Good Stuff Happens in 1:1 Meetings: Why you need them and how to do them well
Networks Chapter5
1. Chapter 5: Positive and Negative Relationships
keep your friends close and your enemies closer
understanding tension in social networks
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2. What is this chapter about?
• structural balance of a social network (both ’balanced’ and
’semi-balanced’)
• problems in a graph may be the relationship between 2 nodes
rather than a node in particular (as in the case of a “broker” in
chapter 3
• could be used to add dimensionality in the social graph
(strong vs weak ties AND friends vs enemies)
• directly applies to online ratings such as Slashdot ’friend’ or
’foe’ or Epinions ’trust’ or ’distrust
• presumptions: everyone knows everyone else, all ’friend’
relationships are equal and all ’enemy’ relationships are equal
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3. Structural Balance Property: For every set of three nodes, if we
consider the three edges connecting them, either all three of
these edges are labeled +, or else exactly one of them is labeled
+.
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4. Cartwright-Harary Theorem: If a labeled complete graph is bal-
anced, then either all pairs of nodes are friends, or else the
nodes can be divided into two groups, X and Y , such that every
pair of people in X like each other, every pair of people in Y like
each other, and everyone in X is the enemy of everyone in Y .
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6. Weak Structural Balance Property: There is no set of
three nodes such that the edges among them consist of
exactly two positive edges and one negative edge.
Characterization of Weakly Balanced Networks: If a
labeled complete graph is weakly balanced, then its
nodes can be divided into groups in such a way that every
two nodes belonging to the same group are friends, and
every two nodes belonging to different groups are
enemies.
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8. Advanced Material
• what if everyone in the graph doesn’t know each other
• if most triangles are balanced can the world be
approximately divided into two factions?
• dividing the graph into positive or negative
supernodes
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9. Structural Balance in Non-Complete Networks
an arbitrary non-complete graph is balanced only if lling in
missing edges achieves balance AND the graph can be divide into
two sets where all people in X are friends and all people in Y are
enemies
if a signed graph contains a cycle with an odd number of
negative edges, then it is not balanced
problems: still based on the assumption that the graph is
balanced – we want to check IF the graph is balanced
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10. Identifying Super-nodes
Divide the graph into X and Y so that all edges inside X and Y are
positive and all edges between X and Y are negative.
it will either succeed in doing this or fail when it nds a cycle with
an odd number of negative nodes
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12. Approximately Balanced Networks
Claim: Let ε be any number such that 0 ≤ε < 1, and de ne δ =
√
3
ε. If at least 1 – ε of all triangles in a labeled complete graph are
balanced, then either
1. there is a set consisting of at least 1 – δ of the nodes in which
at least 1 – δ of all pairs are friends, or else
2. the nodes can be divided into two groups, X and Y , such that
(a) at least 1 – δ of the pairs in X like each other,
(b) at least 1 – δ of the pairs in Y like each other, and
(c) at least 1 – δ of the pairs with one end in X and the other
end in Y are enemies.
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13. Things I hoped to get out of this chapter but didn’t
• identifying sources of con ict in a social graph
• ways theories of balance can be used to understand social
websites by way of people’s subjective evaluations of each
other
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