Molecular model of dynamic social network based on e-mail communication

2.1 Dynamics of social networks

In the past few years the problem of predicting the future interactions between users in social networks has become an important research challenge. Due to the availability of datasets of online activities and communication between people, scientists try to describe both structure and evolution of such networks. Most of the approaches addressing the complex networks growth take into consideration a limited set of global characteristics of the networks and develop models that reproduce only these characteristics, e.g. node degree distribution (Barabasi 2003), clustering coefficient (Watts 2002) or network diameter (Bollobas 1985).

There are some approaches that aim at developing specific models for online social networks and take into consideration some information characteristic to such networks (Kumar et al. 2006; Lescovec et al. 2008; Bringmann et al. 2010; Braha and Bar-Yam 2006; Liben-Nowell and Kleinberg 2007; Davis et al. 2012; Kashoob et al. 2012). Different models propose different methods of network growth. In (Kumar et al. 2006), on the basis of the analysis of real-world networks such as Flickr and Yahoo 360!, the users have been divided into three different types: passive, linkers and inviters. The members of the first group (passive users) join the network out of curiosity or because of being invited by a friend. These users, as their name suggests, never engage in any significant activities within the network and do not interact with other users. Inviters on the other hand, are interested in migrating the group that they have in the real world into a virtual world; thus their actions focus on inviting their friends to participate in an online social network. Linkers actively connect themselves to other members within the online social network. Based on the analysis of datasets the authors define a rule-based system that follows specific rules used for evolution of the social network. The method that describes the network growth can be defined as the set of steps: (1) at each time step, a node arrives, and one of the statuses: passive, linker or inviter is randomly assigned to it; (2) during the same time step, x edges arrive and the source of each of the edges is chosen at random from the existing inviters and linkers in the network using preferential attachment. Depending on the chosen type of the source node (inviter or linker) different actions are performed. If the source is an inviter, then it invites a non-member to join the network, and so the destination is a new node. If the source is a linker, then the destination is chosen from among the existing linkers and inviters, again using preferential attachment (Kumar et al. 2006). This model represents the growth of a network, i.e. it takes into account adding new nodes and edges. However, the problem of link prediction covers not only the creation of new links but also fading of existing relations.

In (Bringmann et al. 2010) the authors have presented another approach that defines a set of rules regarding how the network evolves. They focus on discovering patterns of interactions between users and their evolution over time. The authors propose to create a single graph that represents social network, which is supplemented with additional information—a time-stamp, added to each relation when it appears in the network for the first time. The experiments were performed on the DBLP database (Bringmann et al. 2010). Similarly to the previous presented study, also this one assumes that the users and the relations between them can only be added to the system and will never be deleted. Moreover, both of approaches presented so far allow to investigate the creation of new edges but do not allow to follow the dynamics of the relationships strengths between users, which is one of their limitations.

Yet another framework for the network growth was developed in (Lescovec et al. 2008) where the authors studied four online social networks: Flickr, Delicious, Answers and LinkedIn. They proposed to apply the maximum-likelihood estimation principle to compare a family of parameterised models in terms of their probability of generating the observed data and as a result to select the model that reflects the data in the best possible way. The task in this framework was to predict which nodes will a new edge connect. For every edge arriving to the network the likelihood that it will connect two given nodes under some model is assessed. The product of these values over all edges gives the likelihood of the model and the model with the highest likelihood is chosen. Similarly to the previously presented methods this one also does not consider the strength of the relation as well as the fact that an edge can disappear from the network.

A survey of other link prediction methods can be found in (Liben-Nowell and Kleinberg 2007), where the approaches like common neighborurs, Jaccard’s coefficient and Adamic/Adar method, preferential attachment, Katz method, PageRank and its variants, low-rank approximation, unseen bigriams and clustering are discussed.

