Voting in social networks

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 • 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

Voting in social networks

by Carlos Castillo on Nov 12, 2009

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