Topology and Resilience

In netrics: Many Ways to Measure and Classify Membership for Networks, Nodes, and Ties

library(netrics)
library(autograph)
library(patchwork)

library(learnr)
knitr::opts_chunk$set(echo = FALSE)
library(netrics)
library(autograph)
stocnet_theme("default")
clear_glossary()
learnr::random_phrases_add(language = "fr",
                           praise = c("C'est génial!",
                                      "Beau travail",
                                      "Excellent travail!",
                                      "Bravo!",
                                      "Super!",
                                      "Bien fait",
                                      "Bien joué",
                                      "Tu l'as fait!",
                                      "Je savais que tu pouvais le faire.",
                                      "Ça a l'air facile!",
                                      "C'était un travail de première classe.",
                                      "C'est ce que j'appelle un bon travail!"),
                          encouragement = c("Bon effort",
                                            "Vous l'avez presque maîtrisé!",
                                            "Ça avance bien.",
                                            "Continuez comme ça.",
                                            "Continuez à travailler dur!",
                                            "Vous apprenez vite!",
                                            "Vous faites un excellent travail aujourd'hui."))
learnr::random_phrases_add(language = "en",
                           praise = c("That's brilliant!",
                                      "Great work!",
                                      "Well done!",
                                      "Awesome!"),
                           encouragement = c("Good effort"))

sys_locale <- Sys.getlocale("LC_CTYPE")
if (grepl("fr", sys_locale, ignore.case = TRUE)) {
  options(tutorial.language = "fr")
} else {
  options(tutorial.language = "en")
}

Today's target

So far the {netrics} tutorials have measured networks from the inside out: which r gloss("nodes","node") are central, which cluster into communities, and which share positions. This tutorial turns the telescope around and asks about the shape of the whole r gloss("network") — its topology. We will create ideal-typical networks by deterministic rule (stars, trees, lattices, and rings), generate others by probabilistic recipe (random, small-world, and scale-free networks), and then use these ideals as yardsticks for empirical questions: Does this network have a core and a periphery? How hierarchical is it? And how many nodes or r gloss("ties","tie") could it lose before falling apart?

These ideal networks exaggerate centrality, clustering, and randomness features, and are thus great for theory-building and investigating the relationship between rules and structure.

::: {.callout} Catching up: This tutorial assumes you can already load, manipulate, and draw a network in R, and that you have met the earlier {netrics} tutorials. If any of that is hazy, work through the earlier {stocnet} tutorials first: run run_tute("Making") and run_tute("Manipulating") for network data, run_tute("Visualising") for graphing, and run_tute("Centrality"), run_tute("Community"), and run_tute("Position") for measuring networks, or read their static versions on the manynet, autograph, and netrics websites. :::

::: {.callout} New to network vocabulary?: Throughout this tutorial, key terms are italicised: hover over them for a definition, and a full glossary of the terms used appears at the end of the tutorial. :::

Aims

By the end of this tutorial, you should be able to:

• [ ]   Create deterministic network topologies with create_*(): empty, complete, star, tree, lattice, and ring networks
• [ ]   Generate probabilistic networks with generate_*(): random, small-world, and scale-free networks
• [ ]   Measure and interpret a network's small-world coefficient and power-law exponent
• [ ]   Partition a network into core and periphery with node_is_core(), and judge the fit with net_by_core()
• [ ]   Peel a network into k-cores with node_by_kcoreness()
• [ ]   Measure how hierarchical a directed network is with net_x_hierarchy()
• [ ]   Assess a network's resilience via its connectedness, cohesion, and adhesion, and locate its cutpoints and bridges

Choose your own data: The first half of this tutorial creates and generates its own toy networks, and the worked examples later use ison_lawfirm, ison_adolescents, and other datasets bundled with {manynet}. But wherever there is an exercise box, you are encouraged to swap in a network that interests you. Remember the three flavours of bundled data as a rough difficulty ladder — Classic (ison_*, small & tidy), Fiction (fict_*, mid-sized & fun), Real-world (irps_*, larger & realistic) — and that you can browse the full list with table_data().

Creating networks

On this page: Empty & complete · Stars · Trees · Lattices · Rings

In this practical, we're going to create/generate a number of ideal-typical network topologies and plot them. We'll first look at some deterministic algorithms for creating networks of different structures, and then look at how the introduction of some randomness can generate a variety of network structures.

::: {.callout} Beginner note: All the create_*() functions take the number of nodes to create as their first argument. You can even pass them an existing network instead, in which case they will create a network of the same dimensions — handy for comparing an empirical network with its ideal-typical counterpart. Many also accept a directed = TRUE argument. :::

Empty and complete {#empty-and-complete}

To begin with, let's create a few 'r gloss("empty")' and full/'r gloss("complete")' graphs. You will want to use some of the create_*() group of functions from {manynet}, because they create graphs following some strict rule(s). The two functions you will want to use here are create_empty() and create_filled(). create_empty() creates an empty graph with the given number of nodes, in this case 50 nodes. For create_filled() we're creating a full graph, where all of the nodes are connected to all of the other nodes.

