Affinity-driven blog cascade analysis and prediction

Abstract

Information propagation within the blogosphere is of much importance in implementing policies, marketing research, launching new products, and other applications. In this paper, we take a microscopic view of the information propagation pattern in blogosphere by investigating blog cascade affinity. A blog cascade is a group of posts linked together discussing about the same topic, and cascade affinity refers to the phenomenon of a blog’s inclination to join a specific cascade. We identify and analyze an array of macroscopic and microscopic content-oblivious features that may affect a blogger’s cascade joining behavior and utilize these features to predict cascade affinity of blogs. Based on these features, we present two non-probabilistic and probabilistic strategies, namely support vector machine (SVM) classification-based approach and Bipartite Markov Random Field-based (BiMRF) approach, respectively, to predict the probability of blogs’ affinity to a cascade and rank them accordingly. Evaluated on a real dataset consisting of 873,496 posts, our experimental results demonstrate that our prediction strategy can generate high quality results ($F1$-measure of 72.5 % for SVM and 71.1 % for BiMRF) comparing with the approaches using traditional or singular features only such as elapsed time, number of participants which is around 11.2 and 8.9 %, respectively. Our experiments also showed that among all features identified, the number of quasi-friends is the most important factor affecting bloggers’ inclination to join cascades.

1 Introduction

The popularity of blogs has been increasing dramatically over the last few years. According to a recent report by Technorati (2008)¹, a popular blog search engine, more than a half of the Internet users read blogs. Technorati have indexed more than 133 million blogs since 2002, and have tracked blogs in 81 languages by June, 2008. Blogs contain diverse variety of information. General topics include personal diaries, experiences, opinions, information technology, and politics to name a few. Due to their accessible and timely nature, many bloggers surveyed have advertisement on their blogs. The mean annual revenue for blogs with advertisement is estimated to be $6,000 (Technorati 2008). This figure jumps to $75,000 for those blogs with 100,000 or more unique visitors per month.

1.1 Motivation

A blog consists of several entries. Each entry within a blog, called a post, is time stamped and the most recent entries always appear at the top. Bloggers can also create hyperlinks to other blogs or websites in their posts. The universe of all these blogs and their interconnections is often referred to as blogosphere (Stewart et al. 2007; Technorati 2008). Blogosphere is an intuitive source for data involving the spread of information and influence within the network of bloggers (Agarwal et al. 2008; Gruhl et al. 2004; Kumar et al. 2003; Stewart et al. 2007). By analyzing the linking patterns from one blog post to another, we can infer the way information is propagated through the blog network over the Web. In particular, a piece of information flows from a post to another along the hyperlink between them. For example, consider Fig. 1a. The ellipses represent different blogs (e.g.,  $b_1,\, b_2,\, b_3,\, b_4,\, b_5$, and $b_6$), and each ellipse contains a set of posts. The edges in the figure indicate hyperlinks between posts. Assume that post $p_1$ in blog $b_1$ contains opinion about recent events related to the spread of H1N1 virus. Some time later, blog $b_2$ visited $b_1$ and wrote a post $p_2$ in response to this topic of discussion and explicitly created a hyperlink to $p_1$. Subsequently, new posts will join this conversation by linking to existing posts. For instance, at time $T_0$, the structure of this conversation related to H1N1 virus containing a group of posts ($p_1,\, p_2,\, p_3$, and $p_4$) is depicted by the dashed rectangular component in Fig. 1a. Aggregating all the linked posts by backtracking the hyperlinks will result in a directed acyclic graph (DAG), where each node is a post. Such a DAG is called a cascade (Leskovec et al. 2007b; Watts 2002) (also known as conversation tree). Cascade is the most common phenomenon of information propagation within blogosphere. All posts in the same cascade typically discuss about a similar topic.

Fig. 1

Blog cascade

Observe that at time $T_0$ there are two blogs, $b_5$ and $b_6$, which did not join the conversation on H1N1 virus by writing a post and linking to the cascade. Now assume that at time $T_1 > T_0$ $b_6$ joined the cascade by writing a post $p_6$ and linking it explicitly to $p_2$. The modified structure of the cascade is now depicted in Fig. 1b. Notice that $b_5$ still did not join the conversation. Why did $b_6$ join the cascade but $b_5$ did not? Is it possible to predict the cascade affinity of $b_5$ and $b_6$ by analyzing the information embedded in the cascade at time $T_0$? In order to provide answers to these question, in this paper we propose probabilistic and non-probabilistic techniques based on Bipartite Markov Random Field (BiMRF) (Shi et al. 2009) and support vector machine (SVM), respectively, to analyze an array of macroscopic and microscopic cascade features for predicting which blogs are highly likely to join the cascade in the future ². We refer to the phenomenon of a blog’s inclination to join a specific cascade as cascade affinity.

Although the notion of information cascade was formally introduced by Sushil Bikhchandani (Bikhchandani et al. 1992), it was first systematically studied in the context of blogosphere by Kumar et al. (2003). Majority of research on blog cascades (Leskovec et al. 2007b) have focused their attention at the macroscopic level. In particular, these efforts investigated information flow in cascades, common shapes of cascades and their frequencies, and performed a series of topological analysis. In contrast, we take a hybrid view by analyzing cascade affinity behavior of individual bloggers from both microscopic and macroscopic level. To the best of our knowledge, this is the first approach that undertakes a systematic study to predict such behavior.

1.2 Applications

The knowledge of a blogger’s affinity to cascades is useful in several applications. It not only facilitates the design of advanced blogging system with more sophisticated personalized recommendations and filters, but also help us to set up intelligent strategies in online advertising. By predicting which blogs have stronger affinity to a cascade, we can make recommendations to those bloggers in case they have not yet read any post in the cascade. Consequently, we can influence the population faster by accelerating the information propagation process. In this way, new services or products can be disseminated and popularized in a shorter time. Furthermore, we can predict to what scale of population a cascade will finally expand so that when disseminating an advertisement along blog cascade we can understand the final effect of the advertisement ahead of time and adjust our advertisement strategies accordingly. For example, assume that the release of “iPhone4” may trigger off a cascade within the blogosphere. At the very beginning, there is no post talking about iPhone4 at time $t_0$ (see Fig. 2). Later at time $t_1$ there are two posts talking about this topic which initiates a cascade over “iPhone4”. By studying the cascade affinity of many candidate blogs who are most probable to join this cascade, we can predict that two new bloggers have high chance to join this cascade at time $t_2$. In Fig. 2, a person with a dashed outgoing arrow represents a blogger who is predicted to join the cascade at current timestamp whereas a solid arrow denotes the blogger who has already joined it. Thus, we may estimate the final scope of the cascade at $t_{n-1}$ by iteratively predicting the set of bloggers who may join it at the next timestamp. Our ability to forecast the final scope of the population involved in this cascade at an early stage paves way to more judicious adjustment of advertisement strategies and budget ahead of time.

Fig. 2

An example application of blog cascade affinity prediction

1.3 Overview

At first glance, it may seem that we can predict a blogger’s affinity to a cascade by analyzing the textual content of existing posts in the cascade and estimating the overlap between the content of the blogger’s previous posts and cascade content. However, such content-aware strategy is computationally expensive and may adversely affect the accuracy of prediction for several reasons. Firstly, the content of posts are often in conversational language containing flavors of abbreviated words and local lingo. Secondly, a blog cascade may consists of posts written in different languages. Thirdly, posts may only contain multimedia objects such as pictures or video clips. Consequently, these factors make content analysis significantly challenging. Hence, we take a content-oblivious strategy to address this issue.

