Inducing Efficient and Equitable Professional Networks Through Link Recommendations

Individuals’ professional networks play a crucial role in their labor market outcomes. They facilitate job placement and lead to higher wages, increased productivity and greater performance [24, 8, 28, 22, 1, 27, 4, 13, 2, 3, 7, 9]. However, they may also exacerbate employment or wage inequality, lead to differential job search costs, yield uneven gains to new technology or otherwise entrench or increase economic or social stratification [15, 21, 30, 5]. Indeed, recent empirical work has demonstrated that minority individuals’ connection requests on platforms are accepted at lower rates [18] and that widespread disconnection across social groups is partially attributable to lack of exposure [12].

Digital professional networking platforms crucially influence network formation by providing link recommendations between individuals. Indeed, a large fraction of all connections are formed through recommendations [31]. Thus, the choices that platforms make with respect to their recommendation systems have far-reaching implications for their millions of users.

In our work, we explore the role a platform may play in increasing social welfare and ameliorating inequality through link recommendations. We show link recommendation policies that favor cross-group connections decrease inequality and improve efficiency relative to exogenous levels. The additional welfare comes from a better use of opportunities on the platform, which is arguably good for the platform itself as well, since a primary consideration for professional networking platforms is that users find opportunities through the platform (and thus have a reason to maintain their accounts). Thus, an increase in welfare may be aligned with the incentives of the platform acting in its long term interests.

We also characterize how, if the platform does not intervene on the network formation process, inequality can increase relative to starting levels. Importantly, these results hold even if individuals do not have exogenous preferences for connecting with members of their own group. In other words, inequality can be self-propagating. This is because inequality can increase as a result of the differential leverage between individuals who come to the network with a starting level of privilege – and therefore can offer greater benefits to potential connections – and those that do not. This confirms and complements prior work, where in-group preferences are assumed to be exogenous.

At a high level, our model consists of a population of individuals, each belonging to one of two groups. Each individual in a group receives a random number of opportunities. Opportunities may be thought of as jobs, referrals, contracts, projects, or other desirable professional outcomes.¹¹1This model may also be suitable to describing the (costly-per-person) transmission of information within a network. This is not to be confused with “fixed-cost broadcasts”, like social media posts that simultaneously go out to all social contacts at once. Our intuition is that costly-per-person transmission is the more relevant kind for professional networking. This is perhaps justified by the fact that fixed-cost broadcasts look similar to network-wide communication, and is therefore less dependent on network structure. They may come from one of two sources: first, exogenously or, second, after exogenous opportunities are drawn, from a contact in a professional network. Exogenous opportunities might come from an individual’s educational or socio-economic background, their family, or other pre-determined sources.²²2We may even extend the model to consider “off-platform” links as having formed exogenously through pre-determined factors like an indivual’s personal history. These off-platform links may be encapsulated into a person’s exogenous opportunity distribution and we may assume that all links considered in this work are “on-platform” links. Our results can easily accommodate this framing of exogenous versus endogenous connections. See Appendix A of the full manuscript for further discussion. The arrival of exogenous opportunities is assumed to be out of the individuals’ control and unrelated to the individual’s ability to make use of an opportunity. If an individual receives more exogenous opportunities than they can use, they pass remaining opportunities to their contacts in a professional social network. Sometimes, two individuals may unwittingly pass more opportunities to a single person than that person can use, in which case the extra opportunities are wasted.

Our focus in this work is on network formation. Prior to the point at which exogenous opportunities are realized, individuals in the model will form connections with others. Forming each connection bears a small cost for the individuals involved but allows for the possibility that one may pass an opportunity to the other. The cost may be imagined to be a search cost (e.g., the difficulty of finding each other) or an attentional burden (e.g., the time necessary to navigate a user interface or communicate). Each individual will choose how many connections to create, and with whom, according to their own interests: balancing the cost of the connection against the probability that they receive an opportunity as a result of the connection and would not have otherwise received opportunities.