Let's say that we want to explore networks of fifty nodes in this script. Graph one empty and one complete network with 50 nodes each, give them an informative title, and plot the graphs together. What would a complete network with half the nodes look like? Add that too.

(graphr(create_empty(50), "circle") + ggtitle("Empty graph"))
(graphr(create_filled(50)) + ggtitle("Complete graph"))
(graphr(create_filled(50/2)) + ggtitle("Complete graph (smaller)"))

These two extremes bookend every topology to come: an empty network has a r gloss("density") of 0 and a complete network a density of 1, and every real network lies somewhere in between — usually much closer to the empty end.

Stars {#stars}

In a r gloss("star", "star") network, there is one node to which all other nodes are connected, otherwise known as the r gloss("universal", "universal") node. There is no transitivity. The maximum path length is two. And r gloss("degree", "degree") centrality is maximised!^[In fact, all centrality measures are maximised, as one node acts as the sole bridge connecting one part of the network to the other.]

Use the create_star() function to graph three star networks:

• an undirected star network
• a out-directed star network
• and an in-directed star network

(graphr(create_star(50)) + ggtitle("Star graph"))
(graphr(create_star(50, directed = TRUE)) + ggtitle("Star out"))
(graphr(to_redirected(create_star(50, directed = TRUE))) + ggtitle("Star in"))

Substantively, a star is what maximal centralisation looks like: everything depends on the hub. That makes stars very efficient at spreading anything from the centre, but also maximally fragile — remove the universal node and the network shatters into isolates.

Trees {#trees}

Trees, or regular trees, are networks with branching nodes. They can be directed or undirected, and tend to indicate strong hierarchy: each node reports up to a single parent, branching down from a root to the r gloss("pendant","pendant") 'leaf' nodes at the bottom. Again graph three networks:

• one undirected with 2 branches per node
• a directed network with 2 branches per node
• the same as above, but graphed using the "tree" layout

# width argument specifies the breadth of the branches
(graphr(create_tree(50, width = 2)) + ggtitle("Tree graph"))
(graphr(create_tree(50, width = 2, directed = TRUE)) + ggtitle("Tree out"))
(graphr(create_tree(50, width = 2, directed = TRUE), "tree") + ggtitle("Tree layout"))

Try varying the width argument to see the result. We will return to trees when we measure hierarchy later in this tutorial.

Lattices {#lattices}

r gloss("Lattices","lattice") reflect highly clustered networks where there is a high likelihood that interaction partners also interact. They are used to show how clustering facilitates or limits diffusion or makes pockets of behaviour stable.

question("Why are lattices considered highly clustered?",
         answer("Because neighbours are likely also neighbours of each other",
                message = learnr::random_praise(),
                correct = TRUE),
         answer("Because all nodes are directly connected to each other",
                message = learnr::random_encouragement()),
         answer("Because there is a single component",
                message = learnr::random_encouragement()),
         answer("Because there is a single community",
                message = learnr::random_encouragement()),
         random_answer_order = TRUE,
         allow_retry = TRUE)

Note that create_lattice() in {manynet} works a little differently to how it works in {igraph}. In {igraph} the number or vector passed to the function indicates the length of each dimension. So c(50) would be a one-dimensional lattice, essentially a chain of 50 nodes connected to their neighbours. c(50,50) would be a two-dimensional lattice, of 50 nodes long and 50 nodes wide. c(50,50,50) would be a three-dimensional lattice, of 50 nodes long, 50 nodes wide, and 50 nodes deep, etc.

But this doesn't help us when we want to see what a lattice representation with the same order (number of nodes) as a given network would be. For example, perhaps we just want to know what a lattice with 50 nodes would look like. So {manynet} instead tries to find the most even or balanced two-dimensional representation with a given number of nodes.

Graph two lattices, one with 50 nodes, and another with half the number of nodes.

(graphr(create_lattice(50)) + ggtitle("One-mode lattice graph"))
(graphr(create_lattice(50/2)) + ggtitle("Smaller lattice graph"))

We can put numbers on what makes a lattice special. Its density is low, but its r gloss("transitivity","transitivity") — the tendency for two nodes with a shared neighbour to be tied themselves — is high. Measure both for a 50-node lattice.

net_by_density(create_lattice(50))
net_by_transitivity(create_lattice(50))

So with only around an eighth of all possible ties present, nearly half of all open two-paths are closed into triangles: lots of local clustering despite few ties overall.

Rings {#rings}

This creates a graph where each node has two separate neighbours which creates a ring graph. Graph three ring networks:

• one with 50 nodes
• one with 50 nodes where they are connected to neighbours two steps away, on a "circle" layout
• the same as above, but on a "stress" layout

(graphr(create_ring(50)) + ggtitle("Ring graph", subtitle = "Starring Naomi Watts"))
# width argument specifies the width of the ring
(graphr(create_ring(50, width = 2), "circle") + ggtitle("The Ring Two", subtitle = "No different?"))
(graphr(create_ring(50, width = 2), "stress") + ggtitle("The Ring Two v2.0"))

The price a ring pays for its regularity is distance. The r gloss("diameter") of a network is the length of its longest shortest path (r gloss("geodesic")), and net_by_length() returns the average shortest path length. Check how far apart nodes in a ring really are, and what widening the ring does about it.

net_by_diameter(create_ring(50))
net_by_length(create_ring(50))
net_by_diameter(create_ring(50, width = 2))

In a 50-node ring, a message travelling neighbour-to-neighbour needs 25 steps to reach the far side of the circle — and about 13 on average to reach anyone. Doubling the width halves these distances, but they remain long: in rings and lattices, distances grow with the number of nodes. Hold that thought for the small-world networks on the next page.

