Detection of strong attractors in social media networks
 Ziyaad Qasem^{1}Email author,
 Marc Jansen^{1},
 Tobias Hecking^{2} and
 H. Ulrich Hoppe^{2}
Received: 9 February 2016
Accepted: 23 November 2016
Published: 7 December 2016
Abstract
Background
Detection of influential actors in social media such as Twitter or Facebook plays an important role for improving the quality and efficiency of work and services in many fields such as education and marketing.
Methods
The work described here aims to introduce a new approach that characterizes the influence of actors by the strength of attracting new active members into a networked community. We present a model of influence of an actor that is based on the attractiveness of the actor in terms of the number of other new actors with which he or she has established relations over time.
Results
We have used this concept and measure of influence to determine optimal seeds in a simulation of influence maximization using two empirically collected social networks for the underlying graphs.
Conclusions
Our empirical results on the datasets demonstrate that our measure stands out as a useful measure to define the attractors comparing to the other influence measures.
Keywords
Introduction
Social media have become an important information resource to gain insights into and acquire knowledge about a wide variety of more or less numerous communities interacting through the internet. Moreover, applying analytic techniques to social media data can support better informed decisionmaking processes in numerous fields, such as marketing, politics and education. One prominent aspect of such analytics is the characterization and detection of influential actors in social networks. There are several studies on social media which have suggested different approaches and specific measures to solve the problem of influential actor detection.
In this paper, we elaborate on a new approach for the detection of influential actors which is based on quantifying the contribution of this actor to increasing the size of the network by attracting new active members of the specific subcommunity [1]. In comparison to weighted or unweighted indegree measures, our new measure would only count those neighbors who were new to the network when the relationship to the actor in focus was first established. In other words, an actor who has a high value in terms of this measure has been an important "target” node for the attraction of new members to the network and this for increasing the overall size of the network. A formal specification of this property (referred to as "T measure”) is given in the first part of the paper.
Our approach can be applied to social networks in which timestamps are attached to edges connecting to actors. In the evaluation section of this paper, we apply our approach first to dataset from the Asterisk open source software developer community (a relatively small community with less than 1400 members and much less active actors) to test whether the influential actors who are already known from the Asterisk mailing list can be also identified using our approach. Second, we use a bigger dataset based on Twitter communication around #EndTaizSiege and #coup_suffocates_Taiz (related to recent events in Yemen). Here, we compare our approach with other standard measures such as indegree, and betweeness in terms of how good these measures are if used to generate seeds for an independent cascade diffusion process. The objective of studying our T measure in the field of information diffusion is to show that T measure is effective to define influential actors who are effective in attracting others to become active in a specific community.
The rest of the paper is organized as follows: "Literature review" section presents related research. An overview of our proposed approach is given in "Approach" section, which also provides the basic formal definitions. "Implementation" section introduces the concept, followed by the description of our datasets and the experimental results in "Experimental results" section. "Information diffusion and T measure" and "Simulation of attraction processes with timerespecting paths" sections deal with the performance of our approach in the influence maximization problem. Finally, conclusions are drawn and an outlook for further research is described in "Conclusion" section.
Literature review
In this section, we review studies of influence in social media such as Twitter and remind the concept of information diffusion and its relation with the type of influence on which our approach is based.
Influence in social media
In the field of social media analysis, there exists a large body of research on modeling and measuring influence and on detecting influential actors. Here, social networking platforms such as Twitter are of special interest. However, regarding the manifestation and identification there are still open questions. Researchers have studied influence in social media networks, and many approaches rank users according to their influence.
Leavitt et al. [2] employ four features to evaluate influence, which are replies, retweets, mentions, and number of followers. They support statistical results related to these measures, but do not present a global influence measure based on all the suggested criteria. In the work of Cha et al. [3], it could be shown that employing different measures can lead to completely different results when it comes to the task of ranking users according to their importance. Results were presented based on Twitter data and three different measures of influence, namely indegree (number of followers of an actor), retweets (number of retweets containing one’s actor name), and mentions (number of mentions containing one’s actor name). They presented an indepth comparison of these measures with the conclusion that different measures can be used to identify different types of influential actors. Indegree tends to be highest for news sites and