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##  Appendix A Approximation for the dependence of characteristic mass for UV
suppression on $z_{\rm rei}$

We use the simulations of Okamoto et al. (2008) to obtain the redshift
evolution for characteristic mass scale $M_{\rm c}$ at reionization redshift
$z_{\rm rei}=9$. Figure 18 shows the data from Okamoto et al. (2008)
simulations over plotted with our approximating function (blue line) described
in eq. 8. The plot is in log-linear scale and thus approximately linear
behaviour of $\log M_{\rm c}$ with $z$ at $z\lesssim 8$ indicates that this
dependence can be approximated by the exponential function. At redshifts
$z\gtrsim z_{\rm rei}=9$ $M_{\rm c}$ decreases sharply with increasing $z$.
Note that the points include simulations at different resolutions, but the
lowest points at high $z$ correspond to the highest resolution simulation,
which our analytic equation aims to approximate.

The characteristic mass scale at time of reionization ($M_{\rm c,rei}$)
according to our approximation function (eq. 8) is $M_{\rm c,rei}=M_{\rm
c}(z=9)\approx 9.3\times 10^{6}\,\,M_{\odot}$. The general expression for
evolution for characteristic mass scale is given by eq. 9 where $\beta$
controls when the sharp increase in characteristic mass scale occurs and
$\gamma$ controls the sharpness of that increase. To be able to flexibly model
the characteristic mass scale evolution for different reionization redshifts,
we we apply the constraint that the corresponding characteristic mass scale
must be equal to $M_{\rm c,rei}$ at the assumed $z_{\rm rei}$. Furthermore,
the characteristic mass scale must be equal to that seen in Okamoto et al.
(2008) simulations at $z=0$. Keeping this sharpness parameter $\gamma$ fixed,
the above constraints give $M_{\rm c,rei}=M_{\rm c}(z_{\rm rei})$ with $M_{\rm
c}(z)$ given by eq. 9. We can also use this equation to get the expression for
$\beta$ (Equation 10) for a given $z_{\rm rei}$ and $\gamma$. Figure 18 shows
the evolution of characteristic mass scale for different $z_{\rm rei}$.

Figure 18: Evolution of the characteristic mass scale $M_{c}$ as a function of
redshift. The data points shown are simulation results from Figure 3 in
Okamoto et al. (2008). The colored lines show the evolution of characteristic
function for reionizaton redshifts $z_{\rm rei}=5,7,9$ according to our
analytical parametrization in Equation 9.

##  Appendix B Sensitivity of gas masses of dwarf galaxies to $\rm H_{2}$
model

Figure 19 shows comparison of the $M_{\star}-M_{\rm g}$ relation for model
galaxies (red dots) to observations similar to that shown in Fig. 10 for the
fiducial model. The two panels two models with all parameters identical to the
fiducial model, but with different models for molecular fraction $f_{\rm
H_{2}}$: the modified Gnedin-Draine model given by eqs. 8a-10a in Gnedin &
Draine (2016) in the left panel, and the model described in Section 2 of
Krumholz & Dekel (2012, namely eqs 18-21 with $c=5$).

Figure 20 shows results of these models for the $M_{\rm
g}-\dot{\mathcal{M}}_{\star}$ correlation. Compared to the fiducial model
results shown in Figure 11, the star formation rates in the modified Gnedin-
Draine model are larger compared to fiducial model, while SFRs in the KD12
model are systematically and considerably lower. The number of model galaxies
is also noticeably smaller in the right panel because in this model a large
fraction of dwarf galaxies does not have $\rm H_{2}$ and thus no ongoing star
formation.

These figures show that gas masses and star formation rates in model galaxies
of $M_{\star}\lesssim 10^{9}\,\,M_{\odot}$ are very sensitive to the details
of $f_{\rm H_{2}}$ model. To be fair the model used KD12 is based on
equilibrium assumption, which is not expected to be valid at low metallicities
(see Krumholz, 2013). The latter paper discusses extensions of the model to
dwarf galaxy regime, but at the expense of additional assumptions. Given the
sensitivity of gas masses to the details of such model, it is not clear that
predictions would be robust.

The differences that can be seen in Figures 10 and left panel of Figure 19 for
the models that use different fitting formulae (Gnedin & Draine, 2016) to the
same numerical simulation results of Gnedin & Draine (2014) show that even the
current calibrations based on simulation results are likely not robust.

Figure 19: Comparison of the $M_{\star}-M_{\rm g}$ relation for model galaxies
(red dots) to observations similar to that shown in Fig. 10 for the fiducial
model. The two panels shows two models with all parameters identical to the
fiducial model, but with different models for molecular fraction $f_{\rm
H_{2}}$: in the model shown in the left panel the modified Gnedin-Draine model
given by eqs. 8a-10a in Gnedin & Draine (2016) is used, while in the right
panel the model described in Section 2 of Krumholz & Dekel (2012, namely eqs
18-21 with $c=5$).

Figure 20: Comparison of the $M_{\rm g}-\dot{\mathcal{M}}_{\star}$ relation
for model galaxies (red dots) to observations similar to that shown in Fig. 11
for the fiducial model. The two panels shows two models with all parameters
identical to the fiducial model, but with different models for molecular
fraction $f_{\rm H_{2}}$: in the model shown in the left panel the modified
Gnedin-Draine model given by eqs. 8a-10a in Gnedin & Draine (2016) is used,
while in the right panel the model described in Section 2 of Krumholz & Dekel
(2012, namely eqs 18-21 with $c=5$).


# Revisiting information cascades in online social networks

Michael Sidorov
Department of Communication Systems Engineering
Ben Gurion University of the Negev
Be’er Sheba, Israel
<EMAIL_ADDRESS>
Dan Vilenchik
Department of Communication Systems Engineering
Ben Gurion University of the Negev
Be’er Sheba, Israel
<EMAIL_ADDRESS>
(Mar, 2021)

###### Abstract

It’s by now folklore that to understand the activity pattern of a user in an
online social network (OSN) platform, one needs to look at his friends or the
ones he follows. The common perception is that these friends exert influence
on the user, effecting his decision whether to re-share content or not.
Hinging upon this intuition, a variety of models were developed to predict how
information propagates in OSN, similar to the way infection spreads in the
population. In this paper, we revisit this world view and arrive at new
conclusions. Given a set of users $V$, we study the task of predicting whether
a user $u\in V$ will re-share content by some $v\in V$ at the following time
window given the activity of all the users in $V$ in the previous time window.
We design several algorithms for this task, ranging from a simple greedy
algorithm that only learns $u$’s conditional probability distribution,
ignoring the rest of $V$, to a convolutional neural network-based algorithm
that receives the activity of all of $V$, but does not receive explicitly the
social link structure. We tested our algorithms on four datasets that we
collected from Twitter, each revolving around a different popular topic in
2020. The best performance, average F1-score of 0.86 over the four datasets,
was achieved by the convolutional neural network. The simple, social-link
ignorant, algorithm achieved an average F1-score of 0.78.