::: {.callout} In brief: The create_*() functions build deterministic topologies from rules: create_empty() and create_filled() set the two density extremes, create_star() maximises centralisation around one universal node, create_tree() branches like a hierarchy, create_lattice() clusters neighbours into grids, and create_ring() chains them into circles. net_by_density(), net_by_transitivity(), net_by_diameter(), and net_by_length() put numbers on the resulting shapes. :::

Generating networks

On this page: Random · Small-world · Scale-free · Degree mixing · More generators

Next we are going to take a look at some probabilistic graphs. These involve some r gloss("random") element, perhaps in addition to specific rules, to stochastically 'generate' networks of certain types of topologies. As such, we'll be using the generate_*() group of functions from {manynet}.

::: {.callout} Beginner note: Because these functions are stochastic, you will get a slightly different network every time you run them. That is the point! If you need the same network twice — say, to reproduce a figure — call set.seed() with some number of your choice first. :::

Random graphs {#random-graphs}

An Erdös-Renyi graph is simply a random graph. You will need to specify the probability of a tie in addition to the number of nodes. An Erdos-Renyi graph on the vertex set $V$ is a random graph which connects each pair of nodes ${i,j}$ with probability $p$, independent. Note that for a r gloss("sparse","sparse") ER graph, $p$ must decrease as $N$ goes up. Generate three random networks of 50 nodes and a density of 0.08.

(graphr(generate_random(50, 0.08)) + ggtitle("Random 1 graph"))
(graphr(generate_random(50, 0.08)) + ggtitle("Random 2 graph"))
(graphr(generate_random(50, 0.08)) + ggtitle("Random 3 graph"))

Keep going if you like... it will be a little different every time. Note that you can also pass the second argument an integer, in which case the function will interpret that as the number of ties/edges rather than the probability that a tie is present. Try generating a random graph with 200 edges/ties now.

(erdren4 <- graphr(generate_random(50, 200)) + ggtitle("Random 1 graph"))

Random graphs are the mirror image of the lattice: their paths are short, but they have next to no clustering — ties are sprinkled independently, so triangles arise only by coincidence. They serve as the customary null model in network analysis: the baseline of what a network with no structuring principle at all would look like.

Small-world graphs {#small-world-graphs}

Remember the ring graph from above? What if we rewire (change) some of the edges at a certain probability? This is how r gloss("small-world","smallworld") networks are generated. Graph three small-world networks, all with 50 nodes and a rewiring probability of 0.025.

(graphr(generate_smallworld(50, 0.025)) + ggtitle("Smallworld 1 graph"))
(graphr(generate_smallworld(50, 0.025)) + ggtitle("Smallworld 2 graph"))
(graphr(generate_smallworld(50, 0.025)) + ggtitle("Smallworld 3 graph"))

With on average 2.5 ties randomly rewired, does the structure look different? This is a small-world network, where clustering/transitivity remains high but path lengths are much lower than they would otherwise be. Remember that in a small-world network, the shortest-path distance between nodes increases sufficiently slowly as a function of the number of nodes in the network. You can also call these networks a Watts–Strogatz toy network. If you want to review this, go back to the reading by Watts (2004). Check both properties for a generated small-world network.

sw <- generate_smallworld(50, 0.025)
net_by_transitivity(sw)
net_by_length(sw)

Compare these against the pages before: transitivity close to the lattice's (and far above a random graph's), yet an average path length a fraction of the ring's. A handful of rewired ties act as shortcuts spanning the circle, which is why "six degrees of separation" can hold even in enormous, highly clustered social networks.

There is also such a thing as a network's small-world coefficient. See the help page for more details, but with the default equation ('omega'), the coefficient typically ranges between 0 and 1, where 1 is as close to a small-world as possible. Try it now on a small-world generated network, but with a rewiring probability of 0.25.

net_by_smallworld(generate_smallworld(50, 0.25))

Substantively, a coefficient near 1 says the network offers the best of both worlds: neighbourhoods cohesive enough to sustain trust and local norms, yet paths short enough for information or contagion to traverse the whole network in a few steps. A coefficient near 0 means the network behaves more like a pure lattice (clustered but slow) or a pure random graph (fast but unclustered).

Scale-free graphs {#scale-free-graphs}

There is another famous model in network science: the r gloss("scale-free","scalefree") model. Remember: "In many real-world networks, the distribution of the number of network neighbours the degree r gloss("distribution","distribution") is typically right-skewed with a "heavy tail". A majority of the nodes have less-than-average degree and a small fraction of r gloss("hubs","hub") are many times better connected than average (2004, p. 250).

