Voting in
Social Networks
Paolo Boldi, Francesco Bonchi, Carlos Castillo
and Sebastiano Vigna

Voting

Voting
• Voting is a set of rules that a community
adopts to take collective decisions

Voting
• Voting is a set of rules that a community
adopts to take collective decisions
• Voters expresses their preferences through
a ballot

Voting
• Voting is a set of rules that a community
adopts to take collective decisions
• Voters expresses their preferences through
a ballot
• The outcome is called the tally

Voting
• Voting is a set of rules that a community
adopts to take collective decisions
• Voters expresses their preferences through
a ballot
• The outcome is called the tally
• Direct/Representative/Delegative/Liquid
democracy

Social Networks

Social Networks
• Electronically mediated social networks
allow people to maintain ties

Social Networks
• Electronically mediated social networks
allow people to maintain ties
• Sometimes, a decision must be taken

Social Networks
• Electronically mediated social networks
allow people to maintain ties
• Sometimes, a decision must be taken
• It is natural to delegate decisions to people
you are actually acquainted with

Social Networks
• Electronically mediated social networks
allow people to maintain ties
• Sometimes, a decision must be taken
• It is natural to delegate decisions to people
you are actually acquainted with
• An ideal setting for delegative/liquid
democracy

The Main Idea

The Main Idea
• We suggest a voting system in which every
voter chooses an acquaintance (or himself)

The Main Idea
• We suggest a voting system in which every
voter chooses an acquaintance (or himself)
• The choice transfers the voting power of the
voter to the chosen acquaintance, but with
an attenuation factor α

The Main Idea
• We suggest a voting system in which every
voter chooses an acquaintance (or himself)
• The choice transfers the voting power of the
voter to the chosen acquaintance, but with
an attenuation factor α
• We avoid in this way an excessively long
vote transfer (friend of a friend of a friend
of...): viscous democracy?

Formally...

Formally...
• A social network is an undirected graph G

Formally...
• A social network is an undirected graph G
• A voting function is a function f:V →V such
G G
that f(x) is x or it is adjacent to x (you vote
for an acquaintance or for yourself)

Formally...
• A social network is an undirected graph G
• A voting function is a function f:V →V such
G G
that f(x) is x or it is adjacent to x (you vote
for an acquaintance or for yourself)
• The delegation graph is the 1-outregular
graph with arcs x→f(x)

Note

Note
• A 1-outregular graph is a set of cycles with
attached trees

Note
• A 1-outregular graph is a set of cycles with
attached trees
• Votes flow from voters in the tree nodes up
to nodes in some cycle

Note
• A 1-outregular graph is a set of cycles with
attached trees
• Votes flow from voters in the tree nodes up
to nodes in some cycle
• Once inside a cycle, votes flow recursively

PageRank!

PageRank!
• This is like computing PageRank on the
delegation graph (attenuation = damping)

PageRank!
• This is like computing PageRank on the
delegation graph (attenuation = damping)
• This is not surprising, as PageRank aims at
computing authoritative pages

PageRank!
• This is like computing PageRank on the
delegation graph (attenuation = damping)
• This is not surprising, as PageRank aims at
computing authoritative pages
• Due to the simple structure of the
delegation graph, we have a closed formula

PageRank!
• This is like computing PageRank on the
delegation graph (attenuation = damping)
• This is not surprising, as PageRank aims at
computing authoritative pages
• Due to the simple structure of the
delegation graph, we have a closed formula
• Note:Yamakava et al. (2007) use similar
techniques with different purposes

To be fair

To be fair
• Spectral techniques for computing “best” entities date
at least 1949 (Seeley’s paper—essentially, PageRank
without damping)

To be fair
• Spectral techniques for computing “best” entities date
at least 1949 (Seeley’s paper—essentially, PageRank
without damping)
• In 1953 Katz introduces its index, which uses damping
on non-stochastic matrices (essentially, PageRank
without normalisation); since the delegation graph is
stochastic, our voting system computes exactly Katz’s
index

To be fair
• Spectral techniques for computing “best” entities date
at least 1949 (Seeley’s paper—essentially, PageRank
without damping)
• In 1953 Katz introduces its index, which uses damping
on non-stochastic matrices (essentially, PageRank
without normalisation); since the delegation graph is
stochastic, our voting system computes exactly Katz’s
index
• If interested, have a look at “Spectral Ranking” [V.]