celebrities, and thus, is suited to model popularity. However, the number of followers (indegree) does necessarily go along with a high number of retweets or mentions. The number of retweets is highest for information aggregation services and the number of mentions for celebrities. Consequently, the way in which a network is extracted from social media content and the measure of influence should be considered carefully with respect to the roles and type of influence one aims to uncover. Azaza et al. [4] proposed a new influence assessment approach depending on belief theory to combine different types of influence markers on Twitter such as retweets, mentions, and replies. They used Twitter dataset of European Election 2014 and deduced the top influential candidates. In our approach, we depend on the retweet relation as a marker to attract others to become active in a specific community in which a specific topic is dealt. As well as, a retweet relation can be understood as a form of information diffusion and as a means of participating in an event in social media [5].
Other researches propose to define influential actors based on link analysis. Twitter User Rank (TURank) [6] is an algorithm which utilizes ranking algorithms to define authoritative actors on Twitter, based on link analysis. TURank introduces actor–tweet graph where nodes are actors and tweets, and edges are follow and retweet relationships. TwitterRank [7] extended PageRank algorithm to measure influential actors in Twitter based on link structure and topical similarity.
Apart from the pure network information, influence can also be modeled additionally taking into account the actions of actors (e.g. on Flickr [8]), similarity of actors [9], and produced content associated with each actor [10].
Our work aims for a clear formulation of social influence and a methodology to produce an exact ranking of the actors according to the definition. In concrete, we provide a new type of influence in online social network to emphasize on those actors who attract many outsiders to join the own community in which a specific topic is dealt. For example, in Twitter those actors spawn many retweets on a certain topic from people who have no previous contributions on that topic. This new type of influence led us to propose a new approach to detect those actors, and compare the results with other standard measures.
Information diffusion
Influence is often related to information diffusion in a network. Information diffusion is the process by which a new idea or innovation spread over the networks by the means of connection among the social network actors [11]. Especially in social media, influential actors can control the diffusion of information through the network to some extent.
There are numerous research on the information diffusion over social network. For instance, Gruhl et al. [12] studied and modeled the dynamic of information diffusion on blogsspace environment. Yang et al. [13] proposed a model to capture the attribute of information diffusion which are related to speed, scale, and range. With spreading of information diffusion models and their variations, Vallet et al. [14] used graph rewriting to compare the different information diffusion models.
Widely used information diffusion models are the independent cascade (IC) [15, 16] and the linear threshold (LT) [17]. The two models describe different aspects of influence diffusion. IC model focuses on influence among neighbors on social network, and LT model focuses on the threshold behavior in influence diffusion [18].
Kempe et al. [19] proposed to use the IC and LT models to solve the influence maximization problem which asks for a set of actors whose aggregated influence in the social network is maximized, whereas Pei et al. [20] provided strategies to search for spreaders based on the following of information flow rather than simulating the spreading dynamics (modeled_dependent results). The study of [19] was followed by several research on the same problem (e.g. [18, 21, 22]). Furthermore, the features of identifying spreaders measures using independent interaction and threshold models through empirical diffusion data from LiveJournal are discussed in [23]. Morone et al. [24] proposed to map the problem of influence maximization in complex networks onto optimal percolation using Collective Influence (CI) algorithm.
In this paper, we evaluated the performance of our measure T in the information diffusion maximization problem by selected sets of top actors based on T measure and other sets which are defined by other standard measures. The advantage of our measure is to consider a new type of influence which refers to actors who attract others to be active in a particular community. Thus, we use the IC model to evaluate the performance of our measure comparing with other standard measures.
Approach
Our approach is based on this premise: the more a certain actor (Actor a) attracts new actors, the more actor a is important to the social network. Thus, in this approach we tried to evaluate the attractiveness value of social media actor which leads us to detect the attractors.
In this section, we will provide some definitions for special terms that help to provide a profound methodology in presenting our approach. This approach is based mainly on the decomposition of data collected from a given social network according to the time period of collection. Let us refer to that period by the term Pperiod. For instance, if the Pperiod of a given social network is 30 days, the social network data collection took 30 days.
Definition 1
(Pperiod) Pperiod is a time duration of the data collection process from social networks.
In this paper, the social networks’ data are depicted by a graph representation. To distinguish this graph in any context, it is defined under the name Pgraph. Thus, we can say that our approach is based on the decomposition of the Pgraph into subgraphs depending on the Pperiod.
Definition 2

