# Echo chamber effects in signed networks

Antoine Vendeville  Corresponding author<EMAIL_ADDRESS>médialab, Sciences Po, 75007 Paris, France  Complex Systems Institute of Paris
Île-de-France (ISC-PIF) CNRS, 75013 Paris, France  Learning Planet Institute,
Research Unit Learning Transitions, 75004 Paris, France  Fernando Diaz-Diaz
<EMAIL_ADDRESS>IFISC (UIB-CSIC), Institute for Cross-
Disciplinary Physics and Complex Systems, Campus Universitat de les Illes
Balears, 07122 Palma de Mallorca, Spain

###### Abstract

Echo chamber effects in social networks are generally attributed to the
prevalence of interactions among like-minded peers. However, recent evidence
has emphasized the role of hostile interactions between opposite-minded
groups. Here, we model information propagation between such groups by
generalizing popular contagion models to signed networks. We show that echo
chambers spontaneously emerge in balanced networks, and in antibalanced ones
for specific parameters. The robustness of our results is confirmed through
simulations on various network topologies, including a real-world dataset.

Echo chambers, Signed networks, Structural balance, Social networks

The study of echo chambers has been central in research about online social
platforms. By continuously reinforcing pre-existing views [1, 2], these
tightly knit communities of like-minded users exacerbate the polarization of
opinions and in turn, the fragmentation of societies [3, 4, 5, 6]. They have
been observed at multiple levels, from the connections established by the
users themselves, to personalised timelines of content curated by engagement-
seeking algorithms [7, 8].

In recent years however, this evidence has been increasingly challenged [9,
10, 11, 12, 13, 14]. Scholars have pointed out that there is communication
between opposite-minded communities, often in the form of hostile interactions
[15, 16, 17]. Opposite-minded users may frame the same piece of information
under different lights, reinforcing ideological divides while still allowing
information to travel through partisan lines. Negative relations might be even
more prevalent than positive ones for some users, such as journalists [16].
Thus, it is primordial that we distinguish between friends and foes when
studying online debates [18].

The presence of hostile interactions among individuals is often modelled as a
signed network with positive and negative links [19]. The inclusion of
negative links gives rise to novel phenomena, such as structural balance [20,
21]. A signed network is perfectly balanced if it can be divided into two
groups, so that all links between agents of the same group are positive and
all links between agents of different groups are negative [20]. In other
words, it contains two groups that are antagonistic towards one another [22,
23]. While perfectly balanced networks are a mathematical idealization,
studies have consistently shown a tendency towards structural balance in
empirical networked systems of various types [24, 25, 26, 27]. On the other
hand, when all links within each group are mostly negative and links between
groups are mostly positive, the network is said to be antibalanced [28]. While
not as common as balanced networks, antibalanced networks can emerge in
historical, social and political contexts, like international relations during
the Thirty Years’ War [29] or the U.S. Congress during the Carter and Reagan
administrations (see [22] and the Appendix, Sec. D).

These findings have motivated the introduction of the notion of degree of
balance, with indices based on triangles [27, 22], random walks [30, 31, 32]
or frustration metrics [33, 34], among others. Recent research has discovered
intriguing connections between the degree of balance of social and political
networks and their tendency towards polarization [22, 34]. The degree of
balance also impacts the evolution of various dynamical processes [35, 36,
32], including contagion models [37, 38, 39], random walks [40], diffusive
dynamics [41] and the voter model [42]. Conversely, structural balance can
spontaneously arise as a result of opinion dynamics on signed networks [43,
44].

Within this context, our contribution is two-fold. First, we extend to signed
networks two popular models of information propagation, the Independent
Cascade Model and the Linear Threshold Model [45, 46]. The key idea driving
these extensions is that a single piece of information may be framed in
different ways between, and within, antagonistic groups. Then, we use those
models to analyze the key role played by the degree of balance in the
emergence of echo chambers. We uncover a sudden transition as the degree of
balance becomes positive, and an anomalous region where echo chambers can
emerge in networks with negative degree of balance. Echoing empirical
observations on the correlation between activity and polarisation [47, 48, 1,
49], we find reinforced echo chambers when information spreads more easily.
The robustness of our results is confirmed on different network topologies,
including a real-life dataset. As a by-product, we also provide a simple
procedure to transform an unsigned network into a signed one with a prescribed
degree of balance.

