Returns

Primary: The number of edges of the node, or their total edge length when weighted.

In an events network, the count ranks who took part in the most gatherings, before any question of how well-placed those gatherings were.

Options

• Centrality: weighs every node on these paths by betweenness centrality (absolute, relative, or normalised), showing which nodes the connection depends on.
• Tolerance: (detour/slack) widens the centrality to near-shortest (ε-shortest) paths: a node is included when the best path via it is at most the tolerance longer than the shortest path (in weights, or steps when unweighted), showing which nodes the connection would run through if the shortest paths were unavailable. The tolerance applies when a Target is set.

Returns

Primary: The node's distance from the From selection: the length of the shortest path to it, in weights or steps when unweighted; From nodes report 0, nodes that are not reached report no value.
Secondary: With the centrality enabled, how strongly the connection depends on the node: the number of best routes between From and Target that run through it, or its betweenness for the paths from the From selection to the whole network when no Target is set.

In a correspondence network, the shortest path from a scholar to the court shows through whose hands a request most plausibly travelled; the tolerance adds who could have passed it on had a connection failed.

Returns

Primary: The number of disjoint routes that run through the node; the From and Target nodes report the number of routes that start or end at them.
Secondary: The node's earliest step along a route, counted from the From side (a position, not a score).
Statistic: Paths: the number of independent routes between the selections. Distance: the total distance of that set of routes.

In a relationship network, disjoint paths show whether the contact between two milieus rests on a single go-between or on several independent chains of acquaintance.

Options

• Measure: sets how shared connections are weighed:
  • Count (Common Neighbours): shared connections as a plain count.
  • Relative (Jaccard): shared connections relative to all connections of both nodes.
  • Selective (Adamic-Adar): shared connections, discounted moderately by their own number of connections.
  • Exclusive (Resource Allocation): shared connections, discounted strongly by spreading their evidence evenly over all their connections, so only sparsely connected intermediate nodes add substantial weight.
• Mode: the plausibility scores as they are (absolute), or rescaled between 0 and 1 so the most plausible node is 1 (normalised).

Returns

Primary: The node's best plausibility score against the From selection; nodes with a documented connection score 0.
Secondary: The number of shared connections behind that score.
Statistic: Candidates: the number of nodes scoring above 0. Best: the highest score found.

In a correspondence network, link prediction flags two scholars who never appear in the same source yet share most of their correspondents: an acquaintance the letters imply but do not state.

Options

• Iterations: the maximum path length that is included.
• Attenuation: how strongly longer paths are discounted; lower values keep the prediction local, higher values include more distant structure.
• Mode: the plausibility scores as they are (absolute), or rescaled between 0 and 1 so the most plausible node is 1 (normalised).

Returns

Primary: The node's best plausibility score against the From selection; nodes with a documented connection score 0.
Secondary: The number of directly shared connections behind that best pair, which can be 0 when the plausibility is based on longer paths only.
Statistic: Candidates: the number of nodes scoring above 0. Best: the highest score found.

In a book network, Katz prediction can surface a printer and an author who share no acquaintances but sit in tightly interlinked circles, a connection worth checking the sources for.

Options

• Mode: 1 divided by the summed distances, in which an unreachable node counts as a distance longer than any real path (absolute); or rescaled so 1 means directly connected to every other node, comparable across networks, where nodes in small disconnected parts rank low (normalised).

Returns

Primary: How near the node is to the whole network: higher means nearer.

In a travel or correspondence network, closeness centrality identifies from whom news could reach everyone in the fewest steps.

Returns

Primary: 1 divided by the distance to the farthest node the node can reach.

In a travel network, eccentricity finds the natural centre: the place from which no destination is truly remote.

Options

• Mode: the plain path count (absolute), the share of all node pairs the node could sit between, comparable across networks (relative), or rescaled between 0 and 1 within this result (normalised).

Returns

Primary: The number of shortest paths between other nodes that run through the node.

In a book-trade network, betweenness centrality finds the agent through whom two otherwise separate circles of printers and buyers had to pass.

Options

• Mode: the summed strengths as they are (absolute), or the incoming and outgoing sums each rescaled between 0 and 1 so the highest node is 1 (normalised).

Returns

Primary: The summed strength of the edges running to the node (in-degree).
Secondary: The summed strength of the edges running from the node (out-degree).

In a correspondence network, the in-degree ranks who received letters from the most correspondents and the out-degree who wrote the most widely; weighted, a sustained exchange counts for more than a single letter.

Options

• Iterations: the number of times the scores are passed on before the result is reported.

Returns

Primary: The node's influence, scaled so the most influential node is 1.
Statistic: Iterations: the number of iterations run. Change: how much the scores changed in the last iteration; near 0 means they have converged, a larger value means more iterations would still change them.

In a patronage network, a courtier with three well-placed patrons outranks a merchant with thirty ordinary contacts.

