Domain-based Latent Personal Analysis and its use for impersonation detection in social media

Abstract

Zipf’s law defines an inverse proportion between a word’s ranking in a given corpus and its frequency in it, roughly dividing the vocabulary into frequent words and infrequent ones. Here, we stipulate that within a domain an author’s signature can be derived from, in loose terms, the author’s missing popular words and frequently used infrequent words. We devise a method, termed Latent Personal Analysis (LPA), for finding domain-based attributes for entities in a domain: their distance from the domain and their signature, which determines how they most differ from a domain. We identify the most suitable distance metric for the method among several and construct the distances and personal signatures for authors, the domain’s entities. The signature consists of both over-used terms (compared to the average) and missing popular terms. We validate the correctness and power of the signatures in identifying users and set existence conditions. We test LPA in several domains, both textual and non-textual. We then demonstrate the use of the method in explainable authorship attribution: we define algorithms that utilize LPA  to identify two types of impersonation in social media: (1) authors with sockpuppets (multiple) accounts and (2) front-users accounts, operated by several authors. We validate the algorithms and employ them over a large-scale dataset obtained from a social media site with over 4000 users. We corroborate these results using temporal rate analysis. LPA  can further be used to devise personal attributes in a wide range of scientific domains in which the constituents have a long-tail distribution of elements.

Appendices

Appendix 1: Distance metrics

A distance metric d on a set X is a function $d:X\times X\rightarrow [0,\infty )$. I.e., it receives two elements in the set and gives the distance between them as a real, non-negative number. To be a true metric, such a function needs to fulfill the following four criteria:

• Non-negativity $d(x,y)\ge 0$. The distance between any two elements is greater or equal to zero.

• Symmetry $d(x,y)=d(y,x)$. Distance is independent of starting point—for any two elements x, y in the set, the distance from x to y is the same as the distance from y to x.

• Identity of indiscernibles $d(x,y)=0\leftrightarrow x=y$. Two elements have a zero distance from each other if and only if they are the same element.

• Triangle inequality $d(x,z)\le d(x,y)+d(y,z)$. The shortest path is a direct one—given two elements, x, z in the set, it is never shorter to go through a third one, y.

We show here that each of our selected metrics, as described in Sect. 3.2.1, is a distance metric by the definition given in Section 8. Specifically, we show that the discussed distance metrics in Sect. 3.2.1 all satisfy the following three properties: Non-negativity, Symmetry and Identity of indiscernibles.

1.1 RBD distance metric

Non-negativity We first notice that $\sum _{d=1}^\infty p^{d-1}$ is the sum of the geometric progression $p^{d-1}$ and is therefore equal to $\frac{1}{1-p}$. since $A_d \le 1$, we get

$$\begin{aligned} \sum _{d=1}^\infty p^{d-1}\cdot A_d\le \frac{1}{1-p} \end{aligned}$$

and therefore

$$\begin{aligned} (1-p)\sum _{d=1}^\infty p^{d-1}\cdot A_d\le (1-p)\frac{1}{1-p} = 1 \end{aligned}$$

As p is in the range [0, 1] and $A_d$ is non-negative, we also get

$$\begin{aligned} (1-p)\sum _{d=1}^\infty p^{d-1}\cdot A_d\ge 0 \end{aligned}$$

Hence RBO is always in the range [0, 1]. As $RBD=1-RBO$ it is also in that range.

Symmetry For two lists, $V_1,V_2$, $A_d$ is defined by the intersection of the lists over the first d terms. This is a symmetrical property—the intersection of S with T is the same as the intersection of T with S. p is an independent parameter, and we therefore have $RBO(V_1,V_2,p)=RBO(V_2,V_1,p)$, and hence this also holds for RBD.

Identity of indiscernibles Let us consider $V_1=V_2$, that is for every term d, $V_1(d),V_2(d)$. Then, $\forall d, X_d=d$ and $A_d=1$. We then have

$$\begin{aligned} \sum _{d=1}^\infty p^{d-1}\cdot A_d=\sum _{d=1}^\infty p^{d-1} \end{aligned}$$

As before, this is the sum of the geometric progression $p^{d-1}$ and is equal $\frac{1}{1-p}$. Therefore, for $V_1=V_2$ we get $RBO(V_1,V_2,p)=1$ and $RBD(V_1,V_2,p)=0$.

