From individuals decisions to emerging social structurepagann/research/IRC_2018_Pagan.pdf · Individuals shape social networks structure. Interested in:Correlation between stable

From individuals decisions to emerging socialstructure

Nicolo Pagan 1 Florian Dorfler 1

1 Automatic Control Laboratory, ETH Zurich, Switzerland

International Conference on Infrastructure Resilience

Zurich, 14.02.2018

How do social networks form?

Social networks influence individual behavior,

Individuals shape social networks structure.

Interested in: Correlation between

stable social network structures,

e.g. star network, bipartite network,complete network,

individual incentives of formingsocial ties.

Nicolo Pagan, Florian Dorfler From individuals decisions to emerging social structure 14.02.2018 2/ 11





























stable social network structures,e.g. star network,

bipartite network,complete network,







stable social network structures,e.g. star network, bipartite network,

complete network,







stable social network structures,e.g. star network, bipartite network,complete network,







stable social network structures,e.g. star network, bipartite network,complete network,



Individual incentives: why do we form social ties?

Popularity Capital:The more people we areconnected to, the morewe can influence them.

Bonding Capital:The more our friends’friends are our friends,the safer we feel.[Heider (1946)], [Coleman(1990)]

Bridging Capital:The more we are on thepath between people,the more we can control.[Burt (1992)]

Degree Centrality High Clustering Betweenness Centrality

Closed triads havepositive externalities(Structural Balancetheory) [Cartwright andHarary (1956)]

Close triads havenegative externalities(Structural Holestheory) [Burt (1992)]



Popularity Capital:

The more people we areconnected to, the morewe can influence them.








Popularity Capital:


Bonding Capital:

The more our friends’friends are our friends,the safer we feel.[Heider (1946)], [Coleman(1990)]







Popularity Capital:


Bonding Capital:

The more our friends’friends are our friends,the safer we feel.[Heider (1946)], [Coleman(1990)]

Bridging Capital:

The more we are on thepath between people,the more we can control.[Burt (1992)]





Individual incentives: why do we form social ties?Popularity Capital:The more people we areconnected to, the morewe can influence them.




























Social Network Formation Model

Directed weighted network G withN ≥ 3 agents.

Homogeneous agents.

Let aij ∈ [0,1] quantify theimportance of the friendship among iand j from i’s point of view.

A typical action of each agent i is:

ai = [ai1, . . . ,ai,i−1,ai,i+1, . . . ,aiN ]

∈ A = [0,1]N−1 ,

Rational agents: i looks for

a?i = argmaxai∈A

Vi(ai,a−i)




Homogeneous agents.



ai = [ai1, . . . ,ai,i−1,ai,i+1, . . . ,aiN ]

∈ A = [0,1]N−1 ,


a?i = argmaxai∈A

Vi(ai,a−i)




Homogeneous agents.



ai = [ai1, . . . ,ai,i−1,ai,i+1, . . . ,aiN ]

∈ A = [0,1]N−1 ,


a?i = argmaxai∈A

Vi(ai,a−i)




Homogeneous agents.



ai = [ai1, . . . ,ai,i−1,ai,i+1, . . . ,aiN ]

∈ A = [0,1]N−1 ,


a?i = argmaxai∈A

Vi(ai,a−i)




Homogeneous agents.



ai = [ai1, . . . ,ai,i−1,ai,i+1, . . . ,aiN ]

∈ A = [0,1]N−1 ,


a?i = argmaxai∈A

Vi(ai,a−i)


From individual incentives to the Payoff Function

Vi(ai,a−i) =

α

Pi(ai,a−i)

β

Bi(ai,a−i)−

γ

Ci(ai), α≥ 0, β ∈ R, γ≥ 0

Popularity capital: social influenceon friends, on friends of friends,

. . .

with δ ∈ [0,1]:

Pi(ai,a−i) = ∑j 6=i

a ji

+δ ∑k 6= j

∑j 6=i

ak ja ji+

+δ2∑l 6=k

∑k 6= j

∑j 6=i

alkak ja ji,

Bonding/Bridging capital: numberof closed triads:Bi(ai,a−i) = ∑ j 6=i ai j

(∑k 6=i, j aikak j

),

Cost of maintaining ties:

Ci(ai) = ∑j 6=i

ai j.



Vi(ai,a−i) =

α

Pi(ai,a−i)

β

Bi(ai,a−i)−

γ

Ci(ai), α≥ 0, β ∈ R, γ≥ 0

Popularity capital: social influenceon friends,

on friends of friends,

. . .

with δ ∈ [0,1]:


a ji

+δ ∑k 6= j

∑j 6=i

ak ja ji+

+δ2∑l 6=k

∑k 6= j

∑j 6=i

alkak ja ji,



),


Ci(ai) = ∑j 6=i

ai j.



