The choice of this parameter controls the level of specificity wanted for the
textual clusters.

######  Choosing $r$

We saw the experiments of Section .18 that the choice of $r$ allows us to
nudge the clustering towards textual-based clustering of temporal-based
clustering. Smaller values of $r$ favour the textual content as the main
information in the creation of the clusters, whereas larger values of $r$ make
their composition rely more on the inferred temporal dynamics.

Figure 51: Timeline of the top inferred clusters from news headlines from
01/2019 to 12/2019 — Each line is normalized with respect to its maximum
value. Each bin accounts for half a day. The darker, the more observations in
the cluster at a given time.

In our experiments, we see that smaller values of $r$ tend to increase the
number of inferred clusters. This can be explained in the following way: the
temporal concentration parameter $\lambda_{0}$ has been fixed so that an
observation $2\sigma$ away from the center of one RBF kernel entry (that is
$\sim$95% chances not to be triggered by it) has $\sim$50% chances to open a
new cluster. Mechanically, it is likely that $\lambda_{0}$ is lesser than the
value of the RBF kernel most of the time. However, for smaller values of $r$,
the gap between $\lambda_{0}$ and the temporal intensity $\lambda(t)$ fades to
0 (see Fig. 29), making it more likely to open new clusters from the temporal
perspective. In the limit $r\rightarrow 0$, we recover a Uniform Prior [124],
making the opening of new clusters entirely governed by the parameter
$\theta_{0}$ –clusters are created and filled based on textual content only.
On the contrary, when $r$ is large, the gap between $\lambda_{0}$ and
$\lambda(t)$ is greater, making it less likely to open new clusters from a
temporal perspective. In the limit $r\rightarrow\infty$, the opening and
filling of clusters is deterministically governed by the temporal information,
as the smallest gap in the intensities leads to a Dirac distribution on the
cluster with the largest temporal intensity.

#####  .20.4.b Visualizing topics over time

In Fig. 51, we plot the timeline of the inferred clusters on a real-time axis
for one of our experiments (kernel “Hour”, $\theta_{0}=0.01$, $r=1$) for
illustration purpose. Each bin represents a half-day period. We can make
several interesting observations from this figure.

Firstly, two clusters seem to be always present. Their intensity does not
follow any visible bursty dynamics. When we look at their composition, we
notice that the first cluster is made of 75% of articles from the subreddit
r/worldnews, which is +20% from what one would expect from chance (55% of the
corpus is from r/worldnews, see Fig. 50). Similarly, the second cluster
comprises 46% of r/news articles, which is also roughly +20% from expected at
random (28% of the corpus is from r/news, see Fig. 50). These two clusters
therefore significantly account for publications from either of these
subreddits, independently from the textual content. Our first intuition is
that there are strong interactions between these subreddits. Both are general
news forums with a large audience; an article that gets posted in one of them
is highly likely to be copy-pasted on the other.

Secondly, topics that are not part of these two clusters appear and fade
quickly in time. This is in line with the expected behaviour of news on the
internet, which typically bursts for a few hours/days before being replaced by
the latest news. We see however from this plot that only a small fraction of
the inferred clusters coexists simultaneously in the dataset. The chance of
spotting an interaction is therefore weak, as noted in Chapter -.

Thirdly, it seems that clusters expand on larger periods at the beginning of
the algorithm. This is due to the cold start of MPDHP. It needs some time
before statistically distinguishing clusters and explores several directions
at once. The early clusters are artefacts of such a cold-start effect.

#####  .20.4.c Quantifying interactions

######  Effective interaction

We introduce the parameters we are going to use in follow-up analyses. The
output of MPDHP consists of a list of clusters comprising timestamped bags of
words –news headlines. Between each pair of clusters, MPDHP inferred a
temporal influence function $\lambda(t)$, that represents the probability for
one cluster to trigger publications from another. Therefore, our model yields
an adjacency matrix $A\in\mathbb{R}^{K\times K\times L}$, where $K$ is the
number of clusters and $L$ the size of the RBF kernel $\vec{\kappa}(t)$. One
entry $a_{i,j,l}$ represents the strength of the influence of $j$ in $i$ due
to the $l^{th}$ entry of $\vec{\kappa}(t)$.

However, we must consider the effective number of interactions to get relevant
metrics. A given triggering function $\lambda(t)$ could be inferred from the
observation of very few observations only; we must weigh these interactions.
To do so, we simply consider a normalized weight matrix
$W\in\mathbb{R}^{K\times K\times L}$, whose entries $w_{i,j,l}$ are the
average of the intensity of $i$ above $\lambda_{0}$ due to $j$ from the kernel
entry $l$ for all observations. Explicitly:

$w_{i,j,l}=\frac{1}{|\mathcal{H}_{i}|}\sum_{t_{i}\in\mathcal{H}_{i}}\sum_{t_{j}<t_{i}}max(a_{i,j,l}\kappa_{l}(t_{i}-t_{j})-\lambda_{0},0)$
(78)

where $t_{x}\in\mathcal{H}_{x}$ is the publication time of an observation from
cluster $x$ and $\lambda_{0}$ the temporal concentration parameter. Note that
we retract $\lambda_{0}$ from the intensity term, because it is considered as
a background probability for a publication to happen –similarly to the
virality in Chapter - and to the background noise Chapter .8.0..5. Note that
$W$ can also be interpreted as the instantaneous increase in probability due
to interactions.

Note that in all the following computations, we do not consider clusters that
comprise less than 10 documents. Such clusters are considered leftovers from
the algorithm.

######  Interactions strength

In Table 12, we investigate the effective impact of interactions in the
dataset. We consider the following metrics:

  * •

$<A>$: the average value of the whole adjacency matrix. It tells us to which
extent pieces of information are connected to each other according to MPDHP.

  * •

$<W>$: the average value of the effective interactions. It tells us the extent
to which the interactions (encoded in $A$) effectively happen in the dataset.

  * •

$<A>_{W}$: the average of the inferred interaction matrix $A$ weighted by the
effective interactions $W$. In this case, $W$ can be interpreted as our
confidence in the corresponding entries of $A$ given the data with which they
were inferred. This value quantifies the overall role of interactions in the
dataset.

  * •

$\frac{<W^{intra}>}{<W^{extra}>}$: ratio of the intra-cluster effective
interactions with the extra-cluster effective interactions. This tells us how
much different clusters influence each other versus how much they influence
themselves.

When computing the means, we discard the entries of $A$ and $W$ equal to 0.
This is because all clusters do not exist simultaneously, and thus should not
be considered. An interaction strictly equal to 0 means that clusters simply
did not exist at the same time.

$\vec{\kappa}(t)$ | $\theta_{0}$ | $r$ |  $<A>$ ($10^{-3}$) |  $<W>$ ($10^{-5}$) |  $<A>_{W}$ ($10^{-3}$) | $\frac{<W^{intra}>}{<W^{extra}>}$
---|---|---|---|---|---|---
Minute |  0.01 | 0.5 | 49(21) | 342(889) | 66(17) | 1.8(62)
1.0 | 48(20) | 478(1124) | 60(17) | 1.4(43)
1.5 | 48(20) | 746(1901) | 60(17) | 1.0(33)
2-7 |  0.001 | 0.5 | 50(22) | 316(882) | 66(17) | 3.1(138)
| 1.0 | 50(21) | 279(752) | 67(16) | 2.6(105)
| 1.5 | 50(22) | 268(665) | 67(16) | 2.3(84)
Hour |  0.01 | 0.5 | 49(18) | 389(843) | 56(17) | 0.5(13)
1.0 | 49(18) | 478(1187) | 56(17) | 0.6(15)
1.5 | 48(17) | 471(789) | 52(15) | 0.7(13)
2-7 |  0.001 | 0.5 | 50(21) | 110(398) | 61(17) | 1.7(67)
| 1.0 | 50(18) | 133(506) | 57(17) | 1.4(60)
| 1.5 | 49(17) | 183(554) | 55(17) | 1.1(37)
Day |  0.01 | 0.5 | 49(18) | 41(97) | 55(17) | 1.2(34)
1.0 | 49(19) | 63(131) | 54(17) | 1.2(31)
1.5 | 49(19) | 91(187) | 53(18) | 1.2(31)
2-7 |  0.001 | 0.5 | 50(20) | 18(90) | 60(19) | 1.1(59)
| 1.0 | 50(19) | 23(101) | 58(17) | 1.0(50)
| 1.5 | 50(19) | 37(111) | 56(18) | 1.0(36)
Table 12: Interaction strength — Overall, interaction between clusters is
weak. The large standard deviations suggest that there is a large variety of
interacting behaviours. Interactions tend to happen within a cluster (self-
interactions).

The main conclusion of the results Table 12 is that most interactions are
weak. The average value of $A$ tells us that the average value of the inferred
parameters is around $0.05$, which is few given the value is bounded between 0
and 1. The metric ${<W>}$ tells us that on all events, the interaction between
clusters rose the probability of publication by 0.1%-1% on average. We can
also note that the values of ${<W>}$ are of the same order of magnitude as
$\lambda_{0}$ (0.01 for the “Minute” kernel, 0.001 for “Hour”, and 0.0001 for
“Day”). We can interpret this as the probability for a new document belonging
to a cluster or being from a new cluster is roughly the same from a temporal
perspective. The metric $<A>_{W}$ tells us that when weighting the average of
$A$ with the effective interaction, the values of $A$ are slightly higher than
$0.05$; we can now be confident in this value, given it has been inferred on a
statistically significant number of observations. However, it still tells us
that only some interactions are significant, which correlates with the
findings of Chapter -. Finally, the last metric
$\frac{<W^{intra}>}{<W^{extra}>}$ finds that most effective interactions take
place more often within the same cluster, meaning that clusters tend to self-
replicate. Only for the “Hour” kernel and $\theta_{0}=0.01$ this value is
lesser than one. It is because in this case, MPDHP infers two large clusters
that exist for the entire year (see the 2 first rows of Fig. 51) and side
topic-specific clusters. These clusters strongly influence each other, and all
the topic-specific clusters can interact with them. In fact, the same effect
explains the decrease of $\frac{<W^{intra}>}{<W^{extra}>}$ as $r$ grows: fewer
clusters are inferred, and the probability of having large clusters that last
for the whole period increases.

Another major observation from Table 12 is that standard deviations of
effective interactions are large. It means that despite most interactions
being weak, some of them play a significant role in the dataset. In Fig. 52,
we plot the distribution of effective interactions for one specific run
(“Hour” kernel, $\theta_{0}=0.01$, $r=1$). Note that we recover the same trend
in all other experiments. The results of this figure are similar to the ones
in Chapter - (Fig. 11): most interactions are weak, and only a few of them are
significant.

Figure 52: Distribution of interaction strength – Most interactions are weak.

######  Interactions range

Finally, in Table 13, we investigate the range of effective interactions in
every experiment. The interaction range studies the persistence of
interactions. To do so, we compute the effective interaction for each kernel
entry individually (and not the aggregated interaction as in Table 12) and
average it over all existing clusters.

