Exploring complex networks by walking on them

An integrative dynamical perspective for graph theory and the analysis of complex networks

Built upon the shoulders of graph theory, the field of complex networks has become a central tool for studying real systems across various fields of research. Represented as graphs, different systems can be studied using the same analysis methods, which allows for their comparison. Here, we challenge the widespread idea that graph theory is a universal analysis tool, uniformly applicable to any kind of network data. Instead, we show that many classical graph metrics-including degree, clustering coefficient, and geodesic distance-arise from a common hidden propagation model: the discrete cascade. From this perspective, graph metrics are no longer regarded as combinatorial measures of the graph but as spatiotemporal properties of the network dynamics unfolded at different temporal scales. Once graph theory is seen as a model-based (and not a purely data-driven) analysis tool, we can freely or intentionally replace the discrete cascade by other canonical propagation models and define new network metrics. This opens the opportunity to design-explicitly and transparently-dedicated analyses for different types of real networks by choosing a propagation model that matches their individual constraints. In this way, we take stand that network topology cannot always be abstracted independently from network dynamics but shall be jointly studied, which is key for the interpretability of the analyses. The model-based perspective here proposed serves to integrate into a common context both the classical graph analysis and the more recent network metrics defined in the literature which were, directly or indirectly, inspired by propagation phenomena on networks.

Unexpected advantages of exploitation for target searches in complex networks

Exploitation universally emerges in various decision-making contexts, e.g., animals foraging, web surfing, the evolution of scientists' research topics, and our daily lives. Despite its ubiquity, exploitation, which refers to the behavior of revisiting previous experiences, has often been considered to delay the search process of finding a target. In this paper, we investigate how exploitation affects search performance by applying a non-Markovian random walk model, where a walker randomly revisits a previously visited node using long-term memory. We analytically study two broad forms of network structures, namely, (i) clique-like networks and (ii) lollipop-like networks and find that exploitation can significantly improve search performance in lollipop-like networks, whereas it hinders target search in clique-like networks. Moreover, we numerically verify that exploitation can reduce the time needed to fully explore the underlying networks using 550 diverse real-world networks. Based on the analytic result, we define the lollipop-likeness of a network and observe a positive relationship between the advantage of exploitation and lollipop-likeness.

Optimal exploration of random walks with local bias on networks

We propose local-biased random walks on general networks where a Markovian walker is defined by different types of biases in each node to establish transitions to its neighbors depending on their degrees. For this ergodic dynamics, we explore the capacity of the random walker to visit all the nodes characterized by a global mean first passage time. This quantity is calculated using eigenvalues and eigenvectors of the transition matrix that defines the dynamics. In the first part, we illustrate how our framework leads to optimal exploration for small-size graphs through the analysis of all the possible bias configurations. In the second part, we study the most favorable configurations in each node by using simulated annealing. This heuristic algorithm allows obtaining approximate solutions of the optimal bias in different types of networks. The results show how the local bias can optimize the exploration of the network in comparison with the unbiased random walk. The methods implemented in this research are general and open the doors to a broad spectrum of tools applicable to different random walk strategies and dynamical processes on networks.

Identifying the essential nodes in network pharmacology based on multilayer network combined with random walk algorithm

Compared with the general complex network, the multilayer network is more suitable for the description of reality. It can be used as a tool of network pharmacology to analyze the mechanism of drug action from an overall perspective. Combined with random walk algorithm, it measures the importance of nodes from the entire network rather than a single layer. Here a four-layer network was constructed based on the data about the action process of prescriptions, consisting of ingredients, target proteins, metabolic pathways and diseases. The random walk algorithm was used to calculate the betweenness centrality of the protein layer nodes to get the rank of their importance. According to above method, we screened out the top 10% proteins that play a key role in treatment. Prescriptions Xiaochaihu Decoction was taken as example to prove our method. The selected proteins were measured with the ones that have been validated to be associated with the treated diseases. The results showed that its accuracy was no less than the topology-based method of single-layer network. The applicability of our method was proved by another prescription Yupingfeng Decoction. Our study demonstrated that multilayer network combined with random walk algorithm was an effective method for pre-screening vital target proteins related to prescriptions.

