Topologies and Laplacian spectra of a deterministic uniform recursive tree

Structural and spectral properties of the generalized corona networks

Let Gr,p be a graph obtained from a path by adjoining a cycle Cr of length r to one end and the central vertex of a star Sp on p vertices to the other end.In this paper, it is proven that unicyclic graph Gr,p with r even is determined by its Laplacian spectrum except for n = p + 4.

We destroy a finite tree of size n by cutting its edges one after the other and in uniform random order. Informally, the associated cut‐tree describes the genealogy of the connected components created by this destruction process. We provide a general criterion for the convergence of the rescaled cut‐tree in the Gromov‐Prohorov topology to an interval endowed with the Euclidean distance and a certain probability measure, when the underlying tree has branching points close to the root and height of order . In particular, we consider uniform random recursive trees, binary search trees, scale‐free random trees and a mixture of regular trees. This yields extensions of a result in Bertoin (Probab Stat 5 (2015), 478–488) for the cut‐tree of uniform random recursive trees and also allows us to generalize some results of Kuba and Panholzer (Online J Anal Combin (2014), 26) on the multiple isolation of vertices. The approach relies in the close relationship between the destruction process and Bernoulli bond percolation, which may be useful for studying the cut‐tree of other classes of trees. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 404–427, 2017

Topological characteristics is the foundation for optimizing topology structure and designing protocols. In this work, we study broadly the topological characteristics of Ad hoc networks through the perspective of complex network and identify the small-world effect, scale-free property, clustering property and hierarchy of Ad hoc networks, we also analyze the relationship among many of topological characteristics. Results shows that the degree distribution, core distribution, path distribution all present an obvious distribution trait. Moreover, the Ad hoc networks have clear small-world and clustering properties. Additionally, we found that a node whose degree is close to average degree usually has higher clustering coefficient, namely this node has a good fault tolerance and survivability. Keywords—ad hoc networks; complex network; topological

We collected the data of the documents and their links of China Education and Research Network’s which construct the complex directed network China Education Network (CEN) with large amount of documents with their edges (URLs). This paper analyzes some statistical properties, including degree distributions, average path length, clustering coefficient, and the community structure of China Education Network basing on the practical data. By analyzing the practical data, we found that the in-degree and out-degree distribution of the CEN has power-law tail and the network displays both properties of small world and scale free. The CEN has a considerably small average path length and its clustering coefficient is in the mediate. As a large scale complex network, China Education Network clearly present its community structure in which the colleges in a school constitute communities generally with a large modularity.

The previous research on the controllability of complex networks is qualitative issue which can only identify whether the network is controllable or not, and it is difficult to assess the controlled ability of network. In this paper, a quantitative measurement index of network controllability is introduced. And the index is used to study the controllability of the IEEE power network system. The relationship between controllability and network parameters (including the average degree, the clustering coefficient, the average path length) of power network is analyzed by comparing with the ER random network, WS small world network, BA scale-free network and CM network. The results show that the power networks have a small average path length and a higher clustering coefficient, which belong to the small world network. And, the controllability of power networks is larger than the corresponding model networks, indicating that power networks are relatively easier to be controlled.

Based on the Koch network constructed using Koch fractals, we proposed a class of expanded Koch networks in this paper. The original triangle is replaced by r-polygon, and each node generates m sub r-polygons by every step, which makes the Koch network more general. We studied the structure and properties of the networks. The exact analytical result of the degree distribution, clustering coefficient and average path length were obtained. When parameters m and r satisfy some certain conditions, the networks follow a power-law distribution and have a small average path length. Finally, we introduced the random walk on the network. Our discussions focused on the trapping problem, particularly the calculation and derivation of mean first passage time (MFPT) and global mean first passage time (GMFPT). In addition, we also gave the relationship between the above results and the network size.

Small-world network is a very common network structure, which is characterized by low average degree, small average path length and high centrality. At the same time, small-world networks have high resilience to random errors and low resilience to targeted attacks. In this study, the importance of nodes is represented by attributes such as degree and centrality, and attacks refer to the removal of important nodes. The network is attacked according to the degree, betweenness and closeness centrality to observe the power distribution. The data is mainly obtained from the open source OpenFlight. Gephi, Python, and Excel are used as tools. Gephi is used for network visualization and analysis. The third-party python libraries Pandas, Matplotlib, and NetworkX were used in this study to deal with the things that Gephi can't compute or represent well, and then plot the corresponding graphs with Matplotlib. The work of cleaning data is mainly done by excel.

