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# Tracking online topics over time: understanding dynamic hashtag communities

- Philipp Lorenz-Spreen
^{1}Email author, - Frederik Wolf
^{2}, - Jonas Braun
^{3}, - Gourab Ghoshal
^{4}, - Nataša Djurdjevac Conrad
^{5}and - Philipp Hövel
^{1, 6}

**Received:**16 February 2018**Accepted:**28 September 2018**Published:**19 October 2018

## Abstract

### Background

Hashtags are widely used for communication in online media. As a condensed version of information, they characterize topics and discussions. For their analysis, we apply methods from network science and propose novel tools for tracing their dynamics in time-dependent data. The observations are characterized by bursty behaviors in the increases and decreases of hashtag usage. These features can be reproduced with a novel model of dynamic rankings.

### Hashtag communities in time

We build temporal and weighted co-occurrence networks from hashtags. On static snapshots, we infer the community structure using customized methods. On temporal networks, we solve the bipartite matching problem of detected communities at subsequent timesteps by taking into account higher-order memory. This results in a matching protocol that is robust toward temporal fluctuations and instabilities of the static community detection. The proposed methodology is broadly applicable and its outcomes reveal the temporal behavior of online topics.

### Modeling topic-dynamics

We consider the size of the communities in time as a proxy for online popularity dynamics. We find that the distributions of gains and losses, as well as the interevent times are fat-tailed indicating occasional, but large and sudden changes in the usage of hashtags. Inspired by typical website designs, we propose a stochastic model that incorporates a ranking with respect to a time-dependent prestige score. This causes occasional cascades of rank shift events and reproduces the observations with good agreement. This offers an explanation for the observed dynamics, based on characteristic elements of online media.

## Keywords

- Online media
- Hashtags
- Temporal community detection
- Random walk
- Memory matching
- Topic dynamics
- Ranking
- Aging model
- Bursts

## Background

Networks of complex systems represent functional or contextual relations that show globally and locally heterogeneous substructures. One important feature is the densely interconnected groups of nodes, which are called communities. Their organizational arrangements can have various characteristics such as overlapping, fuzziness or hierarchical structure and require diverse detection algorithms [1–4].

Time-resolved data of online content has become increasingly available and is of great importance for understanding the dynamics of content, including the emergence and lifetime of topics or trends. The development of methods, which capture these temporal communities is a subject of current research [5–7]. Moving from a static to a temporal picture requires tracking the communities in time. This naturally raises the question of a temporal matching of communities resulting from static snapshots [8–12]. By incorporating higher orders of memory [13] in a method proposed in [14], long-term developments can be tracked reliably.

The temporal aspect of this approach is independent from the choice of static community detection algorithm and provides a free parameter to define the timescale of a thread in order to meaningfully define a topic. The proposed method can track trajectories of content on various timescales that can occur for instance in the highly dynamical world of online media. Especially long-term developments can be followed well by canceling out noise and by memorizing topics even with interruptions due to daily or weekly periodicities.

Previously in [14], we introduced a random-walk approach for hashtag community detection and a subsequent memory-based matching scheme on temporal networks. We had demonstrated that the lifetime of small communities can be increased with this approach. This paper serves as a substantial extension of the said conference paper. This paper provides a more detailed motivation of the approach, a detailed description of the matching procedure, including a discussion of different memory kernels and in a completely new part, focuses on analyzing the empirical dynamics of the groups that we can trace over time. Furthermore, we will elaborate on a mechanistic model to reproduce and understand their features. The resulting trajectories of hashtag groups allow us to analyze the way these groups grow and shrink. If a topic is new and widely discussed, people start inventing hashtags for its description. They are combined with established and popular hashtags, for the posts to appear in many queries and reach many users. This leads to an imitation or preferential attachment behavior as often observed in other social settings [15]. Simultaneously the total volume of hashtags that are posted within a topic decreases after some time. Other topics come up and the discussion will eventually switch to new subjects and their corresponding hashtags, leading to a cycle similar to the news media [16].

This behavior leads to a fat-tailed distribution of increases of group sizes in agreement with observations in other systems [17, 18]. It has been described based on a ranking model for network growth [19] via exogenous and random shifts. We, however, observe also bursty behavior in the decreases of the communities. To account for this, we extend the existing models by a recency ranking and gain a deeper understanding of the complex dynamics of the ever-changing usage of hashtags.

