The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. 
Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. 
The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. 
The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology.

/pdf/statistical-mechanics-of-complex-networks-1zfjpvxipu.pdf

Statistical mechanics of complex networks

Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

/pdf/the-structure-and-function-of-complex-networks-34deompc9l.pdf

The Structure and Function of Complex Networks

http://www.maths.qmul.ac.uk/%7Elatora/report_06.pdf

Complex networks: Structure and dynamics

Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution—the part of the distribution representing large but rare events—and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov (KS) statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data, while in others the power law is ruled out.

/pdf/power-law-distributions-in-empirical-data-4dybihkltl.pdf

Power-Law Distributions in Empirical Data

A key aim of postgenomic biomedical research is to systematically catalogue all molecules and their interactions within a living cell. There is a clear need to understand how these molecules and the interactions between them determine the function of this enormously complex machinery, both in isolation and when surrounded by other cells. Rapid advances in network biology indicate that cellular networks are governed by universal laws and offer a new conceptual framework that could potentially revolutionize our view of biology and disease pathologies in the twenty-first century.

/pdf/network-biology-understanding-the-cell-s-functional-589nl2mnk6.pdf

Network biology: understanding the cell's functional organization

Network generators that capture the Internet's large-scale topology are crucial for the development of efficient routing protocols and modeling Internet traffic. Our ability to design realistic generators is limited by the incomplete understanding of the fundamental driving forces that affect the Internet's evolution. By combining several independent databases capturing the time evolution, topology, and physical layout of the Internet, we identify the universal mechanisms that shape the Internet's router and autonomous system level topology. We find that the physical layout of nodes form a fractal set, determined by population density patterns around the globe. The placement of links is driven by competition between preferential attachment and linear distance dependence, a marked departure from the currently used exponential laws. The universal parameters that we extract significantly restrict the class of potentially correct Internet models and indicate that the networks created by all available topology generators are fundamentally different from the current Internet.

https://www.pnas.org/content/pnas/99/21/13382.full.pdf

Modeling the Internet's large-scale topology

The elucidation of the cell’s large-scale organization is a primary challenge for post-genomic biology, and understanding the structure of protein interaction networks offers an important starting point for such studies. We compare four available databases that approximate the protein interaction network of the yeast, Saccharomyces cerevisiae, aiming to uncover the network’s generic large-scale properties and the impact of the proteins’ function and cellular localization on the network topology. We show how each database supports a scale-free, topology with hierarchical modularity, indicating that these features represent a robust and generic property of the protein interactions network. We also find strong correlations between the network’s structure and the functional role and subcellular localization of its protein constituents, concluding that most functional and/or localization classes appear as relatively segregated subnetworks of the full protein interaction network. The uncovered systematic differences between the four protein interaction databases reflect their relative coverage for different functional and localization classes and provide a guide for their utility in various bioinformatics studies.

http://path.upmc.edu/people/faculty/oltvai-lab/04-Proteomics.pdf

Functional and topological characterization of protein interaction networks

Many biological, ecological, and economic systems are best described by weighted networks, as the nodes interact with each other with varying strength. However, most evolving network models studied so far are binary, the link strength being either 0 or 1. In this paper we introduce and investigate the scaling properties of a class of models which assign weights to the links as the network evolves. The combined numerical and analytical approach indicates that asymptotically the total weight distribution converges to the scaling behavior of the connectivity distribution, but this convergence is hampered by strong logarithmic corrections.

https://repository.library.northeastern.edu/files/neu:331105/fulltext.pdf

Weighted Evolving Networks

Networks with complex topology describe systems as diverse as the cell or the World Wide Web. The emergence of these networks is driven by self-organizing processes that are governed by simple but generic laws. In the last three years it became clear that many complex networks, such as the Internet, the cell, or the world wide web, share the same large-scale topology. Here we review recent advances in the characterization of complex networks, focusing the emergence of the scale-free and the hierarchical architecture. We also present empirical results to demonstrate that the scale-free and the hierarchical property are shared by a wide range of complex networks. Finally, we discuss the impact of the network topology on our ability to stop the spread of viruses in complex networks.

/pdf/scale-free-and-hierarchical-structures-in-complex-networks-2l2vge0nnk.pdf

Scale‐Free and Hierarchical Structures in Complex Networks

Intracellular signal transduction occurs through cascades of reactions involving dozens of proteins that transmit signals from the cell surface, through a crowded cellular environment filled with organelles and a filamentous cytoskeleton, to specific targets. Numerous signaling molecules are immobilized or transiently bound to the cytoskeleton, yet most models for signaling pathways have no specific role for this mesh, which is often presumed to function primarily as a scaffold that determines cell mechanics but not information flow. We combined analytical tools with several recently established large-scale protein-protein interaction maps for Saccharomyces cerevisiae to quantitatively address the role of the cytoskeleton in intracellular signaling. The results demonstrate that the network of signaling proteins is intimately linked to the cytoskeleton, suggesting that this interconnected filamentous structure plays a crucial and distinct functional role in signal transduction.

/pdf/role-of-the-cytoskeleton-in-signaling-networks-2tsqnnuzyo.pdf

Soon-Hyung Yook

Papers

Modeling the Internet's large-scale topology

Functional and topological characterization of protein interaction networks

Weighted Evolving Networks

Scale‐Free and Hierarchical Structures in Complex Networks

Role of the cytoskeleton in signaling networks.