A set of approaches that take into consideration the fact that links can disappear from the network have been proposed in (Hill et al. 2010; Braha and Bar-Yam 2006) where the authors have detected a dramatic time dependance in network centrality and the role of nodes, something not apparent from static analysis of node connectivity and network topology. Their experiments studied large-scale email networks consisting of 57,000 users based on data gathered over a period of 113 days. They found that although the daily networks were scale-free, the well-connected nodes in these networks changed from day to day.

A recent method also accounting for the disappearing links has been proposed and investigated in (Juszczyszyn et al. 2011a, b, c), where based on the changes in the local structure, a 1st order probabilistic model of transitions between various triad types has been derived. The model results from an observation that there exist distinctive patterns which drive the evolution of connections between nodes. Node disappearance has also been addressed in (Sarr et al. 2012), but in a somewhat different the context of disruption of the information flow.

Our approach differs from these presented above as we do not propose a model for network growth per se but we investigate the limitations of sociodynamic model verified on data coming from real system. Our proposition takes into account both creation and vanishing of the relationships. Additionally, the network investigated in this work is a structure where strength of the relationships changes over time, which is an important factor in social networks due to the cognitive limitations of people (Hill and Dunbar 2002). In our approach, we do not assign roles to users as this may be misleading. We rather assess, based on the current interactions between users, how the relations strength and the structure of the network may look like in the future.

2.2 Distance preserving graph embedding

Following the in-depth discussion presented in (Watts 2002) we cannot expect the social space to be metric, i.e. the triangle inequality between any three nodes does not hold. On the other hand, as it was mentioned above, molecular modelling assumes the interaction between the particles embedded in the Euclidean space. For this reason, to apply molecular modelling we must first perform embedding of the social network graph in a metric, Euclidean space. Numerous embedding methods exist whose overview is presented below.

The Big Bang embedding algorithm (BBE) presented in (Shavitt and Tankel 2004) simulates an explosion of particles that represent network users under a force field that is derived from the embedding error. Each particle is the geometric image of a vertex. The force field reduces the potential energy of the particles which is related to the total embedding error of all particle pairs. In the Big Bang Simulation (BBS) all particles are initially placed in the same location in space. The whole process is performed in an iterative manner and each iteration moves the particles in discrete time intervals. Every iteration begins with calculation of the field force on each particle at the current particles’ positions (for the first iteration forces are chosen randomly). As it was mentioned, the forces are derived from the potential energy. In the next step, the positions and velocities at the next time step are calculated. The final step of each iteration is to evaluate the new potential energy. This method allows to embed the network into a freely selected number of dimensions.

Another method that can be used to embed a graph in Euclidean space is called the Multidimensional Scaling (MDS) (Torgerson 1965). MDS defines a suite of methods often used in information visualization and exploration of similarities or dissimilarities in data. There are two variations of MDS, i.e. classical multidimensional scaling (CMDS) algorithm and standard MDS (Bronstein and Kimmel 2006; Kruskal et al. 1978). Classical metric MDS develops the metric as a symmetric bilinear form and calculates the leading d eigenvalues of the corresponding matrix (Torgerson 1965). An MDS algorithm starts with a matrix of similarities between objects (similarity relation does not have to be symmetrical) and then assigns a location of each item in a low-dimensional space. It hence estimates the coordinates of a set of objects in a space of specified dimensionality on the basis of measuring the distances (which, however do not have to be metric) between pairs of objects. A variety of models can be used that include different ways of computing distances and various functions relating the distances to the actual data. Both methods allow to embed graph into different numbers of dimensions. However, the problem that we faced during our experiments with MDS was that the computational overhead was very high and we were not able to obtain results within reasonable time.

In High–Dimensional Embedding (Harel and Koren 2004), which is a fast method for creating 2D representations of large graphs, the graph is first embedded into a very high dimensional space—usually associated with the number of nodes—and then projected into a 2D plane using Principal Components Analysis. This method is used for undirected graphs. It will not be useful from the perspective of our experiments as one of the goals of this study is to embed the graph into different dimensions and verify which number of dimensions helps to achieve best results from the link prediction perspective.