We propose a group of content-oblivious macroscopic and microscopic features of a blog cascade that may influence a blog’s affinity to the cascade. The macroscopic features are associated with the overall structure of a cascade such as time elapsed since the genesis of the cascade, number of participants in the cascade, and the shape of the cascade defined by the star-likeness ratio. On the other hand, the microscopic features (number of quasi-friends of a blogger in the cascade, popularity of participants in the cascade, citing factor, and initiator-media link) are related to the blogs or posts of a cascade. Note that all these features are computed by analyzing only the link structure and topology of the cascade. For each of the proposed features, we investigate how it influences a blog’s affinity to the given cascade and performed a one-way analysis of variance (ANOVA) to test the significance of each feature’s influence. Then we present two non-probabilistic and probabilistic methods, namely SVM classification-based approach and Bipartite Markov Random Field-based (BiMRF) (Shi et al. 2009) approach, respectively, that exploit these features to predict the probability of blogs’ affinity to a cascade and rank them accordingly. We did not exploit the content of the posts, our experimental results demonstrated that our prediction strategy can generate high quality results ($F1$-measure of 72.5 % for SVM and 71.1 % for BiMRF). In summary, the main contributions in this paper are as follows.

• We propose an array of content-oblivious macroscopic and microscopic features that influence a blog’s inclination to join a cascade. To the best of our knowledge, these features have not been studied together in the context of a blog network earlier. Further, we present different measures to calculate each feature’s effect on the cascade affinity phenomenon.

• We formulate the task of predicting cascade affinity of blogs into a standard classification problem. We take two different methods to evaluate the probability of a blog’s affinity to a particular cascade and rank them accordingly, namely SVM-based and BiMRF-based classification strategies.

• We present an exhaustive evaluation of our proposed prediction and ranking methods demonstrating their effectiveness and practical significance using real-world datasets. In particular, our proposed techniques perform the best when all features except citing factor is used. Further, our results demonstrate that the number of quasi-friends feature is the most important factor affecting bloggers’ inclination to join cascades.

The rest of this paper is organized as follows. Section 2 presents a brief review of related work. In Sect. 3, we introduce the dataset as well as the cascade extraction process. The macroscopic and microscopic cascade features and their analysis are described in Sects. 4 and 5, respectively. Section 6 describes our proposed models to measure and rank the probability of a blog to join a cascade. In Sect. 7, we conduct an exhaustive empirical study to evaluate many aspects of our proposed techniques and their effectiveness. The last section concludes the paper.

2 Related work

2.1 Community affinity

Much work have been done in the field of information flow modeling and word-of-mouth effect. The work in this area can be traced back to the epidemic research in virus propagation problem (Satorras and Vespignani 2001; Strang and Soule 1998; Newman 2002; Pastor-Satorras and Vespignani 2002; Wang et al. 2003; Dodds and Watts 2004; Clements et al. 2010). Similar work have been done within large online social networks recently focusing on modeling the word-of-mouth effect in different social networks. Backstrom et al. (2006) showed that the probability of joining a social community depends on the number of acquaintances already in it. Leskovec et al. (2006; 2007a) reported that an individual’s probability of buying a DVD increases with the number of recommendation he has received. There is a saturation point at the value of 10, which means after a person receives 10 recommendations on buying a particular DVD, the probability of buying does not increase anymore. Cha et al. (2009) conducted a study on Flickr over the same problem. They showed that the probability for a user to become a fan of a photo increases with the number of her friends who are already fans of the photo. These above work all focused on the number of quasi-friends feature, which can be considered as a microscopic feature of a cascade. Hence, this feature is also used in our work to model the probability of a blog’s affinity to a cascade. However, in contrast to the aforementioned work, we exhaustively examine several additional microscopic as well as macroscopic cascade features that may affect this behavior.

2.2 Information diffusion

Several recent papers have focused on modeling the information diffusion patterns within social networks, which is considered to play a significant role in political science and viral marketing (Watts 2002; Rogers 2003; Gruhl et al. 2004; Iribarren and Moro 2009; Chen et al. 2009b; Lerman and Hogg 2010). In particular, several algorithms are proposed to find a set of nodes which have the most influence on the others so that by selecting those nodes as seeds we can make our piece of information spread over a large population (Kempe et al. 2003; Hartline et al. 2008; Wang et al. 2010; Chen et al. 2009a; Kimura et al. 2009; Adams et al. 2010; Lee et al. 2010). Gruhl et al. (2004) modeled the information diffusion within blogosphere by defining a read probability and copy probability for each blogger, and iteratively computed the two and finally converged to the best solution. Agarwal et al. (2008) proposed a ranking function for the blogs according to their influence based on the influence of posts appeared in each blog. The influence of a post is computed based on its length, comments, and a propagation factor which is the aggregated influence from the posts that linked to and from the current one. Another research by Ma et al. (2008) focused on finding a set of $k$ candidates as target for marketing strategy using heat diffusion models. Recently, Bao et al. (2010) proposed AdHeat, which diffuses hint words of influential users to others and then matches advertisements for each user with aggregated hints.

Our research differs from the aforementioned studies in two key ways. Firstly, existing approaches mainly focused on finding the most influential blogs in blogosphere (Agarwal et al. 2008) whereas the goal of this research is to discover blogs that are most probably to be influenced by other blogs. Hence, our work is orthogonal to these efforts. A recent study showed that large-scale changes in public opinion are not driven by highly influential people who influence everyone else but by easily influenced people influencing other easily influenced people (Watts and Dodds 2007). The authors investigated at a global scale the average size of cascades that are initiated by influential nodes and average nodes using different influence models. They showed that early adopters enrolled in a cascade is more important to affect the final cascade size than the initiators. Our work differs from it in that we study in detail under what situation a blogger will be influenced as well as retrieval of most easily influenced individuals. Secondly, in our work we propose a group of features of blogs and cascades to model the probability of a blog to join a cascade.

Karagiannis et al. (2009) studied the human behavior (Davidson et al. 2012) related to email responses. They showed that the email replying probability depends on a series of factors. By conditioning on each individual factor, they can achieve moderate prediction gains with respect to predicting replied emails. Putting together all the factors achieves a significant prediction gain. In contrast, our work analyzed the joining behavior of each individual blogger using a group of features. Additionally, we proposed two different models for ranking the blogs according to their probabilities of joining a cascade.

2.3 Retweeting in microblogging

In a different media, Pal and Counts (2011) used the count of original tweets, conversational tweets, and re-tweets of a tweeter as features to rank the authority of each tweeter in the context of different topics. They employed a Gaussian Mixture Model to compute the authority score of each tweeter. Formally, the authority score for twitter $i$ can be computed as the following.

$$\begin{aligned} R_G(x_i)=\prod \limits _{f=1}^{d}[\int _{-\infty }^{x_i^f}N(x;\mu _f,\sigma _f)]^{w_f} \end{aligned}$$

(1)

In the above equation, $w_f$ is the weight that is put on feature $f$; $x_i^f$ is the associated value of node $i$ on feature $f$; $N(x;\mu _f,\sigma _f)$ is the univariate Gaussian distribution with model parameters as $\mu _f$ and $\sigma _f$. The authority score defined above helps in devising a total ordering under “$\le$” over all the users. To validate their results, they conducted a survey to rate the authority of the tweeters and use it as the ground truth for authority ranking. Blog posts may contain much more information than microblogs. A blog post may consist of text in different languages, urls, tables, photos, video clips, et al. these information with different formats, which can introduce plenty of unsolvable problems. One main contribution of this paper is to resolve the aforementioned problems that multimedia content and multi-language text introduced. We show that cascade behavior can be predicted using only link-based features which avoid these problems.

Recently, Goyal et al. (2012) proposed a credit distribution (CD) model that leverages on historical action logs of a network to learn how influence flows in the network and use this to estimate influence spread. An action log is a set of triples ($u,\, a,\, t$) which says user $u$ performed action $a$ at time $t$. The basic idea is that if user $v$ takes action $a$ and later on $v$’s friend $u$ does the same, then the authors assume that action $a$ have propagated from $v$ to $u$. Based on this assumption the CD model assigns “credits” to the possible influencers of a node $u$ whenever $u$ performs an action. The sophisticated variant of this model distinguishes between different influenceability of different users by incorporating a user influenceability function. It is defined as the fraction of actions that $u$ performs under the influence of at least one of its neighbors (e.g., $v$) and is learnt from the historical log data. In contrast to our approach, this model suffers from two key limitations. Firstly, it depends on the availability of large amount of historical action logs to compute influence probability as well as user influenceability. Unfortunately, historical action logs may not be available to end-users in many real-world social networks.

