Measuring income inequality in social networks

Stark, Oded; Bielawski, Jakub; Falniowski, Fryderyk Article — Published Version Measuring income inequality in social networks The Journal of Economic Inequality Provided in Cooperation with: Springer Nature Suggested Citation: Stark, Oded; Bielawski, Jakub; Falniowski, Fryderyk (2023) : Measuring income inequality in social networks, The Journal of Economic Inequality, ISSN 1573-8701, Springer US, New York, NY, Vol. 22, Iss. 2, pp. 333-356, https://doi.org/10.1007/s10888-023-09589-3 This Version is available at: https://hdl.handle.net/10419/317834 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ The Journal of Economic Inequality (2024) 22:333–356 https://doi.org/10.1007/s10888-023-09589-3 ORIGINAL RESEARCH Measuring income inequality in social networks Oded Stark1,2 ·Jakub Bielawski3·Fryderyk Falniowski3 Received: 21 March 2022 / Accepted: 20 July 2023 © The Author(s) 2023 Abstract We present a new index for measuring income inequality in networks. The index is based on income comparisons made by the members of a network who are linked with each other by direct social connections. To model the comparisons, we compose a measure of relative deprivation for networks. We base our new index on this measure. The index takes the form of a ratio: the network’s aggregate level of relative deprivation divided by the aggregate level of the relative deprivation of a hypothetical network in which one member of the network receives all the income, and it is with this member that the other members of the network compare their incomes. We discuss the merits of this representation. We inquire how changes in the composition of a network affect the index. In addition, we show how the index accommodates specific network characteristics. Keywords: Income inequality in networks; Relative deprivation in networks; An index of income inequality in networks; Compositional changes of networks JEL classification: D31; D63; I31; L14 1. Introduction Because by its very nature a network is a social architecture that is different from a population at large, the usual way in which income inequality is measured in a population cannot seamlessly be applied to networks. This is not all that surprising because the protocols of income comparisons differ. Whereas in a population a standard measure of income inequality such as the Gini coefficient counts all the income comparisons between pairs (an approach that can be supported because every member is “connected” with every other member), in a network there is no such uniformity: the group of people with whom an We are indebted to Matthew O. Jackson for kind words and illuminating comments. We gained considerably from insights provided by several reviewers. Oded Stark [email protected] 1University of Bonn, Bonn, Germany 2University of Warsaw, Warsaw, Poland 3Krakow University of Economics, Krakow, Poland / Published online: 24 October 2023 Oded Stark, Jakub Bielawski, and Fryderyk Falniowski individual is connected, or with whom he compares his income, is specific to that individual and, typically, this group is smaller than the population at large. Individuals have comparison groups, consisting of other individuals with whom they compare their incomes, and those comparators have their own comparison groups, and so on; the loops need not close or overlap. This chain of comparisons is at the heart of the structure of a network. When people change their jobs and the places in which they live, the social matrices of their colleagues, acquaintances, and friends change, as does the set of their links. The structure of a network affects the choices of its members. This influence has been noted, for example, by Calvo-Armengol and Jackson (2004) in an employment model, and by Tsvetkova et al. (2018) in an experimental study. Melamed et al. (2022)haveshownthat wealth inequality induces individuals to form new connections, resulting in a network structural change. A recurrent theme in the existing network literature is that individuals forge and discard links so as to improve their overall position (consult, for example, Goyal and Vega-Redondo, 2007). The position of an individual in a network and the architecture of the network influence the individual’s perception of inequality. The study of networks is guided by the observation that, as underscored in a large literature on social networks, individuals typically interact with other individuals in small groups (Ghiglino and Goyal, 2010; Jackson, 2010;Goyal,2011; Immorlica et al., 2017; Jackson et al., 2020; and others). This characteristic has long been emphasized in studies of human evolution, a context in which the “Dunbar’s number” (Dunbar, 1992)comesto mind: there is a cognitive limit to the number of people with whom an individual can maintain stable social relationships and, presumably, bring into his orbit of comparisons. That individuals compare their incomes only with their network neighbors can have critical implications for their behavior (see Calvo-Armengol and Jackson, 2004; Melamed et al., 2022). The network neighbors of an individual are the members of the network with whom the individual is connected. Several indices of inequality already feature in studies of social networks. These indices are of two types: those that are based on differences