V is the set of all actors who joined the community during Pperiod.

E is the set of all connections that have been established between the actors V during Pperiod.
Decomposition of a Pgraph based on Pperiod requires decomposition of the Pperiod into slices of time so that every subgraph is related to a slice. In our approach, we refer to each slice as Pslice.
Definition 3
(Pslice) Pslice is a time slice of Pperiod.
EPslice values for Pperiod of 30 days
EPslice  Value 

\(s_1\)  6 
\(s_2\)  12 
\(s_3\)  18 
\(s_4\)  24 
\(s_5\)  30 
Definition 4
(EPslice) EPslice is a Pslice such that all Pslices are equidistant.
To facilitate the definition of subgraphs of this approach, we will define some terms related to actors according to Pslices.
Definition 5
(Pactors) Let \(s_1,s_2,\ldots s_n\) be the Pslices. For every i such that \(0 < i \le n\), the Pactors \(A_i\) is a set of all actors that joined the social network between 0 and \(s_i\).
Definition 6
(\(P_s\)actors) Let \(s_1,s_2,\ldots s_n\) be the Pslices. For every i such that \(0 < i \le n\), the \(P_s\)actors \(A_{s_i}\) are a set of all actors that joined the social network between the Pslices \(s_{i1}\) and \(s_i\).
Figure 1 shows how the Pactors and \(P_s\)actors are taken with respect to Pslice in our approach. The figure displays the Pactors \(A_3\) and \(P_s\)actors \(A_{s_3}\) as an example. \(A_3\) joined the social network between Pslices \(s_0\) and \(s_3\) whereas \(A_{s_3}\) joined between Pslices \(s_2\) and \(s_3\).
After discussing the terms mentioned above, now it is easy to provide the definitions for the different types of subgraphs which will be used in this approach with. These definitions will be helpful on our way to reach the goal of this approach.
Definition 7

\(A_i\) is the Pactors \(A_i\).

\(E_i= \{(a,b) : a,b\in A_i\}\)
By this, we focus on the connections by which the actors attracted the new actors; hence, we can easily measure the actors’ attractiveness. The next definition will discuss this issue in formal way.
Definition 8

\(A_i\) is the Pactors \(A_i\).

\(E_{si}= \{(a,b) : a\in A_{i1} \ {\rm and} \ b\in A_{s_i}\} \ \cap E_i\)
From Definition 8, we notice that Ssubgraph \(S_i\) contains the new connections by which the new actors \(A_{s_i}\) joined the network. The number of these connections refers to the attractiveness value of the actors \(A_{i1}\). Later in the implementation section, Definition 8 is used to facilitate the calculation of the attractiveness value T. Figure 2 shows the difference between Psubgraph and Ssubgraph in our approach where n is the number of Pslices and \(1<i\le n\). Psubgraph \(G_{i1}\) is the Psubgraph of the Pslice \(s_{i1}\), and Psubgraph \(G_{i}\) and Ssubgraph \(S_{i}\) are of the Pslice \(s_{i}\).
What if the Pgraph is a directed graph? The Psubgraph would be directed with the same properties of Psubgraph in Definition 7; however, the definition of the Ssubgraph would be slightly different.
Definition 9

\(A_i\) is the Pactors \(A_i\).