##  1 Introduction

The propagation of ideas and innovations has led to political, cultural and
economic changes. Social scientists found that ideas tend to flow along social
links, in a similar manner that an epidemic infects a population [11, 15].
Understanding how information propagates in online social network (OSN)
platforms has significant real-world impact, from the political stability of
states to marketing. Significant efforts have been invested in predicting
various properties of information cascades, such as size [25], temporal growth
[27], and virality [7].

Researchers mainly studied transient-copy propagation protocols (a term coined
in [8]), where content is simply replicated in a network, like a re-tweet
(contrasted with nomination cascades for example, such as the famous ALS ice
water bucket challenge).

It is tempting to think that information propagates in OSNs according to
epidemiological models such as Bass [3] or SIR [23]. However several
“cautionary tales” were told by various researchers regarding the validity of
this image. For example, one important assumption in epidemiological models is
that all newly infected individuals arise from the susceptible set (neighbors
of infectious people). However, mass media marketing efforts rely on a
”broadcast” mechanism, where a large number of individuals can receive the
information directly from the same source, which need not be their neighbor.
Indeed, [5] who studied information cascades in Flickr, found that out of 10
million total favorite markings, 47% were propagated not through the social
links, but via other broadcasting mechanisms that Flickr employs. Also [14]
found that 33% of the retweets in their dataset credit users that the
retweeters do not follow.

Furthermore, unlike the epidemiological picture of a disease spreading in
waves, it was observed that most cascades are extremely shallow and wide. For
example, [18] found that the average size of the diffusion trees in Twitter is
1.3, and that the vast majority contain only 1 node. This phenomenon is also
observed in other platforms such as Digg and Flickr [5, 25, 2, 17], email
forwarding [35] and recommendation chains [26].

The majority of works focus on macro prediction tasks such as predicting the
depth of a cascade, its size, or estimating other macro properties such as the
transmission rate. Micro-level prediction tasks, such as whether a tweet $t$
will be retweeted or not, received considerably less attention. Even when
microscopic properties were computed, like the probability that user $u$ will
transmit information to $v$, it was used to derive macroscopic properties such
as cascade size, and not for prediction in the standard machine-learning
setting (train-test on a specific database of events). In [14], two model-
based algorithms for the task of url retweet prediction were proposed, based
on the At-Least-One and Linear-Threshold schemes, and tested for a set of 100
popular URLs, whether they will be retweeted in the following time interval or
not. In [31], machine learning was used for the task of predicting for a
stream of tweets which will get retweeted.

We study a different setting, which is user-centric and not post-centric.
Given a user $u\in V$, $V$ is some fixed set of users, and a time interval
$[t_{0},t_{0}+\Delta T]$, the task is to predict whether user $u$ is going to
react (retweet, reply, or mention) to an existing post by some user $v\in V$
(which $u$ doesn’t necessarily follow). The input to the prediction algorithm
is a snapshot of the activity of all users in $V$ at time $[t_{0}-\Delta
T,t_{0}]$, namely a binary vector indicating whether each user was active or
not. The user-centric task coincides with the post-centric task if all users
react to at most one post. This is indeed the case for most users in our
datasets, as Figure 4 demonstrates.

In this work we address some fundamental questions regarding the nature in
which information propagates in OSNs: (1) Can we perform the prediction task
without any other features besides the time series of the users’ postings (in
particular, no linguistic features) (2) Is there an algorithm which performs
this task, but is model-agnostic? namely, it does not rely on epidemiological
models, and (3) Does the algorithm need to know the social links (which user
follows/mentions which user)? Or can the algorithm implicitly learn this
information from the snapshot it is given as input?

Our Contribution. We answer these questions, providing new insights into the
way information propagates in online social media platforms. We design several
prediction algorithms, each using a different approach, including a neural
network (NN) based algorithm, $TWCRN$, which does not use the adjacency matrix
of $V$. We evaluate our approach on four datasets, spanning a total of 77M
tweets, which we collected from Twitter in 2020, following four different
major events in that year. Our algorithm $TWCRN$ performed the prediction task
with an average F1 score of 0.86 over all datasets, which were highly
imbalanced (on average, only 10% of users in a time slice of width $\Delta
T=12h$ reacted to some post).

We were able to verify that indeed $TWCRN$ implicitly learns information about
the social links using a permutation test, where we randomly permuted the
input to the NN at train time, contaminating social links information. The
algorithm performed very poorly in this setting.

We further designed a very intuitive algorithm, which for every user $u\in V$
learns the conditional probability that $u$ will re-share something in this
time interval given $u$’s activity in the previous time interval (ignoring
information about all other users). Intuitively, we are learning the “trends”
of $u$, is he a lazy-twitter (taking breaks between tweets) or a chain-
tweeter. The performance of this algorithm was extremely competitive to
$TWCRN$ (with respect to simplicity and running time), with an average
F1-score of 0.78.

Thus we arrive at two new insights. The first is that a relatively accurate
prediction at the user’s level can be obtained by simply learning the user’s
habits, regardless of the social links. The second is that a higher accuracy
can be achieved by integrating information about social links, but, this
information need not be served explicitly, but rather the algorithm can learn
the relevant links by itself.

From a technical point of view, we are able to train a rich algorithm with
many parameters, $TWCRN$, on a large set of users, $10^{4}$. Previous work
either trained a simple model (perceptron) on a large set [31], or a rich
algorithm (Linear Threshold with many parameters) but trained on a small
dataset (of size 100). To overcome the computational challenge we wrap the NN
with an encoder-decoder pair, to reduce dimensionality.

As far as we know, we are the first to use neural networks to perform such a
prediction task, with such a success rate. The F1-score achieved in [31],
using many more features, and the passive-aggressive algorithm of [9] was
0.46, albeit for a somewhat different problem and setting. Neural networks
were used very recently for the first time to predict macro properties of
cascades (cascade structure), [21, 27], or to predict the influence of a given
user based on activity features [32].

Our final contribution is to make all datasets openly available to the
community, along with the Python code that we wrote for all algorithms.

##  2 Related Work

Users’ influence in OSNs and the diffusion processes that stand behind
information propagation in them were extensively studied in the past years.
The works we survey here are by no means a concise review of the huge body of
work available on this topic.

One line of work tries to maximize the size of the cascade by picking a good
set of influential users that will be the starting seed. In [28], the authors
tried to predict users’ influence by construction of differential equations
extended from the Susceptible-Infected model. In [33], the authors
incorporated user-specific topic distribution and network structure, and in
[22, 6] by means of a greedy algorithm. In [12], the authors introduce the
idea of novelty decay into the Independent Cascade model from [1].

