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Building and inspecting a temporal
network
This vignette describes how to construct and inspect temporal
networks with dynet(). It covers four relational data
formats, vertex attributes, sessions, observation periods, and vertex
activity spells. It also introduces network editing and descriptive
measures. vignette("dynet") provides a worked analysis of a
simulated classroom network.
The four input formats
dynet() accepts interval, contact, threaded, and
co-presence data in tidy format. These formats differ in how relational
endpoints and timing are recorded. The constructor selects a format from
the supplied arguments and recognised timing columns, or uses the format
specified explicitly by the user.
Interval data
Interval data record the onset and termination of each relationship.
Each relational spell identifies two endpoints and the period during
which their connection is active.
school_contacts contains 240 simulated face-to-face
contacts among fourteen students over approximately three weeks. The
supplied variables from and to identify the
initiating and receiving students. start and
end record onset and termination in days since the
beginning of observation. Decimal values allow contacts to begin and end
within a day.
head(school_contacts, 4)
#> from to start end
#> 1 Jonas Dan 0.00 1.10
#> 2 Gita Ana 0.14 0.98
#> 3 Leo Mira 0.15 0.42
#> 4 Leo Iris 0.15 0.96
dynet() constructs the network directly from these data.
Printing the result reports its format, direction, vertex and spell
counts, distinct pairs, observation period, and measurement grid,
followed by the first relational spells.
school <- dynet(school_contacts)
school
#> # Temporal network (interval format, directed) | a cograph netobject
#> # 14 vertices | 240 edge spells | 110 distinct pairs
#> # observed from 0 to 21.52 step, binned every 1
#>
#> from to start end duration weight
#> Jonas Dan 0.00 1.10 1.10 1
#> Gita Ana 0.14 0.98 0.84 1
#> Leo Mira 0.15 0.42 0.27 1
#> Leo Iris 0.15 0.96 0.81 1
#> Kira Ben 0.33 0.69 0.36 1
#> Leo Iris 0.38 0.50 0.12 1
#> # 234 more spells. summary() describes the network; plot() draws it.
Column recognition is case-insensitive. The endpoint aliases
from/to,
sender/receiver, and
source/target are equivalent, as are
start/end and
onset/terminus for interval boundaries.
Explicit column specification is needed only when names do not match
recognised aliases or their interpretation is ambiguous. A
duration column may replace end; termination
is then calculated as start + duration.
The constructor standardises the supplied variables and derives
additional quantities where needed. Here, it calculates
duration = end - start and assigns weight = 1
because no multiplicity variable is supplied or recognised.
The 240 spells connect 110 distinct ordered pairs. With fourteen
vertices and self-links excluded, there are \(14 \times 13 = 182\) possible ordered
pairs. Approximately 60% are connected at least once during observation.
This aggregate proportion does not indicate how many pairs are connected
within any particular interval.
Threaded data
Threaded data contain timestamps and discussion identifiers. A
discussion-based duration represents the period during which a post
remains part of an ongoing exchange, as subsequent interactions respond
to or address that discussion. Following the approach of Saqr and Nouri
(2020), Dynet treats a post as active from its timestamp until the last
retained post in the same thread. For a post at time \(t_i\) in thread \(T\), the relational spell is \([t_i, \max_{j \in T} t_j)\). A final post
has zero duration. This is a modelling assumption about discussion
activity, rather than a directly observed contact duration.
Specifying thread selects this construction. The
optional nodes argument supplies vertex attributes.
forum <- dynet(forum_posts, thread = "thread", nodes = forum_people)
forum
#> # Temporal network (threaded format, directed) | a cograph netobject
#> # 20 vertices | 241 edge spells | 172 distinct pairs
#> # observed from 0 to 54.96387 days, binned every 1
#> # vertex attributes: role, achievement
#>
#> from to start end duration weight thread
#> student_14 student_05 0.0000000 2.296969 2.296969 1 thread_47
#> student_10 student_05 0.7257216 2.296969 1.571247 1 thread_47
#> teacher_A student_09 0.8559934 3.235993 2.380000 1 thread_11
#> student_04 student_05 1.1715344 2.296969 1.125435 1 thread_47
#> student_02 student_09 1.2382611 3.235993 1.997732 1 thread_11
#> student_06 teacher_A 1.9797595 3.235993 1.256234 1 thread_11
#> # 235 more spells. summary() describes the network; plot() draws it.
The network retains the same 241 spells, twenty vertices, and 172
ordered pairs as the contact representation. Each spell now carries its
thread identifier and a derived termination time. The 62 thread-closing
posts have zero duration; the remaining spells have positive
duration.
summary(..., temporal_density = TRUE) compares the two
representations using both mean snapshot density and temporal density.
Temporal density is optional because its calculation integrates activity
over eligible vertex pairs and can be more computationally
demanding.
summary(clicks, temporal_density = TRUE)
#> property value
#> 1 format contact
#> 2 directed yes
#> 3 vertices 20
#> 4 edge spells 241
#> 5 distinct pairs 172
#> 6 time unit days
#> 7 observed from 0
#> 8 observed to 54.96387
#> 9 span 54.96387
#> 10 bin width 1
#> 11 time bins 55
#> 12 mean snapshot density 0.0112
#> 13 temporal density 0
#> 14 sessions none
#> 15 vertex attributes none
summary(forum, temporal_density = TRUE)
#> property value
#> 1 format threaded
#> 2 directed yes
#> 3 vertices 20
#> 4 edge spells 241
#> 5 distinct pairs 172
#> 6 time unit days
#> 7 observed from 0
#> 8 observed to 54.96387
#> 9 span 54.96387
#> 10 bin width 1
#> 11 time bins 55
#> 12 mean snapshot density 0.0248
#> 13 temporal density 0.0139
#> 14 sessions none
#> 15 vertex attributes role, achievement
Mean snapshot density averages the proportion of
ordered pairs connected at some point within each measurement bin.