in social connections (Horvath and Zhang, 2018), and those that are based on income (payoff) comparisons. Among the indices belonging to the second, better populated, type are the ratio of the highest income to the lowest income (Gagnon and Goyal, 2017); Atkinson’s class of inequality measures (Ambrus and Elliott, 2021); and second-order stochastic dominance of income (consumption) distributions (Bourles et al., 2017). Arguably, the most popular index of income inequality featuring in studies of social networks is the Gini coefficient (Cavalcanti and Giannitsarou, 2017; Tsvetkova et al., 2018; Plotnikova and Ulceluse, 2022;andToth et al., 2021). The existing indices of inequality in networks do not utilize the complete data of a network’s income distribution and of the same network’s architecture of connections, although both these factors impact on the level of the income inequality experienced by members of the network. For example, consider an individual who is connected (compares his income) with (the income of) a much richer individual. If the first individual severs this link, then, naturally, he will experience a considerably lower level of income inequality. The same argument applies when his income increases measurably while the incomes of others do not. Therefore, both the network architecture and income distribution matter to an individual. A proper index of income inequality should cater for these dependencies. In this paper we construct an index of income inequality that is tailor-made for networks. We base this index on a measure of relative deprivation. We say that an individual is relatively deprived when his income is lower than the incomes of his comparators (these are the individuals with whose incomes the individual compares his own income). We compose 334 Measuring income inequality in social networks a measure of relative deprivation for networks. Following Stark et al. (2017), we show that the index in Yitzhaki (1979), which is based on Atkinson’s index (Atkinson, 1970), can serve as a measure of relative deprivation adjusted to networks. We aggregate the levels of relative deprivation experienced by the members of a network to obtain the network’s aggregate level of relative deprivation, which we refer to as total relative deprivation, TRD.Weshow that a fitting index of income inequality in networks is the following ratio: TRD divided by a “variant” of TRD calculated for a network in which one person with whose income the other members of the network compare their incomes receives all the income of the network. We inquire how our new index “behaves” in a dynamic setting, that is, we consider situations in which, while the incomes of the individuals in a network are held constant, the individuals are repositioned in the network. We formulate conditions under which, when a new link is added, the value of the index decreases / increases, and conditions under which, when an existing link is severed, the value of the index decreases / increases. These conditions have the nice feature that they yield an association between a decrease of the level of the index of income inequality in networks and an increase in each of three network characteristics: the average degree of a network, the network density, and the clustering coefficient. (In Section 2we introduce and explain these terms.) In addition, we show that holding constant the incomes of the individuals, the index attains its maximal value when the network architecture is such that the individuals who are relatively deprived form an independent set. The “behavior” of the index in a changing environment is of natural interest in studies of networks, both because quite often the structure of the links (relations, comparisons) in a social network is dynamic rather than static, and because of a preference to reduce inequality while keeping incomes intact. We proceed as follows. In Section 2, we present notation and characterizations of networks. In Section 3, we compose a measure of relative deprivation for networks, and we highlight properties that differentiate this measure from the standard measure of relative deprivation. In addition, we introduce a measure of total relative deprivation, TRD.Drawing on TRD, in Section 4we construct our index of income inequality in networks, and we list properties of this index. In Section 5, we investigate how the index changes when the architecture of the network changes. In Section 6, we conclude. 2. A network building blocks: Notation, and characterizations In this section we present several terms that will be needed for our study of networks. Throughout this paper we use a number of terms that feature in the standard vocabulary of network research. For ease of reference, we briefly explain these terms. (For definitions, usages, and examples of terms, Jackson (2010), among others, can gainfully be consulted.) Drawing on the standard terminology of network topology, we define a network as a pair N=(V , E) such that V={1,2, ..., n},wheren≥3, is a set of nodes that corresponds to a set of individuals, and E⊂V×Vis a set of links. We denote by ij the link that connects nodes i∈Vand j∈V.1We make two assumptions. Assumption 1. The network N=(V , E) has no self-loops (no links that connect a node with itself), that is, there are no ii or jj links. 