Hierarchical Scale-free Network Research Articles

AVERAGE FERMAT DISTANCE OF A PSEUDO-FRACTAL HIERARCHICAL SCALE-FREE NETWORK

Fermat point of a triangle is the point with the minimal total distance from the three vertices in a triangle. In this paper, we discuss the average Fermat distance for a class of hierarchical networks. First, the unweighted hierarchical scale-free network is established in an iterative way. Applying the recursive method, we deduce the analytical expression of average Fermat distance and average geodesic distance. Then we reveal the linear relation of the leading terms for average Fermat distance and average geodesic distance. Finally, we obtain the small-world property of the hierarchical scale-free network, which indicates that average Fermat distance can be a valuable index of small-word property.

Hierarchical deposition and scale-free networks: A visibility algorithm approach.

The growth of an interface formed by the hierarchical deposition of particles of unequal size is studied in the framework of a dynamical network generated by a horizontal visibility algorithm. For a deterministic model of the deposition process, the resulting network is scale free with dominant degree exponent γ_{e}=ln3/ln2 and transient exponent γ_{o}=1. An exact calculation of the network diameter and clustering coefficient reveals that the network is scale invariant and inherits the modular hierarchical nature of the deposition process. For the random process, the network remains scale free, where the degree exponent asymptotically converges to γ=3, independent of the system parameters. This result shows that the model is in the class of fractional Gaussian noise through the relation between the degree exponent and the series' Hurst exponent H. Finally, we show through the degree-dependent clustering coefficient C(k) that the modularity remains present in the system.

Conflict Dynamics in Scale-Free Networks with Degree Correlations and Hierarchical Structure.

We present a study of the dynamic interactions between actors located on complex networks with scale-free and hierarchical scale-free topologies with assortative mixing, that is, correlations between the degree distributions of the actors. The actor's state evolves according to a model that considers its previous state, the inertia to change, and the influence of its neighborhood. We show that the time evolution of the system depends on the percentage of cooperative or competitive interactions. For scale-free networks, we find that the dispersion between actors is higher when all interactions are either cooperative or competitive, while a balanced presence of interactions leads to a lower separation. Moreover, positive assortative mixing leads to greater divergence between the states, while negative assortative mixing reduces this dispersion. We also find that hierarchical scale-free networks have both similarities and differences when compared with scale-free networks. Hierarchical scale-free networks, like scale-free networks, show the least divergence for an equal mix of cooperative and competitive interactions between actors. On the other hand, hierarchical scale-free networks, unlike scale-free networks, show much greater divergence when dominated by cooperative rather than competitive actors, and while the formation of a rich club (adding links between hubs) with cooperative interactions leads to greater divergence, the divergence is much less when they are fully competitive. Our findings highlight the importance of the topology where the interaction dynamics take place, and the fact that a balanced presence of cooperators and competitors makes the system more cohesive, compared to the case where one strategy dominates.

ANALYSES OF SOME STRUCTURAL PROPERTIES ON A CLASS OF HIERARCHICAL SCALE-FREE NETWORKS

Hierarchical networks as fundamental models to describe the complex networks, have many applications in networks science, engineering technology and so on. In this paper, we first propose a new class of hierarchical networks with fractal structure, which are the networks with triangles compared to traditional hierarchical networks. Second, we study the precise results of some structural properties to derive small-world effect and scale-free feature. Third, it is found that the constructed network is sparse through the average degree and density. Fourth, it is also demonstrated that the degree distributions of hub nodes and the bottom nodes are the power law and exponential, respectively. Finally, we prove that clustering coefficient with a definite value [Formula: see text] tends to stabilize at a lower bound as [Formula: see text] iterates to a certain number, and the average distance of [Formula: see text] has an increasing relationship along with the value of [Formula: see text].

A general model of hierarchical fractal scale-free networks.

We propose a general model of unweighted and undirected networks having the scale-free property and fractal nature. Unlike the existing models of fractal scale-free networks (FSFNs), the present model can systematically and widely change the network structure. In this model, an FSFN is iteratively formed by replacing each edge in the previous generation network with a small graph called a generator. The choice of generators enables us to control the scale-free property, fractality, and other structural properties of hierarchical FSFNs. We calculate theoretically various characteristic quantities of networks, such as the exponent of the power-law degree distribution, fractal dimension, average clustering coefficient, global clustering coefficient, and joint probability describing the nearest-neighbor degree correlation. As an example of analyses of phenomena occurring on FSFNs, we also present the critical point and critical exponents of the bond-percolation transition on infinite FSFNs, which is related to the robustness of networks against edge removal. By comparing the percolation critical points of FSFNs whose structural properties are the same as each other except for the clustering nature, we clarify the effect of the clustering on the robustness of FSFNs. As demonstrated by this example, the present model makes it possible to elucidate how a specific structural property influences a phenomenon occurring on FSFNs by varying systematically the structures of FSFNs. Finally, we extend our model for deterministic FSFNs to a model of non-deterministic ones by introducing asymmetric generators and reexamine all characteristic quantities and the percolation problem for such non-deterministic FSFNs.

