# Delayed and Indirect Impacts of Link Recommendations

Han Zhang  University of California, Berkeley  Shangen Lu  The Wharton School
Yixin Wang  University of Michigan  Mihaela Curmei  University of California,
Berkeley

###### Abstract

The impacts of link recommendations on social networks are challenging to
evaluate, and so far they have been studied in limited settings. Observational
studies are restricted in the kinds of causal questions they can answer and
naive A/B tests often lead to biased evaluations due to unaccounted network
interference. Furthermore, evaluations in simulation settings are often
limited to static network models that do not take into account the potential
feedback loops between link recommendation and organic network evolution. To
this end, we study the impacts of recommendations on social networks in
dynamic settings. Adopting a simulation-based approach, we consider an
explicit dynamic formation model—an extension of the celebrated Jackson-Rogers
model—and investigate how link recommendations affect network evolution over
time. Empirically, we find that link recommendations have surprising _delayed_
and _indirect_ effects on the structural properties of networks. Specifically,
we find that link recommendations can exhibit considerably different impacts
in the immediate term and in the long term. For instance, we observe that
friend-of-friend recommendations can have an immediate effect in decreasing
degree inequality, but in the long term, they can make the degree distribution
substantially more unequal. Moreover, we show that the effects of
recommendations can persist in networks, in part due to their indirect impacts
on natural dynamics even after recommendations are turned off. We show that,
in counterfactual simulations, removing the indirect effects of link
recommendations can make the network trend faster toward what it would have
been under natural growth dynamics.

††1Corresponding author<EMAIL_ADDRESS>
_K_ eywords recommendation systems $\cdot$ link recommendation $\cdot$ dynamic
graph modeling $\cdot$ social dynamics in graphs

##  1 Introduction

Link recommendation algorithms such as Facebook’s "People You May Know",
Twitter’s "Who to Follow" and LinkedIn’s "Recommended for You" have an ever-
increasing influence on the evolution of social networks, with some accounts
crediting over 50% of links to algorithmic recommendations (Team, 2022). This
can cause downstream effects on information flow, opinion dynamics, and
resource allocation. For instance, recommendations can be a polarizing force
increasing network segregation, which in turn may re-inforce opinion echo-
chambers (Cinus et al., 2022) or restrict access to information and resources
for less connected communities (Dandekar et al., 2013; Auletta et al., 2022;
Santos et al., 2021). At the same time, recommendation systems can also
surface “long-range” connections between nodes that would not otherwise be
exposed to each other, and thus, increase network integration by promoting and
maintaining diverse links (Rajkumar et al., 2022; D’Angelo et al., 2019).
Given the ubiquity of algorithmic recommendations on social media, studying
their impacts is seeing increased academic and regulatory interest. However,
such studies are challenging for a variety of reasons, such as lack of
normative framing (Daly et al., 2010) and limited access to large-scale
platforms. In this work, we explore a more foundational evaluation challenges
stemming from the fact that social networks have underlying dynamics that
interfere with algorithmic recommendations.

Existing real-world evaluations of link recommendation algorithms rely on A/B
tests (experimental) or longitudinal data (observational). However, both
experimental and observational evaluations can yield misleading conclusions.
The validity of A/B tests relies on the Stable Unit Treatment Value Assumption
(SUTVA) (Imbens and Rubin, 2015; Gui et al., 2015), which is violated in case
of network interference. Observational studies may fail to assess causal
impacts as longitudinal evaluation lacks counterfactual measurements on what
the network evolution would have been without algorithmic recommendations.
These challenges have motivated the use of simulation-based evaluations of
link recommendations. In simulation studies, one can make explicit network
modeling assumptions and then evaluate the impact of recommenders. Despite a
growing number of works in this space (Stoica et al., 2018; Cinus et al.,
2022; Fabbri et al., 2020, 2022; Ferrara et al., 2022; Santos et al., 2021),
existing simulation-based evaluation falls short of providing insights into
the mechanism through which recommendations impact social networks. Existing
simulation studies primarily consider static networks, and thus do not take
into account feedback loops between link recommendation and organic network
evolution.

In dynamic settings, the main challenge is to measure impacts relative to a
"baseline" network. Given this, existing works often measure the effects
relative to the initial networks; they are rarely measured relative to a
plausible counterfactual based on the natural evolution of the graph without
link recommendations. Such evaluations can lead to qualitatively wrong
conclusions. For instance, Abebe et al. (2022) shows that triadic closure –
the most common type of friend-of-friend recommendations – can reduce
segregation with respect to the initial network before intervention. However,
triadic closure can, at the same time, increase network segregation in
relative terms with respect to a natural evolution dynamic which assumes the
addition of random edges.

In this work, we initiate a study to explore _dynamic_ impacts of link
recommendations through simulations. We find that link recommendations can
have surprising delayed and indirect effects on the structural properties of
networks. For instance, the effects of friend-of-friend recommendations can
vary in the short-term and long-term: it may alleviate degree inequality in
the short term, but increase degree inequality in the long term; the perceived
increase in degree inequality can even be significantly more pronounced after
the recommendations are discontinued. Moreover, we demonstrate that the
effects of link recommendations can persist in the network even after
recommendations have been discontinued. This phenomenon is due to the fact
that recommendations impact the network in two ways: directly through the
creation of algorithmic edges, and indirectly by influencing the natural
growth dynamics. Indirect effects amplify the direct effects, contributing to
the persistent impact of link recommendation algorithms. A stylized
illustration of the indirect and delayed effects can be found in Figure 1.

