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89-1043: Social network analysis Small-world networks Centrality Motifs Graph theory Scaling Robustness Systems biology Dynamic networks Evolutionary computation Genetic algorithms Genetic programming Artificial life Machine learning Evolutionary developmental biology Artificial intelligence Evolutionary robotics Reaction–diffusion systems Partial differential equations Dissipative structures Percolation Cellular automata Spatial ecology Self-replication Conversation theory Entropy Feedback Goal-oriented Homeostasis Information theory Operationalization Second-order cybernetics Self-reference System dynamics Systems science Systems thinking Sensemaking Variety Ordinary differential equations Phase space Attractors Population dynamics Chaos Multistability Bifurcation Rational choice theory Bounded rationality In mathematics and science ,

178-411: A ( x ) {\displaystyle L_{a}(x)} becomes y = f ( a ) + M ( x − a ) {\displaystyle y=f(a)+M(x-a)} . Because differentiable functions are locally linear , the best slope to substitute in would be the slope of the line tangent to f ( x ) {\displaystyle f(x)} at x =

267-441: A {\displaystyle a} is f ′ ( a ) {\displaystyle f'(a)} . To find 4.001 {\displaystyle {\sqrt {4.001}}} , we can use the fact that 4 = 2 {\displaystyle {\sqrt {4}}=2} . The linearization of f ( x ) = x {\displaystyle f(x)={\sqrt {x}}} at x =

356-409: A {\displaystyle x=a} is y = a + 1 2 a ( x − a ) {\displaystyle y={\sqrt {a}}+{\frac {1}{2{\sqrt {a}}}}(x-a)} , because the function f ′ ( x ) = 1 2 x {\displaystyle f'(x)={\frac {1}{2{\sqrt {x}}}}} defines the slope of

445-444: A {\displaystyle x=a} . While the concept of local linearity applies the most to points arbitrarily close to x = a {\displaystyle x=a} , those relatively close work relatively well for linear approximations. The slope M {\displaystyle M} should be, most accurately, the slope of the tangent line at x = a {\displaystyle x=a} . Visually,

534-475: A ) ) {\displaystyle y=(f(a)+f'(a)(x-a))} For x = a {\displaystyle x=a} , f ( a ) = f ( x ) {\displaystyle f(a)=f(x)} . The derivative of f ( x ) {\displaystyle f(x)} is f ′ ( x ) {\displaystyle f'(x)} , and the slope of f ( x ) {\displaystyle f(x)} at

623-485: A polynomial equation such as x 2 + x − 1 = 0. {\displaystyle x^{2}+x-1=0.} The general root-finding algorithms apply to polynomial roots, but, generally they do not find all the roots, and when they fail to find a root, this does not imply that there is no roots. Specific methods for polynomials allow finding all roots or the real roots; see real-root isolation . Solving systems of polynomial equations , that

712-539: A global optimum . In multiphysics systems—systems involving multiple physical fields that interact with one another—linearization with respect to each of the physical fields may be performed. This linearization of the system with respect to each of the fields results in a linearized monolithic equation system that can be solved using monolithic iterative solution procedures such as the Newton–Raphson method . Examples of this include MRI scanner systems which results in

801-467: A linear map (or linear function ) f ( x ) {\displaystyle f(x)} is one which satisfies both of the following properties: Additivity implies homogeneity for any rational α , and, for continuous functions , for any real α . For a complex α , homogeneity does not follow from additivity. For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in

890-516: A nonlinear system (or a non-linear system ) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers , biologists , physicists , mathematicians , and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems , describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems . Typically,

979-582: A free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) phase portraits and approximate periods. Social network analysis 1800s: Martineau · Tocqueville  ·  Marx ·  Spencer · Le Bon · Ward · Pareto ·  Tönnies · Veblen ·  Simmel · Durkheim ·  Addams ·  Mead · Weber ·  Du Bois ·  Mannheim · Elias Social network analysis ( SNA )

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1068-422: A frictionless pendulum under the influence of gravity . Using Lagrangian mechanics , it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation where gravity points "downwards" and θ {\displaystyle \theta } is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation

1157-434: A given network. Homophily : The extent to which actors form ties with similar versus dissimilar others. Similarity can be defined by gender, race, age, occupation, educational achievement, status, values or any other salient characteristic. Homophily is also referred to as assortativity . Multiplexity: The number of content-forms contained in a tie. For example, two people who are friends and also work together would have

1246-419: A group of people who are unlikely to change their opinions of the other people in the group. Unbalanced graphs represent a group of people who are very likely to change their opinions of the people in their group. For example, a group of 3 people (A, B, and C) where A and B have a positive relationship, B and C have a positive relationship, but C and A have a negative relationship is an unbalanced cycle. This group

1335-425: A line, given a point ( H , K ) {\displaystyle (H,K)} and slope M {\displaystyle M} . The general form of this equation is: y − K = M ( x − H ) {\displaystyle y-K=M(x-H)} . Using the point ( a , f ( a ) ) {\displaystyle (a,f(a))} , L

1424-627: A means of qualitatively assessing networks by varying the visual representation of their nodes and edges to reflect attributes of interest. Social network analysis has emerged as a key technique in modern sociology . It has also gained significant popularity in the following: anthropology , biology , demography , communication studies , economics , geography , history , information science , organizational studies , physics , political science , public health, social psychology , development studies , sociolinguistics , and computer science , education and distance education research, and

1513-458: A more accurate picture of collaborative learning experiences. A number of research studies have combined other types of analysis with SNA in the study of CSCL. This can be referred to as a multi-method approach or data triangulation , which will lead to an increase of evaluation reliability in CSCL studies. Linearization In mathematics , linearization ( British English : linearisation )

1602-428: A multiplexity of 2. Multiplexity has been associated with relationship strength and can also comprise overlap of positive and negative network ties. Mutuality/Reciprocity: The extent to which two actors reciprocate each other's friendship or other interaction. Network Closure : A measure of the completeness of relational triads. An individual's assumption of network closure (i.e. that their friends are also friends)

1691-420: A network relative to the total number possible. Distance: The minimum number of ties required to connect two particular actors, as popularized by Stanley Milgram 's small world experiment and the idea of 'six degrees of separation'. Structural holes: The absence of ties between two parts of a network. Finding and exploiting a structural hole can give an entrepreneur a competitive advantage. This concept

1780-586: A network, and the relatively small role played by an instructor in an asynchronous learning network. Although many studies have demonstrated the value of social network analysis within the computer-supported collaborative learning field, researchers have suggested that SNA by itself is not enough for achieving a full understanding of CSCL. The complexity of the interaction processes and the myriad sources of data make it difficult for SNA to provide an in-depth analysis of CSCL. Researchers indicate that SNA needs to be complemented with other methods of analysis to form

1869-476: A point p ( a , b ) {\displaystyle p(a,b)} is: The general equation for the linearization of a multivariable function f ( x ) {\displaystyle f(\mathbf {x} )} at a point p {\displaystyle \mathbf {p} } is: where x {\displaystyle \mathbf {x} } is the vector of variables, ∇ f {\displaystyle {\nabla f}}

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1958-464: A positive relationship (friendship, alliance, dating) and a negative edge between two nodes denotes a negative relationship (hatred, anger). Signed social network graphs can be used to predict the future evolution of the graph. In signed social networks , there is the concept of "balanced" and "unbalanced" cycles. A balanced cycle is defined as a cycle where the product of all the signs are positive. According to balance theory , balanced graphs represent

2047-412: A tool to understand behavior between individuals or organizations through their linkages on social media websites such as Twitter and Facebook . One of the most current methods of the application of SNA is to the study of computer-supported collaborative learning (CSCL). When applied to CSCL, SNA is used to help understand how learners collaborate in terms of amount, frequency, and length, as well as

2136-418: A wide range of applications and disciplines. Some common network analysis applications include data aggregation and mining , network propagation modeling, network modeling and sampling, user attribute and behavior analysis, community-maintained resource support, location-based interaction analysis, social sharing and filtering, recommender systems development, and link prediction and entity resolution. In

2225-637: Is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at θ = π {\displaystyle \theta =\pi } , corresponding to the pendulum being straight up: since sin ⁡ ( θ ) ≈ π − θ {\displaystyle \sin(\theta )\approx \pi -\theta } for θ ≈ π {\displaystyle \theta \approx \pi } . The solution to this problem involves hyperbolic sinusoids , and note that unlike

