Random walk-based ranking in signed social networks: model and algorithms

Abstract

How can we rank nodes in signed social networks? Relationships between nodes in a signed network are represented as positive (trust) or negative (distrust) edges. Many social networks have adopted signed networks to express trust between users. Consequently, ranking friends or enemies in signed networks has received much attention from the data mining community. The ranking problem, however, is challenging because it is difficult to interpret negative edges. Traditional random walk-based methods such as PageRank and random walk with restart cannot provide effective rankings in signed networks since they assume only positive edges. Although several methods have been proposed by modifying traditional ranking models, they also fail to account for proper rankings due to the lack of ability to consider complex edge relations. In this paper, we propose Signed Random Walk with Restart (SRWR), a novel model for personalized ranking in signed networks. We introduce a signed random surfer so that she considers negative edges by changing her sign for walking. Our model provides proper rankings considering signed edges based on the signed random walk. We develop two methods for computing SRWR scores: SRWR-Iter and SRWR-Pre which are iterative and preprocessing methods, respectively. SRWR-Iter naturally follows the definition of SRWR, and iteratively updates SRWR scores until convergence. SRWR-Pre enables fast ranking computation which is important for the performance of applications of SRWR. Through extensive experiments, we demonstrate that SRWR achieves the best accuracy for link prediction, predicts trolls $4\times$ more accurately, and shows a satisfactory performance for inferring missing signs of edges compared to other competitors. In terms of efficiency, SRWR-Pre preprocesses a signed network $4.5 \times$ faster and requires $11 \times$ less memory space than other preprocessing methods; furthermore, SRWR-Pre computes SRWR scores up to $14 \times$ faster than other methods in the query phase.

Appendix

1.1 Details of the hub-and-spoke reordering method

SlashBurn [19, 28] is a node reordering algorithm which concentrates nonzero entries of the adjacency matrix of a given graph based on the hub-and-spoke structure. Let n be the number of nodes in a graph, and t be the hub selection ratio whose range is between 0 and 1 where $\lceil tn \rceil$ indicates the number of nodes selected by SlashBurn as hubs. For each iteration, SlashBurn disconnects $\lceil tn \rceil$ high-degree nodes, called hub nodes, from the graph; then the graph is split into the giant connected component (GCC) and the disconnected components. The nodes in the disconnected components are called spokes, and each disconnected component forms a block in $\mathbf {|H|}_{11}$ (or $\mathbf {T}_{11}$) in Fig. 6. Then, SlashBurn reorders nodes such that the hub nodes get the highest ids, the spokes get the lowest ids, and the nodes in the GCC get the ids in the middle. SlashBurn repeats this procedure on the GCC recursively until the size of GCC becomes smaller than $\lceil tn \rceil$. After SlashBurn is done, the reordered adjacency matrix contains a large and sparse block diagonal matrix in the upper left area, as shown in Fig. 6. Figure 14 depicts the procedure of SlashBurn when $\lceil tn \rceil =1$.

1.2 Properties and lemmas

1.2.1 Sum of positive and negative SRWR scores

Property 3

Consider the recursive equation $\mathbf {p}= (1-c)|{{\tilde{\mathbf {A}}}}|^{\top }\mathbf {p} + c\mathbf {q}$ where $\mathbf {p}= \mathbf {r}^{+}+ \mathbf {r}^{-}$ and $|{{\tilde{\mathbf {A}}}}|^{\top }$ is a column stochastic matrix. Then $\mathbf {1}^{\top }\mathbf {p}= \sum _{i}\mathbf {p}_{i} = 1$.