The following generates a scale-free graph according to the Barabasi-Albert (BA) model that rests upon the mechanism of preferential attachment. More on this in the Watts paper (2005, p.51) and Merton (1968). The BA model rests on two mechanisms: population growth and preferential attachment. Population growth: real networks grow in time as new members join the population. Preferential/cumulative attachment means that newly arriving nodes will tend to connect to already well-connected nodes rather than poorly connected ones.

Generate and graph three scale-free networks, with alpha parameters of 0.5, 1, and 1.5.

(graphr(generate_scalefree(50, 0.5)) +
    ggtitle("Scalefree 1 graph", subtitle = "Power = .5"))
(graphr(generate_scalefree(50, 1)) +
    ggtitle("Scalefree 2 graph", subtitle = "Power = 1"))
(graphr(generate_scalefree(50, 1.5)) +
    ggtitle("Scalefree 3 graph", subtitle = "Power = 1.5"))

You can also measure the degree to which a network has a degree distribution that fits a power-law distribution. With an alpha/power-law exponent between 2 and 3, one generally cannot reject the hypothesis that the observed data comes from a power-law distribution. Fit the power-law exponent to a scale-free network generated with an alpha of 2.

net_by_scalefree(generate_scalefree(50, 2))

Why does this matter? A power-law degree distribution means the network is dominated by a few hubs, and such networks behave distinctively: they are remarkably robust to random failure (a random node is almost surely peripheral), yet fragile to targeted attack on their hubs — a theme we return to on the Resilience page.

::: {.callout} Going further: The exponent reported by net_by_scalefree() is fitted to the empirical degree distribution, so it is most informative for networks large enough to have a meaningful tail. For a visual check, plot() the network's degree distribution (see the Centrality tutorial) and look for the heavy right tail. :::

Degree mixing {#degree-mixing}

Once a network has hubs, two further topological questions arise about how degrees are arranged with respect to one another. Do the hubs stick together? — the rich-club coefficient (net_by_richclub()) asks whether high-degree nodes are more densely tied to each other than their numbers alone would predict. And do similar nodes attach to similar nodes? — degree assortativity (net_by_assortativity()) correlates the degrees at the two ends of each tie: positive means high-degree nodes pair with high-degree nodes (assortative), negative means hubs mostly attach to the periphery (disassortative). Both are purely structural, needing no node attributes. Contrast a scale-free network with a random one.

set.seed(123)
sf <- generate_scalefree(50, 2)
net_by_richclub(sf)          # hubs interconnected?
net_by_assortativity(sf)     # do like degrees attach?
net_by_assortativity(generate_random(50, 0.08))

The scale-free network is strongly disassortative (a negative score): its preferential-attachment hubs are joined mostly to the many low-degree nodes, not to each other — the signature of technological and biological networks. Many social networks, by contrast, are mildly assortative, as popular people tend to know one another. A random graph sits near zero, having no degree-mixing tendency at all.

More generators {#more-generators}

Beyond the three classics above, {manynet} offers a few further generative models worth knowing about, each motivated by how real networks grow:

• generate_fire() implements the forest-fire model, in which each arriving node "burns" outward from a random contact along existing ties, producing the heavy tails, high clustering, and shrinking diameters seen in many evolving networks (Leskovec et al. 2007).
• generate_islands() stitches together several dense islands (à la communities) joined by a few bridges — handy for pairing this tutorial's topology measures with the community structure of the earlier tutorial.
• generate_citations() grows a citation network in which new nodes cite existing ones with a recency bias, the classic model of directed, (near-)acyclic reference networks.

Generate and glance at all three.

set.seed(123)
graphr(generate_fire(50)) + ggtitle("Forest fire")
graphr(generate_islands(50, islands = 3, bridges = 2)) + ggtitle("Islands")
graphr(generate_citations(50)) + ggtitle("Citations")

Each model leaves its own visual fingerprint: the forest-fire network grows a few dominant hubs (like a scale-free network, but with more clustering around them), the islands network shows several dense blobs joined by a handful of lone bridging ties, and the citation network fans out as a directed, tree-like cascade from the earliest nodes. These are exactly the ideal patterns you have learned to measure in this tutorial — so you could, for instance, confirm the islands model with net_by_modularity() (from the community tutorial) or the citation model's one-directional flow with net_by_reciprocity().

::: {.callout} Going further: These three are more advanced and we will not dwell on them in class. Each takes further arguments controlling its growth (e.g. islands, bridges, or the forest-fire burn probabilities); see ?generate_fire, ?generate_islands, and ?generate_citations, and their underlying igraph samplers, for the details. :::

::: {.callout} In brief: The generate_*() functions add randomness: generate_random() places ties by probability or count (Erdös-Renyi), generate_smallworld() rewires a ring so paths shorten while clustering survives (Watts–Strogatz), and generate_scalefree() grows hubs through preferential attachment (Barabasi-Albert); generate_fire(), generate_islands(), and generate_citations() add forest-fire, modular, and citation growth models. net_by_richclub() and net_by_assortativity() then describe how a network's degrees mix — whether hubs cluster together, and whether like attaches to like. :::

Core-Periphery

On this page: Ideal graphs · Assignment · Coreness

Core-periphery graphs {#core-periphery-graphs}

Lastly, we'll take a look at some r gloss("core-periphery","core") graphs. The most common definition of a core-periphery network is one in which the network can be partitioned into two groups such that one group of nodes (the core) has dense interactions among themselves, moderately dense interactions with the second group, and the second group (the periphery) has sparse interactions among themselves.

question("Can a single network have both a community structure and a core-periphery structure?",
         answer("No", message = learnr::random_encouragement()),
         answer("Yes",
                message = "That's right. For example, the core and periphery might represent two or more communities.",
                correct = TRUE),
         random_answer_order = TRUE,
         allow_retry = TRUE)

We can visualise extreme versions of such a network using the create_core() function. Graph a core-periphery network of 50 nodes (which, unless a core-periphery membership assignment is given, will be split evenly between core and periphery partitions).