$\vec{\kappa}(t)$ | $\theta_{0}$ | $r$ | $\kappa_{1}$ | $\kappa_{2}$ | $\kappa_{3}$ | $\kappa_{4}$ | $\kappa_{5}$ | $\kappa_{6}$ | $\kappa_{7}$ | $\kappa_{8}$ | $\kappa_{9}$
---|---|---|---|---|---|---|---|---|---|---|---
|  |  | 0m | 10m | 20m | 30m | 40m | 50m | 60m | 70m | 80m
Minute (m) |  0.01 | 0.5 | 266 | 421 | 407 | 451 | 428 | 403 | 395 | 345 | 121
1 | 396 | 591 | 580 | 607 | 532 | 575 | 521 | 507 | 224
1.5 | 616 | 937 | 893 | 914 | 955 | 840 | 808 | 810 | 304
2-12 |  0.001 | 0.5 | 436 | 509 | 457 | 424 | 371 | 340 | 313 | 178 | 52
| 1 | 284 | 435 | 396 | 388 | 343 | 327 | 272 | 187 | 45
| 1.5 | 208 | 388 | 366 | 353 | 326 | 333 | 290 | 215 | 61
|  |  | 0h | 2h | 4h | 6h | 8h | - | - | - | -
Hour (h) |  0.01 | 0.5 | 494 | 430 | 502 | 456 | 324 |  |  |  |
1 | 658 | 538 | 549 | 542 | 451 |  |  |  |
1.5 | 558 | 615 | 532 | 526 | 411 |  |  |  |
2-8 |  0.001 | 0.5 | 124 | 149 | 119 | 137 | 92 |  |  |  |
| 1 | 154 | 164 | 172 | 149 | 111 |  |  |  |
| 1.5 | 208 | 244 | 223 | 197 | 156 |  |  |  |
|  |  | 0d | 1d | 2d | 3d | 4d | 5d | 6d | - | -
Day (d) |  0.01 | 0.5 | 44 | 45 | 47 | 46 | 47 | 47 | 37 |  |
1 | 70 | 71 | 72 | 68 | 68 | 70 | 61 |  |
1.5 | 102 | 100 | 101 | 105 | 98 | 105 | 82 |  |
2-10 |  0.001 | 0.5 | 18 | 20 | 21 | 21 | 21 | 21 | 17 |  |
| 1 | 24 | 26 | 24 | 27 | 28 | 26 | 22 |  |
| 1.5 | 21 | 41 | 42 | 41 | 41 | 41 | 35 |  |
Table 13: Interaction range — All the values for effective interaction are
given in ten-thousandth ($10^{-5}$). Influence tends to decrease over time for
all the kernels considered.

Importantly, the raw values of effective interaction corresponding to the
first entry of the triggering kernel $\kappa_{1}$ are consistently smaller
than subsequent values. This is induced by our kernel choice, because
$\kappa_{1}$ is always centered around $t=0$, which makes half of the
associated Gaussian function account for (impossible) backwards influence.
Therefore, where other kernels can contribute on both sides of their means,
$\kappa_{1}$ cannot. In Table 13, we extrapolate their value as twice the
computed one.

We see in Table 12 that influence tends to decrease over time for all the
kernels considered, after reaching a first peak. Overall, the interaction
between documents seems to play a marginal role still. We did not plot the
standard deviation for visualization purposes, but they are as large as in
Table 12. Therefore, most interactions do not play a significant role in the
publication of subsequent documents over time, but it greatly helps identify
the right cluster for some of them. Overall, the increase in probability for a
new document to belong to a cluster due to interactions is within 0.1%-1% (we
recall that $\lambda(t)=\lim_{\Delta t\rightarrow\infty}\frac{P(\text{event in
}\Delta t)}{\Delta t}$).

#####  .20.4.d Visualizing topical interactions

In this section, we plot the temporal interaction network between the inferred
clusters both on the global and the monthly scale. The clusters’ composition
is given explicitly. Each edge between a pair of clusters represents two
metrics: the inferred interaction strength $A$, and the effective interaction
$W$. The inferred interaction strength $A$ is plotted using a colour code
(darker is stronger). The effective interaction is represented using
transparency (the less transparent the stronger the effective interaction).
Therefore, a barely visible dark edge means that MPDHP inferred a strong
interaction, but that few of them were effectively observed.

######  Experiment considered for subsequent analyses

In the following paragraphs, we consider the “Hour” kernel, with
$\theta_{0}=0.01$ and $r=1$. We justify this choice for easing the
interpretation of our results. This way, we restrict our analysis to clusters
that do not contain fragments of a whole. For instance, we prefer to have only
one cluster about the Notre-Dame cathedral fire and related news instead of
three clusters containing fragments of the news, such as the initial fire, the
political reactions, funds raising, etc. Therefore, we choose to consider
$\theta_{0}=0.01$, which avoids clusters to be overly specific. We choose the
“Hour” kernel, which spans over long enough periods so that news fragments
about a similar topic are considered as possibly related. Besides, from direct
observation, it seldom happens for news to stick around for more than a few
days, which the “Hours” kernel is fit to capture. The choice of $r$ is based
on an arbitrary trade-off between textual and temporal information. We do not
want to consider extreme values ($r=0$ or $r>2$) so that we exploit both
pieces of information. Besides, we saw in Table 12 and Table 13 that only
slight variations are observed across the range of $r\in\\{0.5,1,1.5\\}$
considered. Finally, we are interested in seeing how the large clusters
spanning over the whole period relate to topic-specific smaller clusters seen
in Fig. 51.

Figure 53: A typical MPDHP output – A set of manually selected clusters along
with the vocabulary of their documents (top), their inferred dynamics (middle)
and the clusters that influenced them on the real-time axis (bottom).

######  Representing individual clusters

In Fig. 53, we represent the raw data that we will use in ulterior
visualizations for some selected clusters. In the top part, we represent their
textual content as a wordcloud. In this case, we chose to pick clusters about
climate change inaction protests, catholic church child abuse scandals,
China’s Uighur detention camps, “Gilets Jaunes” protests in France, and
Brexit. For each cluster, we represent the top temporal influence that other
clusters may exert on them. Transparency accounts for the effective
interaction $W$ discussed in the previous sections. The bottom plot represents
the influence exerted on these clusters by all other clusters at all times
–which does not mean this influence led to a publication.

This way of representing the data fits well in the univariate case, as in
[30], but does not allow to capture the complexity of the inferred mechanisms
at stake. In the following sections, we propose alternative visualizations in
the form of temporal networks.

######  Globally

In Fig. 54, we plot the clusters interaction network over the whole period we
considered (12 months). This figure uses the same data as Fig. 51. Both the
adjacency matrix and the transparency matrix are normalized by their highest
value. We see that the strongest interaction happens between the blue and the
orange clusters. Besides, this interaction seems to be stronger at smaller
times. The interaction between other clusters is essentially directed toward
these two clusters. Shortly after a news cluster appears, it is likely to find
an echo on either r/news or r/worldnews. After existing for a while,
subsequent publications are merged into one of these clusters depending on
where they got published. This explains why most clusters last so few over
time in Fig. 51.

We see that there are very few interactions between the clusters which account
for actual news –in opposition to the clusters that account for a subreddit,
see Section .20.4.b.

Figure 54: Global clusters temporal interaction from 01/2019 to 12/2019 — Each
node represents one cluster, and each edge represents the interaction strength
at various times – the darker and the less transparent the stronger the
effective interaction. The clusters composition is given below the network.
Figure 55: Monthly clusters temporal interaction from 01/2019 to 12/2019 —
Each line stands for one month, each node represents one cluster, and each
edge represents the interaction strength at various times – the darker and the
less transparent the stronger the effective interaction. The cluster
composition is given below the network.

######  Monthly

In Fig. 55, we plot the same interaction between clusters as in Fig. 54, but
for each month. This figure uses the same data as Fig. 51, except it is now
broken down into temporal slices. From this figure, we recover that at the
early times of the algorithm, MPDHP has not converged yet. It makes the
interaction plot messy, with several interactions that are likely to be
inaccurate. As time goes by, interactions are better defined –in particular
between the blue and orange clusters. Another interesting fact is that most
interactions happen in a 1h time frame: the publication of a document on
r/news is highly likely to trigger a similar publication on r/worldnews within
the hour. Here again, we see that most side clusters do not interact
significantly with each other.

####  .20.5 Conclusion

######  A real-world application

In this section, we conducted extensive experiments on a single real-world
large-scale dataset from Reddit. We described how we gather a year worth of
news data and pre-processed it. We then argued how the dataset was fit to
apply to MPDHP. We extensively discussed the various hyper-parameters that had
to be tuned to run our experiments (in particular $\kappa(t)$, $r$ and
$\theta_{0}$). In the end, we conducted 6 different experiments, one for each
combination of parameters. The choice of other hyper-parameters has also been
justified.

######  Interactions do not appear to play a significant role in this dataset

We proposed several ways to assess the role of interactions in the dataset. In
particular, we introduced the notion of effective interaction as a way to
evaluate how confident we can be in MPDHP’s output. On this basis, we analysed
the importance of interactions in general, as well as from a temporal
perspective. We recovered our conclusions from Chapter -: interactions are
sparse. We also recovered our conclusions from Chapter .8.0..5: interaction
strength decays over time. These observations emphasize the adequateness of
our approach to model interactions. Building on these considerations and
looking at the global effective interaction average, we conclude that
interactions play a minor role in such a dataset. Overall, they only increase
the instantaneous probability for a new observation to appear by 1%. Even the
most extreme values represented in Fig. 52 seem to only increase this
probability by 12% top.

######  Perspectives

However, despite intending our study as exhaustive, there is room for
improvement in interaction modelling using MPDHP. In particular, there are two
biases that we could not explore yet. Firstly, the parameter $\lambda_{0}$ has
been set according to a heuristic (so that a new cluster is opened with fifty
percent chances when we are 95% sure that it does not match the existing one).
Its direct inference would robustify the approach, despite seeming
complicated. Indeed, $\lambda_{0}$ does not account for individual events
realizations, but for Hawkes processes starts, which is not a trivial
difference. Secondly, another possible improvement would be to allow clusters
to passively replicate –without the need for an interaction. We expect that
this would boil down to adding a time-independent kernel entry to
$\vec{\kappa}$, in a similar fashion as in Chapter .8.0..5. However, non-
technical questions may arise from such modification: when to consider a
cluster as extinct given a non-fading kernel? How should this kernel relate to
the temporal concentration parameter $\lambda_{0}$?

We believe such improvements would make MPDHP more robust and interpretable,
and find applications beyond interaction modelling.

##  Chapter Conclusion

###  .21 Contributions

####  .21.1 Overview

Throughout this manuscript, we developed seven original models that allowed us
to tackle various aspects of interactions between pieces of information in
their spread:  IMMSBM,  SIMSBM,  SDSBM,  InterRate,  PDP,  PDHP,  MPDHP. Along
the road, this research led us to study seemingly distant fields of machine
learning. A first venture in  stochastic block modelling provided us with
insights on the frequency at which interactions happen. An exploration of
convex network inference methods allowed us to conclude on the time range of
interactions. Driven by the need for more global models, a dive into
fundamental bricks of machine learning led us to explore and modify canonical
Dirichlet processes. Finally, answering our problematic required a  junction
between Dirichlet process and point processes. We recall the summary of our
contributions in Table 14 –this table is identical to Table 2 and is simply
recalled here.

Table 14: Contributions presented in this manuscript — Our contributions are
listed below in the order of their appearance in the text.

SIMSBM IMMSBM SDSBM InterRate PDP PDHP MPDHP Chap. - Chap. - Chap. - Chap.
.8.0..5 Chap. .14.0..4 Chap. .14.0..4 Chap. .14.0..4 Self-interactions x x x x
x x x Pair-interactions x x x x x x N-interactions x x x x Clustering x x x x
x x Discrete time x x x x Continuous time x x x Online inference x x x

####  .21.2 Answers to our problematic

#####  .21.2.a Q1: How frequent are interactions?  $\bullet$

To answer our first question Q1, we considered Stochastic Block Modelling to
be a good approach.

Indeed, a straightforward way to represent interactions is to embed them as a
network. Entities are nodes of this network, and interactions between these
entities are link between these nodes. These links can account for distinct
types of relations that we can associate with different labels. Given the
number of possibly interacting entities can be high in some real-world
systems, we need to group them to see whether groups of entities interact in a
similar way instead of dealing with each entity individually. In practice,
typically only a few of these possible interactions actually happen, but we
did not know this beforehand. Besides, we considered that entities could
interact as pairs, triplets or more, and that they could interact with
different elements of their environment (time, user’s metadata, etc.). After a
careful review of the literature, we noticed that several state-of-the-art
models could be expressed as special cases of a more global framework: the
Serialized Interacting Mixed membership Stochastic Block Model (SIMSBM,
Section .6.2.a). This generalization allows modelling an arbitrarily large
context (number of input pieces of information) as well as an arbitrarily high
order of interactions.

Along with this generalization, we introduced a simple procedure to
incorporate time modelling into any iteration of SIMSBM. This extension takes
the form of a prior on the model’s parameters. This approach relies on the
single assumption that dynamics are not abrupt.

We considered a special case of SIMSBM, SIMSBM(2) or Interacting Mixed
Membership SBM (IMMSBM), and used it to model interactions between entities in
several real-world datasets. We investigated the role of interactions between
different spreading entities (hashtags, words, memes, etc.) and quantified
their importance in several corpora –see Section .6.3.