Proof

By multiplying both sides by $\mathbf {1}^{\top }$, the equation is represented as follows:

$$\begin{aligned} \mathbf {p}= (1-c)|{{\tilde{\mathbf {A}}}}|^{\top }\mathbf {p} + c\mathbf {q}\Leftrightarrow \mathbf {1}^{\top }\mathbf {p}= (1-c)\mathbf {1}^{\top }|{{\tilde{\mathbf {A}}}}|^{\top }\mathbf {p} + c\mathbf {1}^{\top }\mathbf {q}\end{aligned}$$

Note that $\mathbf {1}^{\top }|{{\tilde{\mathbf {A}}}}|^{\top } = (|{{\tilde{\mathbf {A}}}}|\mathbf {1})^{\top }$, and $|{{\tilde{\mathbf {A}}}}|$ is a row stochastic matrix; thus, $(|{{\tilde{\mathbf {A}}}}|\mathbf {1})^{\top } = \mathbf {1}^{\top }$. Hence, the above equation is represented as follows:

$$\begin{aligned} \mathbf {1}^{\top }\mathbf {p}= (1-c)\mathbf {1}^{\top }|{{\tilde{\mathbf {A}}}}|^{\top }\mathbf {p} + c\mathbf {1}^{\top }\mathbf {q}\Leftrightarrow \mathbf {1}^{\top }\mathbf {p}= (1-c)\mathbf {1}^{\top }\mathbf {p}+ c \Leftrightarrow \mathbf {1}^{\top }\mathbf {p}= 1 \end{aligned}$$

$\square$

Fig. 14

Node reordering based on hub-and-spoke method when $\lceil tn \rceil =1$ where $\lceil tn \rceil$ indicates the number of selected hubs at each step, and t is the hub selection ratio ($0< t < 1$). Red nodes are hubs; blue nodes are spokes that belong to the disconnected components; green colored are nodes that belong to the giant connected component. At Step 1 in (a), the method disconnects a hub node, and assigns node ids as shown in (b). The hub node gets the highest id (14), the spoke nodes get the lowest ids (1–7), and the GCC gets the middle ids (8–13). The next iteration starts on the GCC in (b), and the node ids are assigned as in (c)

1.2.2 Analysis on number of iterations of SRWR-Iter

Lemma 3

Suppose $\mathbf {h}= [\mathbf {r}^{+};\mathbf {r}^{-}]^{\top }$, and $\mathbf {h}^{(k)}$ is the result of kth iteration in SRWR-Iter. Let $\delta ^{(k)}$ denote the error $||\mathbf {h}^{(k)} - \mathbf {h}^{(k-1)} ||_{1}$. Then $\delta ^{(k)} \le 2(1-c)^{k}$, and the estimated number T of iterations for convergence is $\log _{1-c}\frac{\epsilon }{2}$ where $\epsilon$ is an error tolerance, and c is the restart probability.

Proof

According to Eq. (4), $\delta ^{(k)}$ is represented as follows:

$$\begin{aligned} \delta ^{(k)} = ||\mathbf {h}^{(k)} - \mathbf {h}^{(k-1)} ||_{1}&= (1-c) ||{{\tilde{\mathbf {B}}}}^{\top } (\mathbf {h}^{(k-1)} - \mathbf {h}^{(k-2)}) ||_{1} \\&\le (1-c) ||{{\tilde{\mathbf {B}}}}^{\top } ||_{1}||\mathbf {h}^{(k-1)} - \mathbf {h}^{(k-2)} ||_{1} \\&= (1-c) ||\mathbf {h}^{(k-1)} - \mathbf {h}^{(k-2)} ||_{1} = (1-c)\delta ^{(k-1)} \end{aligned}$$

Note that $||{{\tilde{\mathbf {B}}}}^{\top } ||_{1} = 1$ since ${{\tilde{\mathbf {B}}}}^{\top }$ is column stochastic as described in Theorem 1. Hence, $\delta ^{(k)} \le (1-c)\delta ^{(k-2)} \le \cdots \le (1-c)^{k}\delta ^{(1)}$. Since $\delta ^{(1)} = ||\mathbf {h}^{(1)} - \mathbf {h}^{(0)} ||_{1} \le ||\mathbf {h}^{(1)}||_{1} + ||\mathbf {h}^{(0)} ||_{1} = 2$, $\delta ^{(k)} \le 2(1-c)^{k}$. Note that when $\delta ^{(k)} \le \epsilon$, the iteration of SRWR-Iter is terminated. Thus, for $k \le \log _{1-c}\frac{\epsilon }{2}$, the iteration is terminated, and the number T of iterations for convergence is estimated at $\log _{1-c}\frac{\epsilon }{2}$. $\square$

