Network-based Learning for Understanding Collective Human ...xdong/talk/SICSS_NetworkLearning.pdf•...

Network-based Learning for Understanding Collective Human Behaviour

Summer Institute in Computational Social Science Oxford, June 2019

Xiaowen Dong Department of Engineering Science

University of Oxford

Introduction

Auguste Comte defined “social physics” as the study of the laws of society or the science of civilisation

Early 1800s

Newtonian physics (society conceptualised as a machine)

http://www.eoht.info/page/Social+physics https://en.wikipedia.org/wiki/Royal_Statistical_Society

Introduction

“...procuring, arranging and publishing facts to illustrate the condition and prospects of society”

Auguste Comte defined “social physics” as the study of the laws of society or the science of civilisation

Early 1800s

Newtonian physics (society conceptualised as a machine)

http://www.eoht.info/page/Social+physics https://en.wikipedia.org/wiki/Royal_Statistical_Society

Introduction

1940-50s

many social indicators have statistical regularities (Zipf distribution, gravity law)

Zipf, “Human behavior and the principle of least effort,” Addison-Wesley Press, 1949. Zipf, “The P1P2/D hypothesis: On the inter-city movement of persons,” American Sociological Review, 1946.

Introduction

1940-50s

many social indicators have statistical regularities (Zipf distribution, gravity law)

Zipf, “Human behavior and the principle of least effort,” Addison-Wesley Press, 1949. Zipf, “The P1P2/D hypothesis: On the inter-city movement of persons,” American Sociological Review, 1946.

statistical regularities within human movement and communication

correlations with economic indicators

1990-2000s

Eagle et al., “Network diversity and economic development,” Science, 2010.

Introduction

new data sources about human behaviour

Introduction

advanced computational methods

new data sources about human behaviour

Introduction

new perspectives from computational social science

Introduction

new perspectives from computational social science current population

management

- demographics - individual records

- static information

the new way

- behavioural traits

- collective behaviour - dynamics

practical impact

Introduction

management

the new way

practical impact

- need new models & tools to understand structure and dynamics - computational methods in the context of social sciences research

Introduction

management

the new way

practical impact

- need new models & tools to understand structure and dynamics - computational methods in the context of social sciences research

- how social network affects decision-making?

- how to infer unobserved social network structure from observed decisions?

Motivation• From “six degrees of separation” to “three degrees of influence”

Milgram’s experiment

7Christakis and Fowler, “Connected,” Little, Brown and Company, 2009.

Milgram’s experimentspreading of obesity,

happiness, smoking behaviour

7Christakis and Fowler, “Connected,” Little, Brown and Company, 2009.

• How influence propagates in social networks and affect decision-making?

Milgram’s experimentspreading of obesity,

happiness, smoking behaviour

Setting• Large-scale mobile phone data during Cirque de Soleil in Andorra • Study influence on individual decision-making: whether or not to attend

the event after receiving phone calls from an attendee

Period 18 days

Attended and observed by mobile phone

16,043

Number of mobile phone users ‘influenced’

161,857

Number of mobile phone users not influenced 71,337

Setting• Information cascade (phone calls) for an observation period

- initial adopters: attendees of event

- influenced/not influenced: connected/not connected to initial adopters in cascade

- hop index: shortest distance from an individual to any initial adopter in cascade

Setting• Information cascade (phone calls) for an observation period

- initial adopters: attendees of event

- influenced/not influenced: connected/not connected to initial adopters in cascade

- hop index: shortest distance from an individual to any initial adopter in cascade

• Observe adoption decisions in “future”

Homophily vs Influence• Decisions on adoption may be attributed to two factors

- homophily: similarity between friends

- influence: which drives contagion

homophily

homophily influence

Matched sample estimation• Quantify effect of influence while controlling for bias due to homophily

treatment control

- matched sample estimation via propensity score matching (PSM)

Matched sample estimation• Quantify effect of influence while controlling for bias due to homophily

treatment control

- matched sample estimation via propensity score matching (PSM)

- matching conditioned on preferences revealed by individual mobility history

Samuelson, “A note on the pure theory of consumer’s behaviour,” Economica, 1938.

explorative exploitative

Matched sample estimation• Study effect of influence w.r.t. communication distance

- one treatment group for each hop index

- one corresponding control group from outside information cascade

treatment control explorative exploitative

Matched sample estimation• Study effect of influence w.r.t. communication distance

- one treatment group for each hop index

- one corresponding control group from outside information cascade

12Aral et al., “Distinguishing influence-based contagion from homophily-driven diffusion in dynamic networks,” PNAS, 2009.

treatment control explorative exploitative

- difference in future adoption rate establishes upper bound of social influence effect

Empirical results

- social influence increases likelihood of future adoption

Empirical results

- long range effect of social influence over communication network

Empirical results

- over-estimation of effect based on random matching

Empirical results

- over-estimation of effect based on random matching

- decaying pattern disappears after random shuffling of hop index

Bayesian learning model

- an individual updates her estimation of “product” characteristics after dynamically aggregating local information