#### Signed social networks.

We consider an undirected signed network with $N$ agents. Each agent is
assigned a latent group membership, which represents the divergent opinions
within the population. If all in-group interactions are friendly and all out-
group interactions are hostile, then the network is perfectly balanced. In the
opposite case, where in-groups are always hostile towards each other and
friendly with out-groups, we call the network perfectly antibalanced [28]. To
interpolate between these two extremes, we employ the triadic degree of
balance $\tau$, commonly used in the literature [50, 27, 51, 22]. It is
defined as:

$\tau=\text{tr}(W^{3})/\text{tr}(|W|^{3}),$ (1)

where $W$ is the signed adjacency matrix of the network and $|W|$ its
entrywise absolute value. In perfectly balanced networks $\tau=+1$, and in
perfectly antibalanced ones $\tau=-1$. Values in-between quantify intermediate
degrees of balance, with $\tau\approx 0$ for networks that are neither
significantly balanced nor antibalanced. The definition of $\tau$ is based on
the product of link signs in triangles, which have been found to be an
accurate and computationally fast proxy for balance 111Triangles with a
positive product of link signs famously respect the old adage “the friend of
my friend is my friend, the enemy of my friend if my enemy, the enemy of my
enemy is my friend”.. Hereafter, we employ the terms balanced and antibalanced
to refer to networks with $\tau>0$ and $\tau<0$ respectively, while we reserve
the terms perfectly (anti)balanced for the cases $\tau=\pm 1$.

The majority of our simulations are run on networks with an underlying Erdös-
Rényi (ER) topology with connection probability $p$. Hence, they do not
exhibit communities from an unsigned point of view. Nevertheless, group
memberships can induce mesoscopic structures affecting the degree of balance.
We thus attribute signs to the links following a procedure that guarantees a
pre-specified triadic degree of balance on average. This procedure is inspired
by [53] and detailed in the Appendix, Sec. A.

#### Information propagation.

We assume that an item is propagating throughout the network: a journal
article, a tweet, a picture, etc. To describe its propagation, we adapt the
archetypal Independent Cascade Model of simple contagion [45]. In this model,
agents are initially all inactive (state 0), except for a few _seeds_ who are
the original emitters of the item (state 1). Their neighbors are thus exposed
to the item, and may choose to share it themselves. Formally, in consecutive
steps, each agent that was activated in the previous step activates each of
its neighbors individually with probability $\lambda$. This probability is
called the activation rate. If all attempted activations in a single step
fail, the process stops. To account for signed links, we add a novel feature
to these dynamics and propose the Signed Independent Cascade Model (SICM).
Active agents can now be in one of two states $\\{-1,+1\\}$, reflecting two
opposite-minded ways to frame the information: a fresh government decision may
spark praise from some and criticism from others. When an agent is activated
through a positive (resp. negative) link, they adopt the same (resp. opposite)
state of the agent responsible for their activation. Once an agent is active,
its state is fixed and never changes. More details can be found in the
Appendix, Sec B.2.

The SICM relies on a simple contagion mechanism: one exposure to information
is enough to trigger a potential activation. In contrast, complex contagion
mechanisms necessitate multiple exposures. This creates a different
phenomenology [54, 55, 56] that has been argued to better model realistic
behavioral dynamics [57, 58, 59]. For this purpose, we propose the Signed
Linear Threshold Model (SLTM), adapted from [45, 46]. Similarly as for the
SICM, agents can be in one of two states $\\{-1,+1\\}$ depending on who
triggered their activation. Agents activate if the aggregated influence of
their neighbors exceeds a threshold $T$—modelling details can be found in the
Appendix, Sec. B2. While most of our results are presented for the SICM, we
show at the end that the SLTM exhibits very similar behaviors.

#### Echo chamber effect

To measure the intensity of the echo chamber effect, we adapt a metric from
[60]. Let $\rho_{ab}^{+}$ denote the expected fraction of agents in state $+1$
among all the active agents in group $b$, at the end of a process with all
seeds belonging to group $a$. Similarly, we define
$\rho_{ab}^{-},\rho_{ba}^{+},\rho_{ba}^{-}$ as well as
$\rho_{aa}^{+},\rho_{aa}^{-},\rho_{bb}^{+},\rho_{bb}^{-}$, excluding the seeds
from the computation of the last four. The echo chamber strength is defined as

$E=\frac{\rho_{aa}^{+}-\rho_{aa}^{-}-\rho_{ab}^{+}+\rho_{ab}^{-}+\rho_{bb}^{+}-\rho_{bb}^{-}-\rho_{ba}^{+}+\rho_{ba}^{-}}{4}.$
(2)

In the Appendix, Sec. C.2. we prove that $E$ is a second-order moment; thus,
we can interpret $E$ as a measure of the correlation between states and group
memberships. The maximum $E=+1$ corresponds to perfect echo chambers, where
everyone inside a group is in the same state, opposite to that of the other
group. The minimum $E=-1$ is the case of an _anti-echo_ chamber, where agents
within the seed group disagree with the seeds, while agents of the opposite
group agree with the seeds. When $E\approx 0$, there is no significant echo
chamber effect. For the sake of conciseness, we refer to $E$ simply as the
echo chamber effect, bearing in mind that it is actually a metric that
quantifies this effect.

