Control and spread of contagion in networks with global effects

• Published date: 01 December 2023
• Author: John Higgins,Tarun Sabarwal
• Date: 01 December 2023
• DOI: http://doi.org/10.1111/jpet.12643

Received: 11 January 2023

|

Accepted: 13 March 2023

DOI: 10.1111/jpet.12643

ORIGINAL ARTICLE

Control and spread of contagion in networks

with global effects

John Higgins

1

|Tarun Sabarwal

2

1

Department of Economics, University of

Wisconsin, Madison, Wisconsin, USA

2

Department of Economics, University of

Kansas, Lawrence, Kansas, USA

Correspondence

Tarun Sabarwal, Department of

Economics, University of Kansas,

Lawrence, KS, USA.

Email: sabarwal@ku.edu

Abstract

We study proliferation of an action in binary action

network coordination games that are generalized to

include global effects. This captures important aspects of

proliferation of a particular action or narrative in online

social networks, providing a basis to understand their

impact on societal outcomes. Our model naturally

captures complementarities among starting sets, network

resilience, and global effects, and highlights inter-

dependence in channels through which contagion

spreads. We present new, natural, computationally

tractable, and efficient algorithms to define and compute

equilibrium objects that facilitate the general study of

contagion in networks and prove their theoretical

properties. Our algorithms are easy to implement and

help to quantify relationships previously inaccessible due

to computational intractability. Using these algorithms,

we study the spread of contagion in scale‐free networks

with 1000 players using millions of Monte Carlo

simulations. Our analysis provides quantitative and

qualitative insight into the design of policies to control

or spread contagion in networks. The scope of application

is enlarged given the many other situations across

different fields that may be modeled using this

framework.

KEYWORDS

algorithmic computation, contagion, coordination games, network

games

J Public Econ Theory. 2023;25:1149–1187. wileyonlinelibrary.com/journal/jpet © 2023 Wiley Periodicals LLC.

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1|INTRODUCTION

Proliferation of contagion in networks is an increasingly central problem with systemic

consequences for society. For example, political divisiveness, anti‐vaccination rhetoric, climate

change skepticism, conspiracy theories, and stereotyping on social networks contribute to

influence broad societal outcomes (Vosoughi et al., 2018). Proliferation of divisive politics on

social media exacerbates a fractious political climate, intensifying divisions across party lines

(Grinberg et al., 2019). Attacks on election integrity undermine trust in the election process

(Frenkel, 2020). Repeated assaults on the science behind disease transmission contribute to

significant long‐term harm to public health, especially during the COVID‐19 pandemic

(Brennen et al., 2020). Anti‐vaccine rhetoric contributes to a decline in vaccination rates, posing

another risk to public health (Broniatowski et al., 2018). Persistent promotion of distrust in

scientific evidence prevents adoption of measures to mitigate impacts of climate change.

Crimes against minorities (ethnic, immigrant, or religious) may be precipitated by posting and

spreading incendiary rhetoric on social networks. Mental health of vulnerable communities

may be worsened by social network pressures (Wells et al., 2021). These problems are

exacerbated by the rapid and widespread diffusion of information in online social networks

(Shao et al., 2018; Varol et al., 2017). These activities can cause systemic upheaval, as evidenced

by the mob attack on the US Capitol on January 6, 2021, and in similar events in other

countries as described in the 2021 Nobel peace prize lecture by Ressa (2021). Collectively, such

events are a significant and growing threat to both democratic institutions and society at large.

Proliferation of an alternative action can also be beneficial for society. For example,

proliferation of welfare‐improving innovations such as more secure transactions, health or

vaccine adoption, environmentally beneficial technology, and collective action for citizen safety

can influence broad societal outcomes.

Even though different campaigns to proliferate an action or narrative in society have

different goals, their structural characteristics are similar. An alternative narrative is promoted

by a small group of participants who rely on social network dynamics for its proliferation. This

is achieved both by person‐to‐person spread through individual connectivity in the network

and by global effects based on similar activity by others in different parts of the network. The

objective of the promoting group is typically achieved with partial proliferation of the narrative

in the network. Once the objective is achieved, the underlying truth or falsity of the narrative

may not matter; it may be undermined or simply swept away.