## Hashtag networks

### Community structure

*M*from 52 snapshots. The bimodal character suggests that mainly two structural types can be found. We hypothesize that this corresponds to different ways of using hashtags: A descriptive usage of hashtags as keywords results in structures with lower clustering coefficient (example: Fig. 2, purple inset), while the usage of high numbers of buzzword hashtags in each post shape strongly clustered groups (example: Fig. 2, green inset).

This picture is supported by investigating the relation of degree and local clustering coefficient. Figure 3 shows the distribution of combinations \((d_i,\;C_i)\) of degree \(d_i\) and clustering coefficient \(C_i\) for each node *i* across all snapshots. A majority of the networks follow the relation: \(C_i \sim d_i^{-1}\) as described in [25] for hierarchical networks. Low clustering coefficients define top level hashtags for broader topics (e.g., ’#summer,’ ’#denim’). The upper part of the distribution contains very specific hashtags, as expected in hierarchical networks, but also less meaningful buzzwords (e.g., ’#fashionblogger,’ ’#effortless’).

Considering their topological position in the network (Fig. 5a), we observe that nodes with high clustering coefficient either lie in the periphery of hubs (Fig. 5b), or they shape strongly intraconnected groups. This leads to the picture of networks that consist of several hierarchically structured subgroups that share nodes in their periphery, which have large clustering coefficients. Figure 5c shows how modularity maximization can be misled in such networks by combining hubs that do not belong together. To separate the topics from each other and possibly filter out the unspecific groups between them, we incorporated our understanding of the data in a customized community detection method.

### Finding hashtag communities

*A*is the weighted adjacency matrix, \(d_i\) is the degree, and \(C_i\) is the clustering coefficient of a node

*i*. Parameter \(\phi >0\) is a constant that is used to regulate the general importance of \(C_i\) depending on the given data. Transition rates from a node

*i*to a node

*j*are given by the off-diagonal elements of \(L_\phi.\) Diagonal elements indicate the metastability of the process within hashtag communities, since the expected waiting time in every node

*i*is given by \(\frac{1}{\Vert L_\phi (i,i)\Vert } = e^{\phi (1-C_i)}.\) Therefore, a process stays longer on average in nodes with smaller values of the clustering coefficient. By taking into account both local measures and topological information, we achieve two things: hubs are naturally often visited, while the densely connected groups between them are not attractive for the random walker and it passes through them quickly.

Now, we can find the hashtag communities \(M_1,\ldots , M_m\) as metastable sets of the RW process given by Eq. (1). For this, we use the Markov state modeling approach [27], as it provides a way to find fuzzy communities and filter out unspecific hashtags. In particular, we obtain clustering into communities \(M_1,\ldots , M_m\) and additionally a transition region \(T = {V\setminus}\left(\bigcup _{l=1}^m M_l\right),\) consisting of the remaining nodes from the set of all nodes *V*, which are not uniquely assigned to exactly one of the communities. A transition region can act as a filter for very unspecific hashtags by accounting for the typical fuzzy character of communities in tag co-occurrence networks, avoiding overlapping areas [28, 29]. For nodes in *T,* we can calculate the affiliation probability to each \(M_1,\ldots , M_m\) by solving sparse, symmetric, and positive definite linear systems [26, 30].

*T*between them becomes small. Increasing \(\phi\) separates the hubs and leads to the highest diversity in the modules and their sizes, where \(\theta =0.9\) and \(\phi =4.0\) (see Fig. 4). A higher value for \(\theta\) increases the size of the transition region even further, shrinking the smaller communities, leading to less homogeneous sizes again.

The described method has two advantages: (1) It can be customized to our needs for the characteristic underlying network structure, by modifying the RW process. (2) It can detect fuzzy communities, which is important property of the hashtag groups under investigation and possibly more general for word groups. It is outside the scope of this work to compare our method to other community detection methods, in part because our method is designed to infer fundamentally different topological structures. The remaining parts of this work are independent of the community detection on the static snapshots, allowing for a customized solution as the one presented above.

## Dynamics of communities

### Matching problem

*A*and

*B*, from the snapshots at \(t-1\) and

*t*, respectively, by considering their Jaccard index:

In this first example, the matching is simple, namely, the one that results in the maximal sum of Jaccard indexes \(J_{\text{max}} = 3/5 + 2/3 = 1.27.\) All groups, for which a matching was found, are then renamed to be consistent with the labeling from the previous timestep (Fig. 7a). Names of communities which could not be matched, like \(F_t,\) are kept. This renaming procedure gives the possibility to track the development of a community over time and to measure its lifetime or the changes of its size.

