+
### Diffusing across a lattice
Let us begin with a lattice network that shows us exactly how such a cascade works.
@@ -65,92 +65,84 @@ graphr(lat)
Ok, great! That's made a nice little lattice network.
Next we want to play a diffusion process _on_ this network.
-To do this, we just need to run `play_diffusion()`,
-and assign the result.
-Then we can investigate the resulting object in a few ways.
-The first way is simply to print the result by calling the object.
-The other way is to unpack the result by calling for its summary.
+To do this, we need to run `play_diffusion()`.
+This function plays a simple contagion process on a network,
+where any contact with an infected/adopter node
+is enough to cause susceptible nodes to adopt/become infected by the following timepoint.
+The function returns a network object that contains information about how the diffusion played out.
+Then we can investigate the diffusion process by extracting this information
+using `as_changelist()` and `as_diffusion()`.
+The first simply extracts the list of adoption/infection events,
+while the second summarises the diffusion process by time point.
```{r lat_diff, exercise = TRUE, exercise.setup = "clattice"}
-lat_diff <- play_diffusion(lat)
-lat_diff
-summary(lat_diff)
+lat_w_diff <- play_diffusion(lat)
+(diff_events <- as_changelist(lat_w_diff))
+(diff_summ <- as_diffusion(lat_w_diff))
```
-The main report from the `lat_diff` object shows the number of nodes that don't
-(yet) have the attribute (S for susceptible, but can also be non-adopter)
-and those that do have the attribute (I for infected, or adopter) at each time point (t).
-We can see a steady growth here, except for a slower initialisation and winding down.
-
-The secondary report, `summary(lat_diff)`,
-presents a list of the events at each time point.
+The changelist presents a list of the events at each time point.
In the event variable, the 'I' indicates that these are all infection/adoption events.
Where 't' is 0, that means that these are the seeds producing the starting condition
for the diffusion.
-The final column, 'exposure', records the number of infected nodes that the adopting
-node was exposed to when it adopted.
-Note that we have no information about the exposure for the seed nodes when they
-were infected, and so this is a missing value.
-The exposure at infection is recorded here to accelerate later analysis.
+The diffusion summary shows the number of nodes that don't
+(yet) have the attribute (S for susceptible, but can also be non-adopter)
+and those that do have the attribute (I for infected, or adopter) at each time point (t).
+We can see a steady growth here, except for a slower initialisation and winding down.
### Visualising cascades
We have several different options for visualising diffusions.
The first visualisation option that we have is to plot the diffusion result itself.
-```{r plotlat, exercise = TRUE, exercise.setup = "lat_diff", purl = FALSE, fig.width=9}
-plot(lat_diff)
-plot(lat_diff, all_steps = FALSE)
+```{r plotlat, exercise = TRUE, exercise.setup = "lat_diff", fig.width=9}
+plot(diff_summ)
+# plot(diff_summ, all_steps = FALSE) # To plot only 'where the action is', use the argument `all_steps = FALSE`.
```
This plot effectively visualises what we observed from the print out of the
-`lat_diff` object above.
+`diff_summ` object above.
The red line traces the proportion of infected;
-the blue line the (inverse) proportion of susceptible.
+the blue line the (inverse) proportion of susceptible nodes in the network.
The grey histogram in the plot shows how many nodes are newly 'infected' at each
time point, or the so-called 'force of infection' ($F = \beta I$).
-We can see that by default the whole simulated period (32 steps) is shown,
-even though there is complete infection after only 10 steps.
-That's because the simulation runs over the number of nodes in the network
-by default.
-If the structure is amenable to diffusion, infection/diffusion will be completed
-before that.
-To plot only 'where the action is', use the argument `all_steps = FALSE`.
-
But maybe we want to also/instead view the diffusion on the actual network.
-Here we can use all the three main graphing techniques offered in `{manynet}`.
+Here we can use all the three main graphing techniques offered in `{autograph}`.
First, `graphr()` will graph a static network where the nodes are coloured
according to how far through the diffusion process the node adopted.
Note also that any seeds are indicated with a triangle.
