In-degree Distribution Research Articles

On the formation of structure in growing networks

Based on the formation of triad junctions, the mechanism proposed in this paper generates networks that exhibit extended rather than single power law behavior. Triad formation guarantees strong neighborhood clustering and community-level characteristics as the network size grows to infinity. The asymptotic behavior is of interest in the study of directed networks in which (i) the formation of links cannot be described according to the principle of preferential attachment, (ii) the in-degree distribution fits a power law for nodes with a high degree and an exponential form otherwise, (iii) clustering properties emerge at multiple scales and depend on both the number of links that newly added nodes establish and the probability of forming triads, and (iv) groups of nodes form modules that feature fewer links to the rest of the nodes.

Scaling laws in critical random Boolean networks with general in- and out-degree distributions

We evaluate analytically and numerically the size of the frozen core and various scaling laws for critical Boolean networks that have a power-law in- and/or out-degree distribution. To this purpose, we generalize an efficient method that has previously been used for conventional random Boolean networks and for networks with power-law in-degree distributions. With this generalization, we can also deal with power-law out-degree distributions. When the power-law exponent is between 2 and 3, the second moment of the distribution diverges with network size, and the scaling exponent of the nonfrozen nodes depends on the degree distribution exponent. Furthermore, the exponent depends also on the dependence of the cutoff of the degree distribution on the system size. Altogether, we obtain an impressive number of different scaling laws depending on the type of cutoff as well as on the exponents of the in- and out-degree distributions. We confirm our scaling arguments and analytical considerations by numerical investigations.

State concentration exponent as a measure of quickness in Kauffman-type networks

We study the dynamics of randomly connected networks composed of binary Boolean elements and those composed of binary majority vote elements. We elucidate their differences in both sparsely and densely connected cases. The quickness of large network dynamics is usually quantified by the length of transient paths, an analytically intractable measure. For discrete-time dynamics of networks of binary elements, we address this dilemma with an alternative unified framework by using a concept termed state concentration, defined as the exponent of the average number of t-step ancestors in state transition graphs. The state transition graph is defined by nodes corresponding to network states and directed links corresponding to transitions. Using this exponent, we interrogate the dynamics of random Boolean and majority vote networks. We find that extremely sparse Boolean networks and majority vote networks with arbitrary density achieve quickness, owing in part to long-tailed in-degree distributions. As a corollary, only relatively dense majority vote networks can achieve both quickness and robustness.

Emergence of Scale-Free Close-Knit Friendship Structure in Online Social Networks

Although the structural properties of online social networks have attracted much attention, the properties of the close-knit friendship structures remain an important question. Here, we mainly focus on how these mesoscale structures are affected by the local and global structural properties. Analyzing the data of four large-scale online social networks reveals several common structural properties. It is found that not only the local structures given by the indegree, outdegree, and reciprocal degree distributions follow a similar scaling behavior, the mesoscale structures represented by the distributions of close-knit friendship structures also exhibit a similar scaling law. The degree correlation is very weak over a wide range of the degrees. We propose a simple directed network model that captures the observed properties. The model incorporates two mechanisms: reciprocation and preferential attachment. Through rate equation analysis of our model, the local-scale and mesoscale structural properties are derived. In the local-scale, the same scaling behavior of indegree and outdegree distributions stems from indegree and outdegree of nodes both growing as the same function of the introduction time, and the reciprocal degree distribution also shows the same power-law due to the linear relationship between the reciprocal degree and in/outdegree of nodes. In the mesoscale, the distributions of four closed triples representing close-knit friendship structures are found to exhibit identical power-laws, a behavior attributed to the negligible degree correlations. Intriguingly, all the power-law exponents of the distributions in the local-scale and mesoscale depend only on one global parameter, the mean in/outdegree, while both the mean in/outdegree and the reciprocity together determine the ratio of the reciprocal degree of a node to its in/outdegree. Structural properties of numerical simulated networks are analyzed and compared with each of the four real networks. This work helps understand the interplay between structures on different scales in online social networks.

Quantifying and analyzing the network basis of genetic complexity.

