Archive for the ‘Dynamic Graphs’ Category

Dynamical Systems on Networks: A Tutorial

Thursday, May 14th, 2015

Dynamical Systems on Networks: A Tutorial by Mason A. Porter and James P. Gleeson.

Abstract:

We give a tutorial for the study of dynamical systems on networks. We focus especially on “simple” situations that are tractable analytically, because they can be very insightful and provide useful springboards for the study of more complicated scenarios. We briefly motivate why examining dynamical systems on networks is interesting and important, and we then give several fascinating examples and discuss some theoretical results. We also briefly discuss dynamical systems on dynamical (i.e., time-dependent) networks, overview software implementations, and give an outlook on the field.

At thirty-nine (39) pages and two hundred and sixty-three references, the authors leave the reader with an overview of the field and the tools to go further.

I am intrigued by the closer by the authors:


Finally, many networks are multiplex (i.e., include multiple types of edges) or have other multilayer features [16, 136]. The existence of multiple layers over which dynamics can occur and the possibility of both structural and dynamical correlations between layers offers another rich set of opportunities to study dynamical systems on networks. The investigation of dynamical systems on multilayer networks is only in its infancy, and this area is also loaded with a rich set of problems [16, 136, 144, 205].

Topic maps can have multiple type of edges and multiple layers.

For further reading on those topics see:

The structure and dynamics of multilayer networks by S. Boccaletti, G. Bianconi, R. Criado, C.I. del Genio, J. Gómez-Gardeñes, M. Romance, I. Sendiña-Nadal, Z. Wang, M. Zanin.

Abstract:

In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal- or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.

Multilayer Networks by Mikko Kivelä, Alexandre Arenas, Marc Barthelemy, James P. Gleeson, Yamir Moreno, Mason A. Porter.

Abstract:

In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such “multilayer” features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize “traditional” network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others. We also survey and discuss existing data sets that can be represented as multilayer networks. We review attempts to generalize single-layer-network diagnostics to multilayer networks. We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks. We conclude with a summary and an outlook.

This may have been where we collectively went wrong in marketing topic maps. Yes, yes it is true that topic maps could do multilayer networks but network theory has made $billions with an overly simplistic model that bears little resemblance to reality.

As computation resources improve and closer to reality models, at least somewhat closer, become popular, something between simplistic networks and the full generality of topic maps could be successful.

Methods for visualizing dynamic networks (Parts 1 and 2)

Thursday, April 16th, 2015

Methods for visualizing dynamic networks Part 1

Methods for visualizing dynamic networks Part 2

From part 1:

The challenge of visualizing the evolution of connected data through time has kept academics and data scientists busy for years. Finding a way to convey the added complexity of a temporal element without overwhelming the end user with it is not easy.

Whilst building the KeyLines Time Bar – our component for visualizing dynamic networks – we spent a great deal of time appraising the existing temporal visualization options available.

In this blog post, we’ve collated some of the most popular ways of visualizing dynamic graphs through time. Next week, we’ll share some of the more creative and unusual options.

Not a comprehensive survey but eight (8) ways to visualize dynamic networks that you will find interesting.

Others that you would add to this list?

A Fast Parallel Maximum Clique Algorithm…

Sunday, March 3rd, 2013

A Fast Parallel Maximum Clique Algorithm for Large Sparse Graphs and Temporal Strong Components by Ryan A. Rossi, David F. Gleich, Assefaw H. Gebremedhin, Md. Mostofa Ali Patwary.

Abstract:

We propose a fast, parallel, maximum clique algorithm for large, sparse graphs that is designed to exploit characteristics of social and information networks. We observe roughly linear runtime scaling over graphs between 1000 vertices and 100M vertices. In a test with a 1.8 billion-edge social network, the algorithm finds the largest clique in about 20 minutes. For social networks, in particular, we found that using the core number of a vertex in combination with a good heuristic clique finder efficiently removes the vast majority of the search space. In addition, we parallelize the exploration of the search tree. In the algorithm, processes immediately communicate changes to upper and lower bounds on the size of maximum clique, which occasionally results in a super-linear speedup because vertices with especially large search spaces can be pruned by other processes. We use this clique finder to investigate the size of the largest temporal strong components in dynamic networks, which requires finding the largest clique in a particular temporal reachability graph.

Thirty-two networks are reported in this paper and a promised online appendix as around eighty (80).

The online appendix is live but as of today (March 2, 2013), it has no content.

No matter, the paper should keep you busy for more than a little while. 😉

I am interested in parallel graph processing in general but the concept of communicating “…changes to upper and lower bounds on the size of maximum clique…” seems applicable to “merging” in topic maps.

That is if some set of topics share some common characteristic that exclude them from consideration for merging, why apply the merging test at all?

Will have to think about that.

On the Treewidth of Dynamic Graphs

Sunday, December 18th, 2011

On the Treewidth of Dynamic Graphs by Bernard Mans, Luke Mathieson.

Abstract:

Dynamic graph theory is a novel, growing area that deals with graphs that change over time and is of great utility in modelling modern wireless, mobile and dynamic environments. As a graph evolves, possibly arbitrarily, it is challenging to identify the graph properties that can be preserved over time and understand their respective computability.

In this paper we are concerned with the treewidth of dynamic graphs. We focus on metatheorems, which allow the generation of a series of results based on general properties of classes of structures. In graph theory two major metatheorems on treewidth provide complexity classifications by employing structural graph measures and finite model theory. Courcelle’s Theorem gives a general tractability result for problems expressible in monadic second order logic on graphs of bounded treewidth, and Frick & Grohe demonstrate a similar result for first order logic and graphs of bounded local treewidth.

We extend these theorems by showing that dynamic graphs of bounded (local) treewidth where the length of time over which the graph evolves and is observed is finite and bounded can be modelled in such a way that the (local) treewidth of the underlying graph is maintained. We show the application of these results to problems in dynamic graph theory and dynamic extensions to static problems. In addition we demonstrate that certain widely used dynamic graph classes naturally have bounded local treewidth.

There are going to be topic maps that are static editorial artifacts, similar to a back-of-book index. Which can be modeled as traditional graphs. But, merging two or more such graphs and/or topic maps that are designed as dynamic information sources, will require different analysis. This paper is a starting place for work on those issues.