Arbitrary Degree Distribution Research Articles

Connecting core percolation and controllability of complex networks.

Core percolation is a fundamental structural transition in complex networks related to a wide range of important problems. Recent advances have provided us an analytical framework of core percolation in uncorrelated random networks with arbitrary degree distributions. Here we apply the tools in analysis of network controllability. We confirm analytically that the emergence of the bifurcation in control coincides with the formation of the core and the structure of the core determines the control mode of the network. We also derive the analytical expression related to the controllability robustness by extending the deduction in core percolation. These findings help us better understand the interesting interplay between the structural and dynamical properties of complex networks.

Spectra of random graphs with community structure and arbitrary degrees.

Using methods from random matrix theory researchers have recently calculated the full spectra of random networks with arbitrary degrees and with community structure. Both reveal interesting spectral features, including deviations from the Wigner semicircle distribution and phase transitions in the spectra of community structured networks. In this paper we generalize both calculations, giving a prescription for calculating the spectrum of a network with both community structure and an arbitrary degree distribution. In general the spectrum has two parts, a continuous spectral band, which can depart strongly from the classic semicircle form, and a set of outlying eigenvalues that indicate the presence of communities.

Copula-based modeling of degree-correlated networks

Dynamical processes on complex networks such as information exchange, innovation diffusion, cascades in financial networks or epidemic spreading are highly affected by their underlying topologies as characterized by, for instance, degree–degree correlations. Here, we introduce the concept of copulas in order to generate random networks with an arbitrary degree distribution and a rich a priori degree–degree correlation (or ‘association’) structure. The accuracy of the proposed formalism and corresponding algorithm is numerically confirmed, while the method is tested on a real-world network of yeast protein–protein interactions. The derived network ensembles can be systematically deployed as proper null models, in order to unfold the complex interplay between the topology of real-world networks and the dynamics on top of them.

Probabilistic Analysis on Minimum <i>s-t</i> Cut Capacity of Random Graphs with Specified Degree Distribution

The capacity (or maximum flow) of an unicast network is known to be equal to the minimum s-t cut capacity due to the max-flow min-cut theorem. If the topology of a network (or link capacities) is dynamically changing or unknown, it is not so trivial to predict statistical properties on the maximum flow of the network. In this paper, we present a probabilistic analysis for evaluating the accumulate distribution of the minimum s-t cut capacity on random graphs. The graph ensemble treated in this paper consists of weighted graphs with arbitrary specified degree distribution. The main contribution of our work is a lower bound for the accumulate distribution of the minimum s-t cut capacity. From some computer experiments, it is observed that the lower bound derived here reflects the actual statistical behavior of the minimum s-t cut capacity of random graphs with specified degrees.

Approximate Master Equations for Dynamical Processes on Graphs

We extrapolate from the exact master equations of epidemic dynamics on fully connected graphs to non-fully connected by keeping the size of the state space N + 1, where N is the number of nodes in the graph. This gives rise to a system of approximate ODEs (ordinary differential equations) where the challenge is to compute/approximate analytically the transmission rates. We show that this is possible for graphs with arbitrary degree distributions built according to the configuration model. Numerical tests confirm that: (a) the agreement of the approximate ODEs system with simulation is excellent and (b) that the approach remains valid for clustered graphs with the analytical calculations of the transmission rates still pending. The marked reduction in state space gives good results, and where the transmission rates can be analytically approximated, the model provides a strong alternative approximate model that agrees well with simulation. Given that the transmission rates encompass information both about the dynamics and graph properties, the specific shape of the curve, defined by the transmission rate versus the number of infected nodes, can provide a new and different measure of network structure, and the model could serve as a link between inferring network structure from prevalence or incidence data.

Percolation on random networks with arbitraryk-core structure

The k-core decomposition of a network has thus far mainly served as a powerful tool for the empirical study of complex networks. We now propose its explicit integration in a theoretical model. We introduce a hard-core random network (HRN) model that generates maximally random networks with arbitrary degree distribution and arbitrary k-core structure. We then solve exactly the bond percolation problem on the HRN model and produce fast and precise analytical estimates for the corresponding real networks. Extensive comparison with real databases reveals that our approach performs better than existing models, while requiring less input information.

The diminishing role of hubs in dynamical processes on complex networks

It is notoriously difficult to predict the behaviour of a complex self-organizing system, where the interactions among dynamical units form a heterogeneous topology. Even if the dynamics of each microscopic unit is known, a real understanding of their contributions to the macroscopic system behaviour is still lacking. Here, we develop information-theoretical methods to distinguish the contribution of each individual unit to the collective out-of-equilibrium dynamics. We show that for a system of units connected by a network of interaction potentials with an arbitrary degree distribution, highly connected units have less impact on the system dynamics when compared with intermediately connected units. In an equilibrium setting, the hubs are often found to dictate the long-term behaviour. However, we find both analytically and experimentally that the instantaneous states of these units have a short-lasting effect on the state trajectory of the entire system. We present qualitative evidence of this phenomenon from empirical findings about a social network of product recommendations, a protein–protein interaction network and a neural network, suggesting that it might indeed be a widespread property in nature.

