8  BA-Bipartite Kernel

Kernel Specification

The BA-Bipartite kernel is the bipartite analogue of the BA kernel: the network has two disjoint node sets (say Part A and Part B), and edges may only occur across parts (A–B), never within a part.

At each event time \(t\), exactly one new node arrives, and it attaches to a subset of already-existing nodes in the opposite part using a normalized degree-based weighting (with optional aging / recency down-weighting).

Concretely:

  • Let \(V_A(t^-)\) and \(V_B(t^-)\) be the sets of existing nodes in Parts A and B just before time \(t\).
  • A single new node arrives in exactly one part (either A or B). Denote the new node by \(v_{\text{new}}\) and its part by \(P \in \{A,B\}\).
  • The candidate attachment targets are the old nodes in the other part: \[ V_{\text{opp}}(t^-) = \begin{cases} V_B(t^-) & \text{if } v_{\text{new}} \in A,\\ V_A(t^-) & \text{if } v_{\text{new}} \in B. \end{cases} \]

For each candidate target \(u \in V_{\text{opp}}(t^-)\), the new node attaches to \(u\) independently with probability \[ p_u(t) \propto (d_u(t^-) + \delta)\,\exp\!\big(-\beta_{\text{edges}}\cdot \text{age}_u(t)\big), \] where \(d_u(t^-)\) is the current (bipartite) degree of \(u\) right before time \(t\), and \(\text{age}_u(t) = t - t_u\) is time since \(u\) arrived (from the stored node time attribute).

These weights are normalised over \(V_{\text{opp}}(t^-)\) to form Bernoulli probabilities, and the probability of observing a specific attachment set is a product of Bernoulli terms over the opposite-part node set: \[ \Pr(\text{attachments at } t) = \prod_{u \in V_{\text{opp}}(t^-)} p_u(t)^{I_u}\,(1-p_u(t))^{1-I_u}, \] where \(I_u=1\) if an edge between \(v_{\text{new}}\) and \(u\) is present at time \(t\), and \(I_u=0\) otherwise.

Intuitively:

  • Nodes in the opposite part with higher degree are more likely to receive a new tie (preferential attachment).
  • If \(\beta_{\text{edges}} > 0\), older nodes are down-weighted (recency/aging).
  • \(\delta\) is a small smoothing constant so degree-zero nodes remain eligible.

Data Expectations

To use the BA-Bipartite kernel, your observed events must represent a bipartite growth process where:

  • Each node belongs to exactly one of two parts (A or B).
  • Each event introduces exactly one new node (in either part).
  • Any edges observed at that event must connect the new node to already-existing nodes in the opposite part.

In particular, the BA-Bipartite kernel assumes:

  • No within-part edges (no A–A or B–B edges).
  • No old–old edges (edges cannot form between two already-existing nodes).
  • No new–new edges (the new node cannot connect to another node that arrives at the same event).
  • Within an event, you should not repeat the same attachment (no duplicate new–old pair).

Edge cases:

  • If the opposite part has no existing nodes yet (e.g. \(V_{\text{opp}}(t^-)=\emptyset\)), then the event must have no edges.

If your data contains edges between two already-existing nodes (e.g. repeated interactions/transactions), or edges within the same part, then this BA-Bipartite mark model is not an appropriate fit without extending the mark space / PMF.

Simulation

Simulate BIP-BA data.

set.seed(1)

time <- 10
params_bip_true <- list(
  mu = 0.5,
  K = 0.5,
  beta = 0.5,
  beta_edges = 2,
  lambda_new = 3.5
)

sim <- sim_hawkesNet(
  params = params_bip_true,
  T_end = time,
  mark_type = "ba_bip"
)
[1] "Simulation took 0.19 seconds"

col_map <- c(
  event = "#377EB8",
  perp  = "#E41A1C"
)

roles <- network::get.vertex.attribute(sim$net, "role")
cols <- col_map[roles]

op <- par(no.readonly = TRUE)
on.exit(par(op), add = TRUE)

# Add a little bottom space for legend
par(mar = c(2.5,1,1,1), xaxs = "i", yaxs = "i")

plot(sim$net, vertex.col = cols)

legend(
  "bottomleft",
  legend = c("Event", "Perp"),
  pch = 21,
  pt.bg = col_map[c("event", "perp")],  # fill colour
  col = "black",                        # border colour
  horiz = TRUE,
  inset = 0.01,
  xpd = TRUE,
  cex = 2,
  pt.cex = 2.5
)

Model fitting

Fit BIP-BA data.

params_bip_init <- list(
  mu = 1,
  K = 1,
  beta = 1,
  beta_edges = 1,
  lambda_new = 3.5
)

fit <- fit_hawkesNet(
      ev = sim$ev,
      params_init = params_bip_init,
      mark_type = "ba_bip"
    )
[1] "Fitting took 0.39 seconds"

Parameter values on the fitted scale:

unlist(fit$par)
        mu          K       beta beta_edges lambda_new 
 1.3977102  0.4811354  0.9347585  2.1233094  3.2400725

Parameter values on the transformed scale:

fit$fit$par
         mu           K        beta  beta_edges  lambda_new 
 0.33483530 -0.73160664 -0.06746709  0.75297592  1.17559572