#### Results.

Figure 1: Echo chamber effect on signed ER networks of various densities $p$.
On the right half of the heatmaps are balanced networks $(\tau>0)$, on the
left half are antibalanced networks $(\tau<0)$.

Figure 2: Echo chamber effect in balanced ER networks. (a) $E$ as a function
of the $\tau$ for various $\lambda$. (b) $E$ as a function of $\lambda$ for
various $\tau$. (c) Activation rates $\lambda$ realising the minimum and
maximum echo chamber effect for various $p$, averaged over all values of
$\tau$. (d) Density $p$ realising the minimum and maximum echo chamber effect
for various $\lambda$, averaged over all values of $\tau$. Shaded areas cover
one standard deviation from the mean.

We assess the echo chamber effect via simulations of the SICM on ER networks
with $N=250$ agents. Figure 1 shows $E$ as function of $\tau$ and $\lambda$ on
networks of various densities $p$. The difference between balanced ($\tau>0$)
and antibalanced ($\tau<0$) networks is immediately striking, with a
transition between the two that gets sharper as $p$ increases. In balanced
networks, $E$ increases monotonically with $\tau$, as shown in Fig.2(a). Thus,
we find that a stronger balance entails a higher echo chamber effect. In other
words, echo chambers, much like polarization [22, 31] and dissensus [41], are
amplified by structural balance in signed social networks.

Furthermore, $E$ is in fact not monotonous with respect to $\lambda$, as shown
in Fig. 2(b). There exists a nontrivial value of $\lambda$ for which $E$ is
minimal, which decreases continuously with the network density (Fig. 2(c)). In
contrast, the argmax of $E$ jumps discontinuously from $\lambda\approx 0$ to
$\lambda\approx 1$. The same phenomenon is observed when computing the argmax
of $E$ over $p$ (Fig. 2(d)). This suggests that minimizing the echo chamber
effect is simpler than maximizing it: the optimal values of $\lambda$ and $p$
regarding minimization span a narrower range of values and are less
susceptible to perturbations, whereas the optimal values for maximization vary
widely between $0$ and $1$ and can change abruptly in the critical region.

In antibalanced networks, one could expect the same behavior, with $E$
decreasing monotonously towards $-1$ as $\tau$ becomes more negative. As we
see in Fig. 1 however, this is not the case. Instead, we find that $E$ changes
non-monotonously with both $p$ and $\lambda$ and $\tau$ (see Appendix, Sec.
D). Specifically, $E$ reaches its minimum value, $E=-1$, both when $p$ and
$\lambda$ are very high (e.g. $p=\lambda=1$) and when they are very low (e.g.
$p=0.025,\lambda=0.01$). Perhaps most surprisingly, certain regions of the
$(\tau<0,\lambda)$ half-plane display positive echo chamber effects. This
anomalous region, corresponding to the red semi-circular area of Fig. 1,
shifts downwards as the network density $p$ increases, and almost vanishes at
high densities. Interestingly, Figure 3(a) reveals that for any given $\tau$,
the boundaries of the anomalous region with respect to $p$ and $\lambda$ do
not move. Instead, the value of $\tau$ only influences the strength of this
anomaly.

Figure 3: (a) Echo chamber effect as a function of network density and
activation rate in antibalanced ER networks, for three different $\tau$. (b)
Two realisations of the SICM on a signed ER network with $p=0.5,\tau=-1$, for
$\lambda=0.08$ (echo chamber) and $\lambda=0.30$ (anti echo chamber). Circles
correspond to agents of group $a$ and triangles to agents of group $b$. Agents
are colored according to their state: grey for state $0$, green for state $+1$
and pink for state $-1$.

To better understand this phenomenon, we compare how the propagation unfolds
for two different values of the activation rate. Figure 3(b) shows simulations
of the SICM on a highly dense, perfectly antibalanced network ($\tau=-1$) with
$\lambda=0.08$ and $\lambda=0.30$. Because $\tau=-1$, agents activated during
odd steps adopt state $-1$ if they are in the seed group and state $+1$
otherwise, while the opposite happens during even steps. For $\lambda=0.30$,
most activations occur in the first step, leading the seeding group to
predominantly adopt state $-1$, resulting in a negative echo chamber effect.
However, with $\lambda=0.08$, fewer activations happen initially, and most
agents are activated in the second step. This causes the seeding group to
adopt state $+1$, and generates a positive echo chamber effect even though the
network is perfectly antibalanced. This alternance mechanism leads us to
believe that this phenomenon is due to the synchronous aspect of the
propagation process. Indeed, we show in the Appendix, Sec. D that these
anomalies disappear with asynchronous activations. Additionally, the exact
shape of the anomalous regions seem to change slightly with the network size
(Appendix, Sec. D).