1.2.3 Time complexity of sparse matrix multiplication

Lemma 4

(Sparse Matrix Multiplication [32]) Suppose that $\mathbf {A}$ and $\mathbf {B}$ are $p \times q$ and $q \times r$ sparse matrices, respectively, and $\mathbf {A}$ has $nnz (\mathbf {A})$ nonzeros. Calculating $\mathbf {C}=\mathbf {A}\mathbf {B}$ using sparse matrix multiplication requires $O( nnz (\mathbf {A})r)$.

1.3 Complexity analysis of proposed methods for SRWR

We analyze the complexity of our proposed methods SRWR-Iter and SRWR-Pre in terms of time and space. The space and time complexities of SRWR-Iter are presented in Lemma 5, and those of SRWR-Pre are in Lemmas 6, 7, and 8 , respectively.

1.3.1 Space and time complexities of SRWR-Iter

Lemma 5

(Space and Time Complexities of SRWR-Iter) Let n and m denote the number of nodes and edges of a signed network, respectively. Then the space complexity of Algorithm 2 is $O(n+m)$. The time complexity of Algorithm 2 is $O(T(n+m))$ where the number T of iterations is $\log _{1-c}\frac{\epsilon }{2}$, c is the restart probability, and $\epsilon$ is an error tolerance.

Proof

The space complexity for ${{\tilde{\mathbf {A}}}_{+}}$ and ${{\tilde{\mathbf {A}}}_{-}}$ is O(m) if we exploit a sparse matrix format such as compressed column storage to save the matrices. We need O(n) for SRWR score vectors $\mathbf {r}^{+}$ and $\mathbf {r}^{-}$. Thus, the space complexity is $O(n+m)$. One iteration in Algorithm 2 takes $O(n+m)$ time due to sparse matrix vector multiplications and vector additions where the time complexity of a sparse matrix vector multiplication is linear to the number of nonzeros of a matrix [7]. Hence, the total time complexity is $O(T(n+m))$ where the number T of iterations is $\log _{1-c}\frac{\epsilon }{2}$ which is proved in Lemma 3. $\square$

1.3.2 Space and time complexities of SRWR-Pre

Lemma 6

(Space Complexity of SRWR-Pre) The space complexity of the preprocessed matrices from SRWR-Pre is $O(n_{2}^{2} + m)$ where $n_{2}$ is the number of hubs and m is the number of edges in the graph.

Proof

The space complexity of each preprocessed matrix is summarized in Table 7. ${{\tilde{\mathbf {A}}}_{-}}$, $\mathbf {|H|}_{12}$, $\mathbf {|H|}_{21}$, $\mathbf {T}_{12}$, and $\mathbf {T}_{21}$ are sparse matrices, and constructed from the input graph; hence, the space complexity is bounded by the number of edges (i.e., O(m)). Note that $\mathbf {|H|}$ and $\mathbf {T}$ have the same sparsity pattern; hence, $\mathbf {|H|}_{11}$ and $\mathbf {T}_{11}$ identified by [19, 28] have the same b blocks. The ith block in $\mathbf {|H|}_{11}^{-1}$ (or $\mathbf {T}_{11}^{-1}$) contains $n_{1i}^{2}$ nonzeros; therefore, $\mathbf {|H|}_{11}^{-1}$ and $\mathbf {T}_{11}^{-1}$ require $O(\sum _{i=1}^{b}n_{1i}^{2})$ space, respectively. Since the dimension of $\mathbf {L}^{-1}_{\mathbf {|H|}}$, $\mathbf {U}^{-1}_{\mathbf {|H|}}$, $\mathbf {L}^{-1}_{\mathbf {T}}$, and $\mathbf {U}^{-1}_{\mathbf {T}}$ is $n_2$, they require $O(n_2^2)$ space. $\square$

Note that the blocks in $\mathbf {|H|}_{11}$ (or $\mathbf {T}_{11}$) are discovered by the reordering method [19, 28] as briefly described in Appendix A.1. In real-world graphs, $\sum _{i=1}^{b}n_{1i}^{2}$ can be bounded by O(m) as shown in [34]. Hence, we assume that the space complexity of $\mathbf {|H|}_{11}^{-1}$ and $\mathbf {T}_{11}^{-1}$ is O(m) for simplicity.