Minimum Volume Embedding (MVE) presented in (Shaw and Jebara 2007, 2009) is an algorithm for non–linear dimensionality reduction that uses semi-definite programming (SDP) and matrix factorization to find a low-dimensional embedding that preserves local distances between points while representing the dataset in fewer dimensions. Authors of MVE emphasise that in all cases MVE in comparison with Semi-definite Embedding and Kernel Principal Component Analysis is able to capture more of the variance of the data in the first two eigenvectors, providing a more accurate 2-dimensional embedding (Shaw and Jebara 2007, 2009). The main features of the minimum volume embedding approach are (1) MVE for a given dataset returns always the same set of coordinates, (2) isolated nodes are neglected in the embedding process and (3) MVE is stable, i.e. adding one node with very weak connections does not influence significantly the positions of the remaining nodes. Enumerated characteristics of MVE means that the graph can be embedded only into 2D space which is not enough from the perspective of the proposed experiments in this paper as one of the goals is to find out what is the best number of dimensions to which the graph should be embedded. Moreover, in the case of not connected graphs the algorithm does not work.

2.3 Dynamic molecular modelling and simulation

by using the definition of the momentum and acceleration a, defined as a second derivative of the position vector s. The above equation must be solved for every particle in every simulation step. In order to start the algorithm, the initial positions of all particles, the formula for the force which is identical for all particle pairs and the interacting potential need to be specified. One of the standard potential functions used to describe the many-particle problems is Lennard–Jones potential which is given by

$$ V(r)=4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right] $$

(4)

where r denotes the distance between particles. The Lennard–Jones potential, which has been depicted in Fig. 1, is fully defined by two parameters: the depth of the potential ε-responsible for the strength of interactions between particles and σ-related to the minimum distance between two particles. As it can be seen the potential has a global minimum equal to ε for r _min = 2^1/6, σ = 1.12σ. An important characteristic of this potential is that the nature of interaction between two particles depends on their distance. Namely, for distances bigger than σ the particles attract each other, while for distances smaller then σ the character of the potential changes to strongly repulsive.

Open image in new window

Having the analytical formula for the interaction potential one can easily obtain the formula for the force by simple differentiation, which should be performed analytically to avoid accumulation of numerical approximations. As a first approximation all particles are assumed to have unit mass. Knowing the forces, the Verlet algorithm may be used to obtain the position and the velocity of each particle in consecutive time steps (Juszczyszyn et al. 2009).

The concept of molecular modelling and simulation is used in this study to model the dynamics of a social network. The users who are the nodes of the network become particles in Euclidean space and the distance between particles will be determined based on the relationship strength between the users. The changes in the distances between particles over time will be the basis for inferring the potential in a purely data-driven way and in consequence for determining the force between particles.

4.1 Data preparation—creating Email-based social network

In consequence, every email with more than one recipient is treated as 1/n of a regular one (n is the number of its recipients). Although ‘to-list’ recipients are likely to be of much greater message-network importance than the ‘cc-list’ recipients, both groups are treated in the same way, i.e. the total number of the recipients of an email is always taken into account. Such approach results from the fact that the obtained data do not include information if the recipient of the email is on the ‘to-list’ or ‘cc-list’.

The resulting social network \(SN = \langle N, I\rangle\) is defined as a tuple consisting of a set of network nodes N and a set of relationships that are described by their mean intensity \({{I : N\times N \rightarrow\ \mathbb{R}^+ \cup \{0\},}}\) given by Eq. 5. Note that the resulting structure is a non-directed graph with intensity I as a label assigned to the relationships.

It should be emphasized that the social network derived from the email logs does not have a static structure. The existence of any link in such a graph (i.e. relationship) is a result of a series of discrete events (email messages) which occur in certain time instants and usually with changing frequency. We may also think of the computed relationships’ intensity as of the social distance between network members (nodes). Greater I reflects smaller distance in the social space. In order to track changes in relationship strength we have used a sliding window approach.