In particular, we focused on the study of a Twitter dataset and investigated
the role of interaction between Twitter URLs on their spreading probability.

The conclusion of this study answered the first question Q1 raised in the
introduction: interactions are sparse. On the Twitter dataset, significant
interactions took place only between a limited fraction of cluster pairs, and
between an even smaller fraction of entity pairs.

#####  .21.2.b Q2: How persistent are interactions?  $\bullet$

Despite the generality of the proposed SIMSBM, this model was not fit for
modelling the time range over which interactions may take place. However,
taking the interaction range into account is fundamental following a simple
consideration: that the influence of entities on someone cannot last
indefinitely as they get gradually forgotten by the users.

We introduced a convex multi-kernel model designed for this task: InterRate
–see Chapter .8.0..5. This model was able to infer the evolution of pair
interactions’ intensity over time, for any pair of entities, called the
interaction profile. It allowed us to study how users exhibit different
behaviours according to combinations of exposures they have been exposed to in
continuous time. We showed that the joint effect of two exposures on a user is
more than the disjoint sum of their individual effect, which means there is an
interaction.

The answer of this study to the second question (Q2) raised in the
introduction is that interactions are brief. A study of the processes at stake
in the Twitter dataset revealed that interaction between two entities is
significant only when those are close to each other in time. Typically, the
intensity of an interaction decreases exponentially with the time separating
the interacting entities.

#####  .21.2.c Q3: Can we efficiently model interactions?  $\bullet$ $\bullet$

Our third question raised the problem of efficiently model interactions.
Designing a dedicated study to get insights into one aspect or the other of
the interacting processes is one thing. Designing a scalable and useful method
that answers the possible challenges raised by the task at hand is quite
another.

The conclusions drawn from Chapter - and Chapter .8.0..5 are that interactions
are sparse (we need clusters to model them) and that interactions are brief
(we need to consider time). Besides, we discussed the fact that interacting
entities may have a short lifespan, and that their semantic content must be
taken into account. In Chapter .14.0..4, we introduced the steps paving our
way to such a model.

Our approach was based on in-depth modifications of an existing Bayesian
prior: the Dirichlet-Hawkes Process. This model was promising regarding our
problem: it allowed us to create clusters ( Q1) that can have a lasting
influence in time on ulterior observations ( Q2). Moreover, it could perform
this task in linear time with the size of the dataset. However, it also
suffered from several limitations. In particular, it was not fit to retrieve
meaningful clusters when textual information conveyed little information or
when temporal dynamics were hard to unveil. Furthermore, it assumed that the
textual content of a document is perfectly correlated to its temporal
dynamics, which cannot be rigorously true in real-world processes.

As an indirect way to overcome these limits, we explored the Powered Dirichlet
Process as an alternative to the Dirichlet Process. This new prior alleviates
the “rich-get-richer” property of vanilla Dirichlet processes.

We then incorporated this Powered Dirichlet Process into the standard
Dirichlet-Hawkes process to create the Powered Dirichlet-Hawkes Process. This
formulation improved the results when temporal information or textual content
were weakly informative and alleviated the hypothesis that textual content and
temporal dynamics were always perfectly correlated. Thus, our approach
eventually allowed us to correctly model self-interactions within a cluster.

As a final improvement on the proposed approach, we extended the Powered
Dirichlet-Hawkes Process to the multivariate case. This way, we can
efficiently model interactions between all pairs of clusters, and by extension
for all pairs of entities they comprise. The model can also run in constant
time, and typically takes 300ms per document on large-scale real-world
datasets.

We therefore provided a way to efficiently model interactions, positively
answering our third introductory question (Q3). Efficiency can be understood
both as technical efficiency and relevance efficiency. This former because our
approach scales well on large real-world datasets and takes a minimal time to
process a consequent amount of data. The latter because the final model can
answer all the crucial challenges raised by interaction modelling, as it…

  * •

considers entities’ content. An entity is no more described as a unique
identifier, but instead by its semantic content. Two entities that convey the
same information are now considered as such and clustered together as a more
global entity–a topic.

  * •

models sparse interactions. Entities are now clustered together into temporal
clusters. It makes it feasible to spot interaction terms between sets of
entities. The lifespan of entities is no more a problem since clusters can
comprise entities spanning over extended periods, which also increases the
data available for each cluster.

  * •

models dynamic interactions. Each cluster is associated with its own intensity
function, which determines its effect on ulterior observations. Eventually,
entities ’ influence fades away as time goes by.

  * •

models multivariate interactions. Every cluster interacts as a pair with other
clusters, enabling the study of interactions between the inferred topics.

#####  .21.2.d Q4: Do interactions play a significant role in spreading
processes?  $\bullet$

We conducted extensive experiments on a real-world large-scale dataset made of
one year of news headlines gathered from Reddit. This dataset seems fit for
interaction modelling: contents are user-generated, dynamic, and mutating. We
proposed several ways to assess the role of interactions in this specific
dataset. The notion of effective interaction has been chosen as a privileged
way to evaluate how much interaction has a role in the assumed data generation
process. Firstly, we recovered our conclusions from Chapter -: interactions
are sparse. Secondly, we recovered our conclusions from Chapter .8.0..5:
interaction strength decays over time. By looking at the average role of
interactions in this dataset, we saw that interactions play a minor role in
such a dataset. On average, interactions increase the instantaneous
probability for a new observation to appear by 1% over its assumed random
appearance probability. Without looking at the aggregated statistics, extreme
values typically increase this probability by $\sim$10%, which does not hint
toward strong interaction effects.

####  .21.3 General uses for our models

Each of the models introduced in this manuscript has been presented as a novel
way to tackle the interaction modelling problem. However, efforts have
systematically been made to illustrate other, more general use cases. As a
final note, we argue and detail some use cases of our approaches outside of
interaction modelling.

#####  .21.3.a Powered Dirichlet Processes

The Dirichlet process (DP) is canonical in Bayesian nonparametric modelling.
Enumerating the models based on DP is not feasible; among the most popular
ones, we find the Latent Dirichlet Allocation model, the Infinite Gaussian
Mixture Model, and the non-parametric MMSBMs and the Dirichlet-Hawkes process,
all of which have seen numerous extensions depending on the use context. The
popularity of Dirichlet processes is explained by the possibility to
automatically infer the number of latent groups along with their composition,
and the possibility to process data sequentially, in the order of arrival.

A current way to extend models based on DP is to make them hierarchical. Using
the hierarchical DP, one can consider a mixture of partitions, by drawing the
base distribution of a DP from another DP. Other extensions such as the
Pitman-Yor process and the nested DP have been discussed in Section .17. By
proposing the Powered DP (PDP), we paved the way for a whole new class of
extensions. We illustrated the impact of redefining DP by adding a nonlinear
dependence term. Possible applications for this advance would already comprise
all existing models based on DP and models based on its extension, as the
Powered DP is not exclusive. In the case of hierarchical DP for instance, the
base distribution from which is drawn a Powered DP would also be drawn from a
Powered DP, making the Powered hierarchical DP. Similar extensions and
combinations are possible for the nested DP, the hierarchical nested DP, etc.

#####  .21.3.b Stochastic Block Models

When working on stochastic block models, our main goal was to design a
framework (SIMSBM, Section .6.2.a) that allows us to model almost any type of
categorical data. It has been designed so that it can take as many pieces of
information as needed as an input context, as many symmetric interactions
between these entities as needed, and consider the temporal dimension to model
dynamic cluster memberships (SDSBM extension, Section .7). The flexibility
provided by this work fits interaction modelling purposes, but also many other
applications. The IMMSBM (Section .6.3.a) has been accepted as a contribution
to recommender systems, which is not explicitly about interactions modelling
[98].

Already discussed use cases in recommender systems include product
recommendations on online retail websites, movies recommendation on streaming
platforms, and songs on music streaming platforms. We also showed these models
could be used to predict players’ next move on real-world datasets and
replicated studies of [102] that identified elementary players’ behaviours. We
argued for a possible application in automated medical diagnosis, where the
combination of a few symptoms leads to high-quality diagnoses. We showed it
could help predict the next tweet retweeted by Twitter users.

Further uses are encouraged in the field of humanities, where data often lies
unexploited due to the lack of explainable, readily usable models. We
illustrated a possible use case using a Latin epigraphy database. Projects
using SIMSBM to infer the gender and age of antic remains given the tools and
objects found in their tombs are currently ongoing. We strongly argued for
such use in the SDSBM section. We paid particular attention to humanities
recurrent challenges on data availability; our approach is shown to work even
with scarce data.

The point is, that the field of applications for (dynamical) SIMSBM ranges way
beyond simply interaction modelling. Dedicated studies using these new models
may provide interesting results in social sciences, medicine and online
recommendation.

#####  .21.3.c Dirichlet-Point processes

In Chapter .14.0..4 and in this Conclusion, we extend the idea of Dirichlet-
Hawkes processes to a broader class of models. The challenges inherent to
Dirichlet-Hawkes processes have been answered by developing the Multivariate
Powered Dirichlet-Hawkes process. This model can specifically be used to
create a time-lined summary of event streams. At a time when internet content
appears at an unprecedented pace, dedicated tools for big data summarizing
will become increasingly necessary in many applications. Understanding ongoing
trends on social platforms would help identify topics of interest, or simply
provide these platform’s users with an overview of what is going on. Several
digital newspapers manually compile trends of interest that appeared over a
day, a week; such tasks could be helped if not automated by such tools.

Another possible application specific to the MPDHP is the understanding of
publication mechanisms. Significant research is being done in identifying and
countering fake-news diffusion on social media. The tool we developed allows
us to get insights into the way topics relate to each other. Dedicated studies
on the interplay between fake-news topics and disclaimers could help develop
countering strategies. Going one step further, we showed Dirichlet-Point
processes could be used to make unsupervised modelling of topic-dependent
spreading subnetworks. In the same line as before, automatically identifying
fake news spreading subnetworks could help to surgically burst opinion bubbles
by encouraging new links to other communities. As a more direct approach, it
could help target specific nodes with disclaimers.

###  .22 Perspectives

####  .22.1 Towards more general block-modelling approaches

In Chapter -, we developed a global framework that allows us to model
interactions of any given order using any size of context. We then proposed an
extension to our method that allows us to consider time. We believe that this
modelling flexibility can serve as a base to develop improved, even more
flexible models to tackle a range of problems. We consider two possible
extensions below.

#####  .22.1.a Considering time as a continuous variable

In Section .7, we proposed a way to model time as an additional constraint to
the stochastic block modelling approach. There, time is discrete and one model
is inferred for each time slice, conditional on models from other time slices.
However, we discussed in subsequent sections how slicing time in discrete
intervals can induce biases in the modelling. A possible lead to alleviate
this problem would be to merge our work on SIMSBM (Section .6.2) with the
Dirichlet-Point processes discussed in Chapter .14.0..4. In particular, SIMSBM
uses a Dirichlet prior as a priori on its membership vectors. It happens that
some works explored this direction to derive a non-parametric version of
MMSBM, by expressing the Dirichlet prior as a Chinese Restaurant process [34]
–as what we did in Chapter .14.0..4. Once the MMSBM is expressed as a
sequential Dirichlet process, it might be possible to include the advances in
Dirichlet-Point processes as an explicit way to model time as a continuous
variable. In general, this approach could make SIMSBM-based approaches fit to
consider continuous data in general, provided it obeys an underlying point
process.

#####  .22.1.b Considering nodes’ metadata

Up to now, we modelled interactions at the level of the interacting entities
themselves. Using our Stochastic Block Modelling framework, it is now possible
to account for the entities ’ content and interaction time. However, this is
not the most elegant way to consider the context in which a piece of
information interacts with others. Recent works based on similar models
proposed to model nodes’ metadata as an additional layer, whose links can be
activated or deactivated [32]. In this case, metadata does not have to be of a
given type, not it is mandatory for it to be useful. The inclusion of these
advances into the proposed framework would make a significant step towards an
SBM that could be applied to any problem at hand with minimal model designing
effort.