Individuals within each group have the same opportunity distributions, and there will be one group with a higher number of expected exogenous opportunities (the privileged group) and one with a lower number (the unprivileged group). Our focus will be on quantifying efficiency, or the number of opportunities that are used, and inequality, the ratio of the number of opportunities used within each group.

We analyze the policies of a platform that has the limited capacity to influence outcomes by making link recommendations. Recommendations have the effect of reducing connection costs by reducing search or attentional effort, and we analyze the effect of different policies on efficiency and inequality. We introduce the model in more detail and define notation in Section 2.

We analyze a population of individuals $i\in [n]$. Individuals belong to one of two groups: a privileged green group, denoted $G$, and an unprivileged blue group, denoted $B$. We assume that $\left|G\right|,\left|B\right|=\mathrm{\Omega }(n)$, but the groups need not be the same size.

Individuals each seek a single opportunity. We assume all opportunities are of equal value to all individuals, normalized to $1$. Opportunities arrive either exogenously from nature or endogenously through the job network. Exogenous opportunities are drawn from group-dependent probability distributions. For each $i\in B$ (resp. $i\in G$), the probability of $i$ receiving $k$ exogenous opportunities is denoted $b_k$ (resp. $g_k$). Exogenous opportunities are drawn IID conditional on group. Privilege relates to the exogenous distribution of opportunities. Namely, we assume that $g$ stochastically dominates $b$ (i.e., for all $m\in {\mathbb{Z}}_{\ge 0}$ it holds ${\sum }_{\mathrm{\ell }\ge m}{g}_{\mathrm{\ell }}\ge {\sum }_{\mathrm{\ell }\ge m}{b}_{\mathrm{\ell }}$ with the inequality strict for some $m$). We will also impose assumptions, to be formalized later, ensuring that the $b_0$ is sufficiently large and that $g_0$ is not too large. These assumptions allow us to ensure that inequality is sufficiently large so that our results hold over all possible equilibria and thus for any equilibrium selection process. If we do not wish to specify a group membership for an individual $i$, we will refer to their probability of receiving $\mathrm{\ell }$ exogenous opportunities as $p_{i\mathrm{\ell }}$. We also assume that the support of the exogenous opportunity distribution is bounded: there is a constant $C$ so that for all $\mathrm{\ell }\ge C$, it holds $p_{i\mathrm{\ell }}=0$ for all $i\in [n]$.

Opportunities also arrive through an endogenously-formed job network, creating a game whose action space for each individual is a selection of potential neighbors. The timing of the game is as follows. A platform first recommends a (potentially empty) subset of links $Q_i\subseteq \{(i,j):j\in [n]\}$ of size $\left|Q_i\right|=k$ for each $i\in [n]$, which have the effect of eliminating the cost of the connection. We assume if $(i,j)\in Q_i$, then $(j,i)\in Q_j$. Define $Q={\cup }_{i\in [N]}{Q}_i$. Individuals – knowing $Q$, $[n]$, and the exogeneous distribution of opportunities for each individual but not their specific realization – then choose neighbors, forming a network $E\subseteq \{(i,j):i,j\in [n]\}$. Let $N_i(E)=\{j:(i,j)\in E\}$ and $d_i(E)=\left|N_i(E)\right|$ (when the edge set $E$ is clear from the context; we drop it from the notation). We will call edges in the network that were not recommended (i.e., $E\setminus Q$) organic or non-recommended connections. Finally, exogenous opportunities are realized and extra ones are distributed to neighbors in the network uniformly at random. Specifically, if $i$ receives $\mathrm{\ell }>1$ exogenously opportunities, then they select $(\mathrm{\ell }-1)$ of their $d_i$ neighbors uniformly at random and pass each of them a single opportunity (discarding leftover opportunities if $\mathrm{\ell }-1>d_i$). Let