Options

• Iterations: the maximum path length over which influence is passed on.
• Attenuation: how strongly influence fades with every step; lower values keep the measure close to counting direct connections, higher values include the influence of the whole network.

Returns

Primary: The node's influence, scaled so the most influential node is 1.
Statistic: Iterations: the number run. Change: how much the scores still moved in the last iteration; near 0 means they have settled.

In a correspondence network, Katz centrality still ranks the peripheral figures: a copyist reached through a single chain of intermediaries keeps a score where eigenvector centrality would give none.

Options

• Iterations: the number of times the rank is passed on.
• Damping: the share of rank that follows the edges; the remainder is the base score every node keeps, which stops rank from pooling in closed loops.

Returns

Primary: The node's rank, on the scale of the original paper: a node without incoming edges keeps the base score of 1 minus the damping (0.15 by default), and the scores average around 1 over the network.
Statistic: Iterations: the number run. Change: how much the scores still moved in the last iteration; near 0 means they have settled.

In a citation or dedication network, PageRank ranks a work named by a few much-named works above one named often by works nobody names.

Options

• Iterations: the number of times the two scores update each other.

Returns

Primary: The node's authority score, scaled so the highest is 1.
Secondary: The node's hub score, scaled so the highest is 1.
Statistic: Iterations: the number run. Change: how much the two scores changed in the last iteration; near 0 means they have converged.

In a correspondence network, HITS separates the great senders from the great receivers: the secretary who writes to every authority is a hub, the authority every hub writes to is an authority.

Returns

Primary: The node's coefficient: 0 means none of its neighbours are connected to each other, 1 means they form a complete cluster.
Statistic: Average: the mean coefficient over the analysed nodes, the overall clustering of the network.

In a relationship network, the clustering coefficient distinguishes a salon, where everyone knows everyone, from a patronage web that only stays together through its patron.

Returns

Primary: The node's core number, a strength level when weighted.
Secondary: The node's degree, or its summed edge strength when weighted.
Statistic: Degeneracy: the deepest core level of the network.

In an events network, the k-core peels the incidental attendees away layer by layer until the standing core of the scene remains.

Returns

Primary: The number of additional pieces the removal of the node would create; 0 means the network remains connected without it.
Secondary: The number of bridges connected to the node.
Statistic: Points: the number of articulation points in the network. Bridges: the number of bridges.

A single documented marriage binding two family networks together is an articulation point: a warning that everything said about their connection rests on a single relationship in the sources.

Returns

Primary: The node's weak component. The components are numbered by size, 1 being the largest; like the community numbers, these numbers are identifiers, not scores.
Secondary: The node's strong component, numbered the same way.

Connected components tell whether the sources describe one connected world or several separate ones; in a correspondence network the strong components show between whom letters actually travelled both ways.

Returns

Primary: The node's effective size: higher means more brokerage.
Secondary: Burt's constraint, on the scale used in the literature; constraint is read inverted: a low value indicates the broker.
Statistic: Average Constraint: the mean constraint over the analysed nodes that have at least one edge.

In a trade network, the bookseller whose buyers and suppliers do not know one another scores a high effective size: every deal runs through the bookseller, and the sources on either side may never mention the other.

Options

• Runs & Seed: every run after the first restarts the analysis with the nodes visited in a different order, and the best result by the mode's own quality is kept; the seed makes the random orders, and so the whole analysis, reproducible. At 0 runs the mode sets its own number: 10 for Label Propagation, which has to draw at random, 1 for the other modes, which choose the best step every time. Result determinism: with a single run of these other modes the seed has no effect: the division follows from the network and the settings alone, so two results only differ where the data or the settings differ, which makes them comparable across versions of a dataset. With more than one run, the runs that are not kept are still used: the modes that keep a best division also report how much the runs agreed on it, as each node's consensus (secondary value) and a figure over the whole network (statistic).
• Significance: a quality score on its own is of limited value, because these measures score any network, including a random one. The setting compares the found quality with the same analysis on the set number of randomised networks, in which every node keeps its number of edges but all other structure is randomised; the resulting statistics report the mean and spread of the randomised qualities and the z-score and p of the found one, and a large z means the communities are actual structure, not a result of the edge counts alone.
• Resolution: a higher resolution gives fewer, larger communities; a lower resolution more, smaller ones. 1 applies the measure as defined in the literature.
• Size: how the size of a node's community is reported as the secondary value: absolute, relative, normalised, or not at all.

Returns

Primary: The community the node belongs to, numbered 1 up to the number of communities; the numbers are to be used as identifiers, not scores.
Secondary: The size of the node's community, as set by the Size option; with more than one run Modularity, Leiden, Leiden CPM and Label Propagation replace it with the node's consensus, how consistently the members of its community stay with it across the runs (1 when they are always in the same community, lower for a node between two circles). Statistical Inference and Infomap replace it as stated in their entries.
Statistic: Communities: the number found, reported by every mode. Consensus: with more than one run, the mean consensus over the nodes, how much the runs agreed on the division as a whole. Additional figures are specific to the mode and stated per entry.

