Reduction of Dilute Ising Spin Glasses

The recently proposed reduction method for diluted spin glasses is investigated in depth. In particular, the Edwards-Anderson model with \pm J and Gaussian bond disorder on hyper-cubic lattices in d=2, 3, and 4 is studied for a range of bond dilutions. The results demonstrate the effectiveness of using bond dilution to elucidate low-temperature properties of Ising spin glasses, and provide a starting point to enhance the methods used in reduction. Based on that, a greedy heuristic call ``Dominant Bond Reduction'' is introduced and explored.Comment: 10 pages, revtex, final version, find related material at http://www.physics.emory.edu/faculty/boettcher

Heterogeneous-k-core versus Bootstrap Percolation on Complex Networks

We introduce the heterogeneous-$k$-core, which generalizes the $k$-core, and contrast it with bootstrap percolation. Vertices have a threshold $k_i$ which may be different at each vertex. If a vertex has less than $k_i$ neighbors it is pruned from the network. The heterogeneous-$k$-core is the sub-graph remaining after no further vertices can be pruned. If the thresholds $k_i$ are $1$ with probability $f$ or $k \geq 3$ with probability $(1-f)$, the process forms one branch of an activation-pruning process which demonstrates hysteresis. The other branch is formed by ordinary bootstrap percolation. We show that there are two types of transitions in this heterogeneous-$k$-core process: the giant heterogeneous-$k$-core may appear with a continuous transition and there may be a second, discontinuous, hybrid transition. We compare critical phenomena, critical clusters and avalanches at the heterogeneous-$k$-core and bootstrap percolation transitions. We also show that network structure has a crucial effect on these processes, with the giant heterogeneous-$k$-core appearing immediately at a finite value for any $f > 0$ when the degree distribution tends to a power law $P(q) \sim q^{-\gamma}$ with $\gamma < 3$.Comment: 10 pages, 4 figure

The entropy of randomized network ensembles

Randomized network ensembles are the null models of real networks and are extensivelly used to compare a real system to a null hypothesis. In this paper we study network ensembles with the same degree distribution, the same degree-correlations or the same community structure of any given real network. We characterize these randomized network ensembles by their entropy, i.e. the normalized logarithm of the total number of networks which are part of these ensembles. We estimate the entropy of randomized ensembles starting from a large set of real directed and undirected networks. We propose entropy as an indicator to assess the role of each structural feature in a given real network.We observe that the ensembles with fixed scale-free degree distribution have smaller entropy than the ensembles with homogeneous degree distribution indicating a higher level of order in scale-free networks.Comment: (6 pages,1 figure,2 tables

Rhythmogenic neuronal networks, pacemakers, and k-cores

Neuronal networks are controlled by a combination of the dynamics of individual neurons and the connectivity of the network that links them together. We study a minimal model of the preBotzinger complex, a small neuronal network that controls the breathing rhythm of mammals through periodic firing bursts. We show that the properties of a such a randomly connected network of identical excitatory neurons are fundamentally different from those of uniformly connected neuronal networks as described by mean-field theory. We show that (i) the connectivity properties of the networks determines the location of emergent pacemakers that trigger the firing bursts and (ii) that the collective desensitization that terminates the firing bursts is determined again by the network connectivity, through k-core clusters of neurons.Comment: 4+ pages, 4 figures, submitted to Phys. Rev. Let

Understanding edge-connectivity in the Internet through core-decomposition

Internet is a complex network composed by several networks: the Autonomous Systems, each one designed to transport information efficiently. Routing protocols aim to find paths between nodes whenever it is possible (i.e., the network is not partitioned), or to find paths verifying specific constraints (e.g., a certain QoS is required). As connectivity is a measure related to both of them (partitions and selected paths) this work provides a formal lower bound to it based on core-decomposition, under certain conditions, and low complexity algorithms to find it. We apply them to analyze maps obtained from the prominent Internet mapping projects, using the LaNet-vi open-source software for its visualization

Organization of modular networks

We examine the global organization of heterogeneous equilibrium networks consisting of a number of well distinguished interconnected parts--``communities'' or modules. We develop an analytical approach allowing us to obtain the statistics of connected components and an intervertex distance distribution in these modular networks, and to describe their global organization and structure. In particular, we study the evolution of the intervertex distance distribution with an increasing number of interlinks connecting two infinitely large uncorrelated networks. We demonstrate that even a relatively small number of shortcuts unite the networks into one. In more precise terms, if the number of the interlinks is any finite fraction of the total number of connections, then the intervertex distance distribution approaches a delta-function peaked form, and so the network is united.Comment: 9 pages, 3 figure

Scale-free models for the structure of business firm networks

We study firm collaborations in the life sciences and the information and communication technology sectors. We propose an approach to characterize industrial leadership using k-shell decomposition, with top-ranking firms in terms of market value in higher k-shell layers. We find that the life sciences industry network consists of three distinct components: a “nucleus,” which is a small well-connected subgraph, “tendrils,” which are small subgraphs consisting of small degree nodes connected exclusively to the nucleus, and a “bulk body,” which consists of the majority of nodes. Industrial leaders, i.e., the largest companies in terms of market value, are in the highest k-shells of both networks. The nucleus of the life sciences sector is very stable: once a firm enters the nucleus, it is likely to stay there for a long time. At the same time we do not observe the above three components in the information and communication technology sector. We also conduct a systematic study of these three components in random scale-free networks. Our results suggest that the sizes of the nucleus and the tendrils in scale-free networks decrease as the exponent of the power-law degree distribution λ increases, and disappear for λ≥3. We compare the k-shell structure of random scale-free model networks with two real-world business firm networks in the life sciences and in the information and communication technology sectors. We argue that the observed behavior of the k-shell structure in the two industries is consistent with the coexistence of both preferential and random agreements in the evolution of industrial networks

Critical phenomena in complex networks

The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, researchers have made important steps toward understanding the qualitatively new critical phenomena in complex networks. We review the results, concepts, and methods of this rapidly developing field. Here we mostly consider two closely related classes of these critical phenomena, namely structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. We also discuss systems where a network and interacting agents on it influence each other. We overview a wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, k-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks. We also discuss strong finite size effects in these systems and highlight open problems and perspectives.Comment: Review article, 79 pages, 43 figures, 1 table, 508 references, extende

Analyzing and Modeling Real-World Phenomena with Complex Networks: A Survey of Applications

The success of new scientific areas can be assessed by their potential for contributing to new theoretical approaches and in applications to real-world problems. Complex networks have fared extremely well in both of these aspects, with their sound theoretical basis developed over the years and with a variety of applications. In this survey, we analyze the applications of complex networks to real-world problems and data, with emphasis in representation, analysis and modeling, after an introduction to the main concepts and models. A diversity of phenomena are surveyed, which may be classified into no less than 22 areas, providing a clear indication of the impact of the field of complex networks.Comment: 103 pages, 3 figures and 7 tables. A working manuscript, suggestions are welcome