Information Based Complexity of Networks

Scientific Reports

Understanding the structure and the dynamics of networks is of paramount importance for many scientific fields that rely on network science. Complex network theory provides a variety of features that help in the evaluation of network behavior. However, such analysis can be confusing and misleading as there are many intrinsic properties for each network metric. Alternatively, Information Theory methods have gained the spotlight because of their ability to create a quantitative and robust characterization of such networks. In this work, we use two Information Theory quantifiers, namely Network Entropy and Network Fisher Information Measure, to analyzing those networks. Our approach detects non-trivial characteristics of complex networks such as the transition present in the Watts-Strogatz model from k-ring to random graphs; the phase transition from a disconnected to an almost surely connected network when we increase the linking probability of Erdős-Rényi model; distinct phases of sc...

Network theory based controllability and observability analysis have become widely used techniques. We realized that most applications are not related to dynamical systems, and mainly the physical topologies of the systems are analysed without deeper considerations. Here, we draw attention to the importance of dynamics inside and between state variables by adding functional relationship defined edges to the original topology. The resulting networks differ from physical topologies of the systems and describe more accurately the dynamics of the conservation of mass, momentum and energy. We define the typical connection types and highlight how the reinterpreted topologies change the number of the necessary sensors and actuators in benchmark networks widely studied in the literature. Additionally, we offer a workflow for network science-based dynamical system analysis, and we also introduce a method for generating the minimum number of necessary actuator and sensor points in the system. Y. Y. Liu et al. started a new trend in network science when they become the first to analyse complex networks as dynamical systems with the maximum matching algorithm 1. They considered nodes as state variables, interpreted networks as linear multivariable dynamical systems and studied the controllability and observability of these models 2. Based on these principles Yan et al. analysed the required energy for controlling a system 3 , Ruths & Ruths determined control profiles for cluster networks 4 , Pósfai, Liu, Slotine & Barabási examined how the degree correlation influences the required inputs 5 , and the robustness of an input configuration was also improved by X. Liu et al. 6. The application of the proposed method is also widespread, for example, Penn, Knight, Chalkias, Velenturf & Lloyd applied this approach on fuzzy cognitive maps as well 7. These studies impressively show the benefits of network science-based analysis of dynamical systems. Despite the groundbreaking successes, some critiques have also been received. Müller & Schuppert determined that in transcriptional networks the method drastically overestimates the number of necessary inputs 8. Sun, Cornelius, Kath & Motter also highlighted that the methodology needs further clarification because the method gives incorrect results for non-linear systems even for small examples 9. This fact has also been evinced by Dunne, Williams, & Martinez 10. Another problem is that researchers examined the correlation between necessary inputs generated by the proposed method, and structural properties, like degree distribution, but they did not take into account that the result of the maximum matching algorithm is not unique 11. The most contestable point of the network-based analysis is that it is based on a static and structural view of the system. We wish to offer a solution to the previously mentioned problems by examining how system dynamics should be represented realistically. The usage of proper topology is important and a crucial part of network analysis, as this is the only way to emphasise dynamics in statical representations. We introduce connection types according to the typical relationships of the state variables. To analyse how the determined connection types influence the controllability and observability of dynamical systems we developed a MATLAB toolbox. We examined 35 example networks used in articles and found that 27 do not represent dynamical systems. By comparing them with 18 independently selected dynamical systems, we revealed significant differences. While in dynamical systems the number of inputs and outputs does not change when the proper topology of the model is studied, in the case of other networks more than 95% of inputs and outputs disappeared because of the determined connection types.

2007

Abstract: I start by reviewing some basic properties of random graphs. I then consider the role of random walks in complex networks and show how they may be used to explain why so many long tailed distributions are found in real data sets. The key idea is that in many cases the process involves copying of properties of near neighbours in the network and this is a type of short random walk which in turn produce a natural preferential attachment mechanism.

