Broad Degree Distributions Research Articles

Stochastic blockmodels and community structure in networks

Stochastic blockmodels have been proposed as a tool for detecting community structure in networks as well as for generating synthetic networks for use as benchmarks. Most blockmodels, however, ignore variation in vertex degree, making them unsuitable for applications to real-world networks, which typically display broad degree distributions that can significantly affect the results. Here we demonstrate how the generalization of blockmodels to incorporate this missing element leads to an improved objective function for community detection in complex networks. We also propose a heuristic algorithm for community detection using this objective function or its non-degree-corrected counterpart and show that the degree-corrected version dramatically outperforms the uncorrected one in both real-world and synthetic networks.

Onion-like network topology enhances robustness against malicious attacks

We develop a method to generate robust networks against malicious attacks, aswell as to substantially improve the robustness of a given network by swappingedges and keeping the degree distribution fixed. The method, based on persistenceof the size of the largest cluster during attacks, was applied to several types ofnetworks with broad degree distributions, including a real network—the Internet. Wefind that our method can improve the robustness significantly. Our results showthat robust networks have a novel ‘onion-like’ topology consisting of a core ofhighly connected nodes hierarchically surrounded by rings of nodes with decreasingdegree.

The Role of Degree Distribution in Shaping the Dynamics in Networks of Sparsely Connected Spiking Neurons

Neuronal network models often assume a fixed probability of connection between neurons. This assumption leads to random networks with binomial in-degree and out-degree distributions which are relatively narrow. Here I study the effect of broad degree distributions on network dynamics by interpolating between a binomial and a truncated power-law distribution for the in-degree and out-degree independently. This is done both for an inhibitory network (I network) as well as for the recurrent excitatory connections in a network of excitatory and inhibitory neurons (EI network). In both cases increasing the width of the in-degree distribution affects the global state of the network by driving transitions between asynchronous behavior and oscillations. This effect is reproduced in a simplified rate model which includes the heterogeneity in neuronal input due to the in-degree of cells. On the other hand, broadening the out-degree distribution is shown to increase the fraction of common inputs to pairs of neurons. This leads to increases in the amplitude of the cross-correlation (CC) of synaptic currents. In the case of the I network, despite strong oscillatory CCs in the currents, CCs of the membrane potential are low due to filtering and reset effects, leading to very weak CCs of the spike-count. In the asynchronous regime of the EI network, broadening the out-degree increases the amplitude of CCs in the recurrent excitatory currents, while CC of the total current is essentially unaffected as are pairwise spiking correlations. This is due to a dynamic balance between excitatory and inhibitory synaptic currents. In the oscillatory regime, changes in the out-degree can have a large effect on spiking correlations and even on the qualitative dynamical state of the network.

Critical effect of dependency groups on the function of networks.

Current network models assume one type of links to define the relations between the network entities. However, many real networks can only be correctly described using two different types of relations. Connectivity links that enable the nodes to function cooperatively as a network and dependency links that bind the failure of one network element to the failure of other network elements. Here we present an analytical framework for studying the robustness of networks that include both connectivity and dependency links. We show that a synergy exists between the failure of connectivity and dependency links that leads to an iterative process of cascading failures that has a devastating effect on the network stability. We present exact analytical results for the dramatic change in the network behavior when introducing dependency links. For a high density of dependency links, the network disintegrates in a form of a first-order phase transition, whereas for a low density of dependency links, the network disintegrates in a second-order transition. Moreover, opposed to networks containing only connectivity links where a broader degree distribution results in a more robust network, when both types of links are present a broad degree distribution leads to higher vulnerability.

Concurrency and Network Disassortativity

The relationship between a network's degree-degree correlation and a loose version of graph coloring is studied on networks with broad degree distributions. We find that, given similar conditions on the number of nodes, number of links, and clustering levels, fewer colors are needed to color disassortative than assortative networks. Since fewer colors create fewer independent sets, our finding implies that disassortative networks may have higher concurrency potential than assortative networks. This in turn suggests another reason for the disassortative mixing pattern observed in biological networks such as those of protein-protein interaction and gene regulation. In addition to the functional specificity and stability suggested by Maslov and Sneppen, a disassortative network topology may also enhance the ability of cells to perform crucial tasks concurrently. Hence, increased concurrency may also be a driving force in the evolution of biological networks.