A spectrum of routing strategies for brain networks

Communication of signals among nodes in a complex network poses fundamental problems of efficiency and cost. Routing of messages along shortest paths requires global information about the topology, while spreading by diffusion, which operates according to local topological features, is informationally “cheap” but inefficient. We introduce a stochastic model for network communication that combines local and global information about the network topology to generate biased random walks on the network. The model generates a continuous spectrum of dynamics that converge onto shortest-path and random-walk (diffusion) communication processes at the limiting extremes. We implement the model on two cohorts of human connectome networks and investigate the effects of varying the global information bias on the network’s communication cost. We identify routing strategies that approach a (highly efficient) shortest-path communication process with a relatively small global information bias on the system’s dynamics. Moreover, we show that the cost of routing messages from and to hub nodes varies as a function of the global information bias driving the system’s dynamics. Finally, we implement the model to identify individual subject differences from a communication dynamics point of view. The present framework departs from the classical shortest paths vs. diffusion dichotomy, unifying both models under a single family of dynamical processes that differ by the extent to which global information about the network topology influences the routing patterns of neural signals traversing the network.

Brain network communication is typically approached from the perspective of the length of inferred paths and the cost of building and maintaining network connections. However, these analyses often disregard the dynamical processes taking place on the network and the additional costs that these processes incur. Here, we introduce a framework to study communication-cost trade-offs on a broad range of communication processes modeled as biased random walks. We control the system’s dynamics that dictates the flow of messages traversing a network by biasing node’s routing strategies with different degrees of “knowledge” about the topology of the network. On the human connectome, this framework uncovers a spectrum of dynamic communication processes, some of which can achieve efficient routing strategies at low informational cost.

MADM-based smart parking guidance algorithm

In smart parking environments, how to choose suitable parking facilities with various attributes to satisfy certain criteria is an important decision issue. Based on the multiple attributes decision making (MADM) theory, this study proposed a smart parking guidance algorithm by considering three representative decision factors (i.e., walk duration, parking fee, and the number of vacant parking spaces) and various preferences of drivers. In this paper, the expected number of vacant parking spaces is regarded as an important attribute to reflect the difficulty degree of finding available parking spaces, and a queueing theory-based theoretical method was proposed to estimate this expected number for candidate parking facilities with different capacities, arrival rates, and service rates. The effectiveness of the MADM-based parking guidance algorithm was investigated and compared with a blind search-based approach in comprehensive scenarios with various distributions of parking facilities, traffic intensities, and user preferences. Experimental results show that the proposed MADM-based algorithm is effective to choose suitable parking resources to satisfy users’ preferences. Furthermore, it has also been observed that this newly proposed Markov Chain-based availability attribute is more effective to represent the availability of parking spaces than the arrival rate-based availability attribute proposed in existing research.

Traffic congestion and the lifetime of networks with moving nodes

For many power-limited networks, such as wireless sensor networks and mobile ad hoc networks, maximizing the network lifetime is the first concern in the related designing and maintaining activities. We study the network lifetime from the perspective of network science. In our model, nodes are initially assigned a fixed amount of energy moving in a square area and consume the energy when delivering packets. We obtain four different traffic regimes: no, slow, fast, and absolute congestion regimes, which are basically dependent on the packet generation rate. We derive the network lifetime by considering the specific regime of the traffic flow. We find that traffic congestion inversely affects network lifetime in the sense that high traffic congestion results in short network lifetime. We also discuss the impacts of factors such as communication radius, node moving speed, routing strategy, etc., on network lifetime and traffic congestion.

Network exploration using true self-avoiding walks

We study the mean first passage time (MFPT) of true self-avoiding walks (TSAWs) on various networks as a measure of searching efficiency. From the numerical analysis, we find that the MFPT of TSAWs, τ^{TSAW}, approaches the theoretical minimum τ^{th}/N=1/2 on synthetic networks whose degree-degree correlations are positive. On the other hand, for biased random walks (BRWs) we find that the MFPT, τ^{BRW}, depends on the parameter α, which controls the degree-dependent bias. More importantly, we find that the minimum MFPT of BRWs, τ_{min}^{BRW}, always satisfies the inequality τ_{min}^{BRW}>τ^{TSAW} on any synthetic network. The inequality is also satisfied on various real networks. From these results, we show that the TSAW is one of the most efficient models, whose efficiency approaches the theoretical limit in network explorations.