####  .22.2 Improving the Multivariate Powered DHP

#####  .22.2.a Accounting for exogenous data generation

A consideration that we factored out from our analysis is the role of
exogenous data generation. It has been underlined on some occasions that
documents can get published according to dynamics external to the dataset. Its
apparition could have been conditioned by other media sources (TV, radio, …),
social links not accounted for on Twitter or simply a demonstration of free
will, and therefore should not be included in our dataset generative
assumption. In [88], the authors model the rate of arrival of documents from
such exogenous influence using temporal point processes. They conclude that
the role of external influence is not trivial. Supporting this claim, [56]
introduced a term that accounts for exogenous events in Hawkes-based
modelling. The fact that both works make extensive use of temporal point
processes in their modelling suggests that the inclusion of their findings
into MPDHP is doable and would certainly yield insightful results.

#####  .22.2.b Going further than Dirichlet-Hawkes processes

In Chapter .14.0..4, we studied in-depth the Dirichlet-Hawkes process and
various models that can be built on them. However, the full picture is broader
than simply the association of Dirichlet processes and Hawkes processes.
Instead, the method described throughout Chapter .14.0..4 can be applied to
merge any Dirichlet process (hierarchical, nested, or powered) or variants
(Indian Buffet Process, Pitman-Yor process, etc.) with any point process
(Hawkes, Cox, Poisson, Determinantal, Geometric, etc.). We believe that the
resulting Dirichlet-Point processes are powerful tools that can adapt to many
modelling problems. This field of such combinations has been little explored
up to now and may offer interesting insights for further studies.

####  .22.3 Considering the network structure

In this manuscript, we considered both the content and the dynamics of
spreading entities to unveil interactions. However, our models are not fit to
consider the structure of the network entities spread on. An interesting lead
to explore is how interactions between pieces of information differ at the
user level, depending on their position in the spreading network.

Both as a final perspective and as an illustration of a broader use for
Dirichlet-Point processes, we propose to develop a model that considers the
network structure based on Dirichlet-Point processes. We sketch a model that
can jointly infer dynamic (Chapter .8.0..5) clusters (Chapter -) of textual
documents (Chapter .14.0..4) spreading online and the subnetworks they spread
along. We did not find any previous attempt to jointly infer these parameters,
by using an iteration of the Dirichlet-Point processes discussed in this
conclusion.

#####  .22.3.a Possible lead: Dirichlet-Survival process

We can bridge the gap between network inference and dynamic clustering models
by defining the Dirichlet-Survival process.

######  Network inference as a survival process

In [47], the authors demonstrate that most of the then-existing underlying
network inference models can be derived as a special case of a global
framework. Without entering the details here, it is shown that using a
specific formulation of Survival processes allows retrieving network inference
models such as NetRate [45], KernelCascade [28], MoNet [125], InfoPath [48],
etc.

Each of these models can be characterized by a single intensity function,
similar to the Hawkes intensity discussed in Chapter .14.0..4, named the
hazard rate. For the interested reader, we detail how to get to this function
for the NetRate model [45] in Appendix .9.3.

We write this intensity function $\lambda(t_{i}^{c}|t_{j}^{c},\alpha_{j,i})$.
It represents the instantaneous probability for a node $i$ to be infected at
time $t_{i}^{c}$ by an infection $c$ because of the node $j$ that got infected
by $c$ at time $t_{j}^{c}$. The strength of the link between $i$ and $j$ is
encoded in $\alpha_{j,i}$. As this intensity function results from a survival
point-process similar to the Hawkes process, we will see that we can
substitute it to the counts of a Dirichlet process to create a Dirichlet-
Survival process.

######  Dirichlet-Survival prior

As in [30, 83, 118] and what we did throughout Chapter .14.0..4, we will
create a new Dirichlet-Point process by merging the Dirichlet Process in its
Chinese Restaurant iterative metaphor, to a Point process such as the one
characterizing NetRate. We write the intensity of such process
$\lambda(t_{i}^{c}|t_{j}^{c},\alpha_{j,i})$.

Doing so essentially breaks down the temporal dynamics at the level of the
network’s nodes’ in-going edges. Each edge is now associated with its own
point process. This makes a yet unexplored bridge between Dirichlet processes
and Survival analysis.

Instead of considering a single network on which data spreads, we assume there
is any number of such subnetworks, each associated with a cluster. Instead of
associating one point process to one cluster as in Chapter .14.0..4, we
associate a collection of point processes to one cluster–one per edge in the
network. We write $\lambda(t_{i}^{c}|t_{j}^{c},\alpha_{j,i}^{(k)})$ the
intensity associated with the edge between $i$ and $j$ given their infection
times by $c$ are separated of a time $\Delta t=t_{j}-t_{i}$ cluster $k$.

An infection event from cascade $c$ is now assumed to have a given probability
of being trigger by any of the $k$ existing clusters (or subnetworks) on which
information spread. By substituting the counts in the Dirichlet process with
the total intensity on node $i$ due to all its neighbours $\mathcal{H}_{i,c}$
that got infected earlier by $c$ in subnetwork $k$, noted
$\Lambda_{i}^{c,(k)}=\sum_{j\in\mathcal{H}_{i,c}}\lambda(t_{i}^{c}|t_{j}^{c},\alpha_{j,i}^{(k)})$.
Let $\lambda_{0}$ be the probability that the observation did not get
triggered by any existing subnetwork –the concentration parameter.

$P(s_{n}=k|\mathcal{H}_{i,c})=\begin{cases}\frac{\Lambda_{i}^{c,(k)}}{\lambda_{0}^{(K+1)}+\Lambda_{i}^{c,(k)}}\text{
if k = 1, ..., K}\\\
\frac{\lambda_{0}^{(K+1)}}{\lambda_{0}^{(K+1)}+\Lambda_{i}^{c,(k)}}\text{ if k
= K+1}\end{cases}$ (79)

Table 15: Numerical results of Dirichlet-Survival process, TopicCascade,
Dirichlet-Hawkes process and NetRate models. The AUC, F1 score and MAE are
computed considering every top cluster’s edges at once so there is no error to
report.

Dir-Surv TC DHP NetRate ER NMI 0.787 0.711 0.638 - ARI 0.631 0.488 0.411 - 2-6
AUC 0.849 0.800 - 0.659 F1 0.263 0.176 - 0.005 MAE 0.229 0.278 - 0.481

######  Some preliminary experimental results

Similarly to what we did in Chapter .14.0..4, we coupled the Dirichlet-
Survival prior to a Dirichlet-Multinomial language model. In this section, we
sketch some experimental results as a way to support the relevance of further
studies using Dirichlet-Point processes, especially for modelling interactions
in information spread.

We run the Dirichlet-Survival process on synthetic data first. We generate an
Erdös-Renye network (ER) [31], on which we simulate information spread using a
Dirichlet-Survival process. Using 500 nodes, we create 5 subnetworks of size
of approximately 250 nodes and 700 edges, one per different topic.

We present the results obtained using Dirichlet-Survival processes in Table
15. We compare to several models: TC [29], DHP [30] and NetRate [45]. The
NetRate model infers edges without taking clusters or textual content into
account. The DHP considers the dynamics and the textual content, but not the
network’s structure. The TC model first infers textual clusters and then
infers one subnetwork for each of them. Overall, we see that accounting for
all three data types (content, time, and structure) increases the model’s
performance.

We finally illustrate a possible use case on real-world data in Fig. 56. We
used data from the Memetracker dataset [77].

Figure 56: Dirichlet-survival process applied to real data. (Top row) The most
frequent words within the inferred cluster. (Bottom row) The inferred
subnetworks associated with each cluster. Mass media nodes are represented in
red, and blogs nodes are represented in blue. The most influenced node is the
one having the highest total in-going transmission rate. The nodes’ positions
are identical for every network to ease comparison.

#####  .22.3.b Perspectives on interaction modelling

We briefly showed a possible use case for Dirichlet-Point processes in
information spread modelling by defining the Dirichlet-Survival process. In
this case, interactions are not modelled since cascades spread independently
from each other.

However, it paves the way for defining even more complex Dirichlet-Point
processes that consider time, content and structure of information spread in
interaction modelling. In our case, substituting the non-interacting cascades
model with a more elaborated underlying network inference model might allow us
to uncover interaction mechanisms. In particular, recent years have seen some
works tackling the network inference problem by assuming a Hawkes process on
each edge of a network. Considering a multivariate Hawkes process instead,
similarly to what we did Section .19 would also be an interesting lead.

It should be noted, however, that the Dirichlet-Survival process introduced in
this chapter may not be the best approach to model such phenomena. In
particular, comparing its performances to [56, 10] is needed to get solid
results on this specific application. However, the mere fact that we can
almost instantly elaborate new models to tackle new problems argues in favour
of the use of Dirichlet-Point processes in a broader context.

###  .23 Final words

The work presented in this manuscript is the outcome of three years of
questioning, exploring, and discovering various aspects of interactions at
stake in information spread. As it seems to be the norm in research, this
initial question served as a guiding thread. A thread that sewed this
manuscript through large and disconnected areas of the machine learning
canvas: stochastic block models, dynamic networks inference, Dirichlet
processes, temporal point processes. Here, our developments over those pieces
are used to answer our problematic. However, we essentially focused on
improving the backbone of these areas, whose specific application to
interaction modelling yields interesting insights. From a broader perspective,
our work is also intended as a contribution to machine learning in general;
Formulating alternative Dirichlet Processes, granting flexibility and dynamism
to block models, merging Dirichlet and Point processes, have implications that
range beyond solely interactions modelling. Our efforts resulted in
explainable, scalable and flexible methods that can tackle a wide range of
problems. It is our sincere hope that the advances presented in this
manuscript will be of greater help for researchers of all horizons, either to
improve over them or to use them as tools for answering concrete real-world
questions.

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##  Chapter Appendix – Stochastic Block Models

##  Appendix A SIMSBM - Additional experimental results

Table 16: Replication results on two datasets used in [42] and [102],
referenced in the main text. The standard error on the last digits over all
100 runs is indicated in standard notation – $0.123(12)\Leftrightarrow
0.123\pm 0.012$. Overall, we retrieve the same results as those presented in
[42] and [102]. The models presented in this chapter are underlined.

F1 P@1 AUCROC AUCPR RAP NCE Imdb User, Movie SIMSBM(1,1) $0.3995(2)$
$0.3558(3)$ $0.7665(1)$ $0.3406(3)$ $0.5805(2)$ $0.1593(1)$ TF $0.2570$
$0.2348$ $0.5031$ $0.1541$ $0.4627$ $0.2573$ KNN $0.2668$ $0.2002$ $0.5558$
$0.1735$ $0.3308$ $0.4834$ NB $0.2585$ $0.2382$ $0.5377$ $0.1660$ $0.4664$
$0.2536$ BL $0.2570$ $0.2349$ $0.5000$ $0.1525$ $0.4647$ $0.2557$ MrBanks Ply,
Full sit SIMSBM(1,1) $0.7126(2)$ $0.6688(4)$ $0.7126(3)$ $0.7180(4)$
$0.8344(2)$ $0.1656(2)$ TF $0.6795$ $0.6037$ $0.5176$ $0.5363$ $0.8019$
$0.1981$ KNN $0.6940$ $0.6433$ $0.6668$ $0.6430$ $0.8217$ $0.1783$ NB $0.6795$
$0.6037$ $0.5907$ $0.5822$ $0.8019$ $0.1981$ BL $0.6795$ $0.6037$ $0.5000$
$0.5215$ $0.8019$ $0.1981$

##  Appendix B IMMSBM - Datasets

####  .2.1 Medical records

The Pubmed dataset collect has been inspired by [146]. Every article on PubMed
is manually annotated by experts with a list of keywords describing the main
topics of the publication. We downloaded a list of 322 symptoms and 4,442
diseases provided by [146]. Then, we used the PubMed API to query each one of
the symptom/disease keywords aforementioned. For each result, we got a list of
every publication in which the keyword is among the main topics. Then, we
build the dataset by considering every publication in which there is at least
one symptom and one disease. Finally, we create the triplets (symptom1,
symptom2, disease) by looking at all the pairs of symptoms in an article and
linking each one of them to all the diseases observed in the same article. In
the end, we are left with a total of 52,833,690 observed triplets, distributed
over 15,809,271 PubMed publications.

