remstats_fourcycles: Compute the Four-Cycles Network Statistic for Event Dyads in...

In dream: Dynamic Relational Event Analysis and Modeling

Compute the Four-Cycles Network Statistic for Event Dyads in a Relational Event Sequence

Description

The function computes the four-cycles network sufficient statistic for a two-mode relational sequence with the exponential weighting function (Lerner and Lomi 2020). In essence, the four-cycles measure captures the tendency for clustering to occur in the network of past events, whereby an event is more likely to occur between a sender node a and receiver node b given that a has interacted with other receivers in past events who have received events from other senders that interacted with b (e.g., Duxbury and Haynie 2021, Lerner and Lomi 2020). The function also allows users to use two different weighting functions, return the counts of past events, reduce computational runtime, and specify a dyadic cutoff for relational relevancy.

Usage

remstats_fourcycles(
  time,
  sender,
  receiver,
  observed,
  sampled,
  halflife = 2,
  counts = FALSE,
  dyadic_weight = 0,
  exp_weight_form = FALSE
)

Arguments

time	The vector of event times from the post-processing event sequence.
sender	The vector of event senders from the post-processing event sequence.
receiver	The vector of event receivers from the post-processing event sequence
observed	A vector for the post-processing event sequence where i is equal to 1 if the dyadic event is observed and 0 if not.
sampled	A vector for the post-processing event sequence where i is equal to 1 if the observed dyadic event is sampled and 0 if not.
halflife	A numerical value that is the halflife value to be used in the exponential weighting function (see details section). Preset to 2 (should be updated by the user based on substantive context).
counts	TRUE/FALSE. TRUE indicates that the counts of past events should be computed (see the details section). FALSE indicates that the temporal exponential weighting function should be used to downweigh past events (see the details section). Set to FALSE by default.
dyadic_weight	A numerical value for the dyadic cutoff weight that represents the numerical cutoff value for temporal relevancy based on the exponential weighting function. For example, a numerical value of 0.01, indicates that an exponential weight less than 0.01 will become 0 and that events with such value (or smaller values) will not be included in the sum of the past event weights (see the details section). Set to 0 by default.
exp_weight_form	TRUE/FALSE. TRUE indicates that the Lerner et al. (2013) exponential weighting function will be used (see the details section). FALSE indicates that the Lerner and Lomi (2020) exponential weighting function will be used (see the details section). Set to FALSE by default

Details

The function calculates the four-cycles network statistic for two-mode relational event models based on the exponential weighting function used in either Lerner and Lomi (2020) or Lerner et al. (2013).

Following Lerner and Lomi (2020), the exponential weighting function in relational event models is:

w(s, r, t) = e^{-(t-t') \cdot \frac{ln(2)}{T_{1/2}} }

Following Lerner et al. (2013), the exponential weighting function in relational event models is:

w(s, r, t) = e^{-(t-t') \cdot \frac{ln(2)}{T_{1/2}} } \cdot \frac{ln(2)}{T_{1/2}}

In both of the above equations, s is the current event sender, r is the current event receiver (target), t is the current event time, t' is the past event times that meet the weight subset (in this case, all events that have the same sender and receiver), and T_{1/2} is the halflife parameter.

The formula for four-cycles for event e_i is:

four cycles_{e_{i}} = \sqrt[3]{\sum_{s' and r'} w(s', r, t) \cdot w(s, r', t) \cdot w(s', r', t)}

That is, the four-cycle measure captures all the past event structures in which the current event pair, sender s and target r close a four-cycle. In particular, it finds all events in which: a past sender s' had a relational event with target r, a past target r' had a relational event with current sender s, and finally, a relational event occurred between sender s' and target r'.

Four-cycles are computationally expensive, especially for large relational event sequences (see Lerner and Lomi 2020 for a discussion on this), therefore this function allows the user to input previously computed target indegree and sender outdegree scores to reduce the runtime. Relational events where either the event target or event sender were not involved in any prior relational events (i.e., a target indegree or sender outdegree score of 0) will close no-four cycles. This function exploits this feature.