Blind and myopic ants in heterogeneous networks

The diffusion processes on complex networks may be described by different Laplacian matrices due to heterogeneous connectivity. Here we investigate the random walks of blind ants and myopic ants on heterogeneous networks: While a myopic ant hops to a neighbor node every step, a blind ant may stay or hop with probabilities that depend on node connectivity. By analyzing the trajectories of blind ants, we show that the asymptotic behaviors of both random walks are related by rescaling time and probability with node connectivity. Using this result, we show how the small eigenvalues of the Laplacian matrices generating the two random walks are related. As an application, we show how the return-to-origin probability of a myopic ant can be used to compute the scaling behaviors of the Edwards-Wilkinson model, a representative model of load balancing on networks.

The structure and dynamics of multilayer networks

In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal- or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.

Navigability of interconnected networks under random failures

Assessing the navigability of interconnected networks (transporting information, people, or goods) under eventual random failures is of utmost importance to design and protect critical infrastructures. Random walks are a good proxy to determine this navigability, specifically the coverage time of random walks, which is a measure of the dynamical functionality of the network. Here, we introduce the theoretical tools required to describe random walks in interconnected networks accounting for structure and dynamics inherent to real systems. We develop an analytical approach for the covering time of random walks in interconnected networks and compare it with extensive Monte Carlo simulations. Generally speaking, interconnected networks are more resilient to random failures than their individual layers per se, and we are able to quantify this effect. As an application--which we illustrate by considering the public transport of London--we show how the efficiency in exploring the multiplex critically depends on layers' topology, interconnection strengths, and walk strategy. Our findings are corroborated by data-driven simulations, where the empirical distribution of check-ins and checks-out is considered and passengers travel along fastest paths in a network affected by real disruptions. These findings are fundamental for further development of searching and navigability strategies in real interconnected systems.

Characteristic times of biased random walks on complex networks

We consider degree-biased random walkers whose probability to move from a node to one of its neighbors of degree k is proportional to k(α), where α is a tuning parameter. We study both numerically and analytically three types of characteristic times, namely (i) the time the walker needs to come back to the starting node, (ii) the time it takes to visit a given node for the first time, and (iii) the time it takes to visit all the nodes of the network. We consider a large data set of real-world networks and we show that the value of α which minimizes the three characteristic times differs from the value α(min)=-1 analytically found for uncorrelated networks in the mean-field approximation. In addition to this, we found that assortative networks have preferentially a value of α(min) in the range [-1,-0.5], while disassortative networks have α(min) in the range [-0.5,0]. We derive an analytical relation between the degree correlation exponent ν and the optimal bias value α(min), which works well for real-world assortative networks. When only local information is available, degree-biased random walks can guarantee smaller characteristic times than the classical unbiased random walks by means of an appropriate tuning of the motion bias.

Dynamic source routing strategy for two-level flows on scale-free networks

Packets transmitting in real communication networks such as the Internet can be classified as time-sensitive or time-insensitive. To better support the real-time and time-insensitive applications, we propose a two-level flow traffic model in which packets are labeled as level-1 or level-2, and those with level-1 have higher priority to be transmitted. In order to enhance the traffic capacity of the two-level flow traffic model, we expand the global dynamic routing strategy and propose a new dynamic source routing which supports no routing-flaps, high traffic capacity, and diverse traffic flows. As shown in this paper, the proposed dynamic source routing can significantly enhance the traffic capacity and quality of time-sensitive applications compared with the global shortest path routing strategy.

Coverage maximization under resource constraints using a nonuniform proliferating random walk

Information management services on networks, such as search and dissemination, play a key role in any large-scale distributed system. One of the most desirable features of these services is the maximization of the coverage, i.e., the number of distinctly visited nodes under constraints of network resources as well as time. However, redundant visits of nodes by different message packets (modeled, e.g., as walkers) initiated by the underlying algorithms for these services cause wastage of network resources. In this work, using results from analytical studies done in the past on a K-random-walk-based algorithm, we identify that redundancy quickly increases with an increase in the density of the walkers. Based on this postulate, we design a very simple distributed algorithm which dynamically estimates the density of the walkers and thereby carefully proliferates walkers in sparse regions. We use extensive computer simulations to test our algorithm in various kinds of network topologies whereby we find it to be performing particularly well in networks that are highly clustered as well as sparse.

Exploring complex networks by means of adaptive walkers

Finding efficient algorithms to explore large networks with the aim of recovering information about their structure is an open problem. Here, we investigate this challenge by proposing a model in which random walkers with previously assigned home nodes navigate through the network during a fixed amount of time. We consider that the exploration is successful if the walker gets the information gathered back home, otherwise no data are retrieved. Consequently, at each time step, the walkers, with some probability, have the choice to either go backward approaching their home or go farther away. We show that there is an optimal solution to this problem in terms of the average information retrieved and the degree of the home nodes and design an adaptive strategy based on the behavior of the random walker. Finally, we compare different strategies that emerge from the model in the context of network reconstruction. Our results could be useful for the discovery of unknown connections in large-scale networks.