1The networks studied in this paper are undirected networks, meaning networks in which the connections between the individuals are mutual. However, our results can easily be adapted to directed networks, meaning networks in which the connections between the individuals are directional: the connections are not mutual. 335 Oded Stark, Jakub Bielawski, and Fryderyk Falniowski Assumption 2. Between any two nodes of N=(V , E) there is at most one link.2 A path between two nodes i,j ∈Vis a sequence of links i1i2,i 2i3, ..., ik−2ik−1,i k−1ik∈E, for a natural number k>1, such that i1=iand ik=j.Bydist(i,j) we denote the distance between nodes i,j ∈V, meaning the length of the shortest path connecting iand j.When no path connects iand j,wesetdist(i,j) =∞.AsetW⊂Vof nodes is an independent set if there are no links between the nodes of W. We refer to N=V,Eas a sub-network of N=(V , E) when V⊂Vand E⊂Eis the set of links between the nodes of V. In this case, we write N⊂N. A sub-network C=(VC,E C),C⊂N, is a connected component of N=(V , E) when the following two properties hold. 1. Between any two nodes of VCthere is a path that consists of EClinks. 2. There is no other sub-network N⊂Nthat satisfies the preceding property, and such that C⊂N. AnetworkN=(V , E) can be represented as the sum of its disjoint connected components such that there are no links between nodes from different components. In the specific case in which a network is complete, there is only one connected component comprising the entire network; and in another specific case of an empty network (a network without links), each node forms a connected component. We consider a network N=(V , E) of n≥3 individuals such that the income of individual i∈Vis xi≥0, and such that there is at least one individual whose income is strictly positive. We denote the income distribution of Vby X,thatis,X=(x1,x 2, ..., xn). To simplify notation, we occasionally refer to N=(V , E) as N. The structure of (N,X) is such that there is a link ij ∈Eif individual icompares his income with the income of individual j. For each node i∈V,letNi≡{j∈V:ij ∈E}be the comparison group of individual i, that is, the nodes with which individual ihas a link. These nodes constitute the set of i’s neighbors in the network. (In line with the conventional terminology of networks, here too the term “neighbors” means the set of nodes with which individual iis connected, so the term means the individual’s comparison group.) We denote by di=|Ni|thedegreeof a node i, that is, the number of links that ihas with other nodes, and by V∗≡{i∈V:di>0} the set of nodes that are connected with at least one other node. We introduce and explain three features of networks, or network characteristics: the average degree of a network; the network density; and the clustering coefficient. Let N=(V , E) be a network of n≥3 nodes. The average degree of a network: this is the sum of the degrees of the nodes divided by the number of the nodes. The network density: this is the number of connections in Ndivided by the number of connections in a complete network of nnodes. Because in a network the sum of the degrees is equal to twice the number of the connections, we can express the network density as |E| n(n −1) 2 = 2 n|E| n−1= 1 n i∈V di n−1, that is, the network density is the average degree of a network divided by n−1. 2When referring to directed networks, we consider a two-way link as one link. 336 Measuring income inequality in social networks The clustering coefficient: this is the fraction of fully connected triples of nodes out of the potential of triples in which at least two links are present. Formally, the clustering coefficient is  i{jk ∈E:j= k,j ∈Ni,k∈Ni}  i{jk ∈V×V:j= k,j ∈Ni,k∈Ni} if ∃i∈V:di≥2, and 0 otherwise, where Niis the comparison group of individual i. Put heuristically, the clustering coefficient indicates how often two comparators of an individual are also comparators of each other. 3. Relative deprivation in networks In this section we formulate the relative deprivation of an individual in a network, and the aggregate or the total relative deprivation in a network. 3.1 A measure of the relative deprivation in a network Our measure of relative deprivation in networks considers the neighborhood of an individual as his comparison group, and computes the individual’s average income shortfall which results from comparisons of his income with the incomes of his neighbors. We show how the position that an individual occupies in a network affects the level of his relative deprivation. In network Nof nindividuals, n≥3, let the income distribution be X=(x1, ..., xn);we refer to the pair (N,X) as an “income network:” that is, an income network is a network associated with an income distribution. We construct a measure of relative deprivation of a member of an income network by drawing on the axioms presented in Stark et al. (2017). We adjust these axioms to fit the setting of a network. Observation 1. In income network (N,X), the relative deprivation of individual iwhose comparison group is Niis given by RD(i, Ni)=⎧ ⎪ ⎨ ⎪ ⎩ 1 di  j∈Ni max{xj−xi,0}if di>0, 0ifdi=0. (1) Observation 1follows from the fact that the individual’s neighbors in the network constitute the individual’s comparison group. The set of axioms presented in Stark et al. (2017) can be adapted to obtain (1). Application