####  .2.2 Spotify

The Spotify dataset has been collected using the Spotify API. We randomly
sampled 2,000 playlists using the keywords "english" and "rock", which
corresponds to a total of 135,100 songs. Then, for each playlist, we used a
running window of 4 songs to build the dataset. The artist of the song
immediately after the running window is the output we aim at predicting, x,
and the artists of the 4 songs within the running window are the interacting
inputs. Once again, we consider all the possible pairs of artists in the
running window and associate them to the output artist x. Note that we only
considered the artists that appear more than 50 times in the whole dataset,
for the sake of statistical relevance. The resulting dataset is consists of
1,236,965 triplets for 2,028 artists.

####  .2.3 Twitter

We gathered the Twitter dataset used in [59]. It consists in a collection of
all the tweets containing URLs posted during the month of October 2010. A
first operation consisted in cleaning the dataset of URLs that are considered
as aggressive advertising. To do so, we considered only the URLs whose
retweets built a chain of length at least 50; this choice comes from the idea
that commercial spams are not likely to be retweeted by actual users and
therefore do not create chains. Secondly, we considered only the users who
have not tweeted a given URL more than 5 times, this behavior being an
activity typical of spamming bots. Doing so, we are left with tweets that are
mostly coming from the non-commercial activity of human users. Then, we follow
a dataset building process similar to [87]. For each user, we slice her feed +
tweets temporal sequence in intervals separated by the tweets of the user.
Every time a user tweets something, the interval ends. An interval therefore
consists of the tweet of the user and all the tweets she has been exposed to
right before tweeting. Following the suggestion of [87], we only consider the
3 last tweets the user has been exposed to before retweeting one of them. Each
one of these intervals form an entry of our message+answer dataset (3 last
entries in the feed + next tweets). In the end, we are left with 284,837
intervals containing a total of 2,110 different tweeted URLs. Our dataset then
consists of 1,181,543 triplets.

####  .2.4 Reddit

Finally, we downloaded the May 2019 Reddit dataset from the data repository
pushhift.io, which stores regular saves of the comments posted on the website.
We chose to consider only the comments made in the subreddit r/news. Reddit’s
comments system work as a directed tree network, where each answer to a given
comment initiates a new branch. We considered pairs of messages such as one
(the answer) has for direct parent the other one (the comment). We then
extracted all the named entities in both of them using the Spacy Python
library. For each pair of named entities in the comment, we associated every
name entity in the answer. We consider here only the named entities that
appear at least 200 times in the subreddit, for the same reason as for the
Spotify dataset. The final dataset results in 35,364,725 triplets for a total
of 1,656 named entities.

##  Appendix C IMMSBM - Upper limit to predictions

We derive an analytical expression for the upper-limit to our model for a
given dataset. Explicitly, we analytically maximize the likelihood according
to each entry of the dataset.

We enforce the constraint that the sum over the output space of probabilities
given any observations made has to sum to 1. To do so, we use a Lagrange
multiplier $\lambda_{obs}$ for every different observation (in the case of our
model: for every different triplet). The log-likelihood then takes the
following form:

$\begin{split}&\ell=\sum_{(obs,x)}\ln
P_{obs}(x)-\sum_{obs}\lambda_{obs}(\sum_{x}P_{(obs,x)}-1)\\\ \Leftrightarrow\
&\frac{\partial\ell}{\partial
P_{obs_{i}}(x_{i})}=\sum_{\partial(obs_{i},x_{j})}\frac{1}{P_{obs_{i}}(x_{j})}-\lambda_{obs_{i}}=0\\\
\Leftrightarrow\
&P_{obs_{i}}(x_{j})=\frac{1}{\lambda_{obs_{i}}}\sum_{\partial(obs_{i},x_{j})}1\end{split}$
(80)

Where $obs_{i}$ correspond to any given couple of inputs (i,j) in the model
presented in the main paper. We use the following notation:
${\partial(obs_{i},x_{j})=\\{obs|(obs_{i},x_{j})\in R^{\circ}\\}}$, with
$R^{\circ}$ the dataset entries. Therefore, we can define
$\sum_{\partial(obs_{i},x_{j})}1\equiv N_{obs_{i},x_{j}}$ the number of times
$obs_{i}$ appears jointly with $x_{j}$ in the dataset. We are now looking for
the $\lambda_{obs_{i}}$ maximizing the likelihood:

$\begin{split}&\frac{\partial\ell}{\partial\lambda_{obs_{i}}}=\sum_{x}P_{obs_{i}}(x)-1=0\\\
\Leftrightarrow\ &\sum_{x}\frac{N_{obs_{i},x}}{\lambda_{obs_{i}}}=1\\\
\Leftrightarrow\ &\lambda_{obs_{i}}=\sum_{x}N_{obs_{i},x}\\\ \end{split}$ (81)

Finally, plugging Eqs.80 and 81 together, we obtain:

$P_{obs_{i}}(x_{j})=\frac{N_{obs_{i},x_{j}}}{\sum_{x}N_{obs_{i},x}}$ (82)

This equation gives the probability maximizing the likelihood for any entry of
the dataset. In the main model, it translates to
$P_{(i,j)}(x)=N_{(i,j),x}/\sum_{x}^{\prime}N_{(i,j),x^{\prime}}$ with
$N_{(i,j),x}$ the number of times output x has been witnessed after a pair of
inputs (i,j). Note that this is simply the frequency of an output given a pair
of input entities.

Keep in mind that the result we just derived gives perfect predictions only
for a particular dataset, and therefore and has no global predictive value.
This tool is useful when it comes to assess the performance of other models’
results, but is unusable in prediction tasks.

##  Appendix D SDSBM - Explicit derivation of the E-step

####  .4.1 Short derivation

This demonstration can be found in [42, 120, 98]. We recall the log-likelihood
as defined in the main paper:

$\displaystyle\log P(\theta,p|R^{\circ})$ $\displaystyle\propto\log
P(R^{\circ}|\theta,p)\prod_{t}\prod_{i}P(\theta_{i}^{(t)})\prod_{k}P(p_{k}^{(t)})$
$\displaystyle=$ $\displaystyle\sum_{(i,o,t)\in R^{\circ}}\log\sum_{k\in
K}\theta_{i,k}^{(t)}p_{k}^{(t)}(o)$ (83) $\displaystyle+\sum_{t}\sum_{i}\log
P(\theta_{i}^{(t)})\sum_{k}\log P(p_{k}^{(t)})$ $\displaystyle\geq$
$\displaystyle\sum_{(i,o,t)\in R^{\circ}}\sum_{k\in
K}\omega_{i,o}^{(t)}(k)\log\frac{\theta_{i,k}^{(t)}p_{k}^{(t)}(o)}{\omega_{i,o}^{(t)}(k)}$
$\displaystyle+\sum_{t}\sum_{i}\log P(\theta_{i}^{(t)})\sum_{k}\log
P(p_{k}^{(t)})$

In Eq..4.1, we introduced a proposal distribution $\omega_{i,o}^{(t)}(k)$,
that represents the probability of one cluster allocation $k$ given the
observation $(i,o,t)$. The last line followed from Jensen’s inequality
assuming that $\sum_{k}\omega_{i,o}^{(t)}(k)=1$. We notice that Jensen’s
inequality holds as an equality when:

$\omega_{i,o}^{(t)}(k)=\frac{\theta_{i,k}^{(t)}p_{k}^{(t)}(o)}{\sum_{k^{\prime}}\theta_{i,k^{\prime}}^{(t)}p_{k^{\prime}}^{(t)}(o)}$
(84)

which provides us with the expectation formula. The prior terms
$P(\theta_{i}^{(t)})$ and $P(p_{k}^{(t)})$ have no effect on the result as
they cancel in the inequality .4.1.

####  .4.2 Full derivation

The derivation presented in this section follows a well-known general
derivation of the EM algorithm, which can be found in C.M. Bishop’s Pattern
Recognition and Machine Learning-p.450 for instance.

We recall that one entry of the dataset $R^{\circ}$ takes the form of a tuple
$(i,o,t)$, where $i$ is the input item and $o$ an associated label at time
$t$. $k\in K$ denotes the latent variable accounting for cluster allocation
among $K$ possible values. The total log-likelihood is the sum of each
observation’s log-likelihood. Without loss of generality, we focus on a single
observation $(i,o,t)$. The expression of the log-posterior distribution for
one observation reads:

$\displaystyle\log P(\theta^{(t)},p^{(t)}|(i,o,t))$ (85)
$\displaystyle\propto\log
P(R^{\circ}|\theta^{(t)},p^{(t)})P(\theta_{i}^{(t)})\prod_{k}P(p_{k}^{(t)})$
$\displaystyle=\log P^{(t)}(i,o|\theta^{(t)},p^{(t)})+\log
P(\theta_{i}^{(t)})+\sum_{k}\log P(p_{k}^{(t)})$

For an observation $(i,o,t)\in R^{\circ}$, we assume a distribution
$Q_{i,o}^{(t)}(k)$ on the latent variables associated to it; this distribution
is yet to be defined. Because $k$ takes values among $K$ possible ones, we
have $\sum_{k\in K}Q_{i,o}^{(t)}(k)=1$. Given this normalization condition, we
can decompose each summed term in Eq.85 for any distribution
$Q_{i,o}^{(t)}(k)$ as:

$\displaystyle\log P^{(t)}(i,o|\theta^{(t)},p^{(t)})$
$\displaystyle=\underbrace{\log P^{(t)}(i,o,k|\theta^{(t)},p^{(t)})-\log
P^{(t)}(k|i,o,\theta^{(t)},p^{(t)})}_{\text{Does not depend on $k$}}$
$\displaystyle=\sum_{k\in K}Q_{i,o}^{(t)}(k)\log
P^{(t)}(i,o,k|\theta^{(t)},p^{(t)})$ $\displaystyle\,\,\,\,-\sum_{k\in
K}Q_{i,o}^{(t)}(k)\log P^{(t)}(k|i,o,\theta^{(t)},p^{(t)})$
$\displaystyle=\sum_{k\in
K}Q_{i,o}^{(t)}(k)\log\frac{P^{(t)}(i,o,k|\theta^{(t)},p^{(t)})}{Q_{i,o}^{(t)}(k)}$
$\displaystyle\,\,\,\,-\sum_{k\in
K}Q_{i,o}^{(t)}(k)\log\frac{P^{(t)}(k|i,o,\theta,p^{(t)})}{Q_{i,o}^{(t)}(k)}$
(86)

We note that the term in the last line of Eq..4.2, $-\sum_{k\in
K}Q_{i,o}^{(t)}(k)\log\frac{P^{(t)}(k|i,o,\theta^{(t)},p^{(t)})}{Q_{i,o}^{(t)}(k)}$,
is the Kullback-Leibler (KL) divergence between $P^{(t)}$ and $Q_{i,o}^{(t)}$,
noted $KL(P^{(t)}||Q_{i,o}^{(t)})$. The KL divergence obeys
$KL(P^{(t)}||Q_{i,o}^{(t)})\geq 0$, and is null iif $P^{(t)}$ equals
$Q_{i,o}^{(t)}$. Therefore, the term in the before-last line of Eq..4.2,
$\sum_{k\in
K}Q_{i,o}^{(t)}(k)\log\frac{P^{(t)}(i,o,k|\theta^{(t)},p^{(t)})}{Q_{i,o}^{(t)}(k)}$,
is interpreted as a lower bound on the log-likelihood $\log
P^{(t)}(i,o|\theta^{(t)},p^{(t)})$.

The aim of the E-step is to find the expression of $Q_{i,o}^{(t)}(k)$ that
maximizes the lower bound of the log-likelihood with respect to the latent
variables $k$. Given that the log-likelihood does not depend on
$Q_{i,o}^{(t)}(k)$ and $KL(P^{(t)}||Q_{i,o}^{(t)})\geq 0$, the lower-bound is
maximized when $KL(P^{(t)}||Q_{i,o}^{(t)})=0$, which occurs when
$Q_{i,o}^{(t)}(k)=P^{(t)}(k|i,o,\theta^{(t)},p^{(t)})$. In this case, the
lower-bound on the log-likelihood equals the likelihood itself and thus
reaches a global maximum with respect to the latent variables $k$ for fixed
parameters $\theta^{(t)}$ and $p^{(t)}$.