Moreover, researchers interested in modeling temporal relevancy (see Quintane, Mood, Dunn, and Falzone 2022; Lerner and Lomi 2020) can specify the dyadic weight cutoff, that is, the minimum value for which the weight is considered relationally relevant. Users who do not know the specific dyadic cutoff value to use, can use the remstats_dyadcut function.

Following Lerner and Lomi (2020), if the counts of the past events are requested, the formula for four-cycles formation for event e_i is:

four cycles_{e_{i}} = \sum_{i=1}^{|S'|} \sum_{j=1}^{|R'|} \min\left[d(s'_{i}, r, t),\ d(s, r'_{j}, t),\ d(s'_{i}, r'_{j}, t)\right]

where, d() is the number of past events that meet the specific set operations, d(s'_{i},r,t) is the number of past events where the current event receiver received a tie from another sender s'_{i}, d(s,r'_{j},t) is the number of past events where the current event sender sent a tie to another receiver r'_{j}, and d(s'_{i},r'_{j},t) is the number of past events where the sender s'_{i} sent a tie to the receiver r'_{j}. Moreover, the counting equation can leverage relational relevancy, by specifying the halflife parameter, exponential weighting function, and the dyadic cut off weight values (see the above sections for help with this). If the user is not interested in modeling relational relevancy, then those value should be left at their default values.

Value

The vector of four-cycle statistics for the two-mode relational event sequence.

Author(s)

Kevin A. Carson [e-mail], Diego F. Leal [e-mail]

References

Duxbury, Scott and Dana Haynie. 2021. "Shining a Light on the Shadows: Endogenous Trade Structure and the Growth of an Online Illegal Market." American Journal of Sociology 127(3): 787-827.

Quintane, Eric, Martin Wood, John Dunn, and Lucia Falzon. 2022. “Temporal Brokering: A Measure of Brokerage as a Behavioral Process.” Organizational Research Methods 25(3): 459-489.

Lerner, Jürgen and Alessandro Lomi. 2020. “Reliability of relational event model estimates under sampling: How to fit a relational event model to 360 million dyadic events.” Network Science 8(1): 97-135.

Lerner, Jürgen, Margit Bussman, Tom A.B. Snijders, and Ulrik Brandes. 2013. "Modeling Frequency and Type of Interaction in Event Networks." The Corvinus Journal of Sociology and Social Policy 4(1): 3-32.

Examples

data("WikiEvent2018.first100k")
WikiEvent2018 <- WikiEvent2018.first100k[1:1000,] #the first one thousand events
WikiEvent2018$time <- as.numeric(WikiEvent2018$time) #making the variable numeric
### Creating the EventSet By Employing Case-Control Sampling With M = 5 and
### Sampling from the Observed Event Sequence with P = 0.01
EventSet <-create_riskset(type = "two-mode",
 time = WikiEvent2018$time, # The Time Variable
 eventID = WikiEvent2018$eventID, # The Event Sequence Variable
 sender = WikiEvent2018$user, # The Sender Variable
 receiver = WikiEvent2018$article, # The Receiver Variable
 p_samplingobserved = 0.01, # The Probability of Selection
 n_controls = 8, # The Number of Controls to Sample from the Full Risk Set
 combine = TRUE,
 seed = 9999) # The Seed for Replication

#Computing the four-cycles statistics for the relational event sequence with
#the exponential weights of past events returned
cycle4_weights <- remstats_fourcycles(
   time = EventSet$time,
   sender = EventSet$sender,
   receiver = EventSet$receiver,
   sampled = EventSet$sampled,
   observed = EventSet$observed,
   halflife = 2.592e+09, #halflife parameter
   dyadic_weight = 0,
   exp_weight_form = FALSE)



#Computing the four-cycles statistics for the relational event sequence with
#the counts of past events returned
cycle4_counts <- remstats_fourcycles(
   time = EventSet$time,
   sender = EventSet$sender,
   receiver = EventSet$receiver,
   sampled = EventSet$sampled,
   observed = EventSet$observed,
   halflife = 2.592e+09, #halflife parameter
   dyadic_weight = 0,
   counts = TRUE)

cbind(cycle4_weights, cycle4_counts)

dream documentation built on Jan. 21, 2026, 1:06 a.m.