### Memory weights

*F*in Fig. 7a) or merge (cf.

*A*) due to small temporal topological changes, but can reunite or separate after only one step. This can lead to unwanted effect, as in the third timestep \(t=3\) from our example, where a pairwise Jaccard index finds no match for

*G*and it is understood as a new development. Similarly,

*I*is identified as

*F*, while there will not be any match for

*C*and its development would stop. However, these events are just temporal fluctuations and should not deter the continuity of groups

*A*and

*C*. To overcome this algorithmic deficiency, we expand the task to a multistep matching. We recursively consider possible matchings from snapshots further in the past within a time window of length

*n*. As similarity measure, we sum up the Jaccard indexes over the

*n*preceding steps, weighted by the inverse temporal distance to compute memory-dependent weights

*W*:

*A*has disappeared but can be rediscovered by the value \(W\left(\lbrace A_{t-2},\;A_{t-1}\rbrace,\;H_t\right) = 1/2,\) which is higher than \(W \left(\lbrace B_{t-2},\;B_{t-1}\rbrace,\;H_t\right) = 2/5.\) The other scenario is the small group

*F*that split off

*C*but merges back afterwards. The memory accounts for that by a high overlap \(W\left(\lbrace C_{t-2},\;C_{t-1}\rbrace,\;J_t\right) = 2/3\) and results in keeping the label

*C*. The choice of the window size

*n*depends on the data, but also on the natural timescales of the dynamical processes that are of interest. If it, for example, is not desirable to relabel a group when it undergoes weekly periodicities, one should choose the window to be longer than a week. Following the goal to capture developments that have timescales of months, we use \(n=4\) weeks to explore the seasonal trends in fashion.

*r*to compute

*W*:

*r*from the data. Calculating the average relative overlap of all hashtags

*H*between adjacent snapshots \(\left\langle \frac{H_t \cap H_{t+1}}{H_t \cup H_{t+1}} \right\rangle _t \approx 0.9\) naturally suggests a choice \(r=0.1.\)

### Testing stabilization

*p*. The obtained randomized snapshots can be assembled one after the other to construct a noisy time series with a stable underlying community structure (Fig. 8). One can then run the matching procedure on this artificial timeseries and quantitate how often the matching algorithm found the underlying (known) groups in the noisy data by the relative success rate

*s*. The resulting values for different shuffling probabilities

*p*, depending on memory lengths

*n*and decay rates

*r*, are compared in Fig. 8. The case of \(n=1\) corresponds to a usual Jaccard index-based matching. By means of only a few steps of memory, the accuracy can be increased significantly, especially for relatively low shuffling probabilities. For strongly randomized matchings, only high memory values can still find the underlying structure. The empirically measured decay rate \(r=0.1\) as well as the finite window size of \(n=4,\) which we use throughout this work, achieve both good success scores, especially in the more realistic regime of low shuffling probabilities.

### Empirical results

## Modeling online dynamics

*i*at time

*t*. Our dataset includes the likes

*L*that are placed on the postings, and we could observe a strong correlation of average likes per hashtag \(\langle L_i/S_i \rangle,\) a community receives and its size \(S_i,\) shown in Fig. 10a (Pearson correlation \(\rho =0.925\)). In the following, we treat its size as a proxy for its popularity, and the quantity, which is simulated by the proposed model, can be understood as conceptual score for popularity.

Figure 10b shows the distributions *P*(*S*(*t*)) of total sizes in each week. They are plotted on top of each other to illustrate their stability and shape. We used the maximum likelihood method from [35] to estimate the exponent and the standard error of the indicated power law. This shape fits to the general picture of many distributions that are related to popularity measures.

*S*. To quantitate this further, we consider the distributions of the logarithmic derivative \(\Delta S / S = (S(t)-S(t-1))/S(t-1)\) (green), as was previously done in [17, 34], which describes the relative gains. In addition, in this work, we describe the relative losses \((\Delta S / S)_r = (S(t-1)-S(t))/S(t)\) (red) from 1 week to the other. Both their distributions are plotted in Fig. 10c, d.

We observe in agreement with Fig. 11a that both distributions are very similar and exhibit a fat tail. A similar behavior has been demonstrated for relative gains in Wikipedia traffic [17], Youtube views [18], hashtag usage on Twitter [33], and citation counts [34]. These fat-tailed distributions of relative changes seem to be a characteristic feature of online popularity dynamics.