of the adapted set yields a one-parameter family of measures of relative deprivation of an individual. The measure RD(i, Ni)in (1) can be obtained by supplementing the Stark et al. (2017) axioms with the Transfer Property, which says that a top-down or a bottom-up transfer of some positive income between two individuals (comparators of individual i) who are wealthier than individual idoes not change RD(i, Ni), provided that following the transfer, the transferor does not become poorer than individual i. Because in making income comparisons the relative deprivation of an individual is a local measure, we see that when calculating (1), only the income distribution and the number of neighbors of the individual (the size of his comparison group) matter. The measure of 337 Oded Stark, Jakub Bielawski, and Fryderyk Falniowski relative deprivation in (1) informs us that individual icompares his income with the incomes of the members of his comparison group Ni, and that when individual ihas no connections, he does not experience relative deprivation. In comparison with a standard formulation of relative deprivation, (1) is more intricate: the structure of the network matters as does the position of the individual (whose level of relative deprivation is measured) in the network. We illustrate this difference with the help of an example. Example 1. We consider a network of four individuals, as displayed in Figure 1.The numbers in the nodes are the names of the individuals. Figure 1. A network of four individuals. We endow the individuals with the following income distribution X=(x1,x 2,x 3,x 4): x1=2, x2=1, x3=2, and x4=8. Then only individuals 2 and 3 experience relative deprivation. In spite of individual 2 having the lowest income, his relative deprivation is not the highest. The level of relative deprivation of individual 3, which is RD(3,{1,2,4})=1 3(8−2)=2, is higher than the level of relative deprivation of individual 2, which is RD(2,{1,3})=1 2(2·(2−1))=1. 3.2 Total relative deprivation, TRD, in networks In order to introduce the aggregate or the total relative deprivation, TRD, in a network, we formulate the following axiom. Additive Decomposition axiom.Let( N,X) be an income network where N=(V , E). Equipping the network Nwith an additional node l(and its connections), we obtain a new network ˆ N=ˆ V, ˆ E,where ˆ V=VY{l}and ˆ E=EY j∈Nl jl.Then TRD ˆ N,XY{xl}=TRD ˆ N\{l},XY{xl}+RD(l, Nl).3 Drawing on this axiom, we see that computing the TRD of a network equipped with an additional node requires updating the levels of relative deprivation of the neighbors of the node, and of adding the level of relative deprivation of the node. 3The TRD ˆ N\{l},XY{xl}term does not include the level of relative deprivation of individual l, while it includes the impact that the income of this individual has on the levels of relative deprivation of his neighbors in network ˆ N. 338 Measuring income inequality in social networks Observation 2. Let the Additive Decomposition axiom hold. Then, the total relative deprivation of income network (N,X) is equal to the sum of the levels of relative deprivation of the members of the network: TRD(N,X)= n  i=1 RD(i, Ni). In addition, given the measure of relative deprivation in (1), TRD(N,X)= i∈V∗⎛ ⎝ 1 di  j∈Ni max{xj−xi,0}⎞ ⎠.(2) It is easy to see that Observation 2follows from an iterative application of the Additive Decomposition axiom and of Observation 1. Because the individuals represented by isolated nodes are not relatively deprived, such nodes do not affect TRD(N,X). Therefore, the summation in (2) is applied to the nodes that have at least one link. We next seek to identify a network architecture and an income distribution that cause TRD in (2) to reach its maximal level. Presumably, what causes TRD to be that high is of concern to a social planner who may want to know what to avoid most. Let nbe the income network of nindividuals, n≥3, that results in the highest level of TRD(N,X). To reach this level, the levels of relative deprivation of essentially all the members of Nhave to be maximal. From formula (1) of relative deprivation, we infer that such a result could be obtained by maximizing the income shortfall of every individual, except for one individual who is the richest, and by redesigning the network architecture in such a way that the individuals who experience income shortfalls are connected only with the richest individual. Consequently, income network nhas to be a star network of n individuals, n≥3, where the individual at the center of the star receives all the income of the network, y,thatis,y= n  i=1 xi, and every other individual is left with no income.4Among the income networks that have the same number of nodes and the same aggregate income as does n, the total relative deprivation attains its maximal value at n.Using(2), we express the total relative deprivation of nas TRD(n)= j∈V\{c} 1 1(y−0)=(n −1)y, (3) where cdenotes the individual who is represented by the node at the center of the star. 4. An index of income inequality in networks We now have in hand the components needed to construct our index of income inequality in networks. This index, (N,X), of an income network (N,X)ofnindividuals, n≥3, is a 4A star network is a network in which one node (the central node) has connections to all other nodes, while every other node has a connection only to the central node. 339 Oded Stark, Jakub Bielawski, and Fryderyk Falniowski ratio: the TRD of the actual income network divided by the TRD of the hypothetical income network n: (N,X)≡TRD(N,X) TRD(n)= n  i=1 RD(i, Ni) (n −1)y =  i∈V∗⎛ ⎝ 1 di  j∈Ni max{xj−xi,0}⎞ ⎠ (n −1) n  i=1 xi .