- she then makes adoption decision based on the perception and her own preference

Bayesian learning model

- measure difference in estimation between individuals in information cascade and initial adopters

- difference increases w.r.t. hop index

- observed decaying influence may be related to bias in estimation which accumulates along cascade

Discussion• Summary & Implication

- revealed preferences via behaviour patterns may provide more robust information than traditional socio-demographic factors

- long range effect of social influence highlights hidden connection among seemingly independent individuals

- implication in viral marketing, public health management, and social mobilisation

Discussion• Summary & Implication

- revealed preferences via behaviour patterns may provide more robust information than traditional socio-demographic factors

- long range effect of social influence highlights hidden connection among seemingly independent individuals

- implication in viral marketing, public health management, and social mobilisation

• Open issues & future directions - identification of initial adopters

- multiple treatment effect via more than one communication path

/35 17

Motivation• Consider a group of students making choices on educational effort

- making effort is costly

- I will benefit from my own effort

- I will also benefit from my friends’ effort

more effort

less effort

Motivation• Consider a group of students making choices on buying a book

- buying a book is costly

- if a friend of mine will buy, then I will not buy

- but if none of my friends will buy, then I will buy

buy a book

do not buy a book

Motivation• Such strategic interactions can be modelled as games on networks

- players, actions, payoffs, interaction network

- payoff of an individual depends on her action as well as her neighbours’ actions

- strategic complements or strategic substitutes- payoff of an individual depends on her action as well as her neighbours’ actions

buy a book

do not buy a book

more effort

less effort

Motivation• Economics

- existence of equilibrium or how action/payoff depends on network structure

- on a given or predefined network

• Computer Science (graphical games) - algorithms for computing equilibrium

- binary or finite discrete action space

• This work - learning network given continuous actions

• Many examples - observe individual decisions (e.g., adoptions), but not social relationship

- observe R&D activities of firms, but not collaboration networks

- observe international policies of countries, but not political alliance

Network games with linear-quadratic payoffs• Consider a graph with edge weights

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• Games with linear-quadratic payoffs

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2a2i + �ai

j2N (i)

Gijaj<latexit sha1_base64="tkq7e4J4le4vaLcszmh/BTFOcLY=">AAACO3icbVBNSwMxEM36bf2qevQSLIIilt0i6EUoetCTqFgtdOsym2Zt2mx2SbJCCfu/vPgnvHnx4kERr95Naw9aHZjw8t4MM/PClDOlXffJGRufmJyanpktzM0vLC4Vl1euVJJJQmsk4Ymsh6AoZ4LWNNOc1lNJIQ45vQ67R339+o5KxRJxqXspbcZwK1jECGhLBcWLLGD4AIf2BZs7fiSBGC83ldz+byp4G/sh1TBQfZXFgelgnwnsx6DbBLg5zTfZVo6PA8M6uS3rBMWSW3YHgf8CbwhKaBhnQfHRbyUki6nQhINSDc9NddOA1Ixwmhf8TNEUSBduacNCATFVTTO4PccblmnhKJE2hcYD9meHgVipXhzayv7GalTrk/9pjUxH+03DRJppKsj3oCjjWCe4byRuMUmJ5j0LgEhmd8WkDdY9be0uWBO80ZP/gqtK2XPL3vluqXo4tGMGraF1tIk8tIeq6ASdoRoi6B49o1f05jw4L8678/FdOuYMe1bRr3A+vwClYKwf</latexit><latexit sha1_base64="tkq7e4J4le4vaLcszmh/BTFOcLY=">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</latexit><latexit sha1_base64="tkq7e4J4le4vaLcszmh/BTFOcLY=">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</latexit><latexit sha1_base64="tkq7e4J4le4vaLcszmh/BTFOcLY=">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</latexit>

individual action

ui = biai �1

2a2i + �ai

j2N (i)

individual actionmarginal benefit

ui = biai �1

2a2i + �ai

j2N (i)

individual actionmarginal benefit network factor

ui = biai �1

2a2i + �ai

j2N (i)

individual effect network effect

ui = biai �1

2a2i + �ai

j2N (i)

Network games with linear-quadratic payoffs

23Jackson and Zezou, “Games on networks,” Handbook of Game Theory with Economic Applications, 2015. Bramoullé and Kranton, “Games played on networks,” The Oxford Handbook of the Economics of Networks, 2016.

