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Evolving Apollonian Networks with Small-world Scale-free topologies  [PDF]
Zhongzhi Zhang,Lili Rong,Shuigeng Zhou
Physics , 2005, DOI: 10.1103/PhysRevE.74.046105
Abstract: We propose two types of evolving networks: evolutionary Apollonian networks (EAN) and general deterministic Apollonian networks (GDAN), established by simple iteration algorithms. We investigate the two networks by both simulation and theoretical prediction. Analytical results show that both networks follow power-law degree distributions, with distribution exponents continuously tuned from 2 to 3. The accurate expression of clustering coefficient is also given for both networks. Moreover, the investigation of the average path length of EAN and the diameter of GDAN reveals that these two types of networks possess small-world feature. In addition, we study the collective synchronization behavior on some limitations of the EAN.
Evolving small-world networks with geographical attachment preference  [PDF]
Zhongzhi Zhang,Lili Rong,Francesc Comellas
Mathematics , 2005, DOI: 10.1088/0305-4470/39/13/005
Abstract: We introduce a minimal extended evolving model for small-world networks which is controlled by a parameter. In this model the network growth is determined by the attachment of new nodes to already existing nodes that are geographically close. We analyze several topological properties for our model both analytically and by numerical simulations. The resulting network shows some important characteristics of real-life networks such as the small-world effect and a high clustering.
Evolving small-world scale-free networks consist of cliques  [PDF]
Zhongzhi Zhang,Shuigeng Zhou,Lichao Chen
Physics , 2007, DOI: 10.1140/epjb/e2007-00229-9
Abstract: We present a family of scale-free network model consisting of cliques, which is established by a simple recursive algorithm. We investigate the networks both analytically and numerically. The obtained analytical solutions show that the networks follow a power-law degree distribution, with degree exponent continuously tuned between 2 and 3. The exact expression of clustering coefficient is also provided for the networks. Furthermore, the investigation of the average path length reveals that the networks possess small-world feature. Interestingly, we find that a special case of our model can be mapped into the Yule process.
Rapidly evolving in humans topologically associating domains  [PDF]
Gennadi Glinsky
Quantitative Biology , 2015,
Abstract: Genome-wide proximity placement analysis of 10,598 HSGRL within the context of the principal regulatory structures of the interphase chromatin, namely topologically-associating domains (TADs) and specific sub-TAD structures termed super-enhancer domains (SEDs) revealed that 0.8%-10.3% of TADs contain more than half of HSGRL. Of the 3,127 TADs in the hESC genome, 24 (0.8%); 53 (1.7%); 259 (8.3%); and 322 (10.3%) harbor 1,110 (52.4%); 1,936 (50.9%); 1,151 (59.6%); and 1,601 (58.3%) HSGRL sequences from four distinct families, respectively. TADs that are enriched for HSGRL and termed rapidly-evolving in humans TADs (revTADs) manifest distinct correlation patterns between HSGRL placements and recombination rates. There are significant enrichment within revTAD boundaries of hESC-enhancers, primate-specific CTCF-binding sites, human-specific RNAPII-binding sites, hCONDELs, and H3K4me3 peaks with human-specific enrichment at TSS in prefrontal cortex neurons (p < 0.0001 in all instances). In hESC genome, 331 of 504 (66%) of SE-harboring TADs contain HSGRL and 68% of SEs co-localize with HSGRL, suggesting that HSGRL rewired SE-driven GRNs within revTADs by inserting novel and/or erasing existing regulatory sequences. Consequently, markedly distinct features of chromatin structures evolved in hESC compared to mouse: the SE quantity is 3-fold higher and the median SE size is significantly larger; concomitantly, the TAD number is increased by 42% while the median TAD size is decreased (p=9.11E-37). Present analyses revealed a global role for HSGRL in increasing both quantity and size of SEs and increasing the number and size reduction of TADs, which may facilitate a convergence of TAD and SED architectures of interphase chromatin and define a trend of increasing regulatory complexity during evolution of GRNs.
Local-world evolving networks with tunable clustering  [PDF]
Zhongzhi Zhang,Lili Rong,Bing Wang,Shuigeng Zhou,Jihong Guan
Physics , 2005, DOI: 10.1016/j.physa.2007.02.045
Abstract: We propose an extended local-world evolving network model including a triad formation step. In the process of network evolution, random fluctuation in the number of new edges is involved. We derive analytical expressions for degree distribution, clustering coefficient and average path length. Our model can unify the generic properties of real-life networks: scale-free degree distribution, high clustering and small inter-node separation. Moreover, in our model, the clustering coefficient is tunable simply by changing the expected number of triad formation steps after a single local preferential attachment step.