(graphr(create_core(50)) + ggtitle("Core"))

Notice the tell-tale shape: a tight knot of mutually-connected core nodes in the middle, a scatter of peripheral nodes around the edge that connect to the core but rarely to each other. This is the ideal that the fitting measures below compare real networks against.

Core-periphery assignment {#core-periphery-assignment}

Let's consider identifying the core and peripheral nodes in a network. Let's use the ison_lawfirm dataset from {manynet}. This dataset involves relations between partners in a corporate law firm in New England. First of all, graph the data and see whether you can guess which nodes might be part of the core and which are part of the periphery. Colour the nodes by Gender, Office, Practice, and School. Any you might think correlate with core status?

lawfirm <- ison_lawfirm |> to_uniplex("friends") |> to_undirected()

lawfirm <- ison_lawfirm |> to_uniplex("friends") |> to_undirected()
graphr(lawfirm, node_color = "school", edge_color = "darkgray")
graphr(lawfirm, node_color = "gender", edge_color = "darkgray")
graphr(lawfirm, node_color = "office", edge_color = "darkgray")
graphr(lawfirm, node_color = "practice", edge_color = "darkgray")

Next, let's assign nodes to the core and periphery blocks using the node_is_core() function. It works pretty straightforwardly. By default it runs down the rank order of nodes by their degree, at each step working out whether including the next highest degree node in the core will maximise the core-periphery structure of the network. Assign core/periphery membership and graph the network coloured by it.

lawfirm |>
  mutate_nodes(nc = node_is_core()) |>
  graphr(node_color = "nc", edge_color = "gray")

This graph suggests that there is a core and a periphery. There might even be two cores here, one on the left and one on the right.

But is it really all that much of a core-periphery structure? We can establish how correlated our network is compared to a core-periphery model of the same dimension using net_by_core(). A value close to 1 would mean the observed ties line up almost perfectly with the ideal blockmodel — dense within the core, sparse within the periphery — while a value near 0 means the partition fits no better than chance. Measure how well a core-periphery model fits this network with net_by_core().

net_by_core(lawfirm, node_is_core(lawfirm))

question("What can we say about this correlation.",
         answer("It is a perfect negative relationship",
                message = learnr::random_encouragement()),
         answer("It is fairly strong",
                message = learnr::random_encouragement()),
         answer("It is positive",
                message = learnr::random_encouragement()),
         answer("There is absolutely no correlation",
                message = learnr::random_encouragement()),
         answer("None of the above", correct = TRUE,
                message = learnr::random_praise()),
         allow_retry = TRUE
)

So despite what the eye suggested, this friendship network is only weakly — indeed slightly negatively — characterised by a single core-periphery structure. That is substantively interesting in itself: the firm's friendships seem to cluster into groups (remember the two suspected "cores") rather than stratify into insiders and outsiders.

Note that node_is_core() also includes a method that descends through the rank order of nodes' eigenvector centralities instead of degree centralities. Why might that not be such a good choice here?

::: {.callout} Going further: A two-way split can be too blunt. node_in_core() classifies nodes into three blocks — "Core", "Semi-periphery", and "Periphery" — for when the world-systems flavour of the concept fits better. :::

Now let's see whether our core-periphery membership vector correlates with any of the three categorical attributes we looked at before. Since we're doing this on categorical variables, we'll use the Chi-squared test in base R. Take a look and see whether there is a statistically significant association between gender and core (or periphery) status.

chisq.test(as.factor(node_is_core(lawfirm)),
           as.factor(node_attribute(lawfirm, "gender")))

chisq.test(as.factor(node_is_core(lawfirm)),
           as.factor(node_attribute(lawfirm, "gender")))
chisq.test(as.factor(node_is_core(lawfirm)),
           as.factor(node_attribute(lawfirm, "office")))
chisq.test(as.factor(node_is_core(lawfirm)),
           as.factor(node_attribute(lawfirm, "school")))
chisq.test(as.factor(node_is_core(lawfirm)),
           as.factor(node_attribute(lawfirm, "practice")))

question("There a statistically significant association between the core assignment and...",
         answer("gender.",
                message = learnr::random_encouragement()),
         answer("office.",
                message = learnr::random_encouragement()),
         answer("school.",
                message = learnr::random_encouragement()),
         answer("practice.",
                message = learnr::random_encouragement()),
         answer("none of the above variables.", correct = TRUE,
                message = "That's right. The p-value for office is close, but no cigar."),
         allow_retry = TRUE
)