\(E_{s_i}= \{(a,b) : ( \ a\in A_{i1} \ and \ b\in A_{s_i} \ ) \ or \ ( \ b\in A_{i1} \ and \ a\in A_{s_i} \ ) \} \ \cap E_i\)
In Fig. 3, the directed Psubgraph and Ssubgraph are shown where n is the number of Pslices and \(1<i\le n\). The directed Psubgraph \(G_{i1}\) is the Psubgraph the Pslice \(s_{i1}\), and the directed Psubgraph \(G_{i}\) and Ssubgraph \(S_{i}\) are of the Pslice \(s_{i}\).
In the next section, we will introduce the implementation of our approach to evaluate the attractiveness value of each actor in online social media.
Implementation
According to the Pslices, the Pgraph in this approach is decomposed into n Psubgraphs \(G_1,G_2,\ldots G_n\) and \((n1)\) Ssubgraphs \(S_{2},S_{3},\ldots S_{n}\) where n is the number of Pslices. To evaluate the attractiveness value of each actor in each Psubgraph, we use the formula in next definition.
Definition 10
From Fig. 2, we notice that the attractiveness value of the actor \(a_1\) in Psubgraph \(G_{i1}\) is equal to 2/3 which is resulted from his/her degree in Ssubgraph \(S_i\) divided by number of \(A_{s_i}\).
Now, we provide the way by which the new measure of attractiveness can be evaluated. Let us call the new measure by T, and it is evaluated as follows:
Definition 11

\(A_{s_2}\) which is the set of the \(P_s\)actors E, F, G, H, I, J, and K.

Psubgraph \(G_2(A_2,E_2)\) where

\(A_2\) is the set of the Pactors A, B, C, D, E, F, G, H, I, J, and K.

\(E_2\) is the set of the connections (B, A), (B, C), (D, C), (E, C), (H, E), (G, C), (F, C), (I, C), and (K, I).


Ssubgraph \(S_2(A_2,E_{s_2})\) where

\(E_{S_2}\) is the set of the connections (E, C), (H, E), (G, C), (F, C), (I, C), and (K, I).
To calculate the attractiveness value of the actor C in the whole Pgraph G, we have to calculate


\(T(C_{G_1})\) which equals the indegree value of the actor C in the Ssubgraph \(S_2\). In this case, it equals 5. In normalized form, we evaluate also the number of \(P_s\)actors \(A_{s_2}\) which equals 7. Thus, \(T(C_{G_1})\) equals 5/7

\(T(C_{G_2})\) which equals the indegree value of the actor C in the Ssubgraph \(S_3\). In this case, it equals 3. In normalized form, \(T(C_{G_2})\) equals 3/6, where 6 is the number of \(P_s\)actors \(A_{s_3}\).
In this section, we will describe the type of our dataset, and the characteristic of each type. Furthermore, the experimental results on the different dataset will be discussed in this section.
Evaluation strategy
Our approach has been applied to three different datasets. First, we chose the open source software development project Asterisk. Here, the dataset originated from the communications in the developer mailing lists during 2006 and 2007. The Asterisk dataset contains 13,542 messages and 4694 threads that were discussed by 1324 developers. Two actors are linked if they participated in the same mailing thread. Figure 5 shows an example of an actor a participating once in the same mailing thread with actor b and having shared two mailing threads with actor c. According to our approach and the timestamps in Asterisk dataset, we decomposed the Pperiod into eight Pslices. According to Definitions 7 and 8, we got eight Psubgraphs and seven Ssubgraphs.
Second, we gathered a dataset from Twitter via Twitter API from December 31, 2015, to January 06, 2016. The collected dataset is the data of hashtag #EndTaizSiege (14,944 actors and 46,552 connections) that comprises a big connected component (containing 84% of actors), singletons (14%), and smaller components (2%). We worked with the biggest component because that our goal is to evaluate the attractiveness of actors; hence, we focus on the biggest component which is considered as a single interaction domain for actors [3]. Applying our approach leads to decompose Pgraph constructed from Twitter dataset into three Psubgraphs and two Ssubgraphs based on three Pslices.
The directed weighted Pgraph of our collected Twitter datasets is constructed based on retweet activities so that actor a gets incoming connection from actor b if actor b retweeted a tweet of actor a. The weight of connection refers to the number of retweets activity between two connected actors. Figure 6 shows an example where actor a retweeted three tweets of actor b whereas the actor c retweeted two tweets of the actor a.
As a matter of fact, the time slicing does not depend on a specific predefined strategy but it has been estimated in accordance to the size of dataset using an equal window size for each slice. For instance, Fig. 7 shows how the Pperiod with Twitter dataset #EndTaizSiege has been decomposed into equal window size so that we get a fair division of the retweet activities for each time slice. (In our ongoing work, we try to find a general overall strategy for the time period decomposition).
Experimental results
Asterisk
Top influential actors according to different influence measures over Asterisk dataset
Rank  T  Degree  Betweenness 

