Link prediction in directed complex networks: combining similarity-popularity and path patterns mining

Abstract

Discovering new relationships between entities in networked data is essential in various applications such as sociology, security, physics, and biology. This paper introduces a novel approach to directed link prediction, filling a notable research gap by acknowledging the importance of the directionality of relationships often overlooked in traditional methods. We present three algorithms: an asymmetric similarity-popularity algorithm, which applies the similarity-popularity paradigm specifically to directed networks; a path exploration algorithm, which utilizes path patterns, closure probabilities, and paths’ exploratory potential to predict new links formation; and a hybrid algorithm that merges the strengths of both approaches. The effectiveness of these methods is rigorously evaluated on real-life networks, demonstrating their robust performance across various types and sizes of networked data. In addition to predictive power and runtime performance assessments, we study the impact of predicted links on network spreading capacity. This perspective provides invaluable insights into the broader implications of our algorithms on network behavior and dynamics.

Appendices

A Data description

 

Table 5 Description of the networks used in the experimental performance analysis

B Proof of Theorem 1

Proof

The proof involves analyzing the hierarchical sub-graph generated from a randomly chosen node i, numbering the levels starting from 0. Let $v_l$ be the number of nodes in level l, $e_l$ the number of edges starting at level l, and $e_{l}^{l'}$ the edges linking levels l and $l'$. We have, by definition:

$$\begin{aligned} e_l&= e_{l}^{l-1} + e_{l}^{l} + e _{l}^{l+1}, \quad \forall l > 0,\end{aligned}$$

(20)

$$\begin{aligned} n_l&= \sum _{k=0}^l v_k, \end{aligned}$$

(21)

$$\begin{aligned} m_l&= \sum _{k=0}^{h-1} e_k^k + e_k^{k+1}. \end{aligned}$$

(22)

At level 0, $\mathbb {E}(v_0)$ = 1, $\mathbb {E}(e_0) = \bar{\kappa }$, and due to degree independence, $\mathbb {E}(v_1)= \bar{\kappa }$, and

$$\begin{aligned} \mathbb {E}(e_1)=\mathbb {E}(\mathbb {E}(e_1|v_1))=\mathbb {E}(\bar{\kappa } v_1) = \bar{\kappa } \mathbb {E}(v_1)= \bar{\kappa }^2, \end{aligned}$$

(23)

Among the $e_1$ edges, $e_0^1$ edges point to level 0, that is, $\mathbb {E}(e_{1}^{0}) = \bar{\kappa }$. Furthermore, by definition of the average clustering coefficient, there is a $\bar{C}$ probability that any two nodes in level 1 connect. Hence,

$$\begin{aligned} \mathbb {E}(e_{1}^{1})=\mathbb {E}(\mathbb {E}(e_{1}^{1}|e_{0}^{1},v_1)) = \mathbb {E}(\bar{C} v_1(e_{0}^{1}-1))= \bar{C} \bar{\kappa }(\bar{\kappa }-1). \end{aligned}$$

(24)

Using (20), this implies that $\mathbb {E}(e_{1}^{2})=\bar{\kappa }(1-c)(\bar{\kappa }-1)$. Since some of these edges will point to the same nodes, $v_2$ is then the expected number of distinct nodes obtained by selecting $e_{1}^{2}$ nodes with repetition from the remaining $n-n_1$ nodes:

$$\begin{aligned} \mathbb {E}(v_2 | e_{1}^{2}, n_1) = (n-n_1) \left( 1- \left( \dfrac{n-n_1-1}{n-n_1}\right) ^{e_{1}^{2}}\right) . \end{aligned}$$

(25)

Since, we can at most choose $e_{1}^{2}$ distinct nodes,

$$\begin{aligned} \mathbb {E}(v_2 | e_{1}^{2}, n_1) \le e_{1}^{2}. \end{aligned}$$

(26)

Note that this upper limit can be reached when n is very large so that the approximation

$$\begin{aligned} \left( \dfrac{n-n_1-1}{n-n_1}\right) ^{e_{1}^{2}} = \left( 1 - \dfrac{1}{n-n_1}\right) = 1 - \dfrac{e_{1}^{2}}{n-n_1} +o\left( \dfrac{1}{(n-n_1)^2}\right) \end{aligned}$$

(27)

is valid. Using this approximation in (25) results in:

$$\begin{aligned} \mathbb {E}(v_2 | e_{1}^{2}, n_1) = e_{1}^{2} - o\left( \dfrac{1}{n-n_1}\right) . \end{aligned}$$