� > 0<latexit sha1_base64="4FuzAc7bz3x7RjoD5RQgDT19KiE=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6EmKXjxWsB/QhrLZTtqlm03YnQil9Ed48aCIV3+PN/+N2zYHbX0w8Hhvhpl5YSqFIc/7dgpr6xubW8Xt0s7u3v5B+fCoaZJMc2zwRCa6HTKDUihskCCJ7VQji0OJrXB0N/NbT6iNSNQjjVMMYjZQIhKckZVa3RCJ3Xi9csWrenO4q8TPSQVy1Hvlr24/4VmMirhkxnR8L6VgwjQJLnFa6mYGU8ZHbIAdSxWL0QST+blT98wqfTdKtC1F7lz9PTFhsTHjOLSdMaOhWfZm4n9eJ6PoOpgIlWaEii8WRZl0KXFnv7t9oZGTHFvCuBb2VpcPmWacbEIlG4K//PIqaV5Ufa/qP1xWard5HEU4gVM4Bx+uoAb3UIcGcBjBM7zCm5M6L86787FoLTj5zDH8gfP5A7iljyY=</latexit><latexit sha1_base64="4FuzAc7bz3x7RjoD5RQgDT19KiE=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6EmKXjxWsB/QhrLZTtqlm03YnQil9Ed48aCIV3+PN/+N2zYHbX0w8Hhvhpl5YSqFIc/7dgpr6xubW8Xt0s7u3v5B+fCoaZJMc2zwRCa6HTKDUihskCCJ7VQji0OJrXB0N/NbT6iNSNQjjVMMYjZQIhKckZVa3RCJ3Xi9csWrenO4q8TPSQVy1Hvlr24/4VmMirhkxnR8L6VgwjQJLnFa6mYGU8ZHbIAdSxWL0QST+blT98wqfTdKtC1F7lz9PTFhsTHjOLSdMaOhWfZm4n9eJ6PoOpgIlWaEii8WRZl0KXFnv7t9oZGTHFvCuBb2VpcPmWacbEIlG4K//PIqaV5Ufa/qP1xWard5HEU4gVM4Bx+uoAb3UIcGcBjBM7zCm5M6L86787FoLTj5zDH8gfP5A7iljyY=</latexit><latexit sha1_base64="4FuzAc7bz3x7RjoD5RQgDT19KiE=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6EmKXjxWsB/QhrLZTtqlm03YnQil9Ed48aCIV3+PN/+N2zYHbX0w8Hhvhpl5YSqFIc/7dgpr6xubW8Xt0s7u3v5B+fCoaZJMc2zwRCa6HTKDUihskCCJ7VQji0OJrXB0N/NbT6iNSNQjjVMMYjZQIhKckZVa3RCJ3Xi9csWrenO4q8TPSQVy1Hvlr24/4VmMirhkxnR8L6VgwjQJLnFa6mYGU8ZHbIAdSxWL0QST+blT98wqfTdKtC1F7lz9PTFhsTHjOLSdMaOhWfZm4n9eJ6PoOpgIlWaEii8WRZl0KXFnv7t9oZGTHFvCuBb2VpcPmWacbEIlG4K//PIqaV5Ufa/qP1xWard5HEU4gVM4Bx+uoAb3UIcGcBjBM7zCm5M6L86787FoLTj5zDH8gfP5A7iljyY=</latexit><latexit sha1_base64="4FuzAc7bz3x7RjoD5RQgDT19KiE=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6EmKXjxWsB/QhrLZTtqlm03YnQil9Ed48aCIV3+PN/+N2zYHbX0w8Hhvhpl5YSqFIc/7dgpr6xubW8Xt0s7u3v5B+fCoaZJMc2zwRCa6HTKDUihskCCJ7VQji0OJrXB0N/NbT6iNSNQjjVMMYjZQIhKckZVa3RCJ3Xi9csWrenO4q8TPSQVy1Hvlr24/4VmMirhkxnR8L6VgwjQJLnFa6mYGU8ZHbIAdSxWL0QST+blT98wqfTdKtC1F7lz9PTFhsTHjOLSdMaOhWfZm4n9eJ6PoOpgIlWaEii8WRZl0KXFnv7t9oZGTHFvCuBb2VpcPmWacbEIlG4K//PIqaV5Ufa/qP1xWard5HEU4gVM4Bx+uoAb3UIcGcBjBM7zCm5M6L86787FoLTj5zDH8gfP5A7iljyY=</latexit>