1  Kevin P. Fleming  Kevin P. Fleming  Kevin P. Fleming 
2  Tilghman Lesher  Olle E. Johansson  Olle E. Johansson 
3  Tzafrir Cohen  Tzafrir Cohen  Tilghman Lesher 
4  Russell Bryant  Tilghman Lesher  Tzafrir Cohen 
5  Olle E. Johansson  Russell Bryant  Russell Bryant 
6  Steven Critchfield  Steven Critchfield  Steven Critchfield 
7  Eric Wieling  Tony Mountifield  Jared Smith 
8  Jared Smith  Jared Smith  Tony Mountifield 
9  Steve Totaro  Eric Wieling  Steve Totaro 
10  Steve Murphy  Anton Vazir  Eric Wieling 
Spearman’s rank correlation coefficient over Asterisk dataset
T  Degree  Betweenness  Closenness  Eigenvalue  

T  –  0.643  0.6930  0.551  0.574 
Degree  –  –  0.869  0.864  0.910 
Betweenness  –  –  –  0.668  0.716 
Closeness  –  –  –  –  0.986 
Eigenvalue  –  –  –  –  – 

The rank correlation between indegree (retweets number) measure and number of followers is very low (\(\rho\) = 0.08). This goes along with the findings of [3]. Thus, we can state that the popularity of actors in terms of the number of followers is not an important factor that affects retweet activities in Twitter.

Furthermore, we found that the rank correlation between T and indegree (retweets number) measures is strong (\(\rho\) = 0.6) and consequently, the correlation with the number of followers is low. This is reasonable since the T measure incorporates the indegree. However, in contrast to the indegree the T measure emphasizes attraction of new actors by not counting relations to actors who are already active in the community. This explains that these two measures are not more strongly correlated.

Furthermore, we notice that the rank correlation between T and authority measures is high (\(\rho\) = 0.5) but not as high as the correlation between the authority measure and indegree, which leads to the conclusion that the T measure also detects influential actors as the tradtional measures, but puts different emphasis on the attractors.
Spearman’s rank correlation coefficient over Twitter dataset #EndTaizSiege
Followers  T  Indegree  Outdegree  Betweenness  Hub  Authoritiy  

Followers  –  0.1057  0.0805  0.0383  0.0871  0.0206  0.0780 
T  –  –  0.6149  0.0027  0.5543  0.0013  0.4579 
Indegree  –  –  –  −0.2600  0.6221  −0.2409  0.7555 
Outdegree  –  –  –  –  0.3030  0.7298  0.2572 
Betweenness  –  –  –  –  –  0.2464  0.4604 
Hub  –  –  –  –  –  –  0.0916 
Authority  –  –  –  –  –  –  – 
Spearman’s rank correlation coefficient over Twitter dataset #coup_suffocates_Taiz
Followers  T  Indegree  Outdegree  Betweenness  Hub  Authoritiy  

Followers  –  0.0921  0.0783  0.197  0.0815  0.0201  0.0639 
T  –  –  0.6273  −0.1657  0.4231  −0.1485  0.4859 
Indegree  –  –  –  −0.4345  0.4865  −0.4325  0.8035 
Outdegree  –  –  –  –  0.2694  0.7878  0.0138 
Betweenness  –  –  –  –  –  0.2169  0.3796 
Hub  –  –  –  –  –  –  −0.1279 
Authority  –  –  –  –  –  –  – 
Kendall’s tau rank correlation coefficient over Twitter dataset #EndTaizSiege
Followers  T  Indegree  Outdegree  Betweenness  Hub  Authoritiy  