Vi(ai,a−i) =

α

Pi(ai,a−i)

β

Bi(ai,a−i)−

γ

Ci(ai), α≥ 0, β ∈ R, γ≥ 0

Popularity capital: social influenceon friends, on friends of friends,

. . .

with δ ∈ [0,1]:


a ji +δ ∑k 6= j

∑j 6=i

ak ja ji

+

+δ2∑l 6=k

∑k 6= j

∑j 6=i

alkak ja ji,



),


Ci(ai) = ∑j 6=i

ai j.



Vi(ai,a−i) =

α

Pi(ai,a−i)

β

Bi(ai,a−i)−

γ

Ci(ai), α≥ 0, β ∈ R, γ≥ 0

Popularity capital: social influenceon friends, on friends of friends, . . .with δ ∈ [0,1]:


a ji +δ ∑k 6= j

∑j 6=i

ak ja ji+

+δ2∑l 6=k

∑k 6= j

∑j 6=i

alkak ja ji,



),


Ci(ai) = ∑j 6=i

ai j.



Vi(ai,a−i) =

α

Pi(ai,a−i)±

β

Bi(ai,a−i)

−

γ

Ci(ai), α≥ 0, β ∈ R, γ≥ 0



a ji +δ ∑k 6= j

∑j 6=i

ak ja ji+

+δ2∑l 6=k

∑k 6= j

∑j 6=i

alkak ja ji,



),


Ci(ai) = ∑j 6=i

ai j.



Vi(ai,a−i) =

α

Pi(ai,a−i)±

β

Bi(ai,a−i)−

γ

Ci(ai),

α≥ 0, β ∈ R, γ≥ 0



a ji +δ ∑k 6= j

∑j 6=i

ak ja ji+

+δ2∑l 6=k

∑k 6= j

∑j 6=i

alkak ja ji,



),


Ci(ai) = ∑j 6=i

ai j.



Vi(ai,a−i) = αPi(ai,a−i)+βBi(ai,a−i)− γCi(ai), α≥ 0, β ∈ R, γ≥ 0



a ji +δ ∑k 6= j

∑j 6=i

ak ja ji+

+δ2∑l 6=k

∑k 6= j

∑j 6=i

alkak ja ji,



),


Ci(ai) = ∑j 6=i

ai j.


Nash stability

Remark (Payoff function).

Vi(ai,a−i) =αPi(ai,a−i)+ βBi(ai,a−i)− γCi(ai),

α≥ 0, β ∈ R, γ≥ 0.

Definition (Nash equilibrium, NE).The network G? is a NE if for all agents i

Vi (ai,a−i?)≤Vi (ai

?,a−i?) , ∀ai ∈ A .

Stability conditions depend on:network G?:

{a?i , i ∈N

},

parameters: {α,β,γ,δ,N}.

Question: For which parameters is acertain network stable?


Nash stability



α≥ 0, β ∈ R, γ≥ 0.



?,a−i?) , ∀ai ∈ A .


{a?i , i ∈N

},




Nash stability



α≥ 0, β ∈ R, γ≥ 0.



?,a−i?) , ∀ai ∈ A .


{a?i , i ∈N

},




Nash stability



α≥ 0, β ∈ R, γ≥ 0.



?,a−i?) , ∀ai ∈ A .


{a?i , i ∈N

},




Nash stability



α≥ 0, β ∈ R, γ≥ 0.



?,a−i?) , ∀ai ∈ A .


{a?i , i ∈N

},




Network Motifs

Figure: Empty Network Figure: Complete Network

Figure: Complete Balanced Bipartite Network Figure: Star Network


Empty/Complete Network stability regionsEmpty Network

Theorem . The empty network GEN isalways a Nash equilibrium.

Complete Network

Theorem . Let GCN be a completenetwork.




Complete Network

Theorem . Let GCN be a completenetwork. Define

γNE :=

αδ(1+δ(2N−3))

d(N−1)d−1 + 2β(N−2)max(d,2)(N−1)d−1 , if β≥ 0

αδ(1+δ(2N−3))d(N−1)d−1 + 2β(N−2)

d(N−1)d−1 , if β < 0

then GCN is a NE if and only if γ≤ γNE .

Remark (Payoff function).Vi(ai,a−i) = αPi(ai,a−i)+βBi(ai,a−i)− γCi(ai), α≥ 0, β ∈ R, γ≥ 0.Nicolo Pagan, Florian Dorfler From individuals decisions to emerging social structure 14.02.2018 8/ 11



Complete Network

Theorem . Let GCN be a completenetwork. GCN is a NE if and only if

β

γ≥max

{max{d,2}2d (N−2)

(d (N−1)d−1−δ(1+δ(2N−3))

α

γ

),

12(N−2)

(d (N−1)d−1−δ(1+δ(2N−3))

α

γ

)}.