(4) Proposition 1.Let( N,X) be an income network of nindividuals, n≥3. Then: 1. 0 ≤(N,X)≤1. 2. (N,X)=0 if and only if in every connected component of the network the component’s income is divided equally between the component’s members. 3. (N,X)=1 if and only if - a relabeling of the nodes notwithstanding - the income network (N,X) is identical to the income network n. 4. In a complete income network (NC,X), the Gini coefficient and the index (NC,X) can be obtained from each other by means of rescaling. Proof. The proof is in the Appendix. The index (4) of income inequality in networks obtains the minimal value if and only if the members of the given connected component have the same income, and it obtains the maximal value if and only if the network is a star network of any number of individuals where only the individual at the center of the star receives income. We present an example which shows how the index (4) can be used to rank by their levels of inequality two networks that have the same number of members and the same income distributions, yet differ in their architectures. Example 2. Figures 2A and 2B portray networks N1and N2, respectively, and the numbers in the nodes are the names of the individuals. Figure 2A. Network N1.Figure 2B. Network N2. We let the two networks have the same income distribution X=(x1, ..., x5)where x1=x3=1, individual 2 has income 9, that is, x2=9, and x4=x5=2. Let Nj idenote the 340 Measuring income inequality in social networks Figure 3. The consequences of creating the link ij . The empty circle marks the level RD(i, Ni); the solid circle marks the level RD(i, Ni)+di+1 dj+1RD(j, Nj); and “Excess income” is the difference xj−xi. Looking at (10) we see that after severing a link, the index (N,X) decreases if and only if the sum of the changes of the levels of relative deprivation of individuals iand j, that is, the numerator of (10), is negative. Whether or not the index decreases depends on the number of the comparators that individuals iand jhave. For instance, when individual iis the only comparator of individual j, severing the link ij has no effect on the level of relative deprivation of individual j. In this case, either the index decreases unconditionally (as per case 1 below), or the decrease is conditioned by the change of the relative deprivation of individual i(as per case 2 below). 1. If individuals iand jconstitute each other’s sole comparators (di=dj=1), then by (7), RD(i, Ni\{j})−RD(i, Ni)=−(xj−xi), and by (8), RD(j, Nj\{i})−RD(j, Nj)=0, so that the numerator in (10) reduces to xi−xj≤0, in which case the severance of ij decreases the index (N,X). 2. If individual ihas several comparators while he is the only comparator of individual j(di>1, dj=1), then the numerator of (10) reduces to RD(i, Ni\{j})−RD(i, Ni)=1 di−1 RD(i, Ni)−(xj−xi), in which case the severance of ij decreases the index (N,X) if and only if as a result of the severance the relative deprivation of individual idecreases. 5.2 Network characteristics In this subsection we look at networks through the lens of network characteristics. As already noted, we refer to the average degree of a network, to the network density, and to the clustering coefficient. Claim 3. Let the number of individuals and their incomes in network N=(V , E) be fixed. Assume that a connection between individuals i, j ∈V, with xi≤xj, is added only when the difference in the incomes of these individuals does not exceed the weighted sum of their levels of relative deprivation, that is, only when xj−xi≤RD(i, Ni)+di+1 dj+1RD(j, Nj). (11) Then, because the average degree or the network density or the clustering coefficient of N increases, the level of the index of income inequality in Ndecreases. Proof. The proof is in the Appendix. Claim 3builds on the results of Subsection 5.1 in that in a network with a fixed number of individuals, the only way to increase a given characteristic is through the addition of 347 Oded Stark, Jakub Bielawski, and Fryderyk Falniowski new links. Condition (11) ensures that the addition of a link that connects individuals iand jdecreases the level of the index of income inequality in networks. Consequently, when a connection is created between two individuals with a difference in incomes that is small relative to the weighted sum of their levels of relative deprivation, our index decreases, while the three network characteristics (weakly) increase. The next example illustrates why condition (11) is essential for obtaining Claim 3. Example 4.LetN1=(V , E1)be a network of three individuals, V={1,2,3}, such that there are links between individuals 1 and 3 and between individuals 2 and 3, but initially there is no link between individuals 1 and 2. The income distribution is X=(x1,x 2,x 3): x1=10, x2=4, and x3=4. Then (N1,X)=RD(1,{3})+RD(2,{3})+RD(3,{1,2}) 2(10 +4+4) =0+0+1 2[(10 −4)+(4−4)] 36 =1 12. The clustering coefficient of N1is zero. We denote by N2=(V , E2)the network obtained from N1=(V , E1)by adding a link between individuals 1 and 2. Because