Properties

Properties
• Small α: undelegable mandates

Properties
• Small α: undelegable mandates
• More precisely: for sufficiently small α,
winners are those with more direct votes

Properties
• Small α: undelegable mandates
• More precisely: for sufficiently small α,
winners are those with more direct votes
• Large α: winners are given by tree size

Properties
• Small α: undelegable mandates
• More precisely: for sufficiently small α,
winners are those with more direct votes
• Large α: winners are given by tree size
• α→1: winners belong to the cycles with
largest average tree size

More properties

More properties
• For sufficiently large α, anybody can win
(ultimate sovereignty)

More properties
• For sufficiently large α, anybody can win
(ultimate sovereignty)
• Bolzano-Weierstrass property: for α 0≤ α1, if
x wins over y with attenuation α0 and
viceversa for α1, then somewhere in between
they tie

Voting Without Voting

Voting Without Voting
• We have to deal with the possibility that
someone will not vote at all

Voting Without Voting
• We have to deal with the possibility that
someone will not vote at all
• The extreme case: nodoby votes

Voting Without Voting
• We have to deal with the possibility that
someone will not vote at all
• The extreme case: nodoby votes
• What can we do?

Voting Without Voting
• We have to deal with the possibility that
someone will not vote at all
• The extreme case: nodoby votes
• What can we do?
• We can compute the expected value of the
voting process when everybody gives their
votes at random

Voting Centrality

Voting Centrality
• This gives a new measure of centrality

Voting Centrality
• This gives a new measure of centrality
• Maybe surprisingly, we can provide a closed
formula based on computations on a large
but finite number of paths in the social
network

Voting Centrality
• This gives a new measure of centrality
• Maybe surprisingly, we can provide a closed
formula based on computations on a large
but finite number of paths in the social
network
• We can use the same approach when some
votes are expressed

The Shifting Surfer

The Shifting Surfer
• Voting centrality is expected PageRank on a
distribution over graphs

The Shifting Surfer
• Voting centrality is expected PageRank on a
distribution over graphs
• Consider a surfer on a family G of graphs on the
same nodes over which there is a distribution P

The Shifting Surfer
• Voting centrality is expected PageRank on a
distribution over graphs
• Consider a surfer on a family G of graphs on the
same nodes over which there is a distribution P
• With probability α, it follows at random an outlink

The Shifting Surfer
• Voting centrality is expected PageRank on a
distribution over graphs
• Consider a surfer on a family G of graphs on the
same nodes over which there is a distribution P
• With probability α, it follows at random an outlink
• With probability 1-α, it changes graph following P and
moves to a random node

This Is Voting Centrality!

This Is Voting Centrality!
• We prove that the average time spent by the
surfer on a node is its expected PageRank

This Is Voting Centrality!
• We prove that the average time spent by the
surfer on a node is its expected PageRank
• In our case, things have a more detailed
interpretation, as there is just one outlink

This Is Voting Centrality!
• We prove that the average time spent by the
surfer on a node is its expected PageRank
• In our case, things have a more detailed
interpretation, as there is just one outlink
• The surfer/voter either follows its only
possible vote, or chooses at random a
delegation graph and a node and then follow
its only possible vote

Experiments

Experiments
• The paper contains results from a number of
experiments

Experiments
• The paper contains results from a number of
experiments
• It is difficult to assess the meaningfulness of
our voting system (in general of any voting
system)

Experiments
• The paper contains results from a number of
experiments
• It is difficult to assess the meaningfulness of
our voting system (in general of any voting
system)
• We analyse DBLP/Flicker data and measure
dependency on α, introduction of noise,
independence from other metrics, etc.

Conclusions

Conclusions
• We have proposed the first voting system
tailored to social networks

Conclusions
• We have proposed the first voting system
tailored to social networks
• The interest of the system stems from the
interplay between the social-network
structure and the voting process

Conclusions
• We have proposed the first voting system
tailored to social networks
• The interest of the system stems from the
interplay between the social-network
structure and the voting process
• There are gazillions of open problems and
directions for research