I No agent has a selfish incentive tocreate a link→ always a NE.

Complete Network

I Piecewise linear relation betweenβ

γand α

γ.



I No agent has a selfish incentive tocreate a link→ always a NE.

Complete Network

I Complete network stability iscorrelated with large values of β

(Bonding capital / highclustering).


Bipartite/Complete Networks stability regionsBipartite Network Complete Network








Bipartite/Complete Networks stability regionsBipartite Network

I Bipartite network stability iscorrelated with small values of β

(Bridging capital / betweennesscentrality).

Complete Network



Remark (Payoff).Vi(ai,a−i) = αPi(ai,a−i)+βBi(ai,a−i)− γCi(ai), α≥ 0, β ∈ R, γ≥ 0.Nicolo Pagan, Florian Dorfler From individuals decisions to emerging social structure 14.02.2018 9/ 11

Bipartite/Star Networks stability regionsBipartite Network



Star Network





Star Network

I Star network stability is correlatedwith large values of α (Popularitycapital / degree centrality).


Phase diagram

Complete network stability andhigh clustering [Buechel andBuskens (2013)],

Balanced Bipartite networkstability and betweennesscentrality [Buskens and van de Rijt(2008)],

Star network stability and degreecentrality [Bala and Goyal (2000)].

Payoff function

Vi(ai,a−i) = α

(∑j 6=i

a ji +δ ∑k 6= j

∑j 6=i

ak ja ji +δ2∑l 6=k

∑k 6= j

∑j 6=i

alkak ja ji

)

+β ∑j 6=i

ai j

(∑

k 6=i, jaikak j

)− γ ∑

j 6=iai j, α≥ 0, β ∈ R, γ≥ 0

PopularityCost

ClusteringCost


Phase diagram




Payoff function

Vi(ai,a−i) = α

(∑j 6=i

a ji +δ ∑k 6= j

∑j 6=i


∑k 6= j

∑j 6=i

alkak ja ji

)

+β ∑j 6=i

ai j

(∑

k 6=i, jaikak j

)− γ ∑

j 6=iai j, α≥ 0, β ∈ R, γ≥ 0

PopularityCost

ClusteringCost


Phase diagram




Payoff function

Vi(ai,a−i) = α

(∑j 6=i

a ji +δ ∑k 6= j

∑j 6=i


∑k 6= j

∑j 6=i

alkak ja ji

)

+β ∑j 6=i

ai j

(∑

k 6=i, jaikak j

)− γ ∑

j 6=iai j, α≥ 0, β ∈ R, γ≥ 0

PopularityCost

ClusteringCost


Phase diagram




Payoff function

Vi(ai,a−i) = α

(∑j 6=i

a ji +δ ∑k 6= j

∑j 6=i


∑k 6= j

∑j 6=i

alkak ja ji

)

+β ∑j 6=i

ai j

(∑

k 6=i, jaikak j

)− γ ∑

j 6=iai j, α≥ 0, β ∈ R, γ≥ 0

PopularityCost

ClusteringCost


Bala, V. and Goyal, S. (2000). A Noncooperative Model of Network Formation.Econometrica, 68(5):1181–1229.

Buechel, B. and Buskens, V. (2013). The Dynamics of Closeness andBetweenness. The Journal of Mathematical Sociology, 37(July):159–191.

Burger, M. J. and Buskens, V. (2009). Social context and network formation: Anexperimental study. Social Networks, 31(1):63–75.

Burt, R. S. (1992). Structural hole. Harvard Business School Press, Cambridge, MA.

Buskens, V. and van de Rijt, A. (2008). Dynamics of Networks if Everyone Strivesfor Structural Holes. American Journal of Sociology, 114(2):371–407.

Cartwright, D. and Harary, F. (1956). Structural balance: a generalization of heider’stheory. Psychological review, 63(5):277.

Coleman, J. (1990). Foundations of social theory. Cambridge, MA: Belknap.

Heider, F. (1946). Attitudes and cognitive organization. The Journal of psychology,21(1):107–112.




(high clustering).




(high clustering).



I α

γ≥ ·· · → non

destroying existinglinks acrossdifferent partitions

Complete Network


(high clustering).



I β

γ≤ ·· · → non

creating new linkswithin the samepartition

Complete Network


(high clustering).




(high betweenness centrality).

Complete Network


(high clustering).





Star Network





Star Network

I α

γ≥ ·· · → non destroying existing

links across different partitions

I β

γ≤ ·· · → non creating new links

within the same partition





Star Network

I Star network stability is correlatedwith large values of α (highPopularity capital).


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From individuals decisions to emerging social structurepagann/research/IRC_2018_Pagan.pdf · Individuals shape social networks structure. Interested in:Correlation between stable