Evolution of network structure by temporal learning

We study the effect of learning dynamics on network topology. A network of discrete dynamical systems is considered for this purpose and the coupling strengths are made to evolve according to a temporal learning rule that is based on the paradigm of spike-time-dependent plasticity. This incorporates necessary competition between different edges. The final network we obtain is robust and has a broad degree distribution.

Signatures of Currency Vertices

Many real-world networks have broad degree distributions. For some systems, this means that the functional significance of the vertices is also broadly distributed, in other cases the vertices are equally significant, but in different ways. One example of the latter case is metabolic networks, where the high-degree vertices -- the currency metabolites -- supply the molecular groups to the low-degree metabolites, and the latter are responsible for the higher-order biological function, of vital importance to the organism. In this paper, we propose a generalization of currency metabolites to currency vertices. We investigate the network structural characteristics of such systems, both in model networks and in some empirical systems. In addition to metabolic networks, we find that a network of music collaborations and a network of e-mail exchange could be described by a division of the vertices into currency vertices and others.

Voter models on heterogeneous networks.

We study simple interacting particle systems on heterogeneous networks, including the voter model and the invasion process. These are both two-state models in which in an update event an individual changes state to agree with a neighbor. For the voter model, an individual "imports" its state from a randomly chosen neighbor. Here the average time TN to reach consensus for a network of N nodes with an uncorrelated degree distribution scales as N mu1 2/mu2, where mu k is the kth moment of the degree distribution. Quick consensus thus arises on networks with broad degree distributions. We also identify the conservation law that characterizes the route by which consensus is reached. Parallel results are derived for the invasion process, in which the state of an agent is "exported" to a random neighbor. We further generalize to biased dynamics in which one state is favored. The probability for a single fitter mutant located at a node of degree k to overspread the population-the fixation probability--is proportional to k for the voter model and to 1k for the invasion process.

Global firing induced by network disorder in ensembles of active rotators

We study the influence of repulsive interactions on an ensemble of coupled excitable rotators. We find that a moderate fraction of repulsive interactions can trigger global firing of the ensemble. The regime of global firing, however, is suppressed in sufficiently large systems if the network of repulsive interactions is fully random, due to self-averaging in its degree distribution. We thus introduce a model of partially random networks with a broad degree distribution, where self-averaging due to size growth is absent. In this case, the regime of global firing persists for large sizes. Our results extend previous work on the constructive effects of diversity in the collective dynamics of complex systems.

Gene duplication models for directed networks with limits on growth

Background: Duplication of genes is important for evolution of molecular networks. Manyauthors have therefore considered gene duplication as a driving force in shapingthe topology of molecular networks. In particular it has been noted that growthvia duplication would act as an implicit means of preferential attachment, andthereby provide the observed broad degree distributions of molecular networks.Results: We extend current models of gene duplication and rewiring by including directionsand the fact that molecular networks are not a result of unidirectional growth.We introduce upstream sites and downstream shapes to quantify potential linksduring duplication and rewiring. We find that this in itself generates the observedscaling of transcription factors for genome sites in prokaryotes. The dynamicalmodel can generate a scale-free degree distribution, , with exponent γ = 1 in thenon-growing case, and with γ>1 when the network is growing.Conclusions: We find that duplication of genes followed by substantial recombination ofupstream regions could generate features of genetic regulatory networks. Our steady statedegree distribution is however too broad to be consistent with data, thereby suggestingthat selective pruning acts as a main additional constraint on duplicated genes. Ouranalysis shows that gene duplication can only be a main cause for the observed broaddegree distributions if there are also substantial recombinations between upstream regionsof genes.

Beyond centrality—classifying topological significance using backup efficiency and alternative paths**Partial results presented in this paper also appear in the IFIP NETWORKING 2007 Conference.