In this paper, we introduce a new method for safety analysis of process industry systems based on recent advances in complex networks. We model the process industry system as a large, complex network and study its topological properties, and then we investigate the cascading failures by constructing a simple model incorporating the loads on nodes and the efficiency of network. We use the Tennessee eastman process (TEP) as a case to illustrate our work. We find the industry system network is an inhomogeneous network and has a high clustering coefficient and a small average path length, which is different from random networks and regular networks. We also find that an initial failure of a device of the system can generate a cascading failure, if the devices have extremely high initial load. The method can help to identify the critical components of the process industry system and can also be available for failure prevention and design safety control system.

The propagation of various opinions in social networks, which influences human inter-relationships and even social structure, and hence is a most important part of social life. We have incorporated social balance into opinion propagation in social networks are influenced by social balance. The edges in networks can represent both friendly or hostile relations, and change with the opinions of individual nodes. We introduce a model to characterize the coevolutionary dynamics of these two dynamical processes on Watts-Strogatz (WS) small-world network. We employ two distinct evolution rules (i) opinion renewal; and (ii) relation adjustment. By changing the rewiring probability, and thus the small-worldness of the WS network, we found that the time for the system to reach balanced states depends critically on both the average path length and clustering coefficient of the network, which is different than other networked process like epidemic spreading. In particular, the system equilibrates most quickly when the underlying network demonstrates strong small-worldness, i.e., small average path lengths and large clustering coefficient. We also find that opinion clusters emerge in the process of the network approaching the global equilibrium, and a measure of global contrariety is proposed to quantify the balanced state of a social network.

The wholeness, conceived and developed by Christopher Alexander, is what exists to some degree or other in space and matter, and can be described by precise mathematical language. However, it remains somehow mysterious and elusive, and therefore hard to grasp. This paper develops a complex network perspective on the wholeness to better understand the nature of order or beauty for sustainable design. I bring together a set of complexity-science subjects such as complex networks, fractal geometry, and in particular underlying scaling hierarchy derived by head/tail breaks—a classification scheme and a visualization tool for data with a heavy-tailed distribution, in order to make Alexander’s profound thoughts more accessible to design practitioners and complexity-science researchers. Through several case studies (some of which Alexander studied), I demonstrate that the complex-network perspective helps reduce the mystery of wholeness and brings new insights to Alexander’s thoughts on the concept of wholeness or objective beauty that exists in fine and deep structure. The complex-network perspective enables us to see things in their wholeness, and to better understand how the kind of structural beauty emerges from local actions guided by the 15 fundamental properties, and by differentiation and adaptation processes. The wholeness goes beyond current complex network theory towards design or creation of living structures.

A complex-network perspective on Alexander’s wholeness

supply networks modeling has gained tremendous attention in the field of supply chain during last decades. However, most existent literatures in supply networks do not pay much attention to the topological characteristics, which, as revealed by Complex Networks, have close relations with their behaviors in various complex systems. This paper introduces a novel topological structure evolving model of complex supply networks from the perspective of Complex Networks. After investigating supply networks typical structural characteristics and formation mechanisms, Complex Supply Networks Evolving Model (CSNEM) are proposed to mimic these structural characteristics. Then, three statistical properties are investigated. To validate CSNEM, three real supply chain networks are introduced. The results indicate that CSNEM is consistent with the real-world supply networks. Further the relations between these properties and supply networks macro behaviors i.e. rapid response capability and robustness are discussed. This work will provide managers and researches a beneficial insight on macroscopic structures characteristics as well as a helpful understanding on the relations between macroscopic structures and global behaviors in supply networks.

Research shows the degree distribution of firms' innovation network in accordance with power-law distribution as continuous evolution of firms' innovation network, which has larger clustering coefficient and smaller average path length. Therefore, the network structure and simulation results of two stages are compared with the data of Taiyuan stainless steel firms' innovation network. The size of Taiyuan stainless steel firms' innovation network is constantly expanding, so is the clustering coefficient, while the average path length is constantly shrinking.

The increased usage of directional methods of communications to improve medium reuse, network capacity, and bandwidth has prompted research into leveraging directionality in every layer of the network stack. Recently, there has been work on bringing the apparent capacity gains on layer 2 using directional communications methods to layer 3 by using directionality to route packets scalably in unstructured, flat networks. In their protocol, Orthogonal Rendezvous Routing Protocol, Cheng et al. [1] showed that by ”drawing” two lines orthogonal to each other at each node, it is possible to provide over 98% connectivity while maintaining only O(N3/2) evenly distributed states at a cost of only 1.2 path stretch. In this paper, we seek to provide more in-depth performance analysis by tuning additional factors such as the number of directions to transmit, the number of interfaces per node, among others, to understand its affect on varying network densities, topologies, connections, and traffic patterns. We show that by sending packets out in more directions, increased connectivity, smaller average path length, better goodput results only up to a point as compared to other routing protocols. The trade-off, however, is added state information maintained at each node and additional control packets received. We also show that the addition of more interfaces generally yields better packet delivery success, average path length, and goodput.