UACD: A Local Approach for Identifying the Most Influential Spreaders in Twitter in a Distributed Environment

Abstract

Efficient spreading of important information through social media can be highly beneficial, while quick spreading of false content is alarming. Finding the users who are the most influential at information spreading can help develop efficient strategies. However, with the increasing growth of gigantic social networks, existing methods either lack accuracy or have high latency, sometimes being infeasible within limited memory. In this study, we find that rich user-specific information can guide us toward designing more effective methods. We propose UACD, a novel method for identifying the most influential spreaders on the Twitter social network by combining both user-specific and topological information. We provide a distributed implementation of our proposed algorithm on the Amazon EC2 and compare our ranking result with the state-of-the-art methods. Results suggest that UACD is scalable and can process a very large network while being on average $\mathbf {12.5}\%$ more accurate and $\mathbf {175}{\times }$ faster.

Appendix A Technical Background

In recent years, the number of users of social network sites have been increased by a large scale. People now can publish their statuses and get connected with other users which we can refer to as social relationship (Garton et al. 1997). As we have already discussed in the earlier section, we can formally represent an online social network using a graph, where each user is represented by a node of the graph and the social relationships among them are represented by edges (directed or undirected) (Newman et al. 2002). In this graph, the nodes play an important role to disseminate information. This knowledge of node spreadability is very significant while it comes to develop an efficient method of accelerating the spreading in the case of information diffusion (Mahajan 2010) as well as decelerate it in the case of diseases (Heesterbeek 2000; Keeling and Rohani 2008). Therefore, in recent years, the microscopic study of spreadability for each node has caught attention of the researchers. Moreover, it can be very useful in case of finding out the initial spreaders of a contagious disease (Anderson et al. 1992) or any information (Fu et al. 2015a). In this section, we define some terminologies related to our work and then we provide a brief description of the existing techniques of finding the most influential spreaders in a network. Finally, we discuss the SIR model (Smith and Moore 2004) which we use to evaluate the merit of our proposed technique.

1.1 Centrality measurement

In graph theory, centrality is a term to describe the importance of an individual vertex within a graph or a network. Centrality measures (Disney 2020) were initially developed for social network analysis since it helps answering the question, “Which vertices are the most influential in a graph?” The popular centrality measures for such analysis can be divided into two types: local centrality measures and global centrality measures, briefly described below.

1.1.1 Local centrality measures

To compute the centrality of a node, local measures generally use the information of a node and its neighborhood only. The number of neighbors (degree of a node) plays the main role in such local measures and they are more suitable for networks modeled as undirected graphs. The two most popular local centrality measures are described below:

(i) Degree centrality Among all the centrality measures, Degree centrality (Disney 2020) is the simplest one. It assumes the nodes with maximum number of neighbors to be the most influential in the network.

If, G(V, E) is a graph with the set of vertices, V and the set of edges, E, then the degree of a vertex v can be denoted by $d_v$, and is defined as the total number of edges incident to it (i.e., the number of neighbors of v). Now, if the number of vertices, |V| is n, then the degree centrality of node v is given as follows:

$$\begin{aligned} DC(v) = \dfrac{d_v}{(n-1)} \end{aligned}$$

(13)

Here, $(n-1)$ is used to normalize the value of degree centrality within 0 and 1. The most important reason for using degree centrality for finding the most influential spreaders is its simplicity and low computational complexity. However, this measure typically fails to identify the most influential spreaders accurately. Despite this, there are several cases where degree centrality can provide surprisingly good performance. For example, if the spreading rate is very small, degree centrality is reported to perform better in finding the spreadability of nodes than other well-known centrality measures (Klemm et al. 2012; Liu et al. 2016).

(ii) K-core decomposition In case of degree centrality, the number of neighbors is solely responsible in determining the influence of any node in the network. Later Kitsak et al. (2010) determined that the location of the nodes in the network can impact more significantly in their spreadability. They identified that nodes located at the center of the network possess higher probability to be the most significant spreaders in a network than the nodes located at the perimeter of the network. To sum up, they suggested that the core number of a node should be considered as a more suitable measure in order to identifying the most influential spreaders of the network, and this core number can be determined by the k-core decomposition (Dorogovtsev et al. 2006; Alvarez-Hamelin et al. 2005) of the network.

At the very beginning of the k-core decomposition method, all the nodes with degree 1 are removed. This process is recursively continued until no node with degree 1 is remaining in the network. These removed nodes are grouped together to form 1-core. After that, in a similar way, all the nodes with residual degree 2 are removed recursively until no node with degree 2 is remaining in the network. All these removed node are then grouped together to form 2-core. This process is continued until all the nodes are assigned to some core group. In this conventional k-core decomposition method, nodes having the largest core number are considered to be located at the center of the network and they are assumed to have the most spreadability. Figure 7 presents a simple network and the core number for the nodes.

Fig. 7

A simple network with core numbers (k) of the nodes

The drawback of k-core decomposition method is that it has a tendency to assign the same core number (k value) to multiple nodes in case of large networks. Therefore, we may end up having a huge number of nodes as the most influential spreaders, which may not be a desired outcome in many cases. However, the simplicity and lower computational complexity make this measure very useful. We suggest that incorporating user-specific information while computing the core decomposition helps to find the influential spreaders more accurately. Based on this, we propose User Attributed Core Decomposition (UACD) method and show that our proposed method significantly improves the accuracy without incurring noticeable computational overhead.