� < 0<latexit sha1_base64="3ASQuhTVsnrPuyrinPf2HkCciQs=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tDoNgFe5E0MIiaGMZwXxAcoS9zVyyZG/v2J0TQsiPsLFQxNbfY+e/cZNcoYkPBh7vzTAzL0ylMOR5305hbX1jc6u4XdrZ3ds/KB8eNU2SaY4NnshEt0NmUAqFDRIksZ1qZHEosRWO7mZ+6wm1EYl6pHGKQcwGSkSCM7JSqxsisRuvV654VW8Od5X4OalAjnqv/NXtJzyLURGXzJiO76UUTJgmwSVOS93MYMr4iA2wY6liMZpgMj936p5Zpe9GibalyJ2rvycmLDZmHIe2M2Y0NMveTPzP62QUXQcTodKMUPHFoiiTLiXu7He3LzRykmNLGNfC3uryIdOMk02oZEPwl19eJc2Lqu9V/YfLSu02j6MIJ3AK5+DDFdTgHurQAA4jeIZXeHNS58V5dz4WrQUnnzmGP3A+fwC1m48k</latexit><latexit sha1_base64="3ASQuhTVsnrPuyrinPf2HkCciQs=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tDoNgFe5E0MIiaGMZwXxAcoS9zVyyZG/v2J0TQsiPsLFQxNbfY+e/cZNcoYkPBh7vzTAzL0ylMOR5305hbX1jc6u4XdrZ3ds/KB8eNU2SaY4NnshEt0NmUAqFDRIksZ1qZHEosRWO7mZ+6wm1EYl6pHGKQcwGSkSCM7JSqxsisRuvV654VW8Od5X4OalAjnqv/NXtJzyLURGXzJiO76UUTJgmwSVOS93MYMr4iA2wY6liMZpgMj936p5Zpe9GibalyJ2rvycmLDZmHIe2M2Y0NMveTPzP62QUXQcTodKMUPHFoiiTLiXu7He3LzRykmNLGNfC3uryIdOMk02oZEPwl19eJc2Lqu9V/YfLSu02j6MIJ3AK5+DDFdTgHurQAA4jeIZXeHNS58V5dz4WrQUnnzmGP3A+fwC1m48k</latexit><latexit sha1_base64="3ASQuhTVsnrPuyrinPf2HkCciQs=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tDoNgFe5E0MIiaGMZwXxAcoS9zVyyZG/v2J0TQsiPsLFQxNbfY+e/cZNcoYkPBh7vzTAzL0ylMOR5305hbX1jc6u4XdrZ3ds/KB8eNU2SaY4NnshEt0NmUAqFDRIksZ1qZHEosRWO7mZ+6wm1EYl6pHGKQcwGSkSCM7JSqxsisRuvV654VW8Od5X4OalAjnqv/NXtJzyLURGXzJiO76UUTJgmwSVOS93MYMr4iA2wY6liMZpgMj936p5Zpe9GibalyJ2rvycmLDZmHIe2M2Y0NMveTPzP62QUXQcTodKMUPHFoiiTLiXu7He3LzRykmNLGNfC3uryIdOMk02oZEPwl19eJc2Lqu9V/YfLSu02j6MIJ3AK5+DDFdTgHurQAA4jeIZXeHNS58V5dz4WrQUnnzmGP3A+fwC1m48k</latexit><latexit sha1_base64="3ASQuhTVsnrPuyrinPf2HkCciQs=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tDoNgFe5E0MIiaGMZwXxAcoS9zVyyZG/v2J0TQsiPsLFQxNbfY+e/cZNcoYkPBh7vzTAzL0ylMOR5305hbX1jc6u4XdrZ3ds/KB8eNU2SaY4NnshEt0NmUAqFDRIksZ1qZHEosRWO7mZ+6wm1EYl6pHGKQcwGSkSCM7JSqxsisRuvV654VW8Od5X4OalAjnqv/NXtJzyLURGXzJiO76UUTJgmwSVOS93MYMr4iA2wY6liMZpgMj936p5Zpe9GibalyJ2rvycmLDZmHIe2M2Y0NMveTPzP62QUXQcTodKMUPHFoiiTLiXu7He3LzRykmNLGNfC3uryIdOMk02oZEPwl19eJc2Lqu9V/YfLSu02j6MIJ3AK5+DDFdTgHurQAA4jeIZXeHNS58V5dz4WrQUnnzmGP3A+fwC1m48k</latexit>

remarks

- continuous actions

- for strategic complements ( ) and substitutes ( )

- can be used to approximate complex non-linear payoffs

- widely adopted in literature [Jackson15,Bramoullé16]

ui = biai �1

2a2i + �ai

j2N (i)

Network games with linear-quadratic payoffs

23Jackson and Zezou, “Games on networks,” Handbook of Game Theory with Economic Applications, 2015. Bramoullé and Kranton, “Games played on networks,” The Oxford Handbook of the Economics of Networks, 2016.