Pure rich-get-richer mechanisms, where \(\Delta S \sim S\) cannot reproduce these broad distributions; however, their shapes suggest self-enhancing process in both directions (gain and loss). Exogenous events can be responsible for small items to rise up quickly as modeled in [17], but also aging of the top items [34] can lead to rising newcomers. The distribution of interevent times (event: \(\Delta S / S > 10\)), as shown in the inset of Fig. 10d, is following a power law, where we also used the method from [35] to estimate the exponent. This is an indicator of cascades of events, which we believe are caused by competition among pieces of content [36], e.g., by the downfall of a popular item, leaving room for others.

In the following, we propose a class of models that is able to explain these observations by an interplay of ephemeral popularity and ranking mechanisms.

### Ranking

*i*according to a prestige score \(\lambda _i(t),\) which depends on time. To order these scores relative to each other, we formulate a general ranking function \(r(\lambda _i(t), \lbrace \lambda _1(t), ... ,\lambda _N(t) \rbrace )\) as the sum of Heaviside functions:

*i*has a high score \(\lambda _i\) relative to the others. By that coupling, the ranking implies a competition between the topics

*i*, due to a limited capacity of the users and websites.

*i*receives is distributed according to the ranks, which result from the last timestep. Following are the attachment probabilities:

*m*scores (e.g., likes) along this feed of posts at every time step. This can be expressed in the following update rule:

### Aging model

*t*is the current time, \(t_i\) the time of introduction. The aging factor

*a*weights the influence that the age has on the ranking. This leads to the following attachment probabilities in the aging model:

### Numerical results

*P*value \(<0.001\)) for the gains and 0.06 (

*p*value \(<0.001\)) for the losses. The distribution is broad due to cascades of rank shifts. The lack of regularity and the burstiness of these jumps become clear in the power-law distribution of interburst times between events of \(\Delta S / S > 10\) as shown in the inset of Fig. 12b.

### Staying on top

*i*in rank

*r*loses its position to item

*k*from the rank below \(r+1{:}\)

*k*from the lower rank \(r+1\) to be very young \(t_k \approx t\) and the score on rank

*r*to be on average \(\langle S(r) \rangle = m \cdot P(r){:}\)

*r*with the given \(m, a, \alpha\) and

*N*, leading directly to a lower bound of the average time \(\tau\) spent in a rank

*r*:

## Conclusion

We have presented an investigation to analyze the dynamic behavior of topics in online media. We have focused on a dataset of hashtags, which were used on the fashion platform: lookbook.nu over the course of 1 year.

We have built timestamped co-occurrence networks and aggregated them to weighted snapshot graphs. We have applied a random-walk-based approach for finding a transition and a community region to obtain a reliable and meaningful clustering.

Independent from the method for community detection, we have proposed a construction of weighted bipartite networks of successive timesteps to track group dynamics over time. For robustness against temporal fluctuations and instabilities, we have extended the Jaccard index, determining the weights, to incorporate higher-order memory.

The resulting dynamics show fat-tailed distributions of relative gains and losses, characterizing bursty behaviors in the increases and also decreases of hashtag groups. In order to describe and understand these developments, we have formulated a ranking model that incorporates gain and loss from a combined attractiveness score of community size and age.

Based on the model results and affirmed by the empirical findings, competition among ranked items with unlasting prestige scores can lead to bursty behaviors in the gains and also losses of popularity. In addition, we found that competition becomes intense for higher ranks so they have to be reached in a young age. The simple model can be further extended and is applicable to other online media where recency plays an important role.

## Declarations

### Authors’ contributions

All authors contributed to the design of the study. PLS, FW, and JB evaluated the data. PLS, FW, and NDC implemented the methodology. PLS, GG, and PH developed the theoretical model. All authors analyzed the results and wrote the manuscript. All authors read and approved the final manuscript.

### Acknowledgements

P. Lorenz and P. Hövel acknowledge the support provided by the Deutsche Forschungsgemeinschaft (DFG) within the framework of Collaborative Research Center 910. These authors thank A. Koher, V. Belik, J. Siebert, and C. Bauer for fruitful discussions.

### Competing interests

The authors declare that none of them have competing interests.

### Availability of supporting data

We provide our matching framework as an open-source project. The source code can be downloaded from our Git repository: https://github.com/philipplorenz/memory_community_matching.

### Publisher’s Note

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**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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