Spatio-Temporal Analysis of Topic Popularity in Twitter

Spatio-Temporal Analysis of Topic Popularity in Twitter

Sebastien Ardon NICTA, Australia, Email {sebastian.ardon,anirban.mahanti,sipat.triukose}@nicta.com.au Amitabha Bagchi Department of CS&E, IIT Delhi, Email {bagchi,aruhela,aseth,tripathy}@cse.iitd.ernet.in Anirban Mahanti NICTA, Australia, Email {sebastian.ardon,anirban.mahanti,sipat.triukose}@nicta.com.au Amit Ruhela Department of CS&E, IIT Delhi, Email {bagchi,aruhela,aseth,tripathy}@cse.iitd.ernet.in Aaditeshwar Seth Department of CS&E, IIT Delhi, Email {bagchi,aruhela,aseth,tripathy}@cse.iitd.ernet.in Rudra Mohan Tripathy Department of CS&E, IIT Delhi, Email {bagchi,aruhela,aseth,tripathy}@cse.iitd.ernet.in Sipat Triukose NICTA, Australia, Email {sebastian.ardon,anirban.mahanti,sipat.triukose}@nicta.com.au
Abstract

We present the first comprehensive characterization of the diffusion of ideas on Twitter, studying more than 4000 topics that include both popular and less popular topics. On a data set containing approximately 10 million users and a comprehensive scraping of all the tweets posted by these users between June 2009 and August 2009 (approximately 200 million tweets), we perform a rigorous temporal and spatial analysis, investigating the time-evolving properties of the subgraphs formed by the users discussing each topic. We focus on two different notions of the spatial: the network topology formed by follower-following links on Twitter, and the geospatial location of the users. We investigate the effect of initiators on the popularity of topics and find that users with a high number of followers have a strong impact on popularity. We deduce that topics become popular when disjoint clusters of users discussing them begin to merge and form one giant component that grows to cover a significant fraction of the network. Our geospatial analysis shows that highly popular topics are those that cross regional boundaries aggressively.

1 Introduction

We live in the era of Twitter. From the shenanigans of pop stars and actors to enduring political transformations, everything is being transacted on microblogging services. Nonetheless, fundamental questions remain unanswered. We know, for instance, that discussions around certain topics “go viral” whereas other topics die an early death. The network propagates some ideas, and some make no headway. In view of the enormous influence of online social networks (OSN), understanding the mechanics of these systems is critical. To characterize the properties of popular and non-popular topics is of surpassing importance to our understanding of how these complex networks are shaping our world.

In this paper we present a large-scale measurement study that attempts to describe and explain the processes that animate microblogging services. We study a large set of popular and non-popular topics derived from a comprehensive data set of tweets and user information taken from Twitter. A key strength of our study is that we observe both popular and not-so-popular topics. This allows us to hypothesize about the temporal and spatial behavior of popular topics and support our hypotheses by showing that non-popular topics display contrary behavior.

Note that we use the more general term popular rather than the more specific term viral. This is to make a clear distinction between those topics that achieve popularity because of processes and situations that lie outside the network and those whose popularity can be attributed to the dynamics that take place within the network. We reserve the term viral for those topics whose vast popularity is a product of the social network’s internal dynamics. These topics could not have gained popularity in the pre-OSN era unless traditional news media decided to promote them. Our study does not focus on these kinds of topics in particular because we intend to study the entire ecosystem.

Our work emphasizes the structural aspects of topic spread. We give the semantic aspect its due importance in the process of topic identification and then proceed to study the fundamental temporal and spatial aspects of the spread of topics. In particular, we study topic movement over two interrelated spatial dimensions: the topology of the Twitter network as formed by “follower” and “following” relationships, and the geospatial embedding of that network in the map of the world.