� > 0<latexit sha1_base64="4FuzAc7bz3x7RjoD5RQgDT19KiE=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6EmKXjxWsB/QhrLZTtqlm03YnQil9Ed48aCIV3+PN/+N2zYHbX0w8Hhvhpl5YSqFIc/7dgpr6xubW8Xt0s7u3v5B+fCoaZJMc2zwRCa6HTKDUihskCCJ7VQji0OJrXB0N/NbT6iNSNQjjVMMYjZQIhKckZVa3RCJ3Xi9csWrenO4q8TPSQVy1Hvlr24/4VmMirhkxnR8L6VgwjQJLnFa6mYGU8ZHbIAdSxWL0QST+blT98wqfTdKtC1F7lz9PTFhsTHjOLSdMaOhWfZm4n9eJ6PoOpgIlWaEii8WRZl0KXFnv7t9oZGTHFvCuBb2VpcPmWacbEIlG4K//PIqaV5Ufa/qP1xWard5HEU4gVM4Bx+uoAb3UIcGcBjBM7zCm5M6L86787FoLTj5zDH8gfP5A7iljyY=</latexit><latexit sha1_base64="4FuzAc7bz3x7RjoD5RQgDT19KiE=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6EmKXjxWsB/QhrLZTtqlm03YnQil9Ed48aCIV3+PN/+N2zYHbX0w8Hhvhpl5YSqFIc/7dgpr6xubW8Xt0s7u3v5B+fCoaZJMc2zwRCa6HTKDUihskCCJ7VQji0OJrXB0N/NbT6iNSNQjjVMMYjZQIhKckZVa3RCJ3Xi9csWrenO4q8TPSQVy1Hvlr24/4VmMirhkxnR8L6VgwjQJLnFa6mYGU8ZHbIAdSxWL0QST+blT98wqfTdKtC1F7lz9PTFhsTHjOLSdMaOhWfZm4n9eJ6PoOpgIlWaEii8WRZl0KXFnv7t9oZGTHFvCuBb2VpcPmWacbEIlG4K//PIqaV5Ufa/qP1xWard5HEU4gVM4Bx+uoAb3UIcGcBjBM7zCm5M6L86787FoLTj5zDH8gfP5A7iljyY=</latexit><latexit sha1_base64="4FuzAc7bz3x7RjoD5RQgDT19KiE=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6EmKXjxWsB/QhrLZTtqlm03YnQil9Ed48aCIV3+PN/+N2zYHbX0w8Hhvhpl5YSqFIc/7dgpr6xubW8Xt0s7u3v5B+fCoaZJMc2zwRCa6HTKDUihskCCJ7VQji0OJrXB0N/NbT6iNSNQjjVMMYjZQIhKckZVa3RCJ3Xi9csWrenO4q8TPSQVy1Hvlr24/4VmMirhkxnR8L6VgwjQJLnFa6mYGU8ZHbIAdSxWL0QST+blT98wqfTdKtC1F7lz9PTFhsTHjOLSdMaOhWfZm4n9eJ6PoOpgIlWaEii8WRZl0KXFnv7t9oZGTHFvCuBb2VpcPmWacbEIlG4K//PIqaV5Ufa/qP1xWard5HEU4gVM4Bx+uoAb3UIcGcBjBM7zCm5M6L86787FoLTj5zDH8gfP5A7iljyY=</latexit><latexit sha1_base64="4FuzAc7bz3x7RjoD5RQgDT19KiE=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6EmKXjxWsB/QhrLZTtqlm03YnQil9Ed48aCIV3+PN/+N2zYHbX0w8Hhvhpl5YSqFIc/7dgpr6xubW8Xt0s7u3v5B+fCoaZJMc2zwRCa6HTKDUihskCCJ7VQji0OJrXB0N/NbT6iNSNQjjVMMYjZQIhKckZVa3RCJ3Xi9csWrenO4q8TPSQVy1Hvlr24/4VmMirhkxnR8L6VgwjQJLnFa6mYGU8ZHbIAdSxWL0QST+blT98wqfTdKtC1F7lz9PTFhsTHjOLSdMaOhWfZm4n9eJ6PoOpgIlWaEii8WRZl0KXFnv7t9oZGTHFvCuBb2VpcPmWacbEIlG4K//PIqaV5Ufa/qP1xWard5HEU4gVM4Bx+uoAb3UIcGcBjBM7zCm5M6L86787FoLTj5zDH8gfP5A7iljyY=</latexit>

