The simplicity of planar networks

Shortest paths are not always simple. In planar networks, they can be very different from those with the smallest number of turns - the simplest paths. The statistical comparison of the lengths of the shortest and simplest paths provides a non trivial and non local information about the spatial organization of these graphs. We define the simplicity index as the average ratio of these lengths and the simplicity profile characterizes the simplicity at different scales. We measure these metrics on artificial (roads, highways, railways) and natural networks (leaves, slime mould, insect wings) and show that there are fundamental differences in the organization of urban and biological systems, related to their function, navigation or distribution: straight lines are organized hierarchically in biological cases, and have random lengths and locations in urban systems. In the case of time evolving networks, the simplicity is able to reveal important structural changes during their evolution.Comment: 8 pages, 4 figure

Fully-dynamic Approximation of Betweenness Centrality

Betweenness is a well-known centrality measure that ranks the nodes of a network according to their participation in shortest paths. Since an exact computation is prohibitive in large networks, several approximation algorithms have been proposed. Besides that, recent years have seen the publication of dynamic algorithms for efficient recomputation of betweenness in evolving networks. In previous work we proposed the first semi-dynamic algorithms that recompute an approximation of betweenness in connected graphs after batches of edge insertions. In this paper we propose the first fully-dynamic approximation algorithms (for weighted and unweighted undirected graphs that need not to be connected) with a provable guarantee on the maximum approximation error. The transfer to fully-dynamic and disconnected graphs implies additional algorithmic problems that could be of independent interest. In particular, we propose a new upper bound on the vertex diameter for weighted undirected graphs. For both weighted and unweighted graphs, we also propose the first fully-dynamic algorithms that keep track of such upper bound. In addition, we extend our former algorithm for semi-dynamic BFS to batches of both edge insertions and deletions. Using approximation, our algorithms are the first to make in-memory computation of betweenness in fully-dynamic networks with millions of edges feasible. Our experiments show that they can achieve substantial speedups compared to recomputation, up to several orders of magnitude

Graph Metrics for Temporal Networks

Temporal networks, i.e., networks in which the interactions among a set of elementary units change over time, can be modelled in terms of time-varying graphs, which are time-ordered sequences of graphs over a set of nodes. In such graphs, the concepts of node adjacency and reachability crucially depend on the exact temporal ordering of the links. Consequently, all the concepts and metrics proposed and used for the characterisation of static complex networks have to be redefined or appropriately extended to time-varying graphs, in order to take into account the effects of time ordering on causality. In this chapter we discuss how to represent temporal networks and we review the definitions of walks, paths, connectedness and connected components valid for graphs in which the links fluctuate over time. We then focus on temporal node-node distance, and we discuss how to characterise link persistence and the temporal small-world behaviour in this class of networks. Finally, we discuss the extension of classic centrality measures, including closeness, betweenness and spectral centrality, to the case of time-varying graphs, and we review the work on temporal motifs analysis and the definition of modularity for temporal graphs.Comment: 26 pages, 5 figures, Chapter in Temporal Networks (Petter Holme and Jari Saram\"aki editors). Springer. Berlin, Heidelberg 201

Scalable Online Betweenness Centrality in Evolving Graphs

Betweenness centrality is a classic measure that quantifies the importance of a graph element (vertex or edge) according to the fraction of shortest paths passing through it. This measure is notoriously expensive to compute, and the best known algorithm runs in O(nm) time. The problems of efficiency and scalability are exacerbated in a dynamic setting, where the input is an evolving graph seen edge by edge, and the goal is to keep the betweenness centrality up to date. In this paper we propose the first truly scalable algorithm for online computation of betweenness centrality of both vertices and edges in an evolving graph where new edges are added and existing edges are removed. Our algorithm is carefully engineered with out-of-core techniques and tailored for modern parallel stream processing engines that run on clusters of shared-nothing commodity hardware. Hence, it is amenable to real-world deployment. We experiment on graphs that are two orders of magnitude larger than previous studies. Our method is able to keep the betweenness centrality measures up to date online, i.e., the time to update the measures is smaller than the inter-arrival time between two consecutive updates.Comment: 15 pages, 9 Figures, accepted for publication in IEEE Transactions on Knowledge and Data Engineerin

Methods to Determine Node Centrality and Clustering in Graphs with Uncertain Structure

Much of the past work in network analysis has focused on analyzing discrete graphs, where binary edges represent the "presence" or "absence" of a relationship. Since traditional network measures (e.g., betweenness centrality) utilize a discrete link structure, complex systems must be transformed to this representation in order to investigate network properties. However, in many domains there may be uncertainty about the relationship structure and any uncertainty information would be lost in translation to a discrete representation. Uncertainty may arise in domains where there is moderating link information that cannot be easily observed, i.e., links become inactive over time but may not be dropped or observed links may not always corresponds to a valid relationship. In order to represent and reason with these types of uncertainty, we move beyond the discrete graph framework and develop social network measures based on a probabilistic graph representation. More specifically, we develop measures of path length, betweenness centrality, and clustering coefficient---one set based on sampling and one based on probabilistic paths. We evaluate our methods on three real-world networks from Enron, Facebook, and DBLP, showing that our proposed methods more accurately capture salient effects without being susceptible to local noise, and that the resulting analysis produces a better understanding of the graph structure and the uncertainty resulting from its change over time.Comment: Longer version of paper appearing in Fifth International AAAI Conference on Weblogs and Social Media. 9 pages, 4 Figure

Temporal Graph Traversals: Definitions, Algorithms, and Applications

A temporal graph is a graph in which connections between vertices are active at specific times, and such temporal information leads to completely new patterns and knowledge that are not present in a non-temporal graph. In this paper, we study traversal problems in a temporal graph. Graph traversals, such as DFS and BFS, are basic operations for processing and studying a graph. While both DFS and BFS are well-known simple concepts, it is non-trivial to adopt the same notions from a non-temporal graph to a temporal graph. We analyze the difficulties of defining temporal graph traversals and propose new definitions of DFS and BFS for a temporal graph. We investigate the properties of temporal DFS and BFS, and propose efficient algorithms with optimal complexity. In particular, we also study important applications of temporal DFS and BFS. We verify the efficiency and importance of our graph traversal algorithms in real world temporal graphs