Role of Clustering Coefficient on Cooperation Dynamics in Homogeneous Networks

Based on previous works, we give further investigations on the Prisoners＇ Dilemma Game （PDG） on two different types of homogeneous networks, i.e. the homogeneous small-world network （HSWN） and the regular ring graph. We find that the so-called resonance-like character can occur on both the networks. Different from the viewpoint in previous publications, we think the small-world effect may be unnecessary to produce this character. Therefore, over these two types of networks, we suggest a common understanding in the viewpoint of clustering coefficient. Detailed simulation results can sustain our viewpoint quite well. Furthermore, we investigate the Snowdrift Game （SG） on the same networks. The difference between the outputs of the PDG and the SG can also sustain our viewpoint.     

Robustness of Complex Networks under Attack and Repair

To study the robustness of complex networks under attack and repair, we introduce a repair model of complex networks. Based on the model, we introduce two new quantities, i.e. attack fraction fa and the maximum degree of the nodes that have never been attacked ~Ka, to study analytically the critical attack fraction and the relative size of the giant component of complex networks under attack and repair, using the method of generating function. We show analytically and numerically that the repair strategy significantly enhances the robustness of the scale-free network and the effect of robustness improvement is better for the scale-free networks with a smaller degree exponent. We discuss the application of our theory in relation to the
understanding of robustness of complex networks with reparability.     

A Robustness Model of Complex Networks with Tunable Attack Information Parameter

We introduce a novel model for robustness of complex with a tunable attack information parameter. The random failure and intentional attack known are the two extreme cases of our model. Based on the model, we study the robustness of complex networks under random information and preferential information, respectively. Using the generating function method, we derive the exact value of the critical removal fraction of nodes for the disintegration of networks and the size of the giant component. We show that hiding just a small fraction of nodes randomly can prevent a scale-free network from collapsing and detecting just a small fraction of nodes preferentially can destroy a scale-free network.     

Emergence of scale-free behavior in networks from limited-horizon linking and cost trade-offs

We study network growth from a fixed set of initially isolated nodes placed at random on the surface of a sphere. The growth mechanism we use adds edges to the network depending on strictly local gain and cost criteria. Only nodes that are not too far apart on the sphere may be considered for being joined by an edge. Given two such nodes, the joining occurs only if the gain of doing it surpasses the cost. Our model is based on a multiplicative parameter λ that regulates, in a function of node degrees, the maximum geodesic distance that is allowed between nodes for them to be considered for joining. For n nodes distributed uniformly on the sphere, and for within limits that depend on cost-related parameters, we have found that our growth mechanism gives rise to power-law distributions of node degree that are invariant for constant . We also study connectivity- and distance-related properties of the networks.     

Robustness of heterogeneous complex networks

In this paper we study the robustness of heterogeneous preferential attachment networks. The robustness of a network measures its structural tolerance to the random removal of nodes and links. We numerically analyze the influence of the affinity parameters on a set of ensemble-averaged robustness metrics. We show that the presence of heterogeneity does not fundamentally alter the smooth nature of the fragmentation process of the models. We also show that a moderate level of locality translates into slight improvements in the robustness metrics, which prompts us to conjecture an evolutionary argument for the existence of real networks with power-law scaling in their connectivity and clustering distributions.     

Local affinity in heterogeneous growing networks

In this paper we present a study of the influence of local affinity in heterogeneous preferential attachment (PA) networks. Heterogeneous PA models are a generalization of the Barabási-Albert model to heterogeneous networks, where the affinity between nodes biases the attachment probability of links. Threshold models are a class of heterogeneous PA models where the affinity between nodes is inversely related to the distance between their states. We propose a generalization of threshold models where network nodes have individual affinity functions, which are then combined to yield the affinity of each potential interaction. We analyze the influence of the affinity functions in the topological properties averaged over a network ensemble. The network topology is evaluated through the distributions of connectivity degrees, clustering coefficients and geodesic distances. We show that the relaxation of the criterion of a single global affinity still leads to a reasonable power-law scaling in the connectivity and clustering distributions under a wide spectrum of assumptions. We also show that the richer behavior of the model often exhibits a better agreement with the empirical observations on real networks.     

Randomness and a step-like distribution of pile heights in avalanche models

This paper considers a one-parameter family of sand-piles. The family exhibits the crossover between the models with deterministic and stochastic relaxation. The mean pile height is used to describe the crossover. The height densities corresponding to the models with relaxation of both types approach one another as the parameter increases. Relaxation is supposed to deal with the local losses of grains by a fixed amount. In that case the densities show a step-like behaviour in contrast to the peaked shape found in the models with the local loss of grains down to a fixed level [S. Lübeck, Phys. Rev. E 62, 6149 (2000)]. A spectral approach based on the long-run properties of the pile height considers the models with deterministic and random relaxation more accurately and distinguishes between the two cases for admissible parameter values.     