Given the definition of our simple model, the derivation of
$P(k|i,o,\theta^{(t)},p^{(t)})$ is straightforward. The probability of one
combination of clusters $k$ among $K$ possible combinations given an input
features vector and an output $o$ is proportional to
$p_{k}^{(t)}(o)\theta_{i,k}$. Therefore:

$\displaystyle
P^{(t)}(k|i,o,\theta^{(t)},p^{(t)})=\frac{p_{k}^{(t)}(o)\theta_{i,k}^{(t)}}{\sum_{k^{\prime}\in
K}p_{k^{\prime}}^{(t)}(o)\theta_{i,k^{\prime}}^{(t)}}$ (87)

which is the expression of $\omega_{i,o}^{(t)}(k)$ in the main article.

##  Appendix E SDSBM - Explicit derivation of the M-step for p

$\displaystyle\frac{\partial\left(\log
P(\theta,p|R^{\circ})-\sum_{k^{\prime},t^{\prime}}\psi_{k^{\prime}}^{(t^{\prime})}(\sum_{o}p_{k^{\prime}}^{(t^{\prime})}(o)-1)\right)}{\partial
p_{k}^{(t)}(o)}=0$ $\displaystyle\Leftrightarrow\sum_{(i,t)\in\partial
o}\frac{\omega_{i,o}^{(t)}(k)}{p_{k}^{(t)}(o)}+\frac{\beta\langle
p_{k}^{(t)}(o)\rangle}{p_{k}^{(t)}(o)}-\psi_{k}^{(t)}=0$
$\displaystyle\Leftrightarrow\sum_{(i,t)\in\partial
o}\omega_{i,o}^{(t)}(k)+\beta\langle
p_{k}^{(t)}(o)\rangle=\psi_{k}^{(t)}p_{k}^{(t)}(o)$
$\displaystyle\Leftrightarrow\sum_{(i,t)\in\partial
o}\sum_{o}\omega_{i,o}^{(t)}(k)+\beta\underbrace{\sum_{o}\langle
p_{k}^{(t)}(o)\rangle}_{=1\text{}}=\psi_{k}^{(t)}$
$\displaystyle\Leftrightarrow\frac{\sum_{(i,t)\in\partial
o}\omega_{i,o}^{(t)}(k)+\beta\langle p_{k}^{(t)}(o)\rangle}{\sum_{(i,o,t)\in
R^{\circ}}\omega_{i,o}^{(t)}(k)+\beta}=p_{k}^{(t)}(o)$ (88)

##  Appendix F SDSBM - Using the prior in related works

Throughout this section, we highlight the changes brought by our method to the
EM equations derived in the mentioned papers. In summary, we see that out
method allows to make these works dynamic with minimal changes of the original
models.

####  .6.1 Bi-MMSBM

In [42], the authors apply a MMSBM to a labeled bipartite network. The nodes
on each side of the bipartite network are associated to their own membership
matrix; membership of nodes $i\in I$ over $K$ clusters is encoded into
$\theta\in\mathbb{R}^{I\times K}$, and membership of nodes $j\in J$ over $L$
clusters is encoded into $\eta\in\mathbb{R}^{J\times L}$. The block-
interaction matrix for the label $o\in O$ is noted $p(o)\in\mathbb{R}^{K\times
L}$.

Assuming a temporal version, items $i$ and $j$ to be linked by a label $o$ at
time $t$ reads:

$P^{(t)}(o|i,j)=\sum_{k\in K}\sum_{l\in
L}\theta_{i,k}^{(t)}\eta_{j,l}^{(t)}p_{k,l}^{(t)}(o)$ (89)

Given the same set of observation s $R^{\circ}$ as in the main article, the
posterior distribution follows:

$\displaystyle P(\theta,\eta,p|R^{\circ})$
$\displaystyle=P(R^{\circ}|\theta,\eta,p)$ (90)
$\displaystyle\times\prod_{t}\left(\prod_{i}P(\theta_{i}^{(t)})\prod_{j}P(\eta_{j}^{(t)})\prod_{k,l}P(p_{k,l}^{(t)})\right)$

such that:

$\displaystyle P(R^{\circ}|\theta,\eta,p)=\prod_{(i,j,t,o)\in
R^{\circ}}\sum_{k\in K}\sum_{l\in
L}\theta_{i,k}^{(t)}\eta_{j,l}^{(t)}p_{k,l}^{(t)}(o)$ (91) $\displaystyle
P(\theta_{i}^{(t)})\propto\prod_{k}{\theta_{i,k}^{(t)}}^{\beta\langle\theta_{i,k}^{(t)}\rangle}$
(92) $\displaystyle
P(\eta_{j}^{(t)})\propto\prod_{l}{\eta_{j,l}^{(t)}}^{\beta\langle\eta_{j,l}^{(t)}\rangle}$
(93) $\displaystyle
P(p_{k,l}^{(t)})\propto\prod_{o}{p_{k,l}^{(t)}}(o)^{\beta\langle
p_{k,l}^{(t)}(o)\rangle}$ (94)

where $\langle x^{(t)}\rangle=\frac{\sum_{t^{\prime}\neq
t}\kappa(t,t^{\prime})x^{(t^{\prime})}}{\sum_{t^{\prime}\neq
t}\kappa(t,t^{\prime})}$. The expectation step is not influenced by the priors
choice and is the same as in [42] for each temporal slice. The new
maximization steps are:

$\displaystyle\theta_{i,k}^{(t)}=\frac{\sum_{l}\sum_{(o,j)\in\partial(i,t)}\omega_{i,j,o}^{(t)}(k,l){\color[rgb]{1,0,0}+\beta\langle\theta_{i,k}^{(t)}\rangle}}{N_{i,t}{\color[rgb]{1,0,0}+\beta}}$
$\displaystyle\eta_{j,l}^{(t)}=\frac{\sum_{k}\sum_{(o,i)\in\partial(j,t)}\omega_{i,j,o}^{(t)}(k,l){\color[rgb]{1,0,0}+\beta\langle\eta_{j,l}^{(t)}\rangle}}{N_{j,t}{\color[rgb]{1,0,0}+\beta}}$
$\displaystyle p_{k,l}^{(t)}(o)=\frac{\sum_{(i,j,t)\in\partial
o}\omega_{i,j,o}^{(t)}(k,l){\color[rgb]{1,0,0}+\beta\langle
p_{k,l}^{(t)}(o)\rangle}}{\sum_{(i,j,o,t)\in
R^{\circ}}\omega_{i,j,o}^{(t)}(k,l){\color[rgb]{1,0,0}+\beta}}$

Here again, $\beta$ is set fixed for demonstration purposes, but can be tuned
at will by the user. This allows to choose the extent to which dynamics shall
be smoothed, or ignored.

####  .6.2 T-MBM

The T-MBM is essentially the same model as [42] but with one type of entry
that can appear twice in one observation. Both entries share the same
membership matrix $\theta$. The probability of a label of type $o$ given
entries $h$; $i$ and $j$ at time $t$ is now written:

$P(o|h,i,j,t)=\sum_{k\in K}\sum_{l\in L}\sum_{m\in
M}\theta_{h,k}^{(t)}\theta_{i,l}^{(t)}\eta_{j,m}^{(t)}p_{k,l,m}^{(t)}(o)$ (95)

The posterior distribution follows the same expression as in Eq.90. The
expectation step is left unchanged by the choice of the priors, and the new
maximization equations are given below:

$\displaystyle\theta_{h,k}^{(t)}=\frac{\sum_{l,m}\sum_{(o,i,j)\in\partial(h,t)}\omega_{h,i,j,o}^{(t)}(k,l,m){\color[rgb]{1,0,0}+\beta\langle\theta_{h,k}^{(t)}\rangle}}{N_{h,t}{\color[rgb]{1,0,0}+\beta}}$
$\displaystyle\eta_{i,l}^{(t)}=\frac{\sum_{k,m}\sum_{(o,h,j)\in\partial(i,t)}\omega_{h,i,j,o}^{(t)}(k,l,m){\color[rgb]{1,0,0}+\beta\langle\eta_{i,l}^{(t)}\rangle}}{N_{i,t}{\color[rgb]{1,0,0}+\beta}}$
$\displaystyle p_{k,l,m}^{(t)}(o)=\frac{\sum_{(h,i,j,t)\in\partial
o}\omega_{h,i,j,o}^{(t)}(k,l,m){\color[rgb]{1,0,0}+\beta\langle
p_{k,l,m}^{(t)}(o)\rangle}}{\sum_{(h,i,j,o,t)\in
R^{\circ}}\omega_{h,i,j,o}^{(t)}(k,l,m){\color[rgb]{1,0,0}+\beta}}$

##  Appendix G SDSBM - Inferring two dynamic matrices of parameters

In the main article, $p$ is provided to the model and only $\theta$ has to be
inferred. Doing so, we can confront inferred membership vectors to the ground
truth while avoiding label-switching issues [82, 76]. When $p$ is also to be
inferred, finding a correspondence between the inferred clusters and the
ground-truth is not a trivial task, and cannot be performed in unbiased ways.
However, the good results yielded by the model, presented in Fig.57, when also
inferring $p$ hints that the membership vectors are correctly inferred.

Figure 57: Experimental results when inferring both $\theta$ and $p$ jointly.
The AUC-ROC is as good as when $p$ is provided to the model.

##  Appendix H SDSBM - Clusters composition for the Epigraphy experiment

  * •

Cluster 0

    * –

Roma (98.0%)

  * •

Cluster 1

    * –

Latium et Campania (32.0%)

    * –

Venetia et Histria (14.0%)

    * –

Samnium (11.0%)

    * –

Umbria (10.0%)

    * –

Apulia et Calabria (8.0%)

  * •

Cluster 2

    * –

Pannonia superior (15.0%)

    * –

Dalmatia (11.0%)

    * –

Noricum (10.0%)

    * –

Hispania citerior (7.0%)

    * –

Gallia Narbonensis (6.0%)

  * •

Cluster 3

    * –

Dacia (24.0%)

    * –

Pannonia inferior (17.0%)

    * –

Moesia inferior (15.0%)

    * –

Syria (6.0%)

    * –

Numidia (6.0%)

    * –

Pannonia superior (5.0%)

  * •

Cluster 4

    * –

Germania superior (24.0%)

    * –

Mauretania Caesariensis (11.0%)

    * –

Asia (11.0%)

    * –

Etruria (11.0%)

    * –

Galatia (9.0%)

##  Chapter Appendix – Temporal diffusion networks

##  Appendix I Implementation of Clash of the Contagions

In this appendix, we provide technical details on the way the Clash of
Contagions baseline is implemented. Following the directions given in the
reference article [87], we implemented a Stochastic Gradient Descent (SGD)
method for parameters inference. Given the small number of entities considered
in the experiments, each iteration of the SGD is computed using the full
dataset instead of slicing it into mini-batches.

####  .9.1 Setup

For each corpus, we run the SGD algorithm 100 times, from which we save the
parameters maximizing the likelihood the most. At the beginning of each run,
parameters M and $\Delta$ are randomly initialized. The stopping condition
makes the algorithm ends when the relative variation of the likelihood
according to the last iteration is been lesser than $10^{-6}$ for more than 30
times in a row; those numbers have been chosen empirically to maximize the
performances of the algorithm. The hyper-parameters have been set to: T=5
(number of clusters) and K=20 (number of considered time steps).

####  .9.2 Update rule

In each iteration, we update the parameters in the direction of the gradient
descent (noted G). However, a major problem when dealing with SGD is to choose
the line step length $\eta$ (the amplitude of the variation of the parameters
in the direction of the gradient G). After each iteration, we compare several
update rules, and we select the one maximizing the likelihood. Those rules are
as follows:

  * •

AdaGrad: $\eta^{AG}\times G$

  * •

AdaDelta: $\eta^{AD}\times G$

  * •

Line search in the direction of the gradient: $\eta^{LS}\times G$

  * •

Line search in the direction of AdaDelta: $\eta^{LS}\times\eta^{AD}\times G$

The line search snippets consider 50 values of $\eta^{LS}$ logarithmically
distributed in the interval $[10^{-6};10^{3}]$.

####  .9.3 Constraints on the parameters

The membership vectors entries $M_{i,t}$ (membership of i to cluster t) must
be positive and sum to 1 over all the clusters ($\sum_{t}M_{i,t}=1$). In order
to enforce this constraint, we consider the following variable change:
$M_{i,t}\rightarrow\frac{\phi_{i,t}^{2}}{\sum_{t^{\prime}}\phi_{i,t^{\prime}}^{2}}$.
This transformation guarantees the membership vector properties with no need
for penalty methods in the implementation.

