Polygon-Based Hierarchical Planar Networks Based on Generalized Apollonian Construction

Experimentally observed complex networks are often scale-free, small-world and have an unexpectedly large number of small cycles. An Apollonian network is one notable example of a model network simultaneously having all three of these properties. This network is constructed by a deterministic procedure of consequentially splitting a triangle into smaller and smaller triangles. In this paper, a similar construction based on the consequential splitting of tetragons and other polygons with an even number of edges is presented. The suggested procedure is stochastic and results in the ensemble of planar scale-free graphs. In the limit of a large number of splittings, the degree distribution of the graph converges to a true power law with an exponent, which is smaller than three in the case of tetragons and larger than three for polygons with a larger number of edges. It is shown that it is possible to stochastically mix tetragon-based and hexagon-based constructions to obtain an ensemble of graphs with a tunable exponent of degree distribution. Other possible planar generalizations of the Apollonian procedure are also briefly discussed.

Partitioning Wide Area Graphs Using a Space Filling Curve

Graph structure is a very powerful tool to model system and represent their actual shape. For instance, modelling an infrastructure or social network naturally leads to graph. Yet, graphs can be very different from one another as they do not share the same properties (size, connectivity, communities, etc.) and building a system able to manage graphs should take into account this diversity. A big challenge concerning graph management is to design a system providing a scalable persistent storage and allowing efficient browsing. Mainly to study social graphs, the most recent developments in graph partitioning research often consider scale-free graphs. As we are interested in modelling connected objects and their context, we focus on partitioning geometric graphs. Consequently our strategy differs, we consider geometry as our main partitioning tool. In fact, we rely on Inverse Space-filling Partitioning, a technique which relies on a space filling curve to partition a graph and was previously applied to graphs essentially generated from Meshes. Furthermore, we extend Inverse Space-Filling Partitioning toward a new target we define as Wide Area Graphs. We provide an extended comparison with two state-of-the-art graph partitioning streaming strategies, namely LDG and FENNEL. We also propose customized metrics to better understand and identify the use cases for which the ISP partitioning solution is best suited. Experimentations show that in favourable contexts, edge-cuts can be drastically reduced, going from more 34% using FENNEL to less than 1% using ISP.

Inverse Space Filling Curve Partitioning Applied to Wide Area Graphs

The most recent developments in graph partitioning research often consider scale-free graphs. Instead we focus on partitioning geometric graphs using a less usual strategy: Inverse Spacefilling Partitioning (ISP). ISP relies on a space filling curve to partition a graph and was previously applied to graphs essentially generated from Meshes. We extend ISP to apply it to a new context where the targets are now Wide Area Graphs. We provide an extended comparison with two state-of-the-art graph partitioning streaming strategies, namely LDG and FENNEL. We also propose customized metrics to better understand and identify the use cases for which the ISP partitioning solution is best suited. Experimentations show that in favourable contexts, edge-cuts can be drastically reduced, going from more 34% using FENNEL to less than 1% using ISP.

Generation of 2-mode scale-free graphs for link-level internet topology modeling

Comprehensive analysis that aims to understand the topology of real-world networks and the development of algorithms that replicate their characteristics has been significant research issues. Although the accuracy of newly developed network protocols or algorithms does not depend on the underlying topology, the performance generally depends on the topology. As a result, network practitioners have concentrated on generating representative synthetic topologies and utilize them to investigate the performance of their design in simulation or emulation environments. Network generators typically represent the Internet topology as a graph composed of point-to-point links. In this study, we discuss the implications of multi-access links on the synthetic network generation and modeling of the networks as bi-partite graphs to represent both subnetworks and routers. We then analyze the characteristics of sampled Internet topology data sets from backbone Autonomous Systems (AS) and observe that in addition to the commonly recognized power-law node degree distribution, the subnetwork size and the router interface distributions often exhibit power-law characteristics. We introduce a SubNetwork Generator (SubNetG) topology generation approach that incorporates the observed measurements to produce bipartite network topologies. In particular, generated topologies capture the 2-mode relation between the layer-2 (i.e., the subnetwork and interface distributions) and the layer-3 (i.e., the degree distribution) that is missing from the current network generators that produce 1-mode graphs. The SubNetG source code and experimental data is available at https://github.com/netml/sonet.

Transfinite fractal dimension of trees and hierarchical scale-free graphs

In this article, we introduce a new concept: the transfinite fractal dimension of graph sequences motivated by the notion of fractality of complex networks proposed by Song et al. We show that the definition of fractality cannot be applied to networks with ‘tree-like’ structure and exponential growth rate of neighbourhoods. However, we show that the definition of fractal dimension could be modified in a way that takes into account the exponential growth, and with the modified definition, the fractal dimension becomes a proper parameter of graph sequences. We find that this parameter is related to the growth rate of trees. We also generalize the concept of box dimension further and introduce the transfinite Cesaro fractal dimension. Using rigorous proofs, we determine the optimal box-covering and transfinite fractal dimension of various models: the hierarchical graph sequence model introduced by Komjáthy and Simon, Song–Havlin–Makse model, spherically symmetric trees and supercritical Galton–Watson trees.

Chromatic transitions in the emergence of syntax networks

The emergence of syntax during childhood is a remarkable example of how complex correlations unfold in nonlinear ways through development. In particular, rapid transitions seem to occur as children reach the age of two, which seems to separate a two-word, tree-like network of syntactic relations among words from the scale-free graphs associated with the adult, complex grammar. Here, we explore the evolution of syntax networks through language acquisition using thechromatic number, which captures the transition and provides a natural link to standard theories on syntactic structures. The data analysis is compared to a null model of network growth dynamics which is shown to display non-trivial and sensible differences. At a more general level, we observe that the chromatic classes define independent regions of the graph, and thus, can be interpreted as the footprints of incompatibility relations, somewhat as opposed to modularity considerations.

Communication-Sensitive Pseudo-Tree Heuristics for DCOP Algorithms

Distributed Constraint Optimization Problem (DCOP) is a powerful paradigm to model multi-agent systems through enabling multiple agents to coordinate with each other to solve a problem. These agents are often assumed to be cooperative, that is, they communicate with other agents in order to optimize a global objective. However, the communication times between all pairs of agents are assumed to be identical in the evaluation of most DCOP algorithms. This assumption is impractical in almost all real-world applications. In this paper, we study the impact of empirically evaluating a DCOP algorithm under the assumption that communication times between pairs of agents can vary. In addition, we evaluate a DCOP algorithm using ns-2, a discrete-event simulator that is widely used in the computer networking community, to simulate the communication times, as opposed to the standard DCOP simulators that are used to evaluate DCOP algorithms in the AI community. Furthermore, we propose heuristics that exploit the non-uniform communication times to speed up DCOP algorithms that operate on pseudo-trees. Our empirical results demonstrate that the proposed heuristics improve the runtime of those algorithms up to 20%. These heuristics are evaluated on different benchmarks such as scale-free graphs, random graphs, and an instance of the smart grid, Customer-Driven Microgrid (CDMG) application.

Complex networks

This article deals with complex networks, and in particular small world and scale free networks. Various networks exhibit the small world phenomenon, including social networks and gene expression networks. The local ordering property of small world networks is typically associated with regular networks such as a 2D square lattice. The small world phenomenon can be observed in most scale free networks, but few small world networks are scale free. The article first provides a brief background on small world networks and two models of scale free graphs before describing the replica method and how it can be applied to calculate the spectral densities of the adjacency matrix and Laplacian matrix of a scale free network. It then shows how the effective medium approximation can be used to treat networks with finite mean degree and concludes with a discussion of the local properties of random matrices associated with complex networks.