Dynamical Models Explaining Social BalanceVincent Traag, Paul Van Dooren, Patrick De Leenheer

Universite Catholique de Louvain

Social Balance

I Social networks may have positive and negative links.

I Sociological balance explains structure of positive and negative links.

I “Enemy of my enemy is my friend”.

I Balanced triads leads to balanced networks.

I Balanced networks can be split in two factions.

I Central question: what dynamics leads to two factions?

Two Models

I First Model X = X 2.

I Gossip about what neighbour k thinks of j .

I Leads to social balance for symmetric initial conditions.

I No social balance for non-symmetric initial conditions.

I Second Model X = XXT .

I Gossip about what neighbour j did to k .

I Exactly same for symmetric initial conditions.

I Leads (almost) always to social balance.

Balanced

Unbalanced

Two Factions

i j

k

X = X 2

The link tobe updated.

What does ithink of k?

What does kthink of j?

X (0) t X (t∗)

i j

k

X = XXT

The link tobe updated.

What does ithink of k?

How did jtreat k?

Evolution of Cooperation

C D

C b − c −c

D b 0

Agents eitherCooperate orDefect

5 10 15 20

0.25

0.5

0.75

1

b

FixationProbab

ility

X = X 2 X = XXT

Against each other

Against defectors

5 10 15 20

0.25

0.5

0.75

1

b

FixationProbab

ility

X = X 2 X = XXT

Against each other

Against defectors