#### Robustness.

To demonstrate the robustness of our approach, we show that similar results
are achieved _(i)_ on networks with different topologies, and _(ii)_ with our
complex contagion model. In Fig. 4, we apply the SICM on a collection of
networks with similar densities but various topological features. The
Stochastic Block Model (SBM) generates networks with community structure, the
model of Watts-Strogatz (WS) generates networks with clustering, and the model
of Barabasi-Albert (BA) generates scale-free networks. Additionally, we
consider a network extracted from a real-life dataset and exhibiting all of
the above, Polblogs [61] (details in the Appendix, Sec. D). The echo chamber
effect emerges when $\tau>0$ for every network, confirming that structural
balance favors echo chambers regardless of the topology. Moreover, the
anomalous region is also present for every network except the Watts-Strogatz
one (Fig. 4f), due to clustering hindering the dominance of even-length paths.

For complex contagion, we rely on the SLTM. Agents can now only activate if a
sufficiently high number of their neighbors are active themselves. Even with
this fundamental difference, our main results regarding the emergence of echo
chambers hold. Figure 5 shows that SLTM dynamics again lead to a positive echo
chamber effect when $\tau>0$, and an anomalous region in the $\tau<0$ half-
plane. Slight differences with the SICM include are the sharper transitions,
an increased presence of the anti-echo chamber effect in the $\tau<0$ half-
plane, and the persistence of a few anomalous regions under asynchronous
activations (Appendix, Sec. D). Overall, these results suggest that the echo
chamber phenomenology is robust with regards to the dynamical model employed.

Figure 4: Echo chamber effect with different networks topologies. (a) ER
network with $p=0.025$. (b) SBM network with in-group link density
$p_{in}=0.04$ and out-group link density $p_{out}=0.01$. Community memberships
match group memberships. (c) WS network with average degree $k=6$ and rewiring
probability $q=0.5$ (low clustering). (d) BA network with average degree
$k=6$. (e) Polblogs network [61]. (f) WS network with average degree $k=6$ and
rewiring probability $q=0.1$ (high clustering). All networks have and $N=250$
agents, except Polblogs with $N=1,222$. Figure 5: echo chamber effect
resulting from a complex contagion process (SLTM) across various topologies.
Same networks as in Fig. 4.

#### Discussion.

Our study makes two key contributions. First, we propose two opinion contagion
models, the SICM and the SLTM, that address how information can be framed
differently among antagonistic groups. Second, and more importantly, we employ
these models to describe the role of structural balance and connectivity in
the emergence of echo chambers. Our findings align with previous results
connecting polarization with structural balance [31, 22, 34] and connectivity
[43]. However, while those studies focused on structural properties, ours is
the first to emphasize the impact of these on information spreading. Our
results hold for both simple and complex contagion and in a variety of network
topologies, as well as in real networks.

This work represents a first step towards incorporating the effect of negative
links in the study of echo chambers, with potential extensions to incorporate
temporal activity of agents [1], non-Markovian dynamics [62] or homophily [60,
63]. Moreover, the SICM and SLTM show nontrivial regimes and transitions that
could be explored more thoroughly. Future works could also consider the
effects of structural balance beyond triangles [51, 30], other information
transmission biases [60] and examine echo chambers at microscopic and
mesoscopic levels [51]. The comparison between theoretical predictions and
empirical data (e.g. [6, 57]) is also key to assess the validity of our
modelling assumptions. Finally, our results could be applied to mitigate echo
chamber effects and polarization in recommendation algorithms [64].

###### Acknowledgements.

We want to thank the young researchers of the Complex Systems Society for
financing a research visit of AV to IFISC through the Bridge Grants program,
that served as the primary basis for this work. We also thank Maxi San Miguel
for insightful discussions and David Abella for suggesting improvements to the
first draft. FDD thanks financial support from MDM-2017-0711-20-2 funded by
MCIN/AEI/ 10.13039/501100011033 and FSE, as well as project APASOS (No.
PID2021-122256NB-C22). In the early stages of this research, AV was funded by
the UK EPSRC grant EP/S022503/1 that supports the Centre for Doctoral Training
in Cybersecurity delivered by UCL’s Departments of Computer Science, Security
and Crime Science, and Science, Technology, Engineering and Public Policy.