Another line of work is focused on understanding information diffusion as they
unfold, without manipulating the seed. These works typically suggest a
generative model that underpins the information cascade and then check how the
model fits either simulation or real-world datasets. Examples include [36, 29]
where a self-exciting point process is used to develop a statistical model to
predict cascade size; cascades in heterogeneous scale-free networks were
studied in [30] under the SIS and SIR model; a cascade generating function was
developed in [16] to compute macroscopic properties of cascades, such as their
size, spread, diameter, number of paths, and average path length. In [27] an
end-to-end deep learning approach was designed to predict the future size of
cascades, without using any hand-crafted features or generative model
assumptions.

The third line of work does not deal with prediction but rather with
descriptive analysis, providing insights to the distribution of cascade size,
depth, width and adoption rate (as time progresses) [5, 25, 34, 18].

Fewer works tried to predict microscopic post-level or user-level events. In
[14], the task was to predict whether a popular URL that was retweeted several
times, will be retweeted in the following time window. The method was again
model-based, upgrading the Linear Threshold (LT) and At-Least-One (ALO) models
with additional features to achieve better prediction (F1 score of nearly
0.7). In [31] machine learning was used to predict retweets of an original
tweet, albeit with a rather low F1 score (0.46). Our work continues this line
of work but differs in two main aspects: we do not have any model-based
assumptions as in [14], and therefore we don’t use any hand-crafted features
(like the virality of the topic, number of followers, etc.), that were used
also in [31]. Furthermore, our approach does not require an explicit
description of the social links.

##  3 Methodology

Previous studies on user influence in social networks in general, and Twitter
specifically [14, 31], point out that the main predictor of users’ activity is
the users he follows (i.e., his friends).

We consider three models where the information about the user’s neighbors is
taken into account to various degrees. In the first model, _Tweet Prior
Network_ ($TWPN$) we completely ignore the user’s social links and only
consider its past activity, see Figure 1. Our second model, the _Tweet Mask
Network_ ($TWMN$), is a neural network that takes into account the user’s
social links and ignores non-adjacent users, Figure 2. Finally, we train a
fully-connected convolutional model, Tweet Convolutional Residual Network
($TWCRN$), that considers all the activity of all the users in the network and
not only the social links, Figure 3.

The input to each NN is the same, a vector $\tau_{i}\in\\{0,1\\}^{|V|}$, which
for every user $u\in V$ specifies whether he was active at time interval
$[t_{i}-\Delta T,t_{i}]$ or not. The output vector,
$\tilde{\tau}_{i+1}\in{\mathbb{R}}^{|V|}$ is the prediction for every user
whether he is going to react to some post at time interval
$[t_{i},t_{i}+\Delta T]$ or not.

###  3.1 Models definition

#### Tweet Prior Network ($TWPN$)

The architecture of this model is just two layers, input and output, as shown
in Figure 1. All output nodes in the network use the $\tanh(w_{u}x_{u})$
activation function, where $w_{u}$ is the weight of user’s $u$ single edge,
and $x_{u}$, the input, is the entry in $\tau_{i}$ corresponding to $u$.

Figure 1: TWeet Prior Network ($TWPN$) schematic representation. The vector
$\tau_{i}$ is the activity vector in the time window $[t_{i}-\Delta T,t_{i}]$,
and $\tilde{\tau}_{i+1}$ is the predicted activity in time window
$[t_{i},t_{i}+\Delta T]$. Figure 2: TWeet Mask Network ($TWMN$) schematic
representation. The vector $\tau_{i}$ is the activity vector in the time
window $[t_{i}-\Delta T,t_{i}]$, and $\tilde{\tau}_{i+1}$ is the predicted
activity in time window $[t_{i},t_{i}+\Delta T]$.

#### MLE

This model is a simpler variant of the $TWPN$ model where for every user $u\in
V$, the conditional probabilities are learned:

$p^{a,b}_{u}=Pr\left[I_{u}^{0}=a\,\big{|}\,I_{u}^{-1}=b\right],$ (1)

where, $I_{u}^{0}$ is the activity indicator of user $u$ at the predicted
interval, $I_{u}^{-1}$ in the activity indicator in the preceding time
interval, and $a,b\in\\{0,1\\}$. The probabilities $p^{a,b}_{u}$ are estimated
solely on the data of user $u$ in the preceding time intervals, using the
Maximum Likelihood Estimator, MLE (counting and averaging).

#### Tweet Mask Network ($TWMN$)

This neural network extends $TWPN$ by taking into account the information at
the user’s neighbors as well. The architecture is a fully connected 2-layer
Feed-Forward Neural Network (FFNN) [13] (i.e., input layer fully connected to
the output layer). Again, all output nodes use the $\tanh(\cdot)$ activation
function. While the network is fully connected, for each output node $u$ we
mask (hence the name) the non-adjacent input nodes.

For each user $u$, the output is given by $\tanh(\sum_{v\in
N(u)}w_{v,u}x_{v})$, where $x_{v}$ is the entry in $\tau_{i}$ corresponding to
$v$. The weights $w_{v,u}$ between the user $u$ and the ones he follows $v\in
N(u)$ can be interpreted as the influences, which the NN learns by itself.

Figure 3: TW ConvResNet ($TWCRN$) schematic representation, where $\tau_{i}$
is the activity vector in the time window $[t_{i}-\Delta T,t_{i}]$,
$\tilde{\tau}_{i+1}$ is the predicted activity in time window
$[t_{i},t_{i}+\Delta T]$. (a) the Encoder part of the Auto Encoder (AE)
network with $K$ layers [20], which is responsible for compressing the
original activity vector. (b) the inflation step, which receives a 1D vector
and transforms it into a 2D matrix by copying it row-wise. (c) the ConvResNet
with $L=18$ or $L=34$ layers, as described in [19]. (d) the Decoder network,
which constitutes the second part of the AE network, with M layers, which is
responsible for the final de-compression of the prediction vector.

#### Tweet Convolutional Residual Network ($TWCRN$)

This model takes into account all possible connections between users, no
restricted to the follower (or mention) social links. The model consists of
four parts as shown in Figure 3, as follows:

Encoder \- A standard encoder, as described in [20], which maps an activity
vector of dimension $D$ (in our case $D=10^{4}$) to dimension $100=d<<D$. Each
layer down samples the dimension of the input by a factor of 2 until the
output size is $d=100$. The point of this layer is to reduce the computational
effort of training the main part, ConvResNet, by reducing dimensionality.

Inflater \- a simple procedure which receives a vector of size $d$, and
transforms it into a $d\times d$ matrix by copying it row-wise. This step is
necessary because ConvResNet [19] is used for images.