Now, consider two distinct lists. that is, for some term d, $V_1(d)\ne V_2(d)$. For that term we have $X_d<d$ and $A_d<1$. We therefore have

$$\begin{aligned} \sum _{d=1}^\infty p^{d-1}\cdot A_d<\frac{1}{1-p} \end{aligned}$$

and therefore $RBO(V_1,V_2,p)<1$ and $RBD(V_1,V_2,p)>0$.

1.2 Cosine Similarity

As we are dealing with frequency vectors, all coordinates are non-negative. All vectors are therefore in the first orthant and the angles between them are in the range $[0,\pi /2]$ radians, and therefore, $cos\theta$ is in the range [0, 1] and so is $1-cos\theta$.

Symmetry Both the dot product and the standard multiplication are commutative operations and therefore $D(V_1,V_2)=\frac{V_1\cdot V_2}{\Vert V_1 \Vert \times \Vert V_2 \Vert }=\frac{V_2\cdot V_1}{\Vert V_2 \Vert \times \Vert V_1 \Vert }=D(V_2,V_1)$, and therefore, also $1-D(V_1,V_2)=1-D(V_2,V_1)$.

Identity of indiscernibles First, we note that for vectors of length n, the standard dot product $V_1\cdot V_2$ is defined as $\sum _{i=1}^n V_{1_i}\times V_{2_i}$ and the standard euclidean norm is defined as $\sqrt{\sum _{i=1}^n (V_i)^2}$. Assume $V_1=V_2$, i.e., for every i $V_1(i)=V_2(i)$. We then have $V_1\cdot V_2=\sum _{i=1}^n V_1(i)\times V_2(i) = \sum _{i=1}^n V_1(i)\times V_1(i) = \sum _{i=1}^n V_1(i)^2$. We also have $\Vert V_1 \Vert = \Vert V_2\Vert$ and therefore $\Vert V_1\Vert \times \Vert V_2\Vert =\Vert V_1\Vert ^2=\sqrt{\sum _{i=1}^n V_1(i)^2}^2=\sum _{i=1}^n V_1(i)^2$. Therefore, for $V_1=V_2$ we have $D(V_1,V_2)=\frac{V_1\cdot V_2}{\Vert V_1 \Vert \times \Vert V_2 \Vert }=\frac{\sum _{i=1}^n V_1(i)^2}{\sum _{i=1}^n V_1(i)^2}=1$ and $1-D(V_1,V_2)=0$.

Assume $1-D(V_1,V_2)=0$, that is $D(V_1,V_2)=1$. As we have seen when proving non-negativity, this implies that the angle $\theta$ between $V_1$ and $V_2$ is zero. Since both vectors are frequency vectors, i.e., $\Vert V_1\Vert =\Vert V_2\Vert =1$ this means they are the same vector.

1.3 L1 norm

Non-negativity It’s enough to prove non-negativity for each element of the sum, but that is assured by the absolute value.

Symmetry Again, it’s enough to show symmetry for each element of the sum. We note that $(V_1(x)-V_2(x))=-(V_2(x)-V_1(x))$ and therefore $\arrowvert (V_1(x)-V_2(x))\arrowvert =\arrowvert (V_2(x)-V_1(x))\arrowvert$

Identity of indiscernibles Assume $V_1=V_2$, i.e for every $x\in X$ we have $V_1(x)=V_2(x)$. Then $V_1(x)-V_2(x)=0$ and we have $L1(V_1,V_2)=0$.

Assume $V_1\ne V_2$ i.e., for some $x\in X$ we have $V_1(x)\ne V_2(x)$. For that x we have $\big [V_1(x)-V_2(x)\big ]>0$ and as all elements in the sum are non-negative we have $L1(V_1,V_2)>0$.