Our study spans several aspects of spatial diffusion, but our primary focus is on characterizing the temporal and spatial underpinnings of popularity. We focus on three important aspects as described in the sequel. First, in Section 4, we study how topic initiators influence popularity of the topic, and make the following observations:

Hypothesis 1. Twitter is a partially democratic medium in the sense that popular topics are generally started by users with high numbers of followers (we call them celebrities); however, for a topic to become popular it must be taken up by non-celebrity users.

Corollary 1a. Regions with large user bases or with large number of heavily followed celebrities and news sources dominate Twitter.

Second, in Section 5, we study the effect of topology and the dynamics of topic spread on popularity. The primary objects of study to this end are the subnetworks formed by users discussing each topic. While it is known that the Twitter network, like most large OSN, contains a giant connected component, a key finding is that the subgraph of users talking about a popular topic on a particular days always contains a giant connected component containing most of the nodes (users) of the subgraph, whereas the subgraphs of non-popular topics tend to be highly disconnected. To summarize, we make the following observations:

Hypothesis 2. Most of the people talking about a popular topic on a given day tend to form a large connected subgraph (giant component) while unpopular topics are discussed in disconnected clusters.

Hypothesis 2a. The giant component forms when many tightly clustered sets of users discussing the topic merge.

Finally, we study the impact of geography on popularity by partitioning the Twitter network according to regional divisions and studying the behavior of popular and non-popular topics.

Hypothesis 3. Popular topics cross regional boundaries while unpopular topics stay within them.

The evidence for this observation is presented in Section 6.

Apart from the highlights mentioned above, we review related work in Section 2. We describe the various methodological issues that needed to be surmounted to perform our study in Section 3. Section 7 concludes the paper with a discussion of the implications of our observations on different aspect of the OSN sphere.

2 Background and Related Work

Leskovec, Backstrom and Kleinberg’s seminal work on the evolution of topics in the news sphere was the starting point for this paper [10]. They studied how the growth of one topic affects the growth of other topics in the blogosphere. They identified and tracked a small number of popular threads, and showed that the growth of the number of posts on a thread negatively impacts the growth of other threads. The basic question that arose on reading that work was this: Can the nuances of the temporal evolution of topics be explained by a more thorough study of their spatial evolution? Working with a data set taken from Twitter we were able to extract the high level of structural and geographical information about the actors of the process that has allowed us to answer this question in the affirmative. This allows us to challenge the line of research that studies only the temporal evolution of topics [20], or seeks to explain this evolution on the basis of content [19].

Following the paper cited above there has been more interest in understanding how information and ideas propagate on OSNs. A pioneering study on these phenomena on Twitter was conducted by Kwak et. al. [9] where several aspects of topic diffusion were studied. Of particular relevance to our work was their study on the topological properties of retweet trees. Since our data set is built on the data set they used (cf. Section 3) for details), our work can easily be compared. Our major contribution is that we work with a more general notion of a topic and that we work with an ecosystem of topics. Also our work views the diffusion of topics through the lens of what we call “topic graphs” (cf. Section 5), that are a significant generalization of retweet trees. Retweet cascades have also been studied specifically for the case of tweets with URLs in them by Galuba et. al. [5] and by Rodrigues et. al. [12]. There is a line of work that seeks to uncover the structural processes behind topic diffusion by studying cascade models (e.g. Ghosh and Lerman [6], Sadikov et. al. [15]) but we feel this is a limited view of the effect of topology and try to view the network structure in a more complex way.

In another work relevant to ours, Sousa el al. [16] investigated whether user interactions on Twitter are based on social ties or on topics, by tracking replies and message exchange on Twitter; their study is focused on only three topics namely sports, religion, and politics. More recently, Romero et al. [13] studied topic diffusion mechanisms on Twitter by focusing on topics identifiable by hashtags. They study the probability of a topic adoption based on repeated exposure, and provide quantitative evidence of a contagion phenomenon made more complex than normal studies of virus-like phenomena by the existence of multiple topics, and briefly report on the graph structure of topic networks. One major limitation of this work we found is that only a very small fraction (approx. 10%) of tweets are tagged with hashtags (see Table 1 in Section 3). Our methodology of using a Natural Language Processor (OpenCalais [1]) allows us to study topic diffusion on a much larger scale than in this work since our topic choices are not limited to hashtags. On the geographical front, Yardi et al. [21] examine information spread along the social network and across geographic regions by analyzing tweets related to two specific events happening at two different geographic locations. As an aside we mention that Krishnamurthy et. al. characterized the geographical properties of the Twitter user base in 2008 [8].