Table 7 Space complexity of each preprocessed matrix from Algorithm 3

Lemma 7

(Time Complexity of Preprocessing Phase in SRWR-Pre) The preprocessing phase in Algorithm 3 takes $O(T(m+n\log {n}) + n_{2}^{3} + mn_{2})$ where $T=\lceil {\frac{n_{2}}{tn}}\rceil$ is the number of iterations, and t is the hub selection ratio in the hub-and-spoke reordering method [19, 28].

Proof

We only consider the main factors of the time complexity of Algorithm 3 in this proof. The hub-and-spoke reordering method takes $O(T(m + n\log {n}))$ time (line 1) where T is $\lceil {\frac{n_{2}}{tn}}\rceil$ which is proved in [19, 28]. Computing the Schur complement of $\mathbf {|H|}_{11}$ takes $O(n_{2}^{2} + mn_{2})$ because it takes $O(mn_{2})$ to compute $\mathbf {P}_{1} = \mathbf {|H|}_{11}^{-1}\mathbf {|H|}_{12}$ and $\mathbf {P}_{2} = \mathbf {|H|}_{21}\mathbf {P}_{1}$ by Lemma 4, and $O(n_{2}^{2})$ to compute $\mathbf {|H|}_{22} - \mathbf {P}_{2}$ (line 6). It takes $O(n_{2}^{3})$ to compute the inverse of the LU factors (line 8). Note that computing $\mathbf {|H|}^{-1}_{11}$(line 4) requires $O(\sum _{i=1}^{b}n_{1i}^{3})$ time where it takes $n_{1i}^{3}$ to obtain the inverse of ith block. In real-world networks, the size $n_{1i}$ of each block is much smaller than the number $n_2$ of hubs; thus, we assume that $\sum _{i=1}^{b}n_{1i}^{3} \ll n_{2}^{3}$ [34]. Hence, the time complexity of preprocessing $\mathbf {|H|}$ is $O(T(m + n\log {n}) + n_{2}^{3} + mn_{2})$. Note that the time complexity of preprocessing $\mathbf {T}$ is included into that of preprocessing $\mathbf {|H|}$ since $\mathbf {T}$ and $\mathbf {|H|}$ have the same sparsity pattern. $\square$

Lemma 8

(Time Complexity of Query Phase in SRWR-Pre) The query phase in Algorithm 4 takes $O(n_{2}^{2} + n + m)$ time.

Proof

We only consider the main factors of the time complexity of Algorithm 4 in this proof. It takes $O(n_{2}^{2} + m)$ to compute $\mathbf {p}_{2}$ since it takes $O(n_{2} + m)$ to compute ${{\tilde{\mathbf{q}}}}_{2} = \mathbf {q}_{2} - \mathbf {|H|}_{21}(\mathbf {|H|}^{-1}_{11}\mathbf {q}_{1})$, and $O(n_{2}^{2}$) to compute $\mathbf {U}^{-1}_{\mathbf {|H|}}(\mathbf {L}^{-1}_{\mathbf {|H|}}{{\tilde{\mathbf{q}}}}_{2})$ (line 2). It takes O(n) time to concatenate the partitioned vectors (lines 4 and 8) and compute $\mathbf {r}^{+}$ and $\mathbf {r}$ (lines 9 and 10 ). Hence, the total time complexity of the query phase is $O(n_{2}^{2} + n + m)$. $\square$

1.4 Detailed limitations of existing random walk-based ranking models in signed networks

In this section, we describe the detailed limitation of existing random walk-based ranking models which are briefly described in Sect. 1.