-```{r graphrlat, exercise = TRUE, exercise.setup = "lat_diff", purl = FALSE, fig.width=9}
-graphr(lat_diff, node_size = 0.3)
+```{r graphrlat, exercise = TRUE, exercise.setup = "lat_diff", fig.width=9}
+graphr(lat_w_diff, node_size = 0.3)
```
Second, `graphs()` visualises the stages of the diffusion on the network.
By default it will graph the first and last wave,
but we can change this by specifying which waves to graph.
-```{r graphslat, exercise = TRUE, exercise.setup = "lat_diff", purl = FALSE, fig.width=9}
-graphs(lat_diff)
-graphs(lat_diff, waves = c(1,4,8))
+```{r graphslat, exercise = TRUE, exercise.setup = "lat_diff", fig.width=9}
+graphs(lat_w_diff)
+graphs(lat_w_diff, waves = c(1,4,7,10))
```
+We can see here exactly how the attribute in question (ideas, information, disease?)
+is diffusing across the network.
+It's like a cascade of red sweeping across the space!
+
Lastly, `grapht()` animates this diffusion process to a gif.
It can take a little time to encode, but it is worth it to see exactly
how the attribute is diffusing across the network!
Note that if you run this code in the console, you get a calming progress bar;
in the tutorial you will just need to be patient.
+> NB: It is not currently working (for diffusions) at the moment,
+but a refactoring soon will fix this and other related bugs.
+
```{r graphtlat, exercise = TRUE, exercise.setup = "lat_diff", purl = FALSE, fig.width=9}
-grapht(lat_diff, node_size = 10)
+# grapht(lat_w_diff, node_size = 10)
```
-We can see here exactly how the attribute in question (ideas, information, disease?)
-is diffusing across the network.
-It's like a cascade of red sweeping across the space!
-
### Varying network structure
While a lattice structure is one way of representing spatially governed diffusion,
@@ -177,8 +169,8 @@ graphr(play_diffusion(generate_smallworld(32, 0.025)))
Which diffusion process completed first?
`graphr()` only colors nodes' relative adoption,
and `graphs()` (at least by default) only graphs the first and last step.
-`grapht()` will show if and when there is complete infection,
-but we need to sit through each 'movie'.
+
+
But there is an easier way.
Play these same diffusions again, this time nesting the call within `net_infection_complete()`.
@@ -211,7 +203,6 @@ You can start the infection in California by specifying `seeds = 5`.
us_diff <- play_diffusion(irps_usgeo, seeds = 5)
plot(us_diff)
graphr(us_diff)
-grapht(us_diff)
net_infection_complete(us_diff)
```
@@ -227,6 +218,8 @@ This is known as a `r gloss("linear threshold", "LTM")` model,
where if infection/influence on a node through some (potentially weighted) network
exceeds some threshold, then they will adopt/become infected.
+
+
### Threshold rising
Let's use the ring network again this time to
@@ -243,9 +236,9 @@ Let's see what the results are if you play four different diffusions:
```{r complex, exercise = TRUE, fig.width=9}
rg <- create_ring(32, width = 2)
-plot(play_diffusion(rg, seeds = 1, thresholds = 1))/
-plot(play_diffusion(rg, seeds = 1, thresholds = 2))/
-plot(play_diffusion(rg, seeds = 1:2, thresholds = 2))/
+plot(play_diffusion(rg, seeds = 1, thresholds = 1))
+plot(play_diffusion(rg, seeds = 1, thresholds = 2))
+plot(play_diffusion(rg, seeds = 1:2, thresholds = 2))
plot(play_diffusion(rg, seeds = c(1,16), thresholds = 2))
```
@@ -303,14 +296,14 @@ on the scale-free networks we have been using here.