Temporal density measures the proportion of available
pair-time occupied by connections. For a fixed population of \(n\) vertices observed continuously for
duration \(\tau\), with self-links
excluded,
\[D_T = \frac{\sum_r U_r}{n(n -
1)\,\tau},\]
where \(U_r\) is the total duration
for which ordered pair \(r\) has at
least one active spell. Overlapping spells on the same pair contribute
their union duration.
The contact representation has mean snapshot density 0.0112 and
temporal density 0: instantaneous contacts count within bins but occupy
no positive duration. The threaded representation has mean snapshot
density 0.0248 and temporal density 0.0139. Its connections occupy 290.8
pair-days across 380 possible pairs and 54.96 observed days. These
differences follow from the specified duration rule, which is applied
when the threaded format is selected.
Co-presence data
Co-presence data record actors’ participation in shared occasions
rather than direct relationships between actors. A projection connects
actors who attend the same occasion. seminar_attendance
records attendance at weekly seminars over one term.
head(seminar_attendance, 3)
#> student seminar date
#> 1 s12 week_01 2024-09-03
#> 2 s22 week_01 2024-09-03
#> 3 s23 week_01 2024-09-03
Specify actor and group to identify the
participant and occasion columns.
seminars <- dynet(seminar_attendance, actor = "student", group = "seminar")
seminars
#> # Temporal network (copresence format, undirected) | a cograph netobject
#> # 24 vertices | 417 edge spells | 224 distinct pairs
#> # observed from 0 to 77 days, binned every 1
#>
#> from to start end duration weight group
#> s03 s10 0 0 0 1 week_01
#> s03 s12 0 0 0 1 week_01
#> s03 s21 0 0 0 1 week_01
#> s03 s22 0 0 0 1 week_01
#> s03 s23 0 0 0 1 week_01
#> s10 s12 0 0 0 1 week_01
#> # 411 more spells. summary() describes the network; plot() draws it.
Each seminar contributes a spell for every pair of attendees, tagged
with that seminar’s identifier. A seminar with \(k\) attendees therefore contributes \(\binom{k}{2}\) spells. Across all seminars,
the resulting network contains 417 spells and 224 distinct pairs among
24 students. These pairs represent approximately 81% of the \(\binom{24}{2} = 276\) possible unordered
pairs.
Co-presence is symmetric, so the constructor creates an undirected
network even if directed = TRUE is supplied. Here, the
input supplies a date without a termination time, so the projected
spells are instantaneous contacts on the seminar date.
Inspecting the network
summary() returns network properties in a tidy
table.
summary(school)
#> property value
#> 1 format interval
#> 2 directed yes
#> 3 vertices 14
#> 4 edge spells 240
#> 5 distinct pairs 110
#> 6 time unit step
#> 7 observed from 0
#> 8 observed to 21.52
#> 9 span 21.52
#> 10 bin width 1
#> 11 time bins 22
#> 12 mean snapshot density 0.0829
#> 13 temporal density not computed
#> 14 sessions none
#> 15 vertex attributes none
vertices reports the size of the vertex set,
edge spells counts relational spells, and
distinct pairs counts the endpoint pairs they connect. The
classroom network has 240 spells on 110 ordered pairs, indicating
repeated contact for at least some pairs.
time unit is step for numeric input or the
selected calendar unit for date-time input. Without explicit observation
bounds, the observed range extends from the earliest onset to the latest
termination. bin width records the construction interval,
and time bins counts the intervals covering that range.
Here, 22 bins cover 21.52 days; the final bin is shorter than one
day.
mean snapshot density is 0.0829: approximately 8.3% of
possible ordered pairs are connected in an average daily bin, compared
with about 60% across the full observation period.
temporal density is calculated only when requested.
as.data.frame() extracts the constructed relational
spells, including duration and weight.
spells <- as.data.frame(school)
head(spells, 4)
#> from to start end duration weight
#> 1 Jonas Dan 0.00 1.10 1.10 1
#> 2 Gita Ana 0.14 0.98 0.84 1
#> 3 Leo Mira 0.15 0.42 0.27 1
#> 4 Leo Iris 0.15 0.96 0.81 1
Use what = "nodes" to extract vertex attributes. For
forum, these include the attributes supplied through
nodes.
forum_nodes <- as.data.frame(forum, what = "nodes")
head(forum_nodes, 4)
#> name role achievement
#> 1 facilitator Facilitator <NA>
#> 2 student_01 Student Middle
#> 3 student_02 Student Low
#> 4 student_03 Student High
Other options are "bins" for the measurement grid,
"network" for the aggregate edge list,
"observations" for the observation calendar,
"observed_edges" for spells clipped to that calendar, and
"vertex_spells" for declared vertex activity.
