Access-Control Prediction in Social Network Sites: Examining the Role of Homophily

Abstract

Often, users of Social Network Sites (SNSs) like Facebook or Twitter have issues when controlling the access to their content. Access-control predictive models are used to recommend access-control configurations which are aligned with the users’ individual privacy preferences. One basic strategy for the prediction of access-control configurations is to generate access-control lists out of the emerging communities inside the user’s ego-network. That is, in a community-based fashion. Homophily, which is the tendency of individuals to bond with others who hold similar characteristics, can influence the network structure of SNSs and bias the users’ privacy preferences. Consequently, it can also impact the quality of the configurations generated by access-control predictive models that follow a community-based approach. In this work, we use a simulation model to evaluate the effect of homophily when predicting access-control lists in SNSs. We generate networks with different levels of homophily and analyse thereby its impact on access-control recommendations.

Appendices

Appendix

In this Appendix we introduce the theoretical foundations used for the definition and implementation of our simulation model of ego-networks.

A Network Evolution with Homophily

Up to now, scholars have proposed several evolution models for constructing scale-free networks. Among them, the prefferential attachment mechanism introduced by Barabassi and Albert [2] is one of the most prominent ones. However, this model does not consider attribute similarity when computing the linking probability between two nodes. Following, we introduce an approach for the simulation of ego-networks that takes homophily explicitly into account.

1.1 A.1 Group-Openness

Many empirical and theoretical studies have shown that people prefer to link to those people with whom they share certain characteristics [13]. Moreover, these studies have also shown that homophily can lead to the emergence of clusters inside a social network in which similar people are linked more densely with each other [9]. In order to study the role of homophily in the evolution of scale-free networks, Kim et al. [10] introduced group-openness characteristics to the preferential attachment mechanism of Barabasi and Albert [1]. In this approach, nodes which share a particular characteristic s belong to group s. Consequently, the group-openness factor $\varLambda _s^t$ between two groups s and t is defined as:

(1)

where the homophily index $\varLambda$ is a real number between 0 and 1 [9]. If $\varLambda = 0$, nodes in group s do not link with nodes in group t ($t \ne s$) but link only with those nodes in the same group. This state describes completely closed groups, in which members prefer to link only with those who hold their same characteristics. Conversely, if $\varLambda = 1$ homophily does not affect the linkage between nodes, independently of whether they belong to the same group or not. This state describes completely open groups that show neutrality when linking to others [10].

1.2 A.2 Preferential Attachment with Homophily

The preferential attachment mechanism introduced by Barabasi and Albert [1] describes the process by which new nodes prefer to link to the more connected nodes in a network (i.e. the hubs). Hence, the probability $\varPi _i$ that a new node connects to node i is proportional to the degree $k_i$ of node i:

(2)

Using the group-openness mechanism defined in Eq. 1, Kim et al. [10] introduced homophily to the preferential attachment model of Eq. 2. As result, the probability $\varPi _i^{pq}$ that a new node of group q is linked to node i of group p is defined as:

(3)

where $k_i^p$ is the degree of node i of group p, and $\varLambda _p^q$ represents the homophily between the group of the new node q and the group of node i. As one can observe, in Eq. 2 the probability that a new node connects to an existing node i is normalized by the sum of degrees of all existing nodes in the network. This is also the case for Eq. 3, only that this time the group-openness factor of each node is considered for the normalization. In other words, if all groups are completely open (i.e. $\varLambda = 1$), Eq. 3 is identical to Eq. 2. On the other hand, if the groups are completely open (i.e. $\varLambda = 0$), Eq. 3 is reduced to Eq. 2 for each particular group [10].

In order to explain the evolution of nodes who are active inside a network for a long period of time, Kim et al. [9] introduced an additional rule which describes the creation of links between existing nodes. This is, the probability $\varPi _{ij}^{pq}$ that node i of group p links to node j of group q is defined as:

$$\begin{aligned} \varPi _{ij}^{pq}&= \frac{k_{i}^{p}.k_{j}^{q}.\varLambda _{p}^{q}}{\sum _{l} \sum _{m>l}k_{l}^{\mu }.k_{m}^{\nu }.\varLambda _{\mu }^{\nu }} \end{aligned}$$

(4)

where $k_i^p$ and $k_j^q$ are the degrees of node i and of node j respectively [10]. Nodes i and j belong to groups p and q respectively, and $\varLambda _p^q$ is the group-openness between these two groups. Like in Eqs. 2 and 3, Eq. 4 is normalized by the sum of all possible combinations of links between existing nodes in the network [10]. In this case, $k_l^\mu$ and $k_m^\nu$ are the degrees of nodes l and m which belong to groups $\mu$ and $\nu$, respectively. Likewise, $\varLambda _\mu ^\nu$ refers to the group-openness between groups $\mu$ and $\nu$ [10].

B Simulation Model

The model introduced in Sect. A.2 of this Appendix generates a network in which nodes link with each other according to attribute similarity. Therefore, it assumes that the values of these attributes have been assigned to the nodes prior to the attachment phase. Following, we define the attributes used to characterise the nodes of our simulated networks and the distribution of their respective values. Likewise, we define the parameters used to set-up the simulation.