Complexity, 2011

Networks in Biology Handling Biological Complexity Requires Novel Inputs into Network Theory T he year 2009 was the tenth anniversary of the first publication on scale-free networks [1] and the fiftieth anniversary of the invention of random graphs [2]. Science magazine devoted a special section to review the present status of network theory, complexity research and its application to different disciplines [3]. Understanding and modeling complex systems without consideration of network topology and network evolution became out-of-date and practically impossible. The Erdös-Rényi model initiated a real breakthrough in the sense that statistical properties of graphs and networks became accessible without knowledge of the connection details. Among many other applications random networks became a useful reference in the biology of macromolecules for mapping polynucleotide sequences into structures [4]. Real world social networks, in particular communication networks, were found to have substantial shorter mean distances between agents than those predicted by the theory of random networks. Watts and Strogatz [5] invented networks that were found to match the available empirical data. These networks are called small-world networks as they have the small-world property 1 [6], and they can be understood as intermediates between a regular lattice and a random graph. Watts and Strogatz characterize small-world networks by a degree of randomness (p), which varies between p 5 0 for the regular and p 5 1 for the random network. Only 1 year later, Barabási and Albert conceived smallworld networks, which are created by the construction principle of preferential attachment [1]: Starting with a fully connected network of three nodes, further nodes are attached one by one with a larger probability for the incoming node to connect to a node that has already more neighbors. The resulting networks are scale-free-besides having the small-world property-and accordingly many of their properties fulfill power laws. 2 Comparison with real world data from a great

Scientific Reports, 2013

We study the influence of noise on information transmission in the form of packages shipped between nodes of hierarchical networks. Numerical simulations are performed for artificial tree networks, scale-free Ravasz-Baraba ´si networks as well for a real network formed by email addresses of former Enron employees. Two types of noise are considered. One is related to packet dynamics and is responsible for a random part of packets paths. The second one originates from random changes in initial network topology. We find that the information transfer can be enhanced by the noise. The system possesses optimal performance when both kinds of noise are tuned to specific values, this corresponds to the Stochastic Resonance phenomenon. There is a non-trivial synergy present for both noisy components. We found also that hierarchical networks built of nodes of various degrees are more efficient in information transfer than trees with a fixed branching factor. A challenging research issue is the understanding of the effect of a network topology on the transfer of information 1-3 . Many biological, technical and social systems posses hierarchical structures, e.g., metabolic networks 4,5 , protein interaction networks 6-8 , the Internet 9 , theWorld WideWeb 10 , and social networks . Topologies of networks evolve over time in order to maximize the systemic efficiency. An important part of this efficiency is the optimal information exchange, even though the meaning and spreading of information varies considerably among such fields as mathematics 15 , physics 16 , biology , social science, communication and computer science . When information is transmitted through a complex network, the process can be represented as an evolution of a distributed information field that is related to the states of network nodes (and/or links) or as paths of localized packages traveling through the network. In the first case, nodes and links can form systems of coupled threshold devices or oscillators 21 that make it possible to propagate a signal from the sender to the receiver. In the second case, the nodes possess some binary variables that can be transmitted from one node to another. Several authors have considered various features of network typologies that influence information transfers (e.g., if the information flow in a scale-free network is more efficient than in a regular lattice) . Furthermore, it is shown to be crucial wether a network consists of heterogeneous (e.g. routers and peripheral nodes) 1,2 or homogeneous nodes with similar connection degrees. Another important attribute is the edge heterogenity in weighted networks or the network clustering coefficient since a lack of loops in a network can impose a less efficient information transfer . There exist several ways to incorporate packet dynamics, from rule simple random walks (e.g. ,) to more elaborate ways, like biased random walks with some local and/or global navigation rules (e.g. 1,3,28-30 ,). It is also possible to implement epidemic models such as SIR (e.g. 1,18,31 ,). Moreover, we can consider single particles (e.g. 1,3,26,28 ,) or interacting particles systems (e.g. 2,27,30 ,). The types of dynamics we choose strongly depends on the interpretation of the modeled information transfer. This paper considers a prototypical model for the flow of information represented by packages travelling on networks with hierarchical structures. We are particularly interested in the dependency between network structures, noise, and information transmission efficiency. We take into account two types of noise in the system. The first one corresponds to a non-deterministic part of system dynamics while the second one to the level of difference between the current network structure in comparison to original one. Numerical simulations show a resonancelike behavior of information transfer efficiency in the presence of both types of noise. This paper is organized as follows. Section Networks topology and topological noise introduces the topology of hierarchical networks. Section Packet navigation rules describes the imposed packet dynamics. Section Results for artificial networks presents the numerical results for the artificial networks, while Section Real network is devoted SUBJECT AREAS: STATISTICAL PHYSICS, THERMODYNAMICS AND NONLINEAR DYNAMICS COMPUTATIONAL SCIENCE INFORMATION THEORY AND COMPUTATION SCIENTIFIC DATA