bins <- as.data.frame(school, what = "bins")
head(bins, 4)
#> bin lo hi time closed
#> 1 1 0 1 0 FALSE
#> 2 2 1 2 1 FALSE
#> 3 3 2 3 2 FALSE
#> 4 4 3 4 3 FALSE
Each bin extends from lo to hi and is
labelled by its starting time. Bins are half-open except
for the final bin, whose closed flag includes an event at
the final observed instant.
pairs <- as.data.frame(school, what = "network")
head(pairs, 4)
#> from to weight
#> 1 Cara Ana 1
#> 2 Dan Ana 2
#> 3 Gita Ana 3
#> 4 Hugo Ana 2
The aggregate edge list groups spells by their relational endpoints
and sums their weights. Because every spell in school has
weight 1, the aggregate weight equals the spell count: Dan
contacted Ana twice and Gita contacted Ana three times. Aggregation
summarises these relationships without retaining their temporal
order.
Direction, loops, weights and attributes
The following dataset contains five spells among three vertices,
including a self-link from A to A. The
supplied posts variable records the number of messages
represented by each spell.
tiny <- data.frame(
from = c("A", "B", "A", "C", "A"),
to = c("B", "A", "C", "A", "A"),
start = c(0, 1, 2, 3, 4),
end = c(2, 3, 5, 4, 6),
posts = c(3, 1, 2, 5, 1)
)
Set directed = FALSE to construct an undirected network
and use weight to identify the multiplicity variable.
Columns named weight, weights, or
strength are recognised automatically; posts
requires explicit specification.
undirected <- dynet(tiny, directed = FALSE, weight = "posts")
#> Dropped 1 self-loop event(s). Use loops = TRUE to keep them.
undirected
#> # Temporal network (interval format, undirected) | a cograph netobject
#> # 3 vertices | 4 edge spells | 2 distinct pairs
#> # observed from 0 to 5 step, binned every 1
#>
#> from to start end duration weight
#> A B 0 2 2 3
#> A B 1 3 2 1
#> A C 2 5 3 2
#> A C 3 4 1 5
The result contains four spells on two unordered pairs. The spells
A -> B and B -> A connect the same
undirected pair but retain their individual onset and termination times;
they overlap during \([1, 2)\). The
constructor removes the self-link because loops = FALSE by
default and records the supplied posts values as
weight.
Set loops = TRUE to retain self-links. Under total
degree, a retained loop contributes twice, once at each endpoint.
with_loops <- dynet(tiny, loops = TRUE)
#> Keeping 1 self-loop event(s); each adds two to its vertex's degree.
with_loops
#> # Temporal network (interval format, directed) | a cograph netobject
#> # 3 vertices | 5 edge spells | 5 distinct pairs
#> # observed from 0 to 6 step, binned every 1
#>
#> from to start end duration weight posts
#> A B 0 2 2 1 3
#> B A 1 3 2 1 1
#> A C 2 5 3 1 2
#> C A 3 4 1 1 5
#> A A 4 6 2 1 1
This call retains direction and all five spells, producing five
distinct ordered pairs. Because weight is not specified and
posts is not a recognised weight alias, posts
remains a spell attribute and the constructor assigns the default weight
of 1.
interval sets the default spacing of measurements in the
network’s time unit.
tiny_dn <- dynet(tiny, interval = 2)
#> Dropped 1 self-loop event(s). Use loops = TRUE to keep them.
summary(tiny_dn)
#> property value
#> 1 format interval
#> 2 directed yes
#> 3 vertices 3
#> 4 edge spells 4
#> 5 distinct pairs 4
#> 6 time unit step
#> 7 observed from 0
#> 8 observed to 5
#> 9 span 5
#> 10 bin width 2
#> 11 time bins 3
#> 12 mean snapshot density 0.3333
#> 13 temporal density not computed
#> 14 sessions none
#> 15 vertex attributes none
An interval of 2 covers the five-unit observation period with three
bins, the last of which is partial. Their active-pair counts are 2, 3,
and 1 out of six possible ordered pairs. Mean snapshot density is
therefore \((2/6 + 3/6 + 1/6)/3 =
1/3\).
The nodes argument supplies a vertex table whose
identifier column is recognised by name. groups selects an
attribute to store as the vertex grouping; cograph::splot()
can then use it for vertex colours.
roles <- dynet(forum_posts, thread = "thread",
nodes = forum_people, groups = "role")
role_nodes <- as.data.frame(roles, what = "nodes")
head(role_nodes, 4)
#> name role achievement groups
#> 1 facilitator Facilitator <NA> Facilitator
#> 2 student_01 Student Middle Student
#> 3 student_02 Student Low Student
#> 4 student_03 Student High Student
Mixing describes connections within and between
groups defined by a vertex attribute. mixing() counts
distinct connected vertex pairs for each ordered group pair and
measurement bin. Connections counted in a bin need not be active
simultaneously.
role_mixing <- mixing(roles, attribute = "role")
head(role_mixing, 4)
#> # Mixing by role (graph-level)
#> # 55 time points, 1 per bin | time in days
#> # measures: Facilitator -> Facilitator, Student -> Facilitator, Teacher -> Facilitator, Facilitator -> Student
#> # first 4 of 495 rows
#> # active binary-dyad counts between vertex groups per time bin
#> time measure value from_group to_group
#> 0 Facilitator -> Facilitator 0 Facilitator Facilitator
#> 0 Student -> Facilitator 0 Student Facilitator
#> 0 Teacher -> Facilitator 0 Teacher Facilitator
#> 0 Facilitator -> Student 0 Facilitator Student
Three roles produce nine ordered group pairs in each of 55 daily
bins, giving 495 observations. In the first bin, no connection involves
a facilitator. These are connection counts, not probabilities or counts
of simultaneous interactions.