Mean first passage time on a family of weighted directed hierarchical scale-free networks

Random walks on binary scale-free networks have been widely studied. However, many networks in real life are weighted and directed, the dynamic processes of which are less understood. In this paper, we firstly present a family of directed weighted hierarchical scale-free networks, which is obtained by introducing a weight parameter [Formula: see text] into the binary (1, 3)-flowers. Besides, each pair of nodes is linked by two edges with opposite direction. Secondly, we deduce the mean first passage time (MFPT) with a given target as a measure of trapping efficiency. The exact expression of the MFPT shows that both its prefactor and its leading behavior are dependent on the weight parameter [Formula: see text]. In more detail, the MFPT can grow sublinearly, linearly and superlinearly with varied [Formula: see text]. Last but not least, we introduce a delay parameter p to modify the transition probability governing random walk. Under this new scenario, we also derive the exact solution of the MFPT with the given target, the result of which illustrates that the delay parameter p can only change the coefficient of the MFPT and leave the leading behavior of MFPT unchanged. Both the analytical solutions of MFPT in two distinct scenarios mentioned above agree well with the corresponding numerical solutions. The analytical results imply that we can get desired transport efficiency by tuning weight parameter [Formula: see text] and delay parameter p. This work may help to advance the understanding of random walks in general directed weighted scale-free networks.

Asymptotic formula on average path length in a hierarchical scale-free network with fractal structure

Abstract In this paper, we study a hierarchical scale-free network with special fractal structure. We study the average path length and associate the average path length with the symbol length. Then using a recursive method, we obtain the asymptotic formula for average path length of our networks.

Rewiring hierarchical scale-free networks: Influence on synchronizability and topology

Many real-world complex networks simultaneously exhibit topological features of scale-free behaviour and hierarchical organization. In this regard, deterministic scale-free (DSF) (Barabási A.-L. et al., Physica A, 299 (2001) 559) and pseudofractal scale-free (PSF) (Dorogovtsev S. N. et al., Phys. Rev. E, 65 (2002) 066122) networks constitute notable models which simultaneously incorporate the aforementioned properties. The rules governing the formation of such networks are completely deterministic. However, real-world networks are presumably neither completely deterministic nor perfectly hierarchical. Therefore, we suggest here initially perfectly hierarchical scale-free networks with subsequently randomly rewired edges as better representatives of practical networked systems. In particular, we preserve the scale-free degree distribution of the deterministic networks but successively relax the hierarchical structure while rewiring them. We utilize the framework of master stability function in investigating the synchronizability of dynamical systems coupled on such rewired networks. Interestingly, this reveals that the process of rewiring is capable of significantly enhancing, as well as, deteriorating the synchronizability of the resulting networks. We investigate the influence of rewiring edges on the topological properties of the rewired networks and, in turn, their relation to the synchronizability of the respective topologies. Finally, we compare the synchronizability of DSF and PSF networks with that of random scale-free networks (generated using the classical Barabási-Albert (BA) model). We find that the BA random scale-free networks promote synchronizability better than the rewired versions of their deterministic counterparts of DSF and PSF networks.

Deterministic hierarchical networks

It has been shown that many networks associated with complex systems are small-world (they have both a large local clustering coefficient and a small diameter) and also scale-free (the degrees are distributed according to a power law). Moreover, these networks are very often hierarchical, as they describe the modularity of the systems that are modeled. Most of the studies for complex networks are based on stochastic methods. However, a deterministic method, with an exact determination of the main relevant parameters of the networks, has proven useful. Indeed, this approach complements and enhances the probabilistic and simulation techniques and, therefore, it provides a better understanding of the modeled systems. In this paper we find the radius, diameter, clustering coefficient and degree distribution of a generic family of deterministic hierarchical small-world scale-free networks that has been considered for modeling real-life complex systems.

Apoptosis regulatory protein-protein interaction demonstrates hierarchical scale-free fractal network.