ConvResNet \- We study separately two architectures, $\times 18$ and $\times
34$, presented in [19] (the number represents the number of layers in the
neural network) which were developed for image processing tasks. This is the
core of the prediction mechanism. The ConvResNet architecture enables
extremely deep neural networks (more than 150 layers deep) to be successfully
trained by a standard stochastic gradient descent with backpropagation,
overcoming the problem of performance degradation present in very deep neural
networks.

Decoder \- Following [20], the decoder maps the compressed vector (of
dimension $d=100$) to its original dimension $D$. It does that by up-sampling
the input vector by factor of 2, until the output size reaches the original
input dimension $D$.

###  3.2 Training and testing the models

A cascade progresses in time, as well as in space (the nodes). In this paper,
we focus on the time-aspect, and we partition the timeline along which the
data was collected into equal-sized slices of $\Delta T=12$ hours each. For
each pair of consecutive time periods $i,i+1$ we define two vectors
$\tau_{i},\tau_{i+1}$ as mentioned already above: $\tau_{i}$ is an indicator
of all activity types and $\tau_{i+1}$ only of reaction activity.

The prediction task is, given the vector $\tau_{i}$, predict the reaction
vector $\tau_{i+1}$.

We train each model in the same manner, taking time period $i=1,\ldots,K$ as
the train, and testing on time periods $i=K+1,\ldots N$, where $N$ is the
total number of time slices, and $K=\lfloor 0.9\cdot N\rfloor$. We further
split the train into train and validation, resulting in a 70-20-10 data split
rule. To avoid over-fitting, we stopped the training if the loss on the
validation set did not improve for 50 epochs.

We used the ADAM optimizer [24] with default settings for all three networks,
$TWPN$, $TWMN$, and $TWCRN$ (the components of the $TWCRN$ model were trained
separately). We chose the Mean Squared Error (MSE) for the loss function. To
enrich the gradient flow in the training step, instead of binary vectors
$\tau_{i}$ we used a technique in which each 1 was replaced with a sample from
a normally distributed random variable $N(1,0.01)$, and 0 with a sample from
$N(-1,0.01)$.

To find the conditional probabilities of the MLE model, $p^{a,b}_{u}$ in Eq.
(1), we counted for user $u$, the fraction of times that each combination
$a,b$ appeared in the pairs $\tau_{i},\tau_{i+1}$, for $i=1,\ldots,K$.

At the test step, we fed each model with $\tau_{i}$, for $i=K+1,\ldots N$, to
receive the predicted vector $\tilde{\tau}_{i+1}$. Each entry in
$\tilde{\tau}_{i+1}$ (either the output of tanh or of the decoder, depending
on the model) is set to 1, if positive, and 0 if negative. The MLE
probabilities were rounded to 0 or 1. This vector is then compared with the
real $\tau_{i+1}$ to compute precision, recall, and F1 score.

##  4 Data

We collected four datasets from Twitter, using the _Tweepy_ API. Each data set
concerns a different major event at that time: _Volcano_ (the eruption of the
Taal volcano in the Philippines on January 12, 2020), _Kobe_ (the death of
Kobe Bryant in a helicopter crash on January 26, 2020), _Princess_ (the
stepping down of the Duke and Duchess of the British royal family in January
2020) and _Beirut_ (the explosion in Beirut on August 4, 2020). Table 1
describes the keywords that were used and the size of each dataset, and Table
2 shows the breakdown according to different types of posts.

Name | Keyword | Posts
---|---|---
Volcano | taal, volcano, philippines | 17,994,283
Kobe | kobe, bryant | 24,562,000
Princess | meghan, markle, harry, prince, princess | 22,353,429
Beirut | beirut, lebanon, explosion | 14,106,557
Unfiltered | - | 56,338,915
Table 1: Description of the five datasets that we collected. The first four
were collected during different ”viral” events, which took place during the
year 2020, and were filtered to include only tweets which were related to
those events. The fifth dataset is an unfiltered tweet stream which was
recorded during the 8/2021.

There are several possible ways of posting and interacting on Twitter. The
simplest post is a _tweet_ , i.e., a post that the user writes himself and is
not related to any other post. Tweets appear on the home timeline of the
sender, and on the home timelines of all his followers, but Twitter’s time-
line algorithm keeps changing frequently. Features like “in case you forgot”
break the synchronous time-line structure, and suggestions of tweets from
users one does not follow break the social link structure.

There are three possible reactions to a tweet: _mention_ is a post that
contains the ” _@username_ ” syntax in the body of the text, _reply_ is a
direct response to another tweet and a _re-tweet_ , which is a propagation of
someone else’s tweet (re-tweet can be done also in an automatic manner using
the 1-click option). All three reactions will be visible on the home timeline
of the reacting user and on the timeline feed of all his followers.

The next key component in the data pre-processing was to recover the social
links between the users. We can infer two types of graphs. The first is the
mention graph, in which a link exists between user $u$ and $v$ if $u$ reacted
to $v$’s post (but $u$ need not follow $v$). Another graph, the followers
graph, is the one recovered using Twitter’s API where a link between $u$ and
$v$ exists if user $u$ follows $v$. Note that there need not be an inclusion
relationship between the two sets of links, but the way our dataset was
collected (using the mention graph), entails that the followers graph is a
subgraph of the mention graph.

We recovered both social networks, and as we shall see, the accuracy of the
classification task differs according to which network we use. Let us note
that querying Twitter’s API is computationally more expensive as Twitter
limits the number of requests. Therefore it is interesting to know what’s the
gain, if any, when using the more costly-obtained information of the Twitter
followers graph.

Name | PO | TW | RT | MT
---|---|---|---|---
Volcano | 17,994,283 | 3,400,415 | 11,810,708 | 2,783,160
Kobe | 24,562,000 | 2,962,387 | 20,768,520 | 831,093
Princess | 22,353,429 | 4,694,745 | 15,457,234 | 2,201,450
Beirut | 14,106,557 | 1,951,752 | 11,418,457 | 736,348
Unfiltered | 56,338,915 | 35,728,226 | 813,013 | 19,797,676
Table 2: Breakdown of the post types for each data set, where PO is a general
post (i.e., tweet, re-tweet or mention), TW is the initial tweet, RT is a re-
tweet using the ” _RT @username_ ” syntax, or by clicking the dedicated button
and MT is a mention.

Figure 4: Histogram of the average activity of users in a single time slice of
$\Delta T=12h$ for the Princess dataset. Top: statistics for the mention graph
dataset. Bottom: statistics for the followers graph dataset. Evidently, most
users tweet or re-share only one post per time slice.

As customary in many works, users that have low activity are filtered out from
the dataset. We created two sets, $A$, the active users, users who post at
least $a$ posts (either tweet or re-share of some sort), and $P$, the popular
users, users that were reacted to at least $p$ times. We chose $a$ and $p$ so
that the number of users in the dataset is roughly 10,000 (due to the
constraint of memory resources).

