Time-stamp based network evolution model for citation networks

Abstract

Citation score has become a very important metric to assess the quality of a publication in the current global ranking scenario. In this context, the study of citation networks gains importance as it helps in understanding the citation process as well as in analyzing citation trends in the research world. Citation networks are modeled as directed acyclic graphs in which publications of the authors are considered as nodes and citations between the papers form the links. In this paper, we propose an additive Time-Stamp based Network Evolution(TNE) model for citation networks, extending Price’s preferential attachment model by including the recency effect on the citation process without neglecting the impact of classical papers. We propose a more meaningful definition of clustering coefficient for citation networks in terms of ’citation triangles’. Further, the network simulated by the TNE model with best-fit parameters is compared with the real-world(DBLP) citation network. The results of various significance tests show that the simulated network matches very well with the DBLP citation network in terms of several network properties.

Appendix Time stamp effect: analytical proof

If the time stamp effect is removed from our model, it will be reduced to the Price’s preferential attachment model. This can be analytically shown as follows:

The master equation of our additive TNE model is given as follows:

$$\begin{aligned} \frac{\partial k_i}{\partial t} = \mu \left[ m.\pi _d(v_i) + m.\pi _t(v_i) + m.\frac{1}{N_0 + t}\right] \end{aligned}$$

Where $\pi _d(v_i) = \frac{k_i}{\sum _j k_j}$ is preferential attachment based on in-degree and $\pi _t(v_i) = \frac{t_i}{t+1}$ is preferential attachment based on time factor.

The resulting master equation after the removal of the timestamp from the TNE model is given below.

$$\begin{aligned} \frac{\partial k_i}{\partial t}= & {} \mu \left[ m.\pi _d(v_i) + m.\frac{1}{N_0 + t}\right] \\ \frac{\partial k_i}{\partial t}= & {} \mu \left[ m.\frac{k_i}{\sum _j k_j} + m.\frac{1}{N_0 + t} \right] \end{aligned}$$

where $\pi _d(v_i) = \frac{k_i}{\sum _j k_j}$

$$\begin{aligned} \frac{\partial k_i}{\partial t}= & {} \mu \left[ m.\frac{k_i}{m_0+m.t} + m.\frac{1}{N_0 + t}\right] \\ \frac{\partial k_i}{\partial t} -\mu m.\frac{1}{N_0 + t}= & {} \mu m.\frac{k_i}{m_0+m.t}\\ \frac{1}{k_i}\frac{\partial k_i}{\partial t}= & {} \frac{\mu m}{m_0+m.t} + \mu m.\frac{1}{k_i(N_0 + t)}\\ \frac{1}{k_i}\partial k_i= & {} \frac{\mu m}{m_0+m.t}.\partial t + \mu m.\frac{1}{k_i(N_0 + t)}.\partial t \end{aligned}$$

Integrating both sides in the range $(t_i, t)$ where $t_i$ is the time at which paper i gets the initial citation. Therefore $k_i(t_i) = b$ where b is a constant.

$$\begin{aligned} \int _{t_i} ^t \frac{1}{k_i}.\partial k_i= & {} \int _{t_i} ^t \frac{\mu m}{m_0+m.t}.\partial t + \int _{t_i} ^t\mu m.\frac{1}{k_i(N_0 + t)}.\partial t\\ \log {\frac{k_i(t)}{k_i(t_i)}}= & {} \frac{\mu m}{m}\log {\left( \frac{t}{t_i}\right) }+\frac{\mu m}{k_i}\log {\left( \frac{N_0+t}{N_0+t_i}\right) }\\ {\frac{k_i(t)}{b}}= & {} {\left( \frac{t}{t_i}\right) }^\frac{\mu m}{m}+{\left( \frac{N_0+t}{N_0+t_i}\right) }^\frac{\mu m}{k_i}\\ \frac{k_i(t)}{b}= & {} {\left( \frac{t}{t_i}\right) }^{\mu +\frac{\mu m }{k_i}} \end{aligned}$$

Since average $k_i = m$

$$\begin{aligned} {\frac{k_i(t)}{b}}= & {} {\left( \frac{t}{t_i}\right) }^{2\mu }\\ k_i(t)= & {} b. {\left( \frac{t}{t_i}\right) }^{2\mu } \end{aligned}$$

Highly connected nodes are the result of the phenomena i.e, older nodes with smaller $t_i$ values increase their in-degree at expense of recent nodes with larger $t_i$. This helps us calculate $p(k_i(t) < k)$ The probability of selecting a node at time t whose degree is less than k.

$$\begin{aligned} p(k_i(t)< k)&= p\left( b. \left( \frac{t}{t_i}\right) ^{2\mu }<k\right) \\&= p\left( t_i>\left( t.b^\frac{-1}{2\mu }.{k}^\frac{-1}{2\mu })\right) \right) \\&= 1- p\left( t_i<\left( t.b^\frac{-1}{w\mu }.{k}^\frac{-1}{2\mu }\right) \right) \end{aligned}$$

Calculating in-degree distribution requires randomly selecting a node at time t with probability $\frac{1}{N_0 + t}$.

$$\begin{aligned} p(k_i(t) < k) = 1- \frac{t b^\frac{-1}{2\mu }}{N_0 +t}.{k}^\frac{-1}{2\mu } \end{aligned}$$

In-degree distribution of nodes having degree k at time t is given by:

$$\begin{aligned} P(k,t) = \frac{\partial }{\partial k}(p(k_i(t) < k)) \end{aligned}$$

Hence

$$\begin{aligned} P(k,t)= & {} \frac{\partial }{\partial k}\left[ 1- \frac{t b^\frac{-1}{2\mu }}{N_0 +t}.{k}^\frac{-1}{2\mu }\right] \\ P(k,t)= & {} \frac{t b^\frac{-1}{2\mu }}{2\mu (N_0+t)} k ^{-\left( \frac{1}{2\mu }+1\right) } \end{aligned}$$

while $t\rightarrow \infty$

$$\begin{aligned} P(k)= & {} \frac{b^\frac{-1}{2\mu }}{2\mu } k ^{-\left( \frac{1}{2\mu }+1\right) }\\ P(k)\sim & {} k^{-\left( \frac{1}{2\mu }+1\right) } \end{aligned}$$

Since $0<\mu <1$ and $\frac{1}{2\mu }+1 >=1$ which means when the time stamp effect is removed from our TNE model, it reduces to Price’s classical PA model.