Discord and Harmony in Networks
DDISCORD AND HARMONY IN NETWORKS
ANDREA GALEOTTI, BENJAMIN GOLUB, SANJEEV GOYAL, AND RITHVIK RAO
Abstract.
Consider a coordination game played on a network, where agents prefer takingactions closer to those of their neighbors and to their own ideal points in action space. Weexplore how the welfare outcomes of a coordination game depend on network structureand the distribution of ideal points throughout the network. To this end, we imagine abenevolent or adversarial planner who intervenes, at a cost, to change ideal points in order tomaximize or minimize utilitarian welfare subject to a constraint. A complete characterizationof optimal interventions is obtained by decomposing interventions into principal componentsof the network’s adjacency matrix. Welfare is most sensitive to interventions proportionalto the last principal component, which focus on local disagreement. A welfare-maximizingplanner optimally works to reduce local disagreement, bringing the ideal points of neighborscloser together, whereas a malevolent adversary optimally drives neighbors’ ideal points apartto decrease welfare. Such welfare-maximizing/minimizing interventions are very differentfrom ones that would be done to change some traditional measures of discord, such as thecross-sectional variation of equilibrium actions. In fact, an adversary sowing disagreement to maximize her impact on welfare will minimize her impact on global variation in equilibriumactions, underscoring a tension between improving welfare and increasing global cohesion ofequilibrium behavior.
Date Printed . March 1, 2021.Joey Feffer and Zo¨e Hitzig provided exceptional research assistance. Andrea Galeotti gratefully acknowledgesfinancial support from the European Research Council through the ERC-consolidator grant (award no.724356) and the European University Institute through the Internal Research Grant. Benjamin Golubgratefully acknowledges financial support from The Pershing Square Fund for Research on the Foundations ofHuman Behavior and the National Science Foundation (SES-1658940, SES-1629446). Galeotti: Department ofEconomics, London Business School, [email protected]. Golub: Departments of Economics and ComputerScience, Northwestern University, [email protected]. Goyal: Faculty of Economics andChrist’s College, University of Cambridge, [email protected]. Rao: School of Engineering and AppliedSciences, Harvard University, [email protected]. a r X i v : . [ ec on . T H ] F e b DISCORD AND HARMONY IN NETWORKS Introduction
Consider a simple coordination game played on a network. Each player takes an action,having an incentive to bring this action closer to both a personal ideal point and to the actionsof neighbors. In the absence of any coordination concerns, each player would set their actionsequal to their ideal points; we thus also call an ideal point a favorite action . Coordinationconcerns typically change this, pulling an agent’s choices in equilibrium toward the idealpoints of network neighbors, as well as of those farther away with whom the agent interactsonly indirectly. A number of examples motivate our setup. The action may be declaringpolitical opinions or values in a setting where it is costly to disagree with friends, but alsocostly to distort one’s true position from the ideal point of sincere opinion. Alternatively,an action might be a choice in a technological space. For instance, in a software company,designer preferences inform tradeoffs between usability and power in the tools they use, butall are better off when their tools are more compatible with those of their colleagues. In thisexample, the network is determined by collaboration relationships, i.e. which designers worktogether. The broad question we are concerned with is how the favorite actions and the networkjointly determine welfare. Given a network, how do changes in agents’ ideal points affectthe efficiency of equilibrium outcomes? When can relatively small changes in these idealpoints have large welfare impacts? We operationalize this question by imagining a plannerwho can, at a cost, change favorite actions. Supposing an adversary can undertake costlyinfluence activities and change people’s views, how would she do so if her goal was to increasemiscoordination? Turning to the organization example, if managers can exert influence,provide encouragement, and offer incentives to change agents’ inclinations, what changeswould a benevolent manager undertake to maximize welfare? By understanding what suchplanners would do, we can understand how the relationship between favorite points and thenetwork determines welfare. Such insights will also be relevant for problems concerning thecomposition of a team; rather than directly manipulating a particular person’s incentives, aplanner may instead choose whom to put in a certain organizational role or position. Suchinterventions require careful analysis of the welfare implications of the joint arrangement ofideal points and network links. Our results shed light on these issues. This interpretation of actions as choices in a technological space aligns with standard models in the literatureon organizations—see, e.g., Calv´o-Armengol, De Mart´ı and Prat (2015). In these examples, and throughout, we take the network to be exogenous to the decisions in question, whichis often realistic in the short run. Endogenous network formation is, as always, an important concern.
ISCORD AND HARMONY IN NETWORKS 3
To analyze this intervention problem, we take a spectral approach. That is, we write therelevant optimization problems in terms of functions of eigenvalues and eigenvectors of thenetwork, which are important invariants often used to capture various aspects of networkstructure. Working in a “principal component” basis permits legible characterizations ofequilibrium outcomes and optimal interventions. (In contrast, in a natural basis the solutionsto our optimization problems would be unwieldy and would not shed much light on therelationship between structural features of the network and the optimal intervention.) Ourmain findings are characterizations of the optimal intervention using certain eigenvectors andsubstantive implications for what such a planner focuses on.Our main result, Theorem 1, is that the most welfare-consequential changes in favoriteactions focus primarily (in a sense we make precise) on the last eigenvector of the network:the one associated with its lowest (typically most negative) eigenvalue. Beyond this, there isa monotonicity to the structure of interventions: principal components with lower eigenvaluesreceive less focus in optimal interventions. In special cases that we describe, the focus on thelowest principal component can be exclusive: at the optimal intervention, all disagreement infavorite actions is loaded onto this one principal component. Our results also imply thatexplicit functions of certain eigenvalues can summarize the sensitivity of equilibrium welfareto optimal perturbations of ideal points. This gives an answer to the question posed at thebeginning about how sensitive welfare is to the configuration of ideal points.Going beyond a characterization in terms of a canonical graph statistic, we interpret theimplications in terms of more intuitive aspects of graph structure. A key distinction weemphasize is between local discord —creating disagreement at the “street level,” betweenneighbors—and global discord —which creates disagreement between separate regions. Ourresult implies that optimal—i.e., welfare-maximizing or minimizing—interventions have avery local focus in a precise sense. An adversary seeks to amplify disagreement betweenneighbors, pushing neighbors’ favorite points apart.Notably, the interventions that best achieve this are quite distinct from those that bestcreate global discord in the network. Indeed, creating global discord is in tension withreducing welfare. When an adversary optimally sows discord in ideal points to reduce welfare,this leads to a low level of variation across the population in equilibrium behavior, in asense we make precise. Relatedly, if there is a certain amount of diversity (cross-sectionalvariation) in ideal points, it turns out that agents are best off when they are arranged so that We will use principal components and eigenvectors interchangeably. Throughout, we use “disagreement” to refer to differences in actions across the network.