```
```{r sfprop-hint, purl = FALSE}
-plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = ____, steps = ____))/
-plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = ____, steps = ____))/
+plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = ____, steps = ____))
+plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = ____, steps = ____))
plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = ____, steps = ____))
```
```{r sfprop-solution}
-plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = 0.1, steps = 10))/
-plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = 0.25, steps = 10))/
+plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = 0.1, steps = 10))
+plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = 0.25, steps = 10))
plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = 0.5, steps = 10))
```
@@ -337,17 +330,14 @@ Something is going around Middle-Earth,
but different races have different resistances (i.e. thresholds).
Let us say that there is a clear ordering to this.
-```{r lotr-resist, exercise=TRUE}
+```{r lotr-resist, exercise=TRUE, fig.width=9}
lotr_resist <- fict_lotr %>% mutate(resistance = dplyr::case_when(Race == "Dwarf" ~ 2,
Race == "Elf" ~ 4,
Race == "Ent" ~ 5,
Race == "Hobbit" ~ 3,
Race == "Human" ~ 1,
Race == "Maiar" ~ 6))
-```
-
-```{r resistdiff, exercise=TRUE, exercise.setup = "lotr-resist", fig.width=9}
-grapht(play_diffusion(lotr_resist, thresholds = "resistance"))
+graphs(play_diffusion(lotr_resist, thresholds = "resistance"))
```
Fun! Now how would you interpret what is going on here?
@@ -362,12 +352,14 @@ Can you rewrite the code above so that fractional thresholds are used?
-## Intervention
+## Make it start
Let's say that you have developed an exciting new policy and
you are keen to maximise how quickly and thoroughly it is adopted.
We are interested here in `r gloss("network intervention", "intervention")`.
+
+
### Choosing where to seed
Since the ring network we constructed is cyclical,
@@ -382,8 +374,8 @@ and see whether the result is any different.
```
```{r ring2-solution}
-plot(play_diffusion(create_ring(32, width = 2), seeds = 1)) /
- plot(play_diffusion(create_ring(32, width = 2), seeds = 16))
+plot(play_diffusion(create_ring(32, width = 2), seeds = 1))
+plot(play_diffusion(create_ring(32, width = 2), seeds = 16))
```
```{r ring2-interp, echo = FALSE, purl = FALSE}
@@ -458,14 +450,14 @@ one with the first node as seed and again one on the middle.
```
```{r lattice-solution}
-plot(play_diffusion(lat, seeds = 1))/
+plot(play_diffusion(lat, seeds = 1))
plot(play_diffusion(lat, seeds = 16))
lat %>%
add_node_attribute("color", c(1, rep(0, 14), 2, rep(0, 16))) %>%
graphr(node_color = "color")
# visualise diffusion in lattice graph
-grapht(play_diffusion(lat, seeds = 16), layout = "grid", keep_isolates = FALSE)
+graphs(play_diffusion(lat, seeds = 16), layout = "grid")
```
```{r lattice-interp, echo = FALSE, purl = FALSE}
@@ -501,14 +493,14 @@ sf %>%
```
```{r scale-solution}
-plot(play_diffusion(sf, seeds = 10, steps = 10)) /
-plot(play_diffusion(sf, seeds = node_is_random(sf), steps = 10)) /
-plot(play_diffusion(sf, seeds = node_is_max(node_degree(sf)), steps = 10)) /
+plot(play_diffusion(sf, seeds = 10, steps = 10))
+plot(play_diffusion(sf, seeds = node_is_random(sf), steps = 10))
+plot(play_diffusion(sf, seeds = node_is_max(node_degree(sf)), steps = 10))
plot(play_diffusion(sf, seeds = node_is_min(node_degree(sf)), steps = 10))
# visualise diffusion in scalefree network
graphs(play_diffusion(sf, seeds = node_is_min(node_degree(sf)), steps = 10))
-grapht(play_diffusion(sf, seeds = 16, steps = 10))
+graphs(play_diffusion(sf, seeds = 16, steps = 10), waves = 1:3)
```
```{r mindeg-interp, echo = FALSE, purl = FALSE}
@@ -566,6 +558,8 @@ But before we get into that,
let's see how we can play and plot several simulations
to see what the range of outcomes might be like.
+
+
### Running multiple simulations
To do this, we need to use `play_diffusions()` (note the plural).