Followers  –  0.0978  0.0612  0.0391  0.0773  0.0321  0.0562 
T  –  –  0.5956  0.0015  0.5401  0.0028  0.4132 
Indegree  –  –  –  −0.2361  0.5980  −0.1812  0.6823 
Outdegree  –  –  –  –  0.2757  0.6077  0.3221 
Betweenness  –  –  –  –  –  0.1944  0.4123 
Hub  –  –  –  –  –  –  0.1088 
Authority  –  –  –  –  –  –  – 
Kendall’s tau rank correlation coefficient over Twitter dataset #coup_suffocates_Taiz
Followers  T  Indegree  Outdegree  Betweenness  Hub  Authoritiy  

Followers  –  0.0671  0.0583  –0.0013  0.0515  –0.0088  0.0458 
T  –  –  0.5993  –0.1466  0.4098  –0.1204  0.4257 
Indegree  –  –  –  –0.3752  0.4605  –0.3433  0.7090 
Outdegree  –  –  –  –  0.2408  0.6630  0.1383 
Betweenness  –  –  –  –  –  0.1777  0.3325 
Hub  –  –  –  –  –  –  –0.0477 
Authority  –  –  –  –  –  –  – 
Description of top influential actors according to different influence measures in Twitter dataset #EndTaizSieg
Rank  Description  

T  Indegree  Betweenness  
1  News account N1  News account N1  ? 
2  Journalist J1  Journalist J1  ? 
3  TV announcer T1  TV announcer T1  ? 
4  Television reporter R1  Journalist J3  Journalist J2 
5  Human rights activist H1  Human rights activist H1  ? 
6  Human rights activist H2  News account N2  ? 
7  News account N2  Human rights activist H2  Human rights activist H3 
8  Political activist P1  ?  TV announcer T1 
9  Journalist J2  Political activist P1  News account N1 
10  Political activist P2  ?  ? 
Description of top influential actors according to different influence measures in Twitter dataset #coup_suffocates_Taiz
Rank  Description  