DISCORD AND HARMONY IN NETWORKS they agree with their immediate neighbors and disagree with those distant from them in thenetwork. This naturally leads to societies sustaining more diversity in equilibrium behaviorand appearing more divided in a global sense. To summarize, our main results deliver starkpredictions about which aspects of the configuration of ideal points matter for welfare, andthese are quite different from what we might expect from standard intuitions about discord(as we elaborate on in our discussion of related literature below).Finally, a conceptual point in our analysis is that intervention problems can be usefulmetaphors for understanding what structural features matter for welfare in a game. Insome cases, a planner may indeed be intervening quite explicitly. For instance, an adversarymay be seeking to use social media to sow division in opinions and cause costly tensionsbetween neighbors. But in many other problems, an analyst may simply be interestedin understanding which shifts in exogenous primitives most affect welfare; hypotheticalintervention problems shed light on this even when an intervention is not literally beingdesigned.1.1.
Related work.
Broadly, we are situated in the economics literature on network games,surveyed, for example by Jackson and Zenou (2014); see also the bibliography of Bramoull´e,Kranton and D’Amours (2014). This literature, in terms of techniques and many of themeasures that are relevant, is also related to the literature on opinion updating and sociallearning in networks, going back to DeGroot (1974) and surveyed by Acemoglu and Ozdaglar(2011) and Golub and Sadler (2016).Within this broad literature, our project is distinguished by two aspects of our substantivefocus. First, we are interested in a welfare objective. While most works in the economicsliterature on network games of course touch on efficiency and welfare considerations, themain outcome of interest is often an overall level of activity or knowledge—as, for instance,in Ballester, Calv´o-Armengol and Zenou (2006) and Kempe, Kleinberg and Tardos (2015). There are fewer that are focused on social welfare. An early contribution, with a price ofanarchy approach, is Bindel, Kleinberg and Oren (2011), who give bounds on the welfaredifference between equilibrium and a social optimum under the Friedkin and Johnsen (1999)social learning model. Another closely related contribution is due to Angeletos and Pavan See U.S. House of Representatives Permanent Select Committee on Intelligence (2018) for a report on suchactivities. Spectral methods play a significant role in the study of global influence, which is closely connected tothe Perron vector ( eigenvector centrality ), as in Ballester, Calv´o-Armengol and Zenou (2006); Acemoglu,Carvalho, Ozdaglar and Tahbaz-Salehi (2012). Different eigenvectors matter in our analysis because we arenot concerned with first moments of behavior but rather variation and disagreement across agents.
ISCORD AND HARMONY IN NETWORKS 5 (2007), who study fundamental structural properties of equilibrium welfare in beauty contestsamong other classes of games. In macroeconomics, the welfare implications of shocks arestudied by Baqaee and Farhi (2019) and Baqaee and Farhi (2020). Galeotti, Golub andGoyal (2020) and King and Allouch (2019) are perhaps the closest in that they considerwelfare-optimal interventions. However, the class of games considered is very different:investment or public goods games. These involve quite different externalities from the onesthat are relevant for coordination games and discord, which is what we focus on (as discussedby Angeletos and Pavan (2007)). Issues of miscoordination and discord are touched on in another thread of literature. Thiswork analyzes how the configuration of agents’ attributes (initial opinions, ideal points, etc.)affects the dynamics and ultimate outcomes of processes in social networks. The connectionof these outcomes to spectral aspects of the network was noted by DeMarzo, Vayanos andZwiebel (2003), and further developed by Golub and Jackson (2012), which highlighted therelation to spectral clustering. An important recent contribution on discord is Gaitonde,Kleinberg and Tardos (2020), which studies maximizing and minimizing particular measuresof discord in Friedkin and Johnsen (1999) updating processes (which, mathematically, areclosely related to our games). Crucially, in all these projects, the notion of discord that is ofinterest is a particular, exogenously given measure, rather than welfare in the game. Criteriaof interest include the duration of disagreement in an updating process, average disagreementacross individuals in the network, etc. In our work, we get the objective from the preferencesof the players themselves, maximizing utilitarian welfare. Thus, while the principal componentapproach overlaps methodologically with many of these studies, the welfare-oriented questionswe ask lead to insights quite different from those in the prior literature. Indeed, a theme inthe prior literature is that global discord between loosely connected regions is most importantin slowing down agreement (DeMarzo, Vayanos and Zwiebel, 2003; Golub and Jackson, 2012).The component of disagreement that most strongly remains after a long period of updatingopinions is proportional to the second eigenvector. As we will show, our results deliver astarkly different message. The classical spectral cut component—the second eigenvector thatpartitions the network into pieces that are relatively loosely connected to each other—is the Targeting of interventions more broadly is studied, e.g., in Albert, Jeong and Barab´asi (2000); Valente(2012); Kempe, Kleinberg and Tardos (2015). Bramoull´e, Kranton and D’Amours (2014) focus on stability of equilibrium rather than targeting, but findthat eigenvectors related to the ones we study matter in public goods games. For more on various segregation measures that come up in various related contexts, see Morris (2000);DeMarzo, Vayanos and Zwiebel (2003); Currarini, Jackson and Pin (2009); Golub and Jackson (2012);Spielman and Teng (2007).
DISCORD AND HARMONY IN NETWORKS least consequential for welfare in our setting. Gaitonde, Kleinberg and Tardos (2020) hasmore subtle results showing that there is no clear ordering of how an adversary focuses efforton various spectral components of disagreement. This is natural in view of their wider classof objectives. We show that for standard welfare-oriented objectives in coordination games,there is a clear ordering, with the last eigenvector being of primary importance. Finally, ourTheorem 1 imposes less structure on the class of possible interventions than, e.g., Golub andJackson (2012) or Gaitonde, Kleinberg and Tardos (2020); we allow perturbations around anarbitrary status quo and, for small interventions, can deal with a large class of interventioncost functions. 2.
Model, Basic Facts, and Definitions
In this section, we state the model and definitions we need. We also mention some standardresults on the structure of equilibrium that serve as a foundation for our subsequent results.2.1.
Coordination game.
We consider a one-shot game played between individuals N = { , . . . , n } , with a typical individual denoted i . Each individual takes an action a i ∈ R . Weare given a favorite action f i ∈ R for each agent and a network with a weighted adjacencymatrix G ∈ R n × n . An agent’s payoff is determined by her favorite action and the actionsof her neighbors in G . We write the vector of actions as a ∈ R n , and the vector of favoriteactions as f ∈ R n . Individual i chooses a i , while f and G are exogenous.We will assume that G is row-stochastic and symmetric, and that each i meets and interactswith j with probability g ij . The payoff to an agent i of interacting with agent j is given by: v i ( a i , a j ) = − β ( a i − a j ) (cid:124) (cid:123)(cid:122) (cid:125) miscoordination − (1 − β )( a i − f i ) (cid:124) (cid:123)(cid:122) (cid:125) distance from favorite action , (1)where β ∈ [0 ,
1) determines the relative payoff weight of miscoordination with other individualsand distance from an individual’s favorite action. The expected payoff of individual i givenaction profile a is V i ( a ) = (cid:88) j g ij v i ( a i , a j ) . Utilitarian welfare is defined by V ( a ) = (cid:88) i V i ( a ) . The one-dimensional space is for simplicity: our analysis extends without much change to actions in anarbitrary Euclidean space.
