My CS specification does NOT allow for marks to contain edges between two old nodes.
The CS (“change-statistic”) kernel is also a new-node attachment model, but instead of choosing attachment targets using degree weights (as in BA), it uses ERGM-style change statistics to score each possible new–old edge.
At each event time \(t\):
A random number of new nodes arrives: \[
M_t \sim \text{Poisson}(\lambda_{\text{node}}),
\] (in the code this is node_lambda). Optionally, arrivals can be forced to stop after a cutoff time max_node_time (in which case \(M_t\) must be 0 for \(t\) past the cutoff).
Let \(V(t^-)\) be the set of existing (“old”) nodes right before time \(t\), and let \(N(t)\) be the set of \(M_t\) new nodes arriving at time \(t\).
The model defines a candidate set of possible edges. In the current implementation, candidate edges are constructed deterministically as \[
\{(u, v): u \in N(t),\; v \in V(t^-)\},
\] with an optional truncation that limits each new node to only consider the first truncation old nodes (according to the current ordering used internally) (Currently not implemented, argument doesn’t do anything but it’s not needed).
For each candidate edge \((u, v)\) (new node \(u\) to old node \(v\)), a vector of change statistics \[
C_{uv}(t)
\] is computed from the current network using the ERNM/ERGM-style specification you provide (via formula_rhs / computeChangeStats()).
These statistics are mapped to a baseline attachment probability using a logistic link: \[ p^{(0)}_{uv}(t) = \text{logit}^{-1}\!\left(C_{uv}(t)^\top \theta\right). \]
Finally, an aging / recency adjustment is applied to down-weight older target nodes: \[ p_{uv}(t) = p^{(0)}_{uv}(t)\,\exp\!\big(-\beta_{\text{edges}}\cdot \text{age}_v(t)\big), \] where \(\text{age}_v(t) = t - t_v\) and \(t_v\) is the stored node arrival time. (In the implementation, probabilities are also clipped away from 0/1 for numerical stability.)
Under this mark model, each candidate edge is treated as an independent Bernoulli trial, so the probability of observing a particular set of edges at time \(t\) is a Bernoulli-product over the candidate set.
To use the CS kernel, your events must match the “new nodes arrive + may connect to existing nodes” pattern:
Importantly (based on the current PMF / validators):
If your data includes edges forming between two existing nodes (e.g., repeated interactions/transactions, new ties between old users, etc.), then the current CS mark model is not appropriate without extending the mark space / PMF.
Simulate CS data.
set.seed(1)
time <- 15
params_cs_true <- list(
mu = 5,
K = 0.5,
beta = 2,
beta_edges = 0.5,
node_lambda = 4,
# CS_edges = -2.5,
# CS_triangles = 0.001,
CS_star.2 = -1,
CS_star.3 = -3
)
sim <- sim_hawkesNet(
params = params_cs_true,
T_end = time,
mark_type = "cs",
# formula_rhs = "edges + triangles()
# + star(c(2,3))",
formula_rhs = "star(c(2,3))"
)[1] "Simulation took 0.99 seconds"
Fit CS data.
Make sure to specify the transformation setting for the CS parameters.
Ideally will default to correct value, but want to discuss how we are implementing the arguments, because lowk I like the old way of passing CS_params better.
params_cs_init <- list(
mu = 3,
K = 1,
beta = 1,
beta_edges = 1,
node_lambda = 5,
# CS_edges = -2.5,
# CS_triangles = 0.001,
CS_star.2 = 0,
CS_star.3 = 0
)
fit <- fit_hawkesNet(
ev = sim$ev,
params_init = params_cs_init,
mark_type = "cs",
transform = list(CS_star.2 = "none", CS_star.3 = "none"),
formula_rhs = "star(c(2,3))"
)[1] "Fitting took 9.31 seconds"Parameter values on the fitted scale:
unlist(fit$par) mu K beta beta_edges node_lambda CS_star.2
4.6384795 0.7236149 3.3119207 0.4046910 4.1888536 -0.9812649
CS_star.3
-3.1691266 Notttt baddddd.
Parameter values on the transformed scale:
fit$fit$par mu K beta beta_edges node_lambda CS_star.2
1.5343866 -0.3234959 1.1975283 -0.9046315 1.4324271 -0.9812649
CS_star.3
-3.1691266