• Random Walk with Restart (RWR): We perform RWR on a given signed network after taking absolute edge weights to obtain $\mathbf {r}$ as follows:

  $$\begin{aligned} \mathbf {r}= (1-c){|{\tilde{\mathbf {A}}}|}^{\top }\mathbf {r}+ c\mathbf {q} \end{aligned}$$

  where ${|{\tilde{\mathbf {A}}}|}$ is the row-normalized matrix of the absolute adjacency matrix in the signed network. RWR does not properly consider negative edges for $\mathbf {r}$.

• Modified Random Walk with Restart (M-RWR) [33]: M-RWR applies RWR separately on both a positive subgraph and a negative subgraph; thus, it obtains $\mathbf {r}^{+}$ on the positive subgraph and $\mathbf {r}^{-}$ on the negative subgraph, and then, computes $\mathbf {r}= \mathbf {r}^{+}- \mathbf {r}^{-}$. The detailed equations for M-RWR are as follows:

  $$\begin{aligned} \mathbf {r}^{+}= (1-c){{\tilde{\mathbf {B}}}}_{+}^{\top }\mathbf {r}^{+}+ c\mathbf {q} \text { and } \mathbf {r}^{-}= (1-c){{\tilde{\mathbf {B}}}}_{-}^{\top }\mathbf {r}^{-}+ c\mathbf {q} \end{aligned}$$

  where ${{\tilde{\mathbf {B}}}}_{+}$ is the row-normalized matrix of the adjacency matrix containing only positive edges, and ${{\tilde{\mathbf {B}}}}_{-}$ is that of the absolute adjacency matrix containing only negative edges. The main limitation of M-RWR is that it does not consider relationships between positive and negative edges due to the separation as shown in the above equations.

• Modified Personalized SALSA (M-PSALSA) [30]: Andrew et al. made a modification on SALSA¹ by introducing the random jump into it, called Personalized SALSA (PSALSA). As similar to M-RWR, we apply PSALSA separately on both positive and negative subgraphs and consider authorities on the positive subgraph as $\mathbf {r}^{+}$, and those scores on the negative subgraph as $\mathbf {r}^{-}$. M-PSALSA also has the same limitation with M-RWR.

• Personalized Signed Spectral Rank (PSR) [23]: Kunegis et al. proposed PSR which is a variant of PageRank by constructing the following matrix similar to Google matrix:

  $$\begin{aligned} \mathbf {M}_{PSR} = (1-c)\mathbf {D}^{-1}\mathbf {A}^{\top } + c\mathbf {e}_{s}\mathbf {1}^{\top } \end{aligned}$$

  where $\mathbf {A}$ is the signed adjacency matrix, $\mathbf {D}$ is the diagonal out-degree matrix, and $\mathbf {e}_{s}$ is the sth unit vector. Then, PSR computes the left eigenvector of $\mathbf {M}_{PSR}$, which induces a relative trustworthy score vector $\mathbf {r}$ including positive and negative values. Although PSR is able to produce $\mathbf {r}$, the equation for PSR is heuristic because $\mathbf {M}_{PSR}$ is not a column stochastic matrix. Also, how the random surfer based on the equation interprets negative edges is veiled.

1.5 Detailed description of evaluation metrics

We describe the details of metrics used in the link prediction and the troll identification tasks. The metrics for the sign prediction task is described in Sect. 5.5.

