Networks Chapter5

Chapter 5: Positive and Negative Relationships

keep your friends close and your enemies closer

understanding tension in social networks

March 11, 2010 Page 1

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 nodesrather than a node in particular (as in the case of a “broker” inchapter 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


Structural Balance Property: For every set of three nodes, if weconsider the three edges connecting them, either all three ofthese edges are labeled +, or else exactly one of them is labeled+.


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 .



Weak Structural Balance Property: There is no set ofthree nodes such that the edges among them consist ofexactly two positive edges and one negative edge.

Characterization of Weakly Balanced Networks: If alabeled complete graph is weakly balanced, then itsnodes can be divided into groups in such a way that everytwo nodes belonging to the same group are friends, andevery two nodes belonging to different groups areenemies.



Advanced Material• what if everyone in the graph doesn’t know each other

• if most triangles are balanced can the world beapproximately divided into two factions?

• dividing the graph into positive or negativesupernodes


Structural Balance in Non-Complete Networks

an arbitrary non-complete graph is balanced only if lling inmissing edges achieves balance AND the graph can be divide intotwo sets where all people in X are friends and all people in Y areenemies

if a signed graph contains a cycle with an odd number ofnegative edges, then it is not balanced

problems: still based on the assumption that the graph isbalanced – we want to check IF the graph is balanced


Identifying Super-nodes

Divide the graph into X and Y so that all edges inside X and Y arepositive and all edges between X and Y are negative.

it will either succeed in doing this or fail when it nds a cycle withan odd number of negative nodes



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 whichat 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 otherend in Y are enemies.


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 socialwebsites by way of people’s subjective evaluations of eachother


Technology

Networks Chapter5