networks
glossary

Glossary for 697E coursework

Published

January 29, 2022

A social network is a set of actors (or points, or nodes, or agents) that may have relationships (or edges, or ties) with one another. Networks can have few or many actors, and one or more kinds of relations between pairs of actors.

## Week 1 - Introduction

### Class lecture 1a: Nodes, Edges and Network Samples

(Note: some terms overlap somewhat or have context-dependent synonyms.)

node / vertex
a junction in a network where two or more lines (edges) intersect; a dot connecting lines.
tie / edge / link / relation
lines connecting nodes, which indicate some sort of connection or relationship.
node / ego / actor
$$i$$ - the node being discussed or focused on; also note “Actors are described by their relations, not by their attributes.”1
alter
$$j$$ - the node that $$i$$ connects to
network population
$$n$$ - the size of the population or count of nodes

With set notation, we define $$i$$ as a set of $$n$$ elements: $i \in 1, 2, 3 \dots n$

and $$j$$ similarly as a set of $$n$$ elements, except that $$j$$ cannot equal $$i$$ (a node cannot connect to itself): $j \in 1, 2, 3 \dots n, i \ne j$

interaction threshold
a measure to determine whether two entities have a sufficient connection to be considered to have a link between them
snowball sample
an entity group formed by starting with “a focal actor or set of actors”2 and “rolling outwards” to its connections until all nodes or actors (in a limited set) are located, or a decision to stop is made
egocentric name generator
a mode of building an entity group (an egocentric network) defined by connection to a single central node; like a snowball sample that doesn’t expand past the first set of connections

### Class lecture 1b: Ties and Adjacency Matrices

an $$n \times n$$ matrix depicting connections between $$n$$ nodes as 0 or 1, where 1 is a connection (or vertex/edge) and 0 is the absence of one:
A B C D
A - 0 1 0
B 0 - 1 0
C 1 1 - 0
D 0 0 0 -
directed tie
a relationship where $$W_{i,j} \ne W_{j,i}$$, as might be used to represent a transfer of resources from one node to another; graphed with arrows
symmetric tie
a relationship where $$W_{i,j} = W_{j,i}$$, with no direction; graphed with lines
binary tie
a tie where $$W_{i,j}$$ is 0 or 1, as in the example above, indicating the absence or presence of a tie (also dichotomous)
valued tie
a tie where $$W_{i,j}$$ is a value $$v$$, indicating a weight or magnitude of the connection; may be graphed with line attributes such as weight, color, etc

Ties are symmetric or directed, and binary or valued.

### Class lecture 1c: Edgelists

edgelist
a table indicating edges in a network, with at least “from” and “to” columns, and possibly additional columns for attributes or values
From To Value
A B 5
A E 2
B A -1
B C 4
B D 2
C A -4

Edgelists are more efficient in sparse networks, as they only list actual connections rather than being a matrix of all possible connections

### Hanneman, Robert A. and Riddle, Mark (2005), chapter 1 (Social network data):

binary measures of relations
undirected relations, 0 or 1 for the absence or presence of a connection
multiple-category nominal measures of relations
directed relations with categories (like “friend, lover, business relationship, kin, or no relationship”)
grouped ordinal measures of relations
measures that reflect a level of intensity or degree; often turned into binary measures by means of a threshold or cut-off
full-rank ordinal measures of relations
an ordering from 1 to $$n$$ of an actor’s relations (uncommon in social networks)
interval measures of relations
continuous measures that express the strength of connections by comparison with others, to be able to say “this tie is twice as strong as that tie”; the “most advanced” level of measurement

### Lazer (2011)

homophily
the idea that individuals who are similar to one another are more likely to form ties
whole network data
relational information about a whole set (or subset) of data, with all of the actors’ relations to each other considered
egocentric data
relational information about a set of nodes connected to one particular node and not to each other
diameter
the maximum degree of separation between any two nodes in the network
one-mode data
ties among one set (or category) of agent, such as nations in the context of trade
two-mode data / bipartite data
ties between different sets (or categories) of agents, such as ties between nations and international organizations; a network split into two parts, each of whose nodes only connect to nodes in the opposite part, not to nodes within its own part; “affiliation network” in Wasserman and Faust (1994)

