Asymmetric evolving random networks

We present a detailed study of the phase diagram of the Ising model in random graphs with arbitrary degree distribution. By using the replica method we compute exactly the value of the critical temperature and the associated critical exponents as a function of the minimum and maximum degree, and the degree distribution characterizing the graph. As expected, there is a ferromagnetic transition provided <k> <= <k^2> < \infty. However, if the fourth moment of the degree distribution is not finite then non-trivial scaling exponents are obtained. These results are analyzed for the particular case of power-law distributed random graphs.

Understanding the relationship between the heterogeneous structure of complex networks and cooperative phenomena occurring on them remains a key problem in network science. Mean-field theories of spin models on networks constitute a fundamental tool to tackle this problem and a cornerstone of statistical physics, with an impressive number of applications in condensed matter, biology, and computer science. In this work we derive the mean-field equations for the equilibrium behavior of vector spin models on high-connectivity random networks with an arbitrary degree distribution and with randomly weighted links. We demonstrate that the high-connectivity limit of spin models on networks is not universal in that it depends on the full degree distribution. Such nonuniversal behavior is akin to a remarkable mechanism that leads to the breakdown of the central limit theorem when applied to the distribution of effective local fields. Traditional mean-field theories on fully-connected models, such as the Curie–Weiss, the Kuramoto, and the Sherrington–Kirkpatrick model, are only valid if the network degree distribution is highly concentrated around its mean degree. We obtain a series of results that highlight the importance of degree fluctuations to the phase diagram of mean-field spin models by focusing on the Kuramoto model of synchronization and on the Sherrington–Kirkpatrick model of spin-glasses. Numerical simulations corroborate our theoretical findings and provide compelling evidence that the present mean-field theory describes an intermediate regime of connectivity, in which the average degree c scales as a power c ∝ N b (b < 1) of the total number N ≫ 1 of spins. Our findings put forward a novel class of spin models that incorporate the effects of degree fluctuations and, at the same time, are amenable to exact analytic solutions.

We analyze maximum entropy random graph ensembles with constrained degrees, drawn from arbitrary degree distributions, and a tuneable number of three-cycles (triangles). We find that such ensembles generally exhibit two transitions, a clustering and a shattering transition, separating three distinct regimes. At the clustering transition, the graphs change from typically having only isolated cycles to forming cycle clusters. At the shattering transition the graphs break up into many small cliques to achieve the desired three-cycle density. The locations of both transitions depend nontrivially on the system size. We derive a general formula for the three-cycle density in the regime of isolated cycles, for graphs with degree distributions that have finite first and second moments. For bounded degree distributions we present further analytical results on cycle densities and phase transition locations, which, while non-rigorous, are all validated via MCMC sampling simulations. We show that the shattering transition is of an entropic nature, occurring for all three-cycle density values, provided the system is large enough.

Social networks generally display a positively skewed degree distribution and higher values for clustering coefficient and degree assortativity than would be expected from the degree sequence. For some types of simulation studies, these properties need to be varied in the artificial networks over which simulations are to be conducted. Various algorithms to generate networks have been described in the literature but their ability to control all three of these network properties is limited. We introduce a spatially constructed algorithm that generates networks with constrained but arbitrary degree distribution, clustering coefficient and assortativity. Both a general approach and specific implementation are presented. The specific implementation is validated and used to generate networks with a constrained but broad range of property values.

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.

Many directed real world networks, such as the WWW, genetic regulation networks and economic networks exhibit significant differences between the properties of the incoming and outgoing edges, while the differences exhibited by other networks, such as Social Netw. are far more limited. This phenomenon is most evident in the differences between the distributions of incoming and outgoing degrees and direct clustering coefficients. There is currently no generic network generation model that would reproduce and tune these observed dissimilarities. We propose and empirically validate a simple and realistic model that can explain the emergence of the dissimilarities between the incoming and outgoing network degrees and clustering coefficients by combining directed triadic closure, random edge addition and directed edge removal. Surprisingly, we find that the difference between in and out degree distributions is attributed to asymmetries in the edge removal, highlighting the neglected yet crucial importance of edge removal mechanisms to the static and dynamic properties of real world networks. The model is governed by only two parameters: the first tunes the randomness of the edge addition mechanism, while the second controls the difference between the in and out degrees. The combination of these parameters reproduces the observed variety of directed degree distributions and clustering coefficients. Further comparisons of the model’s microscopic dynamics against the empirically observed evolution of real world social network confirms that while quite simple, the model properly describes both the edge addition and deletion processes in directed networks.

Service location based on greedy search is studied. A model is built to analyze the influence of the network topologies on this location method, and different network topologies are constructed to validate the model by way of simulation. The results show that, though the model assumes a uniform degree distribution, it agrees with the simulation result well if the average degree of the network is reasonably big. Hops and relative QoS index of the node found in a service location process are used to evaluate the effectiveness of the location method as well as the probability of locating the first 5% nodes with higher QoS level. Both model and simulation result show that the performance of greedy search based service location improves significantly with the increase of the average degree of the network. If changes of both topology and the QoS level of nodes can be ignored during a location process, it can be guaranteed that the location has high probability of finding the nodes with relatively high QoS level in small number of hops in a big service community. Model extension under arbitrary network degree distribution is further studied. A "two-phase" topology construction and service location based on greedy-search is also put forward.