(28)

From (26):

$$\begin{aligned} \mathbb {E}(v_2) = \mathbb {E}(\mathbb {E}(v_2 | e_{1}^{2}, n_1)) \le \mathbb {E}(e_{1}^{2})=\bar{\kappa }(1-c)(\bar{\kappa }-1). \end{aligned}$$

(29)

Using a similar argument for the next stage, we can show that:

$$\begin{aligned} \mathbb {E}(e_{2}) = \bar{\kappa } \mathbb {E}(v_2) \le \bar{\kappa }^2(1-\bar{C})(\bar{\kappa }-1). \end{aligned}$$

(30)

The number of edges internal to level 2 is minimized when each node in level 2 has a single parent in level 1. Hence:

$$\begin{aligned} \mathbb {E}(e_2^2) \ge \bar{C}\mathbb {E}(v_2) (\bar{\kappa }-1) =\bar{\kappa }\bar{C}(1-\bar{C})(\bar{\kappa }-1)^2. \end{aligned}$$

(31)

It follows then:

$$\begin{aligned} e_2^3 = e_2 - e_2^2 - e_1^2 \le \bar{\kappa }(1-\bar{C})^2(\bar{\kappa }-1)^2. \end{aligned}$$

(32)

More generally, for all $l \ge 1$:

$$\begin{aligned} \mathbb {E}(e_{l})&\le \bar{\kappa } v_{l} \le \bar{\kappa }^2(1-\bar{C})^{l-1}(\bar{\kappa }-1)^{l-1},\end{aligned}$$

(33)

$$\begin{aligned} \mathbb {E}(e_l^l)&\ge (e_l - v_l) \bar{C} \le \bar{\kappa }\bar{C}(1-\bar{C})^{l-1}(\bar{\kappa }-1)^l,\end{aligned}$$

(34)

$$\begin{aligned} \mathbb {E}(e_l^{l+1})&\le (e_l - v_{l}) (1-\bar{C}) \le \bar{\kappa }(1-\bar{C})^l(\bar{\kappa }-1)^l,\end{aligned}$$

(35)

$$\begin{aligned} \mathbb {E}(v_{l+1})&\le e_l^{l+1} \le \bar{\kappa }(1-\bar{C})^{l}(\bar{\kappa }-1)^{l}. \end{aligned}$$

(36)

The expected number of nodes in the sub-graph can then be bounded by:

$$\begin{aligned} \mathbb {E}(n_h) = \sum _{l=0}^h \mathbb {E} \left( v_l \right) \le 1+\sum _{l=1}^h \left( \bar{\kappa }(1-\bar{C})^{l-1}(\bar{\kappa }-1)^{l-1} \right) \\ = 1+ \bar{\kappa } \frac{(1-\bar{C})^h(\bar{\kappa }-1)^h - 1}{(1-\bar{C})(\bar{\kappa }-1) - 1} \in O\left( \left( (1-\bar{C})\bar{\kappa }\right) ^h\right) . \end{aligned}$$

(37)

The expected number of edges in this sub-graph is bounded by:

$$\begin{aligned} \mathbb {E}(m_h) = \sum _{l=0}^{h-1} \mathbb {E}\left( e_l^l+e_l^{l+1}\right) \le \bar{\kappa } + \sum _{l=1}^{h-1} \left( \bar{\kappa }(1-\bar{C})^{l-1}(\bar{\kappa }-1)^{l} \right) \\ = \bar{\kappa } + \bar{\kappa } \frac{(1-\bar{C})^{h-1}(\bar{\kappa }-1)^{h} - 1}{(1-\bar{C})(\bar{\kappa }-1) - 1} \in O\left( \left( (1-\bar{C})\bar{\kappa }\right) ^h\right) . \end{aligned}$$

(38)

$\square$

C Detailed results

This section presents the detailed results of the performance evaluation experiments for each network (See Tables 6–10). For each performance measure, results attaining the best significant rank at a significance level of $p=0.05$ are highlighted in bold.

Table 6 Effect of the horizon depth limit h on the proposed algorithms

Table 7 Effect of the sample size k and the horizon depth limit h on the proposed algorithms

Table 8 Effect of weight strategy on the performance of Algorithm 3

Table 9 Performance results on small networks

Table 10 TPR results on large networks