Burstiness and memory of each vertex's activity

Whether a vertex acts in bursts or at a steady pace. Burstiness compares the spread of the gaps between a vertex's events with their average: it approaches 1 for increasingly heterogeneous sequences, has theoretical reference value 0 for a Poisson process, and is -1 for a metronome. The memory coefficient asks a different question – whether a short gap tends to be followed by another short gap.

Two vertices can post the same number of times and differ entirely on both.

Usage

burstiness(
  dn,
  measure = c("burstiness", "memory", "events"),
  sessions = c("bounded", "collapse", "separate"),
  plot = FALSE
)

Arguments

dn: A temporal network from dynet().
measure: One or more of "burstiness", "memory", "events" and "mean_gap". Defaults to the first three. Anything else raises a dynet_unknown_measure error.
sessions: How to treat sessions: "bounded" (the default), "collapse" or "separate", as in centrality_series().
plot: Whether to draw the result as well as return it. Drawing is a side effect in the manner of graphics::hist(): the verb still returns its tidy table, invisibly when it has drawn, so plot = TRUE saves the wrapping plot() call without changing what comes back. Use plot() on the result when the figure needs arguments of its own.

Value

A dynet_metric at node level with no time column: one row per vertex and measure. Attributes record the event identity, dispersion, memory, loop, weight, and session-gap conventions as event_identity = "incident_spell_start", dispersion = "population", memory = "lag1_pearson", loop_contribution = "one_event", weights = "ignored", and mode-specific session_gaps.

Details

One raw spell row contributes its start time once to each distinct incident vertex. A self-loop is one event, equal-time rows remain distinct events, direction does not alter incidence, and interval ends and weights are ignored. Sorted equal times therefore create legitimate zero gaps. Explicitly onset-censored limits are not observed onset events and are excluded; terminus censoring does not affect this onset sequence.

If the usable interevent gaps are $\tau_1,\ldots,\tau_k$, burstiness is $$B=(\sigma-\mu)/(\sigma+\mu),$$ where $\mu$ is their mean and $\sigma=\sqrt{k^{-1}\sum_i(\tau_i-\mu)^2}$ is the population standard deviation of the equal-mass empirical gap distribution. mean_gap needs at least one gap. Burstiness needs at least two and is NA if every usable gap is zero. Its finite-sample range is [-1, 1).

Memory is the ordinary Pearson correlation between consecutive gaps. It needs at least two adjacent-gap pairs and nonzero variation on both sides; otherwise it is NA. In sessions = "bounded", primitive gaps and adjacent pairs are formed within each session and then pooled, so no cross-session gap is introduced. Collapse includes calendar gaps after erasing labels; separate returns session-local blocks over the fixed vertex universe.

References

Goh, K.-I., & Barabasi, A.-L. (2008). Burstiness and memory in complex systems. Europhysics Letters, 81(4), 48002, equations 1 and 4. doi:10.1209/0295-5075/81/48002

Examples

dn <- dynet(school_contacts)
burstiness(dn)
#> # Burstiness (node-level)
#> # 14 vertices | time in step
#> # measures: burstiness, memory, events
#> # 1 is the bursty limit, 0 is the Poisson reference, -1 is regular
#>   node    measure       value
#>    Ana burstiness  0.24575324
#>    Ben burstiness  0.03288611
#>   Cara burstiness -0.05610924
#>    Dan burstiness -0.05061772
#>    Eve burstiness  0.09217906
#>   Finn burstiness  0.02938507
#>   Gita burstiness  0.06104200
#>   Hugo burstiness  0.09885443
#>   Iris burstiness -0.05968068
#>  Jonas burstiness -0.02750549
#>   Kira burstiness -0.07048265
#>    Leo burstiness -0.01020265
#> # 30 more rows. summary() aggregates them; plot() draws them.