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Evolution Of A Random Network
Evolution of a random network is a dynamical process, usually leading to emergence of giant component accompanied with striking consequences on the network topology. To quantify this process, there is a need of inspection on how the size of the largest connected cluster within the network, , varies with the average degree .Albert-László Barabási. Network Science: Chapter 3 Networks change their topology as they evolve, undergoing phase transitions. Phase transitions are generally known from physics, where it occurs as matter changes state according to its thermal energy level, or when ferromagnetic properties emerge in some materials as they are cooling down. Such phase transitions take place in matter because it is a network of particles, and as such, rules of network phase transition directly apply to it. Phase transitions in networks happen as links are added to a network, meaning that having N nodes, in each time increment, a link is placed between a randomly chosen pair of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giant Component
In network theory, a giant component is a connected component of a given random graph that contains a finite fraction of the entire graph's vertices. Giant component in Erdős–Rényi model Giant components are a prominent feature of the Erdős–Rényi model (ER) of random graphs, in which each possible edge connecting pairs of a given set of vertices is present, independently of the other edges, with probability . In this model, if p \le \frac for any constant \epsilon>0, then with high probability all connected components of the graph have size , and there is no giant component. However, for p \ge \frac there is with high probability a single giant component, with all other components having size . For p=p_c = \frac, intermediate between these two possibilities, the number of vertices in the largest component of the graph, P_ is with high probability proportional to n^.. Giant component is also important in percolation theory. When a fraction of nodes, q=1-p, is removed ran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree (graph Theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is denoted \deg(v) or \deg v. The maximum degree of a graph G, denoted by \Delta(G), and the minimum degree of a graph, denoted by \delta(G), are the maximum and minimum of its vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, n-1. In a signed graph, the number of positive edges connected to the vertex v is called positive deg(v) and the number of connected negative edges is entitled negative deg(v). Handshaking lemma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Rényi Model
In the mathematical field of graph theory, the Erdős–Rényi model is either of two closely related models for generating random graphs or the evolution of a random network. They are named after Hungarian mathematicians Paul Erdős and Alfréd Rényi, who first introduced one of the models in 1959, while Edgar Gilbert introduced the other model contemporaneously and independently of Erdős and Rényi. In the model of Erdős and Rényi, all graphs on a fixed vertex set with a fixed number of edges are equally likely; in the model introduced by Gilbert, also called the Erdős–Rényi–Gilbert model, each edge has a fixed probability of being present or absent, independently of the other edges. These models can be used in the probabilistic method to prove the existence of graphs satisfying various properties, or to provide a rigorous definition of what it means for a property to hold for almost all graphs. Definition There are two closely related variants of the Erdős–R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cluster Analysis
Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some sense) to each other than to those in other groups (clusters). It is a main task of exploratory data analysis, and a common technique for statistics, statistical data analysis, used in many fields, including pattern recognition, image analysis, information retrieval, bioinformatics, data compression, computer graphics and machine learning. Cluster analysis itself is not one specific algorithm, but the general task to be solved. It can be achieved by various algorithms that differ significantly in their understanding of what constitutes a cluster and how to efficiently find them. Popular notions of clusters include groups with small Distance function, distances between cluster members, dense areas of the data space, intervals or particular statistical distributions. Clustering can therefore be formulated as a multi-object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Transition
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of matter: solid, liquid, and gas, and in rare cases, plasma. A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium, certain properties of the medium change as a result of the change of external conditions, such as temperature or pressure. This can be a discontinuous change; for example, a liquid may become gas upon heating to its boiling point, resulting in an abrupt change in volume. The identification of the external conditions at which a transformation occurs defines the phase transition point. Types of phase transition At the phase transition point for a substance, for instance the boiling point, the two phases involved - liquid and vapor, have identic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Freezing
Freezing is a phase transition where a liquid turns into a solid when its temperature is lowered below its freezing point. In accordance with the internationally established definition, freezing means the solidification phase change of a liquid or the liquid content of a substance, usually due to cooling. For most substances, the melting and freezing points are the same temperature; however, certain substances possess differing solid-liquid transition temperatures. For example, agar displays a hysteresis in its melting point and freezing point. It melts at 85 °C (185 °F) and solidifies from 32 °C to 40 °C (89.6 °F to 104 °F). Crystallization Most liquids freeze by crystallization, formation of crystalline solid from the uniform liquid. This is a first-order thermodynamic phase transition, which means that as long as solid and liquid coexist, the temperature of the whole system remains very nearly equal to the melting point due to the slow re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ferromagnetism