#### Broadcasticity.

Another parameter that is of interest is the depth of the cascade. As noted,
most cascades are shallow. Various measures were used to compute depth,
including the Wiener index in [18], the average distance between all pairs of
nodes in a diffusion tree. We suggest a much simpler criterion, which does not
require the laborious task of computing the diffusion tree. We call it the
broadcasticity measure, and it measures how much the diffusion resembles a
broadcast rather than a diffusion. It’s computed using the Jaccard index of
the two sets $A$ and $P$:

$B=1-J(A,P)=1-\frac{|A\cap P|}{|A\cup P|}$ (2)

High broadcasticity means that $A$ and $P$ are almost disjoint, namely users
either tweet or react, but very few do both. This is exactly the case of a
shallow 1-hop diffusion. If a diffusion on the other hand is deep and wide
(like an epidemic spreading in the population), then many users belong to both
$A$ and $P$, and the broadcasticity is low. Note that a shallow diffusion with
few long paths still attains large broadcastisity.

Table 3 describes the various features of each dataset with respect to $A$,
$P$ and the broadcastisity.

Name | $a$ | $|A|$ | $p$ | $|P|$ | $Posts$ | #slices | $B$
---|---|---|---|---|---|---|---
Volcano | 103 | 3,189 | 207 | 7,757 | 2,710,915 | 108 | 0.97
Kobe | 167 | 4,674 | 167 | 4,835 | 1,380,543 | 87 | 0.98
Princess | 234 | 5,151 | 234 | 7,866 | 2,710,915 | 87 | 0.92
Beirut | 105 | 3,830 | 210 | 5,778 | 960,228 | 117 | 0.93
Unfiltered | 100 | 3,943 | 201 | 5,238 | 721,301 | 28 | 0.99
Table 3: Datasets are filtered to remove low-activity users. The set $A$ of active users that (re-) share at least $a$ posts (over the entire time period), and popular users $P$ whose tweets were re-shared at least $p$ times. The number of time slices $\tau_{i}$ is given in column 7, and the broadcastiscity in the last column. Name | Network Crawl | Tweet Log
---|---|---
| N | $E_{c}[L]$ | $E[L]$ | N | $E_{c}[L]$ | $E[L]$
Volcano | 8,990 | 3.953 | 2.943 | 10,440 | 4.019 | 4.648
Kobe | 8,339 | 3.921 | 2.368 | 9,225 | 3.965 | 2.593
Princess | 9,793 | 3.991 | 2.869 | 11,891 | 4.075 | 3.579
Beirut | 7,943 | 3.9 | 2.498 | 8,876 | 3.948 | 3.103
Unfiltered | 7,208 | 3.86 | 6.23 | 7,363 | 3.87 | 6.23
Table 4: Mean shortest path ($E[L]$) between any two randomly chosen nodes in
the social networks. The $E_{c}[L]=log(N)$, where $N$ is the number of valid
users in the social networks, and $E_{c}[L]$ is the critical mean shortest
path (i.e., the network should satisfy the condition $E[L]\leq E_{c}[L]$, to
be considered a small world (SW) network). Green - the criterion is satisfied,
red - the criterion is not satisfied.

##  5 The Experiment

Recall the classification task described in Section 3.2. Given a vector
$\tau_{i}\in\\{0,1\\}^{|V|}$, in which the $j^{th}$ entry is 1 if user $j$
tweeted in time window $i$, our goal is to predict the vector $\tau_{i+1}$, in
which the $j^{th}$ entry is 1 if user $j$ reacted to some post in time
interval $i+1$.

###  5.1 Other baselines

We benchmarked our model against four other popular models. The first two are
random guess models $RND_{p=0.5}$ and $RND_{p=\pi}$ which predict
$\tau_{i+1}[j]$ using a fair random coin flip, or the fraction of users that
were active in the previous time window, $\pi$. The two last models are
variations of the SI model, At-Least-One ($ALO$) and Linear-Threshold ($LT$).
We follow the $LT$ and $ALO$ models that were developed in [14] as we found
them the most general. The intuition of both is that if the neighbors of the
user were active in the preceding time window, it may stir the user’s decision
to become active himself in the next time window. Each algorithm takes a
different try at estimating the probability that user $u$ will become active
in the next time window; [14] use the following expression for $LT$:

$p_{u}^{LT}=A\Big{(}\alpha_{v_{1},u},\ldots,\alpha_{v_{k},u},\beta_{u},\gamma\Big{)}T\Big{(}\mu_{u},\sigma_{u}^{2},t_{u}^{post}\Big{)}$
(3)

The $\alpha_{v_{i},u}\in[0,1]$ are the influence of a neighbor of $u$,
$v_{i}$, on $u$. The $\beta_{u}\in[0,1]$ is a prior probability of user $u$ to
become active, $\gamma\in[0,1]$ is the virality of the topic that is being
discussed. $A(\cdot)$ is an a-temporal component, i.e., the probability that
the user will respond to some post due to the influence that was exerted on
him from his social circle. Concretely, this component is given by

$A(\cdot)=\sigma_{a,b}\bigg{(}\gamma\Big{(}\beta_{u}+\sum_{v:v\rightarrow
u}\gamma\alpha_{v,u}p_{u}\Big{)}\bigg{)},$

where $\sigma_{a,b}(x)=\frac{1}{1+e^{-a(b-x)}}$ is a sigmoid function.

The second component of Eq. 3, $T(\cdot)$, is a temporal component of the
predicted activity probability, and it represents an empirical observation
regarding the time that passes from a moment when any user in the social
network initiates a new post until the first of his followers responds to it
($\mu_{u}$ and $\sigma_{u}^{2}$ are the mean and SD of the latter). This
component is unique for each social network, and is given by

$T\Big{(}\mu_{u},\sigma_{u}^{2},t_{u}^{post}\Big{)}=\frac{1}{2}erfc\Big{(}-\frac{ln(t_{u_{i}}^{post})-\mu_{u}}{\sigma_{u}\sqrt{2}}\Big{)}$

The parameter $t_{u_{i}}^{post}$ represents the time of post’s initiation
relative to the start of the time window, and
$erfc(x)=1-\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt$, is a complementary
Gauss error function.
For $ALO$ the following is used in [14]:

$p_{u}^{ALO}=1-\Big{(}1-\gamma\beta_{u}\Big{)}\prod_{v:v\rightarrow
u}\Big{(}1-\gamma\alpha_{v,u}p_{u}\Big{)}$ (4)

In both cases, we followed the rule in [14], where a user is predicted to
react if $p_{u}>0.5$

###  5.2 Training the baseline models

The base-line models were trained similarly to our models, using 90% of the
time intervals for train and 10% for test. Table 4 shows the total number of
time intervals per dataset.