On a more general level, we note that it is implicitly assumed that the attention of users on a platform like Twitter is elastic but bounded (see e.g. [11]) and hence the diffusion process is essentially a competitive one, even if it is not explicitly adversarial. The study of competitive diffusion has largely revolved around the application domain of viral marketing where there is competition between different products [2, 4, 17, 7]. Budak et al. [3] consider the problem of diffusion of mis-information, where opposing ideas are competing and propagating in a social network. The study of processes by which rumor spread may be combated is another example of competitive diffusion [18]. Our work provides an important input into this area of study, articulating the properties of a complex system that requires extensive study to model correctly and comprehensively.

3 Methodology

3.1 Data Set Description

We used a portion of the ‘tweet7’ data set crawled by Yang et al. [20]. This data set contains 467 million tweets, collected over a period of seven months, from June to December, 2009. The tweets emanated from over 17 million users and are estimated to constitute about 20-30% of all tweets posted during that time period. For our analysis, we used the first three month’s tweets of this data set.

Tweets 196,985,580
Users 9,801,062
Hashtags 1,341,733
Tweets with Hashtags 19,043,104
Retweets 15,126,588
Tweets with URLs 54,443,857
Direct (@) Tweets 41,951,786
Table 1: Data set summary.

The high-level description of the data we used is shown in Table 1. As shown, out of approximately millions tweets, only approximately millions tweets contain a hashtag. Next, we augmented this data set with network information, in particular, the Twitter follower-following relationship between users, using a separate data set collected from Twitter during the same time period by Kwak et al. [9]. This allowed us to construct a directed graph of Twitter users, where a node represents a Twitter user and an edge between two nodes , , represented , represents that is a follower of . This captures the fact that if follows then ’s tweets are visible directly on ’s timeline.

(a) In-degree distribution (b) Out-degree distribution (c) In-degree VS Out-degree
Figure 1: Basic properties of the Twitter social network.

Figure 1 shows the basic properties of the collected Twitter graph. Both the in- and out-degree distributions follow a power-law [9]. Most users follow a few people and a few users are followed by a large number of people (“celebrities”). We notice that the in- and out-degree values are positively correlated, as shown in Figure 1(c) by an overall clustering of points around the line. In addition, we observe an additional cluster of points on the top-left quadrant of this graph, which are “celebrities”.

3.2 Geospatial Data

We augmented the tweet7 data set by querying Twitter to obtain user profile information, which often contains the user’s location. Using the Twitter User ID available in the tweet7 data set, we queried Twitter using its API to extract location information from user profiles. The major issue we faced was that of rate limiting by Twitter. Using a white-listed Twitter account allowed 20,000 queries per hour, and running parallel instances of our crawler to reach up to this limit, we were able to retrieve location information of approximately 7.4 million Twitter users in approximately 200 hours. This is roughly 75% of the user population available in the tweet7 data set. The rest of the users had been banned by Twitter or had deleted their accounts in the time elapsed between our data collection June, 2011 and the time the tweet7 data was generated June, 2009.

Only 62% of the 7.4 million users in the data set provided location information. A further problem was that location was specified in several different formats. Hence we first converted the location information into latitude-longitude pairs and then reverse-geocoded the coordinates into a city, stat,e and country format, using the Yahoo! PlaceFinder service which is part of the Yahoo Geo API.

After processing all available location information, countries with the largest number of users in the data set are the USA (57.6% users), followed by UK (7.7%), Brazil (7.1%), and Canada (3.7%). As the USA contains large number of users, we sub-divide it into five commonly used regions: Northeast (10.7%), Southeast (13.5%), Midwest (10.4%), Southwest (6.8%), and West (16%).