3 Data preparation

In this section, we first introduce the real-world data set we have used for our study. Then, we present our approach of cascade extraction from the data set. In the sequel, we shall use the notations shown in Table 1 to represent different concepts. Generally, we shall use superscript to denote a cascade identifier and subscript to denote a blog identifier.

Table 1 Definitions of symbols

3.1 Dataset

We extracted our blog dataset in September, 2008 using Technorati API³. The data set contains blog posts published from June, 2008 to September, 2008. We first selected the group of top 100 blogs indexed by Technorati as seeds. From these seeds, we retrieved the blogs that had linked to these seeds in their posts, and then we iteratively retrieve the posts that linked to the previous level till the sixth level which has been shown as the upper boundary size for most chain cascades (Leskovec et al. 2007b). From the XML collection of blogs, we can get the post-to-post relationships. Notice that a post of blog $b_i$ linking to another post of blog $b_j$ does not always indicate a friendship that author of $b_i$ knows author of $b_j$ or $b_i$ regularly reads $b_j$’s blog. So we additionally extracted blog-to-blog relationships with weighted edges where the weight of an edge from $b_i$ to $b_j$ indicates the number of times $b_i$ has cited $b_j$’s posts. Such a case, to some extent, indicates that $b_i$ does not read $b_j$’s blog by chance. We use this weighted graph as an indication of friends by filtering out the edges with weight less than a friendship threshold ${\mathcal{K }}$. The characteristics of the dataset is shown in Table 2. For each blog, the posts that do not participate in any cascade are excluded from our dataset. Figure 3a shows the in-degree distribution of blogs indexed by Technorati till September, 2008. This figure is plotted using the information extracted from our data set. It is shown to follow a power law distribution⁴ with exponent equal to $-1.505$ whereas in (Leskovec et al. 2007b) this exponent is reported to be $-1.7$. Such a phenomenon indicates a few blogs are more connected than the rest. It is consistent with the result of “preferential attachment” model (rich gets richer) (Barabasi and Albert 1999).

Fig. 3

a Blog in-degree distribution. b Cascade size distribution

3.2 Cascade extraction

Recall that each blog participates in a cascade by writing a post which links to another post that is already in the cascade. We denote a set of cascades as $\mathcal{C }=\{c^1,c^2,\ldots ,c^s\}$. The algorithm for extracting cascades from our data set is outlined in Algorithm 1.

Note that the proposed cascades extraction procedure is slightly different from the one described in (Leskovec et al. 2007b). Let us elaborate on this further. Consider the scenario in Fig. 4a, depicting blog posts and hyperlinks between them. Based on (Leskovec et al. 2007b), each cascade should have only one initiator (top-most post). Hence, the scenario illustrated in Fig. 4a have to be considered as two different cascades (have two initiators $p_1$ and $p_2$) as depicted in Fig. 4b. In contrast, we treat the scenario in Fig. 4a as one cascade. The intuitive justification for this is as follows. Observe that the posts in Fig. 4a are all linked together. That is, both $p_1$ and $p_2$ share some common posts in the conversation (e.g., $p_5$). This may indicate that all these posts are discussing about a common topic. Hence, it makes sense to consider them as part of a single cascade instead of separating them into different ones.

The next step is to post-process the extracted cascades to eliminate the ones which have been there not more than a month till the time $T^{*}$. The set of “matured” cascades extracted after the post-processing is represented as: $\mathcal{C }=\{c^i\big |T^i\le T^{*}-30\}$. The number of cascades detected after filtering out the immature ones is shown in Table 2. The reason for post-processing the cascade set is as follows. We need to ensure that the extracted cascades can provide a robust and accurate framework for feature extraction and subsequent prediction. However, quantifying values of different features based on immature cascades (cascades which have not absorbed all potential participants) will distort the prediction accuracy of cascade affinity. Many participants may join these cascades after time $T^{*}$ and consequently adversely affect the modeling of the ground truth based on the features set. Obviously, this may result in a deviation between our knowledge about the participants of these cascades and the ground truth. It is worth mentioning that it is not possible to justify the prediction performance without knowing the ground truth.

Fig. 4

Different approaches for cascades extraction: a observed post-to-post relationship, it is also the cascade identified by our approach; b the cascades identified from (a) using the approach in (Leskovec et al. 2007b)

Figure 3b shows the distribution of cascade size extracted from our dataset. It is defined as the number of blogs within a cascade. The $X$ and $Y$-axes represent different sizes of cascades and the number of cascades, respectively. The minimum size of cascades is defined as $2$ which is the trivial case, while the maximum size of a cascade is found to be $34$ in our dataset. The distribution of cascade size also follows a power law. The exponent found in our dataset is $-3.1$ whereas this exponent is found to be $-2$ in the dataset used by Leskovec et al. (2007b). This deviation is primarily due to the differences between the characteristics of the two datasets and different definition of a cascade in these two approaches.

4 Macroscopic features

We now present an array of content-oblivious cascade features that may influence a blog’s affinity to a cascade. We classify these features into two types, namely macroscopic and microscopic features. The former refers to features that are associated with the entire cascade whereas the latter refers to features of the blogs or posts of a cascade. In this section, we begin with three macroscopic features, namely elapsed time, number of participants, and star-likeness ratio. In the next section, we shall elaborate on the microscopic features.

4.1 Elapsed time

First we present the role of the elapsed time. Informally, it refers to the difference between the time a blogger joins a cascade and the cascade creation time. We use day as the unit of elapsed time as most bloggers write posts once per day. The distribution of this feature is shown in Fig. 5a. The $X$-axis represents the time elapsed in days, while the $Y$-axis represents the number of blogs that join cascades at a specific elapsed time. Observe that 91 % bloggers join a cascade during the first week. After that affinity to cascades drops almost exponentially with elapsed time. Note that the above results deviate from other types of social networks, shown in (Leskovec et al. 2008), where the authors found that the average number of edges attached to each node did not change much over the lifetime of the node.

Fig. 5

a Number of posts joining versus days elapsed. b Cascade affinity probability versus cascade size

4.2 Number of participants

Intuitively, a blogger may have stronger affinity to a cascade which has absorbed a lot of participants. Hence, we now conduct an analysis using number of participants in a cascade as a feature. We compute the probability of joining a cascade as a function of the number of participants existing in the cascade. Interestingly, we observed that in only a very small fraction (0.2 %) of cascades the number of posts is more than the number blogs. It indicates that bloggers seldom re-posts in the same cascade such that the number of posts is always the same as the number of blogs in a cascade. Consequently, in the sequel we uniformly use number of blogs to represent the size of a cascade. The number of participants is formally defined as follows.

Definition 1

Let $t^i(j)$ be the time when a blog $b_j$ joins a cascade $c^i$. Then, the number of participants in $c^i$ at time $t^i(j)$, denoted as $N_j(c^i)$, is defined as:

$$\begin{aligned} N_j(c^i)=|\phi ^i(t^i(j))| \end{aligned}$$

Figure 5b shows the probability of joining a cascade as a function of the number of participants in that cascade. The number of blogs inside a cascade ranges from 1 to 33. The probability of joining a cascade with $\beta$ participants, referred to as cascade affinity probability (denoted as $Pro(\beta )$), can be computed as follows.

$$\begin{aligned} Pro(\beta ) = \frac{\sum _{c^i}|\{b_j|N_j(c^i)=\beta ,b_j\in \phi ^i\}|}{\sum _{c^i}|\{b_j|N_j(c^i)=\beta \}|} \end{aligned}$$

We separate the cascades size range into 11 bins each with length 3. The height of each bar denotes the mean of the three cascade affinity probability values inside that bin. Notice that at the beginning, as the number of participants grows, the probability slightly grows, but after some point, the probability drops down. There is a peak at the point of cascades with the size 13–15. It indicates that before a cascade absorbed 13–15 participants, the probability for a blog to join this cascade increases. This represents the cascade initiation period where many new blogs keep on joining the cascade. However, after the number of participants in the cascade has reached a value between $13$ and $15$, the probability of a blog joining this cascade drops down to a stable value. This represents the stable period after a cascade has got enough attention.