1.5.1 Link prediction

• GAUC (Generalized AUC): Song et al. [35] proposed GAUC which measures the quality of link prediction in signed networks. An ideal personalized ranking w.r.t. a seed node s needs to rank nodes with positive links to s at the top, those with negative links at the bottom, and other unknown status nodes in the middle of the ranking. For a seed node s, suppose that $\mathbf {P}_{s}$ is the set of positive nodes potentially connected by s, $\mathbf {N}_{s}$ is that of negative nodes, and $\mathbf {O}_{s}$ is that of the other nodes. Then, GAUC of the personalized ranking w.r.t. s is defined as follows:

  $$\begin{aligned} \text {GAUC}_{s}&= \frac{\eta }{|\mathbf {P}_{s}|(|\mathbf {O}_{s}| + |\mathbf {N}_{s}|)}\left( \sum _{p \in \mathbf {P}_{s}}\sum _{i \in \mathbf {O}_{s} \cup \mathbf {N}_{s}} {\mathbb {I}}(\mathbf {r}_{p} > \mathbf {r}_{i}) \right) \\&\quad + \frac{1-\eta }{|\mathbf {N}_{s}|(|\mathbf {O}_{s}| + |\mathbf {P}_{s}|)} \left( \sum _{i \in \mathbf {O}_{s} \cup \mathbf {P}_{s}} \sum _{n \in \mathbf {N}_{s}} {\mathbb {I}}(\mathbf {r}_{i} < \mathbf {r}_{n}) \right) \end{aligned}$$

  where $\eta = \frac{|\mathbf {P}_{s}|}{|\mathbf {P}_{s}| + |\mathbf {N}_{s}|}$ is the relative ratio of the number of positive edges and that of negative edges, and ${\mathbb {I}}(\cdot )$ is an indicator function that returns 1 if a given predicate is true, or 0 otherwise. GAUC will be 1.0 for the perfect ranking list and 0.5 for a random ranking list [35].

• AUC (Area Under the Curve): AUC of the personalized ranking scores $\mathbf {r}$ w.r.t. seed node s in signed networks is defined as follows [35]:

  $$\begin{aligned} \text {AUC}_{s} = \frac{1}{|\mathbf {P}_{s}||\mathbf {N}_{s}|}\sum _{p\in \mathbf {P}_{s}}\sum _{n \in \mathbf {N}_{s}}{\mathbb {I}}(\mathbf {r}_{p} > \mathbf {r}_{n}) \end{aligned}$$

  where $\mathbf {P}_{s}$ is the set of positive nodes potentially connected by s, and $\mathbf {N}_{s}$ is the set of negative nodes. ${\mathbb {I}}(\cdot )$ is an indicator function that returns 1 if a given predicate is true, or 0 otherwise. With an ideal ranking list, AUC should be 1 representing each positive sample is ranked higher than all the negative samples. For a random ranking, AUC will be 0.5. However, AUC is not a satisfactory metric for the link prediction task in signed networks because AUC is designed for two classes (positive and negative), while the link prediction in signed networks should consider three classes (positive, unknown, and negative) as described in the above.

1.5.2 Troll identification

Suppose that we have a personalized ranking ${\mathcal {R}}$ in the ascending order of the trustworthiness scores w.r.t. a seed node (i.e., a node with a low score is ranked high) to have the same effect of searching trolls in the bottom of the original ranking in the descending order of those scores.

• MAP@k (Mean Average Precision): MAP@k is the mean of average precisions, AP@k, for multiple queries. Suppose that there are l trolls to be captured. Then, AP@k is defined as follows:

  $$\begin{aligned} \text {AP@}k = \frac{1}{\min (l, k)}\left( \sum _{t \in \mathbf {T}} \text {Precision@}t\right) \end{aligned}$$

  where $\text {Precision@}t$ is the precision at the cutoff t. Note that $\mathbf {T} = \{t | {\mathbb {I}}({\mathcal {R}}[t]) = 1 \text { for } 1 \le t \le k \}$ where ${\mathcal {R}}[t]$ denotes the user ranked at position t in the ranking ${\mathcal {R}}$, and ${\mathbb {I}}({\mathcal {R}}[t])$ is 1 if ${\mathcal {R}}[t]$ is a troll. For N queries, MAP@k is defined as follows:

  $$\begin{aligned} \text {MAP@}k = \frac{1}{N}\left( \sum _{i=1}^{N}\text {AP@}k\right) \end{aligned}$$

• NDCG@k (Normalized Discount Cumulative Gain): NDCG is the normalized value of Discount Cumulative Gain (DCG), which is defined as follows:

  $$\begin{aligned} \text {DCG}@k = rel_{1} + \sum _{i=2}^{k}\frac{rel_i}{log_{2}{(i)}}, \quad \text {and}\quad \text {NDCG}@k = \frac{\text {DCG}@k}{\text {IDCG}@k} \end{aligned}$$

  where $rel_{i}$ is the user-graded relevance score for the ith ranked item. Then, NDCG@k is obtained by normalizing using Ideal DCG(IDCG) which is the DCG for the ideal order of ranking.