� < 0<latexit sha1_base64="3ASQuhTVsnrPuyrinPf2HkCciQs=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tDoNgFe5E0MIiaGMZwXxAcoS9zVyyZG/v2J0TQsiPsLFQxNbfY+e/cZNcoYkPBh7vzTAzL0ylMOR5305hbX1jc6u4XdrZ3ds/KB8eNU2SaY4NnshEt0NmUAqFDRIksZ1qZHEosRWO7mZ+6wm1EYl6pHGKQcwGSkSCM7JSqxsisRuvV654VW8Od5X4OalAjnqv/NXtJzyLURGXzJiO76UUTJgmwSVOS93MYMr4iA2wY6liMZpgMj936p5Zpe9GibalyJ2rvycmLDZmHIe2M2Y0NMveTPzP62QUXQcTodKMUPHFoiiTLiXu7He3LzRykmNLGNfC3uryIdOMk02oZEPwl19eJc2Lqu9V/YfLSu02j6MIJ3AK5+DDFdTgHurQAA4jeIZXeHNS58V5dz4WrQUnnzmGP3A+fwC1m48k</latexit><latexit sha1_base64="3ASQuhTVsnrPuyrinPf2HkCciQs=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tDoNgFe5E0MIiaGMZwXxAcoS9zVyyZG/v2J0TQsiPsLFQxNbfY+e/cZNcoYkPBh7vzTAzL0ylMOR5305hbX1jc6u4XdrZ3ds/KB8eNU2SaY4NnshEt0NmUAqFDRIksZ1qZHEosRWO7mZ+6wm1EYl6pHGKQcwGSkSCM7JSqxsisRuvV654VW8Od5X4OalAjnqv/NXtJzyLURGXzJiO76UUTJgmwSVOS93MYMr4iA2wY6liMZpgMj936p5Zpe9GibalyJ2rvycmLDZmHIe2M2Y0NMveTPzP62QUXQcTodKMUPHFoiiTLiXu7He3LzRykmNLGNfC3uryIdOMk02oZEPwl19eJc2Lqu9V/YfLSu02j6MIJ3AK5+DDFdTgHurQAA4jeIZXeHNS58V5dz4WrQUnnzmGP3A+fwC1m48k</latexit><latexit sha1_base64="3ASQuhTVsnrPuyrinPf2HkCciQs=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tDoNgFe5E0MIiaGMZwXxAcoS9zVyyZG/v2J0TQsiPsLFQxNbfY+e/cZNcoYkPBh7vzTAzL0ylMOR5305hbX1jc6u4XdrZ3ds/KB8eNU2SaY4NnshEt0NmUAqFDRIksZ1qZHEosRWO7mZ+6wm1EYl6pHGKQcwGSkSCM7JSqxsisRuvV654VW8Od5X4OalAjnqv/NXtJzyLURGXzJiO76UUTJgmwSVOS93MYMr4iA2wY6liMZpgMj936p5Zpe9GibalyJ2rvycmLDZmHIe2M2Y0NMveTPzP62QUXQcTodKMUPHFoiiTLiXu7He3LzRykmNLGNfC3uryIdOMk02oZEPwl19eJc2Lqu9V/YfLSu02j6MIJ3AK5+DDFdTgHurQAA4jeIZXeHNS58V5dz4WrQUnnzmGP3A+fwC1m48k</latexit><latexit sha1_base64="3ASQuhTVsnrPuyrinPf2HkCciQs=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tDoNgFe5E0MIiaGMZwXxAcoS9zVyyZG/v2J0TQsiPsLFQxNbfY+e/cZNcoYkPBh7vzTAzL0ylMOR5305hbX1jc6u4XdrZ3ds/KB8eNU2SaY4NnshEt0NmUAqFDRIksZ1qZHEosRWO7mZ+6wm1EYl6pHGKQcwGSkSCM7JSqxsisRuvV654VW8Od5X4OalAjnqv/NXtJzyLURGXzJiO76UUTJgmwSVOS93MYMr4iA2wY6liMZpgMj936p5Zpe9GibalyJ2rvycmLDZmHIe2M2Y0NMveTPzP62QUXQcTodKMUPHFoiiTLiXu7He3LzRykmNLGNfC3uryIdOMk02oZEPwl19eJc2Lqu9V/YfLSu02j6MIJ3AK5+DDFdTgHurQAA4jeIZXeHNS58V5dz4WrQUnnzmGP3A+fwC1m48k</latexit>

remarks

- continuous actions

- for strategic complements ( ) and substitutes ( )

- can be used to approximate complex non-linear payoffs

- widely adopted in literature [Jackson15,Bramoullé16]

ui = biai �1

2a2i + �ai

j2N (i)

examples - education: action is educational effort, utility is achievement

- collaboration: action is joint R&D activities, utility is firm profit

- urban dynamics: action is mobility behaviour, utility is convenience/satisfaction

Network games with linear-quadratic payoffs• Pure-strategy Nash equilibrium (PSNE)

ui = biai �1

2a2i + �ai

j2VGijaj

@ai= bi � ai + �(Ga)iui = biai �

2a2i + �ai

j2VGijaj

a = (I� �G)�1 b

2a2i + �ai

j2VGijaj

a = (I� �G)�1 b

Ballester et al., “Who’s who in networks. Wanted: The key player,” Econometrica, 2006.

assumption: spectral radius is smaller than 1

- guarantees matrix inversion

- ensures uniqueness and stability of equilibrium action [Ballester06]