ISCORD AND HARMONY IN NETWORKS 7
Nash equilibrium: A formula and a few basic properties.
Here we review a fewstandard facts about the Nash equilibrium.Fixing f and G , the first-order condition characterizing the Nash equilibrium action profileis given by a ∗ i = β (cid:88) j g ij a ∗ j + (1 − β ) f i , and this can be rewritten in vector notation to show that any Nash equilibrium action profile a ∗ must satisfy a ∗ = (1 − β )( I − β G ) − f . (2)We make the following two assumptions, the first of which has already been mentionedabove. Assumption 1.
The adjacency matrix G is row-stochastic and symmetric.Assumption 1 is implied by the description of G as meeting probabilities. It implies thatthe largest eigenvalue of G is 1 and ensures that (2) characterizes a unique, stable Nashequilibrium (Ballester, Calv´o-Armengol and Zenou, 2006; Bramoull´e, Kranton and D’Amours,2014). Indeed, we have the following fact: Fact 1.
The game has a unique Nash equilibrium, which is in pure strategies and given by(2). In this equilibrium, each a i is a (possibly different) weighted average of the f j . Proof.
It is straightforward to check that the second-order conditions for optimization hold,so the first-order condition is necessary and sufficient. Assumption 1 ensures β G has spectralradius less than 1 and so we may rewrite (2) by the Neumann series as a ∗ = (cid:32) ∞ (cid:88) t =0 (1 − β ) β t G t (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) W f . (3)Letting W be the matrix in parentheses, we see that it is a weighted average (with weights(1 − β ) β t ) of stochastic matrices G t , so W is itself stochastic. Thus, a i = W i • f , where W i • is row i of W . (cid:3) To illustrate the implications of Fact 1, consider Figure 1. There, we take a particularvector f where half of the agents (those in the bottom left) have a favorite action of +1,while those in the top right have a favorite action of −
1. We calculate equilibrium using (2)for a particular value of β . We can then see the structure of equilibrium asserted in Fact 1: DISCORD AND HARMONY IN NETWORKS Figure 1.
An illustration of equilibrium for a given network (the circle, where g ij = 0 . i and j are adjacent). The node labels are as shown,continuing clockwise. On the left we depict a particular vector f . When wedepict a node-indexed vector (such as f or a ) visually, our convention is thatentries with positive sign are indicated by making the node green, while entrieswith negative sign are indicated by making the node red. The size of eachnode corresponds to the magnitude of its entry. On the left side we write andillustrate f , while on the right side we calculate a ∗ using Fact 1 and illustrateit in the same type of diagram.everyone’s action is a weighted average of +1’s and − While the favorite actions exhibit a very stark difference between groups, the equilibriumactions attenuate the diversity of favorite actions. “Boundary” agents average togetherroughly as many +1’s as − f , illustrating the attenuation property of bestresponses. Note also from the form of (3) that the Nash equilibrium can be seen as the average of G t f for t ∈ { , , , . . . } ,which are the outcomes of DeGroot (1974) or Friedkin and Johnsen (1999) learning or myopic updating atvarious times t ; see Golub and Jackson (2012). This explains the close connection between properties ofequilibria in network games and the dynamics of certain updating/learning processes in networks; see alsoGaitonde, Kleinberg and Tardos (2020). This can be seen by noting from (3) that in a connected graph, each agent puts positive weight on all others,and thus even the most extreme agents become less extreme. The higher β is, the stronger the attenuation. ISCORD AND HARMONY IN NETWORKS 9
We will make a final, technical, assumption to simplify the statement of some results. Thisholds generically (over the choice of weights in the symmetric matrix).
Assumption 2.
All eigenvalues of G are distinct.2.3. Planner interventions and objective.
Our main interest is in understanding how, inexamples such as the one just discussed, welfare is affected by changes in the favorite actionsof various players. We investigate this question by considering a planner who can modify thevector of favorite actions: the favorite actions ˆ f are modified by some perturbation vector δ ∈ R n . Formally, the planner’s problem is given bymax δ γV ( a ∗ )s.t. f = ˆ f + δa ∗ = (1 − β )( I − β G ) − f ,c ( δ ) ≤ C. (4)The parameter γ scaling the objective is +1 or −
1, corresponding to the planner beingbenevolent or malevolent, respectively. The constraint c ( δ ) ≤ C limits the feasible interven-tions. The cost function c ( · ) is for now taken to be arbitrary. For various results, we will givespecific cost functions: for example, constraining interventions to a ball of fixed size aroundthe status quo. In our most general results in Section 4.2, we study classes of cost functionssatisfying certain assumptions, such as that interventions have (at least locally) convex costs.The number C ≥ budget .2.4. Principal components: Definitions and notation.
We introduce notation for thekey objects that play a role in our approach: the principal components of the network G .We write the spectral decomposition of G as follows: G = | | u . . . u n | | (cid:124) (cid:123)(cid:122) (cid:125) U : eigenvectors λ
0. . .0 λ n (cid:124) (cid:123)(cid:122) (cid:125) Λ : eigenvalues — ( u ) (cid:124) —...— ( u n ) (cid:124) — (cid:124) (cid:123)(cid:122) (cid:125) U (cid:124) : eigenvectors . (5)Here, U gives an orthonormal basis of eigenvectors. We adopt the convention that theeigenvectors and eigenvalues are arranged so that λ ≥ λ ≥ · · · ≥ λ n . We will refer to theeigenvector corresponding to λ (cid:96) as u (cid:96) . For any vector z ∈ R n , let z = U T z . We will refer to Figure 2.