Figure 1: Delayed and indirect effects: The image displays two counterfactual
evolutions of the network. The solid blue line represents the trajectory for
an intervention interval $[\underline{t},\overline{t}]$ in which the full
network receives algorithmic recommendations. The dashed black trajectory
corresponds to a counterfactual network evolution without recommendations. The
total causal effect of recommendation at time $t$; $\text{Effect}_{t}$ can be
computed as the difference between the counterfactual trajectories at time $t$
(solid blue line and dashed black line). The delayed effect at some time
$T\geq\overline{t}$ is the difference:
$\text{Effect}_{T}-\text{Effect}_{\overline{t}}$. The solid purple trajectory
illustrates the counterfactual evolution of the network in which the indirect
effects are removed. The difference between the purple and dashed curve
captures the direct effects and the difference between the blue and purple
line captures the indirect effects.

#### Contributions.

We enumerate our contribution along modeling, evaluation, and experimental
findings.

  * •

_Modeling:_ We propose a dynamic network formation model which extends upon
the Jackson-Rogers model (Jackson and Rogers, 2007) to incorporate algorithmic
recommendations. Unlike the classic model, our dynamic model includes not only
the original two phases —dubbed "meeting strangers" and "meeting friends"—but
also a third phase of “meeting recommendations.” Additionally, we consider
latent node representations, enabling us to model community structure and node
activity levels in a flexible manner.

  * •

_Evaluation:_ We monitor the progression of network metrics over different
intervention windows. We compare the immediate impacts observed during
intervention with the delayed impacts observed after the intervention has
ended. Further, we measure the indirect impacts that recommendations have on
network properties. To tease out the indirect impact, we compare the observed
network evolution with a counterfactual baseline network that discounts the
influence of recommendations on natural growth.

  * •

_Experimental findings:_ Our study reveals diverse qualitative patterns for
delayed and indirect effects; we find that different durations of the
intervention and/or different times of measurement can lead to drastically
different conclusions about _"How do recommendations impact networks?."_
Furthermore, we find that indirect effects can be substantial; they can
significantly amplify the impact of recommendations and persist even after the
intervention has ended.

##  2 Related Work

#### Simulation studies.

Simulation studies typically analyze the impact of link recommendations on
static networks. Some works focus on a single round and examine biases
observed in proposed recommendations, finding that homophily — the preference
for within-group links — leads to exposure bias toward the more homophilous
group, even if it’s a minority (Karimi et al., 2018; Fabbri et al., 2020;
Espín-Noboa et al., 2022; Stoica et al., 2018). Similarly, (Fabbri et al.,
2022; Ferrara et al., 2022) find that homophily and node degree are strong
indicators of which nodes will receive disproportionate visibility. Other
works make explicit behavioral assumptions on how nodes accept link
recommendations and consider cumulative effects of link recommendations over
multiple rounds. These studies reveal algorithmic amplification of biases via
"rich get richer" effects and increases in observed homophily over time
(Fabbri et al., 2022; Ferrara et al., 2022). A shortcoming of existing
simulation based evaluations is that they implicitly assume that the addition
of algorithmic edges is the only change in the network. Conversely, in our
work we make explicit modelling assumptions about the underlying network
dynamics. This evaluation setup allows us to measure the impact of link
recommendations with respect to counterfactual natural evolution. Furthermore,
by emphasizing underlying temporal dynamics we can pose more subtle evaluation
questions such as: "How does the effect of algorithmic intervention fade over
time once recommendations are stopped?" or "How do algorithmic recommendations
bias the underlying network growth?".

#### Platform studies.

There is limited publicly available research evaluating link recommendation
algorithms on real social networking platforms. In one experimental study
(Daly et al., 2010), several recommendation algorithms were compared on IBM’s
SocialBlue network and found to reduce group homophily. The study also
revealed that friend-of-friend recommendations had the highest rate of
acceptance but the lowest level of edge activity. On the other hand, (Rajkumar
et al., 2022) found that recommending more distant connections or "weak ties"
through LinkedIn’s "People You May Know" algorithm led to higher transmission
of job opportunities. In observational settings, a longitudinal study
comparing the Twitter network before and after the introduction of the "Who To
Follow" recommender in 2010 (Su et al., 2016) found that while recommendations
increased the number of connections for all users, the highest gains were
achieved by the most popular nodes. Conversely, a study comparing links formed
naturally and links formed via recommendations on Flickr and Tumblr (Aiello
and Barbieri, 2017) found that the recommended links were more diverse and
less biased towards popular users. So far, existing observational platform
studies provide limited understanding of the underlying mechanisms and lack
access to counterfactual network evolution. A/B tests, on the other hand, may
produce wrong estimates of the effects due to network interference. Our
counterfactual simulations allow us to articulate in stylized settings the
source of bias in both longitudinal and A/B evaluations.

#### Theoretical investigations.

The interaction between homophily and friend-of-friend recommendations was
studied theoretically in a number of works. Under choice homophily, which
captures the setting when nodes preferentially accept recommendations to
within group nodes, (Stoica et al., 2018; Asikainen et al., 2020) show that
such interventions lead to more exposure gains for the homophilous group which
exacerbates observed homophily. In contrast, (Abebe et al., 2022) shows that,
when the closure of triangles via friend-of-friend recommendations is not
biased in favor of in-group edges, recommendations can in fact improve network
integration. In our work, we investigate friend-of-friend recommendations and
their impacts on network segregation and show that the impacts further depend
on the length of intervention as well as the time of measurement.

#### Delayed algorithmic impacts.

Temporal dynamics associated with algorithmic interventions have been
previously studied in the context of fairness in Machine Learning. These works
showcase broad settings where algorithmic interventions designed to improve
fairness (Liu et al., 2018; Daly et al., 2010; Akpinar et al., 2022),
robustness (Milli et al., 2019) or diversity (Curmei et al., 2022) in the
short term, lead to the opposite effect in the long run. Our work uncovers
similar surprising temporal dynamics in the case of link recommendations.