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials are the familiar metals noticeably attracted to a magnet, a consequence of their large magnetic permeability. Magnetic permeability describes the induced magnetization of a material due to the presence of an ''external'' magnetic field, and it is this temporarily induced magnetization inside a steel plate, for instance, which accounts for its attraction to the permanent magnet. Whether or not that steel plate acquires a permanent magnetization itself, depends not only on the strength of the applied field, but on the so-called coercivity of that material, which varies greatly among ferromagnetic materials. In physics, several different types of material magnetism are distinguished. Ferromagnetism (along with the similar effect ferrimagnetis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin (physics)
Spin is a conserved quantity carried by elementary particles, and thus by composite particles (hadrons) and atomic nucleus, atomic nuclei. Spin is one of two types of angular momentum in quantum mechanics, the other being ''orbital angular momentum''. The orbital angular momentum operator is the quantum-mechanical counterpart to the classical angular momentum of orbital revolution and appears when there is periodic structure to its wavefunction as the angle varies. For photons, spin is the quantum-mechanical counterpart of the Polarization (waves), polarization of light; for electrons, the spin has no classical counterpart. The existence of electron spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum. The existence of the electron spin can also be inferred theoretically from the spin–statistics theorem and from th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetic Field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field. A permanent magnet's magnetic field pulls on ferromagnetic materials such as iron, and attracts or repels other magnets. In addition, a nonuniform magnetic field exerts minuscule forces on "nonmagnetic" materials by three other magnetic effects: paramagnetism, diamagnetism, and antiferromagnetism, although these forces are usually so small they can only be detected by laboratory equipment. Magnetic fields surround magnetized materials, and are created by electric currents such as those used in electromagnets, and by electric fields varying in time. Since both strength and direction of a magnetic field may vary with location, it is described mathematically by a function assigning a vector to each point of space, cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polymer
A polymer (; Greek '' poly-'', "many" + ''-mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic and natural polymers play essential and ubiquitous roles in everyday life. Polymers range from familiar synthetic plastics such as polystyrene to natural biopolymers such as DNA and proteins that are fundamental to biological structure and function. Polymers, both natural and synthetic, are created via polymerization of many small molecules, known as monomers. Their consequently large molecular mass, relative to small molecule compounds, produces unique physical properties including toughness, high elasticity, viscoelasticity, and a tendency to form amorphous and semicrystalline structures rather than crystals. The term "polymer" derives from the Greek word πολύς (''polus'', meaning "many, much") and μέρος (''meros'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cellulose
Cellulose is an organic compound with the formula , a polysaccharide consisting of a linear chain of several hundred to many thousands of β(1→4) linked D-glucose units. Cellulose is an important structural component of the primary cell wall of green plants, many forms of algae and the oomycetes. Some species of bacteria secrete it to form biofilms. Cellulose is the most abundant organic polymer on Earth. The cellulose content of cotton fiber is 90%, that of wood is 40–50%, and that of dried hemp is approximately 57%. Cellulose is mainly used to produce paperboard and paper. Smaller quantities are converted into a wide variety of derivative products such as cellophane and rayon. Conversion of cellulose from energy crops into biofuels such as cellulosic ethanol is under development as a renewable fuel source. Cellulose for industrial use is mainly obtained from wood pulp and cotton. Some animals, particularly ruminants and termites, can digest cellulose with the help of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lenka Zdeborová
Lenka Zdeborová (born 24 November 1980) is a Czech physicist and computer scientist who applies methods from statistical physics to machine learning and constraint satisfaction problems. She is a professor of physics and computer science and communication systems at EPFL (École Polytechnique Fédérale de Lausanne). Life Zdeborová was born in Plzeň and attended a local grammar school where she excelled in math and physics. After living in France with her family and working at the Centre National de la Recherche Scientifique (CNRS), she and her partner moved to Switzerland in 2020. They are currently raising their two children there. Education and career Zdeborová earned a master's degree in physics at Charles University in 2004, and 2008, completed an international dual doctorate ("en cotutelle") at both Charles University and University of Paris-Sud. Her doctoral advisors were Václav Janiš at Charles University, and Marc Mézard at Paris-Sud. After postdoctoral re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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