Local-based semantic navigation on a networked representation of information

The size and complexity of actual networked systems hinders the access to a global knowledge of their structure. This fact pushes the problem of navigation to suboptimal solutions, one of them being the extraction of a coherent map of the topology on which navigation takes place. In this paper, we present a Markov chain based algorithm to tag networked terms according only to their topological features. The resulting tagging is used to compute similarity between terms, providing a map of the networked information. This map supports local-based navigation techniques driven by similarity. We compare the efficiency of the resulting paths according to their length compared to that of the shortest path. Additionally we claim that the path steps towards the destination are semantically coherent. To illustrate the algorithm performance we provide some results from the Simple English Wikipedia, which amounts to several thousand of pages. The simplest greedy strategy yields over an 80% of average success rate. Furthermore, the resulting content-coherent paths most often have a cost between one- and threefold compared to shortest-path lengths.

Smart random walkers: the cost of knowing the path

In this work we study the problem of targeting signals in networks using entropy information measurements to quantify the cost of targeting. We introduce a penalization rule that imposes a restriction on the long paths and therefore focuses the signal to the target. By this scheme we go continuously from fully random walkers to walkers biased to the target. We found that the optimal degree of penalization is mainly determined by the topology of the network. By analyzing several examples, we have found that a small amount of penalization reduces considerably the typical walk length, and from this we conclude that a network can be efficiently navigated with restricted information.

Extreme events and event size fluctuations in biased random walks on networks

Random walk on discrete lattice models is important to understand various types of transport processes. The extreme events, defined as exceedences of the flux of walkers above a prescribed threshold, have been studied recently in the context of complex networks. This was motivated by the occurrence of rare events such as traffic jams, floods, and power blackouts which take place on networks. In this work, we study extreme events in a generalized random walk model in which the walk is preferentially biased by the network topology. The walkers preferentially choose to hop toward the hubs or small degree nodes. In this setting, we show that extremely large fluctuations in event sizes are possible on small degree nodes when the walkers are biased toward the hubs. In particular, we obtain the distribution of event sizes on the network. Further, the probability for the occurrence of extreme events on any node in the network depends on its "generalized strength," a measure of the ability of a node to attract walkers. The generalized strength is a function of the degree of the node and that of its nearest neighbors. We obtain analytical and simulation results for the probability of occurrence of extreme events on the nodes of a network using a generalized random walk model. The result reveals that the nodes with a larger value of generalized strength, on average, display lower probability for the occurrence of extreme events compared to the nodes with lower values of generalized strength.

Influence of zero range process interaction on diffusion

We study the aspects of diffusion for the case of zero range process interaction on scale-free networks, through statistical quantities such as the mean first passage time, coverage, mean square displacement etc., and pay attention to how the interaction, especially the resulted condensation, influences the diffusion. By mean-field theory we show that the statistical quantities of diffusion can be significantly reduced by the condensation and can be figured out by the waiting time of a particle staying at a node. Numerical simulations have confirmed the theoretical predictions.

Weak ties: subtle role of information diffusion in online social networks

As a social media, online social networks play a vital role in the social information diffusion. However, due to its unique complexity, the mechanism of the diffusion in online social networks is different from the ones in other types of networks and remains unclear to us. Meanwhile, few works have been done to reveal the coupled dynamics of both the structure and the diffusion of online social networks. To this end, in this paper, we propose a model to investigate how the structure is coupled with the diffusion in online social networks from the view of weak ties. Through numerical experiments on large-scale online social networks, we find that in contrast to some previous research results, selecting weak ties preferentially to republish cannot make the information diffuse quickly, while random selection can achieve this goal. However, when we remove the weak ties gradually, the coverage of the information will drop sharply even in the case of random selection. We also give a reasonable explanation for this by extra analysis and experiments. Finally, we conclude that weak ties play a subtle role in the information diffusion in online social networks. On one hand, they act as bridges to connect isolated local communities together and break through the local trapping of the information. On the other hand, selecting them as preferential paths to republish cannot help the information spread further in the network. As a result, weak ties might be of use in the control of the virus spread and the private information diffusion in real-world applications.