Sandpile on directed small-world networks

We numerically investigate the avalanche dynamics of the Bak-Tang-Wiesenfeld sandpile model on directed small-world networks. We find that the avalanche size and duration distribution follow a power law for all rewiring probabilities p. Specially, we find that, approaching the thermodynamic limit (L→∞), the values of critical exponents do not depend on p and are consistent with the mean-field solution in Euclidean space for any p>0. In addition, we measure the dynamic exponent in the relation between avalanche size and avalanche duration and find that the values of the dynamic exponents are also consistent with the mean-field values for any p>0.     

Preferential attachment renders an evolving network of populations robust against crashes

We study a model for the evolution of chemical species under a combination of population dynamics on a short time scale, and a selection mechanism on a longer time scale. Least fit nodes are replaced by new nodes whose links are attached to the nodes of the given network via preferential attachment. In contrast to a random attachment of newly incoming nodes that was used in previous work, this preferential attachment mechanism accelerates the generation of a so-called autocatalytic set after a start from a random geometry, and the growth of this structure, until it saturates in a stationary phase in which the whole system is an autocatalytic set. Moreover, the system in the stationary phase becomes much more stable against crashes in the population size as compared to random attachment. We explain in detail, in terms of graph theoretical notions, which structure of the resulting network is responsible for this stability. Essentially it is a very dense core with many loops and less nodes playing the role of a keystone that prevents the system from crashing, almost completely.     

Connectivity degrees in the threshold preferential attachment model

In this paper we present a study of the connectivity degrees of the threshold preferential attachment model, a generalization of the Barabási-Albert model to heterogeneous complex networks. The threshold model incorporates the states of the nodes in its preferential linking rule and assumes that the affinity between network nodes follows an inverse relationship with the distance between their states. We numerically analyze the connectivity degrees of the model, studying the influence of the main parameters on the distribution of connectivity degrees and its statistics, the average degree and highest degree of the network. We show that such statistics exhibit markedly different behaviors in the dependence on the model parameters, particularly as regards the interaction threshold. Nevertheless, we show that the two statistics converge in the limit of null threshold and often exhibit scaling that can be described by power laws of the model parameters.     

Agreement Dynamics of Memory-Based Naming Game with Forgetting Curve of Ebbinghaus

We propose a memory-based naming game （MBNG） model which is like some previous opinion propagation models. with two-state variables in full-connected networks, It is found that this model is deeply affected by the memory decision parameter, and then its dynamical behaviour can be partly analysed by using numerical simulation and analytical argument. We also report a modified MBNG model with the forgetting curve of Ebbinghaus in the memory. With deletion of one parameter in the MBNG model, it can converge to success rate S（t） = 1 and the average sum E（t） is decided by the size of network N.     

Dynamics-driven evolution to structural heterogeneity in complex networks

The mutual influence of dynamics and structure is a central issue in complex systems. In this paper we study by simulation slow evolution of network under the feedback of a local-majority-rule opinion process. If performance-enhancing local mutations have higher chances of getting integrated into its structure, the system can evolve into a highly heterogeneous small-world with a global hub (whose connectivity is proportional to the network size), strong local connection correlations and power-law-like degree distribution. Networks with better dynamical performance are achieved if structural evolution occurs much slower than the network dynamics. Structural heterogeneity of many biological and social dynamical systems may also be driven by various dynamics-structure coupling mechanisms.     

Networks that optimize a trade-off between efficiency and dynamical resilience

In this Letter we study networks that have been optimized to realize a trade-off between communication efficiency and dynamical resilience. While the first is related to the average shortest pathlength, we argue that the second can be measured by the largest eigenvalue of the adjacency matrix of the network. Best efficiency is realized in star-like configurations, while enhanced resilience is related to the avoidance of short loops and degree homogeneity. Thus crucially, very efficient networks are not resilient while very resilient networks lack in efficiency. Networks that realize a trade-off between both limiting cases exhibit core-periphery structures, where the average degree of core nodes decreases but core size increases as the weight is gradually shifted from a strong requirement for efficiency and limited resilience towards a smaller requirement for efficiency and a strong demand for resilience. We argue that both, efficiency and resilience are important requirements for network design and highlight how networks can be constructed that allow for both.     

Lamellar pattern formation in small-world media

Numerical and analytical techniques are used to investigate the effects of quenched disorder of small-world networks on the phase ordering dynamics of lamellar patterns as modeled by the Swift-Hohenberg equation. Morphologies for small and large values of the network randomness are quite different. It is found that addition of shortcuts to an underlying regular lattice makes the growth of domains evolving from random initial conditions much slower at late times. As the randomness increases, the evolution is eventually frozen.     