We study these characteristics of proliferation of an action in a network using a binary

action network coordination game. As coordination incentives arise in many socioeconomic

scenarios with interdependent decision‐making, our model and analysis can be applied to many

additional situations, including regime change, technology adoption, bank runs, currency

crises, run on groceries in a pandemic, marketing new products, segregation and desegregation,

success of social platforms, agglomeration in urban economics, and others.

Consider a network in which each person has a preference to coordinate with their friends

(or network neighbors) on the choice of two actions. Suppose this preference dictates that I find

it beneficial to choose an action if a sufficiently large fraction of my friends also choose that

action. This causes me to choose an action based on the choices of some of my friends, which

may cause some of my other friends to choose the same action, and in turn, cause additional

friends of friends to choose the same action, propagating a chain reaction and causing

contagion.

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HIGGINS and SABARWAL

Theoretical properties of contagion arising from individual decisions in coordination games

on networks in the manner above have been studied in the existing literature. A central result

of interest is the following. Suppose there are finitely many players in a connected network.

Each player can take one of two actions, 0 or 1, and each player is initially playing 0. Suppose

an initial subset of players

S

is exogenously infected to play 1 and we want to characterize when

best response dynamics starting from

S

lead to all players in the network playing 1. As a

parameter of the game, suppose that action 1 is a best response for a player, if, and only if,

fraction

∈

q

[0, 1]

of the player's neighbors play 1. (The parameter

q

is derived analytically from

an underlying tradeoff between benefit of coordination

b

and cost of miscoordination

c

to yield

q

=c

bc+. In this sense, it may be viewed as relative cost of miscoordination as well as a measure

of network resilience to contagion. The higher is

q

, the harder it is for a player to play 1). An

adaptation of Morris (2000) in Jackson (2008) shows that contagion occurs from

S

to the entire

network if, and only if, the complement of

S

is uniformly no more than

q

(

1−

)

‐cohesive. Recall

that a set

A

of players is

q

(

1−

)

‐cohesive if each player in

A

has at least fraction

q

1

−

of their

neighbors in

A

. Set

A

is uniformly no more than

q

(

1−

)

‐cohesive if every nonempty subset of

A

is at most

q

(

1−

)

‐cohesive.

To apply this result, we need to check that every subset of the complement of

S

is at most

q

(

1−

)

‐cohesive. Such a task is exponentially expensive, making its application infeasible

beyond very small networks.

In addition to the computational obstacle, the existing result applies only in coordination

games with local network effects, that is, where a player's payoff depends solely on the actions

of their neighbors (and neighbors have uniform unit weights). It does not apply when a player's

best response depends on actions of both their neighbors and the rest of the network. This

dependence arises naturally in the types of phenomena we want to study.

We solve both problems in this paper. We extend the existing theoretical model of a

network coordination game to include a new, flexible, and tractable formulation of global

effects, which capture the notion that each player's decision depends not just on decisions of

their neighbors but also on an aggregate based on decisions taken by others in the network. We

provide new, natural, tractable, and efficient algorithms that can be applied to an arbitrary

network coordination game with heterogeneous local and global effects and arbitrary starting

set

S

to compute equilibrium depth of contagion starting from

S

and

q

, and also compute all

q

for which contagion spreads from

S

to the entire network in equilibrium.

Global effects are included as an additive component to a player's payoff, making them

transparent and tractable while still allowing for natural interactions with local effects in the

best response and facilitating computation of equilibrium. These effects are otherwise flexible

in that we only require global effects to be weakly increasing. There are no restrictions related

to continuity, convexity, concavity, differentiability, and so on, and we allow for heterogeneity

in global effects for different players. Moreover, we allow for heterogeneity in local effects as

well (different weights for different neighbors of a player, asymmetric weights for different

players, and unidirectional links). We also include players who may be exogenously infected to

play 1. Our formulation subsumes the existing formulation with local effects (and uniform unit

weights on neighbors) as a natural special case. Our formulation allows a natural parametric

flexibility to study the differential impact of global effects on each player's decision and the

corresponding equilibrium.

Our algorithms are natural and tractable because they are based on best response dynamics.

They are efficient because we reduce the number of subsets checked to no more than the

number of players in the complement of

S

, reducing the computational burden from

HIGGINS and SABARWAL

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