Six eigenvectors of a circle network. The eigenvector u (cid:96) correspond-ing to the (cid:96) th -highest eigenvalue λ (cid:96) , is depicted using the same visual conventionwe introduced in Figure 1. Note that the eigenvectors higher eigenvalues (higher (cid:96) ) vary “more slowly” over the circle than those with lower eigenvalues (higher (cid:96) ). z (cid:96) as the projection of z onto the (cid:96) th principal component, or the magnitude of z in thatcomponent.Figure 2 illustrates some principal components of an example network.Throughout, we use (cid:104) y , z (cid:105) = (cid:80) i ∈N y i z i to denote the Euclidean dot product, and we let (cid:107) z (cid:107) = (cid:104) z , z (cid:105) / denote the Euclidean norm. Since the eigenvectors are normalized, theysatisfy (cid:107) z (cid:107) = 1.3. Two Simple Planner Problems and Two Distinguished PrincipalComponents
Certain principal components will play an important role in our analysis. For instance, the last principal component, the eigenvector u n corresponding to the lowest eigenvalue λ n willcorrespond to the direction in which interventions are most consequential for welfare. It willalso be helpful to contrast it with another eigenvector, u , the one that corresponds to thesecond-highest eigenvalue λ . This eigenvector, which has been important in prior studiesof segregation and homophily (DeMarzo, Vayanos and Zwiebel, 2003; Golub and Jackson,2012), turns out to describe least welfare-consequential interventions, and so it will serve asan important foil or contrast for some of our results.To show the role these eigenvectors play in optimization problems, we define a specialcase of the planner’s problem, in which the planner chooses any f on a sphere of radius 1 ISCORD AND HARMONY IN NETWORKS 11 to maximize or minimize welfare. This corresponds to holding the cross-sectional variationof favorite actions fixed, and distributing a “fixed” amount of disagreement to achieve theobjective. In this section, we dispense with δ and work with choosing the vector f directly,since the simplicity of the problem makes this change straightforward. Thus, we can simplyconsider how the planner decides to allocate disagreement in her choice of f , subject to anorm constraint.The optimization problem of interest is defined bymax f γV ( a ∗ )s.t. a ∗ = (1 − β )( I − β G ) − f (cid:107) f (cid:107) = 1 . (6) Proposition 1.
Fixing β , there is an increasing function ζ : R → R such that:(1) The optimum of (6) for the malevolent planner ( γ = −
1) is achieved by f ∗ = u n andis equal to ζ ( λ n ).(2) The optimum of (6) for the benevolent planner ( γ = 1) is achieved by f ∗ = u and isequal to ζ ( λ ). Proof.
We begin by writing the formula for equilibrium welfare in terms of an inner productexpression depending on f and G . V ∗ = − (cid:88) i (cid:32) (1 − β )( a ∗ i − f i ) + (cid:88) j g ij β ( a ∗ i − a ∗ j ) (cid:33) = −(cid:104) a ∗ , ((1 + β ) I − β G ) a ∗ (cid:105) + (1 − β ) (cid:104) f − a ∗ , f (cid:105) = − (1 − β ) (cid:2) (cid:104) f , f (cid:105) + (1 − β ) (cid:104) ( I − β G ) − f , ((1 + β ) I − β G )( I − β G ) − f − f (cid:105) (cid:3) We now switch into the basis of principal components. Recall z = U (cid:124) z . Then a = (1 − β )( I − β G ) − f if and only if a = (1 − β )( I − β Λ ) − f . Moreover, we may replace all vectors and matrices in the above expression for − W ∗ bytheir versions in the new basis. All matrices involved are diagonal, so this greatly simplifiesthe expression; indeed, as shown in Lemma A.1 in the appendix, this yields the following expression V ∗ = n (cid:88) (cid:96) =1 ζ ( λ (cid:96) ) f (cid:96) , for some increasing, nonnegative function ζ ( λ ), with ζ (1) = 0 (so that the λ term drops out,since λ = 1). Note also that because the change of basis is orthonormal, the constraint setfor f does not change.Because ζ is increasing in λ , the optimum for γ = − f ∗ = u n and is equalto ζ ( λ n ). The optimum for γ = 1 is achieved by f ∗ = u and is equal to ζ ( λ ). (cid:3) Proposition 1 shows that when f is constrained to a sphere, extremal welfare in theminimization problem depends on G only through an extreme eigenvalue, λ n or λ . Indeed,it remains true if we replace the constraint by (cid:107) f (cid:107) ≤ C , for C >
0, as long as we make theadjustment that ζ ( · ) is replaced by Cζ ( · ). Thus, ζ ( λ n ) captures the sensitivity of welfare tothe size of the invervention when the intervention is chosen optimally. In terms of the form of intervention, loading all the diversity in favorite actions onto thelast principal component is the most effective way of reducing welfare subject to an upperbound on the norm of the favorite actions. This is the first manifestation of the idea that thelast principal component is the one to which welfare is most sensitive.In contrast, the second part of the result highlights that welfare is, in a sense, least sensitiveto disagreement along the second principal component. For fixed norm of f , if we loadall disagreement onto u , welfare turns out to be the least negative—least changed from abaseline of 0 when there is no disagreement.Finally, it is worth remarking on the fact that u plays no role in the characterization.Note that in this problem u is a constant vector, because G is row-stochastic. Thus,changes in f correspond to constant shifts in favorite actions, which, by Fact 1 translateinto the same constant shifts in equilibrium actions. These shifts do not affect welfare, andso are never used by the planner. We reproduce the function ζ here for convenience, from Lemma A.1: ζ ( λ ) = − β (1 − β ) (1 − λ )[2 − β (1 + λ )](1 − βλ ) . In general, this vector gives the agents’ eigenvector centralities in the network, which measure the globalinfluence of each agent. Because of the symmetry of interactions and the fact that each agent has the sameendowment of total interaction probability, there is no heterogeneity in this, but our analysis can be extendedto settings where there is heterogeneity in interaction quantity.
ISCORD AND HARMONY IN NETWORKS 13
Figure 3.
The eigenvectors of a circle network corresponding to the n th largest and 2 nd largest eigenvalues, respectively. The former maximizes localheterogeneity and separates neighbors, while the latter finds a global cut.3.1. Local and global disagreement at the optima.
Next, we are interested in describinghow the eigenvectors identified in Proposition 1 relate to the network, and what qualitativecomparisons we can make between equilibrium behavior at the two configurations analyzed.We will show that, in a suitable sense, the last eigenvector u n is the one that maximizes localdisagreement, while the second eigenvector u maximizes global disagreement, subject to aconstraint on norm.We make a few definitions. Let D R be the uniform distribution on the set { ( i, j ) ∈N × N s.t. i (cid:54) = j } . This corresponds to drawing a random pair. Let D G be the distributionon the same set obtained by drawing the pair ( i, j ) with probability g ij /n . Definition 1.
Fix a vector z ∈ R n such that (cid:80) i ∈N z i = 0.(1) The covariance of a random pair for z is defined to be E ( i,j ) ∼D R [ z i z j ](2) The covariance of neighbors for z is defined to be E ( i,j ) ∼D G [ z i z j ].Now we use the covariance of the actions of a pair of neighbors selected at random as ameasure of local disagreement, and the covariance of the actions of a random pair of agentsas a measure of global disagreement. In each case, the more negative the number, the moredisagreement there is of the relevant kind. Proposition 2.