### Borgatti et al. (2009)

sociometry
“a technique for eliciting and graphically representing individuals’ subjective feelings toward one another”
strength of weak ties (SWT) theory
the theory that one is likelier to hear new information from people they aren’t closely connected to in a network (c.f. homiphily)
centrality
a family of positional properties of nodes in a network
Freeman’s betweenness
a type of centrality where a node is frequently along the shortest path between pairs of nodes, giving control over flow or power
opportunity-based antecedents
“the likelihood that two nodes will come into contact” - when considering the formation of ties
benefit-based antecedents
“some kind of utility maximization or discomfort minimization that leads to tie formation”
node homogeneity
a category of node outcomes referring to the similarity of nodes
performance
a category of node outcomes referring to some good (such as strong performance)

## Week 2 - Network Structure

### Class lecture 2a: Network Structure - Walks, Paths, and Distance

Connections between nodes:

a direct connection between nodes; not necessarily bilateral in directed networks. A leading to B (“A adj B”) does not mean that B leads to A (“B adj A)
reachable
whether a node is reachable from another node, regardless of distance
distance
the number of ties (steps, edges) that must be traversed to reach a target node
walk
sequence (not path, see below) that connects two nodes, consisting of the nodes and edges
trail
a walk that can only go through each edge / tie once, but can hit the same node more than once
path
a trail that only hits each node and edge once; the start and end node may be the same
geodesic distance
the shortest path between two nodes; by definition, not a trail or walk because repeated segments wouldn’t be the shortest; with binary data, the number of edges between the nodes; with weighted data, might be a sum or some other calculation of “effort”

### Class lecture 2b: Graph Substructures and Components

Network substructures

two, three, or four-or-more connected nodes
complete graph
a network where every node is directly connected to every other node
connected graph
a network where every node is indirectly connected to every other node
unconnected graph
a network where at least one node is unreachable
component
the set of all points that constitutes a connected subgraph within a network
main component
the largest component within a network
minor component
a smaller one, possibly one of many
pendant
a node with only one link or edge to a network, “dangling”
isolate
an unconnected node

a dyad where both nodes connect to each other, as in an undirected network
a dyad (in a directed network) where one node connects to another, but non-reciprocally (one way only)
null
a dyad of two unconnected nodes
(empty / one edge / two path / triangle) triad
a triad with zero, one, two or three edges between three nodes (all four possible permutations)
balance theory
the theory that two nodes connected to a common node will also develop connections to each other
global transitivity index
the proportion of triads in a network that are complete (with 3 connections between them)

There is a vocabulary for triads in directed networks describing the 16 possible permutations of ties (or the absence thereof) among 3 nodes, counting the number of mutual, asymmetric and null dyads, with direction indicators, like 003 or 120D - see the slide at 3:45

a triad where (to be continued…) (5 of 16 possibilities)
a triad where (to be continued…) (7 of 16 possibilities)
a triad where (to be continued…) (4 of 16 possibilities)

(See Alhazmi, Gokhale, and Doran (2015))

### Class lecture 2d: Transitivity and Clustering Coefficient

local transitivity / local clustering coefficient
the likelihood that the neighbors of a node are also connected to each other; the number of connections that do exist over the number of connections that could exist

In the example above, there are 8 nodes that Homer (center) could connect to; among those 8 nodes, there are $$7 + 6 + 5 + 4 + 3 + 2 + 1 = 28$$ possible undirected connections (not connecting to Homer), and 9 of those 28 do exist, for a local clustering coefficient of $$9/28 \approx 0.32$$.

average clustering coefficient
the average of the local clustering coefficient of all nodes in the network

## Week 3 - Network Degree

### Class lecture 3a: Degree

(vertex) degree
the number of links that a node has; the number of nodes it’s connected to
degree distribution
a distribution showing the number of nodes of a network that have each degree level
indegree
the number of links that a node receives in a directed network
outdegree
the number of links that a node sends in a directed network