3.3 Topic Identification

The Twitter service allows users to identify topics in messages using hashtags, a single word starting with ’#’. This allow users to follow not only other users but also conversations around specific tags. Using hashtags to identify topics in tweets has been used in several past contributions [13, 9], but results in sparse data sets, as the majority of users do not use hashtags. From Table 1, we can see that only about 10% of the tweets contain a hashtag.

In order to determine topics for tweets without hashtags, we used the OpenCalais [1] text analysis engine to extract entities, fact, and events from tweet texts in English, French, and Spanish languages. We have, in a sense, “outsourced” the semantic analysis of tweets to the OpenCalais service. This service extracts entities and tags. We have assumed that each distinct tag or entity is a separate topic, although, as we describe subsequently, the output provided by OpenCalais is just a starting point in the process of settling on a reasonable set of topics.

OpenCalais also has a rate limit of 50,000 requests per day. This made sending one tweet a time a virtual impossibility. Using a small sample of text data, we computed the topic coverage value of a bundle of tweets, by comparing the number of identified topics when bundling, with the number of identified topic without bundling. We vary the bundle size from 1 tweet to 100KB worth of tweets. Using this technique, we determine an optimum bundling size of 40K, which allows a tractable topic extraction time of 2 weeks, while resulting in a topic coverage value of approximately 94%.

Using this method, we were able to extract nearly 6.2M topics in 52M tweets. The remaining tweets were discarded as no topic could be identified. Reasons for this include short tweet, unsupported language, or no clearly identifiable topic. Examples of topics identified and their occurrence count are ’IRANELECTION’ (341674), ’TWITPOCALYPSE’ (1031), ’SONG OF NICK JONAS’ (16).

(a) Actual topics (b) Reduced topics
Figure 2: Frequency distribution of topics.

Figure 2(a) shows the frequency distribution of these 6.2M topics. The distribution follows a power-law shape: most topics are talked by very few users (¡ 10 users). These topics can be considered as noise in our study, as they are useless to study the diffusion and popularity of topics in the network. We therefore apply a threshold-based filtering, by removing topics used by less than 15 users. After filtering, we are left with 0.9M topics which is still quite a large set. We further reduce this set by sampling 4135 topics which were manually examined to ensure that duplicates were merged and that the topic set comprised sensible concepts, taking care to include both popular and non-popular topics. Figure 2(b) shows the frequency distribution of this reduced topic set. We can see in Figure 2(b) that, the frequency distribution follows a power-law. In the rest of the paper we refer to this set of 4135 as the base set of topics. The analysis done in Sections 5 and 6 is done using this set of topics.

To identify and measure the popularity diversity of topics, Figure 3 compare the number of users to the number of tweets, by plotting those two variables on a scatter plot, for each of the 4135 topics. This graph effectively shows the difference in popularity for all topics. From the graph, we can see that users of unpopular topics typically tweets more than one time on that topic. Popular topics on the other hand, are typically tweeted once by most users. In the following section, we observe and explain why some topics become popular, and other not.

Figure 3: Topic popularity diversity.

One problem with the base set is that it contains topics that already existed in the network at the beginning of the time window over which the data set was collected. While this still allows us to observe the temporal and spatial variation of the topics during the window, it makes it difficult for us to estimate the effect of initiators on the popularity of a topic. In order to address this problem filtered out all topics whose first tweet appears within the first 7 days of the window. We called this reduced set of topics the filtered set. The analysis of the effect of initiators (in Section 4) is done using only the topics in the filtered set.

We note that an initial report on the data engineering process described in this section has been accepted for presentation [14]. However, there have been significant extensions done for the purposes of this work, including the querying of follower-following relationships of several million users whose links were missing from the original data set, and the creation of the filtered set for the purposes of studying initiator influence.