be the expected number of opportunities an individual of degree $d$ passes to their neighbors. Let ${\mu }_B(d)$ (resp. ${\mu }_G(d)$) be this expectation for a generic (i.e., any) blue (resp. green) group member. We will overload notation and write ${\mu }_i={\mu }_i(\mathrm{\infty })$ when we wish to refer to the expected number of extra opportunities a member receives (sim. ${\mu }_G$ and ${\mu }_B$). I.e., ${\mu }_i(\mathrm{\infty })$ is the expected number of opportunities a person with arbitrarily large (hence, $\mathrm{\infty }$) degree will pass to their neighbors. To avoid trivialities where there is no chance a member of a group receives any extra opportunities, we will assume throughout that ${\sum }_{\mathrm{\ell }\ge 1}(\mathrm{\ell }-1)g_{\mathrm{\ell }}$ and ${\sum }_{\mathrm{\ell }\ge 1}(\mathrm{\ell }-1)b_{\mathrm{\ell }}$ are positive.

Given a network and realization of exogenous and endogenous opportunities, an individual $i$ receives a utility of $1$ if they receive an (exogenous or endogenous) opportunity and pays a cost of $\gamma$ for each of their unrecommended neighbors in the network.³³3It would be an interesting direction for future work to explore differential connection costs across different pairs (either where one member of a pair must exert more effort to connect or where different pairs pay different costs based on uncertainty about their value to each other). We imagine that most useful connections in the real world impose some costs on their participants, since relationships in which one exerts no maintenance effort do not typically function as conduits of information about job opportunities. Thus their ex ante expected utility from a network $E$ is:

where the first term is the probability of receiving an opportunity and the second is the cost of their organic connections. Social welfare will be defined, depending on the context, as the sum of individuals’ expected utilities or the minimum utility over the population.

We study the equilibria of the induced game. We will use defection-free pairwise Nash (DFPN) as our equilibrium concept, as in Dwork et al. (2024) [17]. We will characterize pure-strategy equilibria in this work (i.e., we will not consider players who select edges non-deterministically). This yields the property that equilibria are uniquely specified by a set of formed edges. DFPN considers defections in which individuals may unilaterally sever links and/or pairs of individuals may add a link if it is mutually beneficial. Notably in contrast to the notion of pairwise stability studied in Jackson and Wolinsky (1996) [23], a single defection may involve both severing and forming links simultaneously. In this sense, DFPN is a refinement of pairwise stability that allows individuals to consider dropping current links when contemplating forming a new one, analogous to stability in stable marriage problems.

Formally, an edge set $E$ is DFPN if

1. (a)

   for all $i,j$ such that $(i,j)\notin E$ and for all $S_i\subseteq N_i(E)$, $S_j\subseteq N_j(E)$

   (2)

2. (b)

   and for each $i,j$ such that $(i,j)\in E$, it holds

   (3)

Intuitively, Equation 2 specifies that no pair of individuals would prefer to form a connection amongst themselves, dropping a (possibly empty) set of their existing connections. Equation 3 specifies that no individual in a connection would rather sever the connection. For fixed parameters, we will write the set of DFPN equilibrium edge sets as $ℰ$. If we wish to contrast different equilibrium edge sets for different parameters, we will specify a particular choice of parameter (e.g., number of recommendations $k$) using subscripts (e.g., ${ℰ}_k$).

Our main results characterize the inequality and welfare properties of different link recommendation policies on equilibrium outcomes. In particular, we will study policies that are anonymous in the sense that they are the same for people who are indistinguishable to the platform. Since in our simple setting, individuals are only distinguished by their group membership, this means that we consider policies where the platform recommends a fixed fraction of cross-group connections to each individual within a group. Since we are not assuming $\left|B\right|=\left|G\right|$, it may not be possible for these fractions to be the same for members of $B$ and $G$. We will denote the fraction of cross-group connections recommended to $B$ as $\rho$ where $\rho \in \{0,1/k,\mathrm{\dots },1\}$ for the number of recommendations $k$. This implies that members of $G$ receive a $\rho \left|B\right|/\left|G\right|$ fraction of cross-group recommendations. Throughout, we will assume that $\rho$ is set so as to ensure $\rho \left|B\right|/\left|G\right|\le 1$. This inequality is trivially satisfied if $\left|B\right|\le \left|G\right|$. Note that Hall’s marriage theorem implies that such a set of cross-group recommendations always exists. The remaining same-group recommendations can be constructed if the relevant populations have even size, which we assume for convenience. See Appendix A of the full manuscript for further explanation.