4.3 Star-likeness ratio

Recent results showed that the type of cascade topology may indicate the genre of the content in a cascade (McGlohon et al. 2007). We now investigate whether the topology of a cascade influences a blog’s affinity to join it. Firstly, we extracted different cascade shapes in the dataset and classified the cascades into $176$ different shapes. We observed that the most common shape contains only two posts. The top-13 frequent cascade shapes which appear at least 25 times in the dataset are depicted in Fig. 6. The shapes are listed according to descending order of their frequencies ($G_1$ is the most frequent cascade shape). Further, the shapes can be classified into two groups, namely chain and star. Informally, a chain has only one leaf node whereas a star is an $n$-order shape having $n-1$ leaves. Notice that in this dataset chains appear more frequently than stars with respect to the same cascade size (i.e., $G_2$ is more frequent than $G_3$ and $G_{11}$). Besides, shapes containing multiple-initiators are less frequent than the single-initiator ones (i.e., $G_3$ is more frequent than $G_{11}$). Moreover, cascade frequency does not necessarily decrease with the increase in cascade size (i.e., $G_6$ is more frequent than $G_7$).

Fig. 6

Common cascade shapes by frequency

Secondly, we investigate the probability of blogs to join a cascade by varying the cascade shapes. Formally, the cascade shape of $c^i$ is defined as the following.

Definition 2

Let $G_1,\ldots ,G_n$ denote the set of cascade shapes extracted from the dataset. If cascade $c^i_t$ (cascade $c^i$ before time $t$) follows the shape $G_s$, then the shape of $c^i_t$, denoted as $g(c^i_t)$, is defined as: $g(c^i_t)=G_s$.

Figure 7a shows the probability of joining a cascade as a function of the cascade shapes. Notice that, the probability of joining a cascade is measured using the temporal shape of each cascade. The probability of joining a cascade of shape $G_s$ can be computed as follows.

$$\begin{aligned} Pro_{top}(G_s) = \frac{\sum _{c^i_t}|\{b_j|g(c^i_t)=G_s,b_j\in \phi ^i(t)\}|}{\sum _{c^i_t}|\{b_j|g(c^i_t)=G_s\}|} \end{aligned}$$

In general, the curve in Fig. 7a is not informative enough to lead to any conclusion on the relationship between cascade shapes and the probability of joining a cascade. However, if we plot the probability curves for star (i.e., $G_3,G_7$) or chain (i.e., $G_{1},\,G_{2},\,G_{4},\,G_{6},\,G_{10},\ldots$) cascades, then it is clear that chain-shaped cascades are more probable to to attract new blogs to join them compared to their star-shaped counterparts. This phenomenon may be due to the fact that new bloggers can easily see all the blogs within a chain cascade by tracking the hyperlinks one by one. In contrast, new bloggers may only see a small portion of a star-shaped cascade due to its topology. Also, observe that blogs are much more probable to join a shape with multiple roots (i.e., $Pro_{top}(G_{11})=0.042$) compared to other shapes (i.e., $Pro_{top}(G_{2})=0.033,\, Pro_{top}(G_{3})=0.029$) with the same size.

Fig. 7

a Joining probability by cascade shape. b Number of quasi-friends versus $\mathcal{K }$

As discussed above, chain cascades are more probable to attract new blogs compared to their star-shaped counterparts. However, shapes other than chain or star are hard to classify and describe. Thus, we propose a new feature called star-likeness ratio to describe how much different a shape is from a star and utilize it in the future prediction of cascade affinity.

Definition 3

Let $root(G)$ and $leaf(G)$ denote the root nodes and leaf nodes of graph $G(V,E)$, respectively. That is, $root(G)=\{v|\not \exists u\in V, \overrightarrow{vu} \in E\}$ and $leaf(G)=\{v|\not \exists u\in V, \overrightarrow{uv} \in E\}$. Then the star-likeness ratio of graph $G$, denoted as $s(G)$, is defined as:

$$\begin{aligned} s(G) = \frac{|leaf(G)|/|root(G)|}{|V|-1}. \end{aligned}$$

The ratio falls within the range $(0,1]$. A shape with ratio close to 1 indicates that it is closest to star topology. All star-shaped cascades exhibit the same ratio value of 1. For example, consider the frequent shapes in Fig. 6. The star-likeness ratio $(s(G))$ as well as the join probability $Pro_{top}(G)$ of these shapes are listed in Table 3. Observe that the three shapes whose star-likeness ratio are the most (i.e., $s(G)=1$) exhibit the least joining probability. This suggests that star-likeness ratio can contribute to the analysis of the cascade joining behavior.

Table 3 Star-likeness ratio of frequent shapes

5 Microscopic features

In this section, we first investigate four microscopic features of blog cascades that may play important role in cascade affinity prediction, namely number of quasi-friends, popularity of participants, citing factor, and initiator-media links. We conclude this section by conducting a one-way variance analysis (ANOVA) on these macroscopic and microscopic features to quantify their significance related to cascade affinity.

5.1 Number of quasi-friends

We introduce the notion of quasi-friend to model friendship within blogosphere based on post citings. Unlike other social media platforms (i.e. Facebook, Youtube, Twitter) where users can set up friendship links or select users to listen to, there is no explicit definition on friendship in blogosphere. Thus, we need to find a way to define quasi-friendship in blogosphere. Similar way has been adopted by other researchers (Guice 1995). Formally, quasi-friend is defined as follows.

Definition 4

Given two blogs $b_1$ and $b_2,\, b_1$ is a quasi-friend of $b_2$ if and only if $b_2$ cites $b_1$’s posts more than $\mathcal{K }$ times.

A quasi-friend indicates that $b_2$ probably often reads $b_1$’s blog. This probability of frequent reading is controlled by the friendship threshold $\mathcal{K }$. Obviously, $\mathcal{K }$ will affect the number of quasi-friends discovered. As shown in Fig. 7b, $\mathcal{K }$ affects the number of quasi-friends exponentially with exponent $\alpha =-3.37$. Notice that if we set $\mathcal{K }$ to a large value then we may extract a very limited number of quasi-friends for a blog. Hence, we set $\mathcal{K }$ to 2 by default. We shall justify this value empirically in Sect. 7.2. Note that quasi-friendship is directed. That is, $b_2$ is not a quasi-friend of $b_1$ unless $b_1$ has cited $b_2$ more than $\mathcal{K }$ times. Given a value of $\mathcal{K }$, we denote the set of quasi-friends of a blog $b_j$ as $F_j=\{f_1,f_2,\ldots ,f_r\}$, where each element $f_r$ is a blog.