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##  Appendix A Networks generation

To generate signed networks in which to run the dynamical processes, we always
follow a three-steps procedure. First, we create an unsigned network through
standard generative algorithms, like the Erdös-Rényi or Barabasi-Albert ones.
Second, we assign a group to each of the agents in the network. In Erdös-
Rényi, Watts-Strogatz and Barabasi-Albert networks, each agent is assigned
randomly to one of two groups of the same size, whereas for the Stochastic
Block Model and Polblogs, the group labels correspond to the communities
existing in the network. Third, we attribute signs to links as follows: links
whose endpoints fall within the same group are negative with probability
$\eta$ and positive otherwise; whereas links whose endpoints fall in different
groups are positive with probability $\eta$ and negative otherwise. The
parameter $\eta$ regulates the degree of balance in the network. For $\eta=0$
the network is (perfectly) balanced, for $\eta=1/2$ link signs are
uncorrelated with group memberships, while for $\eta=1$ we obtain an
antibalanced network. Intermediate values allow us to interpolate between the
different degrees of balance. One may then think of $\eta$ as a noise
parameter.

To generate networks with a specific triadic degree of balance $\tau$, we make
use of the following nonlinear relation, proven in Sec. C.1 below:

${\tau}=(1-2\eta)^{3}$ (3)

on expectation. Therefore, setting $\eta=(1-\tau^{1/3})/2$ will yield an
ensemble of networks with average degree of balance $\tau$. This result is
topology-independent and holds for any given network. We use it to control
precisely the degree of balance of the networks we generate for our
experiments. The relation is illustrated in Fig. 6 for networks of various
topologies.

Figure 6: Degree of balance as a function of noise, for various networks: ER
with density $p=0.1$, ER with density $p=1.0$, BA with average degree $k=3$,
and Polblogs [61]. Each point is averaged over 10 random assignments of link
signs. Vertical bars indicate one standard deviation from the mean.

Polblogs is a real-life dataset describing connections between political blogs
in the US in 2003 [61]. Each node corresponds to a blog and is in one of two
communities, Democrat or Republican. The presence of a link between two agents
indicates that one mentioned the other. We set group memberships to correspond
to community memberships. We restrict ourselves to the largest connected
component of the network, which has $1,222$ agents and $19,021$ links.

Table 1 summarizes the main characteristics of the networks. We highlight the
fact that Polblogs exhibits both a community structure as the SBM, and local
clustering as the WS model. While it is not exactly scale-free like the BA
model, its degree distribution is well-fitted by a power-law with exponential
cutoff (see [61]), which implies the presence of few hubs and many leaves.

| ER | BA | WS $(q=0.1)$ | WS $(q=0.5)$ | SBM | Polblogs
---|---|---|---|---|---|---
Average density | 0.025 | 0.024 | 0.024 | 0.024 | 0.025 | 0.0224
Hubs and leaves | - | ✓ | - | - | - | ✓
Communities | - | - | - | - | ✓ | ✓
Average modularity | - | - | - | - | 0.84 | 0.41
Average clustering coefficient | 0.02 | 0.08 | 0.45 | 0.1 | 0.03 | 0.32
Table 1: Basic statistics of the networks considered.

##  Appendix B Simulation details

###  B.1 Signed Independent Cascade Model (SICM)

The simulation process is as follows. To compute $E$ for Erdös-Rényi networks
with density $p$ and degree of balance $\tau$, we first generate $m=20$ such
networks via the process described in Sect. A above. Then we perform $2m$ of
the SICM with activation rate $\lambda$ on each of these networks, with $m$
simulations seeded in group $a$, and $m$ simulations seeded in group $b$. The
number of seeds is chosen to be $10\%$ of the seeding group every time. The
remaining agents are initially inactive. At each step, each agent that was
activated in the previous step tries to activate independently each of its
neighbours. The probability that a given neighbor is activated this way, is
the activation rate $\lambda$. If an agent becomes activated through a
positive link, it inherits the state ($+1$ or $-1$) of the agent that
activated it. If it is activated through a negative link, it instead adopts
the opposite state. Once an agent has tried to activate all its inactive
neighbors, it ceases to try to activate them in future steps—in other words,
each agent tries to activate its neighbors a single time.

For each simulation, we use a breadth-first search algorithm. Contrary to the
traditional Independent Cascade, here the specific agent that activates
another one matters in the SICM. For example, consider a inactive agent with
two neighbors in state $+1$, where one of the neighbors is connected through a
positive link and the other through a negative one. The final state of the
agent will change depending on which of the agents activates it. Moreover,
since the agent cannot update its state once activated, the fist agent that
tries to activate has a better chance of propagating its state. To circumvent
this issue, we randomize the order in which neighbors try to activate the
agent. Finally, we restart the simulation if no agents were activated in a
single community. This way, the values of $\rho$ and $E$ are well defined.