Sessions
Sessions identify contexts within which time-respecting paths may be
constrained, such as courses, terms, or class periods. By default, a
path must use spells assigned to a single session. Session labels do not
reset the clock.
Use session during construction to identify an existing
session column. Alternatively, set_tie_sessions() can
assign sessions from spell onset times. The following call assigns
spells to weeks using boundaries at days 7 and 14.
sessioned <- set_tie_sessions(school, breaks = c(7, 14),
labels = c("week_1", "week_2", "week_3"))
summary(sessioned)
#> property value
#> 1 format interval
#> 2 directed yes
#> 3 vertices 14
#> 4 edge spells 240
#> 5 distinct pairs 110
#> 6 time unit step
#> 7 observed from 0
#> 8 observed to 21.52
#> 9 span 21.52
#> 10 bin width 1
#> 11 time bins 22
#> 12 mean snapshot density 0.0829
#> 13 temporal density not computed
#> 14 sessions 3
#> 15 vertex attributes none
The summary now reports three sessions. The sessions
argument controls how path searches use these assignments.
collapsed <- paths(sessioned, from = "Ana", sessions = "collapse")
summary(collapsed)
#> property value
#> 1 source Ana
#> 2 direction forward
#> 3 reachable 13
#> 4 reachable share 1
#> 5 median latency 7.51
#> 6 max latency 11.66
#> 7 median hops 2
#> 8 max hops 4
With sessions = "collapse", session labels are ignored
and the search uses the complete temporal sequence. Ana reaches all
thirteen other students, with median latency 7.51 days and a maximum of
four hops.
inside <- paths(sessioned, from = "Ana", sessions = "bounded")
summary(inside)
#> property value
#> 1 source Ana
#> 2 direction forward
#> 3 reachable 13
#> 4 reachable share 1
#> 5 median latency 8.21
#> 6 max latency 13.21
#> 7 median hops 2
#> 8 max hops 5
With sessions = "bounded", each path uses spells from a
single session. The search compares session-specific results and retains
the best result for each destination. Ana still reaches all thirteen
students, but median latency increases to 8.21 days, maximum latency to
13.21 days, and maximum hop count to five. Earlier paths that combined
spells assigned to different weeks are no longer admissible.
per_session <- paths(sessioned, from = "Ana", sessions = "separate")
summary(per_session)
#> session property value
#> 1 week_1 source Ana
#> 2 week_1 direction forward
#> 3 week_1 reachable 6
#> 4 week_1 reachable share 0.462
#> 5 week_1 median latency 6.36
#> 6 week_1 max latency 6.96
#> 7 week_1 median hops 1.5
#> 8 week_1 max hops 2
#> 9 week_2 source Ana
#> 10 week_2 direction forward
#> 11 week_2 reachable 13
#> 12 week_2 reachable share 1
#> 13 week_2 median latency 3.36
#> 14 week_2 max latency 6.19
#> 15 week_2 median hops 3
#> 16 week_2 max hops 6
#> 17 week_3 source Ana
#> 18 week_3 direction forward
#> 19 week_3 reachable 5
#> 20 week_3 reachable share 0.385
#> 21 week_3 median latency 6.42
#> 22 week_3 max latency 6.67
#> 23 week_3 median hops 2
#> 24 week_3 max hops 3
With sessions = "separate", results are reported
separately for each session. Ana reaches six students in week 1,
thirteen in week 2, and five in week 3, corresponding to proportions of
0.462, 1, and 0.385. The longest path within week 2 uses six hops.
"bounded" is the default. Without session assignments,
it gives the same result as "collapse".
"separate" requires session assignments and otherwise
raises dynet_bad_input.
Observation windows
Without explicit bounds, the observation period extends from the
earliest spell onset to the latest termination. These event-derived
limits may differ from the study’s actual observation period. Specifying
observation_start and observation_end defines
the measurement horizon, including periods when no interaction was
recorded.
bounded <- dynet(school_contacts, observation_start = 0, observation_end = 14)
summary(bounded)
#> property value
#> 1 format interval
#> 2 directed yes
#> 3 vertices 14
#> 4 edge spells 240
#> 5 distinct pairs 110
#> 6 time unit step
#> 7 observed from 0
#> 8 observed to 14
#> 9 span 14
#> 10 bin width 1
#> 11 time bins 14
#> 12 mean snapshot density 0.0922
#> 13 temporal density not computed
#> 14 sessions none
#> 15 vertex attributes none
Restricting observation to days 0–14 produces fourteen bins instead
of 22. Mean snapshot density is 0.0922, compared with 0.0829 across the
full period. This difference reflects the connections observed during
the selected period.
Observation bounds change the measurement period without deleting the
original spells: as.data.frame() still returns them.
Positive-duration spells contribute their intersection with the
observation window, and instantaneous events on either observation
boundary are retained.