x1−x2=6>0+2 20=RD(2,{3})+d2+1 d1+1RD(1,{3}), condition (11) is violated. And because N2is a complete network of three individuals, the clustering coefficient of N2is equal to 1. At the same time, (N2,X)=RD(1,{2,3})+RD(2,{1,3})+RD(3,{1,2}) 2(10 +4+4)= 1 2(0+6+6) 36 =1 6. Thus, while when transitioning from N1to N2the clustering coefficient increases, the level of the index of income inequality increases - it is higher in N2than in N1. Corollary 1. Let the number of individuals and their incomes in a network N=(V , E) be fixed. If the centrality of the position of the wealthiest individual (this individual’s degree centrality) increases, then so does the level of the index of income inequality in networks. This corollary follows from (11): by connecting with the wealthiest individual, we always obtain (11) but with the inequality sign reversed. 5.3 Configurations that maximize the index of income inequality in networks We next ask which configurations of the individuals in a network maximize the index of income inequality in networks, assuming that while the individuals’ incomes are fixed, the individuals can be repositioned. Claim 4. Let the incomes of the individuals in network Nbe fixed. Then the configuration (N,X) maximizes the index of income inequality in the network if and only if: (i) any individual other than the richest individual in (N,X) has at least one connection; (ii) the set of individuals other than the richest in (N,X) is an independent set. Proof. The proof is in the Appendix. 348 Measuring income inequality in social networks Claim 4informs us that when a configuration of individuals maximizes the index of income inequality in a network, then the individuals other than the richest do not have connections between them. These individuals can be connected only with the richest individual(s). When several individuals are the richest, then there are no restrictions regarding the configuration of the links between them. In particular, a configuration that maximizes the index of income inequality in networks can be a star network in which a single individual (the richest) occupies the center of the star, or a bipartite network with several unconnected equally rich individuals.7 6. Conclusion Having introduced measures of relative deprivation and total (aggregate) relative deprivation in networks, we constructed an index of income inequality in networks, tailor-made for this type of social architecture. We presented features of the index: we identified when the index attains a maximal value, and when it attains a minimal value; we studied the sensitivity of the index to transfers of income between members of the network; and we remarked that in a complete network the Gini coefficient can be derived from the index of income inequality in networks. We showed that the architecture of the network has a decisive impact on the total relative deprivation of the network and, consequently, on the level of income inequality, and we also showed how changes in the architecture of the network affect the income inequality in the network. We identified conditions under which the index decreases or increases when a new link is added or an existing link is severed. For example, we provided a condition under which an increase in the network density, that is to say, an increase in the number of links in the network, decreases the level of inequality in the network. We studied which income distributions with a fixed network architecture and which network architectures with a fixed income distribution result in the highest level of the index of income inequality in networks. And we also provided a sufficient condition for the Pigou-Dalton transfer principle to hold in a network setting. Our approach provides a nuanced insight into the measurement of income inequality, and hints at a procedure by which a social planner could influence the extent of income inequality, not only by redistributing incomes, but also by reshaping the architecture of networks and their underlying social relations. Our inquiry in general, and our study of how changing the architecture of a network affects the level of income inequality of the network in particular, help to chart a path for studies of the dynamics of social networks. Appendix. Proofs of Proposition 1,andofClaims1through 4 For ease of reference, the proposition and claims proved in this appendix are replicated. Proposition 1.Let( N,X) be an income network of nindividuals, n≥3. Then: 1. 0 ≤(N,X)≤1. 2. (N,X)=0 if and only if in every connected component of the network the component’s income is divided equally between the component’s members. 7In a bipartite network the nodes are divided into two disjoint groups. Then this network has connections between the groups but there are no connections within the groups. 