We propose a model of the evolution of the networks of scientific citations. The model takes an out-degree distribution (distribution of number of citations) and two parameters as input. The parameters capture the two main ingredients of the model, the aging of the relevance of papers and the formation of triangles when new papers cite old. We compare our model with three network structural quantities of an empirical citation network. We find that an unique point in parameter space optimizing the match between the real and model data for all quantities. The optimal parameter values suggest that the impact of scientific papers, at least in the empirical data set we model is proportional to the inverse of the number of papers since they were published.
Acyclic digraphs arise in many natural and artificial processes. Among the broader set, dynamic citation networks represent a substantively important form of acyclic digraphs. For example, the study of such networks includes the spread of ideas through academic citations, the spread of innovation through patent citations, and the development of precedent in common law systems. The specific dynamics that produce such acyclic digraphs not only differentiate them from other classes of graphs, but also provide guidance for the development of meaningful distance measures. In this article, we develo…
Extensive literature demonstrates how the copying of references (links) can lead to the emergence of various structural properties (e.g., power-law degree distribution and bipartite cores) in bibliographic and other similar directed networks. However, it is also well known that the copying process is incapable of mimicking the number of directed triangles in such networks; neither does it have the power to explain the obsolescence of older papers. In this paper, we propose RefOrCite, a new model that allows for copying of both the references from (i.e., out-neighbors of) as well as the citatio…
For the study of citation networks, a challenging problem is modeling the high clustering. Existing studies indicate that the promising way to model the high clustering is a copying strategy, i.e., a paper copies the references of its neighbour as its own references. However, the line of models highly underestimates the number of abundant triangles observed in real citation networks and thus cannot well model the high clustering. In this paper, we point out that the failure of existing models lies in that they do not capture the connecting patterns among existing papers. By leveraging the know…
We introduce a framework for network analysis based on random walks on directed acyclic graphs where the probability of passing through a given node is the key ingredient. We illustrate its use in evaluating the mutual influence of nodes and discovering seminal papers in a citation network. We further introduce a new similarity metric and test it in a simple personalized recommendation process. This metric's performance is comparable to that of classical similarity metrics, thus further supporting the validity of our framework.
Citation networks represent the flow of information between agents. They are constrained in time and so form directed acyclic graphs which have a causal structure. Here we provide novel quantitative methods to characterise that structure by adapting methods used in the causal set approach to quantum gravity by considering the networks to be embedded in a Minkowski spacetime and measuring its dimension using Myrheim-Meyer and Midpoint-scaling estimates. We illustrate these methods on citation networks from the arXiv, supreme court judgements from the USA, and patents and find that otherwise sim…
Some temporal networks, most notably citation networks, are naturally represented as directed acyclic graphs (DAGs). To detect communities in DAGs, we propose a modularity for DAGs by defining an appropriate null model (i.e., randomized network) respecting the order of nodes. We implement a spectral method to approximately maximize the proposed modularity measure and test the method on citation networks and other DAGs. We find that the attained values of the modularity for DAGs are similar for partitions that we obtain by maximizing the proposed modularity (designed for DAGs), the modularity f…
In many complex networks the vertices are ordered in time, and edges represent causal connections. We propose methods of analysing such directed acyclic graphs taking into account the constraints of causality and highlighting the causal structure. We illustrate our approach using citation networks formed from academic papers, patents, and US Supreme Court verdicts. We show how transitive reduction reveals fundamental differences in the citation practices of different areas, how it highlights particularly interesting work, and how it can correct for the effect that the age of a document has on …
Geometric approaches to network analysis combine simply defined models with great descriptive power. In this work we provide a method for embedding directed acyclic graphs into Minkowski spacetime using Multidimensional scaling (MDS). First we generalise the classical MDS algorithm, defined only for metrics with a Euclidean signature, to manifolds of any metric signature. We then use this general method to develop an algorithm to be used on networks which have causal structure allowing them to be embedded in Lorentzian manifolds. The method is demonstrated by calculating embeddings for both ca…
We study random graph models for directed acyclic graphs, an important class of networks that includes citation networks, food webs, and feed-forward neural networks among others. We propose two specific models, roughly analogous to the fixed edge number and fixed edge probability variants of traditional undirected random graphs. We calculate a number of properties of these models, including particularly the probability of connection between a given pair of vertices, and compare the results with real-world acyclic network data finding that theory and measurements agree surprisingly well -- far…
There are fundamental differences between citation networks and other classes of graphs. In particular, given that citation networks are directed and acyclic, methods developed primarily for use with undirected social network data may face obstacles. This is particularly true for the dynamic development of community structure in citation networks. Namely, it is neither clear when it is appropriate to employ existing community detection approaches nor is it clear how to choose among existing approaches. Using simulated data, we attempt to clarify the conditions under which one should use existi…
In this work we consider the topological analysis of symbolic formal systems in the framework of network theory. In particular we analyse the network extracted by Principia Mathematica of B. Russell and A.N. Whitehead, where the vertices are the statements and two statements are connected with a directed link if one statement is used to demonstrate the other one. We compare the obtained network with other directed acyclic graphs, such as a scientific citation network and a stochastic model. We also introduce a novel topological ordering for directed acyclic graphs and we discuss its properties…
The feed-forward relationship naturally observed in time-dependent processes and in a diverse number of real systems -such as some food-webs and electronic and neural wiring- can be described in terms of so-called directed acyclic graphs (DAGs). An important ingredient of the analysis of such networks is a proper comparison of their observed architecture against an ensemble of randomized graphs, thereby quantifying the {\em randomness} of the real systems with respect to suitable null models. This approximation is particularly relevant when the finite size and/or large connectivity of real sys…
The rate at which nodes in evolving social networks acquire links (friends, citations) shows complex temporal dynamics. Preferential attachment and link copying models, while enabling elegant analysis, only capture rich-gets-richer effects, not aging and decline. Recent aging models are complex and heavily parameterized; most involve estimating 1-3 parameters per node. These parameters are intrinsic: they explain decline in terms of events in the past of the same node, and do not explain, using the network, where the linking attention might go instead. We argue that traditional characterizatio…
We introduce a new method for temporal pattern analysis of scientific collaboration networks. We investigate in particular virus research activities through five epidemic or pandemic outbreaks in the recent two decades and in the ongoing pandemic with COVID-19. Our method embodies two innovative components. The first is a simple model of temporal collaboration networks with time segmented in publication time and convolved in citation history, to effectively capture and accommodate collaboration activities at mixed time scales. The second component is the novel use of graphlets to encode topolo…