For interrupted observation, supply observation_spells
with the start and end of each observed period. Overlapping or adjacent
periods are merged. what = "observations" extracts the
resulting calendar.
gapped <- dynet(
school_contacts,
observation_spells = data.frame(start = c(0, 12), end = c(8, 21))
)
as.data.frame(gapped, what = "observations")
#> observation start end duration instant
#> 1 1 0 8 8 FALSE
#> 2 2 12 21 9 FALSE
summary(gapped)
#> property value
#> 1 format interval
#> 2 directed yes
#> 3 vertices 14
#> 4 edge spells 240
#> 5 distinct pairs 110
#> 6 time unit step
#> 7 observed from 0
#> 8 observed to 21
#> 9 span 21
#> 10 bin width 1
#> 11 time bins 17
#> 12 mean snapshot density 0.0824
#> 13 temporal density not computed
#> 14 sessions none
#> 15 vertex attributes none
The two periods contain eight and nine observed days. The measurement
grid restarts within each period, yielding seventeen bins and none
during the four-day gap. Exposure calculations use the seventeen
observed days, and events within the gap are excluded from
measurements.
set_observations() replaces the observation calendar
after construction. clear_observations() restores
continuous observation from the earliest raw onset to the latest raw
termination.
narrowed <- set_observations(school, start = 2, end = 10)
as.data.frame(narrowed, what = "observations")
#> observation start end duration instant
#> 1 1 2 10 8 FALSE
continuous <- clear_observations(gapped)
as.data.frame(continuous, what = "observations")
#> observation start end duration instant
#> 1 1 0 21.52 21.52 FALSE
Vertex activity spells
Observation periods describe when data collection occurred. Vertex
activity spells describe when individual vertices were eligible to
participate, for example after enrolment or before departure. This
distinction separates an eligible participant with no connections from a
participant who was absent.
Supply vertex_spells as a table containing
node, start, and end. A vertex
without an explicit activity declaration is treated as eligible
throughout observation.
arrivals <- data.frame(
node = c("Ana", "Ben"),
start = c(0, 7),
end = c(21.52, 21.52)
)
scheduled <- dynet(school_contacts, vertex_spells = arrivals)
as.data.frame(scheduled, what = "vertex_spells")
#> vertex_spell node start end duration instant session onset_censored
#> 1 1 Ana 0 21.52 21.52 FALSE <NA> FALSE
#> 2 2 Ben 7 21.52 14.52 FALSE <NA> FALSE
#> terminus_censored
#> 1 FALSE
#> 2 FALSE
Ben is declared eligible from day 7. His degree is NA in
earlier bins, rather than zero. The following calls compare degree with
and without this declaration.
school_degree <- centrality_series(school, measure = "degree")
head(school_degree, 4)
#> # Degree (node-level)
#> # 14 vertices | 22 time points, 1 per bin | time in step
#> # first 4 of 308 rows
#> time node measure value
#> 0 Ana degree 1
#> 0 Ben degree 1
#> 0 Cara degree 1
#> 0 Dan degree 1
scheduled_degree <- centrality_series(scheduled, measure = "degree")
head(scheduled_degree, 4)
#> # Degree (node-level)
#> # 14 vertices | 22 time points, 1 per bin | time in step
#> # first 4 of 308 rows
#> time node measure value
#> 0 Ana degree 1
#> 0 Ben degree NA
#> 0 Cara degree 1
#> 0 Dan degree 1
summary() excludes missing values when summarising the
trajectories, so periods before declared arrival no longer contribute to
the mean.
school_degree_summary <- summary(school_degree)
head(school_degree_summary, 3)
#> node measure n mean sd min max peak_time
#> 1 Ana degree 22 2.181818 2.015095 0 7 6
#> 2 Ben degree 22 2.000000 1.234427 0 4 4
#> 3 Cara degree 22 2.227273 1.342770 0 5 4
scheduled_degree_summary <- summary(scheduled_degree)
head(scheduled_degree_summary, 3)
#> node measure n mean sd min max peak_time
#> 1 Ana degree 22 2.181818 2.015095 0 7 6
#> 2 Ben degree 15 2.133333 1.125463 0 4 11
#> 3 Cara degree 22 2.227273 1.342770 0 5 4
Ana’s mean degree remains 2.18 across 22 bins. Ben’s mean changes
from 2.00 across 22 bins to 2.13 across fifteen eligible bins. His
standard deviation decreases from 1.23 to 1.13, and his peak moves from
day 4 to day 11. The eleven spells recorded for Ben before day 7 remain
in the raw data but are excluded from the eligible measurement period.
This example illustrates the effect of an activity declaration; in an
analysis, declarations should reflect the study’s participation
criteria.