Returns

Statistic: Modularity: the achieved modularity, both as Newman defines it (roughly 0.3-0.7 signals clear community structure) and at the chosen resolution.

In a correspondence network, modularity divides the writers into circles that write mostly among themselves, the epistolary equivalent of schools and milieus.

Returns

Statistic: Modularity: the achieved modularity, both as Newman defines it (roughly 0.3-0.7 signals clear community structure) and at the chosen resolution.

In a correspondence network, Leiden draws the same circles as modularity but guarantees each circle is actually connected, so no reported community consists of two unrelated groups.

Returns

Statistic: CPM: the achieved quality. Modularity: of the same division, for comparison. Absolute Resolution: the density threshold that results from the resolution, which is the resolution as stated in the literature and by other software.

In an events network spanning decades, the Constant Potts Model keeps recognising a small persistent circle of five regulars that modularity would merge into a larger period cluster.

Returns

Secondary: As it always runs several times, Label Propagation reports the node's consensus across those runs by default (see the shared Returns above), in place of the community size.
Statistic: Modularity: of the kept (best) division.

On a network of hundreds of thousands of records, label propagation still returns readable circles in the time the other modes need for a fraction of the data.

Options

• Teleportation: the share of steps the walker jumps to a random node instead of following an outgoing edge, required to prevent a directed flow from accumulating in dead ends; 0.15 is the rate of the original paper, and it applies to directed flow only.

Returns

Secondary: The node's visit rate instead of the community size: the share of the walker's steps spent at the node, its standing within the flow.
Statistic: Codelength: the description length of the walker's movements, in bits per step. Saving: the reduction in codelength by the modules compared to no modules, the evidence of this mode; a saving of 0 means no flow structure was found. Direction: whether the walker followed the direction of the edges or read the network undirected. Teleportation: the rate applied, on directed flow only.

In a travel or citation network, Infomap groups the places or works among which the traffic actually circulates, where the density based modes only see who is connected to whom.

The graph is read as directed, unless every edge is mutual and has the same strength in both directions. On a weighted graph the edges remain unchanged and the connection strengths are described in addition, per pair of groups. On a weighted graph, groups with stronger connections can be found on their strength alone, while weights without a pattern do not add groups and can remove the support for a weak group.

Options

• Model: the kind of division the evidence is weighed for:
  • Planted Partition: ordinary communities, groups denser inside than between.
  • General: weighs every pair of groups on its own, so it also finds structure that is not shaped as communities, such as two sides that only ever connect to each other.
  • Nested: fits the general model on the nodes and then groups the communities themselves in turn, level upon level: circles within circles, which describes a large network more compactly than a single level of communities does.

  None of the three accounts for the number of connections of a node (the degree correction): they only consider how those connections are distributed across the groups, so a node with unusually many connections is read as an ordinary hub, not as a community of its own.

• Samples: sweeps of a sampler around the best division, replacing the community size with the membership certainty per node; the sampler applies to the flat models only, so Nested does not use it.

Returns

Secondary: The size of the node's community; with Samples set, its membership certainty instead: the share of the sampled divisions that agree on its community, so a node between two circles is shown as uncertain rather than assigned to one of them without indication.
Statistic: Model: which model weighed the evidence. Description Length: how compactly the division explains the network (in 'natural units'/nats); on a weighted graph it can be below zero when the strengths are small, as results from a normalisation or a distance weight well above 1 (this is not an error, as only the differences between divisions are meaningful). Direction: whether the connections were fitted directed or undirected. Weights: stated on a weighted graph, as the strengths are described alongside the structure rather than counted as edges. Evidence: what the communities save over describing the network as a single community; at or below 0 the model found no communities to report; the single-community length is the same under every model, so the evidence compares across them. Levels: with Nested, the communities per level, from the node division up to the single top level. With Samples set: the number of sweeps run, the mean certainty over the nodes, and the acceptance rate of the sampler's merge-split moves.

Where every other mode divides a network of chance co-occurrences into "circles", statistical inference reports one community: the sources provide no evidence of any grouping.

Options

• Approach: sets what the distance is measured between:
  • Connected: use the Scope to build a network between the vectors and measure only along its edges, so the distance reflects similarity where the network already connects the vectors; the target vector has to be last in the Scope selection. Weighting only applies to this approach, read as strength.
  • Disconnected: disregard the network and measure between the selected source and target vectors directly.
• Calculate: the distance measure: Euclidean, the straight-line distance between the vectors, or Cosine, the angle between them regardless of their magnitude.
• Mode: the summed distances as they are (absolute), each as its share of the summed distance over all nodes (relative), or rescaled between 0 and 1 within this result (normalised).

Returns

Primary: The node's summed distance to the target vectors, each weighted by its edge's strength when a weighted mode is set; a smaller value is nearer, more similar.

In a corpus of letters embedded by a language model, vector distance ranks which letters read most alike, surfacing near-duplicates and shared sources the catalogue never linked.