@@ -698,14 +692,16 @@ set.seed(123)
plot(play_diffusion(rando, seeds = 10, latency = 0.25, recovery = 0.2))
# visualise diffusion with latency and recovery
-grapht(play_diffusion(rando, seeds = 10, latency = 0.25, recovery = 0.2))
+graphs(play_diffusion(rando, seeds = 10, latency = 0.25, recovery = 0.2), waves = c(1,5,10))
```
-### Make it stop
+## Make it stop
In this section, we are interested in how to most effectively _halt_
a diffusion process.
+
+
An attribute's reproduction number, or $R_0$, is a measure of the rate of infection
or how quickly that attribute will reproduce period over period.
It is calculated as $R_0 = \min\left(\frac{T}{1/L}, \bar{k}\right)$,
@@ -762,7 +758,7 @@ But then...
### How many people do we need to vaccinate?
We can identify how many people need to be vaccinated through
-the `gloss("Herd Immunity Threshold", "hit")` or HIT.
+the `r gloss("Herd Immunity Threshold", "hit")` or HIT.
HIT indicates the threshold at which the reduction of susceptible members
of the network means that infections will no longer keep increasing,
allowing herd immunity to be achieved.
@@ -849,6 +845,8 @@ usually rely on nodes' voluntary participation:
they must accept that vaccination, medication, or behavioral change is necessary
to combat the contagion.
+
+
Lastly, we're going to consider a rather simple type of learning model:
a DeGroot learning model.
A question often asked of these kinds of models is whether,
diff --git a/inst/tutorials/tutorial7/diffusion.html b/inst/tutorials/tutorial7/diffusion.html
index cfc44860a..7a9ee72ae 100644
--- a/inst/tutorials/tutorial7/diffusion.html
+++ b/inst/tutorials/tutorial7/diffusion.html
@@ -146,6 +146,7 @@
Let us begin with a lattice network that shows us exactly how such a @@ -165,33 +166,34 @@
Ok, great! That’s made a nice little lattice network. Next we want to
-play a diffusion process on this network. To do this, we just
-need to run play_diffusion(), and assign the result. Then
-we can investigate the resulting object in a few ways. The first way is
-simply to print the result by calling the object. The other way is to
-unpack the result by calling for its summary.
play_diffusion(). This function plays a simple
+contagion process on a network, where any contact with an
+infected/adopter node is enough to cause susceptible nodes to
+adopt/become infected by the following timepoint. The function returns a
+network object that contains information about how the diffusion played
+out. Then we can investigate the diffusion process by extracting this
+information using as_changelist() and
+as_diffusion(). The first simply extracts the list of
+adoption/infection events, while the second summarises the diffusion
+process by time point.
lat_diff <- play_diffusion(lat)
-lat_diff
-summary(lat_diff)lat_w_diff <- play_diffusion(lat)
+(diff_events <- as_changelist(lat_w_diff))
+(diff_summ <- as_diffusion(lat_w_diff))The main report from the lat_diff object shows the
-number of nodes that don’t (yet) have the attribute (S for susceptible,
-but can also be non-adopter) and those that do have the attribute (I for
-infected, or adopter) at each time point (t). We can see a steady growth
-here, except for a slower initialisation and winding down.
The secondary report, summary(lat_diff), presents a list
-of the events at each time point. In the event variable, the ‘I’
-indicates that these are all infection/adoption events. Where ‘t’ is 0,
-that means that these are the seeds producing the starting condition for
-the diffusion. The final column, ‘exposure’, records the number of
-infected nodes that the adopting node was exposed to when it adopted.
-Note that we have no information about the exposure for the seed nodes
-when they were infected, and so this is a missing value. The exposure at
-infection is recorded here to accelerate later analysis.