Our model extends the base model in Dwork et al. (2024) [17]. The key differences are that (1) we model multiple social groups, (2) we consider a platform making link recommendations and (3) we allow for the possibility for each individual to receive more than one extra opportunity. We add (3) only for the sake of generality: all of our results hold in the special case that each individual may receive no more than one extra opportunity.

The equilibria of our game satisfy several useful properties, also potentially of independent interest. The first is a type of balance condition. It states that almost all individuals who are connected have similar probabilities of passing each other opportunities. In other words, individuals more-or-less get what they give in their relationships.

The formal lemma, stated below and proved in the appendix, extends the results of Dwork et al. (2024) [17] to our more complex setting with recommendations and two groups. Recall that ${\mu }_i(d)/d$ is the probability that an individual $i$ of degree $d$ passes one of their connections an opportunity. Define constant $C(\gamma ,k)=2{({\gamma }^{-1}+k)}^2{({\gamma }^{-1}+k+1)}^2$.

Intuitively, part (a) of Lemma 1 says that each individual $i$ participating in a connection $(i,j)$ in equilibrium satisfies an inequality where their connection $j$ has probability of passing them an opportunity no less than the probability $i$ would pass $j$ an opportunity if they had degree one larger. It additionally says that, for connected members of the same group, each individual exactly satisfies the condition that they have probability of passing their connection an opportunity no more than if their connection had one fewer connection, respectively.⁴⁴4This condition is not trivially satisfied by the fact that ${\mu }_j(d_j)/d_j\ge {\mu }_i(d_i+1)/(d_i+1)-\epsilon$ since the $d_i,d_j$ are determined by equilibrium conditions, and may not continue to hold if individuals’ degrees change. Part (b) further states that each individual has an outside option: they could be (but are not) connected to someone else of their same degree with an approximately equal expected number of extra opportunities.

The result above is quantified by a slack term $\epsilon$. It holds for all $\epsilon$, but there is a trade-off between the tightness of Equation 4 and the size of $S$. Making $\epsilon$ smaller makes the inequality tighter but decreases the size of the set $S$. Making $\epsilon$ larger makes the inequality looser but increases the size of the set $S$. Depending on the situation, it may be desirable to choose different values of $\epsilon$, but the choice of $\epsilon$ is not a parameter influencing the set of possible equilibria. Instead, it is a analyst-defined choice for finding a $n-O(1)$ (as long as $\epsilon$ is not dependent on $n$) set of individuals in any equilibrium who satisfy the balance conditions in the lemma.

We also observe that by the fact that $j\in S$, Equation 4 holds when we switch the roles of $i$ and $j$. That is, ${\mu }_j(d_j)/d_j\ge {\mu }_i(d_i+1)/(d_i+1)-\epsilon$. Together, Equation 4 and the inequality above ensure that all neighbors of $i$ satisfy a balance condition ensuring that neither individual in a connection will have substantially lower probability of passing the other an opportunity.

It is also not hard to see that the form of the utility function as well as the equilibrium conditions imply the following two facts:

We omit the proofs of these elementary facts and remark that, although we assume that recommendations eliminate connection costs in our model, we could relax this modeling choice and assume instead that recommendations only reduce (but do not eliminate) the cost of connections. Our results in this paper continue to hold as long as recommended connection costs are sufficiently small (i.e., small enough that members of the privileged group can be incentivized to form connections with members of the unprivileged group in equilibrium). We also note that central planners might extract even greater social benefits in cases where they can subsidize links by more than the connection cost, yielding payments (negative costs) for links formed. We leave exploration of these topics for future work.

Finally, we establish that there always exists an equilibrium, as long as exogenous inequality is large enough.