We also compare the results with an asynchronous version the process. Whenever
an agent becomes activated, each of its neighbors are placed in a queue. Then
at each step, an inactive agent is picked at random from the queue and
activates with probability $\lambda$. If it succeeds, it adopts the same state
as the agent that triggered this activation. This has a clear impact on the
results, as the anomalous regimes in the $\tau<0$ region disappear (Figs. 10,
11).

###  B.2 Signed Linear Threshold Model (SLTM)

We propose the SLTM to adapt the archetypal Threshold Model [45] for signed
networks. As for the SICM, all agents are initially in state $0$, except a few
_seeds_ in state $+1$, who all belong to the same group. At each step, every
inactive agent $i$ is selected. The average influence from its neighbours is
computed as:

$\phi_{i}(t)=\frac{1}{k_{i}}\sum_{j=1}^{k_{i}}w_{ij}x_{j},$ (4)

where $k_{i}$ is the degree of agent $i$ (independent of the signs of the
links), $x_{j}\in\\{-1,0,+1\\}$ is the state of agent $j$ and $w_{ij}$ the
sign of the link between $i$ and $j$. Notice that both $w_{ij}$ and $x_{j}$
can be negative, so $\phi_{i}\in[-1,1].$ We then update the state of $i$ as
follows.

$x_{i}\leftarrow\begin{cases}+1&\text{if}\quad\phi_{i}(t)>T,\\\
+1&\text{if}\quad\phi_{i}(t)=T\quad\text{with probability }1/2,\\\
-1&\text{if}\quad\phi_{i}(t)<-T,\\\
-1&\text{if}\quad\phi_{i}(t)=-T\quad\text{with probability }1/2,\\\
0&\text{otherwise}.\end{cases}n$ (5)

The threshold $T\in(0,1)$ quantifies the difficulty with which information
flows through the network. Thus, $1-T$ is akin to $\lambda$ in the SICM, as it
measures how easily information travels. Note that all activations happen
simultaneously each step. In Fig. 12 we compare the results with an
asynchronous version the process, where agents are activated one by one in a
random order. This has a clear impact on the results, as the anomalous regimes
in the $\tau<0$ region (almost) disappear.

The simulation process is similar as for the SICM. To compute $E$, we first
generate $m=20$ such networks via the process described in Sect. A above. Then
we perform $2m$ of the SLTM with threshold $T$ on each of these networks, with
$m$ simulations seeded in group $a$, and $m$ simulations seeded in group $b$.
The number of seeds is chosen to be $10\%$ of the seeding group every time.

##  Appendix C Mathematical proofs

###  C.1 Triadic balance as a function of noise

We prove that for any network,

$\mathbb{E}[\tau]=(1-2\eta)^{3},$ (6)

where $\tau$ is the degree of triadic balance and $\eta$ is the level of noise
as defined in Sec. A above. Given a signed network, consider a triangle $T$ in
this network. The triangle has its own degree of balance $\tau(T)$. In the
absence of noise ($\eta=0$), we have $\tau(T)=1$. If $\eta>0$, each link in
this perfectly balanced version of $T$ is flipped with probability $\eta$.

  * •

With probability $(1-\eta)^{3}$ no link is flipped, and $T$ stays balanced.

  * •

With probability $3(1-\eta)^{2}\eta$ exactly one link is flipped, and $T$
becomes antibalanced.

  * •

With probability $3(1-\eta)\eta^{2}$ two links are flipped, and $T$ stays
balanced.

  * •

With probability: $\eta^{3}$ all three links are flipped, and $T$ becomes
antibalanced.

The expected degree of balance of $T$ after the flips is

$\displaystyle\mathbb{E}[\tau(T)]$
$\displaystyle=(1-\eta)^{3}-3(1-\eta)^{2}\eta+3(1-\eta)\eta^{2}-\eta^{3}$
$\displaystyle=(1-2\eta)^{3}.$ (7)

The expected degree of balance $\mathbb{E}[\tau]$ of the whole graph is the
average of $\mathbb{E}[\tau(T)]$ over all triangles $T$ after the flips, hence
$\mathbb{E}[\tau]=(1-2\eta)^{3}$ as well.