$RND_{p=0.5}$ and $RND_{p=\pi}$ have no tunable parameters. The parameters of
$ALO$ and $LT$ were set as follows. Optimization was performed with Python’s
Hyperopt library, with the F1 score as the function to maximize. This library
uses the Tree of Parzen Estimators hyper-parameter optimization algorithm from
[4]. The parameter $\gamma$, topic virality, was set to 1, since all tweets
come from the same topic. Also the temporal part $T(\cdot)$ was set to 1,
since we are not interested in the event that a tweet $t$ is re-shared
sometime in the future, but whether user $u$ re-shared some content in a very
limited time interval. Due to the computational constrains, the optimization
was performed only on the prior vector of the $p_{u}$’s. The dataset in [14]
included 100 URLs for which retweeting was predicted, ours include 10,000
events, and the number of $\alpha_{u,v}$’s is even larger. Therefore, the
influence matrix $\alpha_{u,v}$ was supplied from the output of the $TWMN$
model, where the weight of the edge in the NN that connects $v$ and $u$ may be
thought of as the influence of $v$ on $u$.

Model | CPU | RAM | Accelerator | Run Time
---|---|---|---|---
$RND_{p=0.5}$ | $CPU_{low}$ | 32 GB | None | $31.75\pm 1.48[s]$
$RND_{p=prop}$ | $CPU_{low}$ | 32 GB | None | $30.00\pm 1.22[s]$
$MLE$ | $CPU_{low}$ | 32 GB | None | $28.38\pm 1.41[s]$
$ALO$ | $CPU_{high}$ | 16 GB | None | $1.45\pm 0.38[H]$
$LT$ | $CPU_{high}$ | 16 GB | None | $1.29\pm 0.31[H]$
$TWMN_{SHUF}$ | $CPU_{high}$ | 16 GB | GPU | $1.18\pm 0.75[H]$
$TWMN_{all1}$ | $CPU_{high}$ | 16 GB | GPU | $54.35\pm 22[m]$
$TWMN$ | $CPU_{high}$ | 16 GB | GPU | $1.12\pm 72[H]$
$TWPN$ | $CPU_{high}$ | 16 GB | GPU | $1.29\pm 54[H]$
$TWCRN_{x18}^{SHUF}$ | $CPU_{high}$ | 16 GB | GPU | $1.1\pm 0.28[H]$
$TWCRN_{x18}$ | $CPU_{high}$ | 16 GB | GPU | $54\pm 11.5[m]$
$TWCRN_{x34}^{SHUF}$ | $CPU_{high}$ | 16 GB | GPU | $1.23\pm 0.32[H]$
$TWCRN_{x34}$ | $CPU_{high}$ | 16 GB | GPU | $1\pm 0.23[H]$
Table 5: Hardware specs with corresponding mean run times, where $CPU_{low}$
is Intel i7-8550U @ 1.80GHz, $CPU_{high}$ is Intel Xeon @ 2.00GHz and GPU is
Tesla P100-PCIE-16GB, [H], [m] and [s] stands for ”Hours”, ”minutes” and
”seconds” respectively.

Metric | P | $F_{1}$ | R | P | $F_{1}$ | R | P | $F_{1}$ | R | P | $F_{1}$ | R | P | $F_{1}$ | R
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$RND_{p=0.5}$ | $0.19$ | $0.28$ | $0.5$ | $0.29$ | $0.34$ | $0.5$ | $0.34$ | $0.4$ | $0.5$ | $0.17$ | $0.23$ | $0.5$ | $0.33$ | $0.4$ | $0.5$
$RND_{p=\pi}$ | $0.19$ | $0.21$ | $0.24$ | $0.29$ | $0.29$ | $0.34$ | $0.34$ | $0.38$ | $0.43$ | $0.17$ | $0.18$ | $0.22$ |  $0.63$ $\pm 0.02$ |  $0.64$ $\pm 0.01$ |  $0.64$ $\pm 0.01$
$MLE$ | $0.42$ | $0.42$ | $0.46$ |  $0.31$ $\pm 0.02$ |  $0.38$ $\pm 0.02$ | $0.56$ |  $0.58$ $\pm 0.02$ | $0.6$ | $0.68$ | $0.22$ | $0.29$ | $0.44$ |  $0.63$ $\pm 0.01$ |  $0.63$ $\pm 0.01$ |  $0.64$ $\pm 0.01$
$ALO$ |  $0.18$ $\pm 0.02$ |  $0.26$ $\pm 0.03$ | $0.52$ |  $0.18$ $\pm 0.06$ |  $0.26$ $\pm 0.07$ | $0.51$ |  $0.31$ $\pm 0.03$ |  $0.38$ $\pm 0.04$ | $0.53$ |  $0.09$ $\pm 0.02$ |  $0.15$ $\pm 0.03$ |  $0.51$ $\pm 0.02$ | $0.4$ | $0.54$ | $0.83$
$LT$ |  $0.18$ $\pm 0.03$ |  $0.26$ $\pm 0.04$ | $0.52$ |  $0.17$ $\pm 0.04$ |  $0.24$ $\pm 0.05$ | $0.5$ |  $0.31$ $\pm 0.05$ |  $0.37$ $\pm 0.06$ |  $0.53$ $\pm 0.01$ |  $0.11$ $\pm 0.03$ |  $0.17$ $\pm 0.05$ |  $0.53$ $\pm 0.01$ | $0.17$ | $0.17$ | $0.17$
$TWMN_{all1}$ | $0.49$ | $0.45$ | $0.49$ |  $0.31$ $\pm 0.02$ |  $0.38$ $\pm 0.01$ | $0.59$ |  $0.57$ $\pm 0.01$ |  $0.57$ $\pm 0.01$ | $0.66$ | $0.21$ | $0.29$ | $0.53$ | $0.39$ |  $0.46$ $\pm 0.02$ |  $0.56$ $\pm 0.02$
$TWMN$ | $0.47$ | $0.48$ | $0.52$ |  $0.32$ $\pm 0.02$ |  $0.44$ $\pm 0.02$ | $0.83$ |  $0.6$ $\pm 0.03$ |  $0.63$ $\pm 0.03$ |  $0.75$ $\pm 0.01$ | $0.26$ | $0.32$ | $0.46$ | $0.64$ | $0.64$ | $0.65$
$TWPN$ | $0.51$ | $0.49$ | $0.5$ |  $0.33$ $\pm 0.01$ |  $0.45$ $\pm 0.01$ | $0.8$ | $0.63$ | $0.66$ | $0.78$ | $0.327$ | $0.36$ | $0.45$ | $0.68$ | $0.68$ | $0.68$
$TWCRN_{x18}^{SHUF}$ | $0.23$ | $0.2$ | $0.18$ | $0.22$ | $0.3$ |  $0.48$ $\pm 0.02$ | $0.41$ | $0.42$ |  $0.43$ $\pm 0.01$ | $0.15$ | $0.19$ |  $0.25$ $\pm 0.02$ | $0.59$ | $0.58$ | $0.58$
$TWCRN_{x18}$ |  $0.54$ $\pm 0.02$ |  $0.51$ $\pm 0.01$ | $0.49$ | $0.38$ |  $0.5$ $\pm 0.02$ |  $0.75$ $\pm 0.07$ | $0.7$ | $0.73$ | $0.77$ |  $0.32$ $\pm 0.02$ |  $0.42$ $\pm 0.02$ |  $0.62$ $\pm 0.02$ | $0.837$ | $0.836$ | $0.835$
$TWCRN_{x34}^{SHUF}$ | $0.23$ | $0.2$ | $0.18$ | $0.21$ |  $0.28$ $\pm 0.02$ |  $0.45$ $\pm 0.03$ | $0.41$ | $0.42$ | $0.44$ | $0.15$ | $0.19$ |  $0.25$ $\pm 0.02$ |  $0.59$ $\pm 0.01$ |  $0.58$ $\pm 0.01$ |  $0.58$ $\pm 0.01$
$TWCRN_{x34}$ |  $0.55$ $\pm 0.01$ | $0.52$ |  $0.49$ $\pm 0.01$ | $0.36$ |  $0.49$ $\pm 0.01$ |  $0.79$ $\pm 0.03$ | $0.71$ | $0.74$ |  $0.777$ $\pm 0.01$ |  $0.32$ $\pm 0.02$ |  $0.419$ $\pm 0.01$ |  $0.61$ $\pm 0.04$ |  $0.838$ $\pm 0.01$ |  $0.835$ $\pm 0.01$ |  $0.832$ $\pm 0.02$