We can construct equilibria in our model by letting each individual in a group have the same degree as every other member of their group. Individuals in different groups will have different degrees (according to their privilege) distributions. Setting $\gamma >b_0(1-b_0-b_1)$ ensures that members of $B$ will only form connections through recommendations, which considerably simplifies the analysis and conforms with the parameter regime considered in this work (where members of $B$ are sufficiently unprivileged).⁵⁵5We remark that our model in this paper concerns economically meaningful connections, i.e., connections that carry with them some possibility that the individuals incident to the connection benefit. Our intuition is that the number of economically meaningful connections an individual has may be much lower than their number of “connections” on professional networking platforms might indicate. Thus, depending on the situation, the number of connections in our model may or may not exactly correspond to connections as defined by professional networking platforms. In general, the assumption that individuals must be sufficiently unprivileged so as to form few organic connections is an analytical convenience that is not necessary for the results to hold. We leave for future work the question of whether there exist equilibria for any possible setting of the parameters.

We analyze social welfare of the network at equilibrium under different link recommendation policies. We quantify social welfare as either the average utility of all individuals in the population (a utilitarian notion) or the minimum utility of any individual in the population (a Rawlsian notion), and our results hold for both definitions of social welfare. In Proposition 5, we show that, for sufficiently differing privileges between groups, recommending $k$ cross-group link recommendations generates higher – both utilitarian and Rawlsian – social welfare, for all equilibria, than providing no recommendations. In Proposition 6, we show that, keeping $k$ constant, it is always better to recommend a larger fraction of cross-group link recommendations than a smaller fraction for all equilibria, provided there is a sufficient gap between the two fractions.⁶⁶6A gap is needed between fractions of cross-group recommendations because of the multiplicity of equilibria: the best equilibrium for a particular setting of $\rho$ may be better than the worst equilibrium for $\rho +\delta$ where $\delta$ is a small constant.

We first formally state Proposition 5. Intuitively, this proposition says that any equilibrium $E_k\in {ℰ}_k$ induced by all cross-group edge recommendations achieves greater social welfare than any equilibrium $E_0\in {ℰ}_0$ induced by a platform that makes no recommendations (where ${ℰ}_k$ is the set of equilibria of the game with $k$ cross-group recommendations per person).

Intuitively, the result holds because an opportunity has a lower chance of being wasted if there are many edges from high privilege to low privilege individuals. The first inequality concerns utilitarian social welfare, and the second concerns Rawlsian social welfare. We will sometimes refer to the utilitarian social welfare as the average endogenous utility to contrast it with the average exogenous utility, which is simply the sum of utilities of individuals in the absence of the platform: ${\sum }_{i=1}^n(1-p_{i0})$. Note that because we assume $\rho \left|B\right|/\left|G\right|\le 1$, the result requires $\left|B\right|\le \left|G\right|$.

We prove the result in the full version of the paper and sketch the intuition here. To prove the statement for utilitarian social welfare, we appeal to Lemma 1, to show that almost all individuals form connections with others who have a similar probability of passing each other opportunities. This allows us to derive upper and lower bounds on the degrees of all but a constant number of individuals for a fixed set of parameters. This, in turn, allows us to derive upper and lower bounds on the utilities of all but a constant number of individuals (since their utility is determined by their neighbors’ opportunity distributions and degrees which Lemma 1 greatly simplifies). We can then compare the upper bound for all $E_0\in {ℰ}_0$ with a lower bound for all $E_k\in {ℰ}_k$ and show that the latter is greater than the former. For the Rawlsian social welfare, Lemma 1 is not useful, since the properties implied by the lemma may not hold for some number of individuals, and one of these individuals could achieve the minimum. Thus, we argue that all members of the unprivileged group form connections, that the lowest-utility member of the unprivileged group has lower utility than that of the privileged group, and finally that the lowest-utility member of the unprivileged group under $k$ cross-group recommendations has utility greater than the corresponding individual under no recommendations.