Let F be the set of vectors f satisfying (cid:80) i ∈N f i = 0 and (cid:107) f (cid:107) = 1. Thevalues of f in this set that maximize and minimize each quantity below are given by thefollowing table: According to the same distribution that selects partners to play the bilateral game in our model.
Statistic for eq’m actions a ∗ ( f ) maximizer minimizercovariance of neighbors u u n covariance of random pair u n u Proof.
We show each covariance result separately.The covariance of neighbors for equilibrium actions a ∗ ( f ) is given by1 n (cid:32) (cid:88) i,j ∈N g ij a ∗ i a ∗ j (cid:33) = 1 n (cid:104) a ∗ , Ga ∗ (cid:105) , because we sample an agent i uniformly at random from N , and a second agent j incident to i (the probability of an agent k being sampled is g ik ). As with Proposition 1, we can rewritethis expression in the principal component basis as n (cid:88) (cid:96) =1 η ( λ (cid:96) ) f (cid:96) , for an increasing function η ( λ ). (This is the content of Lemma A.1 in the appendix.) Becauseeach summand is increasing in λ (cid:96) , this expression achieves its minimum at f ∗ = u n and itsmaximum at f ∗ = u .The covariance of a random pair for equilibrium actions a ∗ ( f ) is given by1 n (cid:32) (cid:88) i,j ∈N a ∗ i a ∗ j − (cid:88) i ∈N ( a ∗ i ) (cid:33) , because we sample an agent i uniformly at random from N , and we sample a second agentuniformly at random from N \ { i } . Because G is row-stochastic, its Perron vector is theall-ones vector, with eigenvalue 1. Thus the projection operator onto the eigenspace associatedwith eigenvalue λ = 1 is P (1) = (cid:124) . We can then rewrite the above expression as (cid:104) a ∗ , P (1) a ∗ (cid:105) − (cid:104) a ∗ , a ∗ (cid:105) = (cid:104) a ∗ , ( P (1) − I ) a ∗ (cid:105) . The average equilibrium action is a constant times the average of f . Thus, P (1) a ∗ = . Itfollows that the covariance-minimizing a maximizes (cid:104) a , a (cid:105) . This expression can be writtenin the principal component basis as n (cid:88) (cid:96) =1 ν ( λ (cid:96) ) f (cid:96) , ISCORD AND HARMONY IN NETWORKS 15
Figure 4.
The 2 nd and n th eigenvectors of a more complex network. In thissetting, u n and u still respectively recover local and global structure. In par-ticular, u n separates neighbors by creating heterogeneity between actions (bothin sign and in magnitude), while u globally creates two nearly-homogenousgroups.for a decreasing function ν ( λ ). (See Lemma A.1 for the explicit function.) Because eachsummand is decreasing in λ (cid:96) , this expression achieves its minimum by f ∗ = u n and itsmaximum by f ∗ = u .Note that u does not optimize either function for the same reason as mentioned previously: u is a constant vector for a row-stochastic G , and the attenuation process keeps it constant,so any such intervention has no effect on welfare. (cid:3) Proposition 2 can be seen as a local-global disagreement tradeoff : the f that maximizeslocal disagreement in equilibrium also minimizes global disagreement. Interpretation and discussion.
Imagine that an adversary manipulates individualideal points in a community to reduce its members’ welfare. Naively, one might expect thatthe consequences of this adversary’s activity would be to cause global discord: to make itlikely that two randomly chosen individuals would disagree strongly. Our results show that, infact, for a given amount of cross-sectional variation in favorite points, the adversary in a senseaccomplishes the opposite. We now explore this somewhat counterintuitive phenomenon. If β is not too small, we can obtain equivalent results defining disagreement as the expectation of ( a i − a j ) under the appropriate distribution of i, j (random pair or neighbor). First, let us formalize what we said in the previous paragraph. Proposition 1 says thatthe malevolent planner chooses f = u n . Proposition 2 says that this choice causes the least global disagreement: it creates the highest possible covariance of equilibrium actions betweenrandom pairs of individuals.To understand the forces behind these results, we again consider the example network inFigure 3, showing u and u n for a circle network. As a warm-up exercise, let us discuss alimit case. Suppose that β is positive but quite small, so that, by (3) in Fact 1, a ∗ ≈ f .Then studying statistics of the favorite points f is the same as studying the correspondingstatistics of a ∗ . Let F be the set of vectors f satisfying (cid:80) i ∈N = 0 and (cid:107) f (cid:107) = 1. We willnow note that the extreme eigenvector u n achieves extreme levels of both covariance betweenneighbors and disagreement disutility. For intuition, consider Figure 3. It is clear that under f = u n , each agent has a favorite point that is the opposite of those of its neighbors. It isthen intuitive that neighbor covariance is as negative as possible: each person disagrees witha random neighbor for sure. Because the costs of disagreement are convex, it is also intuitivethat this configuration creates maximum disutility from miscoordination (relative to onewhere neighbors were closer to each other, as in u ). Indeed, by making f “vary gradually”(changing as little as possible between connected nodes), as in u , we achieve the oppositeeffect and minimize both disutility and covariance.These effects are intuitive. However, they do not exhaust the story: to understand howmuch disutility players experience, we must understand their actions in equilibrium. And,as we have already remarked in presenting Fact 1 and in Figure 1, for β not too close to 0,these involve substantial attenuation relative to favorite actions. We now turn to explainingthis aspect of the result.Using (2) and rewriting the condition in the principal component basis, we have a ∗ (cid:96) = 1 − β − βλ (cid:96) f (cid:96) . When only one principal component is represented in the favorite actions, as when f = u n or f = u , the same is true for equilibrium actions. In other words, in these cases a ∗ is a scaling of f . But the scaling is nontrivial: in best-responding to each other, the disagreementin favorite points is attenuated to a smaller disagreement in equilibrium actions. Indeed,because players best-respond to their neighbors, under f = u n they have a strong reason tobring actions closer to zero, in order to coordinate with neighbors. ISCORD AND HARMONY IN NETWORKS 17
Thus, the result involves both forces described above having to do with the structure of f alone (which are present even in the β ≈ a (which has a substantial effect only when β is far from zero); these forces may pull inopposite directions.Our result shows that attenuation is not enough to overcome the harm done by the stronglocal disagreement induced by u n . One reason for this is that even when players benefit fromattenuation by miscoordinating less with neighbors, under f = u n they also suffer by beingfarther from ideal points. It turns out that the planner maximizes their pain by making nearneighbors disagree strongly. This pattern, presented in an exteremely simple way for thecircle, generalizes to more complex networks as shown in Figure 4.On the other hand, a planner who is concerned with creating global disagreement (i.e.,minimizing the covariance of a random pair for equilibrium actions) is not at all concernedwith making neighbors disagree. For this planner, minimizing attenuation turns out to be thedominant consideration: the planner wants to make sure that as much of the “size” of initialdisagreement remains in the final equilibrium actions. It is intuitive that this is accomplishedby