Table 6: Summary of a 5-fold validation results on the test set for the
datasets acquired through the mention graph technique, i.e., each cell is the
average of five train-test executions ((most of the mean values were rounded
up to the second decimal point, unless a higher precision was needed to decide
on the best value, and standard deviations smaller then $0.01$ are omitted).
The best values for each metric are represented by the green color.

Model Data Set | Volcano | Kobe | Princess | Beirut | Unfiltered
---|---|---|---|---|---
Metric | P | $F_{1}$ | R | P | $F_{1}$ | R | P | $F_{1}$ | R | P | $F_{1}$ | R | P | $F_{1}$ | R
$RND_{p=0.5}$ | $0.1$ | $0.16$ | $0.5$ | $0.1$ | $0.17$ | $0.5$ | $0.11$ | $0.18$ | $0.5$ | $0.09$ | $0.15$ | $0.5$ | $0.31$ | $0.39$ | $0.5$
$RND_{p=\pi}$ | $0.1$ | $0.1$ | $0.1$ | $0.1$ | $0.1$ | $0.11$ | $0.11$ | $0.12$ | $0.12$ | $0.09$ | $0.09$ | $0.1$ | $0.32$ | $0.4$ | $0.55$
$MLE$ | $0.8$ | $0.81$ | $0.86$ |  $0.73$ $\pm 0.03$ |  $0.78$ $\pm 0.02$ | $0.88$ | $0.88$ | $0.89$ | $0.97$ |  $0.59$ $\pm 0.01$ | $0.65$ | $0.79$ | $0.63$ | $0.64$ | $0.64$
$ALO$ | $0.09$ |  $0.16$ $\pm 0.01$ | $0.51$ | $0.08$ |  $0.14$ $\pm 0.03$ |  $0.5$ $\pm 0.01$ |  $0.11$ $\pm 0.1$ |  $0.18$ $\pm 0.02$ |  $0.53$ $\pm 0.02$ |  $0.07$ $\pm 0.01$ |  $0.12$ $\pm 0.02$ |  $0.54$ $\pm 0.01$ | $0.32$ | $0.47$ | $0.87$
$LT$ |  $0.1$ $\pm 0.01$ |  $0.17$ $\pm 0.02$ | $0.56$ |  $0.09$ $\pm 0.02$ |  $0.15$ $\pm 0.03$ | $0.56$ |  $0.12$ $\pm 0.01$ |  $0.19$ $\pm 0.02$ | $0.55$ |  $0.07$ $\pm 0.01$ |  $0.12$ $\pm 0.02$ |  $0.56$ $\pm 0.01$ | $0.31$ | $0.2$ | $0.15$
$TWMN_{all1}$ | $0.81$ | $0.81$ | $0.85$ |  $0.74$ $\pm 0.02$ |  $0.8$ $\pm 0.01$ | $0.93$ | $0.86$ | $0.86$ | $0.97$ |  $0.6$ $\pm 0.02$ |  $0.66$ $\pm 0.17$ | $0.79$ |  $0.37$ $\pm 0.03$ |  $0.44$ $\pm 0.02$ |  $0.55$ $\pm 0.01$
$TWMN$ | $0.8$ | $0.86$ | $0.97$ |  $0.74$ $\pm 0.02$ |  $0.83$ $\pm 0.02$ | $0.997$ | $0.86$ | $0.87$ | $0.99$ | $0.56$ | $0.69$ | $0.96$ | $0.47$ | $0.54$ | $0.62$
$TWPN$ | $0.8$ | $0.86$ | $0.97$ |  $0.73$ $\pm 0.03$ |  $0.82$ $\pm 0.02$ | $0.98$ | $0.86$ | $0.87$ | $0.99$ |  $0.58$ $\pm 0.02$ |  $0.71$ $\pm 0.02$ | $0.96$ | $0.69$ | $0.68$ | $0.68$
$TWCRN_{x18}^{SHUF}$ | $0.09$ | $0.01$ | $0.1$ | $0.09$ |  $0.1$ $\pm 0.01$ |  $0.12$ $\pm 0.01$ | $0.1$ | $0.1$ | $0.1$ | $0.08$ |  $0.09$ $\pm 0.01$ |  $0.12$ $\pm 0.01$ | $0.56$ |  $0.55$ $\pm 0.01$ | $0.55$
$TWCRN_{x18}$ | $0.892$ | $0.927$ | $0.966$ | $0.785$ | $0.877$ | $0.995$ | $0.975$ | $0.9832$ | $0.992$ |  $0.71$ $\pm 0.01$ | $0.82$ | $0.97$ | $0.842$ | $0.83$ | $0.82$
$TWCRN_{x34}^{SHUF}$ | $0.11$ | $0.11$ | $0.12$ | $0.1$ | $0.11$ | $0.12$ | $0.11$ | $0.11$ | $0.11$ | $0.1$ |  $0.1$ $\pm 0.01$ |  $0.12$ $\pm 0.01$ | $0.56$ | $0.56$ | $0.551$
$TWCRN_{x34}$ | $0.891$ | $0.929$ | $0.97$ | $0.78$ | $0.876$ | $0.996$ | $0.974$ | $0.9833$ | $0.993$ |  $0.68$ $\pm 0.02$ |  $0.8$ $\pm 0.01$ |  $0.977$ $\pm 0.01$ | $0.84$ | $0.827$ |  $0.81$ $\pm 0.01$

Table 7: Summary of results on the test set for the datasets acquired through
the follower graph technique, i.e., each cell is the average of five train-
test executions (most of the mean values were rounded up to the second decimal
point, unless a higher precision was needed to decide on the best value, and
standard deviations smaller then $0.01$ are omitted). The best values for each
metric are represented by the green color.