The World as Evolving Information  [PDF]
Carlos Gershenson
Mathematics , 2007, DOI: 10.1007/978-3-642-18003-3_10
Abstract: This paper discusses the benefits of describing the world as information, especially in the study of the evolution of life and cognition. Traditional studies encounter problems because it is difficult to describe life and cognition in terms of matter and energy, since their laws are valid only at the physical scale. However, if matter and energy, as well as life and cognition, are described in terms of information, evolution can be described consistently as information becoming more complex. The paper presents eight tentative laws of information, valid at multiple scales, which are generalizations of Darwinian, cybernetic, thermodynamic, psychological, philosophical, and complexity principles. These are further used to discuss the notions of life, cognition and their evolution.
A new local-world evolving network model

Qin Sen,Dai Guan-Zhong,

中国物理 B , 2009,
Abstract: In some real complex networks, only a few nodes can obtain the global information about the entire networks, but most of the nodes own only local connections therefore own only local information of the networks. A new local-world evolving network model is proposed in this paper. In the model, not all the nodes obtain local network information, which is different from the local world network model proposed by Li and Chen (LC model). In the LC model, each node has only the local connections therefore owns only local information about the entire networks. Theoretical analysis and numerical simulation show that adjusting the ratio of the number of nodes obtaining the global information of the network to the total number of nodes can effectively control the valuing range for the power-law exponent of the new network. Therefore, if the topological structure of a complex network, especially its exponent of power-law degree distribution, needs controlling, we just add or take away a few nodes which own the global information of the network.
Multiple Scales in Small-World Graphs  [PDF]
Rajesh Kasturirangan
Physics , 1999,
Abstract: Small-world architectures may be implicated in a range of phenomena from disease propagation to networks of neurons in the cerebral cortex. While most of the recent attention on small-world networks has focussed on the effect of introducing disorder/randomness into a regular network, we show that that the fundamental mechanism behind the small-world phenomenon is not disorder/randomness, but the presence of connections of many different length scales. Consequently, in order to explain the small-world phenomenon, we introduce the concept of multiple scale graphs and then state the multiple length scale hypothesis. Multiple scale graphs form a unifying conceptual framework for the study of evolving graphs. Moreover, small-world behavior in randomly rewired graphs is a consequence of features common to all multiple scale graphs. To support the multiple length scale hypothesis, novel graph architectures are introduced that need not be a result of random rewiring of a regular graph. In each case it is shown that whenever the graph exhibits small-world behavior, it also has connections of diverse length scales. We also show that the distribution of the length scales of the new connections is significantly more important than whether the new connections are long range, medium range or short range connections.
Evolving Graph Representation and Visualization  [PDF]
Anurat Chapanond,Mukkai S. Krishnamoorthy,G. M. Prabhu,J. Punin
Computer Science , 2010,
Abstract: The study of evolution of networks has received increased interest with the recent discovery that many real-world networks possess many things in common, in particular the manner of evolution of such networks. By adding a dimension of time to graph analysis, evolving graphs present opportunities and challenges to extract valuable information. This paper introduces the Evolving Graph Markup Language (EGML), an XML application for representing evolving graphs and related results. Along with EGML, a software tool is provided for the study of evolving graphs. New evolving graph drawing techniques based on the force-directed graph layout algorithm are also explored. Our evolving graph techniques reduce vertex movements between graph instances, so that an evolving graph can be viewed with smooth transitions
Nonequilibrium Phase Transitions in Directed Small-World Networks  [PDF]
Alejandro D. Sanchez,Juan M. Lopez,Miguel A. Rodriguez
Physics , 2001, DOI: 10.1103/PhysRevLett.88.048701
Abstract: Many social, biological, and economic systems can be approached by complex networks of interacting units. The behaviour of several models on small-world networks has recently been studied. These models are expected to capture the essential features of the complex processes taking place on real networks like disease spreading, formation of public opinion, distribution of wealth, etc. In many of these systems relations are directed, in the sense that links only act in one direction (outwards or inwards). We investigate the effect of directed links on the behaviour of a simple spin-like model evolving on a small-world network. We show that directed networks may lead to a highly nontrivial phase diagram including first and second-order phase transitions out of equilibrium.
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