1.1.2 Global centrality measures

Global measures consider the whole network topology while computing the centrality of the nodes. Some of the most popular global centrality measures are briefly described below:

(i) Closeness centrality Closeness centrality (Disney 2020) is a measurement which identifies the closeness of a node from all the other nodes in the network. If the network can be represented by a connected graph, then the normalized version of closeness centrality of any node u of the graph is calculated as the average of all the shortest paths between u and all other nodes of the network.

Let $l_{ij}$ be the length of shortest path between any two nodes ($v_i$, $v_j$), and n be the number of nodes in the network, then the average shortest distance of node $v_i$ from all other nodes, i.e., closeness centrality can be defined as (Sabidussi 1966),

$$\begin{aligned} L_{v_i} = \dfrac{1}{n-1} \sum _{i \ne j}l_{ij} \end{aligned}$$

(14)

The closeness centrality of node $v_i$ can be thought as the inverse of the average shortest distance, $L_i$ and defined as follows:

$$\begin{aligned} CC(v_i) = \dfrac{1}{L_{v_i}} = \dfrac{n-1}{\sum _{i \ne j}l_{ij}} \end{aligned}$$

(15)

However, this equation will not be suitable to use for a disconnected graph where some nodes may be unreachable from a node $v_i$. For graphs with multiple connected components, Wasserman and Faust (Opsahl et al. 2010) revise the definition of closeness centrality. The closeness centrality of node $v_i$ is now measured as the ratio of “the fraction of the nodes in the network which are reachable from $v_i$, ” to the “average shortest distance of $v_i$ from the reachable nodes.” Let $n_i$ be the number of reachable nodes in the network from node $v_i$, then the modified formula of measuring closeness centrality is given as follows:

$$\begin{aligned} CC(v_i) = \dfrac{n_i-1}{n-1} \dfrac{n_i-1}{\sum _{i \ne j, v_j \text { is reachable from } v_i} (l_{ij})} \end{aligned}$$

(16)

(ii) Betweenness centrality Betweenness centrality (Disney 2020) determines how many times a node falls along the shortest path between two different nodes (i.e., acts as a bridge between those two nodes). Freeman (1977) introduces this measure for quantifying the control of a user on the communication between other users in a social network.

Let $v_s$, $v_f$, and $v_i$ are three different nodes in a network represented by G(V, E). Now we define $n_{sf}^i = 1$ if node $v_i$ lies on the shortest path between $v_s$ and $v_f$, and 0 otherwise. The betweenness centrality of node $v_i$ is defined as:

$$\begin{aligned} BC(v_i) = \sum _{(s, f) \in V} n_{sf}^i \end{aligned}$$

(17)

However, there can be more than one shortest path between $v_s$ and $v_f$ and that may count a node for centrality measure more than once. For this reason, if total number of shortest paths between $v_s$ and $v_f$ is $g_{sf}$, the equation for finding betweenness centrality of node $v_i$ is updated as follows:

$$\begin{aligned} BC(v_i) = \sum _{(s, f) \in V} \dfrac{n_{sf}^i}{g_{sf}} \end{aligned}$$

(18)

(iii) Eigenvector centrality Eigenvector centrality (Disney 2020) computes the centrality of a node in a network based on the centrality of its neighbors. The weights of the neighbors are assigned in such a way that a high scoring neighbor contributes more to the centrality of the node. Assume that, a graph G(V, E) is represented by an adjacency matrix $A = \{a_{ij}\}$, where $\{a_{ij}\} = 1$ if nodes $v_i$ and $v_j$ are neighbors and 0 otherwise.

If $\lambda$ is the eigenvalue of graph G, the eigenvector centrality for node $v_i$ is the $i^{th}$ element of the vector $\mathbf {x}$ defined by the equation:

$$\begin{aligned} A\mathbf {x} = \lambda \mathbf {x} \end{aligned}$$

(19)

There can be multiple eigenvalues, i.e., multiple values of $\lambda$ and for which multiple solutions can be found. However, for the largest eigenvalue of the adjacency matrix A, according to the Perron–Frobenius theorem, there exists a unique solution of x which contains all positive entries (Newman 2010).

1.2 Susceptible-infected-recovered (SIR) model

The SIR model (Smith and Moore 2004) is one of the widely used models in epidemiology. In an SIR model, a node in the network can be at one of the three states: Susceptible, Infected, and Recovered. For better understanding, we explain these states using “people” instead of “nodes.”

• S (Susceptible): The group of people who have not been infected with the disease yet. Additionally, they are not immune to the disease, and therefore, they are under the threat of being infected in the future.

• I (Infected): The group of people who have already been infected with the disease. Moreover, they can transmit the disease to the susceptible neighbors with a probability of $\beta$.

• R (Recovered): The group of people who have either recovered from the disease or dead. The recovered people are immune to the disease and no longer can transmit the disease to the susceptible neighbors.

The SIR model is capable of finding the spreadability of a node accurately by considering the node as the only infected node at the beginning, running the simulation multiple times, and considering the average. If the network contains n nodes, one needs to simulate the entire network n times (with multiple runs) to find the spreadability of all the nodes. Although this is the most accurate method of finding the most influential spreaders, it is not realizable for a large network due to its high computational cost.