Genotype-to-phenotype maps exhibit complexity. This genetic complexity is mentioned frequently in the literature, but a consistent and quantitative definition is lacking. Here, we derive such a definition and investigate its consequences for model genetic systems. The definition equates genetic complexity with a surplus of genotypic diversity over phenotypic diversity. Applying this definition to ensembles of Boolean network models, we found that the in-degree distribution and the number of periodic attractors produced determine the relative complexity of different topology classes. We found evidence that networks that are difficult to control, or that exhibit a hierarchical structure, are genetically complex. We analyzed the complexity of the cell cycle network of Sacchoromyces cerevisiae and pinpointed genes and interactions that are most important for its high genetic complexity. The rigorous definition of genetic complexity is a tool for unraveling the structure and properties of genotype-to-phenotype maps by enabling the quantitative comparison of the relative complexities of different genetic systems. The definition also allows the identification of specific network elements and subnetworks that have the greatest effects on genetic complexity. Moreover, it suggests ways to engineer biological systems with desired genetic properties.

COMPLEX STRUCTURES AND SEMANTICS IN FREE WORD ASSOCIATION

We investigate the directed and weighted complex network of free word associations in which players write a word in response to another word given as input. We analyze in details two large datasets resulting from two very different experiments: On the one hand the massive multiplayer web-based Word Association Game known as Human Brain Cloud, and on the other hand the South Florida Free Association Norms experiment. In both cases, the networks of associations exhibit quite robust properties like the small world property, a slight assortativity and a strong asymmetry between in-degree and out-degree distributions. A particularly interesting result concerns the existence of a characteristic scale for the word association process, arguably related to specific conceptual contexts for each word. After mapping, the Human Brain Cloud network onto the WordNet semantics network, we point out the basic cognitive mechanisms underlying word associations when they are represented as paths in an underlying semantic network. We derive in particular an expression describing the growth of the HBC graph and we highlight the existence of a characteristic scale for the word association process.

DIRECT COUNTING ANALYSIS ON NETWORK GENERATED BY DISCRETE DYNAMICS

A detail study on the in-degree distribution (ID) of cellular automata is obtained by exact enumeration. The results indicate large deviation from multi-scaling and classification according to various IDs are discussed. We further augment the transfer matrix such that the distributions for more complicated rules are obtained. Dependence of ID on the lattice size have also been found for some rules including R50 and R77.

Using modified Barabási and Albert model to study the complex logistic network in eco-industrial systems

Abstract In recent years, both researchers and practitioners have devoted attention to environmental sustainability issues in the green supply industry, but, how to design the green logistic network in the eco-industrial system is missing. This article reports on the study of designing green logistic network in an eco-industrial system by modifying the Barabasi and Albert (BA) model based on the intrinsic properties of eco-industrial systems. After verifying that the modified BA model is more accurately to interpret the formation mechanism of an eco-industrial network by the mean-field approach, paper extended and generalized some basic rules of the In-degree and Out-degree distributions of an EIN under the influences of different factors to assist eco-manufactures choose their supplier or customer, even design their own supplying networks by application of complex networks theory with the aid of the Jinjie EIS case example.

Earthquake correlations and networks: A comparative study

We quantify the correlation between earthquakes and use the same to extract causally connected earthquake pairs. Our correlation metric is a variation on the one introduced by Baiesi and Paczuski [M. Baiesi and M. Paczuski, Phys. Rev. E 69, 066106 (2004)]. A network of earthquakes is then constructed from the time-ordered catalog and with links between the more correlated ones. A list of recurrences to each of the earthquakes is identified employing correlation thresholds to demarcate the most meaningful ones in each cluster. Data pertaining to three different seismic regions (viz., California, Japan, and the Himalayas) are comparatively analyzed using such a network model. The distribution of recurrence lengths and recurrence times are two of the key features analyzed to draw conclusions about the universal aspects of such a network model. We find that the unimodal feature of recurrence length distribution, which helps to associate typical rupture lengths with different magnitude earthquakes, is robust across the different seismic regions. The out-degree of the networks shows a hub structure rooted on the large magnitude earthquakes. In-degree distribution is seen to be dependent on the density of events in the neighborhood. Power laws, with two regimes having different exponents, are obtained with recurrence time distribution. The first regime confirms the Omori law for aftershocks while the second regime, with a faster falloff for the larger recurrence times, establishes that pure spatial recurrences also follow a power-law distribution. The crossover to the second power-law regime can be taken to be signaling the end of the aftershock regime in an objective fashion.