349 Oded Stark, Jakub Bielawski, and Fryderyk Falniowski 3. (N,X)=1 if and only if - a relabeling of the nodes notwithstanding - the income network (N,X) is identical to the income network n. 4. In a complete income network (NC,X), the Gini coefficient and the index (NC,X) can be obtained from each other by means of rescaling. Proof. For ease of exposition, we introduce the notation xmin =min i∈Vxi,andxmax =max i∈V xi. Part 1: 0 ≤(N,X)≤1. That (N,X)≥0 holds is obvious because the numerator and the denominator of (N,X) are both non-negative. In order to show that (N,X)≤1, we look at the difference between the denominator (D) and the numerator (N)of(4): D−N=(n −1) n  i=1 xi− i∈V∗⎛ ⎝ 1 di  j∈Ni max{xj−xi,0}⎞ ⎠ =(n −1) n  i=1 xi− i∈V∗ xi<xmax ⎛ ⎜ ⎝ 1 di  j∈Ni xj>xi (xj−xi)⎞ ⎟ ⎠ =(n −1) n  i=1 xi− i∈V∗ xi<xmax ⎛ ⎜ ⎝ 1 di  j∈Ni xj>xi xj⎞ ⎟ ⎠+ i∈V∗ xi<xmax ⎛ ⎜ ⎝ 1 di  j∈Ni xj>xi xi⎞ ⎟ ⎠. (12) From the fact that  i∈V∗ xi<xmax ⎛ ⎜ ⎝ 1 di  j∈Ni xj>xi xj⎞ ⎟ ⎠= i∈V∗ xi>xmin ⎛ ⎜ ⎝xi j∈Ni xj<xi 1 dj⎞ ⎟ ⎠, it follows from (12)that D−N=(n −1) n  i=1 xi− i∈V∗ xi>xmin ⎛ ⎜ ⎝xi j∈Ni xj<xi 1 dj⎞ ⎟ ⎠+ i∈V∗ xi<xmax ⎛ ⎜ ⎝ 1 di  j∈Ni xj>xi xi⎞ ⎟ ⎠ =(n −1) i∈V\V∗ xi+(n −1) i∈V∗ xi=xmin xi+ i∈V∗ xi>xmin xi⎛ ⎜ ⎝(n −1)− j∈Ni xj<xi 1 dj⎞ ⎟ ⎠ + i∈V∗ xi<xmax ⎛ ⎜ ⎝ 1 di  j∈Ni xj>xi xi⎞ ⎟ ⎠. (13) We note that if a node iis not isolated, then di≥1. Therefore,  j∈Ni xj<xi 1 dj ≤n−1. Combining this observation with our initial assumption that incomes are non-negative, we conclude from (13)thatD−N≥0. Thus, 0 ≤(N,X)≤1. Part 2: (N,X)=0 if and only if in every connected component of the network the component’s income is divided equally between the component’s members. 350 Measuring income inequality in social networks We note that (N,X)=0 if and only if TRD(N,X)=0. Moreover, if xi= xjfor any i,j members of the same connected component, then TRD(N,X)>0. Thus, TRD(N,X)=0 if and only if members of the same connected component have the same income. Part 3: (N,X)=1 if and only if (N,X) is identical to n. Because (N,X)=1 if and only if D=N, it follows from (13)that (N,X)=1⇔(n −1) i∈V\V∗ xi+(n −1) i∈V∗ xi=xmin xi+ i∈V∗ xi>xmin xi⎛ ⎜ ⎝(n −1)− j∈Ni xj<xi 1 dj⎞ ⎟ ⎠ + i∈V∗ xi<xmax ⎛ ⎜ ⎝ 1 di  j∈Ni xj>xi xi⎞ ⎟ ⎠=0. Therefore, (N,X)=1 if and only if the following conditions hold. 1. If iis an isolated node, then xi=0. 2. xmin =0. 3. If xi>x min,then  j∈Ni xj<xi 1 dj =n−1. 4. If xi<x max and di>0, then xi=0. From conditions 1, 2, and 4, almost all incomes, except the highest, are equal to zero. From condition 3, if an individual has an income that is not the lowest, then he has n−1 individuals in his comparison group, all of whom are poorer than he is, and for whom he is the only comparator. Because our characterization excludes isolated nodes, this configuration is possible only if (N,X) is a star network such that the central individual receives all the income. Part 4: In a complete income network (NC,X), the Gini coefficient and the index (NC,X) can be obtained from each other by means of rescaling. We choose a representation of the Gini coefficient that can straightforwardly be applied to a complete network: G(X) = 1 n−1 n  i=1  j∈V\{i} max{xj−xi,0} n  i=1 xi .(14) From computing the level of the index of income inequality of (NC,X),weobtain (NC,X)= n  i=11 n−1 j∈V\{i} max{xj−xi,0} (n −1) n  i=1 xi .(15) 351 Oded Stark, Jakub Bielawski, and Fryderyk Falniowski Then, from (14)and(15)weget G(X) = 1 n−1 n  i=1  j∈V\{i} max{xj−xi,0} n  i=1 xi =(n −1) n  i=1  j∈V\{i} max{xj−xi,0} (n −1)2 n  i=1 xi =(n −1)(NC,X). Q.E.D. Claim 1.Lety>0 be the aggregate income of network N=(V , E),andletIbe the set of all the individuals for whom the term  j∈Ni 1 dj is maximal. We then choose any nonempty independent set J⊂I. Then, the distribution of the network’s income y,whereyis divided (in any manner) between the members of the set J, maximizes the index (N,X).In particular, if |I|=1, then (N,X) records its maximal value when this one member of Ireceives all the income of the network. Proof. Given the aggregate income of the network, the denominator of the index (N,X) in (4) does not depend on the income distribution of the network. Thus, we have analyzed the numerator of (N,X), that is, the total relative deprivation of the network. Let ybe the aggregate income of the network. From formula (2) of the total relative deprivation of an income network, we obtain TRD(N,X)= i∈V∗⎛ ⎝ 1 di  j∈Ni max{xj−xi,0}⎞ ⎠≤ i∈V∗⎛ ⎝ 1 di  j∈Ni xj⎞ ⎠= i∈V∗⎛ ⎝xi j∈Ni 1 dj⎞ ⎠ ≤ i∈V⎛ ⎝ximax ⎧ ⎨ ⎩  j∈Ni 1 dj :i∈V∗⎫ ⎬ ⎭⎞ ⎠= i∈V xi·max ⎧ ⎨ ⎩  j∈Ni 1 dj :i∈V∗⎫ ⎬ ⎭ =y·max ⎧ ⎨ ⎩  j∈Ni 1 dj :i∈V∗⎫ ⎬ ⎭ . (16) From (16), it follows that for every income distribution of income network (N,X), the level of TRD(N,X)is bounded from above by y·max ⎧ ⎨ ⎩  j∈Ni 1 dj :i∈V∗⎫ ⎬ ⎭ . Let I≡arg max ⎧ ⎨ ⎩  j∈Ni 1 dj :i∈V∗⎫ ⎬ ⎭ . We then choose a nonempty independent set J⊂I. We assume that Jhas k≥1 elements, that is, J=i∗ 1,i∗ 2, ..., i∗ k. We allocate the aggregate 352 Measuring income inequality in social networks income of the network between the members of Jso that xi∗ 1+xi∗ 2+... +xi∗ k=y. Then, all the neighbors of the individuals from J(and only these individuals) experience relative deprivation. The level of TRD of this network is TRD(N,X)= l∈J  j∈Nl 1 dj (xl−0)= l∈J⎛ ⎝xl j∈Nl 1 dj⎞ ⎠= l∈J xl⎛ ⎝ j∈Nl 1 dj⎞ ⎠=y j∈Nl 1 dj . (17) The penultimate equality in (17) follows from the feature that for the members of J(and I), all the expressions  j∈Nl 1 dj for l∈Jare equal to each other. From the definition of the set I, we obtain that the level of TRD(N,X) in (17) is equal to the upper bound from (16). Therefore, by allocating (in any manner) the aggregate income of the network to the members of the set J, we obtain the maximal levels of TRD(N,X) and (N,X). For the case in which Ihas only one element, its only nonempty independent subset is J=I. Then, the levels of TRD(N,X) and (N,X) are maximal when the