Use set_vertex_spells() to replace declared activity and
add_vertex_spells() to add periods of activity.
activity <- set_vertex_spells(school, arrivals)
extended <- add_vertex_spells(activity,
data.frame(node = "Cara", start = 3, end = 12))
as.data.frame(extended, what = "vertex_spells")
#> vertex_spell node start end duration instant session onset_censored
#> 1 1 Ana 0 21.52 21.52 FALSE <NA> FALSE
#> 2 2 Ben 7 21.52 14.52 FALSE <NA> FALSE
#> 3 3 Cara 3 12.00 9.00 FALSE <NA> FALSE
#> terminus_censored
#> 1 FALSE
#> 2 FALSE
#> 3 FALSE
Editing a network
Editing functions return a new network and leave their input
unchanged. They update the relational spells and associated network
representation together. Use these functions for temporal edits so that
timing and network structure remain consistent.
add_nodes() adds vertices and attributes.
add_ties() adds relational spells whose endpoints already
exist in the vertex set.
step1 <- add_nodes(school, data.frame(name = "Nova", role = "exchange"))
step2 <- add_ties(step1, data.frame(
from = "Ana", to = "Nova", start = 4, end = 6
))
summary(step2)
#> property value
#> 1 format interval
#> 2 directed yes
#> 3 vertices 15
#> 4 edge spells 241
#> 5 distinct pairs 111
#> 6 time unit step
#> 7 observed from 0
#> 8 observed to 21.52
#> 9 span 21.52
#> 10 bin width 1
#> 11 time bins 22
#> 12 mean snapshot density 0.0723
#> 13 temporal density not computed
#> 14 sessions none
#> 15 vertex attributes role
The edited network contains fifteen vertices, 241 spells, and 111
distinct pairs. Original vertices have NA for the newly
introduced role attribute. Mean snapshot density decreases
from 0.0829 to 0.0723 despite the added connection: the additional
vertex increases the number of possible ordered pairs from 182 to
210.
The original network remains unchanged.
summary(school)
#> property value
#> 1 format interval
#> 2 directed yes
#> 3 vertices 14
#> 4 edge spells 240
#> 5 distinct pairs 110
#> 6 time unit step
#> 7 observed from 0
#> 8 observed to 21.52
#> 9 span 21.52
#> 10 bin width 1
#> 11 time bins 22
#> 12 mean snapshot density 0.0829
#> 13 temporal density not computed
#> 14 sessions none
#> 15 vertex attributes none
remove_ties() selects spells by their endpoints and
onset, while rename_nodes() updates vertex names using a
mapping from old to new names.
step3 <- remove_ties(step2, from = "Ana", to = "Nova", start = 4)
summary(step3)
#> property value
#> 1 format interval
#> 2 directed yes
#> 3 vertices 15
#> 4 edge spells 240
#> 5 distinct pairs 110
#> 6 time unit step
#> 7 observed from 0
#> 8 observed to 21.52
#> 9 span 21.52
#> 10 bin width 1
#> 11 time bins 22
#> 12 mean snapshot density 0.0719
#> 13 temporal density not computed
#> 14 sessions none
#> 15 vertex attributes role
renamed <- rename_nodes(step2, c(Nova = "Nova B."))
renamed_nodes <- as.data.frame(renamed, what = "nodes")
tail(renamed_nodes, 3)
#> name role
#> 13 Mira <NA>
#> 14 Nils <NA>
#> 15 Nova B. exchange
Removing the added spell restores the original 240 spells and 110
connected pairs, but Nova remains as an isolated vertex. Mean density is
therefore 0.0719, using the enlarged denominator of 210 possible
pairs.
induce_subgraph() restricts the network to selected
vertices or spells. nodes accepts vertex names, and
ties can specify a condition over the spell table.
five <- induce_subgraph(school, nodes = c("Ana", "Ben", "Cara", "Dan", "Eve"))
summary(five)
#> property value
#> 1 format interval
#> 2 directed yes
#> 3 vertices 5
#> 4 edge spells 18
#> 5 distinct pairs 11
#> 6 time unit step
#> 7 observed from 3.17
#> 8 observed to 21.33
#> 9 span 18.16
#> 10 bin width 1
#> 11 time bins 19
#> 12 mean snapshot density 0.0684
#> 13 temporal density not computed
#> 14 sessions none
#> 15 vertex attributes none
The five selected students share eighteen spells on eleven of twenty
possible ordered pairs. Without explicit observation bounds, the
subgraph’s observation period follows its own spell boundaries, here
days 3.17–21.33. Declare observation_start and
observation_end when the original study period should be
retained.
Descriptive measures
Dynet provides graph-level measures describing the network as a whole
and vertex-level measures describing individual positions. For
window-based calculations, spells active within each window form a
snapshot on which the selected measures are computed. Graph-level
trajectories are obtained with metrics(); vertex centrality
trajectories use centrality_series().
The following call requests three graph-level measures in a tidy
result identified by time and measure.
basics <- metrics(school, measure = c("density", "edges", "active_nodes"))
head(basics, 6)
#> # Graph structure (graph-level)
#> # 22 time points, 1 per bin | time in step
#> # measures: density, edges, active_nodes
#> # first 6 of 66 rows
#> time measure value
#> 0 density 0.05494505
#> 0 edges 10.00000000
#> 0 active_nodes 13.00000000
#> 1 density 0.04395604
#> 1 edges 8.00000000
#> 1 active_nodes 8.00000000
edges counts distinct connected ordered pairs,
density divides that count by the possible pairs, and
active_nodes counts vertices with at least one connection.