### Class lecture 3b: Centrality vs. Centralization

centrality
a measure of the prominence of one node relative to others; can be variously defined
(degree) centrality
proportional the the number of other nodes to which a node is linked
centralization
a property of a graph or network, referring to its overall cohesion; comparing most central point to all other points; ratio of the actual sum of differences to the maximum possible sum of differences

### Class lecture 3c: Network Density

network density
number of ties as a proportion of the maximum possible number of ties; varies from 0 to 1, calculation will vary by whether network is undirected or directed (twice as many potential connections)

## Week 4 - Status & Eigenvectors

### Class lecture 4a: Status and Hierarchy

closeness centrality
the sum of geodesic distances (shortest paths) to all other points in the graph. Divide by $$n-1$$, then invert. A measure of how close a node is to all of the other nodes in the network.

### Class lecture 4b: Status and Prestige

prestige
signal of the quality of a node, of a node’s visibility within the network
eigenvector centrality
“takes into account the centrality of other nodes to which a node is connected. That is, being connected to other central nodes increases centrality.” Takes into account the centrality of the nodes a node is connected to, considering the importance of the connected nodes and not just their quantity or path length. A node with high eigenvector centrality is connected to significant numbers of other highly central nodes. $\lambda \text{c} = \text{Wc}$
Bonacich Power
“A closely related concept, but includes a weighting factor that emphasizes global vs. local structure (and negative connections)” In contrast to eigenvector centrality, penalizes nodes connected to other well-connected nodes, under the theory that being connected to strong or powerful nodes means a node has less influence and power than it would if it were connected to fewer or weaker nodes. $\text{c}(\alpha, \beta) = \alpha(\text{I} - \beta\text{W})^{-1}\text{W1}$

### Class lecture 4c: Hubs and Bridges

bridge
a node with few ties to central actors
hub
a node with many ties to peripheral actors
reflected centrality
the degree of centrality that a node receives from a connected node that is due to that node’s centrality. If A has influential friend B, then A’s eigenvector centrality will be higher; reflected centrality is the portion of A’s centrality that comes from B’s centrality.
derived centrality
the remainder of the eigenvector centrality that A receives from B in the example above, that is not due to B’s centrality but just from B being a connected node

Reflected and derived centrality can be represented in the following matrix by Mizruchi et al:

High reflected centrality Low reflected centrality
High derived centrality Cosmopolitans Pure bridges
Low derived centrality Pure hubs Peripherals

The four prototypes are:

cosmopolitans
well-connected nodes that are connected to other well-connected nodes; the “cool kids” table
pure hubs
well-connected nodes that are connected to nodes that are not well-connected; the cool kid who talks to uncool kids
pure bridge
a node connected to few but high-centrality nodes; a friend of the cool kid with few other friends
peripheral
a node with few and low-centrality connections; me and my D&D nerd friends in high school

### Class lecture 4d: Status and Core/Periphery Networks

betweenness centrality
a count of the number of shortest paths between nodes that pass through another node

## References

Alhazmi, Huda, Swapna S. Gokhale, and Derek Doran. 2015. “Understanding Social Effects in Online Networks.” In 2015 International Conference on Computing, Networking and Communications (ICNC), 863–68. Garden Grove, CA, USA: IEEE. https://doi.org/10.1109/ICCNC.2015.7069459.
Borgatti, Stephen P., Ajay Mehra, Daniel J. Brass, and Giuseppe Labianca. 2009. “Network Analysis in the Social Sciences.” Science, February. https://doi.org/10.1126/science.1165821.
Hanneman, Robert A., and Riddle, Mark. 2005. “Introduction to Social Network Methods.” Introduction to Social Network Methods. http://faculty.ucr.edu/~hanneman/nettext/.
Lazer, David. 2011. “Networks in Political Science: Back to the Future.” PS: Political Science & Politics 44 (1): 61–68. https://doi.org/10.1017/S1049096510001873.
Wasserman, Stanley, and Katherine Faust. 1994. Social Network Analysis: Methods and Applications. Structural Analysis in the Social Sciences. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511815478.

## Footnotes

1. Hanneman, Robert A. and Riddle, Mark (2005)↩︎

2. Hanneman, Robert A. and Riddle, Mark (2005)↩︎