We validate Proposition 5 in Figure 1. In each of the plots in this paper, bounds on utilities are asymptotic (as $n\to \mathrm{\infty }$) and set $\left|G\right|=\left|B\right|$. The non-asymptotic bounds are within additive $O(n^{-1})$ of the asymptotic bounds. For simplicity, we also set $p_{i\mathrm{\ell }}=0$ for all $\mathrm{\ell }\notin \{0,2\}$ and $i\in [n]$ so that exogenous opportunity distributions collapse to a one-dimensional parameter. In Figure 1, we plot upper and lower bounds on average utilities for different numbers of recommendations. Upper bounds are in red and lower bounds are in blue, and different shades indicate different values of $g_0$. On the vertical axis of Figure 1, we plot the difference between the average endogenous (${\sum }_{i=1}^n{u}_i(E)$) and exogenous (${\sum }_{i=1}^n(1-p_{i0})$ utilities. We see that as the number of recommendations increases, the lower bound on average utility increases so that it is above the upper bound at $k=0$. (I.e., for a given $g_0$, the red line at $k=0$ is below the blue line at $k>0$.) However, in general, increasing the number of recommendations – of any fraction cross-group – is not always social welfare increasing, since increased connectivity can lead to more inefficiency. We can see this by the fact that for values of $g_0$ where the upper and lower bounds are tight towards the right of the plot, the average endogenous minus exogenous utility is decreasing in $k$ past a certain point.

We next state our second result of this section, which concerns the comparison between different fractions of cross-group recommendations for a fixed number of recommendations. (This is in contrast with the result above, which is about comparing $k$ cross-group edges with no recommendations at all.) Our result states that a higher fraction of cross-group recommendations always achieves greater utility, for all equilibria, than a lower fraction, as long as the gap between the lower and higher fractions is sufficiently large. Given a positive integer $k$ and $\rho \in [0,1]$, we use ${ℰ}_{\rho }$ to denote the set of all equilibria induced by any $k$ edge recommendations per person among which $\rho k$ recommendations per person are cross-group.

The proof of Proposition 6 is deferred to the full version of the paper and appeals to the same upper and lower bounds on utilities used in the proof of the previous result. We validate our result in Figure 2: for particular values of $k$ and $\rho$, we determine the largest ${\rho }^{\prime }$ for which the inequalities in the result hold. We also demonstrate how, if inequality is not sufficiently large, the inequalities in Proposition 6 may not hold. That is, there are some parameter regimes where there exists $\rho$ such that the worst-case social welfare under a $\rho$-fraction of cross-group recommendations is not greater than the best-case social welfare for any policy with a fraction of cross-group recommendations.

In Figure 2, we show upper and lower bounds on the difference between average endogenous utility and average exogenous utility as $\rho$ varies. The setting is the same as in Figure 1, except that we fix $k=5$ and vary the proportion of cross-group link recommendations $\rho$. Note that since $k=5$, the only valid values of $\rho$ are multiples of $1/5$, the ticks on the horizontal axis. The average endogenous minus exogenous utility is increasing, in general, as the fraction of cross-group link recommendations increases, although there are some values of $\rho$ where the change is indeterminate (where the upper and lower bounds are too far apart). These parts of the parameter space, e.g., the bounds when $\rho \le 0.4$ and $g_0=0.5$ demonstrate that inequality was not sufficiently large so that for every tick on the horizontal axis greater than 0, the blue line at the tick is above the red line for some tick to its left. (By contrast, if we set the exogenous utility ratio to be sufficiently large, then Proposition 6 tells us that this would occur.) Also, note that when $g_0$ is higher (at darker shades), the benefits of increasing $\rho$ are more modest. This is because there is a larger chance that members of $G$ receive no opportunities, so integrating the network can only go so far in increasing the prospects of members of $B$.

We next explore inequality at equilibrium in our model. In Section 5.1, we show how the existence of a platform can result in increased inequality, even without pre-existing homophily in the network formation process. Then, in Section 5.2, we analyze the ability for platforms to ameliorate the extra inequality they created by selectively subsidizing cross-group links.

In this work, we explore the effect of platform link recommendation policies on inequality and inefficiency. We show that cross-group link recommendations decrease inequality and increase efficiency in terms of the total fraction of opportunity used. We also show that laissez-faire platforms, which do not intervene on the link recommendation process, increase inequality if inequality is large enough, for all equilibria. Our work has a number of natural extensions that would be interesting to pursue in future work.