making neighbors agree as often as possible. Then strategic forces will not lead themto moderate their behavior by much relative to f . Of course, the requirement (imposed bydefinition of F ) that f have a positive norm, along with the normalization that the averageof f is equal to zero, requires heterogeneity across society in favorite points. The best wayfor a planner to place this heterogeneity is to put the polarization along a “cut” such asthat depicted in the vectors u of Figure 3. Here disagreement is designed to be as small aspossible across most links, and at the optimum, f (and, consequently a ∗ ) will be quite similarfor most nodes at short distances. As we have already noted, the configuration u findscohesive areas in the network and keeps their f similar, while making relatively “faraway”regions disagree with each other. Especially in networks that have good cuts, with largegroups that interact fairly little, this is natural: if the global disagreement in f is experiencedacross few links, then it makes little difference to welfare. The vector u can be seen in anetwork more interesting than the circle in Figure 4.We have spoken informally of u n tending to make neighbors take opposite signs, whereas u divides the network into cohesive regions. These notions have been extensively formalizedin the graph theory literature: see Desai and Rao (1994), Alon and Kahale (1997), andUrschel (2018) for some examples. Generalizations: General Initial Conditions and Cost Functions
We return to the general case of the planner’s problem stated in (4):max δ γV ( a ∗ )s.t. f = ˆ f + δa ∗ = (1 − β )( I − β G ) − f ,c ( δ ) ≤ C. The previous section showed that for very simple planner’s constraints, there is a simpledescription of the most welfare-consequential interventions. However, we worked under manysimplifying assumptions: ˆ f was taken to be , and the constraint on interventions was tochoose one in a ball or on a sphere.It is worthwhile to relax both restrictions: we want to consider a status quo that is moreflexible. We want to understand to what extent the intuitions extend to more general costfunctions. In this section, we address these issues.To state results, we need to make a definition measuring the similarity of various vectorsto principal components of the underlying network. For this, we use the notion of cosinesimilarity . Definition 2 (Cosine Similarity) . The cosine similarity of two nonzero vectors y and z is ρ ( y , z ) = y · z (cid:107) y (cid:107) (cid:107) z (cid:107) . A canonical interpretation of cosine similarity is that it gives the cosine of the angle betweenthe vectors y and z in the plane determined by y and z . When ρ ( y , z ) = 1 (resp., − z is a positive (resp., negative) rescaling of y . A cosine similarity of 0 implies that y is orthogonal to z .4.1. A monotonicity result.
We are now ready to characterize optimal interventions for aquadratic planner’s adjustment cost and arbitrary status quo vector.Recall the earlier finding that in the simple planner’s problem with γ = − (cid:107) f (cid:107) ≤
1, the planner focused only on the lowestprincipal component. The substance of the next result is that in a suitable sense, this findinggeneralizes: the planner intervenes more on the principal components with lower eigenvalues.
ISCORD AND HARMONY IN NETWORKS 19
Theorem 1 (Characterization of Optimal Interventions) . Suppose c ( δ ) = (cid:107) δ (cid:107) . Alsosuppose that either γ = − C is small enough that W ( a ∗ ) = is not feasible for theplanner. For generic ˆ f , the similarity between δ ∗ and principal component u (cid:96) ( G ) satisfies,for (cid:96) ≥ ρ ( δ ∗ , u (cid:96) ) = ρ ( ˆ f , u (cid:96) ) · m ( λ (cid:96) ) , where the multiplier function m is such that | m ( λ ) | is decreasing in λ . Proof.
Let f ∗ give the optimal choice of f , so that δ ∗ = f ∗ − ˆ f . Define x (cid:96) = f (cid:96) − ˆ f (cid:96) ˆ f (cid:96) . Then we can rewrite the optimization problem in the principal component basis as follows,for an increasing, negative function ζ ( λ ):max x γ (cid:88) (cid:96) ζ ( λ (cid:96) )(1 + x (cid:96) ) ˆ f (cid:96) s.t. (cid:88) (cid:96) ˆ f (cid:96) x (cid:96) ≤ C. (7)By our assumption that either γ = − µ be the Lagrange multiplier on the budget constraint,the Karush-Kuhn-Tucker necessary condition for optimization is2 γ ˆ f (cid:96) · ζ ( λ (cid:96) )(1 + x ∗ (cid:96) ) = 2ˆ f (cid:96) · µx ∗ (cid:96) . Solving for x ∗ (cid:96) , we get γζ ( λ (cid:96) ) = x ∗ (cid:96) ( µ + γζ ( λ (cid:96) )), and since the left-hand side is clearly nonzerowhenever λ (cid:96) (cid:54) = 1, it follows that the right-hand side is nonzero too, and we may write γζ ( λ (cid:96) ) µ + γζ ( λ (cid:96) ) = x ∗ (cid:96) . (8)We note a few facts about the solution. From (7) it follows that the x (cid:96) are all positive at anoptimum if γ = − γ = 1 (by the same argument as in theproof of Theorem 1 of Galeotti, Golub and Goyal (2020)). Lemma A.1 gives us that ζ isa negative, increasing function of its argument. Thus, the denominator µ + γζ ( λ (cid:96) ) in thesolution for x ∗ (cid:96) is always positive, and | x ∗ (cid:96) | is decreasing in λ (cid:96) . Note that we can accommodate any scaling of such a function by suitably adjusting C . The intuition is that at x (cid:96) = 0, the marginal returns of increasing any x (cid:96) are nonzero, while the marginalcosts are arbitrarily low. Note that x ∗ (cid:96) = (cid:107) δ ∗ (cid:107) ρ (cid:0) δ ∗ , u (cid:96) ( G ) (cid:1)(cid:13)(cid:13)(cid:13) ˆ f (cid:13)(cid:13)(cid:13) ρ (cid:16) ˆ f , u (cid:96) ( G ) (cid:17) by definition of cosine similarity, so the previous display (8) becomes γζ ( λ (cid:96) ) µ + γζ ( λ (cid:96) ) = (cid:107) δ ∗ (cid:107) ρ (cid:0) δ ∗ , u (cid:96) ( G ) (cid:1)(cid:13)(cid:13)(cid:13) ˆ f (cid:13)(cid:13)(cid:13) ρ (cid:16) ˆ f , u (cid:96) ( G ) (cid:17) . Rearranging the previous expression gives ρ (cid:0) δ ∗ , u (cid:96) ( G ) (cid:1) = ρ (cid:16) ˆ f , u (cid:96) ( G ) (cid:17) · γζ ( λ (cid:96) ) µ + γζ ( λ (cid:96) ) (cid:13)(cid:13)(cid:13) ˆ f (cid:13)(cid:13)(cid:13) (cid:107) δ ∗ (cid:107) . By our earlier remark about the monotonicity of x ∗ (cid:96) the claim of the proposition follows. (cid:3) It is worth remarking on a few features of the key expression ρ ( δ ∗ , u (cid:96) ) = ρ ( ˆ f , u (cid:96) ) · m ( λ (cid:96) ) . First, the “status quo term” ρ ( ˆ f , u (cid:96) ) reflects that the nature of interventions depends onthe status quo. For example, if the planner is benevolent and ˆ f (cid:96) is zero or nearly zero, thenthere is very little disagreement in that principal component and thus very little to remove;therefore, the planner will not devote a lot of resources to reducing disagreement in thatcomponent. The multiplier term captures that components with lower eigenvalues have abigger welfare impact, and so a planner will care more about adjusting them.Crucially, under the assumptions of the theorem, this is true whether the planner ismalevolent or benevolent. In the malevolent case, the intuition is exactly the same as that ofProposition 1: intensifying disagreement in that component has the greatest impact on thedisutility of miscoordination, and the planner will want to take advantage of that to increasethis disutility. But, under our assumptions that a benevolent planner cannot reach her blisspoint of no misscoordination, the intuition applies in the other direction, too: reducingdisagreement in the lowest-eigenvalue component is the most effective use of resources to reduce disutility. General cost functions and small budgets.