Oriented and degree-generated block models: generating and inferring communities with inhomogeneous degree distributions

The stochastic block model is a powerful tool for inferring community structure from network topology. However, it predicts a Poisson degree distribution within each community, while most real-world networks have a heavy-tailed degree distribution. The degree-corrected (DC) block model can accommodate arbitrary degree distributions within communities. But since it takes the vertex degrees as parameters rather than generating them, it cannot use them to help it classify the vertices, and its natural generalization to directed graphs cannot even use the orientations of the edges. In this paper, we present variants of the block model with the best of both worlds: they can use vertex degrees and edge orientations in the classification process, while tolerating heavy-tailed degree distributions within communities. We show that for some networks, including synthetic networks and networks of word adjacencies in English text, these new block models achieve a higher accuracy than either standard or DC block models.

Inducing effect on the percolation transition in complex networks

Percolation theory concerns the emergence of connected clusters that percolate through a networked system. Previous studies ignored the effect that a node outside the percolating cluster may actively induce its inside neighbours to exit the percolating cluster. Here we study this inducing effect on the classical site percolation and K-core percolation, showing that the inducing effect always causes a discontinuous percolation transition. We precisely predict the percolation threshold and core size for uncorrelated random networks with arbitrary degree distributions. For low-dimensional lattices the percolation threshold fluctuates considerably over realizations, yet we can still predict the core size once the percolation occurs. The core sizes of real-world networks can also be well predicted using degree distribution as the only input. Our work therefore provides a theoretical framework for quantitatively understanding discontinuous breakdown phenomena in various complex systems.

What exactly are the properties of scale-free and other networks?

The concept of scale-free networks has been widely applied across natural and physical sciences. Many claims are made about the properties of these networks, even though the concept of scale-free is often vaguely defined. We present tools and procedures to analyse the statistical properties of networks defined by arbitrary degree distributions and other constraints. Doing so reveals the highly likely properties, and some unrecognised richness, of scale-free networks, and casts doubt on some previously claimed properties being due to a scale-free characteristic.

Pseudo-likelihood methods for community detection in large sparse networks

Many algorithms have been proposed for fitting network models with communities, but most of them do not scale well to large networks, and often fail on sparse networks. Here we propose a new fast pseudo-likelihood method for fitting the stochastic block model for networks, as well as a variant that allows for an arbitrary degree distribution by conditioning on degrees. We show that the algorithms perform well under a range of settings, including on very sparse networks, and illustrate on the example of a network of political blogs. We also propose spectral clustering with perturbations, a method of independent interest, which works well on sparse networks where regular spectral clustering fails, and use it to provide an initial value for pseudo-likelihood. We prove that pseudo-likelihood provides consistent estimates of the communities under a mild condition on the starting value, for the case of a block model with two communities.

SEIR Epidemic Dynamics in Random Networks

Predicting disease transmission on complex networks has attracted considerable recent attention in the epidemiology community. In this paper, we develop a low-dimensional system of nonlinear ordinary differential equations to model the susceptible-exposed-infectious-recovered (SEIR) epidemics on random network with arbitrary degree distributions. Both the final size of epidemics and the time-dependent behaviors are derived within our simple framework. The underlying network is represented by the configuration model, which appropriately accounts for the heterogeneity and finiteness of the degree observed in a variety of real contact networks. Moreover, a generalized model where the infectious state of individual can be skipped is treated in brief.

Random walks on weighted networks

Random walks constitute a fundamental mechanism for a large set of dynamics taking place on networks. In this article, we study random walks on weighted networks with an arbitrary degree distribution, where the weight of an edge between two nodes has a tunable parameter. By using the spectral graph theory, we derive analytical expressions for the stationary distribution, mean first-passage time (MFPT), average trapping time (ATT), and lower bound of the ATT, which is defined as the average MFPT to a given node over every starting point chosen from the stationary distribution. All these results depend on the weight parameter, indicating a significant role of network weights on random walks. For the case of uncorrelated networks, we provide explicit formulas for the stationary distribution as well as ATT. Particularly, for uncorrelated scale-free networks, when the target is placed on a node with the highest degree, we show that ATT can display various scalings of network size, depending also on the same parameter. Our findings could pave a way to delicately controlling random-walk dynamics on complex networks.