For the experiments the data from a period of 84 days were selected and divided into frames covering 7 days each. This allowed to create 12 social network graphs \({\hbox{SN}}(t_0), {\hbox{SN}}(t_1), \ldots {\hbox{SN}}(t_n)\) where \(t_0, t_1,\ldots, t_n\) are discrete instants of time. Each network is created according to the procedure defined above on the basis of 7-day period starting in \(t_0, t_1, \ldots, t_n.\) The networks \({\hbox{SN}}(t_0), {\hbox{SN}}(t_1), \ldots {\hbox{SN}}(t_n)\) are temporal images of evolving social structure which was built on the basis of email communication. In addition, only users who were active in all time windows were taken into account as they constitute the core of the network.

4.2 Distance matrix creation

As a result the distance will always fall into the (1,2) interval for directly connected nodes, (2,3) if there is one intermediate node, etc. Note, that in this setting the number of edges in the shortest path contributes the most, while the additional information given by the edge weights is also taken advantage of. Equation 6 also assigns some finite distance value to all pairs of nodes not connected by any path, as one of the requirements imposed by the embedding algorithm we have used was that the distance should be defined for every pair of nodes.

4.3 Embedding networks in the Euclidean space

With the distance matrices in place the graphs \({\hbox{SN}}(t_0), {\hbox{SN}}(t_1),\ldots,{\hbox{SN}}(t_n)\) can be embedded in the Euclidean space (two or more dimensional), where each node is represented by a point with given coordinates. The resulting sets of points \({\hbox{SN}}_0, {\hbox{SN}}_1,\ldots,{\hbox{SN}}_n\) represent the temporal network images.

An important issue, which should be discussed here, is the dimensionality of the embedding space. Most embedding algorithms have been designed for the purpose of graph visualization. This naturally implies a two- or three-dimensional embedding. However, the higher the dimensionality of the embedding, the more accurately the social distances are mapped into the Euclidean space. Figure 3 depicts the average distance distortion¹ of the embedding as a function of dimensionality for the WrUT email network and for (a) BBS, (b) CMDS methods. As expected, in both cases the accuracy of the embedding grows with dimensionality. Please note that the scales on the vertical axes in the Fig. 3 are different. It should be emphasized that the pace of accuracy growth with increasing dimensionality is much faster in the case of BBS than CMDS. Intuitively we should choose the number of dimensions to be as high as possible. There is a limit, however, which results from the so called ‘curse of dimensionality’ (Bishop et al. 1995), and especially the ‘distance concentration‘ phenomenon, which as demonstrated in (Budka et al. 2011) is particularly relevant in the context of dynamic molecular simulation of potential fields in the Euclidean space.

Open image in new window

It has been observed that as the number of dimensions grows, the Euclidean distance loses its discriminative power, regardless of the characteristics of the dataset (Aggarwal et al. 1973; Francois et al. 2005). The reason for this is that under a broad set of conditions the mean value of the L ₂-norm distribution grows with data dimensionality while the variance remains approximately constant (Fig. 4) (Francois et al. 2005). As a result, the nearest and furthest neighbours of any molecule appear to be at approximately the same distance, which makes the ratio of distances to the nearest and farthest neighbour tend to converge to 1. As argued in (Beyer et al. 1999), it can occur even for sets with as few as ten dimensions and the decrease in the ratio between the farthest and nearest neighbour distance is steepest in the first 20 dimensions. The effect is additionally magnified by the limited precision of calculations a computer can handle and often leads to the molecular simulation failing to converge (Budka et al. 2011). Hence in practice the embedding dimensionality needs to be a compromise between the distance distortion and negative effects of high dimensionality. For this reason we have decided to embed each graph into \(2,3,\ldots,20\) dimensions to investigate the mapping between graph distances and distances in embedded graph.

Open image in new window

Embedding algorithm has to assure that the Euclidean distances between points (nodes) fit in the best possible way the distances in a social space (relation strengths in original graphs). As a result one obtains the representation of social system in which the network is seen as an assembly of N particles, representing the nodes of a social network.