##  3 Methodology for evaluating delayed and indirect impacts

To illustrate temporal complexities of evaluating link recommendation
algorithms in dynamic setting, we consider a stylized network evolution model.
We propose an extension to the the classic Jackson-Rogers (JR) network
evolution model (Jackson and Rogers, 2007). The JR model has been validated
empirically and shown to display proprieties of real network such as
decreasing diameter over time, increased edge densification and emergence of
community structure (Jackson and Rogers, 2007; Ruiz and Finke, 2017). Our
extension adds an optional recommendation phase to model the feedback loop
between natural network evolution and algorithmic recommendations.

We analyze two simple baseline recommendation algorithms: a neighborhood-based
and an affinity-based algorithm. While recommendation algorithms deployed in
practice use both types of information, with potentially additional learning
components, we opt for simplicity in recommendation algorithms to understand
the underlying mechanisms behind evaluation phenomena in dynamic networks.
Algorithm 1 summarizes the simulation and evaluation procedure.

###  3.1 Dynamic network model

Let $G^{t}=(V^{t},E^{t})$ be an undirected network where $V^{t}$ and $E^{t}$
are the set of nodes and edges at time $t$. Nodes are characterized by their
group identity $g_{i}$ and latent representation $v_{i}$. We assume group
identities are static over time and latent representations are sampled from a
group-specific distribution $v_{i}\sim\mathcal{D}_{g_{i}}$. At each time step,
the network evolves via new nodes which first connect "organically" the
existing nodes. Further at each step, the network evolves "algorithmically"
via connections mediated by a link recommender.

#### Natural growth.

Similar to the classic JR model, upon the arrival of a new node, the network
evolves in two phases: "Meeting Strangers" and "Meeting Friends". In the
"Meeting Strangers" phase, the new node makes connections with existing nodes
at random. Unlike the classic model where connections are made with a fixed
probability, we model connections probabilities based on latent representation
of the nodes $v_{i}$. This additional modelling assumption allows us to
consider various community structure. In the "Meeting Strangers" phase,
$N_{s}$ candidate nodes are sampled uniformly from the existing network. The
arriving node $i$ connects with each candidate node $j$ with probability
proportional to the inner product of their respective latent embeddings:
$p_{i,j}=\sigma(\langle v_{i},v_{j}\rangle)$ where $\sigma(\cdot)$ is a scaled
and translated sigmoid function111We consider
$\sigma(x)=\frac{1}{1+e^{-ax+b}}$ where we set $a$ and $b$ to match desired
average linkage probabilities..

In the subsequent "Meeting Friends" phase, the incoming node $i$ makes
additional connections based on neighborhood proximity. Here $N_{f}$ candidate
nodes are sampled from the set of nodes at distance 2 (neighbors of neighbors)
from the node $i$. The node $i$ connects with each candidate node $j$ with
constant probability. Optionally, we model attrition effects by considering
node departures from the network with a hazard function
222$h(\mathbf{a})=cd^{\mathbf{a}}+k$ where $\mathbf{a}$ is the age of the node
and $c,d,k$ are tuned to model mean and variance of lifespans in the network.
that increases with the age of the node.

#### Algorithmic intervention.

We introduce algorithmic intervention as a third "meeting recommendations"
phase, whereby nodes in the network receive link recommendation. This phase
applies not only to incoming nodes but also to existing nodes in the network.
Given recommendation, nodes accept them according to a behavioral model. We
consider neighborhood and affinity-based recommendations. The prototypical
neighborhood recommendation is the friend-of-friend (FoF) recommendation where
candidate nodes are selected uniformly from the set of nodes at distance 2:
$\mathbb{P}(\textbf{rec}_{i}=j)=\frac{\textbf{1}(dist(i,j)=2}{\sum_{j^{\prime}}\textbf{1}(dist(i,j^{\prime})=2}$.
Conversely, affinity-based (Latent) recommendations utilize the latent
structure rather than local neighborhood structure to make recommendations.
Specifically, the Latent recommender computes affinity scores for all the
nodes in the network in terms of inner products and recommends candidates with
probability proportional to the scores: $\mathbb{P}(\textbf{rec}_{i}=j)\propto
e^{\beta s_{j}}$, where $s_{j}=\langle v_{i},v_{j}\rangle$ is the embedding
similarity between node $i$ and node $j$. High values of softmax temperature
parameter $\beta$ correspond to more deterministic recommenders. Lower values
of $\beta$ lead to more random recommendations and qualitatively capture the
effects of estimation noise for learning-based recommenders.

#### Behavioral models.

We consider two behavioral models for accepting link recommendations: the
constant probability baseline and the embedding-based probability where nodes
accept new links based on proximity in latent space. Embedding based
probabilities model more granular notions of choice homophily (Asikainen et
al., 2020). We consider an additional behavioral option in which upon
acceptance of a new recommended link, an edge is removed at random from the
set of existing edges. This option is in line with several works which model
recommendations as a _rewiring_ (Asikainen et al., 2020; Fabbri et al., 2021;
Santos et al., 2021) process that does not impact the average degree.