A quadratic adjustment cost is arestrictive assumption. Here we show that we can relax this assumption and obtain a version Note that the result in Proposition 1(2) was about a constraint with a fixed amount of disagreement, andthus there is no conflict between that result and this intuition.
ISCORD AND HARMONY IN NETWORKS 21 of our result for small budgets C , with a simpler characterization of the multiplier function m .We first make a few assumptions on the structure of the cost function c ( · ). Assumption 3 (Properties of the Cost Function) . The cost function c ( · ) satisfies the followingassumptions: it is twice differentiable; invariant to permutations of the entries of its argument δ ; nonnegative on its domain; has the value c ( ) = ; and has nonsingular Hessian at δ = .Making these assumptions implies by standard arguments the approximation c ( δ ) = k (cid:107) δ (cid:107) + o ( (cid:107) δ (cid:107) ) . Proposition 3 (Characterization of Small Interventions) . Suppose Assumption 3 holds.Then for generic ˆ f , the similarity between δ ∗ and principal component u (cid:96) ( G ) satisfies, for (cid:96) ≥ ρ ( δ ∗ , u (cid:96) ) = ρ ( ˆ f , u (cid:96) ) · m ( λ (cid:96) )where lim C → m ( λ (cid:96) ) m ( λ (cid:96) (cid:48) ) = ζ ( λ (cid:96) ) ζ ( λ (cid:96) (cid:48) ) . The result follows immediately from Theorem 1 by the same argument as in Galeotti,Golub and Goyal (2020, OA3.3).Because we have an explicit form for ζ in Lemma A.1, this result gives a complete descriptionof the optimal intervention. All the cosine similiarities for an orthonormal basis fully pindown the direction of the intervention, and its magnitude is found by exhausting the budget.4.3. An implication for networks with homophily.
We emphasized in Section 3.2 thatinterventions for global discord are extremely different in their form from those for welfarereasons. We can now sketch an application of this to assess whether an intervention is in factoptimal in a practical setting. Our point will be that the characterization permits some simpleinsights, building on what is known about the spectral structure of real social networks.Suppose a planner faces a network such as the one shown in Figure 5, with a certain valueof λ , say λ ≥ . Because ζ ( λ ) is small for large λ , theproposition immediately implies a bound on the cosine similarity ρ ( δ ∗ , u (cid:96) ): if m ( λ (cid:96) ) is small,then ρ ( δ ∗ , u (cid:96) ) is small irrespective of the value of ˆ f , since the ρ ( ˆ f , u (cid:96) ) factor in Proposition3 is bounded by 1. See Golub, Jackson et al. (2012) for more details.
Figure 5.
A school network from Currarini, Jackson and Pin (2009).It follows that if a purportedly optimal intervention has a substantial correlation with u , it is not in fact optimal. In other words, welfare-optimal interventions cannot havesignificant correlation with the main spectral cut of a homophilous network ( u ).5. Conclusion
There is a useful duality between the theory of network games and the study of networkstructure. A familiar pattern goes as follows. We fix a game—e.g., a canonical coordinationgame—and ask a natural economic question about it, such as what perturbations of agents’ideal points result in large welfare changes. Sometimes, a particular family of networkstatistics (in this case, the lowest eigenvalue and its associated eigenvector) emerges as animportant part of a characterization. Then we have learned both an answer to our economicquestion and a new interpretation of certain statistics—as well as a new reason to be attentiveto the statistics in some situations.In this paper, the statistics that emerge from this procedure are λ n and u n , as well asother low eigenvalues and eigenvectors. The eigenvalue λ and the eigenvector u have beenmade famous in both applied mathematics and economics by studies of spectral clustering,homophily, and opinion polarization (Spielman and Teng, 2007; DeMarzo, Vayanos andZwiebel, 2003). But we have spent less time with λ n , u n , and their friends at the low In practice, u is highly correlated with demographic covariates (in this example, race), as discussed inGolub, Jackson et al. (2012). So a substantial correlation with race would imply a substantial correlationwith u . Thus, one can refute that an intervention is optimal even without detailed network data, as long aswe know that racial homophily is strong. ISCORD AND HARMONY IN NETWORKS 23 end of the spectrum. Our analysis here has emphasized their importance for coordination,complementing the findings of some recent studies such as Bramoull´e, Kranton and D’Amours(2014); King and Allouch (2019) and Galeotti, Golub and Goyal (2020). More generally,the spectral method for analyzing welfare functionals should be useful for enriching ourunderstanding of the interplay between economic interactions and the networks in which theyare embedded.
ReferencesAcemoglu, D., V. M. Carvalho, A. Ozdaglar, and A. Tahbaz-Salehi (2012):“The Network Origins of Aggregate Fluctuations,”
Econometrica , 80, 1977–2016, https://doi.org/10.3982/ECTA9623.
Acemoglu, D., and A. Ozdaglar (2011): “Opinion dynamics and learning in socialnetworks,”
Dynamic Games and Applications , 1, 3–49.
Albert, R., H. Jeong, and A.-L. Barab´asi (2000): “Error and attack tolerance ofcomplex networks,”
Nature , 406, 378–382.