• Precision@k and Recall@k: Precision@k (Recall@k) is the precision (recall) at the cutoff k in a ranking. Precision@k is the ratio of identified trolls in top-k ranking, and Recall@k is the ratio of identified trolls in the total trolls.

• MRR (Mean Reciprocal Rank): MRR@k is the mean of the reciprocal rank (RR) for each the top-k query response. RR is the multiplicative inverse of the rank of the first correct answer. Hence, for N multiple queries, MRR@k is defined as follows:

  $$\begin{aligned} \text {MRR}@k = \frac{1}{N}\sum _{i=1}^{N}\frac{1}{rank_{i}} \end{aligned}$$

  where $rank_{i}$ is the rank position of the first relevant item in the top-k ranking. If there is no relevant item in the ranking for the ith query, the inverse of the rank, ${rank_{i}}^{-1}$, becomes zero.

1.6 Discussion on relative trustworthiness scores of SRWR

In Sect. 4.1, we define the relative trustworthiness $\mathbf {r}= \mathbf {r}^{+}- \mathbf {r}^{-}$ where $\mathbf {r}^{+}$ is for positive SRWR scores, and $\mathbf {r}^{-}$ is for negative SRWR ones. We show that $\mathbf {r}^{+}$ and $\mathbf {r}^{-}$ are measures, and $\mathbf {r}$ is a signed measure using definitions from measure theory [38]. We first introduce the definition of measure as follows:

Definition 5

(Measure [38]) A measure $\mu$ on a (finite) set $\Omega$ with $\sigma$-algebra ${\mathcal {A}}$ is a function $\mu : {\mathcal {A}} \rightarrow {\mathbb {R}}_{\ge 0}$ such that

• 1. (Nonnegativity) $\mu (E) \ge 0$$\forall E \in {\mathcal {A}}$,

• 2. (Null empty set) $\mu (\emptyset ) = 0$,

• 3. (Countable additivity) $\mu (\bigcup _{i=1}^{\infty }E_{i})=\sum _{i=1}^{\infty }E_{i}$ for any sequence of pairwise disjoint sets, $E_1, E_2, \ldots \in {\mathcal {A}}$

where $\sigma$-algebra ${\mathcal {A}}$ on $\Omega$ is a collection ${\mathcal {A}}\subseteq 2^{\Omega }$ s.t. it is nonempty, and closed under complements (i.e., $E \in {\mathcal {A}} \Rightarrow E^{c} \in {\mathcal {A}}$) and countable unions (i.e., $E_1, E_2, \cdots \in {\mathcal {A}} \Rightarrow \bigcup _{i=1}^{\infty }E_{i}\in {\mathcal {A}}$). The pair of $(\Omega , {\mathcal {A}})$ is called measurable space. $\square$

In probability theory, $\sigma$-algebra ${\mathcal {A}}$ describes all possible events to be measured as probability. Note that $\mathbf {r}^{+}$ and $\mathbf {r}^{-}$ are joint probabilities of nodes and signs, i.e., $\mathbf {r}^{+}_{u}=P(N={u}, S=+)$ and $\mathbf {r}^{-}_{u}=P(N=u, S=-)$ where N is a random variable of nodes, and S is a random variable of the surfer’s sign. Note that N takes an item from $\sigma$-algebra ${\mathcal {A}}$. The following property shows that $\mathbf {r}^{+}$ and $\mathbf {r}^{-}$ are (nonnegative) measures.