⇢(�G)<latexit sha1_base64="QwOy1LrktTiQ+Q4HlAxixf/fWkU=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksQr2URAQ9Fj3osYL9gKaUzXbTLt1kw+5EKCEX/4oXD4p49Wd489+4aXPQ1gcDj/dmmJnnx4JrcJxvq7Syura+Ud6sbG3v7O7Z+wdtLRNFWYtKIVXXJ5oJHrEWcBCsGytGQl+wjj+5yf3OI1Oay+gBpjHrh2QU8YBTAkYa2EeeGktc83wGBHshgbEfpLfZ2cCuOnVnBrxM3IJUUYHmwP7yhpImIYuACqJ1z3Vi6KdEAaeCZRUv0SwmdEJGrGdoREKm++nsgQyfGmWIA6lMRYBn6u+JlIRaT0PfdOYn6kUvF//zegkEV/2UR3ECLKLzRUEiMEicp4GHXDEKYmoIoYqbWzEdE0UomMwqJgR38eVl0j6vu07dvb+oNq6LOMroGJ2gGnLRJWqgO9RELURRhp7RK3qznqwX6936mLeWrGLmEP2B9fkDqjKVzA==</latexit><latexit sha1_base64="QwOy1LrktTiQ+Q4HlAxixf/fWkU=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksQr2URAQ9Fj3osYL9gKaUzXbTLt1kw+5EKCEX/4oXD4p49Wd489+4aXPQ1gcDj/dmmJnnx4JrcJxvq7Syura+Ud6sbG3v7O7Z+wdtLRNFWYtKIVXXJ5oJHrEWcBCsGytGQl+wjj+5yf3OI1Oay+gBpjHrh2QU8YBTAkYa2EeeGktc83wGBHshgbEfpLfZ2cCuOnVnBrxM3IJUUYHmwP7yhpImIYuACqJ1z3Vi6KdEAaeCZRUv0SwmdEJGrGdoREKm++nsgQyfGmWIA6lMRYBn6u+JlIRaT0PfdOYn6kUvF//zegkEV/2UR3ECLKLzRUEiMEicp4GHXDEKYmoIoYqbWzEdE0UomMwqJgR38eVl0j6vu07dvb+oNq6LOMroGJ2gGnLRJWqgO9RELURRhp7RK3qznqwX6936mLeWrGLmEP2B9fkDqjKVzA==</latexit><latexit sha1_base64="QwOy1LrktTiQ+Q4HlAxixf/fWkU=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksQr2URAQ9Fj3osYL9gKaUzXbTLt1kw+5EKCEX/4oXD4p49Wd489+4aXPQ1gcDj/dmmJnnx4JrcJxvq7Syura+Ud6sbG3v7O7Z+wdtLRNFWYtKIVXXJ5oJHrEWcBCsGytGQl+wjj+5yf3OI1Oay+gBpjHrh2QU8YBTAkYa2EeeGktc83wGBHshgbEfpLfZ2cCuOnVnBrxM3IJUUYHmwP7yhpImIYuACqJ1z3Vi6KdEAaeCZRUv0SwmdEJGrGdoREKm++nsgQyfGmWIA6lMRYBn6u+JlIRaT0PfdOYn6kUvF//zegkEV/2UR3ECLKLzRUEiMEicp4GHXDEKYmoIoYqbWzEdE0UomMwqJgR38eVl0j6vu07dvb+oNq6LOMroGJ2gGnLRJWqgO9RELURRhp7RK3qznqwX6936mLeWrGLmEP2B9fkDqjKVzA==</latexit><latexit sha1_base64="QwOy1LrktTiQ+Q4HlAxixf/fWkU=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksQr2URAQ9Fj3osYL9gKaUzXbTLt1kw+5EKCEX/4oXD4p49Wd489+4aXPQ1gcDj/dmmJnnx4JrcJxvq7Syura+Ud6sbG3v7O7Z+wdtLRNFWYtKIVXXJ5oJHrEWcBCsGytGQl+wjj+5yf3OI1Oay+gBpjHrh2QU8YBTAkYa2EeeGktc83wGBHshgbEfpLfZ2cCuOnVnBrxM3IJUUYHmwP7yhpImIYuACqJ1z3Vi6KdEAaeCZRUv0SwmdEJGrGdoREKm++nsgQyfGmWIA6lMRYBn6u+JlIRaT0PfdOYn6kUvF//zegkEV/2UR3ECLKLzRUEiMEicp4GHXDEKYmoIoYqbWzEdE0UomMwqJgR38eVl0j6vu07dvb+oNq6LOMroGJ2gGnLRJWqgO9RELURRhp7RK3qznqwX6936mLeWrGLmEP2B9fkDqjKVzA==</latexit>