Physica A-statistical Mechanics and Its Applications, 2008

Generalized mutual entropy is defined for networks and applied for analysis of complex network structures. The method is tested for the case of computer simulated scale free networks, random networks, and their mixtures. The possible applications for real network analysis are discussed.

Figure 1: Schematic levels of spatial integration in the nervous system. The spatial scale at which anatomical organizations can be identified varies over many orders of magnitude. Icons to the right represent structures at distinct levels in a bottom-up fashion: (bottom) a chemical synapse, (middle-bottom) a network model of how thalamic afferent cells could be connected to simple cells in visual cortex, (middle-top) maps of orientation preference and ocular dominance in a primary visual area; (top) the subset of visual areas forming visual cortex and their interconnections (adapted from ref. [1]).

Journal of Statistical Physics

Higher order networks are able to characterize data as different as functional brain networks, protein interaction networks and social networks beyond the framework of pairwise interactions. Most notably higher order networks include simplicial complexes formed not only by nodes and links but also by triangles, tetrahedra, etc. More in general, higher-order networks can be cell-complexes formed by gluing convex polytopes along their faces. Interestingly, higher order networks have a natural geometric interpretation and therefore constitute a natural way to explore the discrete network geometry of complex networks. Here we investigate the rich interplay between emergent network geometry of higher order networks and their complexity in the framework of a non-equilibrium model called Network Geometry with Flavor. This model, originally proposed for capturing the evolution of simplicial complexes, is here extended to cell-complexes formed by subsequently gluing different copies of an arbitrary regular polytope. We reveal the interplay between complexity and geometry of the higher order networks generated by the model by studying the emergent community structure and the degree distribution as a function of the regular polytope forming its building blocks. Additionally, we discuss the underlying hyperbolic nature of the emergent geometry and we relate the spectral dimension of the higher-order network to the dimension and nature of its building blocks.

2016

Complexity is a widely used parameter in network design, yet there is no generally accepted definition of the term. Complexity metrics exist in a wide range of research papers, but most of these address only a particular aspect of a network, for example the complexity of a graph or software. There is a desire to define the complexity of a network as a whole, as deployed today to provide Internet services. This document provides a framework to guide research on the topic of network complexity. Status of This Memo This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts.

Complex Networks, 2004

Complex networks are characterized by highly heterogeneous distributions of links, often pervading the presence of key properties such as robustness under node removal. Several correlation measures have been defined in order to characterize the structure of these nets. Here we show that mutual information, noise and joint entropies can be properly defined on a static graph. These measures are computed for a number of real networks and analytically estimated for some simple standard models. It is shown that real networks are clustered in a well-defined domain of the entropynoise space. By using simulated annealing optimization, it is shown that optimally heterogeneous nets actually cluster around the same narrow domain, suggesting that strong constraints actually operate on the possible universe of complex networks. The evolutionary implications are discussed.