In complex networks characterized by broad degree distribution, node significance is often associated with its degree or with centrality metrics which relate to its reachability and shortest paths passing through it. Such measures do not consider availability of efficient backup of the node and thus often fail to capture its contribution to the functionality and resilience of the network operation. In this paper, we suggest the quality of backup (QoB) and alternative path centrality (APC) measures as complementary methods which enable analysis of node significance in a manner which considers backup. We examine the theoretical significance of these measures and use them to classify nodes in social interaction networks and in the Internet AS (autonomous system) graph while applying the valley-free routing restrictions which reflect the economic relationships between the AS nodes in the Internet. We show that both node degree and node centrality are not necessarily evidence of its significance. In particular, we show that social structures do not necessarily depend on highly central nodes and that medium degree nodes with medium centrality measure prove to be crucial for efficient routing in the Internet AS graph.

Degree landscapes in scale-free networks

We generalize the degree-organizational view of real-world networks with broad degree distributions in a landscape analog with mountains (high-degree nodes) and valleys (low-degree nodes). For example, correlated degrees between adjacent nodes correspond to smooth landscapes (social networks), hierarchical networks to one-mountain landscapes (the Internet), and degree-disassortative networks without hierarchical features to rough landscapes with several mountains. To quantify the topology, we here measure the widths of the mountains and the separation between different mountains. We also generate ridge landscapes to model networks organized under constraints imposed by the space the networks are embedded in, associated to spatial or in molecular networks to functional localization.

A model for social networks

Social networks are organized into communities with dense internal connections, giving rise to high values of the clustering coefficient. In addition, these networks have been observed to be assortative, i.e., highly connected vertices tend to connect to other highly connected vertices, and have broad degree distributions. We present a model for an undirected growing network which reproduces these characteristics, with the aim of producing efficiently very large networks to be used as platforms for studying sociodynamic phenomena. The communities arise from a mixture of random attachment and implicit preferential attachment. The structural properties of the model are studied analytically and numerically, using the k-clique method for quantifying the communities.

Systematic Series Expansions for Processes on Networks

We use series expansions to study dynamics of equilibrium and nonequilibrium systems on networks. This analytical method enables us to include detailed nonuniversal effects of the network structure. We show that even low order calculations produce results which compare accurately to numerical simulation, while the results can be systematically improved. We show that certain commonly accepted analytical results for the critical point on networks with a broad degree distribution need to be modified in certain cases due to disassortativity; the present method is able to take into account the assortativity at sufficiently high order, while previous results correspond to leading and second order approximations in this method. Finally, we apply this method to real-world data.

The network analysis of urban streets: A dual approach

The application of the network approach to the urban case poses several questions in terms of how to deal with metric distances, what kind of graph representation to use, what kind of measures to investigate, how to deepen the correlation between measures of the structure of the network and measures of the dynamics on the network, what are the possible contributions from the GIS community. In this paper, the author considers six cases of urban street networks characterized by different patterns and historical roots. The authors propose a representation of the street networks based firstly on a primal graph, where intersections are turned into nodes and streets into edges. In a second step, a dual graph, where streets are nodes and intersections are edges, is constructed by means of a generalization model named Intersection Continuity Negotiation, which allows to acknowledge the continuity of streets over a plurality of edges. Finally, the authors address a comparative study of some structural properties of the dual graphs, seeking significant similarities among clusters of cases. A wide set of network analysis techniques are implemented over the dual graph: in particular the authors show that the absence of any clue of assortativity differentiates urban street networks from other non-geographic systems and that most of the considered networks have a broad degree distribution typical of scale-free networks and exhibit small-world properties as well.

Networks and Cities: An Information Perspective

Traffic is constrained by the information involved in locating the receiver and the physical distance between sender and receiver. We here focus on the former, and investigate traffic in the perspective of information handling. We replot the road map of cities in terms of the information needed to locate specific addresses and create information city networks with roads mapped to nodes and intersections to links between nodes. These networks have the broad degree distribution found in many other complex networks. The mapping to an information city network makes it possible to quantify the information associated with locating specific addresses.

Scale-free networks from a Hamiltonian dynamics.

Contrary to many recent models of growing networks, we present a model with fixed number of nodes and links, where a dynamics favoring the formation of links between nodes with degree of connectivity as different as possible is introduced. By applying a local rewiring move, the network reaches equilibrium states assuming broad degree distributions, which have a power-law form in an intermediate range of the parameters used. Interestingly, in the same range we find nontrivial hierarchical clustering.