Subgroups Level
Subgraph:
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A graph G' = (V', E') is a subgraph of a graph G = (V, E) iff:
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V' is a subset of V and E' is a subset of E
A lot of network methods are designed to split a graph into subgraphs
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Subgraphs can be used to represent the social network concepts of
groups​, c​ liques​, or ​communities
Connected subgraph:​A subgraph is ​connected ​if every node within it can reach
every other one via a path of some length
Disconnected subgraph:​If otherwise, it is disconnected.
Clique:​A maximal c​ omplete subgraph​(density = 1.0). The s​ ize​of a clique is the
number of nodes in it: 3-clique, 4-clique, 5-clique … k-clique. (E.g. clique of size 4)
N-cilque:
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Some subgraphs are not complete graphs (cliques) but they are almost there!
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They are quasi-groups
We can relax the definition to accommodate
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An n
​ -clique​is a ​subgraph​where each node is at least​ n steps away
from each other node
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3-clique (3 steps or less), 4-clique (4 steps or less), 5-clique (5 steps or less)
■
steps or less)
...​k-​clique (​k-
Connected graph:​A graph is connected if there exists a (finite) path between each pair of nodes.
Disconnected graph:​Otherwise the graph is disconnected.
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We can disconnect a graph by removing e​ dges.
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In some graphs removing just one edge leads to disconnection
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Edges with the property of being able to disconnect the graph upon removal
are called ​bridges
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We can also disconnect a graph by removing n
​ odes.
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It can be easier to disconnect a graph this way because nodes take their links with
them!
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Some nodes are more “important” because they serve as the connectors
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Can you guess what kind of centrality “connector” nodes are more likely to
have?
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Betweenness
Component:​A maximal connected subgraph is called a component. If a graph has
more than one component it is by definition disconnected.
Giant Component:​The connected subgraph containing the l​ argest​number of
vertices in a graph is called the ​giant component.
K-connectivity​: A component is k
​ -connected​if every node in the component can
reach any other one via a path of length k​ ​or less.
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A graph is k-connected if it takes removing k nodes to disconnect it
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Menger’s theorem states that: If a graph is k-connected there are at least k
paths that don’t repeat third nodes connecting each non-adjacent pair of nodes ( important result in
graph theory)
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K-connectivity (graph theory) can represent the concept of social cohesion (network theory):
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The larger the (node) cutset = more cohesive (integrated) groups are with one another
■
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More people would have to be removed to disrupt the group
The smaller the (node) cutset = less cohesive and more potentially fragmented the group
structure is
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Any graph can be split into subgraphs with any given level of k-connectivity
Whole Network Level
Degree Sequence:
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A chart that documents the degrees (# of edges a node has) of all nodes
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Maximum Degree:​The most amount of nodes/links adjacent to a specific node.
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Minimum Degree:​The least amount of nodes/links adjacent to a specific node.
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Average Degree:​(​ should know how to calculate!)
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The sum of the degrees of nodes divided by the number of nodes
Degree Distribution (​p​k​)​:​​The probability of observing a node with degree k for each value of degree in the
network!
Degree Centralization​(should know how to calculate!)​:
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Compares an observed network to one that is maximally centralized (the star graph of the same order)
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It is given by the sum of the differences between maximum degree and the degree of each node
in the observed network to the same number in the star graph of the same size.
Path-based Network Properties:
Geodesics: I​n a network, shortest paths between two nodes are also
called ​geodesics.
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Graph Diameter​:
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The length of the longest shortest path between two nodes
Average Shortest Path (L):
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The average shortest path length between all pairs of nodes in a network; c​ an only be computed for a
connected graph or the giant component of disconnected ones
Reachability Matrix:
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The ​reachability matrix ​of a graph has a one in each cell if the corresponding pair of nodes are
connected via a path of some length
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Graph Hierarchy:
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Example: ​H = 1 - (5/(5(5-1)/2) = 1 - (5/(20/2) = 1 - (5/10) = 1 - 0.5 =​ 0.5
Graph Reciprocity:
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Reciprocity (R) is given by counting the number of reciprocal links (V) and dividing by the
total number of ties (E) : R=V/E
Node Based Network Properties
Average Clustering Coefficient​:
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The sum of each node’s clustering coefficient divided by the total number of nodes
Properties of Small World
Graph (Phenomenon):
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A given (large-scale)
network is a “small world” if:
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Large:​The o
​ rder o​f the graph (N) is ​large.
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Sparse:
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The graph is s​ parse​(any node is on average connected to only a relatively small
number of other nodes k)
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k ~ 10​2 ​in human social networks (Pool and Kochen 1978)
Decentralization:
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Graph is d
​ ecentralized​(max degree is much smaller than the order of the graph ​N)​
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Clustered:
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The network exhibits c​ lustering​(friends of friends tend to be friends)
Affiliation Networks
Two mode network: A
​ ka. Affiliation Networks: Are composed of relationships betweeen ​two types o​f
entities.