T  Indegree  Betweenness  
1  Journalist 1  Political activist  Journalist 1 
2  Political activist  Journalist 1  Human rights activist 
3  Joutnalist 3  Journalist 2  Journalist 2 
4  News account 1  Joutnalist 3  ? 
5  Journalist 2  News account 1  ? 
6  Journalist 4  Human rights activist  ? 
7  Human rights activist  Politician  Joutnalist 3 
8  ?  ?  ? 
9  Politician  ?  ? 
10  News account 2  News account 2  ? 
Information diffusion and T measure
The IC model is an information diffusion model where the information flows over the network through cascade. Actors in the IC model can have two states, either active or inactive. Active means the actor is influenced by the information, and inactive means the actor is not influenced. The IC model calculation starts with an initial set of activated actors. In step t, an actor a will get a single chance to activate each currently inactive neighbor b. Actually, the activation process depends on the propagation probability P of the actors connection. The propagation probability P of a connection is the probability by which an actor can influence the other actors. In Twitter, we have proposed that actor a is influenced by actor b if he/she retweeted from actor b in proportion to the tweets number of actor b. So, the propagation probability P on IC model is based in our Twitter dataset on the connection weight divided by tweets number of target actor. The reason why we use the IC model instead of the LT model is that the linear threshold model is receiver oriented. This means an actor becomes active if a certain fraction of its neighbors are active. This does not account for our purpose where we want to find strong attractors who are likely to attract others. The IC model is sender oriented, and thus, is better suited to simulate attraction processes.
Algorithm 1 shows the pseudo code of IC model simulator which takes the seed set S as a parameter, and then evaluates the activated actors for the each actor v in the set S. Finally, it returns the total number of activated actors by whole actors in the set S.
Simulation of attraction processes with timerespecting paths
In addition to the statistical comparison between the T measure and other standard network measures, we also report results based on simulated attraction processes. To do so, we adapt the IC model that is known to simulate the diffusion of information through a network as described above. Information diffusion and attraction processes have some commonalities but differ in various aspects. In traditional information diffusion models such as the IC model, the network is usually considered as stable in the sense that the set of nodes and the set of edges do not change over time. However, the nodes changes their states "inactive” and "active” during the information diffusion process. Attraction, as it is studied in this paper, is similar in the sense that actors who are not part of the community (i.e. do not have contributed a tweet) are inactive while others are considered as active. On the other hand, the original IC model does not account for the fact that the network grows when new actors become attracted to the community. Thus, the IC model was adapted to take into account the creation times of the edges. These timevarying networks have special characteristics regarding reachability of node pairs since a walk on the graph can only take edges with increasing timestamp, which is known as the timerespecting property (see [27, 28]). In this aspect, we added a new activation rule to the IC model which is: the actor who is activated in time t cannot activate those actors who have been linked with him/her before the time t. To explain this activation rule in more detail, we define the following terms:
Definition 12
(Pathtime) The path time of each link in the network is the Pslice number in which this link has been created.
Definition 13
(Activation time) The activation time of each activated actor is the path time of the link by which this actor has been activated.
Now, we can state that the actor a cannot activate the actor b if the link from b to a has a path time later than the activation time of the actor a.
Using this activiation rule, the simulation can be interpreted as an attraction process where actors who are already part of the communities can attract others only if their activity starts after the activator has become active.
Previous studies [1] have shown that a seed selection strategy based on indegree yields similar results as a selection strategy based on the T measure. This is also expected with respect to the high correlation between these two measures. However, the benefit of the T measure that distinguishes it from other measures is that time is explicitly taken into account. The experimental results in the next section support the assumption that the T measure can identify important attractors in timevarying networks while it boils down to indegree if time is neglected.
Experimental results
T test verification for simulation results in case of seed sizes n (n > 13) among T and indegree measures in the dataset #EndTaizSiege
Seed size  t  df  Sig. (2tailed)  95% confidence interval  Mean difference  Mean  Std. deviation  

Lower  Upper  
14  14.4854  19  0.0000  1422.1408  1502.0592  276.55  1462.1  85.3802 
15  10.5787  19  0.0000  1415.4421  1476.6579  154.7  1446.05  65.3996 
16  14.7604  19  0.0000  1424.0960  1509.2040  300.1  1466.65  90.9247 
17  18.2705  19  0.0000  1482.1069  1565.4931  363.95  1523.8  89.0852 
18  11.6923  19  0.0000  1501.8185  1590.4815  247.65  1546.15  94.7225 
19  26.9261  19  0.0000  1598.1139  1660.4861  401.2  1629.3  66.6350 
20  16.3709  19  0.0000  1632.5976  1702.9024  274.95  1667.75  75.1097 
21  17.4834  19  0.0000  1784.6586  1850.7414  276  1817.7  70.5990 
22  12.2143  19  0.0000  1768.7146  1840.0854  208.25  1804.4  76.2485 
23  6.8975  19  0.0000  1766.6357  1827.2643  99.9  1796.95  64.7720 
24  17.6846  19  0.0000  1885.3439  1939.6561  229.45  1912.5  58.0240 
25  17.5075  19  0.0000  1933.0513  1987.9487  229.6  1960.5  58.6493 
Here, we considered the dataset #EndTaizSiege which is related to an organized event in Yemen. Hence, we got a highly connected component that is suitable for the application of our approach which is basically aimed to identify those actors who contribute to attract others to participate in a specific organized event. We simulated the information diffusion based on the IC model with timerespecting paths for seed sets of sizes \(n = 1 \ldots 25\) which are generated from different influence measures. Figure 9 shows the results of applying the IC model to seeds generated from T, indegree, and betweenness measures. We notice that the T measure yields the best performance in information diffusion under the IC model with timerespecting paths for the seed sizes bigger than 13. Additionally, we statistically verified the results of simulation for each seed set using T test. In case of n (n > 13), the differences among T and indegree measures are significant. For example, results for the seed set 14 show that there is a significant difference in the score of T measure \((M = 1462.1, SD = 85.3802 \, {\,\text{conditions}};\, t (19) = 14.4854, P = 0.0000)\). Table 10 presents the relevant descriptive statistics.
T test verification for simulation results in case of seed sizes n (n > 7) among T and indegree measures in the dataset #coup_suffocates_Taiz
Seed size  t  df  Sig. (2tailed)  95% confidence interval  Mean difference  Mean  Std. deviation  