Global dynamic routing for scale-free networks

Traffic is essential for many dynamic processes on networks. The efficient routing strategy [G. Yan, T. Zhou, B. Hu, Z. Q. Fu, and B. H. Wang, Phys. Rev. E 73, 046108 (2006)] can reach a very high capacity of more than ten times of that with shortest path strategy. In this paper, we propose a global dynamic routing strategy for network systems based on the information of the queue length of nodes. Under this routing strategy, the traffic capacity is further improved. With time delay of updating node queue lengths and the corresponding paths, the system capacity remains constant, while the travel time for packets increases.

Pheromone routing protocol on a scale-free network

This paper proposes a routing strategy for network systems based on the local information of "pheromone." The overall traffic capacity of a network system can be evaluated by the critical packet generating rate R(c). Under this critical generating rate, the total packet number in the system first increases and then decreases to reach a balance state. The system behaves differently from that with a local routing strategy based on the node degree or shortest path routing strategy. Moreover, the pheromone routing strategy performs much better than the local routing strategy, which is demonstrated by a larger value of the critical generating rate. This protocol can be an alternation for superlarge networks, in which the global topology may not be available.

Biased random walks in complex networks: the role of local navigation rules

We study the biased random-walk process in random uncorrelated networks with arbitrary degree distributions. In our model, the bias is defined by the preferential transition probability, which, in recent years, has been commonly used to study the efficiency of different routing protocols in communication networks. We derive exact expressions for the stationary occupation probability and for the mean transit time between two nodes. The effect of the cyclic search on transit times is also explored. Results presented in this paper provide the basis for a theoretical treatment of transport-related problems in complex networks, including quantitative estimation of the critical value of the packet generation rate.

Searching method through biased random walks on complex networks

Information search is closely related to the first-passage property of diffusing particle. The physical properties of diffusing particle is affected by the topological structure of the underlying network. Thus, the interplay between dynamical process and network topology is important to study information search on complex networks. Designing an efficient method has been one of main interests in information search. Both reducing the network traffic and decreasing the searching time have been two essential factors for designing efficient method. Here we propose an efficient method based on biased random walks. Numerical simulations show that the average searching time of the suggested model is more efficient than other well-known models. For a practical interest, we demonstrate how the suggested model can be applied to the peer-to-peer system.

Optimal navigation in complex networks

Recent literature has presented evidence that the study of navigation in complex networks is useful to understand their dynamics and topology. Two main approaches are usually considered: navigation of random walkers and navigation of directed walkers. Unlike these approaches ours supposes that a traveler walks optimally in order to minimize the cost of the walking. If this happens, two extreme regimes arise-one dominated by directed walkers and the other by random walkers. We try to characterize the critical point of the transition from one regime to the other in function of the connectivity and the size of the network. Furthermore, we show that this approach can be used to generalize several concepts presented in the literature concerning random navigation and direct navigation. Finally, we defend that investigating the extreme regimes dominated by random walkers and directed walkers is not sufficient to correctly assess the characteristics of navigation in complex networks.

Random-walk access times on partially disordered complex networks: an effective medium theory

An analytic effective medium theory is constructed to study the mean access times for random walks on hybrid disordered structures formed by embedding complex networks into regular lattices, considering transition rates F that are different for steps across lattice bonds from the rates f across network shortcuts. The theory is developed for structures with arbitrary shortcut distributions and applied to a class of partially disordered traversal enhanced networks in which shortcuts of fixed length are distributed randomly with finite probability. Numerical simulations are found to be in excellent agreement with predictions of the effective medium theory on all aspects addressed by the latter. Access times for random walks on these partially disordered structures are compared to those on small-world networks, which on average appear to provide the most effective means of decreasing access times uniformly across the network.

An algorithm for modularization of MAPK and calcium signaling pathways: comparative analysis among different species

Signaling pathways are large complex biochemical networks. It is difficult to analyze the underlying mechanism of such networks as a whole. In the present article, we have proposed an algorithm for modularization of signal transduction pathways. Unlike studying a signaling pathway as a whole, this enables one to study the individual modules (less complex smaller units) easily and hence to study the entire pathway better. A comparative study of modules belonging to different species (for the same signaling pathway) has been made, which gives an overall idea about development of the signaling pathways over the taken set of species of calcium and MAPK signaling pathways. The superior performance, in terms of biological significance, of the proposed algorithm over an existing community finding algorithm of Newman [Newman MEJ. Modularity and community structure in networks. Proc Natl Acad Sci USA 2006;103(23):8577-82] has been demonstrated using the aforesaid pathways of H. sapiens.