input : initial $G_{0}$; time-steps $T$; hazard function; communities:
prevalence $c_{g}$, latent distribution $\mathcal{D}_{g}$;

natural growth: $N_{s}$, $N_{f}$, dist-2 connect prob $p_{2}$; intervention:
window $[\underline{t},\overline{t}]$, recommender, behavior;

for  _$t$ in $1\ldots T$_ do

Natural growth;

sample incoming node $i$ according to group prevalence $c_{g}$ and latent
distribution $\mathcal{D}_{g}$;

strangers $\leftarrow$ samples $N_{s}$ nodes;

for  _$s$ in strangers_ do

add edge $i-s$ with probability $\propto\langle v_{i},v_{s}\rangle$

end for

friends $\leftarrow$ samples $N_{f}$ neighbors of neighbor nodes;

for  _$f$ in friends_ do

add edge $i-f$ with probability $p_{2}$

end for

if  _$t\in[\underline{t},\overline{t}]$_ then

Algorithmic intervention;

for  _node $j$ in treatment group $G^{treatment}_{t}$_ do

candidate = recommender($j,G_{t}$);

if  _behavior($j$, candidate) = accept_ then

add edge $j-$candidate

end if

end for

end if

nodes to remove $\leftarrow$ hazard function

end for

Algorithm 1 Simulating network evolution

###  3.2 Structural metrics

_Clustering coefficient_ : Clustering coefficient of a node is the ratio of
triangles it forms $\Delta_{i}$ and the maximum number of triangles it could
have formed given it’s current degree $d_{i}$:
$c_{i}=\frac{2\Delta_{i}}{d_{i}(d_{i}-1)}$. The clustering coefficient
captures the micro-level cohesion in the network (Ferrara et al., 2022). The
average clustering coefficient is the metric averaged over all the nodes.
Averaging the metric over communities measure clustering at the community
level.

_Gini coefficient_ : We measure the inequality in the degree distribution via
Gini coefficient. An increase of the Gini coefficient resulting from the use
of recommendations is often used to demonstrate biases in link recommendation
(Ferrara et al., 2022; Fabbri et al., 2021, 2022). The Gini coefficient can be
computed for an ordered list of node degrees as:
$G=\frac{2\sum_{i=1}^{n}id_{i}}{n\sum_{i=1}^{n}d_{i}}-\frac{n+1}{n}$.
Similarly, this metric can be computed for the entire graph or restricted to
communities.

_Homophily_ : We define a monocromatic and bichromatic edges to be an edges
that links two nodes from the same community or nodes from different
communities, respectively. The homophily of a community measures the
propensity of nodes to favor within group connections compared to a non-
preferential baseline. We compute the homophily of a community $g$ as follows:
$H_{g}=\frac{|E_{gg}|}{|E_{g}|}-\frac{n_{g}}{n}$, where $|E_{gg}|$ denotes the
number of monochromatic edges within $g$, $|E_{g}|$ denotes the number of
total edges that have at least one node in community $g$. Finally $n_{g}$
denotes the size of $g$, and $n$ denotes the total size of the network, the
ratio $\frac{n_{g}}{n}$ is the baseline fraction of within-group links when
nodes have no group-based preferences. The vast majority of existing
simulation based evaluation study changes in homophily or the closely related
notions of within-community and between-community edges (Fabbri et al., 2020;
Abebe et al., 2022; Daly et al., 2010).

###  3.3 Evaluation

We denote by $\mathcal{G}$ the evolution of the network under natural dynamics
and by $\mathcal{G}(\texttt{rec},[\underline{t},\overline{t}])$ the evolution
when the network receives link recommendation according to rec algorithm over
the $[\underline{t},\overline{t}]$ intervention interval. The subscript $T$ in
$\mathcal{G}_{T}(\cdot)$ is used to denote the snapshot of the network at time
$T$. For a structural metric $m$, we denote by
$m(\mathcal{G}_{t}(\texttt{rec},[\underline{t},\overline{t}]))$ the value of
this metric evaluated at time $t$. Given a network evolution model, we
simulate the counter-factual trajectories of the metrics for intervention
intervals $[\underline{t},\overline{t}]$ of different lengths.

#### Total effects.

The total effect of intervening in a network evolution $\mathcal{G}$ with
recommender rec for intervention interval $[\underline{t},\overline{t}]$ on
the metric $m$ at time $T$ is defined as the difference between two
counterfactual measurements:

$\displaystyle\text{Effect}_{T}(\mathcal{G},\texttt{rec},[\underline{t},\overline{t}]):=m(\mathcal{G}_{T}(\texttt{rec},[\underline{t},\overline{t}]))-m(\mathcal{G}_{T}).$

This definition compares between the “treatment” universe where the whole
network received an algorithmic intervention and a counterfactual “control”
universe where the network evolved without algorithmic recommendations.

#### Delayed effects.

When the measurement occurs during the intervention interval, i.e.
$T\in[\underline{t},\overline{t}]$ we call the effect _immediate_. Conversely,
when the measurement occurs after the intervention stops, i.e.
$T>\overline{t}$, we can define the notion of _delayed_ impact as:

$\text{DelayedEffect}_{T}(\mathcal{G},\texttt{rec},[\underline{t},\overline{t}])=\text{Effect}_{T}(\mathcal{G},\texttt{rec},[\underline{t},\overline{t}])-\text{Effect}_{\overline{t}}(\mathcal{G},\texttt{rec},[\underline{t},\overline{t}])$

The notion of delayed impacts allows us to characterize the impact of link
recommendations into three broad categories: diminishing, amplifying and
persistent based on the sign of
$\text{DelayedEffect}_{T}(\mathcal{G},\texttt{rec},[\underline{t},\overline{t}])$.
Note that the delayed impacts measure the difference in effect sizes between
time $\overline{t}$ and time $T$, rather than the difference in the metric
$m(\mathcal{G}_{T}(\texttt{rec},[\underline{t},\overline{t}]))-m(\mathcal{G}_{\overline{t}}(\texttt{rec},[\underline{t},\overline{t}]))$.
This two notions are equivalent only when the metric remains constant from
time $\overline{t}$ to time $T$ under natural network evolution dynamics.