Alon, N., and N. Kahale (1997): “A Spectral Technique for Coloring Random 3-ColorableGraphs,”
SIAM Journal on Computing , 26, 1733–1748.
Angeletos, G.-M., and A. Pavan (2007): “Efficient Use of Information and Social Valueof Information,”
Econometrica , 75, 1103–1142, https://doi.org/10.1111/j.1468-0262.2007.00783.x.
Ballester, C., A. Calv´o-Armengol, and Y. Zenou (2006): “Who’s Who in Networks.Wanted: The Key Player,”
Econometrica , 74, 1403–1417, https://doi.org/10.1111/j.1468-0262.2006.00709.x.
Baqaee, D. R., and E. Farhi (2019): “The macroeconomic impact of microeconomicshocks: beyond Hulten’s Theorem,”
Econometrica , 87, 1155–1203.(2020): “Productivity and misallocation in general equilibrium,”
The QuarterlyJournal of Economics , 135, 105–163.
Bindel, D., J. Kleinberg, and S. Oren (2011): “How Bad is Forming Your OwnOpinion?” in ,57–66, 10.1109/FOCS.2011.43.
Bramoull´e, Y., R. Kranton, and M. D’Amours (2014): “Strategic Interaction andNetworks,”
American Economic Review , 104, 898–930, 10.1257/aer.104.3.898.
Calv´o-Armengol, A., J. De Mart´ı, and A. Prat (2015): “Communication andinfluence,”
Theoretical Economics , 10, 649–690.
Currarini, S., M. O. Jackson, and P. Pin (2009): “An Economic Model of Friendship:Homophily, Minorities, and Segregation,”
Econometrica , 77, 1003–1045, https://doi.org/10.3982/ECTA7528.
DeGroot, M. H. (1974): “Reaching a Consensus,”
Journal of the American StatisticalAssociation , 69, 118–121.
DeMarzo, P. M., D. Vayanos, and J. Zwiebel (2003): “Persuasion Bias, SocialInfluence, and Unidimensional Opinions,”
The Quarterly Journal of Economics , 118,909–968, 10.1162/00335530360698469.
Desai, M., and V. Rao (1994): “A Characterization of the Smallest Eigenvalue of aGraph,”
Journal of Graph Theory , 18, 181–194.
Friedkin, N. E., and E. C. Johnsen (1999): “Social Influence Networks and OpinionChange,”
Advances in Group Processes , 16, 1–29.
Gaitonde, J., J. Kleinberg, and E. Tardos (2020): “Adversarial Perturbations ofOpinion Dynamics in Networks,” in
Proceedings of the 21st ACM Conference on Economicsand Computation , EC ’20, 471–472, New York, NY, USA: Association for ComputingMachinery, 10.1145/3391403.3399490.
Galeotti, A., B. Golub, and S. Goyal (2020): “Targeting Interventions in Networks,”
Econometrica , 88, 2445–2471, https://doi.org/10.3982/ECTA16173.
Golub, B., M. O. Jackson et al. (2012): “Does homophily predict consensus times?Testing a model of network structure via a dynamic process,”
Review of Network Economics ,11, 1–31.
Golub, B., and M. O. Jackson (2012): “How Homophily Affects the Speed of Learningand Best-Response Dynamics,”
The Quarterly Journal of Economics , 127, 1287–1338,10.1093/qje/qjs021.
Golub, B., and E. Sadler (2016): “Learning in social networks,” in
The Oxford Handbookof the Economics of Networks ed. by Bramoull´e, Y., Galeotti, A., Rogers, B., and Rogers,B.: Oxford University Press, Chap. 19, 504–542.
Jackson, M. O., and Y. Zenou (2014): “Games on Networks,” in
Handbook of GameTheory ed. by Young, P., and Zamir, S.: Elsevier Science, Chap. 3, 95–163.
Kempe, D., J. Kleinberg, and E. Tardos (2015): “Maximizing the Spread of Influencethrough a Social Network,”
Theory of Computing , 11, 105–147, 10.4086/toc.2015.v011a004.
King, M., and N. Allouch (2019): “A network approach to welfare,”
BSGWorking Paper Series , ISCORD AND HARMONY IN NETWORKS 25
BSG-WP-2019-027.pdf . Morris, S. (2000): “Contagion,”
The Review of Economic Studies , 67, 57–78, 10.1111/1467-937X.00121.
Spielman, D. A., and S.-H. Teng (2007): “Spectral partitioning works: Planar graphsand finite element meshes,”
Linear Algebra and its Applications , 421, 284–305, https://doi.org/10.1016/j.laa.2006.07.020, Special issue in honor of Miroslav Fiedler.
Urschel, J. C. (2018): “Nodal decompositions of graphs,”
Linear Algebra and its Applica-tions , 539, 60–71, https://doi.org/10.1016/j.laa.2017.11.003.
U.S. House of Representatives Permanent Select Committee on Intelligence (2018): “Exposing Russia’s Effort to Sow Discord Online: The Internet Research Agencyand Advertisements,” https://intelligence.house.gov/social-media-content/ . Valente, T. W. (2012): “Network Interventions,”
Science , 337, 49–53, . Appendix A. Functions Used in Spectral Forms of Objectives
Lemma A.1.
The following functions give the welfare, covariance of neighbors, and covarianceof a random pair of agents in the principal component basis.(1) Welfare is given by n (cid:88) (cid:96) =1 ζ ( λ (cid:96) ) f (cid:96) , where ζ ( λ ) = − β (1 − β ) (1 − λ )[2 − β (1 + λ )](1 − βλ ) . (2) Covariance of neighbors is given by n (cid:88) (cid:96) =1 η ( λ (cid:96) ) f (cid:96) , where η ( λ ) = (1 − β ) λ (1 − βλ ) n . (3) Covariance of a random pair is given by n (cid:88) (cid:96) =1 ν ( λ (cid:96) ) f (cid:96) , where ν ( λ ) = − (1 − β ) (1 − βλ ) n . Proof.
The welfare function is given by V ∗ = − (1 − β ) (cid:2) (cid:104) f , f (cid:105) + (1 − β ) (cid:104) ( I − β G ) − f , ((1 + β ) I − β G )( I − β G ) − f − f (cid:105) (cid:3) . The covariance of neighbors is given by1 n (cid:104) a ∗ , Ga ∗ (cid:105) = 1 n (cid:104) (1 − β )( I − β G ) − f , (1 − β ) G ( I − β G ) − f (cid:105) . The covariance of a random pair is given by − n (cid:104) a ∗ , a ∗ (cid:105) = − n (cid:104) (1 − β )( I − β G ) − f , (1 − β )( I − β G ) − f (cid:105) . The ζ , η , and ν functions are then immediate by calculation.functions are then immediate by calculation.