The changelist presents a list of the events at each time point. In +the event variable, the ‘I’ indicates that these are all +infection/adoption events. Where ‘t’ is 0, that means that these are the +seeds producing the starting condition for the diffusion. The diffusion +summary shows the number of nodes that don’t (yet) have the attribute (S +for susceptible, but can also be non-adopter) and those that do have the +attribute (I for infected, or adopter) at each time point (t). We can +see a steady growth here, except for a slower initialisation and winding +down.
plot(lat_diff)
-plot(lat_diff, all_steps = FALSE)plot(diff_summ)
+# plot(diff_summ, all_steps = FALSE) # To plot only 'where the action is', use the argument `all_steps = FALSE`.This plot effectively visualises what we observed from the print out
-of the lat_diff object above. The red line traces the
+of the diff_summ object above. The red line traces the
proportion of infected; the blue line the (inverse) proportion of
-susceptible. The grey histogram in the plot shows how many nodes are
-newly ‘infected’ at each time point, or the so-called ‘force of
-infection’ (\(F = \beta I\)).
We can see that by default the whole simulated period (32 steps) is
-shown, even though there is complete infection after only 10 steps.
-That’s because the simulation runs over the number of nodes in the
-network by default. If the structure is amenable to diffusion,
-infection/diffusion will be completed before that. To plot only ‘where
-the action is’, use the argument all_steps = FALSE.
But maybe we want to also/instead view the diffusion on the actual
network. Here we can use all the three main graphing techniques offered
-in {manynet}. First, graphr() will graph a
+in {autograph}. First, graphr() will graph a
static network where the nodes are coloured according to how far through
the diffusion process the node adopted. Note also that any seeds are
indicated with a triangle.
graphr(lat_diff, node_size = 0.3)graphr(lat_w_diff, node_size = 0.3)Second, graphs() visualises the stages of the diffusion
@@ -235,24 +232,28 @@
graphs(lat_diff)
-graphs(lat_diff, waves = c(1,4,8))graphs(lat_w_diff)
+graphs(lat_w_diff, waves = c(1,4,7,10))We can see here exactly how the attribute in question (ideas, +information, disease?) is diffusing across the network. It’s like a +cascade of red sweeping across the space!
Lastly, grapht() animates this diffusion process to a
gif. It can take a little time to encode, but it is worth it to see
exactly how the attribute is diffusing across the network! Note that if
you run this code in the console, you get a calming progress bar; in the
tutorial you will just need to be patient.
+NB: It is not currently working (for diffusions) at the moment, but a +refactoring soon will fix this and other related bugs.
+
grapht(lat_diff, node_size = 10)# grapht(lat_w_diff, node_size = 10)We can see here exactly how the attribute in question (ideas, -information, disease?) is diffusing across the network. It’s like a -cascade of red sweeping across the space!
Which diffusion process completed first? graphr() only
colors nodes’ relative adoption, and graphs() (at least by
-default) only graphs the first and last step. grapht() will
-show if and when there is complete infection, but we need to sit through
-each ‘movie’. But there is an easier way. Play these same diffusions
-again, this time nesting the call within
+default) only graphs the first and last step.
+
+ But there is an easier
+way. Play these same diffusions again, this time nesting the call within
net_infection_complete().
us_diff <- play_diffusion(irps_usgeo, seeds = 5)
plot(us_diff)
graphr(us_diff)
-grapht(us_diff)
net_infection_complete(us_diff)What’s happening here? Can you interpret this?
@@ -343,6 +343,7 @@
Let’s use the ring network again this time to illustrate the impact @@ -362,9 +363,9 @@
rg <- create_ring(32, width = 2)
-plot(play_diffusion(rg, seeds = 1, thresholds = 1))/
-plot(play_diffusion(rg, seeds = 1, thresholds = 2))/
-plot(play_diffusion(rg, seeds = 1:2, thresholds = 2))/
+plot(play_diffusion(rg, seeds = 1, thresholds = 1))
+plot(play_diffusion(rg, seeds = 1, thresholds = 2))
+plot(play_diffusion(rg, seeds = 1:2, thresholds = 2))
plot(play_diffusion(rg, seeds = c(1,16), thresholds = 2))plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = ____, steps = ____))/
-plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = ____, steps = ____))/
+plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = ____, steps = ____))
+plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = ____, steps = ____))
plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = ____, steps = ____))
plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = 0.1, steps = 10))/
-plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = 0.25, steps = 10))/
+plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = 0.1, steps = 10))
+plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = 0.25, steps = 10))
plot(play_diffusion(generate_scalefree(32, 0.025), seeds = 1:2, thresholds = 0.5, steps = 10))
grapht(play_diffusion(lotr_resist, thresholds = "resistance"))Fun! Now how would you interpret what is going on here? Can you @@ -492,13 +488,14 @@
Let’s say that you have developed an exciting new policy and you are keen to maximise how quickly and thoroughly it is adopted. We are interested here in network intervention .