4 The influence of initiators

The sudden rise in importance of Twitter as a global communication medium has made it important to study who are the entities that wield most influence on this medium. In this section we make an initial contribution by finding, as stated in Observation 1, that popular topics are generally initiated by users with very high follower counts. These users are usually either traditional or web-based news media outlets or media personalities (pop stars, politicians, writers etc.); we will refer to such users as celebrities. Finding that the mean number of followers of a user in our data set’s Twitter network was 65.7 and the standard deviation of this quantity was 1291.7, we decided to designate any user with more than 3,000 followers as a celebrity.

As stated earlier, we conducted our study on the influence of initiators on popularity only on the filtered set of topics (i.e., those whose first tweet appeared at least 7 days after the beginning of our data set’s time window). Popularity here refers to the number of users tweeting about a topic on a day. For each such topic we considered the first 5% of all tweets on that topic in the window and designated the set of users who posted these tweets as the initiators of the topic.

2
(a) Total #followers (b) Average #followers
Figure 4: #followers of initiators versus popularity I.

In Figure 4 we present two scatter plots: the first shows the relationship between popularity and the total number of followers of the initiators. We note that highly popular topics have very high aggregate followers of the initiators. But we note in Figure 4(b) that the average number of followers of the initiators of highly popular topics is in the hundreds rather than the thousands. This indicates that there are some initiators with very large number of followers involved in these popular topics.

(a) Maximum #followers (b) 2nd highest #followers
Figure 5: #followers of initiators versus popularity II.

When we examined the maximum and second highest number of followers of the initiators of each topic (cf. Figure 5) we found that it was indeed the case that celebrity users were involved in initiating highly popular topics while most unpopular topics were initiated by users with a low number of followers.

An interesting observation can be made by looking at the points plotted near the bottom right corner of all plots in Figures 4 and 5. These are topics started by a few celebrities that did not achieve any popularity. Hence we see that while it is the case that celebrities drive the popularity of topics, it is not the case that every topic promoted by celebrities becomes popular. This helps us establish Hypothesis 1: Celebrities influence the spread of topics, but cannot make a topic popular unless common users pick up on them.

We expect that regions containing larger numbers of Twitter users will influence the topics discussed to a greater extent. To determine this we tabulated the number of topics for which each region has at least one user in the initiator set (cf. Figure 6).

Figure 6: Users vs topics initiated per country.

We find that there is a linear relationship between the number of topics initiated and the size of the regions user base. While this does lead us to conclude that regions with larger user bases have more influence on the network, at least the relationship is not super-linear, implying that an increase in users in a given region could potentially lead to an increase in the share of the topics initiated therein.

Figure 7: Topics started vs Celebrities count per country.

However, a cautionary note is struck in Figure 7. Here we again plotted the number of topics started by a region on the -axis, but on the -axis instead of plotting the size of the user group from the region (as done in Figure 6), plotted the number of celebrities contained in those regions. Further, we did not consider all topics for this plot, instead focusing on only the top 500 topics (by topic user count) in the filtered set. We found that countries with a greater number of celebrities initiated a disproportionately high number of these topics.

We see a kind of continuity with the past here. The cultural and political dominance of certain regions that existed before Twitter came into being is reflected in the presence of a greater number of celebrity users in those regions, and consequently translates into a greater impact for those regions in terms of popular topics.

5 Graph theoretic properties of topics

In this section we establish Hypothesis 2 and argue towards Hypothesis 2a. To do this we study three different types of graphs associated with each topic. First, we study the lifetime graph of a topic; this is the subgraph induced on the Twitter network by all the users who have tweeted on that topic at any time in our window. Second, we study the evolving graphs of a topic. In particular, we partition the tweets related to a particular topic by day and for each day we construct the subgraph induced on Twitter by the users who have tweeted on that topic on that particular day. Finally, we study the cumulative evolving graphs of a topic. We denote by , the cumulative evolving graph for topic on day and define it as follows:

  • The vertex set of comprises the users who tweet about on day 0. The edge set is empty.