Several recent papers have shown that personal behavior in a social network is highly affected by the person’s neighbors (Backstrom et al. 2006; Cha et al. 2009; Leskovec et al. 2007a). Hence, the number of quasi-friends a blogger may have in a cascade is an important feature that may influence her decision to join the cascade. Naïvely, the number of quasi-friends a blogger has in a cascade can be computed at any time after she has joined the cascade. However, this may mislead us from the actual phenomenon as the number of quasi-friends is highly influenced by the temporal state of the cascade. Let us elaborate on this further. Consider the Fig. 8a. Each node is a blog and the dashed rectangle denotes a cascade at a particular time. Edges represent hyperlinks related to this cascade. Assume that a blog $d$ joined it at time $T_0$. Note that at time $T_0,\, d$ did not have any quasi-friend in that cascade. We refer to $T_0$ as joining time. Now assume that at time $T_0+{\varDelta }T$ node $h$ became a quasi-friend of $d$ as shown in Fig. 8b. We refer to this time when a friendship is created as friendship creation time. Observe that the number of quasi-friends $d$ had during joining time and friendship creation time may be different. However, if we discard these two different phenomenons, then at any time after $T_0+{\varDelta } T$ it may seem that $d$ had a quasi-friend $h$ in this community when she joined it (Fig. 8c). Obviously, this is not an accurate reflection of the ground truth. Note that existing work ignore these two types of temporal features while modeling number of quasi-friends in a social network.

Fig. 8

Effect of friendship creation time

There is another problem if we ignore the above temporal behavior. Consider the Fig. 9a, which represents the same scenario as depicted in Fig. 8a. Now assume that another blog $i$, who is a quasi-friend of $d$, joined this community at time $T_0+{\varDelta } T$ as shown in Fig. 9b. If we do not distinguish between times $T_0$ and $T_0+{\varDelta } T$, then it may seem that $d$ had a quasi-friend $i$ in the cascade when she joined it (Fig. 9c). However, the truth is that when $d$ joined this cascade at time $T_0$, she did not have any quasi-friend. Hence in our approach, we distinguish between the joining time and the friendship creation time to accurately reflect the ground truth. As we shall see in Sect. 7.2, this distinction improves the cascade affinity prediction performance significantly.

Fig. 9

Effect of friendship creation time (contd.)

In our approach, we represent the set of blogs having $\alpha$ quasi-friends in a cascade $c^i$ using $\Gamma ^i(\alpha )$ taking into consideration the time $t^i(j)$. It is computed as follows.

$$\begin{aligned} \Gamma ^i(\alpha )=\{b_j\big ||F_j{(t^i(j))}\bigcap \phi ^i(t^i(j))|=\alpha \} \end{aligned}$$

$F_j{(t^i(j)})$ denotes the set of blogs that became a quasi-friend of $j$’s before time $t^i(j),\, \phi ^i(t^i(j))$ is the set of blogs that appeared in $c^i$ before time $t^i(j)$. Note that by incorporating $t^i(j)$ in our approach, we make a contribution to address the above issues (Figs. 8c, 9c). Based on $\Gamma ^i(\alpha )$, we define the notion of cascade affinity ratio with respect to the number of quasi-friends.

Definition 5

Given the set of $\Gamma ^i(\alpha )$, the cascade affinity ratio, denoted as $P_\alpha$, is defined as:

$$\begin{aligned} P_\alpha = \frac{\sum \limits _i{|\Gamma ^i(\alpha )}\bigcap \phi ^i|}{\sum \limits _i{|\Gamma ^i(\alpha )}|} \end{aligned}$$

We computed $P_\alpha$ for the whole collection of cascades, and plotted the values in Fig. 10a. The $X$-axis is the $\alpha$ value (number of quasi-friends in a cascade). The $Y$-axis represents the values of $P_\alpha$. From the figure, we observe a diminishing return phenomenon. That is, beyond a number each additional quasi-friends in the cascade will contribute less to the probability of joining that cascade. This number is around $7$ in this figure. Note that the curve showed in the figure follows similar trend as found in other social networks in  (Backstrom et al. 2006), and also in (Leskovec et al. 2007a) where the author found a saturation point in the probability of buying a DVD by the number of recommendations received.

Fig. 10

a Cascade affinity ratio versus number of quasi-friends. b Joining probability by cascade rank

5.2 Popularity of participants

Next we study the effect of popularity of cascade participants on the cascade affinity of a blogger. The idea is similar to the preferential attachment model which was first proposed in (Barabasi and Albert 1999). In this model, whenever a new vertex arrives in a network it attaches an edge to an existing vertex with a probability proportional to that of the old vertex’s degree. Newman performed a series of analysis on the model in  (Newman 2003). Leskovec et al. (2008) also showed a similar pattern in some real-world data sets. Here we conduct an analysis based on this model. However, in our study when a blog joins a cascade we consider the model at the cascade-level whereas the above approaches consider it at the node level. Then, the popularity of a cascade $c^i$ is the highest rank of the blogs in the cascade. Formally, it is defined as follows.

Definition 6

Let $D(b)$ be the rank of a blog $b$. Then the popularity rank of a cascade $c^i$ that $b_j$ wants to join, denoted as $D_j(c^i)$, is defined as:

$$\begin{aligned} D_j(c^i)=\min \limits _{b\in \phi ^i(t^i(j))}(D(b)) \end{aligned}$$

Note that the rank of each blog is based on its in-degree (indexed by Technorati). A blog having the largest in-degree has the highest rank as 1. Observe that the above definition can be intuitively explained from the social aspect. When a blogger $b_j$ reads a post $p_r$ she can also see other posts in the same cascade by tracing back the hyperlinks. If there is a popular blog which has a large in-degree in that cascade, then $b_j$ will probably join this cascade. Interestingly, this effect is not so obvious in our result shown in Fig. 10b. The $X$-axis in the figure is the popularity rank of cascades. A cascade having lower rank means it contains a more popular blog. We plot the numbers of blogs that join a cascade (“positive count”) and those who do not (“negative count”) by varying the ranks. The curve labeled “probability” represents the ratio: $\frac{\text{ positive } \text{ count }}{\text{ positive } \text{ count }+\text{ negative } \text{ count }}$. As shown in the figure although the values along $X$-axis is in log-scale, the number of joined blogs in each bin do not vary much. This phenomenon indicates that a minority of cascades which have high popularity ranks influence a large number of bloggers to join.

5.3 Citing factor

The features discussed above are all related to the cascade that a blog is inclined to join. Here we analyze a personal characteristics related to the joining behavior of each blogger. The reason for analyzing this feature is based on the hypothesis that a blogger $b_j$ is more inclined to join a cascade if $b_j$ likes to cite others’ posts.

Definition 7

Let $out(\cdot )$ be the number of outlinks of $\cdot$. Then the citing factor of a blogger $b_j$, denoted as $H_j(c^i)$, is defined as:

$$\begin{aligned} H_j(c^i)=|out(post_j(t^i(j)))| \end{aligned}$$

We can compute the probability for a blog $b_j$ with $p$ citations to join a cascade as follows.

$$\begin{aligned} Pro_{cf}(p)&= \frac{\sum _{c^i}|\{b_j|H_j(c^i)=p,b_j\in \phi ^i\}|}{\sum _{c^i}|\{b_j|H_j(c^i)=p\}|} \end{aligned}$$

The result is shown in Fig. 11a. It is distributed almost uniformly with the change to the number of out-links. It is evident that this feature is not very informative as far as cascade affinity is concerned.

Fig. 11

a $Pro_{cf}(p)$ versus number of citations. b Joining probability by initiator rank

5.4 Initiator-media link

Lastly, we investigate a feature that is associated with the initiator(s) of a cascade. Recall that the initiator of a cascade is the first blogger who initiates discussion in the cascade. We conduct a series of analysis involving the cascade initiators in order to study whether there is any correlation between the cascade affinity and the cascade initiators. We first formally define cascade initiator.

Definition 8

Let $c^i(B,E)$ be a cascade containing blogs $B=\{b_1,b_2,\ldots ,b_s\}$ with post–post links $E$. The initiator $ini(c^i)$ of the cascade $c^i$ is defined as follows

$$\begin{aligned} ini(c^i) = \{b_j|\not \exists b_{i}\in B, \overrightarrow{b_{j}b_{i}} \in E\}. \end{aligned}$$

Notice that it is possible to have more than one initiator in a cascade. For example, consider the cascade in Fig. 4a. Both $p_1$ and $p_2$ are initiators according to the above definition.