The nucleation and growth of surface islands in the pre-coalescence stage has previously been studied by different methods, including the rate equation approach and kinetic Monte Carlo simulations. However, full understanding of island growth kinetics and the scaling properties of their size distributions is still lacking. Here, we investigate rate equations for the irreversible homogeneous growth of islands in the continuum limit, and derive a general island-size distribution whose shape is fully determined by the dynamics of the monomer concentration at a given size dependence of the capture coefficients. We show that the island-size distribution acquires the Family-Viscek scaling shape in the large time limit if the capture coefficients are linear in size for large enough islands. We obtain analytic solutions for the time-dependent monomer concentration, island density, average size and island-size distribution, which are valid for all times, and the analytic scaling function in the large time limit. These results can be used for modeling growth kinetics in a wide range of systems and shed more light on the general properties of the size distributions of different nano-objects.

We consider the phenomenon of Bose-Einstein condensation in a random growing directed network. The network grows by the addition of vertices and edges. At each time step the network gains a vertex with probability p and an edge with probability 1-p. The new vertex has a fitness (a,b) a,b>0, with probability f(a,b). A vertex with fitness (a,b), with in-degree i and out-degree j, gains a new incoming edge with rate a(i+1) and an outgoing edge with rate b(j+1). The Bose-Einstein condensation occurs as a function of fitness distribution f(a,b).

We study the influence of network topology on retrieval properties of recurrent neural networks, using replica techniques for diluted systems. The theory is presented for a network with an arbitrary degree distribution $p(k)$ and applied to power law distributions $p(k) \sim k^{-\gamma}$, i.e. to neural networks on scale-free graphs. A bifurcation analysis identifies phase boundaries between the paramagnetic phase and either a retrieval phase or a spin glass phase. Using a population dynamics algorithm, the retrieval overlap and spin glass order parameters may be calculated throughout the phase diagram. It is shown that there is an enhancement of the retrieval properties compared with a Poissonian random graph. We compare our findings with simulations.

In this paper the linear theory of nonequilibrium thermodynamics, developed by Onsager and others, is applied to random networks with arbitrary degree distribution. Using the well-known methods of nonequilibrium thermodynamics we identify thermodynamic forces and their conjugated flows induced in networks as a result of single node degree perturbation. The forces and the flows can be understood as a response of the system to events, such as random removal of nodes or intentional attacks on them. Finally, we show that cross effects (such as thermodiffusion, or thermoelectric phenomena), in which one force may not only give rise to its own conjugated flow, but to many other flows, can be observed also in complex 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.

The challenges presented by the COVID-19 epidemic have created a renewed interest in the development of new methods to combat infectious diseases, and it has shown the importance of preparedness for possible future diseases. A prominent property of the SARS-CoV-2 transmission is the significant fraction of asymptomatic transmission. This may influence the effectiveness of the standard contact tracing procedure for quarantining potentially infected individuals. However, the effects of asymptomatic transmission on the epidemic threshold of epidemic spreading on networks have rarely been studied explicitly. Here we study the critical percolation transition for an arbitrary disease with a nonzero asymptomatic rate in a simple epidemic network model in the presence of a recursive contact tracing algorithm for instant quarantining. We find that, above a certain fraction of asymptomatic transmission, standard contact tracing loses its ability to suppress spreading below the epidemic threshold. However, we also find that recursive contact tracing opens a possibility to contain epidemics with a large fraction of asymptomatic or presymptomatic transmission. In particular, we calculate the required fraction of network nodes participating in the contact tracing for networks with arbitrary degree distributions and for varying recursion depths and discuss the influence of recursion depth and asymptomatic rate on the epidemic percolation phase transition. We anticipate recursive contact tracing to provide a basis for digital, app-based contact tracing tools that extend the efficiency of contact tracing to diseases with a large fraction of asymptomatic transmission.

In this paper, a Poisson-Moved-Robust Solition Distribution (PMRSD) degree distribution is proposed based on Poisson distribution and the Moved Robust Solition Distribution (Moved-RSD). In the encoder, Poisson distribution is utilized to construct enough encoded data packets of small-degree, and Moved Robust Solition Distribution is used to generate encoded data packets of big-degree. The decoder can achieve completely decoding without interrupted because it is assured that the encoded data packets can contain all information of source data packets. The simulation results show that, comparing to New-Switch degree distribution-based LT code, the new PMRSD degree distribution-based LT code has advantages in improved decoding efficiency, reduced decoding overhead and bit error rate (BER).

We study a simple model of dynamic networks, characterized by a set preferred degree, $κ$. Each node with degree $k$ attempts to maintain its $κ$ and will add (cut) a link with probability $w(k;κ)$ ($1-w(k;κ)$). As a starting point, we consider a homogeneous population, where each node has the same $κ$, and examine several forms of $w(k;κ)$, inspired by Fermi-Dirac functions. Using Monte Carlo simulations, we find the degree distribution in steady state. In contrast to the well-known Erdős-Rényi network, our degree distribution is not a Poisson distribution; yet its behavior can be understood by an approximate theory. Next, we introduce a second preferred degree network and couple it to the first by establishing a controllable fraction of inter-group links. For this model, we find both understandable and puzzling features. Generalizing the prediction for the homogeneous population, we are able to explain the total degree distributions well, but not the intra- or inter-group degree distributions. When monitoring the total number of inter-group links, $X$, we find very surprising behavior. $X$ explores almost the full range between its maximum and minimum allowed values, resulting in a flat steady-state distribution, reminiscent of a simple random walk confined between two walls. Both simulation results and analytic approaches will be discussed.