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Persons​and g​ roups​they belong to
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Actors​and the ​movies​they appear in
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Scientists​and p
​ apers​they write
Bipartite graph​:
Affiliation matrix​:
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An affiliation matrix (A) has number of rows equals to the number of people and number of columns
equal to the number of groups!
Co-membership (co-affiliation) matrix​:
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​
A X A​T returns
the c​ o-membership matrix (C)
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Each o
​ ff-diagonal​c​ell (C​ij​) tell us the number of affiliations that person i has in common with
person j
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Each d
​ iagonal​​cell (C​ii​) tells us the number of total memberships person i has
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The matrix represents a ​weighted one-mode​network
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Strength of tie between i and j = number of common memberships
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A “similarity” relation
Group overlap matrix​:
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A​T​ X A​ ​returns the g​ roup overlap matrix (G)
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Each o
​ ff-diagonal​c​ell (G​ij​) tell us the number of members groups i and j have in common
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Each d
​ iagonal​​cell (G​ii​) tells us the number of total members in group i
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The matrix represents a ​weighted one-mode​network
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Two groups are more strongly connected if they have lots of members in common
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This is called the o
​ ne-mode projection o​f the original two-mode network
Matrix transpose​: The transpose of a matrix is the same matrix with the rows and columns switched around.
Social ​Positions ​in Networks
Relational similarity: ​Two actors occupy the same social position if they have the same (or similar) sets of ties
to other actors in the network.
Positional Analysis:​i​s the branch of social network analysis in charge of defining different notions of network
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The position of actor ​b i​n the network is given by their pattern of relations to other actors.
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Steps in ​positional analysis:
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Define a network-based notion of social position using graph theory
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This is usually an ​equivalence relation
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Translate that notion into a matrix representation
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Cluster actors (nodes) in the network based on whether they share the same position
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Rearrange the adjacency matrix so that actors in the same position are near one another
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Check to see how actors in the same position are connected to:
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Other actors that share the same position
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Other actors in different positions
Come up with a simplified version of the network in which positions (cluster of actors) are the
“nodes”position (using graph theory) and developing techniques (using matrix algebra) to
measure them.
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This simplified structure is a network of r​ elations between positions​!
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But it is still based on the original relations between actors
Structural Equivalence: ​Two actors are similar if they have similar relationships to the ​same​actors or objects.
Regular Equivalence:​​Two actors are similar if they have similar relationships to s​ imilar​actors or objects​.
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Theoretically, the notion of role implies similar patterns of connectivity to similarly positioned actors
not to literally the same actors (If you lose the label, nodes that cannot be told apart are regularly
equivalent. )
Image Matrix and Image Graph:
Blockmodeling:
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What is a blockmodel?
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A procedure to reorder the rows of matrix (representing nodes and their relations) such that:
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Rows that are close together are closer to being (structural or regularly) equivalent
than rows that are far apart
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The matrix can then be divided into ​blocks:
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Blocks contain disjoint set of actors and thus define “positions” in
the network
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Actors in block have connections among themselves
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Actors in a block also have connections with actors in other blocks
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Diagonal blocks
Off-diagonal blocks
We can think of blockmodeling as an attempt to reduce a matrix of N x N actors and their relations to a
reduced matrix of B x B blocks and their relations
Diffusion
S-curve:
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Typical S-shaped pattern of cumulative adoptions
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Initial slow uptake
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Rapid spread
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Final slow saturation
Influence:
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Eigenvector centrality (People who can influence popular people)
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Cohesion-based network models presume that diffusion only happens between connected actors
(direct influence)
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Positional equivalence (indirect influence)
Comparison:
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Mechanism: Comparison to similarly positioned others
Imitation:
Source:​Who started it.
Adopter:​Who adopted it.
Similarity Heuristic:​I should adopt the innovations that people like me adopt (homophily).
Direct Cohesion Model:​Focus on the connectivity of adopters to their sources via network ties.
Two-Step Model:​Focus on the connectivity of adopters to their sources via network sources as
well as characteristics of influentials.
● Nike sends Kim K. shoes, Kim wears shoes, shows fnas, fans buy shoes.
○ Nike to Kim, Kim to fans
Global Threshold Model:
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Focus on susceptibility of adopters to the behavior of multiple sources
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If 2 people start dancing, then I’ll jump in too
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Each actor (i) has a “threshold” for adoption q(i)
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qi is between zero and one
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0 > q(i) < 1
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(i) adopts when the proportion of people who have adopted (p) is equal to q(i)
Positional Equivalence Model:
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Focus on indirect influence and comparison between sources and positionally equivalent adopters
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Positionally equivalent actors don’t have to be necessarily connected to one another!