  • The vertex set of is the set of all users who have tweeted on a topic in days 0 through . An edge is added to if and tweets about on day .

5.1 Lifetime Graphs

Figure 8: Lifetime graphs: Density vs popularity.

We constructed lifetime graphs for each of the 4135 topics in the base set. Our first observation from these graphs is that popular topics tend to occupy the more well-connected portions of the network. To establish this we studied the relationship of the total number of users who have tweeted on a topic (referred to as the topic user count) to the density of the lifetime graph of the topic (defined as the number of edges per user in the graph). In Figure 8, we note something important: the lifetime graphs of non-popular topics (e.g., user count ¡ 1000) do not have densities greater than 10, and in fact many tend to have a density less than 1. A density of less than one for a subgraph of a reasonably well-connected graph like the Twitter network clearly indicates a high number of small isolated clusters. This isolation is observed even in the lifetime graph which establishes relationships between users even where they may not exist, for example, by putting an edge from to although may have tweeted on the topic before . Hence Figure 8 strongly supports one side of Hypothesis 2: less popular topics generally exist in highly disconnected clusters.

Figure 9: Lifetime graphs: Largest connected component vs popularity.

It is difficult to establish the other direction of Hypothesis 2 from lifetime graphs because of the optimistic selection of edges mentioned earlier. Nonetheless we get strong indicative evidence for our hypothesis that a popular topic tends to be discussed in one large cluster that contains most of the users that have tweeted on that topic. This evidence comes from studying the relationship of the topic user count to the size of the largest connected component of the lifetime graph, as shown in Figure 9. From Figure 9, notice that for more popular topics there is a clear linear relationship between the popularity of the topic and the size of the largest component of the lifetime graph. This is strongly indicative of Hypothesis 2, although it cannot be used as conclusive evidence.

Figure 10: Initiators in giant component.

As an aside, we investigated how many of the initiator nodes are present in the largest connected component for the topics in the filtered set used in Section 4. In Figure 10, we see that the more popular topics tend to contain almost all their initiators in their largest connected component (which we know from above is very large). Hence popular topics tend to spread around their initiators, implying that initiators have a topological impact as well as a geographical one for popular topics. Another way of putting this is that those people who are not connected to the “cool” people may in fact miss out on the “hot” topics: an unfortunate conclusion that leads us to think that greater awareness gives some users a jumpstart in the process of collecting cultural capital on OSNs like Twitter.

5.2 Evolving Graphs

It is in the study of evolving graphs that we are able to establish that most users tweeting on popular topics form one large connected component (we will refer to this large component as the giant component from now on). To establish this we began by splitting our base set into three categories, depending on the topic user count : popular (), medium popular () and non-popular (). For each category, we randomly chose topics, and computed evolving graphs for each. For each day’s graph, we then computed the ratio between the sizes of the largest and the second largest component, and also the ratio between the radii of the largest and second largest component.

(a) Ratio of components size
(b) Ratio of components radius
Figure 11: Median value of the ratios.

In Figure 11(a) and (b) we present histograms for the ratios of component and radii sizes. The buckets divide the ranges of ratios observed for the size and the radius. We find the median ratio for each topic and display the percentages of these medians that land in each bucket. Note that only the highly popular topics populate the buckets with size ratio greater than sixteen, and that the median size ratio for these popular topics goes all the way up into the range of and this is just the median, the maximum tends to be much higher but we study the median here because it is a more robust statistic. Most unpopular topics stay below 4 showing a remarkable evenness in the distribution of component sizes. The radii ratios similarly show that the width of the reach of the popular topics comes from the width of one large component rather than from a large number of small components. This effectively establishes Hypothesis 2.

