# JARUA: Joint Embedding of Attributes and Relations for User Alignment across Social Networks
## 3 Problem Formulation  
This section formalizes the problem addressed in this paper. Basically, a social network can be represented in a three-tuple format \(G=(U,R,A)\), where \(U=\{u_{1},u_{2},\ldots,u_{n}\}\) is the set of \(n\) users; \(R=\{(u_{i},u_{j})|u_{i},u_{j}\in U\}\) represents the set of relations among users, e.g., friendships on Facebook; and \(A=\{a_{u_{1}},a_{u_{2}},\ldots,a_{u_{n}}\}\) is the set of users' text attributes, e.g., name, affiliation, and education experiences. Each text attribute of a user is represented as a key-value pair, denoted by \(p=\langle key:value\rangle\). The relationships among users are considered undirected in this paper. Definition 1: Multi-type attributes: Let \(G=(U,R,A)\) be a social network with a set of text attributes \(A\). Each user has multiple attributes, denoted by \(a_{u_{i}}=\{p_{1},p_{2},\ldots,p_{m}\}\). If \(\exists p_{j},p_{k}\in a_{u_{i}}\), \(p_{j}=\langle key_{j}:value_{j}\rangle\), \(p_{k}=\langle key_{k}:value_{k}\rangle\), and \(value_{j}\) and \(value_{k}\) are different in terms of text composition, then \(a_{u_{i}}\) is called the multi-type attributes for the user \(u_{i}\). A concrete example of multi-type attributes is as follows. A user registers their name \"[PERSON]\" on a social network. Then, he has the attribute \(p=\langle\)name:[PERSON]\(\rangle\), where \"name\" is the key (type) of the attribute, and \"Michael\" is the attribute value. The user may also have other attributes of different types, such as nickname, age, affiliations, etc. We suppose that [PERSON] filled out the following information on a social media platform: \(p_{1}=\langle\)name:[PERSON]\(\rangle\), \(p_{2}=\langle\)nickname:[PERSON]\(\rangle\), \(p_{3}=\langle\)age:28\(\rangle\), and \(p_{4}=\langle\)affiliation:Bjerly Hall, 8 Garden Street, Cambridge\(\rangle\). That is to say, he has the multi-type attributes \(a_{u_{i}}=\{p_{1},p_{2},p_{3},p_{4}\}\). Definition 2: Word granularity factor: Let \(\mathcal{V}\) be a vocabulary set composed of all words in a certain language. Given a piece of text \(T\) consisting of basic tokens (e.g., words, letters, numbers, and other symbols) in that language, the word granularity factor \(\omega(T)\) is the proportion of the count number of word tokens to that of all basic tokens in \(T\), defined as:  
\[\omega(T)=\frac{|\mathcal{V}|\cap|f(T)|}{|f(T)|}, \tag{1}\]  
where \(f(\cdot)\) is a tokenization function that splits texts into basic tokens. Table 1 gives some examples of calculating the word granularity factors. We take the first sample to illustrate the calculation. \(T=\)\"\(\blacktriangle\)Joshi\(\blacktriangle\)\" is a piece of text that composed of a word \"Joshi\" and two same symbols \"\(\blacktriangle\)\". Hence, we have the basic tokens \(f(T)=\{\blacktriangle,\text{Joshi},\blacktriangle\}\) and the word token Joshi. The word granularity factor can be obtained according to \(\omega(T)=(1/3)=0.33\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample Texts** & **Basic Tokens** & \(\omega\) \\ \hline \(\blacktriangle\)Joshi\(\blacktriangle\) & \(\blacktriangle\), [PERSON], \(\blacktriangle\) & 0.33 \\ [PERSON] & Joshi, 0001 & 0.50 \\ University of Florida & University, of, Florida & 1.00 \\ A hierarchical three way interconnect architecture for hexagonal processors & A, hierarchical, three, way, interconnect, architecture, for, hexagonal, processors & 1.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Examples of the word granularity factor This paper considers such a scenario: some users are involved in two different social networks which form the anchor links across the networks. The source network is denoted as \(G^{X}=(U^{X},R^{X},A^{X})\), the target as \(G^{Y}=(U^{Y},R^{Y},A^{Y})\), and vice versa. This investigation intends to identify, if any, the counterpart in \(G^{Y}\) (or \(G^{X}\)) for each user in \(G^{X}\) (or \(G^{Y}\)). The UA problem can be formalized as follows.