Dysregulation or inhibition of apoptosis favors cancer and many other diseases. Understanding of the network interaction of the genes involved in apoptotic pathway, therefore, is essential, to look for targets of therapeutic intervention. Here we used the network theory methods, using experimentally validated 25 apoptosis regulatory proteins and identified important genes for apoptosis regulation, which demonstrated a hierarchical scale-free fractal protein-protein interaction network. TP53, BRCA1, UBIQ and CASP3 were recognized as a four key regulators. BRCA1 and UBIQ were also individually found to control highly clustered modules and play an important role in the stability of the overall network. The connection among the BRCA1, UBIQ and TP53 proteins was found to be important for regulation, which controlled their own respective communities and the overall network topology. The feedback loop regulation motif was identified among NPM1, BRCA1 and TP53, and these crucial motif topologies were also reflected in high frequency. The propagation of the perturbed signal from hubs was found to be active upto some distance, after which propagation started decreasing and TP53 was the most efficient signal propagator. From the functional enrichment analysis, most of the apoptosis regulatory genes associated with cardiovascular diseases and highly expressed in brain tissues were identified. Apart from TP53, BRCA1 was observed to regulate apoptosis by influencing motif, propagation of signals and module regulation, reflecting their biological significance. In future, biochemical investigation of the observed hub-interacting partners could provide further understanding about their role in the pathophysiology of cancer.

Degree-ordered percolation on a hierarchical scale-free network.

We investigate the critical phenomena of the degree-ordered percolation (DOP) model on the hierarchical (u,v) flower network with u ≤ v. Highest degree nodes are linked directly without intermediate nodes for u=1, while this is not the case for u ≠ 1. Using the renormalization-group-like procedure, we derive the recursion relations for the percolating probability and the percolation order parameter, from which the percolation threshold and the critical exponents are obtained. When u ≠ 1, the DOP critical behavior turns out to be identical to that of the bond percolation with a shifted nonzero percolation threshold. When u=1, the DOP and the bond percolation have the same vanishing percolation threshold but the critical behaviors are different. Implication to an epidemic spreading phenomenon is discussed.

Hierarchical scale-free network is fragile against random failure

We investigate site percolation in a hierarchical scale-free network known as the Dorogovtsev-Goltsev-Mendes network. We use the generating function method to show that the percolation threshold is 1, i.e., the system is not in the percolating phase when the occupation probability is less than 1. The present result is contrasted to bond percolation in the same network of which the percolation threshold is zero. We also show that the percolation threshold of intentional attacks is 1. Our results suggest that this hierarchical scale-free network is very fragile against both random failure and intentional attacks. Such a structural defect is common in many hierarchical network models.

Origin of the hub spectral dimension in scale-free networks

The return-to-origin probability and the first-passage-time distribution are essential quantities for understanding transport phenomena in diverse systems. The behaviors of these quantities typically depend on the spectral dimension d(s). However, it was recently revealed that in scale-free networks these quantities show a crossover between two power-law regimes characterized by d(s) and the so-called hub spectral dimension d(s)((hub)) due to the heterogeneity of connectivities of each node. To understand the origin of d(s)((hub)) from a theoretical perspective, we study a random walk problem on hierarchical scale-free networks by using the renormalization group (RG) approach. Under the RG transformation, not only the system size but also the degree of each node changes due to the scale-free nature of the degree distribution. We show that the anomalous behavior of random walks involving the hub spectral dimension d(s)((hub)) is induced by the conservation of the power-law degree distribution under the RG transformation.

Optimal scale-free network with a minimum scaling of transport efficiency for random walks with a perfect trap

Average trapping time (ATT) is central in the trapping problem since it is a key indicator characterizing the efficiency of the problem. Previous research has provided the scaling of a lower bound of the ATT for random walks in general networks with a deep trap. However, it is still not well understood in which networks this minimal scaling can be reached. Particularly, explicit quantitative results for ATT in such networks, even in a specific network, are lacking, in spite that such networks shed light on the design for optimal networks with the highest trapping efficiency. In this paper, we study the trapping problem taking place on a hierarchical scale-free network with a perfect trap. We focus on four representative cases with the immobile trap located at the root, a peripheral node, a neighbor of the root with a single connectivity, and a farthest node from the root, respectively. For all the four cases, we obtain the closed-form formulas for the ATT, as well as its leading scalings. We show that for all the four cases of trapping problems, the dominating scalings of ATT can reach the predicted minimum scalings. This work deepens the understanding of behavior of trapping in scale-free networks, and is helpful for designing networks with the most efficient transport process.