A Coupled Model for the Indegree and Outdegree Analysis of the Web

Abstract We introduce a mixed model for the Web graph that simultaneously describes the inlink and outlink distributions by taking into account the interconnection of the two processes. We derive an expression for the steady-state distribution of indegrees (outdegrees) among vertices with fixed outdegree (indegree) in terms of sums of beta functions. Experimentation on subsets of the real Web shows that the proposed distributions well reproduce the behavior of the observed data.

A Study of User Behaviors Based on MMS

Multimedia Messaging Service (MMS) is an interactive communication process between people. So,we treat this network as a social network. In this paper we analyze user behaviors of MMS system. Using a province's dataset of China Mobile, we obtain almost entirely of the messages has been sent via GSM；VAS applications were very popular and profitable；The size and the storage time of message is very important for a MMS system；The in-degree and out-degree distribution both obey power law form. A big found is that users have acknowledgement effects between each other.

Random digraphs with given expected degree sequences: A model for economic networks

Abstract Building upon the growing interest for complex network theory, the main ambition of this paper is to contribute to a more effective application of network theory to economic phenomena, and particularly to financial networks. We depart from recent contributions on credit networks in two respects. In the first place, we constrain diversification with the out- and in-degree distribution of nodes, by adopting a suitable extension of the expected degree model for random weighted digraphs. In the second place, we focus on real networks by using this extension as null model for statistical analysis. With the help of statistical inference, we provide a rigorous answer to the following question: do real networks obey our null model? Further, we show that this answer is tightly connected to the existence of clusters or modules, as defined by Newman and Girvan (2004), within networks.

The community Analysis of User Behaviors Network for Web Traffic

Understanding the structure and dynamics of the user behavior networks for web traffic (To be convenient in next sections, we refer to replace it as UBNWT) that connect users with servers across the Internet is a key to modeling the network and designing future application. The Web-visited bipartite networks, called the user behavioral networks, display a natural bipartite structure: two kinds of nodes coexist with links only between nodes of different types. We obtained the result that the out-degree distribution of clients (the host initiating the connection), the in-degree distribution of servers (the host receiving the connection) and the strength distribution (the exchange bytes between clients and servers) are approximately power-law, whose exponential is between 1.7 and 3.4. The clustering coefficient of clients and servers is larger than that in randomized, degree preserving versions of the same graph, which indicate a modular structure of UBNWT. Finally, based on the algorithm of finding the community structure in bipartite network, we divided the clients into different communities, through manual examination of hosts in these communities, the typical normal (interest) and abnormal (DOS) communities were found. Interestingly, the loyalty of clients belonging to the same community in different time is higher than 80%. The structure analysis of UBNWT is very helpful for the network management, resource allocation, traffic engineering and security.

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.

Information Diversity in Structure and Dynamics of Simulated Neuronal Networks

Neuronal networks exhibit a wide diversity of structures, which contributes to the diversity of the dynamics therein. The presented work applies an information theoretic framework to simultaneously analyze structure and dynamics in neuronal networks. Information diversity within the structure and dynamics of a neuronal network is studied using the normalized compression distance. To describe the structure, a scheme for generating distance-dependent networks with identical in-degree distribution but variable strength of dependence on distance is presented. The resulting network structure classes possess differing path length and clustering coefficient distributions. In parallel, comparable realistic neuronal networks are generated with NETMORPH simulator and similar analysis is done on them. To describe the dynamics, network spike trains are simulated using different network structures and their bursting behaviors are analyzed. For the simulation of the network activity the Izhikevich model of spiking neurons is used together with the Tsodyks model of dynamical synapses. We show that the structure of the simulated neuronal networks affects the spontaneous bursting activity when measured with bursting frequency and a set of intraburst measures: the more locally connected networks produce more and longer bursts than the more random networks. The information diversity of the structure of a network is greatest in the most locally connected networks, smallest in random networks, and somewhere in between in the networks between order and disorder. As for the dynamics, the most locally connected networks and some of the in-between networks produce the most complex intraburst spike trains. The same result also holds for sparser of the two considered network densities in the case of full spike trains.