1.1 B.1 Node-Attributed Ego-Networks

In our model, the ego and its alters are characterized with the attributes gender, workplace and location where gender can take the values male or female, workplace the values Starbucks, Google or Ikea, and location the values Leeds or York. These attributes and their respective values are conditionally distributed following the probability tree of Fig. 4. According to this distribution, a node in the network is generated with 50% chance of being female and 50% chance of being male (i.e. $P(male)=P(female)=0.5$). Then, the values for workplace are assigned with 33% chance according to the gender value of the node. For instance, if $gender=female$, then $P(Starbucks|female)=P(Google|female)=P(Ikea|female)=0.33$. Likewise, the values for location are assigned with 50% chance given the gender and workplace values of the node. This means that in the case of a node whose gender and location attributes are female and Ikea, then $P(York| female \cap Ikea)=P(Leeds| female \cap Ikea)=0.5$.

Fig. 4.

Attributes probability distribution.

Each attribute value represents a group. Therefore, our model consists of 7 groups (i.e. male, female, Starbucks, Google, Ikea, Leeds and York) together with the corresponding group-openness factors between them. If we consider group-openness a symmetric relation between two groups s and t, then $\varLambda _s^t=\varLambda _t^s$. This means that for 7 groups one must define $C_{7,2}=\frac{7!}{2!(7-2)!}=21$ different group-openness factors. This information can be expressed through a group-openness matrix $\varvec{\varLambda }_{7\,\times \,7}$ in which each cell represents a factor $\varLambda _s^t$ as shown in Fig. 5. As one can observe, this matrix is symmetric and contains ones on its main diagonal. This is because $\varLambda _s^t=\varLambda _t^s$ and, according to Eq. 1, the group-openness factor $\varLambda _s^t$ is 1 when $s=t$.

Fig. 5.

Group-openness matrix.

Nodes are described in terms of one value per attribute and, consequently, belong to more than one group at the same time (e.g. a node whose gender is female, works in Ikea and lives in York, belongs to the groups female, Ikea and York respectively). Therefore, the total homophily factor between two nodes i and j depends on more than one group-openness factor. In other words, one should compute the homophily between i and j considering the group-openness factors of all possible combinations among the groups to which i and j belong. For instance, if i belongs to the group-set male, Starbucks and York, and j to the group-set female, Google and York, then one should consider $\varLambda _{female}^{male}$, $\varLambda _{Google}^{male}$, $\varLambda _{York}^{male}$, $\varLambda _{female}^{Starbucks}$, $\varLambda _{Google}^{Starbucks}$, $\varLambda _{York}^{Starbucks}$, $\varLambda _{female}^{York}$, $\varLambda _{Google}^{York}$, and $\varLambda _{York}^{York}$. Consequently, the total homophily factor between two group-sets P and Q is defined as:

(5)

where P and Q are the groups to which nodes i and j belong, respectively. According to the definition above, the preferential attachment model described in Eqs. 3 and 4 can be re-defined. That is, the probability $\varPi _i^{PQ}$ that a new node of group-set Q is linked to node i of group-set P is defined as:

(6)

where $k_i^P$ is the degree of node i from group-set P, and $\mathcal {H}{_P^Q}$ represents the total homophily factor between the group-set of the new node Q and the group-set of node i. Likewise, the probability $\varPi _{ij}^{PQ}$ that node i of group-set P links to node j of group-set Q is defined as:

$$\begin{aligned} \varPi _{ij}^{PQ}&= \frac{k_{i}^{P}.k_{j}^{Q}.\mathcal {H}{_P^Q}}{\sum _{l} \sum _{m>l}k_{l}^{M}.k_{m}^{N}.\mathcal {H}{_M^N}} \end{aligned}$$

(7)

where $k_i^P$ and $k_j^Q$ are the degrees of node i and of node j respectively and $\mathcal {H}{_P^Q}$ is the total homophily factor between group-sets P and Q.

1.2 B.2 Simulation Set-up

Our simulation model for ego-networks comprises Eqs. 6 and 7 together with the attribute probability distribution introduced in Appendix B. According to the Pew Research Center, the average size of an ego-network in Facebook was of 338 friends/nodes in 2014² (this number scaled up to 425 in a study focused on adolescents and online privacy in 2013 [12]). Therefore, we will consider an average ego-network consisting of 500 nodes and execute our simulation for $F=500$ time units. The initial set-up for all simulations consists of an network of two nodes ($N(0)=2$) and one link ($K=1$). It is also assumed that only one node enters the network at time t ($b=2$) and generates only one link ($\beta =1$). On the other hand, the number of new links between existing nodes at time t is given by $\lfloor N(t)\cdot \alpha \rfloor$ where $0 \le \alpha < 1$ and N(t) is the number of links at t. In order to preserve the degree distribution in our simulated networks we adopt $\alpha =0.001$ as suggested by Kim et al. [9].