###  5.3 Results

We turn to describe the results of evaluating both the algorithms described in
Section 3 and the baseline algorithms, Section 5.1, on the data that we
collected, Section 4.

Our raw dataset spans the mention graph. Namely, the social links are derived
from users re-sharing posts of other users. We further removed low-activity
users from the dataset, remaining with roughly $10^{4}$ users as described in
Table 3. In our experiments, we ran the algorithms also on the followers graph
dataset, which we obtained from the mention graph dataset by removing users
which no one follows or they follow no one, by querying Twitter’s API.

To assess the impact of using the adjacency matrix $A$ on the performance of
$TWMN$ we trained a second variant, $TWMN_{all-1}$, in which the masking is
removed, namely the NN becomes fully connected (the same as replacing $A$ with
the all one matrix).

The convolutional network $TWCRN$ does not receive explicitly the social
links. To understand whether it implicitly learns them or not, we performed
the following permutation test, which resulted in a model we call
$TWCRN_{SHUF}$. At train time, the input vectors $\tau_{i}$ that are fed into
the model are randomly shuffled, anew for every $i$, while the labels of the
result vectors $\tau_{i+1}$ were kept without change.

Table 5 describes the hardware and the running time it took to train each
model. The Table 7 describes the results for the mention graph dataset, while
the Table 6: describes the results for the followers graph dataset. Since the
training of the algorithms contains random choices, we repeated each execution
five times. The standard deviations in both tables are over these five
executions.

Note that our dataset is imbalanced. There are more zeros than ones in each
$\tau_{i}$. The average density of $\tau_{i}$ is given by the precision of the
$RND_{p=0.5}$ algorithm.

We summarize our main findings that we read from Tables Table 6 and 7:

  * •

Our first observation is that the results obtained from the followers graph
dataset are significantly better for all four datasets, with F1 score nearly
twice as good in three out of the four (for the leading method). This means
that the signal embedded in the followers graph is stronger than the one
embedded in the mention graph. Trying to understand why, we ran Botometer [10]
on the users that were removed from the mention graph, and found that their
bot score was on average much lower than the remaining users (e.g. for the
Princess dataset, the score was 0.97 vs 1.62, out of 5).

  * •

The best results, across all datasets, were achieved by $TWCRN$ (both for the
follower graph and the mention graph) with an average F1 score of 0.86. This
NN does not take into account the social links, at least not explicitly
(perhaps the NN learns them implicitly). This network is also completely
agnostic to epidemiological models. A clue to that $TWCRN$ learns the network
structure in some implicit way lies in the fact that $TWCRN_{SHUF}$ fails
miserably in all datasets. As we’ve mentioned in the introduction, many works
have pointed out that information spreads differently in social networks than
epidemics spread in the population. Thus, it may be that $TWCRN$ is able to
learn this more complicated model of infection, which eludes simple human
intuition.

  * •

Having said that, we see that the simple $MLE$ algorithm performs surprisingly
well on all datasets, achieving an average F1 score of 0.78 on the followers
graph dataset. The $MLE$ follows a very simple intuition - it learns the
user’s “trends”, whether the user takes a break between tweets, or is a chain-
twitter. Note that the $MLE$ algorithm completely ignores the social links.

  * •

$TWMN$ is the only of our algorithms that takes into account the social links.
Indeed it outperforms $MLE$, with an average F1 score of 0.81, but its
performance is just slightly better than $TWMN_{all-1}$ (at an average F1
score of 0.8), where again the social links are not explicitly served. Hence
we may conclude that the NN is able to learn by itself the important social
links, and does not need it as an explicit input.

  * •

$TWPN$ performs better than the MLE, and the same as $TWMN$, at an average F1
score of 0.81. The intuition behind $TWPN$ is similar to $MLE$, with the
difference that $TWPN$ is basically learning a logistic regression per user
rather than estimating the probabilities by averaging (although the tanh
function is used and not the logit).

  * •

The two baseline models $ALO$ and $LT$ perform poorly, slightly improving over
the random guess. There are several ways to explain these poor results. This
may be attributed to the way we have constructed the classification task,
which may not be suitable for such models. Another possibility is that the
influence weights $\alpha_{u,v}$ that are used in Eq. (4) and (3), which we
derived from the weights of $TWPN$, were not a good choice.

##  6 Discussion

In this work we checked two premises, the first is whether one can develop a
prediction model which is completely agnostic to the motivation of information
cascades as viewed from an epidemiological standpoint. If yes, then how much
does the social link structure is a vital input to that prediction algorithm,
or, can it be learned implicitly. We answered both questions positively,
showing that a high F1 score is obtainable with a neural network that does not
receive the social link structure explicitly, but does make use of it (as the
random shuffle test shows). We also show that a slightly lower F1 score can be
obtained by two simple algorithms, $MLE$ and $TWPN$, that completely ignore
the social links. The role of the social links in predicting user activity
should be further studied in future work, in other OSNs, and in other
settings.

If we take out the Beirut dataset, which was our smallest dataset, then we
discover that the ranking of F1 scores is inverse to the ranking of
broadcasticity score, Princess is leading with highest F1 score and lowest
broadcasticity, and Kobe vice-a-versa.

We found that the results are significantly better in followers graph dataset,
obtained by removing users that interact only with users that are not in their
followers or friends list. Our first guess that these users are bots turned
out false, and we leave this point as well for future work.

Finally, let us discuss the limitations of our approach. Our prediction was
done for datasets that were collected around a single topic. We have not
checked how well this method performs for a database that is not topic
oriented. Our prediction task is time-constrained, namely, the input to the
prediction algorithm is the activity in the last 12h, and prediction is made
for the following 12h. While we played with these time slices and note a minor
effect on the results, we did not try other configurations such as training
for a week and predicting for the next day. We did not evaluate our models on
platforms other than Twitter. Nevertheless, we expect them to generalize well
on any platform which provides the same mechanisms for communication and
social interaction since we did not use any domain-specific features such as
linguistic features or other meta-data features.

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