#### Indirect effects.

The temporal evolution of the network in the presence of link recommendations
is affected _directly_ by the addition of algorithmic edges; but also
_indirectly_ as the addition of algorithmic edges biases the natural growth
dynamics. Recommendations have an indirect impact on natural dynamics in the
"Meeting Friends" phase of the network evolution model. In this phase, the
arriving nodes connect with nodes at distance 2 (neighbors of neighbors). In
the presence of algorithmic edges, there can be nodes at distance 2 that are
only reachable via algorithmic edges. If the incoming node connects to such a
node, the resulting edge is _mediated_ by recommendation. Figure 2 illustrates
the mechanism through which recommendations mediate the organic creation of
new edges.

To measure indirect impacts, we design a counterfactual experimental procedure
to remove the influence of algorithmic edges on natural growth, and thus
discount mediated edges. The counterfactual procedure is defined as follows:
upon arrival of a new node $i$, we modify the "Meeting Friends" phase to only
consider node candidates that are at distance 2 from $i$ via non-algorithmic
edges. For this counterfactual, the network contains no mediated edges, and
therefore it isolates the _direct_ effects. Formally, we refer to this
unmediated evolution as
$\tilde{\mathcal{G}}(\texttt{rec},[\underline{t},\overline{t}])$. Finally we
can define direct effects as the difference between the metric evaluated for
the unmediated counterfactual and the natural growth trajectory. The indirect
effects are the difference between total effect and direct effect:

$\displaystyle\text{DirectEffect}_{T}(\mathcal{G},\texttt{rec},[\underline{t},\overline{t}])$
$\displaystyle=m(\tilde{\mathcal{G}}_{T}(\texttt{rec},[\underline{t},\overline{t}]))-m(\mathcal{G}_{T}),$
$\displaystyle\text{IndirectEffect}_{T}(\mathcal{G},\texttt{rec},[\underline{t},\overline{t}])$
$\displaystyle=\text{Effect}_{T}(\mathcal{G},\texttt{rec},[\underline{t},\overline{t}])-\text{DirectEffect}_{T}(\mathcal{G},\texttt{rec},[\underline{t},\overline{t}]).$

Figure 2: Indirect effects of recommendation though mediated edges: Edge $A-B$
is a recommended edge and contributes to the direct effect of the
recommendation. Edge $F-A$ is a mediated edge as node $A$ would not be
reachable from $F$ in the absence of $A-B$.

#### Longitudinal evaluation.

In a simulated setting, one has access to both counterfactuals: the evolution
of the network in the presence of recommendations and in their absence. In
longitudinal evaluation which are commonly used in observational studies,
effects are measured as the difference in a metric before and after the
intervention:

$\widehat{\text{Effect}}^{obs}_{T}(\mathcal{G},\texttt{rec},[\underline{t},\overline{t}]):=m(\mathcal{G}_{T},\texttt{rec},[\underline{t},\overline{t}])-m(\mathcal{G}_{\underline{t}})$

This measure is biased whenever $m(\mathcal{G}_{T})\neq
m(\mathcal{G}_{\underline{t}})$.

#### A/B evaluation.

The lack of valid counterfactuals motivates the use of A/B testings in
settings with interventional access. In an A/B test, nodes are divided into
two groups: treatment and control; the treatment group receives
recommendations, while the control group does not. However, in networks, the
assumptions of Stable Unit Treatment Value Assumption (SUTVA) (Imbens and
Rubin, 2015) do not hold due to network interference. There are various
methods for choosing treatment nodes, as well as methods to correct for
network interference in estimation procedures.

Let a scheme for choosing treatment nodes exist and let
$\mathcal{G}^{AB}(\texttt{rec},[\underline{t}],\overline{t})$ be the network
evolution where a group of nodes are assigned to the treatment group and
receive recommendations. To estimate the impact of recommendations on a metric
of interest, we compare the metric’s values on the treatment and control
groups:

$\widehat{\text{Effect}}^{AB}_{T}(\mathcal{G},\texttt{rec},[\underline{t},\overline{t}]):=m^{treatment}(\mathcal{G}^{AB}_{T},\texttt{rec},[\underline{t},\overline{t}])-m^{control}(\mathcal{G}^{AB}_{T},\texttt{rec},[\underline{t},\overline{t}]).$

In evaluating the impact of recommendations, we consider both naive estimators
that do not correct for network interference, and more sophisticated ones that
take externalities into account.

##  4 Counterfactual evaluation

We simulate counterfactual scenarios by applying different recommendation
interventions for various time periods. Although accessing counterfactuals in
reality is infeasible, these simulations offer insights into dynamic and
temporal phenomena and provide grounded foundation for subsequent sections.

###  4.1 Setup

The results in sections 4.2 and 4.3 consider a simple experimental setup to
illustrate delayed and indirect effects of FoF and Latent333For Latent
recommendations we use softmax temperature $\beta=10$ recommenders. In the
baseline setup we consider two equally sized communities. We sample latent
embeddings for the nodes independently from $\mathcal{N}(\mu_{c},\sigma I)$,
with $\mu_{1}=[0,1]$ and $\mu_{2}=[1,0]$; for both communities the variance of
the embeddings is set to $\sigma=0.05$. We sample 50 nodes in each group and
initialize edges by connecting pairs of nodes $i-j$ with probability
proportional to the inner product of their embeddings. Then, for each node in
the network, we consider their neighbors at distance 2 and connect to them
with probability $p_{1}=0.05$. This results in a slightly homophilic initial
network with homophily $h1=h2\approx 0.1$. Upon initialization, at each time
step $5$, new nodes arrive. For natural growth we consider $N_{s}=N_{f}=100$
and the connection probability of connecting to a candidate node in the
"Meeting Friends" phase: $p_{2}=0.05$. Further, we assume a behavioral model
where nodes accept recommended edges with a constant probability, $0.5$. At
each time step we measure structural metrics of the network such as clustering
coefficient, Gini coefficient and homophily. We repeat each trajectory for 5
random seeds and report average results along with confidence bands. 444Code
for reproducing experimental results can be found here.