The emergence of scale-free networks with a seceding mechanism

In order to explore further the underlying mechanism of scale-free networks, we study stochastic secession as a mechanism for the creation of complex networks. In this evolution the network growth incorporates the addition of new nodes, the addition of new links between existing nodes, the deleting and rewiring of some existing links, and the stochastic secession of nodes. To random growing networks with preferential attachment, the model yields scale-free behavior for the degree distribution. Furthermore, we obtain an analytical expression of the power-law degree distribution with scaling exponent γ ranging from 1.1 to 9. The analytical expressions are in good agreement with the numerical simulation results.     

Analysis of cascading failure in electric grid based on power flow entropy

Large-scale blackouts are an intrinsic drawback of electric power transmission grids. Here we propose a concept of power flow entropy to quantify the overall heterogeneity of load distribution and then investigate the relationship between the power flow entropy and cascading failure. Simulation results, from the small-world 300-node test system and the IEEE 300-bus system, show that the power flow entropy has close relations with the cascading failure in terms of both the dynamic propagation course and the static blackout size. Particularly, at the early stage of failure spreading the potential large blackout can be identified according to the power flow entropy. The power flow entropy can serve as an index not only for long-term planning, but also for short-term operational defense to large-scale blackouts.     

Opinion Dynamics on Complex Networks with Communities

The Ising or Potts models of ferromagnetism have been widely used to describe locally interacting social or economic systems. We consider a related model, introduced by Sznajd to describe the evolution of consensus in the scale-free networks with the tunable strength （noted by Q） of community structure. In the Sznajd model, the opinion or state of any spins can only be changed by the influence of neighbouring pairs of similar connection spins. Such pairs can polarize their neighbours. Using asynchronous updating, it is found that the smaller the community strength Q, the larger the slope of the exponential relaxation time distribution. Then the effect of the initial upspin concentration p as a function of the final all up probability E is investigated by taking different initialization strategies, the random node-chosen initialization strategy has no difference under different community strengths, while the strategies of community node-chosen initialization and hub node-chosen initialization are different in fina/probability under different Q, and the latter one is more effective in reaching final state.     

Random walks and diameter of finite scale-free networks

Dynamical scalings for the end-to-end distance R_ee and the number of distinct visited nodes N_v of random walks (RWs) on finite scale-free networks (SFNs) are studied numerically. 〈R_ee〉 shows the dynamical scaling behavior , where is the average minimum distance between all possible pairs of nodes in the network, N is the number of nodes, γ is the degree exponent of the SFN and t is the step number of RWs. Especially, in the limit t→∞ satisfies the relation , where d is the diameter of network with for γ≥3 or for γ<3. Based on the scaling relation 〈R_ee〉, we also find that the scaling behavior of the diameter of networks can be measured very efficiently by using RWs.     

Effect of imitation in evolutionary minority game on small-world networks

The system performance in an evolutionary minority game with imitation on small-world networks is studied. Numerical results show that system performance positively correlates with the clustering coefficients. The domain structure of the agents’ strategies can be used to give a qualitative explanation for it. We also find that the time series of the reduced variance σ²/N could have a phasic evolution from a metastable state (two crowds are formed but the distribution of their probabilities does not peak at p≈0 and p≈1) to a steadystate (the two crowds evolve into a crowd and an anticrowd with the distribution of their probabilities peaking at p≈0 and p≈1).     

New tools for characterizing swarming systems: A comparison of minimal models

We compare three simple models that reproduce qualitatively the emergent swarming behavior of bird flocks, fish schools, and other groups of self-propelled agents by using a new set of diagnosis tools related to the agents’ spatial distribution. Two of these correspond in fact to different implementations of the same model, which had been previously confused in the literature. All models appear to undergo a very similar order-to-disorder phase transition as the noise level is increased if we only compare the standard order parameter, which measures the degree of agent alignment. When considering our novel quantities, however, their properties are clearly distinguished, unveiling previously unreported qualitative characteristics that help determine which model best captures the main features of realistic swarms. Additionally, we analyze the agent clustering in space, finding that the distribution of cluster sizes is typically exponential at high noise, and approaches a power-law as the noise level is reduced. This trend is sometimes reversed at noise levels close to the phase transition, suggesting a non-trivial critical behavior that could be verified experimentally. Finally, we study a bi-stable regime that develops under certain conditions in large systems. By computing the probability distributions of our new quantities, we distinguish the properties of each of the coexisting metastable states. Our study suggests new experimental analyses that could be carried out to characterize real biological swarms.