Lower  Upper  
8  3.272474738  19  0.004  154.0691524  169.9308476  12.4  162  16.94573382 
9  8.159694936  19  0.000  167.6259751  179.3740249  22.9  173.5  12.5509488 
10  5.02467484  19  0.000  191.1521152  206.1478848  18  198.65  16.02062815 
11  3.22614144  19  0.004  190.3718072  202.8281928  9.6  196.6  13.30769465 
12  21.28977767  19  0.000  222.9571772  231.9428228  45.7  227.45  9.599753286 
13  11.30200169  19  0.000  219.9183376  232.3816624  33.65  226.15  13.31510816 
14  13.11109148  19  0.000  226.1143374  236.9856626  34.05  231.55  11.61430606 
15  12.14162861  19  0.000  230.4304495  243.6695505  38.4  237.05  14.14390328 
16  8.499171278  19  0.000  246.3088375  265.5911625  39.15  255.95  20.60014052 
17  10.90348517  19  0.000  245.934543  258.565457  32.9  252.25  13.49415078 
18  19.29415746  19  0.000  272.7273155  285.9726845  61.05  279.35  14.15059976 
19  20.77030073  19  0.000  269.3351504  279.7648496  51.75  274.55  11.14249807 
20  13.09099585  19  0.000  284.2110899  298.0889101  43.4  291.15  14.82627469 
21  7.505556897  19  0.000  283.7953814  293.3046186  17.05  288.55  10.15912864 
22  2.617290607  19  0.002  285.5219614  295.6780386  6.35  290.6  10.85017584 
23  1.160344906  19  0.003  285.6721647  302.6278353  4.7  294.15  18.11447517 
24  5.3893686  19  0.000  306.6255244  319.6744756  16.8  313.15  13.94075812 
25  0.946308607  19  0.004  307.8905455  322.7094545  3.35  315.3  15.83168043 
26  3.909066253  19  0.001  313.7701177  322.2298823  7.9  318  9.037931762 
27  8.922128329  19  0.000  325.3910277  333.1089723  16.45  329.25  8.245413398 
28  11.85052393  19  0.000  336.3281838  346.2718162  28.15  341.3  10.62321192 
29  9.295528476  19  0.000  333.9635987  344.6364013  23.7  339.3  11.40221585 
30  13.14909142  19  0.000  345.7835107  356.4164893  33.4  351.1  11.35967012 
Conclusion
In this paper, we introduced a new approach to detect influential actors based on a new type of influence. Influential actors who are detected by our approach are those actors whose tweets spawn many retweets in a way that leads to an increase in the size of social network. We presented through experiment results how our proposed measure T referred to the influential actors in Asterisk and Twitter datasets. Furthermore, we introduced the relation between T measure and other influence measures using Spearman’s rank correlation. Finally, we showed through experiment and statistical tests that the best performance has been yielded by T measure in maximization of influence problem when we took the time into account.
Our current work in extending and improving this approach focuses on a differentiation of the role of the actors and different types of communication networks based on the T measure. As well as, we plan to describe our approach on multilayer networks. Furthermore, we are going to study an efficient general strategy to define the size of pslice depending on the premise: the pslice is the time that the most tweets get the most of their retweets. Moreover, we intend to study the role of time slicing in making T measure far better than existing measures.
Declarations
Authors' contributions
This work is the result of a close joint effort in which all authors contributed almost equally to defining and shaping the problem definition, formulas, algorithms design, implementation, computational data analysis, and manuscript. ZQ, as the first author, took the lead in composing the first draft of the manuscript, while MJ, TH and HH edited it. All authors have read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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