We now investigate two interesting properties of initiators of blog cascades using the Technorati dataset. We extract all the initiators for each cascade. Firstly, we investigated the correlation between the probability of a blog to join the cascade $c^i$ and its initiators’ popularity. Similar to Sect. 5.2, the popularity of initiators of a cascade $c^i$ is the highest rank of the initiators in that cascade: $\min \limits _{b\in {ini(c^i)}}(D(b))$. We plot the probability of a blog to join a cascade by varying the initiators’ popularity in Fig. 11b. Observe that there does not exist any correlation between these two factors.

Secondly, we examine a property of the initiators related to their out-links to other media resources. Initiators in a cascade are the posts that do not reference any other post in that cascade. However, they may reference non-blog media sources such as Flickr, Youtube, etc. We refer to these links as initiator-media links. Formally, it is defined as follows.

Definition 9

Let $c^i$ denote the cascade. Then, the initiator-media link of $c^i$, denoted as $I(c^i)$, is defined as:

$$\begin{aligned} I(c^i)=\left\{ \begin{array}{ll}1 &{} \text{ if } \exists b\in {ini(c^i)} ,\,\,b \text{ hyperlinks } \text{ to } \text{ non-blog } \text{ URLs } \\ 0 &{} \text{ otherwise. }\end{array}\right. \end{aligned}$$

Table 4 reports the statistics of initiators that link to these media resources against those which do not. Observe that the average size of cascades containing initiator-media links is larger than those which do not have such link. It means that the initiators who reference other media resources are probable to generate larger cascades than the ones that do not. This phenomenon suggests that bloggers are more inclined to write post on a topic when they have found related resources from many different media.

Table 4 Initiator data

Thus, it indicates that bloggers are more probable to join the cascades whose initiators referenced other media resources. Hence, we propose to use this property as another feature to predict the cascade affinity. For each cascade, we first identify the initiators within it. After that, we test whether the initiators have hyperlinks to web pages that belong to non-blog domains.

5.5 ANOVA Test

In this section, we conduct a one-way variance analysis (ANOVA) on each of the above macroscopic and microscopic features to quantify their significance related to cascade affinity. For each feature, we compare the values between blogs which finally joined a cascade and those did not using the one-way analysis of variance (ANOVA) to test whether the difference is really caused by the feature values or just by noise in the data. The F and p-values of each feature is shown in Table 5. The result shows that the p-value for citation factor is $0.968$ while other features are all less than $0.05$. It indicates that the different values of citation factor in both groups should only be considered as noise. The remaining six cascade features are all significant for predicting cascade affinity of a blogger.

Table 5 ANOVA test on cascade features

6 Cascade affinity prediction

In this section, we describe how the features discussed in previous sections can be exploited to predict bloggers who may join a cascade. The prediction involves two steps, namely candidate blog extraction and cascade joining prediction. We elaborate on these steps in turn.

6.1 Candidate blog extraction

For a given cascade, all blogs in the blogosphere are potential blogs that may join the cascade in the future. Nevertheless, many of these potential blogs have no interaction (e.g., read the posts) with the blogs/posts already in the cascade and are unlikely to join the cascade. We therefore only consider a much smaller set of candidate blogs that are likely to read one or more posts in the cascade. The candidate blogs are those that have at least one quasi-friend in the given cascade. Formally, for a given cascade $c^i$, the candidate blogs $c$ and $(c^i)$ that may join $c^i$ is given by the following equation.

$$\begin{aligned} cand(c^i)=\{j\big |F_j\cap \phi ^i\ne \emptyset \} \end{aligned}$$

(2)

The algorithm for extracting candidate blogs is outlined in Algorithm 2. Recall that quasi-friend is defined based on the number of times (i.e., $\mathcal{K }$) a blog cites posts from another blog. Hence, the number of candidate blogs extracted for a given cascade naturally depends on the threshold $\mathcal{K }$. In our experiments, we set $\mathcal{K }=2$ by default.

For all cascades in our dataset, there are $312,414$ candidate blogs extracted by Algorithm 2. On average, $43$ candidates are extracted for each cascade. Naturally, the number of candidate blogs increases along the number of participants in a cascade. Particularly, for a cascade having fewer than 10 participants, there are $39$ candidate blogs on average; for a cascade having 11–20 participants, this value increases to $64$ candidates on average; for a cascade having more than $20$ participants, there are $81$ candidates on average. From the numbers reported, candidate blog extraction greatly reduces the number of blogs to be considered in the prediction with respect to the total number of blogs in our data set. As an evaluation of candidate extraction, Table 6 shows 76.1 % blogs that join a cascade have at least a quasi-friend in it when we set $\mathcal{K }=2$.

Table 6 Effect of different values of $\mathcal{K }$

6.2 SVM-based cascade affinity prediction

We now present two techniques that exploits the macroscopic and microscopic features to predict blogs that may join a cascade. One takes a non-probabilistic approach whereas the other is probabilistic in nature. In this subsection, we present the former approach first. We discuss the probabilistic approach in the next subsection. In Sect. 7, we shall empirically compare these two strategies.

The prediction task can be naturally formulated as a binary classification task. Many existing classifiers (e.g., Naïve Bayes, $k$-Nearest Neighbors, and Support Vector Machines) indeed return a category relevance score for each data instance to be classified indicating its likelihood of belonging to a pre-defined category. We adopted a non-probabilistic binary classifier SVMs (Chang and Lin 2001) due to its promising results reported in many data mining/machine learning tasks. SVM models all the samples including both positive and negative ones as points in high dimensional space. The training of SVM learns a hyperplane in the space. The hyperplane learned from SVM model should be able to separate the positive training examples from the negative ones with the largest margin.

Formally, the training data in this paper is represented as a set of points in a 7-dimensional space: $\mathcal D =\{(\mathbf x _i,y_i)|\mathbf x _i\in \mathbb{R ^7},y_i\in {\{-1,1\}}\}$. $y_i$ is either $1$ or $-1$, indicating the class to which the point belongs. $\mathbf x _i$ is a 7-dimensional vector. Our target is to find a hyperplane that divides the points having $y_i=1$ from those having $y_i=-1$ with the maximal margin. Actually, any hyperplane can be written as the set of points $\mathbf x$ that satisfies the following equation:

$$\begin{aligned} \mathbf w \cdot \mathbf x -b=0 \end{aligned}$$

where $\mathbf w$ is normal vector and perpendicular to the hyperplane. The vector $\mathbf w$ and a parameter $b$ to be learned from the training data by minimizing the function:

$$\begin{aligned} \mathbf w ^\top \cdot \mathbf w \end{aligned}$$

subject to

$$\begin{aligned} y_i(\mathbf w \cdot \mathbf{x _i}-b)\ge {1}. \end{aligned}$$

In order to learn an SVM classifier, those candidate blogs that eventually joined and did not join the target cascades were used as positive and negative examples, respectively. Moreover, all the candidates extracted using Algorithm 2 are formulated as vectors in order to fit in the model. For example, a candidate sample $b_j$ is represented as 7-dimensional vector:

$$\begin{aligned} \overrightarrow{b_j}=[b_{j1},b_{j2},\ldots ,b_{j7}]^\top \end{aligned}$$

Within the vector, each entry is the value of one of the seven macroscopic and microscopic features discussed in Sects. 4 and 5.

Given the learned model, we compute a score for an unlabeled object $b_j$ using its decision function:

$$\begin{aligned} f(b_j) = \mathbf w \cdot {b_j} - b. \end{aligned}$$

In our setting, a larger $f(b_j)$ indicates more likelihood of $b_j$ joining the target cascade.