Besides, as stated in [87], it can happen that a probability is larger then 1
or lesser than 0. In the absence of complementary details in the main paper,
we implemented our own method to force the probabilities to stay within
reasonable bounds. Here it is impossible to make a simple variable change to
enforce this constraint, since the probability results of a non-linear
combination of the model’s parameters. We added to the likelihood an
exponential penalty term. Let P denote a quantity we want to constrain between
0 and 1. The penalty term equals $e^{-\lambda P}+e^{\lambda(P-1)}$. $\lambda$
here is a parameter that tunes the intensity of the penalty, and is
empirically set to $\lambda=75$. This penalty function has the form of a well
with very steep walls in x=0 and x=1. In this way, it seldom happens that
probabilities are larger than 1 or lesser than 0, as said in the main article.
When such cases happen, we simply set it back to the closest bound for methods
comparisons.

##  Chapter Appendix – Dirichlet-Survival Process

##  Appendix J Deriving NetRate

The input of NetRate is a collection of observed cascades
$C=\\{\vec{c}\\}_{\vec{c}=\vec{c_{1}},\vec{c_{2}},...}$. Each cascade is a
collection of events with timestamps
$\vec{c}=\\{(u_{i}^{c},t_{i}^{c})\\}_{i}$, where $u_{i}^{c}$ is the node on
which the $i^{th}$ event occurred and $t_{i}^{c}$ the time at which it
happened in cascade $c$. Using survival analysis, [45] denotes the likelihood
of an infection of node $u_{i}^{c}$ at time $t_{i}^{c}$ in cascade $c$ by any
other node $u_{j}^{c}$ previously infected at time $t_{j}^{c}$ in the same
cascade as $f(t_{i}^{c}|t_{j}^{c},\alpha_{u_{j}^{c},u_{i}^{c}})$, where
$\alpha_{u_{j}^{c},u_{i}^{c}}$ is an entry of the objective network’s
adjacency matrix. In survival analysis’ framework,
$f(t_{i}^{c}|t_{j}^{c},\alpha_{u_{j}^{c},u_{i}^{c}})$ is linked to the
instantaneous infection rate (or hazard rate) of $u_{i}^{c}$ at time
$t_{i}^{c}$ by $u_{j}^{c}$ previously infected at time $t_{j}^{c}$, noted
$\lambda(t_{i}^{c}|t_{j}^{c},\alpha_{u_{j}^{c},u_{i}^{c}})$, and the
probability of non-infection of $u_{i}^{c}$ up to time $t_{i}^{c}$ by
$u_{j}^{c}$ previously infected at time $t_{j}^{c}$, noted
$S(t_{i}^{c}|t_{j}^{c},\alpha_{u_{j}^{c},u_{i}^{c}})$, by the following
relation:

$f(t_{i}^{c}|t_{j}^{c},\alpha_{u_{j}^{c},u_{i}^{c}})=\lambda(t_{i}^{c}|t_{j}^{c},\alpha_{u_{j}^{c},u_{i}^{c}})S(t_{i}^{c}|t_{j}^{c},\alpha_{u_{j}^{c},u_{i}^{c}})$
(96)

Within a cascade, the likelihood that a node get infected by only one
neighbour $u_{j}^{c}$ previously infected at time $t_{j}^{c}$ can be written
as the likelihood of infection at time $t_{i}^{c}$ by $u_{j}^{c}$ previously
infected at time $t_{j}^{c}$ times its probability of survival to every
previously infected node:

$f(t_{i}^{c}|t_{j}^{c},\alpha_{u_{j}^{c},u_{i}^{c}})\prod_{k\neq
j,t_{k}^{c}<t_{i}^{c}}S(t_{i}^{c}|t_{k}^{c},\alpha_{u_{k}^{c},u_{i}^{c}})$
(97)

The likelihood of an infection by any neighbour then becomes the sum of those
candidate disjoint events:

$\begin{split}&\sum_{j,t_{j}^{c}<t_{i}^{c}}f(t_{i}^{c}|t_{j}^{c},\alpha_{u_{j}^{c},u_{i}^{c}})\prod_{k\neq
j,t_{k}^{c}<t_{i}^{c}}S(t_{i}^{c}|t_{k}^{c},\alpha_{u_{k}^{c},u_{i}^{c}})\\\
\stackrel{{\scriptstyle\text{Eq.\ref{eq-
survival}}}}{{=}}&\sum_{j,t_{j}^{c}<t_{i}^{c}}\lambda(t_{i}^{c}|t_{j}^{c},\alpha_{u_{j}^{c},u_{i}^{c}})\prod_{t_{k}^{c}<t_{i}^{c}}S(t_{i}^{c}|t_{k}^{c},\alpha_{u_{k}^{c},u_{i}^{c}})\end{split}$
(98)

The likelihood of a cascade then becomes the product of the likelihood of
every event it contains, and the total likelihood $\mathcal{L}(C|A)$ of every
cascade the product over every cascade. Let $A$ be the network’s adjacency
matrix whose entries $\vec{\alpha_{i,j}}$ are directed edges from $i$ to $j$.
Then:

$\mathcal{L}(C|A)=\prod_{\vec{c}\in
C}\prod_{t_{i}^{c}\in\vec{c}}\sum_{j,t_{j}^{c}<t_{i}^{c}}\lambda(t_{i}^{c}|t_{j}^{c},\alpha_{u_{j}^{c},u_{i}^{c}})\prod_{t_{k}^{c}<t_{i}^{c}}S(t_{i}^{c}|t_{k}^{c},\alpha_{u_{k}^{c},u_{i}^{c}})$
(99)

###### List of Figures

  1. 1 Intro - Information spread and interaction
  2. 2 Intro - Independent spread
  3. 3 Intro - Illustration of the definitions
  4. 4 Intro - Definition of an interaction
  5. 5 SBM - Illustration of the stochastic block modelling
  6. 6 SBM - Illustration of the principle
  7. 7 SIMSBM - Illustration of the SIMSBM
  8. 8 IMMSBM - Graphical representation of the model
  9. 9 IMMSBM - Illustration of dataset generation
  10. 10 IMMSBM - Choosing the number of clusters
  11. 11 IMMSBM - Histogram of relative impact of interactions
  12. 12 IMMSBM - Importance of interactions
  13. 13 SDSBM - Users’ attachment to groups can vary over time
  14. 14 SDSBM - Illustration of the base model SIMSBM(1)
  15. 15 SDSBM - Illustration of dynamic prior probability
  16. 16 SDSBM - Prior probability’s variance according to time
  17. 17 SDSBM - Results on synthetic data
  18. 18 SDSBM - Status geographic distribution (Europe, 100BC - 500AD)
  19. 19 InterRate - Interaction profiles between entities pairs
  20. 20 InterRate - Illustration of the interacting process
  21. 21 InterRate - Noise in the data
  22. 22 InterRate - Real-world interaction profiles
  23. 23 SotA - Time slicing bias
  24. 24 Hawkes - A realization of a Hawkes process
  25. 25 PDP - Effect of $r$ on the PDP
  26. 26 PDP - Numerical validation of theorems
  27. 27 PDP - Experimental results on synthetic data
  28. 28 PDP - Application to spatial clustering on geolocated Latin inscriptions
  29. 29 PDHP - Illustration of the PDHP
  30. 30 PDHP - Sequential Monte Carlo algorithm
  31. 31 PDHP - Effect of $r$ on cluster selection probabilities
  32. 32 PDHP - Overlaps
  33. 33 PDHP - PDHP yields good NMI values on synthetic data
  34. 34 PDHP - Experimental comparison to DHP
  35. 35 PDHP - Results when content and dynamics are decorrelated
  36. 36 PDHP - Varying $r$ to choose between textual or temporal clustering
  37. 37 PDHP - Varying $r$ allows to better capture the dynamics at stake
  38. 38 PDHP - Results for the cluster about Sri Lanka 2019 bombings
  39. 39 PDHP - Favouring text or time based clusters on real world datasets
  40. 40 PDHP - Textual clusters are more informative for low values of $r$
  41. 41 PDHP - PDHP allows for modelling bursty events
  42. 42 PDHP - Limit case of encouraging bursty events clustering
  43. 43 MPDHP - Illustration of the MPDHP
  44. 44 MPDHP - Choosing the concentration parameter
  45. 45 MPDHP - Numerical results on synthetic datasets
  46. 46 MPDHP - MPDHP handles scarce textual or temporal information
  47. 47 MPDHP - MPDHP handles many coexisting clusters
  48. 48 MPDHP - How much data to use MPDHP
  49. 49 MPDHP - How complex should the algorithm be
  50. 50 MPDHP x News - Characteristics of the News dataset
  51. 51 MPDHP x News - Timeline news 2019
  52. 52 MPDHP x News - Distribution of interaction strength
  53. 53 MPDHP x News - Typical output
  54. 54 MPDHP x News - Global clusters temporal interaction graph 2019
  55. 55 MPDHP x News - Monthly clusters temporal interaction graph 2019
  56. 56 DSP - Dirichlet-survival Process on real-world data
  57. 57 SDSBM - Experimental results when inferring both $\theta$ and $p$ jointly

List of figures

###### List of Tables

  1. 1 Intro - Contributions
  2. 2 Intro - Codes and datasets
  3. 3 SIMSBM - Notations
  4. 4 SIMSBM - Datasets considered
  5. 5 SIMSBM - Experimental results on real-world datasets
  6. 6 IMMSBM - Experimental results on real-world datasets
  7. 8 InterRate - Experimental results on synthetic data
  8. 9 InterRate - Experimental results on real-world data
  9. 10 PDP - Numerical results on synthetic datasets
  10. 11 MPDHP x News - Results for each experiment
  11. 12 MPDHP x News - Interaction strength
  12. 13 MPDHP x News - Interaction range
  13. 14 Conclusion - Contributions
  14. 15 DSP - Numerical results of Dirichlet-Survival process on synthetic data
  15. 16 SIMSBM - Experimental results on real-world datasets

List of tables

List of equations

## Acronyms

A.R.Prof.

    Associate Research Professor
AP

    Average Precision
AUCPR

    Area Under the Precision-Recall Curve
AUCROC

    Area Under the Receiving Operator Curve
Bi-MMSBM

    Bipartite Mixed Membership Stochastic Block Model
BL

    Baseline
CP

    Canonical Polyadic Decomposition
CRP

    Chinese Restaurant Process
D.R.

    Director of Research
DHP

    Dirichlet-Hawkes Process
DP

    Dirichlet Process
EM

    Expectation-Maximization Algorithm
IMMSBM

    Interacting Mixed Membership Stochastic Block Model
JS

    Jensen-Shannon divergence
KNN

    K-nearest-neighbours
MAE

    Mean Absolute Error
MMSBM

    Mixed Membership Stochastic Block Model
MPDHP

    Multivariate Powered Dirichlet-Hawkes Process
MSE

    Mean Squared Error
NB

    Naive Bayes
NCE

    Normalized Coverage Error
NMF

    Non-negative Matrix Factorization
NMI

    Normalized Mutual Information
P@

    Precision at
PDHP

    Powered Dirichlet-Hawkes Process
PDP

    Powered Dirichlet Process
Prof.

