6  BA Kernel

Kernel Specification

The BA kernel is a new-node attachment model: at each event time, exactly one new node arrives, and it connects to a subset of the already-existing nodes.

In the current hawkesNet implementation, the set of attachments is modelled as follows:

  • Let \(V(t^-)\) be the set of existing (“old”) nodes right before the event at time \(t\).
  • For each old node \(u \in V(t^-)\), the new node attaches to \(u\) independently with probability \[ p_u(t) \propto (d_u(t^-) + \delta)\,\exp(-\beta_{\text{edges}}\cdot \text{age}_u(t)), \] where \(d_u(t^-)\) is the current degree of \(u\), and \(\text{age}_u(t) = t - t_u\) is how long ago node \(u\) arrived (using its stored node “time” attribute).
  • The probabilities are normalised over all old nodes so they sum to 1 before forming the Bernoulli probabilities used in the product model.

Intuitively:

  • Higher-degree nodes are more likely to receive a new connection (preferential attachment).
  • If \(\beta_{\text{edges}} > 0\), older nodes are down-weighted via the exponential term (recency/aging effect).
  • \(\delta\) is a small smoothing constant so degree-zero nodes remain eligible.

Under this mark model, the probability of observing a particular set of attached old nodes is a product of Bernoulli terms over the old node set: \[ \Pr(\text{attachments at } t) = \prod_{u \in V(t^-)} p_u(t)^{I_u}\,(1-p_u(t))^{1-I_u}, \] where \(I_u = 1\) if the new node attached to \(u\), and \(I_u=0\) otherwise.

Data Expectations

To use the BA kernel, your observed events must match the “new node arrives + connects to existing nodes” pattern:

  • Exactly one new node per event (one row in the node-arrivals data for each event time).
  • Any edges at that event must connect that new node to existing nodes.
  • Within an event, you should not have the same old node attached multiple times (i.e., no duplicate new–old pair within the same event).

Edge cases:

  • If there are no existing nodes yet, the event must have no edges (there is nobody to attach to).
  • If your data contains edges between two already-existing nodes, or repeated transactions/multi-edges between the same pair, then this BA mark model is not the right fit without extending the mark space / PMF.

Simulation

Simulate BA data:

set.seed(1)

time <- 50
params_ba_true <- list(
  mu = 0.5,
  K = 0.5,
  beta = 0.5,
  beta_edges = 2
)

sim <- sim_hawkesNet(
  params = params_ba_true,
  T_end = time,
  mark_type = "ba"
)
[1] "Simulation took 2.13 seconds"

Model fitting

Fit BA data.

params_ba_init <- list(
  mu = 1,
  K = 1,
  beta = 1,
  beta_edges = 1
)

fit <- fit_hawkesNet(
      ev = sim$ev,
      params_init = params_ba_init,
      mark_type = "ba"
    )
[1] "Fitting took 5.55 seconds"

Parameter values on the fitted scale:

unlist(fit$par)
        mu          K       beta beta_edges 
 1.4169583  0.4623329  0.5102294  2.0408620

Not too bad.

And, we can also grab parameter values on the transformed scale:

# Yes I know this is a jank way to structure it right now, will fix
fit$fit$par
        mu          K       beta beta_edges 
 0.3485126 -0.7714700 -0.6728948  0.7133722