Exploring complex networks through random walks

Most real complex networks--such as protein interactions, social contacts, and the Internet--are only partially known and available to us. While the process of exploring such networks in many cases resembles a random walk, it becomes a key issue to investigate and characterize how effectively the nodes and edges of such networks can be covered by different strategies. At the same time, it is critically important to infer how well can topological measurements such as the average node degree and average clustering coefficient be estimated during such network explorations. The present article addresses these problems by considering random, Barabási-Albert (BA), and geographical network models with varying connectivity explored by three types of random walks: traditional, preferential to untracked edges, and preferential to unvisited nodes. A series of relevant results are obtained, including the fact that networks of the three studied models with the same size and average node degree allow similar node and edge coverage efficiency, the identification of linear scaling with the size of the network of the random walk step at which a given percentage of the nodes/edges is covered, and the critical result that the estimation of the averaged node degree and clustering coefficient by random walks on BA networks often leads to heavily biased results. Many are the theoretical and practical implications of such results.

A comparison of glycans and polyglycans using solid-state NMR and X-ray powder diffraction

Individual polyglycans and their corresponding monomers have been studied separately for several decades. Attention has focused primarily on the modifications of these polyglycans instead of the simple relationship between the polyglycans themselves and their corresponding monomers. Two polyglycans, chitin and chitosan, were examined along with their respective monomeric units, N-acetyl-D-glucosamine (GlcNAc) and (+)D-glucosamine (GlcN) using solid-state proton decoupling Magic Angle Turning (MAT) techniques and X-Ray Powder Diffraction (XRPD). A down-field shift in isotropic (13)C chemical shifts was observed for both polymers in Cross Polarization/Magic Angle Spinning (CP/MAS) spectra. An explanation of misleading peak assignments in previous NMR studies for these polyglycans was determined by comparing sideband patterns of the polymers with their corresponding monomers generated in a 2D FIve pi REplicated Magic Angle Turning (FIREMAT) experiment processed by Technique for Importing Greater Evolution Resolution (TIGER). Structural changes in the crystalline framework were supported by XRPD diffraction data.

Ring structures and mean first passage time in networks

In this paper we address the problem of the calculation of the mean first passage time on generic graphs. We focus in particular on the mean first passage time on a node for a random walker starting from a generic, unknown, node x. We introduce an approximate scheme of calculation which maps the original process in a Markov process in the space of the so-called rings, described by a transition matrix of size O(ln N/ln (k) x ln N/ln (k)), where is the size of the graph and (k) the average degree in the graph. In this way one has a drastic reduction of degrees of freedom with respect to the size of the transition matrix of the original process, corresponding to an extremely low computational cost. We first apply the method to the Erdös-Renyi random graphs for which the method allows for almost perfect agreement with numerical simulations. Then we extend the approach to the Barabási-Albert graph, as an example of scale-free graph, for which one obtains excellent results. Finally we test the method with two real-world graphs, Internet and a network of the brain, for which we obtain accurate results.

Traffic dynamics based on local routing protocol on a scale-free network

We propose a packet routing strategy with a tunable parameter based on the local structural information of a scale-free network. As free traffic flow on the communication networks is key to their normal and efficient functioning, we focus on the network capacity that can be measured by the critical point of phase transition from free flow to congestion. Simulations show that the maximal capacity corresponds to alpha= -1 in the case of identical nodes' delivering ability. To explain this, we investigate the number of packets of each node depending on its degree in the free flow state and observe the power law behavior. Other dynamic properties including average packets traveling time and traffic load are also studied. Inspiringly, our results indicate that some fundamental relationships exist between the dynamics of synchronization and traffic on the scale-free networks.

Cycles and clustering in bipartite networks

We investigate the clustering coefficient in bipartite networks where cycles of size three are absent and therefore the standard definition of clustering coefficient cannot be used. Instead, we use another coefficient given by the fraction of cycles with size four, showing that both coefficients yield the same clustering properties. The new coefficient is computed for two networks of sexual contacts, one bipartite and another where no distinction between the nodes is made (monopartite). In both cases the clustering coefficient is similar. Furthermore, combining both clustering coefficients we deduce an expression for estimating cycles of larger size, which improves previous estimations and is suitable for either monopartite and multipartite networks, and discuss the applicability of such analytical estimations.