6.3 Bipartite Markov Random Field-based (BiMRF) cascade affinity prediction

Other than SVM, we then adopted a probabilistic approach to predict the likelihood of joining a cascade which is based on Bipartite Markov Random Field (BiMRF). BiMRF (Shi et al. 2009) models the group joining behavior as a bipartite graph where the vertices at one side of the graph are associated with the variables $B=\{b_i\}_{i=1}^N$ which represent users, and the vertices at the other side of the graph are associated with variables $C=\{c^j\}_{j=1}^M$ which represent cascades. The advantage of using BiMRF model in this problem is that it can explicitly incorporate the relationship between bloggers and cascades. Moreover, it has been proved to be more effective than other approaches in modeling the group joining behavior (Shi et al. 2009). Similar to SVM-based approach, based on the observed value of each feature, we study the joining behavior at different time step using BiMRF model. In the model, each user is a $7$-dimensional feature vector $b_i=[b_{i1},b_{i2},\ldots ,b_{i7}]^\top$, the values of which may change over time. Let $O$ denote all the observations, including users and their features, cascades and their features as well as the connections of users. Let $E=\{E_{ij}^t:1\le i\le N, 1\le j\le M$ and $1\le t\le T\}$ be a set of random variables where $e_{ij}=1$ if the user $b_i$ joins cascade $c^j$ at time $t$; otherwise it is 0. Let $\{e\}$ denote an instance of $E$. Then given the observations, BiMRF defines a conditional distribution as follows:

$$\begin{aligned} p(\{e_t\}|O)=\frac{1}{Z(w)}\exp (\sum \limits _{k=1}^K{\omega _{k}f_{k}(\{e_t\},O)}) \end{aligned}$$

where $f_k$ are feature functions and $\omega _k$ are their weights which will be learned. Given the observed features, $p(\{e\}|O)=\prod _{t=1}^{T}p(\{e\}|O)$.

Formally, the dataset is a pairing of observations and joining behaviors (i.e., $\mathcal D =\{\langle \{e\},O\rangle \}$). The best model to fit the data is the one with the maximum conditional likelihood: $\mathcal L =\log {p(\{e\}|O)}$. We define feature functions to compute the feature values $f_{k}(\{e\},O)$ in the above equation. For example, the feature function to compute the feature value of number of participants is as follows.

$$\begin{aligned} f_{nop}(e_{ij}^{t}=1,c^i,b_j,t)=N_j(c^i). \end{aligned}$$

Thus, the optimized $\omega _k$ is learned by maximizing the likelihood:

$$\begin{aligned} \mathcal L =\log {p(\{e\}|O)}. \end{aligned}$$

In line with (Shi et al. 2009), the optimization is achieved using L-BFGS (Limited Memory Broyden-Fletcher-Goldfarb-Shanno) algorithm (Liu and Nocedal 1989).

With the weights $\omega _k$ learned from the training data, the probability that user $b_j$ joins cascade $c^i$ at time $t$ is given by computing the marginal probability

$$\begin{aligned} p(e^t_{ij}=1|O). \end{aligned}$$

7 Experiments

7.1 Experimental setting

For both prediction models, we conducted experiments on our data set using 5-fold cross validation to evaluate the effectiveness of the features in predicting cascade affinity of candidate blogs. That is, the data set was randomly partitioned into 5 parts and in each evaluation, 4 parts were used as training data and the remaining part was used as test data. The results reported are averaged over the 5 runs.

The commonly used performance evaluation measures in classification tasks are precision, recall and $F_1$. Precision, denoted by $Pr$, is the percentage of blogs that eventually joined the target cascade among all blogs predicted to be joining. Recall, denoted by $Re$, is the percentage of the correct predictions among all blogs that eventually joined the target cascade. Note that, recall is computed with respect to all blogs that finally joined the target cascade regardless of whether the blogs are identified as candidate blogs or otherwise. $F_1=\frac{2\times Pr \times Re}{Pr+Re}$ is the harmonic mean of precision and recall. However, both precision and recall are threshold-dependent. A higher threshold leads to higher precision but lower recall. In our experiments, we are more interested in the effectiveness of the features in ranking the candidate blogs according to the likelihood of joining the target cascade. We therefore adopted the area under Precision-Recall curve (AUC-PR) as the evaluation metric.

7.2 Experimental results

7.2.1 Justification of candidate set

Recall that the number of candidate blogs is affected by the parameter $\mathcal{K }$. As $\mathcal{K }$ increases, the number of quasi-friends identified decreases. Consequently, the candidate blog set shrinks. As a result, the maximum recall decreases, but the prediction performance may not. To determine the optimum value for $\mathcal{K }$, we conducted the prediction using different values of $\mathcal{K }$. Table 6 shows the sizes of candidate blog sets for different $\mathcal{K }$ as well as the highest $F_1$-measures achieved by selecting the best thresholds in both models. Observe that in both models the best $F_1$-measures are achieved at $\mathcal{K }=2$. Hence, in the subsequent experiments we shall set $\mathcal{K }=2$.

7.2.2 Comparison of feature sets

Firstly, we compare the prediction performance using either macro features or micro features in order to find which groups of features are more important in affinity prediction.

Secondly, recall that we have identified seven features for cascade affinity prediction, namely number of quasi-friends, popularity of participants, number of participants, citing factor, elapsed time, star-likeness ratio and initiator-media link. To evaluate the effectiveness of these features, we conducted 8 sets of experiments. The first set of experiments used all 7 features for prediction. This feature set is denoted by “ALL” in Table 7. In each of the following seven experiments, one feature is removed. For instance, “A-NF” denotes that the feature number of quasi-friends is removed and the remaining four features were used for prediction. In Table 7, a ‘$\checkmark$’ indicates that the feature is used and ‘-’ otherwise.

Table 7 Feature set notations and prediction performance in AUC-PR

The prediction performances measured by AUC-PR are reported in the last two rows in Table 7. Using all the seven features, the prediction achieved AUC-PR of $0.615$ for SVM and $0.610$ for BiMRF. We can make the following observations:

• Removal of number of quasi-friends resulted in significant drop in prediction performance to $0.046$ in both models indicating that number of quasi-friends is the most important factor that affects a blogger’s cascade affinity. We validated this by performing another experiment using only the number of quasi-friends as feature. The AUC-PR in this case is $0.445$ for SVM and $0.434$ for BiMRF.

• Removal of either popularity of participants, number of participates, star-likeness ratio, elapsed time or initiator-media link led to a small performance degradation. These five features indeed contributed to the cascade affinity modeling.

• An interesting observation is that removal of the citing factor in both models led to better AUC-PR than using all the seven features. This result clearly indicates that the citing factor introduced noise in the prediction, which is consistent with our ANOVA test results reported in Sect. 5.5. The remaining six features: number of quasi-friends, popularity of participants, number of participates, elapsed time, star-likeness ratio and initiator-media link achieved the best performance.

• We conducted additional two experiments using only macroscopic or microscopic features. Microscopic features achieved AUC-PR of 0.533 for SVM ($0.519$ for BiMRF) whereas macroscopic ones only exhibit AUC-PR of 0.064 for SVM ($0.059$ for BiMRF). This is because the number of quasi-friends is one of the microscopic features.

For the completeness of the results, Figure 12 plots the Precision-Recall curves of using eight different feature sets. Under SVM model, all the seven runs (except for “A-NF”) achieved almost perfect precision before recall reached $0.57$. Sharp drop of precision is then observed along with the increase of recall. In contrast, all the seven runs of BiMRF model (except for “A-NF”) achieved almost perfect precision before recall reached $0.48$. As the recall increases, precision in BiMRF drops down more smoothly than that of SVM model. However, SVM and BiMRF show almost the same AUC-PR. Thus, both models can be applied in cascade affinity prediction.

Fig. 12

Precision-Recall Curves for different features

Figure 13 shows the precision, recall and $F_1$-measure by varying the threshold for the feature set “A-CF”, which has the best prediction performance among all the approaches. Both precision and recall of BiMRF model change more smoothly than those of SVM model as the thresholds increase, indicating that SVM model results in a clearer margin between the score of positive samples and negative ones.