Coreness {#coreness}

An alternative route is to identify 'core' nodes depending on their r gloss("k-coreness","kcoreness"). In {manynet}, we can return nodes k-coreness with node_by_kcoreness() instead of the node_is_core() used for core-periphery. Run the code to colour the law-firm network by each node's k-coreness.

lawfirm |>
  mutate_nodes(ncn = node_by_kcoreness()) |>
  graphr(node_color = "ncn")

Where node_is_core() forces a yes/no answer, k-coreness grades how deep each node sits in the network: a node with coreness k survives even after all nodes of degree less than k have been successively peeled away. High-coreness nodes are thus embedded in a densely interlocked middle, which matters for processes like diffusion — what starts in a high k-core is far more likely to spread widely than what starts among the peelable outer layers.

question("Which has more than two classes/groups.",
         answer("node_kcoreness()", correct = TRUE,
                message = learnr::random_praise()),
         answer("node_is_core()",
                message = learnr::random_encouragement()),
         random_answer_order = TRUE,
         allow_retry = TRUE
)

question("Select the correct definitions:",
         answer("The k-core of a network is a maximal subgraph in which each vertex has at least degree k.", correct = TRUE,
                message = learnr::random_praise()),
         answer("The coreness of a node is k if it belongs to the k-core but not to the (k+1)-core.", correct = TRUE,
                message = "Yes -- a node's coreness is the highest k for which it still sits inside the k-core, so it tells you how deep in the network's core the node lies."),
         answer("The coreness of a node is equal to its degree.",
                message = paste(learnr::random_encouragement(), "Coreness is usually less than or equal to degree: a node needs at least degree k to be in the k-core, but many high-degree nodes have lower coreness because their neighbours don't all qualify.")),
         random_answer_order = TRUE,
         allow_retry = TRUE
)

::: {.callout} In brief: create_core() draws the ideal core-periphery network; node_is_core() assigns each node to core or periphery, and net_by_core() reports how well that bipartition actually fits the network (1 = perfectly, ~0 = not at all). node_by_kcoreness() offers a graded alternative, peeling the network into successively deeper k-cores. :::

Hierarchy

On this page: Dimensions · Measuring · Comparing

What sometimes drives our interest in whether a network resembles a core-periphery network is that it offers a generalisation of hierarchy: the core are the rule-makers; the periphery are the rule-takers. But where we have a directed network, we may be able to measure hierarchy explicitly.

Graph theoretic dimensions of hierarchy {#graph-theoretic-dimensions-of-hierarchy}

Recall that measuring hierarchy directly is difficult, as the concept incorporates several different aspects. Can you recall which of the following are aspects of the graph theoretic dimensions of hierarchy?

question("Select the measures included in the graph theoretic dimensions of hierarchy:",
         answer("reciprocity", correct = TRUE,
                message = paste(learnr::random_praise(), "The hierarchy dimension is measured through (a lack of) reciprocity: in a perfect hierarchy relations run one way, never back.")),
         answer("connectedness", correct = TRUE,
                message = "Yes -- connectedness asks whether every pair of nodes is joined by some (undirected) path, the first of Krackhardt's four dimensions."),
         answer("efficiency", correct = TRUE,
                message = "Yes -- efficiency asks whether the structure uses no more ties than necessary (no redundant links beyond those needed to hold it together)."),
         answer("least upper bound", correct = TRUE,
                message = "Yes -- least upper bound asks whether every pair of nodes shares a common superior they both ultimately report up to."),
         answer("components", message = learnr::random_encouragement()),
         answer("degree", message = learnr::random_encouragement()),
         answer("density", message = learnr::random_encouragement()),
         random_answer_order = TRUE,
         allow_retry = TRUE
)

Measuring GTDH {#measuring-gtdh}

Ok, so let's now take a closer look at how to investigate the degree of hierarchy in a given network. The classic example would be to look at a tree network, like the one constructed earlier. Run the code to try the function net_x_hierarchy() on a directed tree of 11 nodes.

treeleven <- create_tree(11, directed = TRUE)
net_x_hierarchy(treeleven)
rowMeans(net_x_hierarchy(treeleven))

We see here four different measures of hierarchy:

• connectedness: the proportion of dyads that can reach each other
• reciprocity: the proportion of ties that are reciprocated (inverse) — in a perfect hierarchy no r gloss("reciprocity") is expected, since deference flows one way
• efficiency: the minimum indegrees required to connect the network over the actual indegrees present
• least upper bound: all nodes in a hierarchy should have a node that can reach both of them

Note that to get an overall score, we can take the average of these four measures. A directed tree is the perfect hierarchy — it scores 1 on all four dimensions — which is exactly why organisational charts are drawn as trees.