On day 0, ten connections involve thirteen students, giving density
0.055. On day 1, eight connections involve eight students, giving
density 0.044.
summary() reports the mean, standard deviation, range,
and peak time for each measure.
six <- metrics(school, measure = c("density", "edges", "active_nodes",
"components", "transitivity",
"reciprocity"))
summary(six)
#> measure n mean sd min max peak_time
#> 1 active_nodes 22 12.13636364 2.07698166 7.00000000 14.0000000 5
#> 2 components 22 3.59090909 2.38365647 1.00000000 9.0000000 21
#> 3 density 22 0.08291708 0.03929021 0.03296703 0.1648352 14
#> 4 edges 22 15.09090909 7.15081807 6.00000000 30.0000000 14
#> 5 reciprocity 22 0.14503815 0.13886108 0.00000000 0.4666667 14
#> 6 transitivity 22 0.11496262 0.12515367 0.00000000 0.4000000 11
Across 22 bins, mean density is 0.083 and reaches 0.165 on day 14,
when thirty pairs are connected. An average of 12.1 students have a
connection within a daily bin, with a minimum of seven.
components counts weakly connected components, ignoring
direction and including isolated vertices. It averages 3.6 and equals 1
on day 14. Its maximum of nine occurs in the final partial bin, when six
connections leave seven students isolated.
reciprocity is the proportion of directed connections
\(i \to j\) for which \(j \to i\) is also present in the window. It
averages 0.145 and reaches 0.467 on day 14. transitivity is
the proportion of two-paths \(i \to j \to
k\) closed by \(i \to k\),
following the weak convention of sna::gtrans(). It averages
0.115 and peaks at 0.4 on day 11. Both measures describe snapshot
structure; they do not establish the temporal order of the constituent
interactions.
step specifies the interval between measurements, while
window specifies the duration covered from each measurement
time. Equal values produce non-overlapping windows. A larger
window produces overlapping, rolling measurements.
rolling <- metrics(school, measure = "density", step = 1, window = 7)
head(rolling, 4)
#> # Density (graph-level)
#> # 22 time points, step 1, window 7 (rolling) | time in step
#> # first 4 of 22 rows
#> time measure value
#> 0 density 0.3241758
#> 1 density 0.3461538
#> 2 density 0.3571429
#> 3 density 0.3791209
The first seven-day window contains 59 connected pairs, giving
density 0.324, compared with ten pairs and density 0.055 in the first
day alone. Wider windows combine more relationships while providing less
detail about changes within each interval.
school_density <- metrics(school, measure = "density")
plot(school_density)
Snapshot measures count a connected pair once within a window
regardless of how long or how often it is connected.
temporal_density instead measures the occupied proportion
of eligible pair-time. onset_intensity divides the number
of spell onsets by eligible pair-time, giving a rate of formation per
unit of relational opportunity. For this fixed population, eligible
pair-time is \(n(n-1)\) multiplied by
observed duration.
integrated <- metrics(school, measure = c("temporal_density", "onset_intensity"),
step = 7, window = 7)
integrated
#> # Graph structure (graph-level)
#> # 4 time points, 7 per bin | time in step
#> # measures: temporal_density, onset_intensity
#> time measure value
#> 0 temporal_density 0.02428571
#> 0 onset_intensity 0.06043956
#> 7 temporal_density 0.03860283
#> 7 onset_intensity 0.08398744
#> 14 temporal_density 0.02352433
#> 14 onset_intensity 0.04395604
#> 21 temporal_density 0.01806847
#> 21 onset_intensity 0.00000000
Each complete week contains \(182 \times 7
= 1274\) eligible pair-days. Onset intensities of 0.0604, 0.0840,
and 0.0440 correspond to 77, 107, and 56 spell onsets, respectively. The
final partial bin contains no onsets.
Temporal density is highest in the second week at 0.0386,
corresponding to 49.2 occupied pair-days. It decreases to 0.0181 in the
final partial bin. These values account for connection duration, whereas
snapshot density records whether a connection occurred at any point in a
bin.
snapshots() lists the connections contributing to each
snapshot. Supplying at selects a measurement time; omitting
it returns the measurement grid.
at_five <- snapshots(school, at = 5)
at_five
#> # Snapshot edges | 1 bin | 16 tie rows | time in step
#> time from to weight n_spells
#> 1 5 Kira Leo 1 1
#> 2 5 Kira Gita 1 1
#> 3 5 Leo Cara 1 1
#> 4 5 Leo Finn 1 1
#> 5 5 Mira Ana 1 1
#> 6 5 Nils Ben 1 1
#> 7 5 Nils Eve 1 1
#> 8 5 Ben Eve 1 1
#> 9 5 Ben Finn 1 1
#> 10 5 Cara Kira 1 1
#> # 6 more rows. summary() counts them by bin.
At day 5, the result contains sixteen connected pairs, matching the
edges measure for that bin. Each pair has a weight and a
count of contributing spells.
events() counts spell onsets and terminations within
each bin.
changes <- events(school)
head(changes, 6)
#> # Edge dynamics (graph-level)
#> # 22 time points, 1 per bin | time in step
#> # measures: formation, dissolution
#> # first 6 of 44 rows
#> time measure value
#> 0 formation 11
#> 0 dissolution 7
#> 1 formation 4
#> 1 dissolution 6
#> 2 formation 8
#> 2 dissolution 6
The first bin contains eleven onsets and seven terminations; the
second contains four onsets and six terminations. These are spell
counts, which may include repeated relationships between the same
endpoints.
durations() summarises the length of relational spells.