Moving towards Hypothesis 2a, we first clarify in the context of evolving graphs what we mean when we say clusters merge. If we visualize the social network as a set of communities connected through users who may belong to multiple communities, our narrative of topic spread says that topics that are going to become very popular witness intense discussion within communities at first. When the level of intensity rises then the users who bridge communities enter the discussion in a big way causing a merging of what were earlier disjoint discussions. If Hypothesis 2a is correct then it can be reinterpreted to mean that the bridge users serve as a baraometer of the topics rising popularity. To investigate the applicability of this narrative we study the conductance of evolving topic graphs. Motivated by a definition widely used in the study of mixing times of random walks in graphs, we define the conductance of a subset of nodes of a directed graph as the ratio of the edges outgoing from the vertices of that land outside :

Clearly, the higher the value of , the more the number of nodes outside that are made aware of a topic being tweeted by the users in the set .

(a) CAMBRIDGE
(b) FOLLOWFRIDAY
(c) IRANELECTION
(d) MICHAEL JACKSON
Figure 12: Evolving graph conductance.

In Figure 12 we plot the evolving value of the conductance of the user set of the day’s graph alongside the evolving topic user count for four topics: one less popular topic “CAMBRIDGE”, one periodically popular topic “FOLLOWFRIDAY”, and two topics that display distinct and very high peaks in their popularity “MICHAEL JACKSON” and “IRANELECTION”. Observing the three popular topics we notice that conductance is very high just before the peak is seen. As soon as the peak is formed the conductance dips down to a low value. This supports Hypothesis 2a because when the users that bridge distinct clusters start tweeting on the topic then a larger number of edges become internal to the day’s topic graph, hence the conductance should dip as it does. Again, we clarify that this result is merely indicative of Hypothesis 2a.

There are a number of other interesting artifacts that can be observed here. The sharp peak in Figure 12(d) comes on the day of Michael Jackson’s demise. The conductance for this topic was uniformly high earlier, indicating a steady level of discussion about Michael Jackson, in tune with his general popularity. But his death leads to a sharp rise in tweets about him, causing an immediate dip in the conductance. After this initial dip the conductance rises again but no peak comparable to the first one is see, indicating that a high sustained level of interest in this topic is accompanied by a high sustained level of disinterest in the followers of the users continuing to tweet about Michael Jackson. Figure 12(c) shows a similar initial behavior accompanying an event, the holding of elections in Iran. Subsequently there is sustained discussion which is more of the nature of a conversation (the latter part of the “IRANELECTION” trajectory shows an unusually high number of tweets per user on this topic). This conversation proceeds in regions of the network that have reasonably high conductance but occasionally show dips in conductance, indicating a higher level of clustering in the user set, something that might be expected of a conversation. We note the the similarly high values of conductance displayed by the topic “CAMBRIDGE” in Figure 12(a) have a different connotation to the high values seen in the other graphs because, like most less popular topics, this too shows highly disconnected daily graphs.

The results on conductance presented here are replicated in other evolving graphs for popular topics, but we omit a wider discussion due to space constraints.

5.3 Cumulative Evolving Graphs

By using a timing relationship to establish edges for the construction of cumulative evolving graphs, we make them a better approximation for the spread of a topic than the lifetime graphs we studied in Section 5.1. In Figure 13 we plot the fraction of nodes in the largest component of the cumulative evolving graph for two highly popular topics “MICHAEL JACKSON” and “IRAN ELECTION” and two less popular topics “INDIANA, UNITED STATES” and “SMARTPHONE”. Note that even at the end of their evolution the two less popular topics have only 25% and 35% of their users in the giant component of the cumulative evolving graph while the two popular topics have half the users in the giant component even before the time window finishes. This supports Hypothesis 2 since the cumulative evolving graph is a better approximation of the spread of a topic.

(a) INDIANA, UNITED STATES
(b) SMARTPHONE
(c) IRANELECTION
(d) MICHAEL JACKSON
Figure 13: Giant component evolution over time.