    Professor
PY

    Pitman-Yor Process
RAP

    Rank Average Precision
RBF

    Radial basis function kernel
RMSE

    Root Mean Squared Error
RSS

    Residual Sum of Squares
SBM

    Stochastic Block Model
SIMSBM

    Serialized Interacting Mixed Membership Stochastic Block Model
SMC

    Sequential Monte-Carlo
SVD

    Singular Value Decomposition
T-MBM

    Tensorial Mixed Membership Stochastic Block Model
TF

    Tensor Factorization
UP

    Uniform Process

## Glossary

cascade

    A series of identical decisions taken by different spreaders about an entity. For instance, a tweet that gets retweeted n times makes a cascade of size n+1.
cluster

    Group formed from a set of objects in such a way that objects in the same group are more similar (in some sense) to each other than to those in other groups.
collaborative filtering

    A method of making automatic predictions (filtering) about the interests of a user by collecting preferences or taste information from many users (collaborating). Its underlying assumption is that if a person A has the same opinion as a person B on an issue, A is more likely to have B’s opinion on a different issue than that of a randomly chosen person.
content

    Mean through which entities carry a semantic meaning: text, pixels, sounds, etc.
decision

    The action of a spreader when facing an entity. The decision can be endogenous (the spreader acts on the entity alone, e.g. by sharing it, liking it, clicking on it, reacting to it, etc.) or exogenous (the spreader creates a new entity as a consequence of the first one, e.g. a denial, an answer to a mail, to a tweet, etc.).
diffusion

    A set of individual decisions. For instance: a retweet cascade, an internet buzz, a Youtube trend, etc.
entity

    Same as “piece of information”. Any object that is susceptible to have an influence on spreaders. It carries a semantic meaning. For instance: a tweet, a meme, a song, a virus, a news article, etc.
independent

    As opposed to interacting, when there is no interaction; the spreading entities do not affect each other. The whole equals the sum of its parts.
infected

    When spreaders take a decision on an entity, they are said to be infected by this entity. Infection denotes the transition between a “Susceptible” state and an “Infected” state.
interacting

    As opposed to independent, when there is an interaction; the spreading entities can affect each other. The whole does not equal the sum of its parts.
interaction profile

    A tool for visualizing the strength of interactions between entities as the time separating them grows.
interaction

    When several entities are interacting, name given when the whole does not equal the sum of its parts.
membership

    The extent to which an entity belongs to a cluster.
outcome

    A possibly exogenous decision that has been taken given a specific context.
piece of information

    Same as “entity”. Any object that is susceptible to have an influence on spreaders. It carries a semantic meaning. For instance: a tweet, a meme, a song, a virus, a news article, etc.
rich-get-richer

    In clustering, a property that gives an entity a higher probability of belonging to the most populated clusters and a lesser probability to belong to the less populated ones.
spread

    An endogenous decision that may infect other spreaders.
type

    Entities of the same type have an identical way of expressing their semantic meaning. For instance, two tweets are of the same type ("Tweet"), two news articles are of the same type ("News"), and a tweet and a news article can be of the same type ("Textual documents") or not depending on modeling choices.
virality

    Probability of a decision on a piece of information in the absence of interactions.

###### Contents

  1. Introduction
    1. .1 General considerations
      1. .1.1 About these large flows of data
      2. .1.2 About the automated means to make sense out large corpora
      3. .1.3 About understanding underlying data-generation mechanisms
    2. .2 Motivations
      1. .2.1 Most existing models do not consider information interaction
        1. .2.1..1 Independent Cascade model
        2. .2.1..2 Extensions and other independent-spreading models
        3. .2.1..3 Inferring the network from independent cascades
      2. .2.2 Should we consider information interaction?
    3. .3 Landscape of information interaction modelling
      1. .3.1 Theoretical studies
        1. .3.1..1 Information overload as the consequence of micro-interactions
        2. .3.1..2 Modelling micro-interactions
        3. .3.1..3 Modelling complex micro-interactions
        4. .3.1..4 We should learn models from the data
      2. .3.2 Data-driven studies
        1. .3.2..1 Clash of the Contagion
        2. .3.2..2 Correlated cascades
        3. .3.2..3 Further learning-based studies?
      3. .3.3 Definitions
    4. .4 About this manuscript
      1. .4.1 Problematic
      2. .4.2 Plan and contributions
        1. .4.2.a Chapter – Stochastic Block Models (Question 1)
        2. .4.2.b Chapter .8.0..5 – Temporal diffusion networks (Question 2)
        3. .4.2.c Chapter .14.0..4 – Dirichlet-Hawkes Processes (Question 3 and Question 4)
      3. .4.3 Reproducible research
  2. Stochastic Block Models – Interactions are rare
    1. .5 Introduction
      1. .5.1 Motivation
      2. .5.2 Overview of the proposed approaches
        1. .5.2..1 Representing interactions as a network
        2. .5.2..2 Spotting regularities in the interactions
        3. .5.2..3 Tensor decomposition approaches
        4. .5.2..4 Limitations of tensor decomposition methods
        5. .5.2..5 Bayesian network modelling
    2. .6 Static interactions
      1. .6.1 State of the art, limitations, and contributions
        1. .6.1.a Modelling static interactions
        2. .6.1.b Stochastic Block Models
          1. .6.1.b.1 Stochastic Block Models
          2. .6.1.b.2 Mixed Membership Stochastic Block Models
          3. .6.1.b.3 The need for a more global framework
        3. .6.1.c Contributions
      2. .6.2 SIMSBM – A global MMSBM framework
        1. .6.2.a SIMSBM – Serialized Interacting MMSBM
          1. .6.2.a.1 Overall idea
          2. .6.2.a.2 Input data
          3. .6.2.a.3 Model parameters
        2. .6.2.b Inference
          1. .6.2.b.1 E-step
          2. .6.2.b.2 M-step
        3. .6.2.c SIMSBM generalizes several state-of-the-art models
          1. .6.2.c.1 Nomenclature
          2. .6.2.c.2 MMSBM
          3. .6.2.c.3 Bi-MMSBM
          4. .6.2.c.4 T-MBM
          5. .6.2.c.5 IMMSBM
          6. .6.2.c.6 Indirect generalizations
        4. .6.2.d Experiments
          1. .6.2.d.1 Range of application
          2. .6.2.d.2 Datasets
          3. .6.2.d.3 Baselines and evaluation
        5. .6.2.e Discussion
        6. .6.2.f Conclusion
      3. .6.3 IMMSBM – A study of pair interactions
        1. .6.3..1 About this section
        2. .6.3.a IMMSBM – Interacting MMSBM
          1. .6.3.a.1 MMSBM to model interactions
          2. .6.3.a.2 Model
        3. .6.3.b Inference of the parameters
          1. .6.3.b.1 Expectation step
          2. .6.3.b.2 Maximization step
          3. .6.3.b.3 Instantaneous derivation as a special case of SIMSBM
        4. .6.3.c Experiments
          1. .6.3.c.1 Datasets and evaluation protocol
          2. .6.3.c.2 Baselines
          3. .6.3.c.3 Results
        5. .6.3.d Discussion
          1. .6.3.d.1 Global impact of interactions
          2. .6.3.d.2 Which clusters interact
          3. .6.3.d.3 Entropy of membership
        6. .6.3.e Conclusion
    3. .7 Dynamic interactions
      1. .7.1 Introduction
        1. .7.1..1 Inferring dynamic, valued and labelled networks
        2. .7.1..2 Overview of the proposed approach
      2. .7.2 State of the art and limitations
        1. .7.2.a Notations
        2. .7.2.b Dynamic unlabelled networks - Single-membership
        3. .7.2.c Dynamic unlabelled networks - Mixed-membership
        4. .7.2.d Static labelled networks - Mixed-membership
      3. .7.3 SDSBM – Simple Dynamic labelled MMSBM
        1. .7.3.a Base model
        2. .7.3.b Simple Dynamic prior
          1. .7.3.b.1 Dirichlet distribution
          2. .7.3.b.2 Prior’s mode
          3. .7.3.b.3 Priors expression
        3. .7.3.c Inference
          1. .7.3.c.1 E step
          2. .7.3.c.2 M step
        4. .7.3.d Discussion
          1. .7.3.d.1 Easy to use
          2. .7.3.d.2 Flexible dynamic modelling
          3. .7.3.d.3 Tuneable temporal dependence
        5. .7.3.e Experiments
          1. .7.3.e.1 Synthetic data
          2. .7.3.e.2 SDSBM unveils complex temporal patterns
          3. .7.3.e.3 SDSBM works with little data
          4. .7.3.e.4 SDSBM handles highly stochastic interaction patterns
            1. .7.3.e.5 Real-world data
              1. .7.3.f Conclusion
                1. .8 Conclusions
                  1. .8.0..1 Global SIMSBM framework
                  2. .8.0..2 Case study with SIMSBM(2)
                  3. .8.0..3 Modelling dynamic memberships of interacting entities
                  4. .8.0..4 Interactions are sparse
                  5. .8.0..5 Time range of interactions?
                    1. Temporal diffusion networks – Interactions are brief
                      1. .9 Introduction
                        1. .9.1 Temporal evolution of interactions
                        2. .9.2 Proposed approach
                        3. .9.3 Workflow
                        4. .9.4 Contributions
                      2. .10 State of the art on temporal interaction network inference
                        1. .10.1 Temporal interactions in general
                        2. .10.2 Modelling interactions
                        3. .10.3 Temporal network inference
                      3. .11 InterRate – Interaction dynamics
                        1. .11.1 Problem definition
                        2. .11.2 Likelihood
                        3. .11.3 Proof of convexity
                      4. .12 Experiments
                        1. .12.1 Experimental setup
                          1. .12.1.a Kernel choice
                            1. .12.1.a.1 Gaussian RBF kernel family (IR-RBF)
                            2. .12.1.a.2 Exponentially decaying kernel (IR-EXP)
                          2. .12.1.b Parameters learning
                          3. .12.1.c Background noise in the data
                          4. .12.1.d Evaluation criteria
                          5. .12.1.e Baselines
                            1. .12.1.e.1 Naive baseline
                            2. .12.1.e.2 Clash of the contagions
                            3. .12.1.e.3 IMMSBM
                            4. .12.1.e.4 ICIR
                        2. .12.2 Results
                          1. .12.2.a Synthetic data
                            1. .12.2.a.1 Data generation
                            2. .12.2.a.2 Numerical results
                          2. .12.2.b Real data
                            1. .12.2.b.1 Datasets
                            2. .12.2.b.2 Numerical results
                      5. .13 Discussion
                        1. .13.1 Exponential interaction profiles
                        2. .13.2 Recovering state of the art conclusions
                      6. .14 Conclusions
                        1. .14.0..1 Modelling the temporal aspect of interactions
                        2. .14.0..2 Recovering conclusions on temporal interactions
                        3. .14.0..3 Interactions are short
                        4. .14.0..4 Towards proper interactions modelling
                          1. Dirichlet-Hawkes Processes - Modelling rare and brief interactions
                            1. .15 Introduction
                              1. .15.1 How to properly model interactions
                              2. .15.2 Objective
                              3. .15.3 Proposed approach
                              4. .15.4 Workflow
                            2. .16 State of the art and limits
                              1. .16.1 A brief overview of temporal clustering of textual documents
                              2. .16.2 Dirichlet-Hawkes Process
                                1. .16.2.a Dirichlet Process
                                2. .16.2.b Hawkes Process
                                3. .16.2.c Dirichlet-Hawkes Process – Expression
                                4. .16.2.d Textual modelling
                              3. .16.3 Limits
                            3. .17 Powered Dirichlet Process – Alleviate the “rich-get-richer” assumption
                              1. .17.1 Introduction
                              2. .17.2 Motivation
                              3. .17.3 Background
                                1. .17.3.a Previous works
                                  1. .17.3.a.1 Dirichlet process
                                  2. .17.3.a.2 Uniform process
                                  3. .17.3.a.3 Pitman-Yor process
                                  4. .17.3.a.4 Other extensions
                                2. .17.3.b Contributions
                              4. .17.4 The model
                                1. .17.4.a The Dirichlet-Multinomial distribution
                                2. .17.4.b Powered conditional Dirichlet prior
                                3. .17.4.c Posterior predictive
                                4. .17.4.d Powered Chinese Restaurant process
                              5. .17.5 Properties of the Powered Chinese Restaurant process
                                1. .17.5.a Convergence
                                2. .17.5.b Expected number of tables
                              6. .17.6 Experiments
                                1. .17.6.a Numerical validation of propositions
                                2. .17.6.b Use case: infinite Gaussian mixture model
                                  1. .17.6.b.1 Synthetic data
                                  2. .17.6.b.2 Real-world data
                              7. .17.7 Conclusion
                            4. .18 Powered Dirichlet-Hawkes Process – Modelling self interacting clusters
                              1. .18.1 Introduction
                                1. .18.1.a PDHP as an answer to DHP’s limits
                                2. .18.1.b Contributions
                              2. .18.2 Model and algorithm
                                1. .18.2.a Dirichlet prior and alternatives
                                2. .18.2.b Hawkes processes
                                3. .18.2.c Powered Dirichlet-Hawkes process
                                4. .18.2.d Textual modelling
                                5. .18.2.e Posterior distribution
                                6. .18.2.f Algorithm for parameters inference