After reviewing several embedding methods, it has been decided that two sets of experiments will be performed: (1) the Big-Bang Simulation and (2) CMDS as these methods enable to embed graph into an arbitrary number of dimensions. Additionally, BBS models the network nodes as a set of particles, which is consistent with the next part of the experiments where molecular modelling approach is used to determine the dynamics of a social network.

Embedding was performed on 12 previously extracted social networks. Each of the networks was embedded into \(2,3,\ldots,20\) dimensions using BBS. CMDS inherently selects the best number of dimensions (in excess of 400 in our case), so in this case the parameter was not set during the experiments, but only first \(2,3,\ldots,20\) dimensions produced by CMDS have been used in our simulations.

For each of the dimensionalities given above we have analysed how well the distances between particles from the social networks (graph) are reflected in the embedded space. To avoid negative effects of high dimensionality we decided to select the lowest number of dimensions that allowed to embed the graph in a way that the mean values of the distances after embedding, which correspond to the graph distances in the ranges <1; 2),  <2; 3),  <3; 4) etc. were well separated. This has been achieved for 12 dimensions, where for both BBS (Fig. 5) and CMDS (Fig. 6) the distributions of distances in the embedding space are approximately unimodal and their expected values are in the required range.

Open image in new window

Open image in new window

Due to the aforementioned, the actual molecular simulation has been performed in 12-dimensional space. However, for the visualisation purposes, where appropriate and to present general idea, the figures were presented for the two-dimensional embedding and molecular simulation.

After selecting the number of dimensions, the next stage of the experiments was to embed the created social networks snapshots into Euclidean space. As discussed in Sect. 3, embedded graphs serve as an input to the molecular simulation process.

4.4 Setting up the dynamic molecular model

Because the sets of network nodes in \({\hbox{SN}}(t_0), {\hbox{SN}}(t_1),\ldots,{\hbox{SN}}(t_n)\) are equal, each point (node) is represented in any of the sets \({\hbox{SN}}_0, {\hbox{SN}}_1,\ldots,{\hbox{SN}}_n\) and is active in each of the windows. We may think of these points as of particles moving as a result of interactions (email communication) between them. At this point we use the formalism of molecular dynamics to associate a potential U with every particle (network node). The actual characteristic of this potential depends on the behaviour of the particles changing their positions in time instants \(t_0, t_1,\ldots, t_n.\)

First experiments were performed using standard Lennard–Jones potential function (Juszczyszyn et al. 2009; Musial et al. 2010). The analysis of server logs has revealed some features of the dynamics of email communication—the growing intensity of communication is always followed by the periods of less frequent email activity. This resembles the repelling force emerging between particles when their distance becomes less than some minimum. We noticed that intense email communication (which results in very small distances in social graph) is never sustained for a longer period of time. On the other hand, fading communication is (in most cases) followed by frequent message exchanges.

It should be stressed that the Lennard–Jones potential was used only for the first experiments and did not accurately fit the underlying data. In the experiments presented in this paper the social network-specific potential function on the basis of available data was developed. In order to do that, first the distance transition probability defined as the probability that a given distance in one window will change into another given distance in the next time frame, was calculated. For the 12 time windows, 11 transition probability matrices were obtained. The matrices were then averaged. The resulting final matrix is presented in Fig. 7a. The force that governs the changes of the location of the particles in the Euclidean space is proportional to the distance change. Hence the third-degree polynomial presented in Fig. 7b, inferred from the distance transition probability, describes the force used in the molecular simulation. Please note that the force will be different for different datasets.

Open image in new window

The presented force allows to simulate the changes between communication patterns in consecutive time instants. The potentials associated with the nodes reflect their abilities and tendency to establish future connections with their neighbours—the nodes which are close in terms of social space (thus changing the distances in social space which is analogous to the behaviour of particles moving under influence of electrical/gravitational forces).