But, more importantly, the sharp rise in the fraction of nodes in the giant component that accompanies a peak in the evolution of the number of tweets stands in support of Hypothesis 2a because a merging of smaller clusters into one large cluster would be accompanied by a sharp rise in this fraction. It could be argued that this rise in the fraction is because of a sharp growth in the number of users in the largest component rather than a merging of clusters, but that seems unlikely given the extent of the rise, and the large number of users already present in the cumulative evolving graph at that point. The less popular topics also show increases in the fraction when their topic user counts drift upwards, but the rise is much less dramatic than that shown by the popular topics, and could possibly be explained by a general growth in the larger component rather than a radical merging of smaller clusters.

6 Geographical analysis

Figure 14: Tweets vs regions count.

This section establishes Hypothesis 3. We argue that the popularity of topics is correlated with their geographical spread. We begin by simply studying the number of regions represented by at least one user talking about a topic and plotted it against the popularity of the topic (see Figure 14). It is quite clear from this plot that the number of regions touched by less popular topics is less than those touched by more popular topics. This plot does not establish our hypothesis but it is indicative of it in the sense that it does not falsify it either.

(a) BARACK (b) CAMBRIDGE (c) HAMBURG
Figure 15: Popularity vs edges crossing geography.

In order to establish the hypothesis, we investigated a geographical property of the cumulative evolving graphs defined in Section 5. For each topic we determined the fraction of edges in the cumulative evolving graph that went from one region to another; that is, we studied the fraction of edges such that belongs to one region and is a user from another region. The evolution of this fraction for three topics, one highly popular, one with a medium level of popularity and one with a low level of popularity (as defined in Section 5) is shown in Figure 15. We observe that the highly popular topic “BARACK” shows a high fraction of edges crossing regional boundaries throughout its evolution, ranging between 0.74 and 0.81. On the other hand the topic with medium popularity, “CAMBRIDGE”, has a low fraction of edges crossing regions. It’s noteworthy that an increase in the popularity of the topic “CAMBRIDGE” is accompanied by an increase in the fraction of edges crossing regional boundaries. This further supports Hypothesis 3. The topic “HAMBURG” which has low popularity shows a very small fraction of edges crossing regional boundaries.

(a) Mean
(b) Median
Figure 16: Fraction of edges cross regional boundaries.

To examine this phenomenon at an aggregate level we took 40 topics from each category (as we had done in Section 5) and computed the mean and median of the fraction of edges crossing regional boundaries for the entire period in our window where the topic is tweeted on. We plotted a histogram using five different ranges for this fraction (see Figure 16). This histogram clearly shows that the most popular topics tend to have a very large fraction of edges crossing regional boundaries while the least popular topics have cumulative graphs that generally evolve within regional boundaries with small fractions of edges going to other regions.

7 Concluding Remarks

The studies we have presented in this paper have wide-ranging implications, some of which, we hope, will be discovered in the future. For now we present a brief discussion of those area we feel our results may have an impact on.

Perhaps the most important implication pertains to the role and impact of highly influential users (and consequently of highly influential geographies). The rise of OSNs has been accompanied by a triumphal narrative of democratization of communication through technology, and while it is true that Twitter and other OSN platforms have played an important role in giving voice to individuals who might otherwise find it difficult to speak to an audience beyond their immediate geography, our study shows that traditional holders of power and influence have not been unseated.

Our hypothesis on how a giant component forms on Twitter–by the merging of smaller tightly clustered sets of users–is an important input into the sociology of how information is transacted on a social network. There is reason to believe that despite the fact that OSN platforms being the world closer, older notions of proximity and community continue to contribute significantly to popularity in the way described. Our study is broad in nature and captures a coarse phenomenon that we hope will excite sociologist and invite them to tease out the finer nuances that lie within such phenomena.

From an engineering standpoint issues of content distribution and caching can be addressed from observing that highly popular topics cross national boundaries. A closer study of which national boundaries are crossed more often than others could underpin efficient content placement methods.

Our results could also be of great interest to those involved in using the vast reach of media like Twitter to advertise their products and services. The notions of trust and reputation inherent in OSNs have been leveraged to a great extent already for marketing purposes. Our study could help advertisers and marketers figure out how best to use these platforms for efficient and well-targeted marketing.

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