1.4 KL divergence

Non-negativity It is enough to show that each element in the sum is non-negative. For every $x\in X$ either $V_1(x)<V_2(x)$, $V_1(x)V_2(x)$ or $V_1(x)=V_2(x)$. If $V_1(x)<V_2(x)$, then $V_1(x)-V_2(x)<0$ and $log\frac{V_1(x)}{V_2(x)}<0$. Therefore, $\big [V_1(x)-V_2(x)\big ]log\frac{V_1(x)}{V_2(x)}>0$. If $V_1(x)>V_2(x)$, then $V_1(x)-V_2(x)>0$ and $log\frac{V_1(x)}{V_2(x)}>0$ and again $\big [V_1(x)-V_2(x)\big ]log\frac{V_1(x)}{V_2(x)}>0$. Lastly, if $V_1(x)=V_2(x)$ then $V_1(x)-V_2(x)=0$ and $log\frac{V_1(x)}{V_2(x)}=0$ and we have $\big [V_1(x)-V_2(x)\big ]log\frac{V_1(x)}{V_2(x)}=0$

Symmetry We note that $(V_1(x)-V_2(x))=-(V_2(x)-V_1(x))$ and likewise $log\frac{V_1(x)}{V_2(x)}=-log\frac{V_2(x)}{V_1(x)}$ and therefore $(V_1(x)-V_2(x))log\frac{V_1(x)}{V_2(x)}=(V_2(x)-V_1(x))log\frac{V_2(x)}{V_1(x)}$. Again, this is true for each element in the sum and therefore holds for the entire sum.

Identity of indiscernibles Assume $V_1=V_2$, i.e for every $x\in X$ we have $V_1(x)=V_2(x)$. Then $V_1(x)-V_2(x)=0$ and $log\frac{V_1(x)}{V_2(x)}=1$ and we have $\big [V_1(x)-V_2(x)\big ]log\frac{V_1(x)}{V_2(x)}=0$.

Assume $V_1\ne V_2$ i.e., for some $x\in X$ we have $V_1(x)\ne V_2(x)$. For that x we have $\big [V_1(x)-V_2(x)\big ]log\frac{V_1(x)}{V_2(x)}>0$ and as all elements in the sum are non-negative we have $D([V_1\Arrowvert V_2]=\sum _{x\in X}\Bigg [\big [V_1(x)-V_2(x)\big ]log\Big [\frac{V_1(x)}{V_2(x)}\Big ]\Bigg ]>0$

Appendix 2: Datasets details

Books dataset The Books dataset is comprised of 30 English language books, taken from the Gutenberg project’s most popular books list. Each book is divided into chapters, varying from 7 to 61 chapters per book. We included only chapters that contain more than 150 words in the text and omitted words that appear less than five times in the book to avoid the extreme consequences of a very long tail. Each text snippet is labeled with an author’s name, and belongs to a book, and more specifically to a chapter in the book. In the Validation section, we use this subdivision to validate whether our method is able to distinguish between two texts written by the same person and two texts written by different people. We also test its ability to distinguish between a text written by one person and a text written by several, by collecting a number of chapters from different books to one virtual book. Table 8 in Appendix 1 lists the book names and relevant statistics.

IMDb reviews dataset IMDb (Internet Movie Database) is among the world’s most popular and authoritative sources for movie, TV and celebrity content. It offers a searchable database of more than 185 million data items, including more than 3.5 million films. This dataset contains 1,406,000 movie reviews, spanning the period of July 1998–June 2016. We use their movie reviews text as our dataset for demonstrating the applications of our method. We define a user/author (contributor) as a registered person who published a movie review on IMDb. Each author is identified by a unique user id and alias. Each review contains a text, a timestamp, and an author ID. The original obtained IMDb dataset contained 467, 961 users. To have a large enough sample of text for each user, we extracted only users who published at least 30 reviews. 3969 users met this criterion. We defined $n=30$ (number of different reviews per user) as our lower threshold to achieve statistical inference. The choice of $n = 30$ for a boundary between small and large samples is already used as a rule of thumb in many research areas: “The number 30 seems to have arisen from the understanding that with fewer than 30 cases, you were dealing with small samples that required specialized handling with small-sample statistics instead of the critical-ratio approach we have been taught” (Cohen 1990).

Table 8 Books datasets characteristics

Table 9 IMDb dataset characteristics