s_t} | y^{t-1}, s_{ \leq t})$ is a vector representing the change in graph statistics from time $t-1$ to $t$ if an edge is present, and $\theta$ is a vector of parameters.

The motivating growth order proceeds 'by vertex.' The network begins 'empty' and then vertices are 'added' to the network sequentially. The order of vertex inclusion may be random or fixed. When a vertex 'enters' the network, each of the edge variables connecting it and vertices already in the network are considered for edge creation in a completely random order.

LOLOG formulas contain a network, DirectedNet or UndirectedNet object on the left hand side. the right hand side contains the model terms used. for example,

net ~ edges

represents and Erdos-Renyi model and

net ~ edges + preferentialAttachment()

represents a Barabasi-Albert model. See lolog-terms for a list of allowed model statistics

Conditioning on (partial) vertex order can be done by placing an ordering variable on the right hand side of the '|' operator, as in

net ~ edges + preferentialAttachment() | order

'order' should be a numeric vector with as many elements as there are vertices in the network. Ties are allowed. Vertices with higher order values will always be included later. Those with the same values will be included in a random order in each simulated network.

offsets and constraints are specified by wrapping them with either offset() or constraint(), for example, the following specifies an Erdos-Renyi model with the constraint that degrees must be less that 10

net ~ edges + constraint(boundedDegree(0L, 10L))

If the model contains any order dependent statistics, additional moment constraints must be specified in auxFormula. Ideally these should be chosen to capture the features modeled by the order dependent statistics. For example, preferentialAttachment

models the degree structure, so we might choose two-stars as a moment constraint.

lolog(net ~ edges + preferentialAttachment(), net ~ star(2))

will fit a Barabasi-Albert model with the number of edges and number of two-stars as moment constraints.

Examples

library(network)
set.seed(1)
data(flo)
flomarriage <- network(flo,directed=FALSE)
flomarriage %v% "wealth" <- c(10,36,27,146,55,44,20,8,42,103,48,49,10,48,32,3)

# A dyad independent model
fit <- lolog(flomarriage ~ edges + nodeCov("wealth"))
summary(fit)

# A dyad dependent model with 2-stars and triangles
fit2 <- lolog(flomarriage ~ edges + nodeCov("wealth") + star(2) + triangles, verbose=FALSE)
summary(fit2)

## Not run:

# An order dependent model
fit3 <- lolog(flomarriage ~ edges + nodeCov("wealth") + preferentialAttachment(),
              flomarriage ~ star(2:3), verbose=FALSE)
summary(fit3)

# Try something a bit more real
data(ukFaculty)

# Delete vertices missing group
delete.vertices(ukFaculty, which(is.na(ukFaculty %v% "Group")))

fituk <- lolog(ukFaculty ~ edges() + nodeMatch("GroupC") + nodeCov("GroupC") + triangles + star(2))
summary(fituk)
plot(fituk$net, vertex.col= ukFaculty %v% "Group" + 2)
## End(Not run)