4 musatov

Extensions of the ’hitchhiker’ game-theoreticnetwork formation model

Sergey Melikov1, Daniil Musatov1,2,3, Alexei Savvateev1,2,3

1Moscow Institute of Physics and Technology (State University)

2New Economic School

3Yandex Company

Moscow, October 24th, 2013

What is a social network?

I As any network, a social network consists of nodes and edges

I Nodes are people (or their accounts)I Edges are some connections between people:

I PersonalI BusinessI Scientific (e.g., coauthorship)I Virtual (in facebook)

I Edges may be directed (e.g., following in online socialnetworks) or undirected (e.g., friendship).

I Only online networks may be analyzed completely!

I And they share many common features

Six degrees of separation

I Stylized fact: every two people on the Earth may beconnected to each other via a chain of at most six“first-name” acquaintances

I 1967: Stanley Milgrom’s “Small world experiment”

I Modern investigations of online networks: average distancebetween two users of Facebook or Twitter is about 4.5

I Diameter is also small (approx. 10)

Clustering among my facebook friends

Powered by TouchGraph

High clustering

I Stylized fact: people having a friend in common are likely tobe friends also (triadic closure)

I Stylized fact: friends of one person are clustered into severalgroups

I Local clustering coefficient

Cu =#{(v ,w) ∈ E | (u, v) ∈ E & (u,w) ∈ E}

#{(v ,w) | (u, v) ∈ E & (u,w) ∈ E}

I Average clustering coefficient C = 1|V |

∑u Cu

I Global clustering coefficient

C =3N4N∧

=#{(u, v ,w) | (u, v) ∈ E , (u,w) ∈ E , (v ,w) ∈ E}

#{(u, v ,w) | (u, v) ∈ E , (v ,w) ∈ E}

I In real social networks clustering coefficients are about 0.1

Scale-freenessI Some people are “hubs” and have a lot of connectionsI One may consider a distribution of degrees of vertices,

P(k) = #{v | deg v = k}I This distribution is well approximated by power-law

distribution: P(k) ≈ ck−γ , typically γ ∈ (2, 3)I Empirically: the degree distribution is approximately a straight

line in log-log scale

Source: Mislove et al, 2006

Summary of observed properties of social networks

I Small average distance and small diameter

I High clustering coefficient

I Power-law degree distribution (scale-freeness)

I Assortativity: “Rich men’s club”

I Existence of densely connected core (most shortest paths gothrough 5% most active vertices — hubs)

I Some dynamic features (not in our focus)

I etc.

The main question

How and whydo all these features emerge?

Types of social network models

I Random graphs (Watts-Strogatz, Barabasi-Albert,Bollobas-Riordan, Buckley-Osthus, Lattanzi-Sivakumar,geometric random graphs etc.)

I Some probabilistic distribution on graphs is consideredI Expectations of different characteristics are computedI May answer how but cannot answer why

I Game-theoretic network formation models (Jackson-Wolinsky,Bala-Goyal, Brautbar-Kearns etc.) where agents formconnections directly

I Agents get utility from connections and may purchase themI Usually a very regular structure emerges in equilibrium

I Game-theoretic approach with “hitchhiking” (Borgs, Chayes,Ding, Lucier 2010): agents may purchase connections forother agents as a side effect

I Many nice properties are supportable in equilibriumI But there are too many equilibria

Affiliation networksNetwork formation through affiliations:

I Initially: bipartite graph between agents and societies;

I Network formation: a connection is formed between twoagents participating in a same society.

Example:

A

B

C

D

E

F

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Workshop

Hiking

TennisA

C D

B

F

E

G




Example:

A

B

C

D

E

F

G

Workshop

Hiking

Tennis

A

C D

B

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Example:

A

B

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D

E

F

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Workshop

Hiking

TennisA

C D

B

F

E

G

Hitchhiking approach: gatherings

I V — the set of agents

I Each agent may organize an arbitrary number of gatheringsP ⊂ V with rates r > 0.

I Gatherings are costly: the fixed cost for gathering of rate r isrb, the marginal cost for one extra participant is rc .

I So, C (Pv ,i ) = rv ,i (c |Pv ,i \ {v}|+ b), where v is an agent andi is the identifier of a gathering. Note that self-inviting iscostless.

Hitchhiking approach: network formation and payoffs

I A link between u and v is formed if they both participate ingatherings with total rate greater than 1.

I No matter who have organized these gatherings: u, v or athird agent.

I Each link delivers utility a to both agents.