Social Network Development using Graph Model in PRI System

The review of current research emphasizes three strands of literature on social networks. The first strand composed of endogenous network models, which highlights the sensitive dependence of the topology on parameters of the behavioural models employed. The second strand draws from the recent econophysics literature reviewing the recent revival of interest in the random graph theory and also allowing a mathematical tool to study social networks. The review explores a number of examples to assess the potential of recent research on random graphs with arbitrary degree distributions. The third strand focuses on a specific model of social networks where random graphs with different dependent edges are present, if edges are incident to the same node, and independent, otherwise. Besides, this paper introduces a new graph model helping to design the interconnected model to connect blocks and gram panchayats in Panchayati Raj Institution (PRI) system at ease and in turn helping in further progress in computer science. In practice, this model is appearing quiet beneficial in network establishment and strongly capable of handling small number of blocks strictly avoiding loss of data resources. The empirical study reveals that this unique dynamic model keeps all the units strongly interconnected. Thus, the present paper is attempting to analyse the benefits of the network structure models and concludes with an assessment of observable consequences of optimising behaviour in networks for the purpose of estimation.

Design of LT Codes with Equal and Unequal Erasure Protection over Binary Erasure Channels

The erasure floor performance of Luby Transform (LT) codes is mainly determined by the minimum variable-node degree. Thus we propose a modified encoding scheme that maximizes the minimum variable-node degree for transmission over binary erasure channels. The proposed scheme leads to an almost-regular variable-node degree distribution. The encoding process is generalized to accommodate arbitrary variable-node degree distributions for additional improved performance. The asymptotic performance is investigated using density evolution and compared with a conventional LT code. The scheme is further extended to enable a higher level of unequal erasure protection.

The Effect of Exogenous Inputs and Defiant Agents on Opinion Dynamics With Local and Global Interactions

Most of the conventional models for opinion dynamics mainly account for a fully local influence, where myopic agents decide their actions after they interact with other agents that are adjacent to them. For example, in the case of social interactions, this includes family, friends, and other immediate strong social ties. The model proposed in this paper embodies a global influence as well where by global we mean that each node also observes a sample of the average behavior of the entire population; e.g., in the social example, people observe other people on the streets, subway, and other social venues. We consider the case where nodes have dichotomous states; examples of applications include elections with two major parties, whether or not to adopt a new technology or product, and any yes/no opinion such as in voting on a referendum. The dynamics of states on a network with arbitrary degree distribution are studied. For a given initial condition, we find the probability to reach consensus on each state and the expected time reach to consensus. To model mass media, the effect of an exogenous bias on the average orientation of the system is investigated. To do so, we add an external field to the model that favors one of the states over the other. This field interferes with the regular decision process of each node and creates a constant probability to lean towards one of the states. We solve for the average state of the system as a function of time for given initial conditions. Then anti-conformists (stubborn nodes who never revise their states) are added to the network, in an effort to circumvent the external bias. We find necessary conditions on the number of these defiant nodes required to cancel the effect of the external bias. Our analysis is based on a mean field approximation of the agent opinions.

Core Percolation on Complex Networks

We analytically solve the core percolation problem for complex networks with arbitrary degree distributions. We find that purely scale-free networks have no core for any degree exponents. We show that for undirected networks if core percolation occurs then it is continuous while for directed networks it is discontinuous (and hybrid) if the in- and out-degree distributions differ. We also find that core percolations on undirected and directed networks have completely different critical exponents associated with their critical singularities.

Replica theory for learning curves for Gaussian processes on random graphs

We use a statistical physics approach to derive accurate predictions for the challenging problem of predicting the performance of Gaussian process regression. Performance is quantified by the learning curve, defined as the average error versus number of training examples. We assume the Gaussian process prior is defined by a random walk kernel, inputs are vertices on a random graph and the outputs are noisy function values. We show that replica techniques can be used to obtain exact performance predictions in the limit of large graphs, after first rewriting the average error in terms of a graphical model. Conventionally, the Gaussian process kernel is only globally normalized, so that the prior variance can differ between vertices. As a more principled alternative we also consider local normalization, where the prior variance is uniform. The normalization constants for the prior then have to be defined as thermal averages in an unnormalized model and this requires the introduction of a second, auxiliary set of replicas. Our results for both types of kernel normalization apply generically to all random graph ensembles constrained by a fixed but arbitrary degree distribution. We compare with numerically simulated learning curves and find excellent agreement, a significant improvement over existing approximations.

Growth of graph states in quantum networks

We propose a scheme to distribute graph states over quantum networks in the presence of noise in the channels and in the operations. The protocol can be implemented efficiently for large graph sates of arbitrary (complex) topology. We benchmark our scheme with two protocols where each connected component is prepared in a node belonging to the component and subsequently distributed via quantum repeaters to the remaining connected nodes. We show that the fidelity of the generated graphs can be written as the partition function of a classical Ising-type Hamiltonian. We give exact expressions of the fidelity of the linear cluster and results for its decay rate in random graphs with arbitrary (uncorrelated) degree distributions.

Theoretical approach and impact of correlations on the critical packet generation rate in traffic dynamics on complex networks

Using the formalism of the biased random walk in random uncorrelated networks with arbitrary degree distributions, we develop theoretical approach to the critical packet generation rate in traffic based on routing strategy with local information. We explain microscopic origins of the transition from the flow to the jammed phase and discuss how the node neighbourhood topology affects the transport capacity in uncorrelated and correlated networks.