::: {.callout} Going further: Each dimension can also be obtained on its own: net_by_connectedness(), net_by_reciprocity(), net_by_efficiency(), and net_by_upperbound(). See ?measure_hierarchy for definitions and the Krackhardt references. :::

Comparing GTDH {#comparing-gtdh}

Because some of these measures (reciprocity), only make sense for a directed network, we'll switch to a couple of directed networks for this exercise. Which one would you consider is more hierarchical: ison_emotions, a directed network of emotional transitions, or fict_thrones, a directed network of kinship ties? Graph each network, then add a net_x_hierarchy() call for both and compare their profiles.

graphr(ison_emotions)
graphr(fict_thrones)

graphr(ison_emotions)
net_x_hierarchy(ison_emotions)
graphr(fict_thrones)
net_x_hierarchy(fict_thrones)

Actually, these two networks have the same average hierarchy score of around 0.525. But they have quite different profiles. Can you make sense of these results?

question("Compared to the emotional transitions network, the Game of Thrones kinship network has:",
         answer("more nodes able to reach each other",
                message = learnr::random_encouragement()),
         answer("fewer reciprocated ties", correct = TRUE,
                message = learnr::random_praise()),
         answer("fewer redundant ties", correct = TRUE,
                message = "Yes -- that shows up as a higher efficiency score: the kinship structure carries fewer redundant ties than the emotional transitions network."),
         answer("at least one boss for all pairs",
                message = "Not here -- that would be the least-upper-bound property, but not every pair in the kinship network shares a common ancestor they both report up to."),
         random_answer_order = TRUE,
         allow_retry = TRUE
)

This is the payoff of a multidimensional measure: a single "hierarchy score" would have called these two networks identical, but their profiles show they are hierarchical in quite different ways — the emotions network is fully connected but saturated with reciprocated, redundant ties, while the kinship network is sparser and more one-directional but does not integrate everyone under common superiors.

::: {.callout} In brief: net_x_hierarchy() returns Krackhardt's four graph-theoretic dimensions of hierarchy — connectedness, (inverse) reciprocity, efficiency, and least upper bound. Average them (e.g. with rowMeans()) for an overall score, but read the whole profile to see how a network is hierarchical. A directed tree scores perfectly on all four. :::

Resilience

On this page: Cohesion · Cutpoints · Bridges

How cohesive is the network? {#how-cohesive-is-the-network}

When investigating a network's resilience, we might think of whether the network will remain connected despite some nodes or ties dropping out. Let's explore how resilient a (core) network of adolescents (ison_adolescents) might be. First, we might be interested in whether the network is r gloss("connected","connected") at all. Check whether ison_adolescents is connected.

net_by_connectedness(ison_adolescents)

This measure gets at the proportion of dyads that can reach each other in the network. In this case, the proportion is 1, i.e. all nodes can reach every other node. Another way to get at this would be to see how many r gloss("components","component") there are in the network.

question("But counting the number of components instead of connectedness can overemphasise:",
         answer("Isolates", correct = TRUE,
                message = learnr::random_praise()),
         answer("Small components", correct = TRUE,
                message = learnr::random_encouragement()),
         answer("Density",
                message = learnr::random_encouragement()),
         random_answer_order = TRUE,
         allow_retry = TRUE
)

A dropped tie can have severe consequences to the topology of a network if it is a bridge, say. The more ties you would need to remove to fragment the network, the greater the r gloss("adhesion","adhesion") of the network. But a dropped node can be even more consequential, as it will take any ties it has with it. Find out the r gloss("cohesion","cohesion"), or how many dropped nodes it would take to (further) fragment the network.

net_by_cohesion(ison_adolescents)

question("The result of this function represents...",
         answer("the minimum number of nodes necessary to remove from the network to increase the number of components.", correct = TRUE,
                message = learnr::random_praise()),
         answer("the number of strong components in the network.",
                message = learnr::random_encouragement()),
         answer("the minimum number of ties necessary to remove from the network to increase the number of components.",
                message = "This is actually the definition of `node_adhesion()`."),
         random_answer_order = TRUE,
         allow_retry = TRUE
)

question("The higher the minimum number of nodes to remove...",
         answer("the more resilient is the network.", correct = TRUE,
                message = learnr::random_praise()),
         answer("the less resilient the network.",
                message = learnr::random_encouragement()),
         random_answer_order = TRUE,
         allow_retry = TRUE
)

A cohesion of 1, as here, is the most fragile a connected network can be: there is at least one single node whose departure would break the network apart. Recall the star and scale-free networks from earlier — highly centralised topologies buy efficiency at exactly this price.

::: {.callout} Going further: Cohesion and adhesion report a minimum count, regardless of network size. For measures that weigh the removals against the fragments they create, see net_by_toughness() (for nodes) and net_by_strength() (for ties). :::

Identifying cutpoints {#identifying-cutpoints}

But which are these nodes? Is there more than one? Nodes that endanger fragmentation of the network are called r gloss("cutpoints","cutpoint"). Find and use a function to identify which, if any, of the nodes in the ison_adolescents network are cutpoints.

node_is_cutpoint(ison_adolescents)

Ok, so this results in a vector identifying which nodes are cutpoints (TRUE) or not (FALSE). Somewhat more useful though would be to highlight these nodes on the network. Can you add a node attribute that highlights which nodes are cutpoints?

ison_adolescents |>
  mutate_nodes(cut = node_is_cutpoint()) |>
  graphr(node_color = "cut")

The highlighted node(s) are the network's single points of failure: each sits astride the only path between two otherwise-separate parts of the network, so its removal would split the adolescents into disconnected groups. These are the nodes you would protect first if you cared about keeping the network together.