aggregate income of the network is allocated to the single member of I. Q.E.D. Claim 2. Consider an income network (N,X)of nindividuals, n≥3. We assume a top-down (respectively, a bottom-up) rank-preserving transfer of income from individual ito individual k. Then, the Pigou-Dalton principle holds, that is, the level of income inequality decreases (respectively, increases) when the following inequality is satisfied:  l∈Nk xl>xk 1 dk − l∈Nk xl<xk 1 dl > (<)  j∈Ni xj>xi 1 di − j∈Ni xj<xi 1 dj . Proof. We note that because a transfer of income between individuals does not affect the denominator of (4), we analyze the numerator of (4), that is, the TRD of the network. We consider a rank-preserving transfer of ε>0 units of income from individual i to individual k. Then, only the neighbors of individual iwho are poorer than himself experience an income shortfall from comparisons of their incomes with the income of individual i. Prior to the transfer, the income shortfall of individual j∈Niwho is poorer than individual iis xi−xj, and following the transfer, the income shortfall of individual j is xi−ε−xj. Thus, the transfer decreases the relative deprivation of individual jby ε dj . At the same time, the level of relative deprivation of individual iincreases by  j∈Ni xj>xi ε di .By aggregating over individual iand his neighbors, we find that the transfer decreases the sum of the levels of relative deprivation of these individuals by  j∈Ni xj<xi ε dj − j∈Ni xj>xi ε di . By performing an analogous reasoning for individual k,weseethatthetransfer decreases the sum of the levels of relative deprivation of individual kand his neighbors by  l∈Nk xl>xk ε dk − l∈Nk xl<xk ε dl . 353 Oded Stark, Jakub Bielawski, and Fryderyk Falniowski In sum, the top-down (bottom-up) transfer decreases (increases) the TRD of the network if  j∈Ni xj<xi ε dj − j∈Ni xj>xi ε di + l∈Nk xl>xk ε dk − l∈Nk xl<xk ε dl > (<) 0. This inequality is equivalent to  l∈Nk xl>xk 1 dk − l∈Nk xl<xk 1 dl > (<)  j∈Ni xj>xi 1 di − j∈Ni xj<xi 1 dj . Q.E.D. Claim 3. Let the number of individuals and their incomes in network N=(V , E) be fixed. Assume that a connection between individuals i, j ∈V, with xi≤xj, is added only when the difference in the incomes of these individuals does not exceed the weighted sum of their levels of relative deprivation, that is, only when xj−xi≤RD(i, Ni)+di+1 dj+1RD(j, Nj). (18) Then, because the average degree or the network density or the clustering coefficient of N increases, the level of the index of income inequality in Ndecreases. Proof. Because the number of individuals in N=(V , E) is fixed, the following observations hold. (i) The only method to increase the average degree of Nor the network density is to add new connections. (ii) The clustering coefficient increases when the number of complete subnetworks of three individuals increases, and this can be obtained only by adding new connections. (iii) From the results of Subsection 5.1 (recalling Figure 3), we infer from condition (18) that the addition of a new connection to the network N=(V , E) decreases the level of the index of income inequality in networks. Q.E.D. Claim 4. Let the incomes of the individuals in network Nbe fixed. Then the configuration (N,X) maximizes the index of income inequality in the network if and only if: (i) any individual other than the richest individual in (N,X) has at least one connection; (ii) the set of individuals other than the richest in (N,X) is an independent set. Proof. We prove the claim by contradiction. First, we assume that in the configuration that maximizes the index of income inequality in networks there is one individual who is not the richest and who is at an isolated node. Then, by connecting this individual with one of the richest individuals, we increase the level of relative deprivation of the former individual (the levels of relative deprivation of the other individuals remain unchanged). Therefore, following this operation, the level of the index of income inequality in networks increases, which contradicts the assumption that this configuration maximizes the index. 354 Measuring income inequality in social networks Second, we assume that in the configuration that maximizes the index of income inequality in networks the set of individuals other than the richest is not an independent set. Thus, we can consider the poorest individual iwho has a link to another individual who is not the richest. We denote the maximal income in this network by xmax.Then RD(i, Ni)=1 di  k∈Ni max{xk−xi,0}<1 di  k∈Ni max{xmax −xi,0}=xmax −xi.(19) From (19), we infer that deleting all the existing links of individual iand connecting him only with one of the richest individuals increases his level of relative deprivation. Moreover, following this operation, the levels of relative deprivation of all the comparators of individual iwill also (weakly) increase (the levels of relative deprivation of the other individuals will not change). As a result, the level of the index of income inequality in networks increases, which contradicts the assumption that the configuration presented at the beginning of this paragraph maximizes the index. Q.E.D. Funding Open Access funding enabled and organized by Projekt DEAL. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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