The Emergence of Cooperative Leadership from Homogenous Random Networks

AbstractWe investigate the game learning skeleton on homogeneous random networks, where an individual plays the Prisoner's Dilemma game with neighbors and the learning frequency during the steady state is counted as edge weight. The learning skeleton is defined as a communication kernel where each individual directs to the most frequently followed neighbor. We show that with the increase of the temptation to defect, the in-degree distribution of the learning skeleton varies from an exponential distribution to a power-law distribution, i.e., the scale-free property can emerge from underlying homogeneous random interaction networks. Moreover, we find that the longer the individual holds on the cooperative strategy, the more followers the individual has, and the easier the leader diffuses its behavior. Therefore, our work will shed some insight into designing reinforcement learning mechanisms in the social system to promote the emergence and persistence of cooperation.

Network of earthquakes and recurrences therein

We quantify the correlation between earthquakes and use the same to distinguish between relevant causally connected earthquakes. Our correlation metric is a variation on the one introduced by Baiesi and Paczuski (Phys Rev E 69:066,106, 2004). A network of earthquakes is constructed, which is time-ordered and with links between the more correlated ones. Data pertaining to the California region has been used in the study. Recurrences to earthquakes are identified employing correlation thresholds to demarcate the most meaningful ones in each cluster. The distribution of recurrence lengths and recurrence times are analyzed subsequently to extract information about the complex dynamics. We find that the unimodal feature of recurrence lengths helps to associate typical rupture lengths with different magnitude earthquakes. The out-degree of the network shows a hub structure rooted on the large magnitude earthquakes. In-degree distribution is seen to be dependent on the density of events in the neighborhood. Power laws are also obtained with recurrence time distribution agreeing with the Omori law.

Evolution of Gene Regulatory Networks by Fluctuating Selection and Intrinsic Constraints

Various characteristics of complex gene regulatory networks (GRNs) have been discovered during the last decade, e.g., redundancy, exponential indegree distributions, scale-free outdegree distributions, mutational robustness, and evolvability. Although progress has been made in this field, it is not well understood whether these characteristics are the direct products of selection or those of other evolutionary forces such as mutational biases and biophysical constraints. To elucidate the causal factors that promoted the evolution of complex GRNs, we examined the effect of fluctuating environmental selection and some intrinsic constraining factors on GRN evolution by using an individual-based model. We found that the evolution of complex GRNs is remarkably promoted by fixation of beneficial gene duplications under unpredictably fluctuating environmental conditions and that some internal factors inherent in organisms, such as mutational bias, gene expression costs, and constraints on expression dynamics, are also important for the evolution of GRNs. The results indicate that various biological properties observed in GRNs could evolve as a result of not only adaptation to unpredictable environmental changes but also non-adaptive processes owing to the properties of the organisms themselves. Our study emphasizes that evolutionary models considering such intrinsic constraining factors should be used as null models to analyze the effect of selection on GRN evolution.

A random intersection digraph: Indegree and outdegree distributions

Let S ( 1 ) , … , S ( n ) , T ( 1 ) , … , T ( n ) be random subsets of the set [ m ] = { 1 , … , m } . We consider the random digraph D on the vertex set [ n ] defined as follows: the arc i → j is present in D whenever S ( i ) ∩ T ( j ) ≠ 0̸ . Assuming that the pairs of sets ( S ( i ) , T ( i ) ) , 1 ≤ i ≤ n , are independent and identically distributed, we study the in- and outdegree distributions of a typical vertex of D as n , m → ∞ .

DChord: An Efficient and Robust Peer to Peer Lookup System

Dynamic Hash Tables (DHTs) are distributed systems that maintain key-value pairs and provide efficient lookup services. Traditional DHTs usually rely on a random ID distribution of the keys to achieve such efficiency. Uniformly random hash functions are typically used to create uniformly random ID distributions from non-random key distributions. However, there are many cases where such random hash functions cannot be applied. For example, those systems that provide range queries over the keys cannot apply random hash functions on the keys, otherwise, the range query is very difficult to support. In this paper, we present a new lookup system called DChord, which does not depend on the randomness assumption to achieve its performance. To show the performance of the proposed system, we provide mathematical analysis and extensive simulation results in a highly non-random USN (Ubiquitous Sensor Network) metadata identifier space. To be specific, we show that DChord has high regularity in terms of in-degree and out-degree distributions. Thus, the system is robust against random node failures. We also show that query processing load is well balanced among nodes and the lookup speed is deterministic in such a way that the number of nodes to visit for a query is at most log2(N).