Personalized PageRank has found many uses in not only the ranking of webpages, but also algorithmic design, due to its ability to capture certain geometric properties of networks. In this paper, we study the diffusion of PageRank: how varying the jumping (or teleportation) constant affects PageRank values. To this end, we prove a gradient estimate for PageRank, akin to the Li–Yau inequality for positive solutions to the heat equation (for manifolds, with later versions adapted to graphs).

Graphlet counting is a widely explored problem in network analysis and has been successfully applied to a variety of applications in many domains, most notatbly bioinformatics, social science, and infrastructure network studies. Efficiently computing graphlet counts remains challenging due to the combinatorial explosion, where a naive enumeration algorithm needs O(Nk) time for k-node graphlets in a network of size N. Recently, many works introduced carefully designed combinatorial and sampling methods with encouraging results. However, the existing methods ignore the fact that graphlet counts and the graph structural information are correlated. They always consider a graph as a new input and repeat the tedious counting procedure on a regular basis even if it is similar or exactly isomorphic to previously studied graphs. This provides an opportunity to speed up the graphlet count estimation procedure by exploiting this correlation via learning methods. In this paper, we raise a novel graphlet count learning (GCL) problem: given a set of historical graphs with known graphlet counts, how to learn to estimate/predict graphlet count for unseen graphs coming from the same (or similar) underlying distribution. We develop a deep learning framework which contains two convolutional neural network models and a series of data preprocessing techniques to solve the GCL problem. Extensive experiments are conducted on three types of synthetic random graphs and three types of real-world graphs for all 3-, 4-, and 5-node graphlets to demonstrate the accuracy, efficiency, and generalizability of our framework. Compared with state-of-the-art exact/sampling methods, our framework shows great potential, which can offer up to two orders of magnitude speedup on synthetic graphs and achieve on par speed on real-world graphs with competitive accuracy.

Measurement errors are omnipresent in network data. Most studies observe an erroneous network instead of the desired error-free network. It is well known that such errors can have a severe impact on network metrics, especially on centrality measures: a central node in the observed network might be less central in the underlying, error-free network. The robustness is a common concept to measure these effects. Studies have shown that the robustness primarily depends on the centrality measure, the type of error (e.g., missing edges or missing nodes), and the network topology (e.g., tree-like, core-periphery). Previous findings regarding the influence of network size on the robustness are, however, inconclusive. We present empirical evidence and analytical arguments indicating that there exist arbitrary large robust and non-robust networks and that the average degree is well suited to explain the robustness. We demonstrate that networks with a higher average degree are often more robust. For the degree centrality and Erdős–Rényi (ER) graphs, we present explicit formulas for the computation of the robustness, mainly based on the joint distribution of node degrees and degree changes which allow us to analyze the robustness for ER graphs with a constant average degree or increasing average degree.

Isolation is a concept originally conceived in the context of clique enumeration in static networks, mostly used to model communities that do not have much contact to the outside world. Herein, a clique is considered isolated if it has few edges connecting it to the rest of the graph. Motivated by recent work on enumerating cliques in temporal networks, we transform the isolation concept to the temporal setting. We discover that the addition of the time dimension leads to six distinct natural isolation concepts. Our main contribution is the development of parameterized enumeration algorithms for five of these six isolation types for clique enumeration, employing the parameter “degree of isolation.” In a nutshell, this means that the more isolated these cliques are, the faster we can find them. On the empirical side, we implemented and tested these algorithms on (temporal) social network data, obtaining encouraging results.

We introduce new definitions of sectional, Ricci, and scalar curvatures for networks and their higher dimensional counterparts, derived from two classical notions of curvature for curves in general metric spaces, namely, the Menger curvature and the Haantjes curvature. These curvatures are applicable to unweighted or weighted and undirected or directed networks and are more intuitive and easier to compute than other network curvatures. In particular, the proposed curvatures based on the interpretation of Haantjes definition as geodesic curvature allow us to give a network analogue of the classical local Gauss–Bonnet theorem. Furthermore, we propose even simpler and more intuitive proxies for the Haantjes curvature that allow for even faster and easier computations in large-scale networks. In addition, we also investigate the embedding properties of the proposed Ricci curvatures. Lastly, we also investigate the behavior, both on model and real-world networks, of the curvatures introduced herein with more established notions of Ricci curvature and other widely used network measures.

This paper investigates the distributions of triangle counts per vertex and edge, as a means for network description, analysis, model building, and other tasks. The main interest is in estimating these distributions through sampling, especially for large networks. A novel sampling method tailored for the estimation analysis is proposed, with three sampling designs motivated by several network access scenarios. An estimation method based on inversion and an asymptotic method are developed to recover the entire distribution. A single method to estimate the distribution using multiple samples is also considered. Algorithms are presented to sample the network under the various access scenarios. Finally, the estimation methods on synthetic and real-world networks are evaluated in a data study.

Critical cascades are found in many self-organizing systems. Here, we examine critical cascades as a design paradigm for logic and learning under the linear threshold model (LTM), and simple biologically inspired variants of it as sources of computational power, learning efficiency, and robustness. First, we show that the LTM can compute logic, and with a small modification, universal Boolean logic, examining its stability and cascade frequency. We then frame it formally as a binary classifier and remark on implications for accuracy. Second, we examine the LTM as a statistical learning model, studying benefits of spatial constraints and criticality to efficiency. We also discuss implications for robustness in information encoding. Our experiments show that spatial constraints can greatly increase efficiency. Theoretical investigation and initial experimental results also indicate that criticality can result in a sudden increase in accuracy.