In section 4.4 we vary these assumptions by considering majority-minority
structure, differentiated homophily, and within group heterogeneity. Finally
in Appendix A.1 and A.2, we consider additional behavioral assumptions,
different variants of the underlying dynamics and modifications to the
recommendation algorithms.

###  4.2 Delayed effects

Figure 3: Trajectory of homophily, global clustering and Gini coefficient for
Latent and FoF recommendations applied over varying intervention intervals.
The solid black line represents the evolution of the metric under natural
growth dynamics. The blue lines represent trajectories for the Latent
recommender whereas the orange lines correspond to FoF. The shaded area
corresponds to the 95% confidence intervals calculated over 5 independent
trajectories.

We find that affinity and neighborhood based recommenders have opposite long
term effects on homophily and clustering coefficients. Latent leads to
increases in homophily as well as global clustering whereas FoF
recommendations decrease them. Both recommenders have diminishing delayed
impacts with respect to homophily and clustering as upon end of the
intervention the trajectory of the metric regresses to the counterfactual
natural growth trajectory. Meanwhile, effects with respect to the Gini
coefficient are qualitatively different. Latent recommendations lead to
increases in degree inequality both in the short and in the long term.
Conversely, FoF recommendations decrease the Gini coefficient in the short
term but increase it in the long term. Figure 3 illustrates these findings.

#### Clustering and homophily.

In the case of Latent recommendations, if nodes $i$, $j$, and $k$ are similar
in embedding space, it is likely they are from the same community and all
three pairs of edges have been proposed as recommendations, leading to a
strong bias for closing triangles within the community. This results in
increased homophily and clustering, especially with the behavioral assumption
that nodes accept links based on embedding similarity (see Fig.
LABEL:fig:bevarior_variants_c,_fig:bevarior_variants_d). The bias is further
exacerbated for high-temperature $\beta$. Conversely, with a low $\beta$
value, which corresponds to a more stochastic recommender, it is less likely
that all three edges of a given triplet will close, thus slightly lowering the
homophily value compared to the high-$\beta$ case.

The FoF recommender intervention mimics the "Meeting Friends" phase of natural
growth, which has a bias towards forming cross-community links, resulting in
decreased homophily. Decreased clustering occurs because random FoF
recommenders lack the bias of connecting nodes with a large common
neighborhood, unlike neighborhood-based models such as Adamic-Adar (Adamic and
Adar, 2003), which favor links with nodes with a large number of common
neighborhoods and thus increase clustering (see Fig. 9).

#### Gini coefficient.

The Gini coefficient for recommendations shows amplifying delayed effects. The
natural growth trend exhibits a slight increase in inequality over time.
Latent recommendations exacerbate the wealth gap between popular and unpopular
nodes through a bias towards nodes with high embedding norms, creating the
"rich-get-richer" effect. FoF recommendations initially reduce degree
inequality as they are less biased towards popular nodes. Surprisingly, once
recommendations stop, the Gini coefficient increases dramatically for both
recommenders, particularly for FoF. This is due to a "relative-rich-minority"
effect caused by differences in edge density between ’older’ and ’newer’
nodes. Over time, as natural growth is continuing to take place, the set of
"rich" nodes becomes relatively smaller, resulting in higher inequality as
measured by the Gini coefficient. If a rewiring behavioral model is assumed
(see Fig. LABEL:fig:bevarior_variants_b,_fig:bevarior_variants_d), where nodes
receiving recommendations do not change their degree, the delayed impacts
diminish, similar to the case of clustering and homophily.

###  4.3 Indirect effects

The "Meeting Friends" phase occurs in feedback loop with algorithmic
recommendations, leading to the mediation phenomenon illustrated in Figure 2.
We find that mediated edges are common, long-lasting and biased. For instance,
in the case of Latent recommendations, the mediated edges are more likely to
connect nodes from the same community, resulting in significant effects on
homophily.

Figure 4: Indirect effects: Solid black line corresponds to natural growth
dynamics. The blue and orange lines represent trajectories for Latent and FoF
recommenders, respectively. Shaded areas corresponds to 95% confidence bands.
Left plot illustrates the prevalence of algorithmically mediated edges in the
"Meeting Friends" phase of the natural growth dynamics. Dashed and dotted
lines correspond to various intervention intervals. Center plot illustrates
the bias in the mediated edges relative to unmediated ones. Solid lines
correspond to fraction of bicromatic edges among unmediated edges created in
the "Meeting Friends" phase; the dashed line records the fraction of
bicromatic mediated edges. Right plot shows the trajectory of homophily
metric. Dotted lines correspond to the trajectory of the metric for the
unmediated counterfactual evolution of the network.

#### Prevalence, persistence and bias of mediated edges.

Mediated edges play a substantial role in the "Meeting Friends" stage of
natural growth dynamics. Figure 4 shows that the proportion of mediated edges
increases with the duration of the intervention period and remains constant
even after recommendations have stopped for both Latent and FoF recommenders.
The bias of mediated edges in terms of bichromaticism is distinct for each
recommender, relative to the proportion of bichromatic edges under natural
growth. For instance, about a third of unmediated "Meeting Friends" edges are
bichromatic for both recommenders. However, the proportion of bichromatic
edges among mediated edges is lower for Latent and higher for FoF. Our
findings about homophily in Section 4.2 suggest that algorithmic edges are
biased towards monochromatic for Latent and bichromatic for FoF. This
experiment supports the hypothesis that recommendations have a compounding
effect by inducing biases in the natural network evolution.