2a2i + �ai

j2VGijaj

a = (I� �G)�1 b

properties

- equilibrium related to Katz-Bonacich centrality

- payoff interdependency spreads indirectly through network

�pGp b<latexit sha1_base64="HtPPVYSPBScOkKqgC6GVwhVIDdI=">AAACHnicbVBNSwMxEM3W7/pV9eglWARBKLui6KVQ9KBHBauF7rZk06yGZrMhmRXKsv4RL/4VLx4UETzpvzGtK2j1QeDlvRlm5oVKcAOu++GUJianpmdm58rzC4tLy5WV1QuTpJqyJk1EolshMUxwyZrAQbCW0ozEoWCXYf9o6F/eMG14Is9hoFgQkyvJI04JWKlb2av7Jo27maq7eSfb9rmMYJBjP2RAOgr7MYHrMMqO8466/f6EebdSdWvuCPgv8QpSRQVOu5U3v5fQNGYSqCDGtD1XQZARDZwKlpf91DBFaJ9csbalksTMBNnovBxvWqWHo0TbJwGP1J8dGYmNGcShrRxuaMa9ofif104hOggyLlUKTNKvQVEqMCR4mBXucc0oiIElhGpud8X0mmhCwSZatiF44yf/JRc7Nc+teWe71cZhEccsWkcbaAt5aB810Ak6RU1E0R16QE/o2bl3Hp0X5/WrtOQUPWvoF5z3T57Yo20=</latexit><latexit sha1_base64="HtPPVYSPBScOkKqgC6GVwhVIDdI=">AAACHnicbVBNSwMxEM3W7/pV9eglWARBKLui6KVQ9KBHBauF7rZk06yGZrMhmRXKsv4RL/4VLx4UETzpvzGtK2j1QeDlvRlm5oVKcAOu++GUJianpmdm58rzC4tLy5WV1QuTpJqyJk1EolshMUxwyZrAQbCW0ozEoWCXYf9o6F/eMG14Is9hoFgQkyvJI04JWKlb2av7Jo27maq7eSfb9rmMYJBjP2RAOgr7MYHrMMqO8466/f6EebdSdWvuCPgv8QpSRQVOu5U3v5fQNGYSqCDGtD1XQZARDZwKlpf91DBFaJ9csbalksTMBNnovBxvWqWHo0TbJwGP1J8dGYmNGcShrRxuaMa9ofif104hOggyLlUKTNKvQVEqMCR4mBXucc0oiIElhGpud8X0mmhCwSZatiF44yf/JRc7Nc+teWe71cZhEccsWkcbaAt5aB810Ak6RU1E0R16QE/o2bl3Hp0X5/WrtOQUPWvoF5z3T57Yo20=</latexit><latexit sha1_base64="HtPPVYSPBScOkKqgC6GVwhVIDdI=">AAACHnicbVBNSwMxEM3W7/pV9eglWARBKLui6KVQ9KBHBauF7rZk06yGZrMhmRXKsv4RL/4VLx4UETzpvzGtK2j1QeDlvRlm5oVKcAOu++GUJianpmdm58rzC4tLy5WV1QuTpJqyJk1EolshMUxwyZrAQbCW0ozEoWCXYf9o6F/eMG14Is9hoFgQkyvJI04JWKlb2av7Jo27maq7eSfb9rmMYJBjP2RAOgr7MYHrMMqO8466/f6EebdSdWvuCPgv8QpSRQVOu5U3v5fQNGYSqCDGtD1XQZARDZwKlpf91DBFaJ9csbalksTMBNnovBxvWqWHo0TbJwGP1J8dGYmNGcShrRxuaMa9ofif104hOggyLlUKTNKvQVEqMCR4mBXucc0oiIElhGpud8X0mmhCwSZatiF44yf/JRc7Nc+teWe71cZhEccsWkcbaAt5aB810Ak6RU1E0R16QE/o2bl3Hp0X5/WrtOQUPWvoF5z3T57Yo20=</latexit><latexit sha1_base64="HtPPVYSPBScOkKqgC6GVwhVIDdI=">AAACHnicbVBNSwMxEM3W7/pV9eglWARBKLui6KVQ9KBHBauF7rZk06yGZrMhmRXKsv4RL/4VLx4UETzpvzGtK2j1QeDlvRlm5oVKcAOu++GUJianpmdm58rzC4tLy5WV1QuTpJqyJk1EolshMUxwyZrAQbCW0ozEoWCXYf9o6F/eMG14Is9hoFgQkyvJI04JWKlb2av7Jo27maq7eSfb9rmMYJBjP2RAOgr7MYHrMMqO8466/f6EebdSdWvuCPgv8QpSRQVOu5U3v5fQNGYSqCDGtD1XQZARDZwKlpf91DBFaJ9csbalksTMBNnovBxvWqWHo0TbJwGP1J8dGYmNGcShrRxuaMa9ofif104hOggyLlUKTNKvQVEqMCR4mBXucc0oiIElhGpud8X0mmhCwSZatiF44yf/JRc7Nc+teWe71cZhEccsWkcbaAt5aB810Ak6RU1E0R16QE/o2bl3Hp0X5/WrtOQUPWvoF5z3T57Yo20=</latexit>

2a2i + �ai

j2VGijaj

a = (I� �G)�1 b

properties

- equilibrium related to Katz-Bonacich centrality

- payoff interdependency spreads indirectly through network

- action is filtered version of marginal benefit on graph

G=�⇤�T

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Learning quadratic games on networks

a = (I� �G)�1 b (I� �G)a = bNash equilibrium

consider K games B = [b(1),b(2), · · · ,b(k)] 2 RN⇥K