By default, it returns the number of spells (events),
summed duration (total), and mean duration
(mean) for each ordered pair. unit selects
pair-level histories ("pair"), individual relational spells
("spell"), vertex activity ("vertex_activity"
or "vertex_spell"), or spells incident to vertices
("node_ties").
tie_durations <- durations(school)
head(tie_durations, 6)
#> # Relationship duration (edge-level)
#> # time in step
#> # first 6 of 330 rows
#> # durations in step
#> from to measure value
#> Ana Cara events 1
#> Ana Dan events 3
#> Ana Gita events 5
#> Ana Iris events 1
#> Ana Jonas events 4
#> Ana Kira events 1
Ana contacted Gita five times and Jonas four times, and Cara, Iris,
and Kira once each. The result contains three measures for each of 110
pairs, giving 330 observations.
spell_durations <- durations(school, unit = "spell")
head(spell_durations, 4)
#> # Relationship duration (edge-level)
#> # time in step
#> # first 4 of 240 rows
#> # durations in step
#> from to raw_spell measure value
#> Ana Cara 71 duration 0.10
#> Ana Dan 143 duration 0.32
#> Ana Dan 168 duration 0.51
#> Ana Dan 228 duration 0.19
With unit = "spell", the result describes the 240
individual spells. Spell identifiers refer to the constructed spell
table. Ana’s three contacts with Dan lasted 0.32, 0.51, and 0.19
days.
node_durations <- durations(school, unit = "node_ties", mode = "all")
head(node_durations, 4)
#> # Incident tie duration (node-level)
#> # 14 vertices | mode all | time in step
#> # first 4 of 28 rows
#> # durations in step
#> node measure value
#> Ana events 36
#> Ben events 34
#> Cara events 35
#> Dan events 35
With unit = "node_ties" and mode = "all",
events counts spells incident to each vertex in either
direction. Ana participates in 36 spells and Ben in 34.
Collapsing to a static network
collapse_network() aggregates a selected observation
period into a static weighted network. start and
end delimit that period. The result contains connected
pairs and their available weight summaries.
flat <- collapse_network(school, start = 0, end = 7)
flat
#> # Collapsed temporal network | 14 vertices | 59 edges | weight: binary
#> # 0 to 7 step
#> from to binary union_duration total_duration duration_fraction spell_count
#> Ana Cara 1 0.10 0.10 0.01428571 1
#> Ana Gita 1 0.33 0.33 0.04714286 1
#> Ana Jonas 1 1.44 1.44 0.20571429 3
#> Ana Mira 1 0.41 0.41 0.05857143 1
#> Ben Eve 1 1.58 1.58 0.22571429 2
#> Ben Finn 1 0.23 0.23 0.03285714 1
#> weight_sum weighted_duration latest_weight first last activity.duration
#> 1 0.10 1 6.67 6.77 0.10
#> 1 0.33 1 6.57 6.90 0.33
#> 3 1.44 1 2.12 7.00 1.44
#> 1 0.41 1 6.36 6.77 0.41
#> 2 1.58 1 3.61 6.24 1.58
#> 1 0.23 1 4.91 5.14 0.23
#> activity.count
#> 1
#> 1
#> 3
#> 1
#> 2
#> 1
The first week contains 59 connected pairs. For each pair, the result
records binary presence (binary), duration with at least
one active spell (union_duration), summed spell duration
(total_duration), and the proportion of observed time
connected (duration_fraction). It also records spell
counts, weight summaries, and first and last contact times.
Ana and Jonas have three spells within the first week, totalling 1.44
days, or approximately 21% of the week. Their first and last observed
boundaries are days 2.12 and 7.00. Union and total duration coincide
because their spells do not overlap.
Use weight to select the summary used as the static edge
weight.
first_week <- collapse_network(school, start = 0, end = 7,
weight = "union_duration")
first_week
#> # Collapsed temporal network | 14 vertices | 59 edges | weight: union_duration
#> # 0 to 7 step
#> from to binary union_duration total_duration duration_fraction spell_count
#> Ana Cara 1 0.10 0.10 0.01428571 1
#> Ana Gita 1 0.33 0.33 0.04714286 1
#> Ana Jonas 1 1.44 1.44 0.20571429 3
#> Ana Mira 1 0.41 0.41 0.05857143 1
#> Ben Eve 1 1.58 1.58 0.22571429 2
#> Ben Finn 1 0.23 0.23 0.03285714 1
#> weight_sum weighted_duration latest_weight first last activity.duration
#> 1 0.10 1 6.67 6.77 0.10
#> 1 0.33 1 6.57 6.90 0.33
#> 3 1.44 1 2.12 7.00 1.44
#> 1 0.41 1 6.36 6.77 0.41
#> 2 1.58 1 3.61 6.24 1.58
#> 1 0.23 1 4.91 5.14 0.23
#> activity.count
#> 1
#> 1
#> 3
#> 1
#> 2
#> 1
The collapsed network supports static analyses such as layouts and
community detection. Its aggregate summaries do not retain the complete
timing of individual spells, so time-respecting paths must be examined
using the temporal network.
References
Holme, P., & Saramäki, J. (2012). Temporal networks. Physics
Reports, 519(3), 97–125.
Kempe, D., Kleinberg, J., & Kumar, A. (2002). Connectivity and
inference problems for temporal networks. Journal of Computer and
System Sciences, 64(4), 820–842.
Saqr, M., & Nouri, J. (2020). High resolution temporal network
analysis to understand and improve collaborative learning. In
Proceedings of the Tenth International Conference on Learning
Analytics & Knowledge (pp. 314–319). ACM.
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