I Formally: (vu) ∈ E if∑

(w ,i) : u,v∈Pw,irw ,i ≥ 1

I Nv = #{u : (vu) ∈ E}.I Uv = aNv −

∑i C (Pv ,i ).

Hitchhiking approach: equilibria

I The situation is a one-shot game

I The solution concept: Nash equilibrium

I The main question: what networks are supportable inequilibrium for what parameters?

I Some results (notation: γ = ac )

Source: Borgs, Chayes, Ding, Lucier, 2010


I Theorem: star configuration is not supportable.

I Theorem: any supportable graph has sufficiently highclustering coefficient.

I Theorem: power-law is supportable.I But...

there are too many equlibria.I Clustering coefficient may not be as high as in real networksI Configurations without the power law may be supportable


I Theorem: star configuration is not supportable.

I Theorem: any supportable graph has sufficiently highclustering coefficient.

I Theorem: power-law is supportable.I But... there are too many equlibria.

I Clustering coefficient may not be as high as in real networksI Configurations without the power law may be supportable

Crucial multiplicity of equilibria

TheoremIf at least one non-empty configuration is supportable then thecomplete graph Kn is supportable.Proof ideaThe least average edge cost is reached when any gatheringcoincides with V . It can be reached for all agents simultaneously ifevery agent organizes the grand gathering with rate 1

n . In anyequilibrium some agent has greater edge cost. So, since it isprofitable for him then the symmetric configuration with grandgatherings would also be profitable.

First idea: introducing distance

I Inspired by economic geography models

I Every agent u lives in a point xu in some metric spacerepresenting geographical location or some taste vector

I The cost of inviting is proportional to the distance betweentwo agents

I It seems that edges should emerge between neighbors

I But in fact things are not so good

I Consider the simplest case: one dimension, no borders, flatdistribution. That is, the uniform distributions on a circle

I A must-be equilibrium: every agent u invites a segment[xu − d , xu + d ] with some rate r

I But it is not rational to invite the closest neighbors!

I Actual symmetric equilibrium: every agent u invites a segment[xu + d1, xu + d2] with rate 1!

I Do we need more dimensions, other metrics or differentequilibrium concepts?

Generalization: heterogeneous costs

I Suppose that for any two agents u and v there is a “distance”ρuv ≥ 0, such that ρuu = 0

I Distance is not necessarily a metric or even a symmetricmeasure

I ρuv is treated as the cost of inviting v for the agent u

I Cv (P) = r(c∑

v∈P ρuv + b)

I The initial model is a subcase for ρuv = 1− δuvI The geometric model is a subcase for metric ρuv

Heterogeneous costs among those who invite

I In reality, some people organize many different gatherings,others organize nothing

I It might be that the cost is varying across inviters: ρuv = ρu

I An analysis shows that conclusions should be pretty similar tothe initial model. For instance, if there is a non-emptyequilibrium then there is a complete-graph equilibrium

I Interesting result: for maximal costs that admit a non-trivialequilibrium all agents bear the same costs in this equilibrium

I This extension may be good for comparative statics: thosewith lower costs organize more gatherings

Extension: heterogeneous profits

I Suppose that for any two agents u and v there is a“potential” πuv ≥ 0

I Potential is not necessarily symmetric

I πuv is treated as the profit of having a connection to v for theagent u.

I Uv = a∑

(uv)∈E πuv −∑

Cv (P)

I The initial formalisation is a subcase for πuv = 1− δuvI In some cases increasing πuv is equivalent to reducing ρuv

Extension: concave profits

I Idea: diminishing marginal utility

I Realization: k-th connection gives utility qk−1 for some q < 1

I There is a bound on maximum clique size

How to reduce the number of equilibria: other ideas

I Upper bound on the size of gatherings

I Convex/superadditive/supermodular costs of gathering

I Harder to make a connection in bigger gatherings

I Costly participation in a gathering

I Profits from not only immediate neighbors

I Dynamic network formation

I Your suggestions?

Plans

I More detailed investigation of the case ρuv = ρu

I Analysis of the case ρuv = ρv (some agents are “slow tomove”, others are more agile)

I In what cases heterogeneous profits and heterogeneous costsare fully equivalent?

I Closer investigation of the case of concave profits

I Some numerical simulations

I Other extensions of the model

Conclusions

I Modeling a social graph is a great challenge

I Social networks usually emerge as affiliation networks, so it isnatural to model such types of networks

I Game-theoretic models seem to be more coherent but arecurrently less explanatory

I It is hard to combine tractability and accordance with reality

Thank you for your attention!Questions? Suggestions?

mailto:[email protected]

4 musatov

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