+
Since the ring network we constructed is cyclical, then no matter @@ -514,8 +511,8 @@
plot(play_diffusion(create_ring(32, width = 2), seeds = 1)) /
- plot(play_diffusion(create_ring(32, width = 2), seeds = 16))plot(play_diffusion(create_ring(32, width = 2), seeds = 1))
+plot(play_diffusion(create_ring(32, width = 2), seeds = 16))plot(play_diffusion(lat, seeds = 1))/
+plot(play_diffusion(lat, seeds = 1))
plot(play_diffusion(lat, seeds = 16))
lat %>%
add_node_attribute("color", c(1, rep(0, 14), 2, rep(0, 16))) %>%
graphr(node_color = "color")
# visualise diffusion in lattice graph
-grapht(play_diffusion(lat, seeds = 16), layout = "grid", keep_isolates = FALSE)
+graphs(play_diffusion(lat, seeds = 16), layout = "grid")plot(play_diffusion(sf, seeds = 10, steps = 10)) /
-plot(play_diffusion(sf, seeds = node_is_random(sf), steps = 10)) /
-plot(play_diffusion(sf, seeds = node_is_max(node_degree(sf)), steps = 10)) /
+plot(play_diffusion(sf, seeds = 10, steps = 10))
+plot(play_diffusion(sf, seeds = node_is_random(sf), steps = 10))
+plot(play_diffusion(sf, seeds = node_is_max(node_degree(sf)), steps = 10))
plot(play_diffusion(sf, seeds = node_is_min(node_degree(sf)), steps = 10))
# visualise diffusion in scalefree network
graphs(play_diffusion(sf, seeds = node_is_min(node_degree(sf)), steps = 10))
-grapht(play_diffusion(sf, seeds = 16, steps = 10))
+graphs(play_diffusion(sf, seeds = 16, steps = 10), waves = 1:3)
To do this, we need to use play_diffusions() (note the
@@ -861,13 +859,15 @@
In this section, we are interested in how to most effectively halt a diffusion process.
+
An attribute’s reproduction number, or \(R_0\), is a measure of the rate of infection or how quickly that attribute will reproduce period over @@ -920,16 +920,16 @@
Ok, let’s start with option 1. After all, those who built up natural immunity (recovered) may have already protected some parts of the network from complete infection. But then…
-We can identify how many people need to be vaccinated through the
-gloss("Herd Immunity Threshold", "hit") or HIT. HIT
-indicates the threshold at which the reduction of susceptible members of
-the network means that infections will no longer keep increasing,
-allowing herd immunity to be achieved. net_immunity() gives
-us the proportion of the population that would need to be recovered or
+
+Herd Immunity Threshold or HIT. HIT indicates the
+threshold at which the reduction of susceptible members of the network
+means that infections will no longer keep increasing, allowing herd
+immunity to be achieved. net_immunity() gives us the
+proportion of the population that would need to be recovered or
vaccinated for the network to have herd immunity.
Lastly, we’re going to consider a rather simple type of learning model: a DeGroot learning model. A question often asked of these kinds of models is whether, despite heterogeneous initial beliefs, those @@ -1017,7 +1018,7 @@
is_connected() marks whether network is weakly
connected if the network is
-
+
undirected or strongly connected if directed.is_aperiodic() marks whether network is aperiodic,
meaning there is no integer k > 1 that divides the length of every
@@ -1147,6 +1148,13 @@ @@ -1213,10 +1220,9 @@