Property 4

Suppose $\Omega$ is the set $\mathbf {V}$ of nodes, and $\sigma$-algebra ${\mathcal {A}}$ on $\Omega$ is $2^\Omega$. Let $\mu ^{+}=P(N, S=+)$ and $\mu ^{-}=P(N, S=-)$. Then, both $\mu ^{+}$ and $\mu ^{-}$ are (nonnegative) measures according to Definition 5.

Proof

For any $E \in {\mathcal {A}}$, $\mu ^{+}(E) \ge 0$ and $\mu ^{+}(\emptyset ) = 0$ are obviously true since $P(N, S=+)$ is a probability; hence, $P(E, S=+) \ge 0$ and $P(\emptyset , S=+) = 0$. Let $(E_n)_{n\in {\mathbb {N}}}$ be a sequence of pairwise disjoint sets where $E_n \in {\mathcal {A}}$. Since the sets in the sequence are mutually disjoint, the following holds:

$$\begin{aligned} P\left( \bigcup _{n \in {\mathbb {N}}}E_n, S=+\right) = \sum _{n \in {\mathbb {N}}} P(E_n, S=+) \end{aligned}$$

Therefore, $\mu ^{+}=P(N, S=+)$ is a measure by Definition 5. Similarly, $\mu ^{-}=P(N, S=-)$ is also a measure. $\square$

Next, we introduce the definition of signed measure, a generalized version of measure by allowing it to have negative values.

Definition 6

(Signed Measure [38]) Given a set $\Omega$ and $\sigma$-algebra ${\mathcal {A}}$, a signed measure on $(\Omega , {\mathcal {A}})$ is a function $\mu : {\mathcal {A}} \rightarrow {\mathbb {R}}$ such that

• 1. (Real value) $\mu (E)$ takes a real value in ${\mathbb {R}}$,

• 2. (Null empty set) $\mu (\emptyset ) = 0$,

• 3. (Countable additivity) $\mu (\bigcup _{i=1}^{\infty }E_{i})=\sum _{i=1}^{\infty }E_{i}$ for any sequence of pairwise disjoint sets, $E_1, E_2, \cdots \in {\mathcal {A}}$$\square$

Note that Shannon entropy and electric charge are representative examples of signed measure. Then, the following lemma indicates the difference between two nonnegative measures is a signed measure.

Lemma 9

(Difference Between Two Nonnegative Measures [38]) Suppose we are given nonnegative measure $\mu ^{+}$ and $\mu ^{-}$ on the same measurable space $(\Omega , {\mathcal {A}})$. Then, $\mu = \mu ^{+} - \mu ^{-}$ is a signed measure.

Proof

Since $\mu ^{+}$ and $\mu ^{-}$ are nonnegative, $\mu$ is located between $-\infty$ and $\infty$. Also, $\mu (\emptyset ) = \mu ^{+}(\emptyset ) - \mu ^{-}(\emptyset ) = 0$. Moreover, $\mu$ is countable additive, i.e.,

$$\begin{aligned} \mu \left( \bigcup _{i=1}^{\infty }E_{i}\right) = \mu ^{+}\left( \bigcup _{i=1}^{\infty }E_{i}\right) - \mu ^{-}\left( \bigcup _{i=1}^{\infty }E_{i}\right) = \sum _{i=1}^{\infty }\left( \mu ^{+}(E_i) - \mu ^{-}(E_i)\right) = \sum _{i=1}^{\infty }\mu (E_i) \end{aligned}$$

Hence, $\mu = \mu ^{+} - \mu ^{-}$ is a signed measure according to Definition 6. $\square$

Lemma 9 implies that the relative trustworthiness $\mathbf {r}= \mathbf {r}^{+}- \mathbf {r}^{-}$ is a signed measure. The trustworthiness $\mathbf {r}_u$ measures a degree of trustworthiness between seed node s and node u: if $\mathbf {r}_{u} > 0$, seed node s is likely to trust node u as much as $\mathbf {r}_{u}$, while if $\mathbf {r}_{u} < 0$, s is likely to distrust u as much as $\mathbf {r}_{u}$.