Identifying bridges {#identifying-bridges}

Let's do something similar now, but with respect to ties rather than nodes. Here we are interested in identifying which ties are r gloss("bridges","bridge"). Report the network's adhesion, then graph it with the bridging ties highlighted.

net_by_adhesion(ison_adolescents)
ison_adolescents |>
  mutate_ties(cut = tie_is_bridge()) |>
  graphr(edge_color = "cut")

The adhesion tells you the minimum number of ties to cut to fragment the network, and the highlighted ties show which ones do that work. As with cutpoints, a network held together by only one or two bridges is fragile: losing a single highlighted tie would break it apart.

We could also investigate the opposite of a bridge, the degree to which ties are deeply embedded in r gloss("triangles","triangle"). This is called (rather confusingly) tie cohesion. Size the ties by their cohesion to see which are most embedded.

ison_adolescents |>
  mutate_ties(coh = tie_by_cohesion()) |>
  graphr(edge_size = "coh")

Where would you target your efforts if you wanted to fragment this network? And conversely, if you wanted to make this network more resilient, the map of bridges and cutpoints shows where redundancy is missing: adding even one tie between the far sides of a bridge would rob both the bridge and its cutpoints of their fragility.

::: {.callout} In brief: net_by_connectedness(), net_by_cohesion(), and net_by_adhesion() summarise how hard a network is to fragment — the share of dyads that can reach each other, and the minimum nodes or ties whose removal would fragment it. node_is_cutpoint() and tie_is_bridge() point to where it would break, and tie_by_cohesion() shows which ties are buttressed by triangles. :::

Free play

Choose another dataset included in {manynet} (browse them with table_data()), and ask of it the questions this tutorial has equipped you to answer. Which ideal topology does it most resemble — is it more lattice-like, small-world, or scale-free? Does it have a discernible core and periphery? And how resilient is it: where are its cutpoints and bridges, and what would it take to fragment it?

If you are not sure where to start, here is one suggestion per flavour:

| Classic (small, tidy) | Fiction (moderate) | Real-world (larger) | |---|---|---| | ison_karateka (the karate club that famously split in two — its degree distribution fits a power-law exponent close to 2, and its cutpoints and bridges foreshadowed where the split would come) | fict_greys (a sparse network of romantic entanglements in several separate components — find the cutpoints, and check the cohesion of its largest component with to_giant()) | irps_books (co-purchases of 105 political books — does polarised buying sort the market into a core and periphery, into communities, or around a few hub titles?) |

Summary

Well done -- you have completed the tutorial on topology and resilience! Along the way, you have learned to use these functions:

| Function | What it does | |---|---| | create_empty(), create_filled() | networks with no ties, or with all possible ties | | create_star() | one universal node connected to all others | | create_tree() | branching networks, with width branches per node | | create_lattice() | balanced two-dimensional grids of a given number of nodes | | create_ring() | circles, with width setting how far along the ring ties reach | | create_core() | an ideal core-periphery network | | generate_random() | ties placed at random, by probability or count (Erdös-Renyi) | | generate_smallworld() | a ring rewired at some probability (Watts–Strogatz) | | generate_scalefree() | growth with preferential attachment (Barabasi-Albert) | | generate_fire(), generate_islands(), generate_citations() | forest-fire, modular-islands, and citation growth models | | net_by_density() | proportion of possible ties present | | net_by_transitivity() | clustering: how often two nodes' shared neighbour closes into a triangle | | net_by_diameter(), net_by_length() | longest and average shortest path lengths | | net_by_smallworld() | small-world coefficient (default 'omega', 1 = maximally small-world) | | net_by_scalefree() | power-law exponent fitted to the degree distribution | | net_by_richclub(), net_by_assortativity() | whether hubs interconnect, and whether like degrees attach to like | | node_is_core(), node_in_core() | assigns nodes to core/periphery (or core/semi-periphery/periphery) | | net_by_core() | correlation of the network with an ideal core-periphery model | | node_by_kcoreness() | each node's k-coreness (depth in the network's successive cores) | | net_x_hierarchy() | Krackhardt's four graph-theoretic dimensions of hierarchy | | net_by_connectedness() | proportion of dyads that can reach each other | | net_by_cohesion(), net_by_adhesion() | minimum nodes / ties to remove to fragment the network | | node_is_cutpoint(), tie_is_bridge() | flags the nodes / ties whose removal would fragment the network | | tie_by_cohesion() | how deeply each tie is embedded in triangles | | to_uniplex(), to_undirected(), to_redirected(), to_giant() | extract one tie type, drop direction, reverse direction, keep the largest component | | mutate_nodes(), mutate_ties(), graphr() | map marks and measures onto the graph | | chisq.test() | (base R) association between core status and a categorical attribute |

This completes the {netrics} tutorials on describing network structure — centrality, community, position, and now topology. Run run_tute() at the console to see all available tutorials, including those in the other {stocnet} packages.

Glossary

Here are some of the terms that we have covered in this tutorial:

r print_glossary()

netrics documentation built on July 24, 2026, 5:07 p.m.