Network Threshold Model:
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relaxes the assumption that persons have global information on all adoptions
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They only know what their direct contacts are doing
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Individuals adopt when some proportion of their contacts adopt
Opinion Leaders:
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Early adopters (innovators) are well-connected
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Central, well-integrated, prominent
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Well connected persons more likely to adopt innovations with low cost/benefit ratios
Simplex Contagion:
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Low risk: Information, gossip, disease
Complex Contagion:
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High risk: social movement participation, religious conversion, migration, etc
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Strategic complementarity: ​(adoption cost declines as a function of existing adopters)
Legitimacy:​ Resistance to challenge, collective acceptance, and taken-for-grantedness
increases in the number of adopters
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Credibility:​ Innovations appears are more trustworthy and less risky when high status
members of the community also adopt
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Emotional contagion: ​Cumulative interaction with a large number of adopters leads persons
to adopt
Tipping Point:
● Point where innovation has been adopted by enough people
Rogers Classification of Adopters:
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Innovators + Early Adopters​: Individuals whose time-of-adoption is greater than one
standard deviation earlier than the average time-of-adoption
Early Majority + Late Majority:​ Individuals whose time-of-adoption is bounded by one
standard deviation earlier and later than the average
Laggard: ​Individuals who adopted later than one standard deviation from the average
Valente Classifications of Adopters:
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Influentials:
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Connected Bandwagon:
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Resistant Rearguard:
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Isolates:
Dominance orders/ Negative Interactions:
Linear hierarchy:​Pecking Order
Transitivity:​Is a relation between node triples
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If ​a​has dominance over ​b,​and b
​ ​dominates c​ , t​hen ​a​necessarily dominates ​c.
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(a>b, b>c, c<a)
Completeness:
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(either a>b or b>a)
Anti-Symmetry:
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(a>b then not b>a)
Cycles:​A path that begins and ends with the same node (like a triangle going in 1 direction). In terms of
hierarchy, this is sooo weird!
Personality/Cognition
Schema-based recall:
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What kinds of networks are easy for people to remember?
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People store network information in a cognitive template called a s​ chema
Schemas influence memory
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We tend to more easily r​ emember​information that is c​ onsistent ​with stored schemas
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We tend to ​forget​information that is ​inconsistent​with stored schemas
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We tend to ​change​schema ​inconsistent​information into a schema ​consistent​form
What kinds of networks are easy to remember?
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Networks with h
​ ierarchical​structure are easier to remember
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Networks with r​ eciprocity​(if the relationship is friendship) are easier to remember
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Networks with t​ ransitivity​are easier to remember
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Networks clustered into ​subgroups​are easier to remember
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Networks featuring ​balance​are easier to remember
The balance schema:
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Friends of friends are friends
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Friends of enemies are enemies
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People find it much easier to remember social structures that respect the balance schema
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People also tend to recall network as obeying the balance schema even if the original network violates
the rules
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They impose “balance” to social structures in their heads
Cognitive Social Structures:
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It is the network of connections between people (including yourself but also between others) as each
person perceives it to be rather than as it “really” is
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We go from asking:
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To asking:
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Is person i connected to person j?
Does person k think that person i is connected to person j?
This means that in a network of size N:
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Each relationship in the network exists N times as recorded inside the head of each the people
in the network!
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We go from a dyadic conception of relations (​R​ij​) to a triadic one:
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R​ijk​(person i is connected to person j according to person k)
Self-monitoring:
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Those in high self-monitoring…
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Regulate the way they present themselves, respond to cues (“SOCIAL CHAMELEONS”)
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Higher indegree in sentiment (e.g. friendship) and exchange (e.g., advice) networks
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More likely to bridge structural holes - betweenness centrality (only in sentiment networks)
○ Use of humor in interaction
○ Appropriate pace in conversation
○ Less likely to do “self-talk”
○ More likely to do “other talk”
○ Likely to collaborate and compromise
Big Five Personality Traits:
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Neuroticism/​Stability​:
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Experiences negative emotions
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Smaller ego-networks
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Tend to be part of weak tie triads
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Are less likely to introduce friends of friends
Stability:
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Extraversion/​Introversion​:
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People High in Extraversion
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Larger networks at all layers
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Tend to be part of a strong tie triads
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More likely to introduce friends of friends
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Add new contacts at a faster rate
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Have higher indegree and outdegree centrality
Introversion:
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Niceness, resolves conflicts
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Have higher in-degree
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Add new contacts as a faster rate
Deta:
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Distant/ emotionally neutral
Openness/​Cautiousness​:
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Like Experiencing new things
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Cautiousness:
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Prefers self-reflection
Agreeableness/​Detachment​:
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Is more emotionally “balanced”
Likes consistency and predictability
Conscientiousness/​Carefreeness​:
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Organization, reliability
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Carefree:
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Spontaneous and disorganized