Phase transition without global ordering in a hierarchical scale-free network

We study the site-bond percolation on a hierarchical scale-free network, namely, the decorated (2,2)-flower, by using the renormalization group technique. The phase diagram essentially depends on the fraction of occupied sites. Surprisingly, when each site is unoccupied even with a small probability, the system permits neither the percolating phase nor the nonpercolating phase, but rather only critical phases. Although the order parameter always remains zero, a transition still exists between the critical phases that is characterized by the value of the fractal exponent, which measures the degree of criticality; the system changes from one critical state to another with the jump of the fractal exponent at the transition point. The phase boundary depends on the fraction of occupied sites. When the fraction of unoccupied sites exceeds a certain value, the transition line between the critical phases disappears, and a unique critical phase remains.

Enrichment and aggregation of topological motifs are independent organizational principles of integrated interaction networks

Topological network motifs represent functional relationships within and between regulatory and protein-protein interaction networks. Enriched motifs often aggregate into self-contained units forming functional modules. Theoretical models for network evolution by duplication-divergence mechanisms and for network topology by hierarchical scale-free networks have suggested a one-to-one relation between network motif enrichment and aggregation, but this relation has never been tested quantitatively in real biological interaction networks. Here we introduce a novel method for assessing the statistical significance of network motif aggregation and for identifying clusters of overlapping network motifs. Using an integrated network of transcriptional, posttranslational and protein-protein interactions in yeast we show that network motif aggregation reflects a local modularity property which is independent of network motif enrichment. In particular our method identified novel functional network themes for a set of motifs which are not enriched yet aggregate significantly and challenges the conventional view that network motif enrichment is the most basic organizational principle of complex networks.

Spectral dimensions of hierarchical scale-free networks with weighted shortcuts

Spectral dimensions have been widely used to understand transport properties on regular and fractal lattices. However, they have received little attention with regard to complex networks such as scale-free and small-world networks. Here, we study the spectral dimension and the return-to-origin probability of random walks on hierarchical scale-free networks, which can be either fractal or nonfractal depending on the weight of the shortcuts. Applying the renormalization-group (RG) approach to a Gaussian model, we obtain the exact spectral dimension. While the spectral dimension varies between 1 and 2 for the fractal case, it remains at 2, independent of the variation in the network structure, for the nonfractal case. The crossover behavior between the two cases is studied by carrying out the RG flow analysis. The analytical results are confirmed by simulation results and their implications for the architecture of complex systems are discussed.

Generating-function approach for bond percolation in hierarchical networks

We study bond percolations on hierarchical scale-free networks with the open bond probability of the shortcuts p and that of the ordinary bonds p. The system has a critical phase in which the percolating probability P takes an intermediate value 0 < P < 1. Using generating function approach, we calculate the fractal exponent ψ of the root clusters to show that ψ varies continuously with p in the critical phase. We confirm numerically that the distribution n(s) of cluster size s in the critical phase obeys a power law n(s) ∝ s(-τ), where τ satisfies the scaling relation τ=1+ψ(-1). In addition the critical exponent β(p) of the order parameter varies as p, from β ≃ 0.164694 at p=0 to infinity at p=p(c)=5/32.

Statistical Analysis of Global Connectivity and Activity Distributions in Cellular Networks

Various molecular interaction networks have been claimed to follow power-law decay for their global connectivity distribution. It has been proposed that there may be underlying generative models that explain this heavy-tailed behavior by self-reinforcement processes such as classical or hierarchical scale-free network models. Here, we analyze a comprehensive data set of protein-protein and transcriptional regulatory interaction networks in yeast, an Escherichia coli metabolic network, and gene activity profiles for different metabolic states in both organisms. We show that in all cases the networks have a heavy-tailed distribution, but most of them present significant differences from a power-law model according to a stringent statistical test. Those few data sets that have a statistically significant fit with a power-law model follow other distributions equally well. Thus, while our analysis supports that both global connectivity interaction networks and activity distributions are heavy-tailed, they are not generally described by any specific distribution model, leaving space for further inferences on generative models. Supplementary Material is available online at www.liebertonline.com.

Average distance in a hierarchical scale-free network: an exact solution

Various real systems simultaneously exhibit scale-free and hierarchical structure. In thispaper, we study analytically average distance in a deterministic scale-free network withhierarchical organization. Using a recursive method based on the network construction, wedetermine explicitly the average distance, obtaining an exact expression for it, which isconfirmed by extensive numerical calculations. The rigorous solution obtainedshows that the average distance grows logarithmically with the network order(number of nodes in the network). We show the similarity and dissimilarity inaverage distance between the network under consideration and some previouslystudied networks, including random networks and other deterministic networks. Onthe basis of the comparison, we argue that the logarithmic scaling of averagedistance with network order could be a generic feature of deterministic scale-freenetworks.