Fig. 13

Precision, Recall and $F1$-measure for “A-CF”

7.2.3 Significance of time for modeling number of quasi-friends

Recall that in Sect. 5.1, we illustrated the significance of time in modeling the number of quasi-friends. To justify the goodness of our solution, we compared it with the approach that ignores time. Specifically, if we discard temporal issue in modeling quasi-friends then the definition of $\Gamma ^i(\alpha )$ (the set of blogs having $\alpha$ quasi-friends in cascade $c^i$) is modified as follows.

$$\begin{aligned} \Gamma ^i(\alpha )=\{b_j\big ||F_j{(T^*)}\bigcap \phi ^i|=\alpha \} \end{aligned}$$

We updated the number of quasi-friends feature in each candidate vector using the above formula. Using the updated feature vectors, we performed the prediction again. The performance of ignoring the time in quasi-friend identification shows a small AUC-PR 0.211 (0.216 for BiMRF) whereas our proposed solution achieves 0.615 (0.610 for BiMRF). The comparison between the Precision-Recall curve of this approach and our proposed solution is shown in Fig. 14. Both of the curves use all the seven features. “FRTM” represents the approach that discards the temporal aspects in number of quasi-friends. It is clear that time is an important factor in quasi-friend identification as it achieved significantly better prediction compared to the approach that ignores time. In fact, whenever K is set to 1, 2, or 3, the experiment result indicates that distinguishing joining time and friend creation time is important, we select not to show the detailed statistics on K=1 and 3 to avoid making the paper to lengthy and tedious.

Fig. 14

Significance of time for modeling number of quasi-friends

7.2.4 Prediction of top-k bloggers

To study the prediction of accuracy of top-$k$ blogs that are inclined to join a cascade, we computed the precision of our approach to retrieve top-$k$ bloggers ranked based on the predicted scores. Specifically, for each cascade $c^i$ having more than $k$ positive samples, we generate the top-$k$ predicted blogs and compute the precision as follows: $Pr^i(k)=\frac{\#\text{ true } \text{ positive }}{k}$. Then for a given $k$, we compute the average precision, denoted as $Pr_{avg}(k)$, using the following formula.

$$\begin{aligned} Pr_{avg}(k)=\frac{\sum _{|\phi ^i|\ge k}{Pr^i(k)}}{\big |\{{c^i\big ||\phi ^i|\ge k}\}\big |} \end{aligned}$$

Table 8 shows average precision values for different $k$ values highlighting the goodness of our approach. Note that $Pr_{avg}(k)$ may not monotonically decrease with increasing $k$ as the number of cascades in the denominator depends on $k$.

Figures 15 and 16 show the top-10 candidate blogs over entire cascades collection. If the candidate is a positive sample, we also showed the corresponding URL of the post that joins the target cascade.

Table 8 Average precision versus top-$k$

Fig. 15

Top 10 candidates that are most probable to join a cascade (SVM)

Fig. 16

Top 10 candidates that are most probable to join a cascade (BiMRF)

7.2.5 Using only the number of quasi-friends as feature

As mentioned above, we found that number of quasi-friends is the most important factor that affects a blogger’s cascade affinity. To further validate this, we conducted another experiment using only the number of quasi-friends as feature. The AUC-PR in this case is $0.445$ for SVM and $0.434$ for BiMRF.

In fact, just as we did in Figs. 15 and 16, if we list the top-10 candidate blogs over the entire collection for the approach of using only the number of quasi-friends in SVM or BiMRF, we can also get 10 positive samples which eventually join the target cascades. However, when we compute the average precision values for for different $k$ values as what we did in Table 8, the approach of using only the number of quasi-friends shows poor result which is shown in Table 9. It indicates that using only the number of quasi-friends does not perform so well as the other approaches combining several informational features. It can only found very limited number of candidate blogs which are most possible to join target cascades over the entire collection but fails to give enough precision in order to predict the affinity in each individual cascade.

Table 9 Average precision versus top-$k$ using only number of quasi-friends

7.2.6 Comparison of the performance between SVM and BiMRF model

Referring to the prediction results shown in Table 7 and Fig. 13, we compare the performance between the SVM and BiMRF models. The observations are as follows.

• In both models, removal of the citing factor perform the best, then is the approach using all the features. However, the removal of either elapsed time or initiator-media link does not perform better than the removal of popularity of participants in BiMRF, which is not the case in SVM.

• In general, both SVM and BiMRF models show satisfactory prediction results with AUC-PR over 0.61 when using the best feature set “A-CF”. However, SVM performs slightly better than BiMRF model for all the other feature sets.

• The precision and recall curves by varying the threshold of both SVM and BiMRF model almost exhibit the same shape, except that the curves of BiMRF model is smoother than those of SVM. It suggests that SVM model tends to produce clearer margin between positive class and negative one.

• Both svm and BiMRF models exhibit the best average precision with 0.970 when predicting the top-1 blogger according to Table 8. In addition, the average precision of top-$k$ ($k=2,\ldots ,5$) of both methods do not vary much. Moreover, 8 out of 10 records in Fig. 16 also appear in Fig. 15, indicating that the results of both models do not vary much.

8 Conclusions and future work

In this paper, we analyzed a large publicly available collections of blog information, to investigate bloggers’ behavior and interaction with blog cascades. We have identified in total seven macroscopic and microscopic features, namely number of quasi-friends, popularity of participants, number of participants, time elapsed since the genesis of the cascade, star-likeness ratio, initiator-media link and citing factor of the blog, that may play important role in predicting blog cascade affinity so as to identify most easily influenced bloggers. Such bloggers play important role in several real-world applications such as viral marketing. Note that our proposed features are derived from structural information of the cascades without any content analysis of posts/blogs. We performed anova test on these features and showed that all of them, except citation factor, have significant impact on cascade affinity. The cascade affinity prediction is then formulated as a classification task and a non-probabilistic (svm-based) and probabilistic (BiMRF classifier) methods are employed. Using the prediction scores from the svm-based approach or the conditional probability from BiMRF, the candidate blogs can be ranked according to their probability of joining a cascade. We have evaluated different combinations of the features and our results on cascade affinity prediction is consistent with the anova test. In general, svm model performs slightly better than BiMRF model in most of the feature sets. Moreover, svm model tends to generate clearer margin between positive class and negative one. The six features that have significant impact on cascade affinity achieved the best prediction accuracy of 0.625 (0.618 for BiMRF) measured by auc-pr. Our experimental results also showed that the number of quasi-friends plays a significant role in blog cascade affinity prediction. The features proposed in this paper may not be independent to each other (i.e., number of participants and number of quasi-friends). As part of future work, we intend to investigate the correlation between different features and how they influence the cascade affinity.

On the other hand, we intend to study micro-blogging behavior using the model and results proposed in this paper. However, the model we proposed in this paper may not be applied directly in micro-blogging scenario, as there exist several differences, which are shown as follows, between retweeting in micro-blog and blog cascade. Firstly, instead of web pages, all micro-blog posts are kept in the server by some operator (i.e., Twitter). An arbitrary blogger can view any blog posts that is published on the Web. However, an arbitrary micro-blogger can only see the content posted by those whom he is listening to. Secondly, micro-blog posts are platform dependent in that a twitter user can never re-tweet a post that is published at another platform such as Tumblr. Thirdly, there is only one initiator in microblogging which is different with blog cascade. Lastly but not the least, collecting data for blog posts study can be done by crawling the static web pages. However, we can only get tweets by querying the database of Twitter through the API provided by Twitter. Twitter API can only provide the tweets that appeared within 2 weeks’ time. Thus, in order to apply out model towards micro-blog domain, definition of some features may be changed, such as quasi-friends, elapsed time, star-likeness ratio, popularity of participants. Moreover, some other features which are specific in micro-blog need to be taken into consider. In fact, this is what we intend to do in future work.