###  C.2 The echo chamber effect is a second order moment

Let $a,b$ denote two complementary subsets of $V$ representing the two groups
in the network, and let $f$ be a vector indicating the group of each agent:
$f_{i}=+1$ if $i\in a$, $f_{i}=-1$ if $i\in b$. Similarly, let $I^{\pm}$
represent the set of agents with positive and negative state at the end of the
contagion process, and let $x$ be a vector defined as follows: $x_{i}=\pm 1$
if $i\in I^{\pm}$, $x_{i}=0$ otherwise. Furthermore, let $s$ denote the set
containing the seeds. Based on these definitions, we can express the fraction
of agents in a final state $\pm 1$, conditioned on the seeds belonging to
group $F_{s}$ and the analyzed agents being active and belonging to group
$F_{t}$, as:

$\displaystyle\rho_{F_{s}F_{t}}^{\pm}=\frac{|\left\\{i\in F_{t}:x_{i}=\pm
1,s\subset F_{s}\right\\}|}{N_{I}(F_{s},F_{t})}$ (8)

where $N_{I}(F_{s},F_{t})$ denotes the final number of active agents belonging
to group $F_{t}$. Note that this quantity also depends on the seed group
$F_{s}$, as the seed group can affect the efficiency of the contagion process
and thus the final number of active agents.

By the law of large numbers, and assuming independent final states (pair
approximation [65]), the fraction of active agents in a final state
$x_{i}\in\\{\pm 1\\}$ is approximately equal to the probability of a final
state $x_{i}$ in agent $i$ conditioned on $i$ being active and belonging to a
given target group $F_{t}$, and the set of seeds belonging to group $F_{s}$:

$\displaystyle\rho_{F_{s}F_{t}}^{\pm}$ $\displaystyle=\frac{|\left\\{i\in
F_{t}:x_{i}=\pm 1,s\subset F_{s}\right\\}|}{N_{I}(F_{s},F_{t})}$
$\displaystyle\approx\mathbb{P}[x_{i}=\pm 1|x_{i}\neq 0,s\subset F_{s},i\in
F_{t}].$ (9)

To obtain an unconditioned expectation of the number of active agents on a
given state, we employ the law of total expectation,
$\mathbb{E}_{X,Y}[X]=\mathbb{E}_{Y}[\mathbb{E}_{X}[X|Y]]$, where the subindex
indicates the random variable over which the expectation is taken. In this
case, the conditional random variable $Y$ corresponds to both the seed and
target groups $F_{s}$ and $F_{t}$:

$\displaystyle\mathbb{E}_{x,F_{s},F_{t}}[x_{i}|x_{i}\neq 0]$
$\displaystyle=\mathbb{E}_{F_{s}}[\mathbb{E}_{F_{t}}[\mathbb{E}_{x}[x_{i}|x_{i}\neq
0,s\subset F_{s},i\in F_{t}]]]$
$\displaystyle=\frac{1}{4}\sum_{F_{s}\in\\{a,b\\}}\sum_{F_{t}\in\\{a,b\\}}\sum_{x_{i}\in\\{\pm
1\\}}x_{i}\mathbb{P}[x_{i}|x_{i}\neq 0,s\subset F_{s},i\in F_{t}]$
$\displaystyle=\frac{1}{4}\sum_{F_{s}\in\\{a,b\\}}\sum_{F_{t}\in\\{a,b\\}}\sum_{x_{i}\in\\{\pm
1\\}}x_{i}\rho^{x_{i}}_{F_{s}F_{t}}$
$\displaystyle=\frac{1}{4}\left(\rho^{+}_{aa}-\rho^{-}_{aa}+\rho^{+}_{ba}-\rho^{-}_{ba}+\rho^{+}_{ab}-\rho^{-}_{ab}+\rho^{+}_{bb}-\rho^{-}_{bb}\right).$
(10)

where the factor $1/4$ comes from assuming equally likely group membership, so
that

$\mathbb{P}(s\subset a)=\mathbb{P}(s\subset b)=\mathbb{P}(i\in
a)=\mathbb{P}(i\in b)=1/2.$ (11)

We have thus proved that the expected value of $x_{i}$ (conditioned on being
active) is linearly dependent on the variables $\rho_{ab}^{\pm}$, accounting
for the sign. We can now do a similar argument to relate the echo chamber
effect to these variables. To do so, we first define a variable $c_{si}$
indicating whether the seed and target groups agree:

$\displaystyle c_{si}=\begin{cases}+1\quad\textrm{if}\quad f_{s}=f_{i}\\\
-1\quad\textrm{if}\quad f_{s}\neq f_{i},\end{cases}$ (12)

where $i$ is an arbitrary agent, $s$ is the set of seeds and all seeds belong
to the same group. Alternatively, we can express $c_{si}$ as a product of the
groups of $i$ and $s$: $c_{si}=f_{s}f_{i}$. Note that, since the group
assignations can be random, $c_{si}$ is a random variable and we may define a
joint probability of $x_{i}$ and $c_{si}$. We now prove that $E$ is a second-
order moment of this joint distribution using the methodology employed before:

$\displaystyle\mathbb{E}_{x,F_{s},F_{t}}[x_{i}c_{si}|x_{i}\neq 0]=$
$\displaystyle=\mathbb{E}_{F_{s}}[\mathbb{E}_{F_{t}}[\mathbb{E}_{x}[x_{i}c_{i}|x_{i}\neq
0,s\subset F_{s},i\in F_{t}]]]$
$\displaystyle=\frac{1}{4}\sum_{F_{s}\in\\{a,b\\}}\sum_{F_{t}\in\\{a,b\\}}\sum_{x_{i}\in\\{\pm
1\\}}x_{i}f_{i}f_{s}\times$ (13)
$\displaystyle\quad\quad\quad\times\mathbb{P}[x_{i}|x_{i}\neq 0,s\subset
F_{s},i\in F_{t}]$
$\displaystyle=\frac{1}{4}\left(\rho^{+}_{aa}-\rho^{-}_{aa}-\rho^{+}_{ba}+\rho^{-}_{ba}-\rho^{+}_{ab}+\rho^{-}_{ab}+\rho^{+}_{bb}-\rho^{-}_{bb}\right)$
(14)

The last expression is the echo chamber effect, so we have proven that $E$ is
a second-order moment of the joint distribution of the final states $x_{i}$
and the seed-target group agreeement variable $c_{si}$.

##  Appendix D Additional figures

Figure 7 shows an example of antibalance in a real-world, signed network from
[22]. The agents are members of the U.S. congress and the sign of links
connecting them is based on bill co-sponsorships. We see that during the
Carter and Reagan administrations (1977-1985), the House of Representatives
exhibits a negative degree of balance, and so does the Senate between 1979 and
1983 (96th-97th congress). Thus, in these instances, congressmembers were co-
sponsoring bills with colleagues from the other party more than within their
own ranks.

Figure 8 gives another perspective on the highly heterogeneous evolution of
$E$ with $\lambda$ and $\tau$ in antibalanced networks and under the SICM.
Also for antibalanced networks and SICM, Fig. 9 illustrates how the anomalous
region changes for simple contagion with a different value of $N$, here $100$
instead of $250$ in the article. The anomalous region has a highly similar
shape but is slightly bigger. Finally, Figs. 10 and 11 show that the anomalous
region disappear with asynchronous updates in the SICM. They also mostly
disappear in the SLTM, except for a tiny one that persists in the Polblogs
network (Fig. 12). This last plot was obtained using the SLTM—see Sec. B.2
above for more details.

Figure 7: Degree of Balance in both chambers of the U.S. congress, from 1973
(93rd congress) to 2017 (111st congress).

(a)

(b)

Figure 8: Echo chamber effect in antibalanced ER networks. (a) $E$ as a
function of the $\tau$ for various $\lambda$. (b) $E$ as a function of
$\lambda$ for various $\tau$.  Figure 9: Echo chamber effect as a function of
network density and activation rate in antibalanced ER networks with $N=100$
agents, for three different $\tau$.  Figure 10: Echo chamber effect in the
SICM with asynchronous updates, on ER networks with various densities $p$.
Figure 11: Echo chamber effect in the SICM with asynchronous updates. (a) ER
network with $p=0.025$. (b) SBM network with in-group link density
$p_{in}=0.4$ and out-group link density $p_{out}=0.1$. Community memberships
match group memberships. (c) WS network with average degree $k=6$ and rewiring
probability $q=0.5$ (low clustering). (d) BA network with average degree
$k=3$. (e) Polblogs network [61]. (f) WS network with average degree $k=6$ and
rewiring probability $q=0.1$ (high clustering). All networks have and a
density between $0.022$ and $0.025$, and $N=250$ agents—except Polblogs with
$N=1,222$. Figure 12: Echo chamber effect in the SLTM with asynchronous
updates. (a) ER network with $p=0.025$. (b) SBM network with in-group link
density $p_{in}=0.4$ and out-group link density $p_{out}=0.1$. Community
memberships match group memberships. (c) WS network with average degree $k=6$
and rewiring probability $q=0.5$ (low clustering). (d) BA network with average
degree $k=3$. (e) Polblogs network [61]. (f) WS network with average degree
$k=6$ and rewiring probability $q=0.1$ (high clustering). All networks have
and a density between $0.022$ and $0.025$, and $N=250$ agents—except Polblogs
with $N=1,222$.