We present a scalable approach for semi-supervised learning on graph-structured data that is based on an efficient variant of convolutional neural networks which operate directly on graphs. We motivate the choice of our convolutional architecture via a localized first-order approximation of spectral graph convolutions. Our model scales linearly in the number of graph edges and learns hidden layer representations that encode both local graph structure and features of nodes. In a number of experiments on citation networks and on a knowledge graph dataset we demonstrate that our approach outperforms related methods by a significant margin.

Semi-Supervised Classification with Graph Convolutional Networks

We propose and study a set of algorithms for discovering community structure in networks-natural divisions of network nodes into densely connected subgroups. Our algorithms all share two definitive features: first, they involve iterative removal of edges from the network to split it into communities, the edges removed being identified using any one of a number of possible "betweenness" measures, and second, these measures are, crucially, recalculated after each removal. We also propose a measure for the strength of the community structure found by our algorithms, which gives us an objective metric for choosing the number of communities into which a network should be divided. We demonstrate that our algorithms are highly effective at discovering community structure in both computer-generated and real-world network data, and show how they can be used to shed light on the sometimes dauntingly complex structure of networked systems.

/pdf/finding-and-evaluating-community-structure-in-networks-1o9f6y8pyr.pdf

Finding and evaluating community structure in networks.

http://brainimaging.waisman.wisc.edu/~chung/neuro.processing/literature/rubinov.2010.NI.network.pdf

Complex network measures of brain connectivity: uses and interpretations.

The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, playing a similar role like, e. g., the tissues or the organs in the human body. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. This problem is very hard and not yet satisfactorily solved, despite the huge effort of a large interdisciplinary community of scientists working on it over the past few years. We will attempt a thorough exposition of the topic, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks.

/pdf/community-detection-in-graphs-34rogm3r6p.pdf

Community detection in graphs

There is no other package around that satisfies all the following requirements: •Ability to handle large graphs efficiently •Embeddable into higher level environments (like R [6] or Python [7]) •Ability to be used for quick prototyping of new algorithms (impossible with “click & play” interfaces) •Platform-independent and open source igraph aims to satisfy all these requirements while possibly remaining easy to use in interactive mode as well.

The igraph software package for complex network research

Motivated by the fast‐growing need to compute centrality indices on large, yet very sparse, networks, new algorithms for betweenness are introduced in this paper. They require O(n + m) space and run in O(nm) and O(nm + n2 log n) time on unweighted and weighted networks, respectively, where m is the number of links. Experimental evidence is provided that this substantially increases the range of networks for which centrality analysis is feasible. The betweenness centrality index is essential in the analysis of social networks, but costly to compute. Currently, the fastest known algorithms require ?(n 3) time and ?(n 2) space, where n is the number of actors in the network.

/pdf/a-faster-algorithm-for-betweenness-centrality-2g52pt39q9.pdf

A faster algorithm for betweenness centrality

This is the beginning of Network Science. The journal has been created because network science is exploding. As is typical for a field in formation, the discussions about its scope, contents, and foundations are intense. On these first few pages of the first issue of our new journal, we would like to share our own vision of the emerging science of networks.

/pdf/what-is-network-science-34c8vflllq.pdf

What is network science

Modularity is a recently introduced quality measure for graph clusterings. It has immediately received considerable attention in several disciplines, particularly in the complex systems literature, although its properties are not well understood. We study the problem of finding clusterings with maximum modularity, thus providing theoretical foundations for past and present work based on this measure. More precisely, we prove the conjectured hardness of maximizing modularity both in the general case and with the restriction to cuts and give an Integer Linear Programming formulation. This is complemented by first insights into the behavior and performance of the commonly applied greedy agglomerative approach.

/pdf/on-modularity-clustering-4kykhxfgop.pdf

On Modularity Clustering

https://kops.uni-konstanz.de/bitstream/handle/123456789/5940/variants.pdf;sequence=1

On variants of shortest-path betweenness centrality and their generic computation ☆

Fundamentals.- I Elements.- Centrality Indices.- Algorithms for Centrality Indices.- Advanced Centrality Concepts.- II Groups.- Local Density.- Connectivity.- Clustering.- Role Assignments.- Blockmodels.- Network Statistics.- Network Comparison.- Network Models.- Spectral Analysis.- Robustness and Resilience.

Ulrik Brandes

Papers

A faster algorithm for betweenness centrality

What is network science

On Modularity Clustering

On variants of shortest-path betweenness centrality and their generic computation ☆

Network Analysis: Methodological Foundations