#### Counterfactual measurements of indirect effects.

To measure the indirect effects on structural metrics, we apply the
counterfactural procedure from Section 3.3 and isolate the direct effects. We
find that without mediation, structural metrics trend faster to their natural
evolution. Figure 4 shows the comparison of homophily under natural growth,
algorithmic growth, and unmediated algorithmic growth. The difference between
the latter two quantifies the magnitude of indirect effects. Additionally,
indirect effects grow relatively stronger over time, even after
recommendations end, highlighting the persistence of indirect impacts.

###  4.4 Impact of group structure

Previous studies have highlighted the unequal impact of recommendations on
different communities, especially when they are divided into majority and
minority groups and display varying levels of homophily (Stoica et al., 2018;
Ferrara et al., 2022; Fabbri et al., 2020). Here, we examine the effects of
differential homophily between majority and minority groups as well as within-
community heterogeneity. majority and minority group and the role of within-
community heterogeneity.

#### Community heterogeneity.

We examine the impacts of homogeneity and heterogeneity in the latent
representation of nodes on the results of link recommendation. We model
within-group heterogeneity by varying the variance of the latent embeddings.
In the heterogeneous setting, we set the variance of the embedding
distribution to $\sigma^{2}=0.1$, and to $\sigma^{2}=0.01$ in the homogeneous
setting; in the extreme case when the variance is 0 this recovers the JR
variant of (Abebe et al., 2022). Figure 5 shows that when there is high
within-group heterogeneity, Latent recommendations greatly increase the global
clustering coefficient, while for high within-group homogeneity, Latent
recommenders have the opposite effect, reducing global clustering. This
unexpected phenomenon is due to Latent recommendations favoring nodes with
high embedding norms. For any existing edge $i-j$, there is a
disproportionately high chance that both $i$ and $j$ will receive
recommendations concentrated on a small subset of large-normed nodes, creating
closed triangles. In the homogeneous setting, such ’collisions’ are less
likely to occur, as the probability of two nodes independently being
recommended with the same node is lower. These findings highlight the
importance of investigating not only between-group differences but also
within-group differences.

Figure 5: Effects of within-group heterogeity. Black trajectory corresponds to
the natural evolution of the global clustering metric. Blue and orange line
indicate the trajectory for Latent and FoF recommender. Left plot: two
heterogenous groups. Center plot: two homogenous group. Right plot: one
homogenous and one heterogeneous group.

#### Homophilic minority and heterophilic majority.

In previous studies, it has been observed that homophilic minority groups
receive an excessive amount of exposure from recommendations, leading to
increased disparities in homophily between groups. We simulate the network
evolution for a majority fraction of $60\%$, $\mu_{1}=[0,1]$ and
$\mu_{2}=[1.2,1]$ and find that our results support this conclusion for Latent
recommendations. In contrast, Figure 6 shows that while FoF increases
homophily for the majority, it decreases it significantly for the minority.
The FoF recommender amplifies the visibility of the majority more than that of
the minority. This phenomenon can be explained by the heterophilic nature of
the majority group, where most of its direct connections are with minority
nodes. Though the minority group is homophilic, there is a larger probability
that a majority node $i$ is recommended with majority nodes at distance 2,
because $i$’s neighbors’ neighbors that are in the minority group are more
likely to also be direct neighbors of $i$.

Figure 6: Effect of homophily: Trajectory of average degree and community
homophily for Latent and FoF recommenders for the heterophilic majority and
homophilic minority.

###  4.5 Evaluation Biases.

As we do not have access to the full range of counterfactual measurements, we
often resort to either longitudinal or A/B evaluations. However, these methods
have their limitations and may fail to accurately capture the impacts of
recommendations. In this section, we explore the potential biases and
limitations of these evaluation procedures.

#### Longitudinal evaluation.

Simulating a longitudinal evaluation mimics an observational study. In this
setting, a single trajectory is observed. The estimated total effect of link
recommendations is the temporal difference between the value of the metric
before and after the intervention. If the underlying dynamics for a metric are
stationary, as is the case for homophily in Figure 3, then a longitudinal
evaluation is unbiased. Conversely, when under natural dynamics a metric is
non-stationary, a naive comparison between the initial network at time
$\underline{t}$ and the network at time $T$ could yield qualitatively
misleading estimates. For instance, in Figure 3, when measuring the impact of
FoF on clustering coefficient, a longitudinal measurement would compare the
solid orange trajectory between time $\underline{t}=50$ and $T=400$, thus
over-estimating about double the size of the true effects. Furthermore, the
naive observational measurement for Latent would incorrectly suggest a
negative effect on network clustering, when in reality it has a positive
impact compared to the natural evolution.

#### A/B evaluation.

A/B evaluations in dynamic networks become complex due to changes of network
structure over time caused by natural dynamics and algorithmic interventions.
We simulate A/B tests aimed at estimating causal effects of Latent and FoF
link recommendations. We perform random treatment ($p=0.5$) on node assignment
for clustering and Gini coefficient and community-based treatment assignment
for measuring homophily. The estimates in Figure 7 are adjusted for network
interference. Detailed computation of the adjusted estimates as well as the
corresponding plots for the naive cases can be found in Appendix A.3. The
quality of A/B estimates varies based on the metric and intervention. For
example, A/B tests overestimate the impact on homophily for Latent but
underestimate it for FoF. The estimation for clustering is accurate for Latent
but greatly underestimated for FoF. For the Gini coefficient, both Latent and
FoF interventions underestimate the metric for the treatment group while
accurately estimating the metric’s evolution under natural growth for the
control group. Accross most settings, the quality of the metric deteriorates
over time, further supporting the existence of dynamic effects that compound
network interference effects.

(a) Latent recommender

(b) FoF recommender

Figure 7: A/B Evaluations: Solid lines correspond to ground truth
counterfactual evaluation of homophily, clustering and Gini coefficient.
Dashed lines correspond to trajectories estimated from running an A/B test.

##  5 Discussion and Future work

In this study, we explored the dynamic effects of link recommendations on
network evolution through simulations. Our proposed extension of the Jackson-
Rogers model provides insight into the impact of link recommendations on
network structure. Emphasizing the importance of temporal dynamics and
measurement timing, our simulations revealed surprising and persistent effects
of link recommendations on network structure.

Using synthetic data and simple network evolution models and recommendation
algorithms, we answered "what-if" scenarios in a controlled setting, providing
a valuable first step in understanding these effects in real networks. Our
results showed that link recommendations can have delayed and indirect impacts
on network structure, with long-lasting effects even after recommendations
have ceased, which result in significant cascading indirect effects over time.
This highlights the need for further research in evaluating link
recommendation algorithms in dynamic networks.

Finally, our study sheds light on the potential biases that can arise in
evaluating link recommendation algorithms in dynamic networks. We find that
evaluating metrics longitudinally or using A/B tests can result in biased
estimates when the underlying network dynamics are not stationary. This
highlights the need for advanced estimation procedures that consider both
network interference and underlying dynamic effects to accurately assess the
impacts of link recommendation algorithms.

We identify several opportunities for future research. To improve the validity
of our conclusions, one avenue is to validate modeling assumptions against
real-world networks. Refining the modeling assumptions, such as allowing for
non-recommender-driven edge creation between existing nodes, could better
reflect reality and potentially result in greater indirect effects. Another
important area of interest is to further develop methods for measuring direct
and indirect effects in different network formation models.

A second direction of future work is to study the downstream impacts of link
recommendation. Modeling the node embeddings as a dynamic property, influenced
by the local neighborhood through biased assimilation (Dandekar et al., 2013)
or mere exposure effects (Curmei et al., 2022), could provide insight into how
opinion formation on networks is influenced by recommendations. Finally,
examining recommendation scenarios where network edges in the social network
are formed by content recommendations, rather than user recommendations, which
is typical of social media sites like Instagram, is a promising area for
investigation.

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##  Appendix A Additional Experiments

###  A.1 Behavioral assumptions

In this section we investigate two behavioral assumptions: edge rewiring and
choice homophily. Under edge rewiring assumption upon accepting a recommended
edge the node will remove at random one of it’s existing edges. Under choice
homophily nodes accept recommendations according to embedding similarity
rather than with a constant probability, assumed in the main body of the
paper. Figure 8 shows the trajectories of homophily, clustering and Gini
coefficient under different behavioral assumption. In the case of edge-
rewiring (Fig. 8(b), 8(d)) the delayed impacts to Gini coefficient diminish.
The behavioral assumption that nodes accept preferentially nodes with high
embedding similarity unsurprisingly leads to further increases in observed
homophily for both Latent and FoF recommenders (Fig 8(c), 8(d)).

(a) Edge addition + Constant recommendation acceptance probability (plot in
Sec. 4.2)

(b) Edge rewiring + Constant recommendation acceptance probability

(c) Edge addition + Choice homophily

(d) Edge rewiring + Choice homophily

Figure 8: Counterfactual trajectory of homophily, global clustering and Gini
coefficient for Latent and FoF recommendations applied over varying
intervention intervals. The solid black line represents the evolution of the
metric under natural growth dynamics. The blue lines represent trajectories
for the Latent recommender whereas the orange lines correspond to FoF. The
shaded area corresponds to the 95% confidence intervals calculated over 5
independent trajectories.

###  A.2 Intervention Variants

In Fig. 9(a) we consider different choices of temperature parameter $\beta$
for the Latent recommender. All recorded metrics increase as the
recommendations become more deterministic (uncreasing $\beta$). Note that
$\beta=4$ and $\beta=10$ are nearly indistinguishable in the homophily graph
whereas $\beta=4$ and $\beta=2$ are close in the graph of the clustering
coefficient. This suggests that different metrics have different sensitivities
to changes in recommendation intervention.

In Fig. 9(b) we compare FoF with a neighborhood based recommendation algorithm
that recommends node pairs with highers Adamic-Adar index. The Adamic-Adar
index assigns a higher similarity score to pairs of low degree nodes that
share many neighbors in common. This change leads to extreme increase in the
clustering coefficient.

(a) Latent recommender

(b) FoF recommender

Figure 9: Counterfactual trajectories for variants of Latent and FoF
recommenders

###  A.3 Naive and Interference Adjusted estimates for A/B evaluation

First we comment on the naive and adjuted estimation procedure for each
metric.

#### Homophily.

Naively, we estimate homophily for an A/B evaluation by assigning one
community as treatment and the other as control and measuring the homophily
for each community. To adjust the estimate for the control group, we discount
the algorithmic edges received by the control group, whereas to adjust for the
treatment, we double the count of the algorithmic edges that the treatment
group forms with the control group.

#### Clustering.

Naively, we estimate the group-level clustering coefficient for treatment and
control by averaging node-level clustering for treatment and control groups
respectively. We adjust the estimate for the treatment/control group by
calculating its clustering coefficient only among the treatment/control
neighbors, i.e., in a sub-network with only the treatment/control nodes.

#### Gini coefficient.

We naively estimate Gini coefficient by simply measuring the value within the
treatment and control groups, respectively. We adjust Gini coefficient by
discounting the algorithmic edges for the control group.

Fig. 10 shows the naive estimates derived from the A/B evaluation.

(a) Latent recommender

(b) FoF recommender

Figure 10: A/B Evaluations: Solid lines correspond to ground truth
counterfactual evaluation of homophily, clustering and Gini coefficient.
Dashed lines correspond to trajectories estimated _naively_ from running an
A/B test.