2314-431: Is always useful whether or not the resulting ordinary differential equation(s) is solvable. Another common (though less mathematical) tactic, often exploited in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem . For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in

2403-421: Is approximately 2 + 4.001 − 4 4 = 2.00025 {\displaystyle 2+{\frac {4.001-4}{4}}=2.00025} . The true value is close to 2.00024998, so the linearization approximation has a relative error of less than 1 millionth of a percent. The equation for the linearization of a function f ( x , y ) {\displaystyle f(x,y)} at

2492-399: Is called transitivity. Transitivity is an outcome of the individual or situational trait of Need for Cognitive Closure . Propinquity : The tendency for actors to have more ties with geographically close others. Bridge : An individual whose weak ties fill a structural hole , providing the only link between two individuals or clusters. It also includes the shortest route when a longer one

2581-502: Is carried out considering the network of words co-occurring in a text. In these networks, nodes are words and links among them are weighted based on their frequency of co-occurrence (within a specific maximum range). Social network analysis has also been applied to understanding online behavior by individuals, organizations, and between websites. Hyperlink analysis can be used to analyze the connections between websites or webpages to examine how information flows as individuals navigate

2670-428: Is collected. Social Networking Potential (SNP) is a numeric coefficient , derived through algorithms to represent both the size of an individual's social network and their ability to influence that network. SNP coefficients were first defined and used by Bob Gerstley in 2002. A closely related term is Alpha User , defined as a person with a high SNP. SNP coefficients have two primary functions: By calculating

2759-491: Is directly tied to every other individual, ' social circles ' if there is less stringency of direct contact, which is imprecise, or as structurally cohesive blocks if precision is wanted. Clustering coefficient : A measure of the likelihood that two associates of a node are associates. A higher clustering coefficient indicates a greater 'cliquishness'. Cohesion: The degree to which actors are connected directly to each other by cohesive bonds . Structural cohesion refers to

Nonlinear system - Misplaced Pages Continue

2848-402: Is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems , linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems . This method

2937-431: Is finding the common zeros of a set of several polynomials in several variables is a difficult problem for which elaborated algorithms have been designed, such as Gröbner base algorithms. For the general case of system of equations formed by equating to zero several differentiable functions , the main method is Newton's method and its variants. Generally they may provide a solution, but do not provide any information on

3026-478: Is near a known differentiable point. The most basic requisite is that L a ( a ) = f ( a ) {\displaystyle L_{a}(a)=f(a)} , where L a ( x ) {\displaystyle L_{a}(x)} is the linearization of f ( x ) {\displaystyle f(x)} at x = a {\displaystyle x=a} . The point-slope form of an equation forms an equation of

3115-537: Is now commonly available as a consumer tool (see the list of SNA software ). Social network analysis has its theoretical roots in the work of early sociologists such as Georg Simmel and Émile Durkheim , who wrote about the importance of studying patterns of relationships that connect social actors. Social scientists have used the concept of " social networks " since early in the 20th century to connote complex sets of relationships between members of social systems at all scales, from interpersonal to international. In

3204-552: Is one-dimensional heat transport with Dirichlet boundary conditions , the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions. First order ordinary differential equations are often exactly solvable by separation of variables , especially for autonomous equations. For example,

3293-408: Is the gradient , and p {\displaystyle \mathbf {p} } is the linearization point of interest . Linearization makes it possible to use tools for studying linear systems to analyze the behavior of a nonlinear function near a given point. The linearization of a function is the first order term of its Taylor expansion around the point of interest. For a system defined by

3382-841: Is the process of investigating social structures through the use of networks and graph theory . It characterizes networked structures in terms of nodes (individual actors, people, or things within the network) and the ties , edges , or links (relationships or interactions) that connect them. Examples of social structures commonly visualized through social network analysis include social media networks , meme proliferation, information circulation, friendship and acquaintance networks , business networks, knowledge networks, difficult working relationships, collaboration graphs , kinship , disease transmission , and sexual relationships . These networks are often visualized through sociograms in which nodes are represented as points and ties are represented as lines. These visualizations provide

3471-480: Is to use d θ / d t {\displaystyle d\theta /dt} as an integrating factor , which would eventually yield which is an implicit solution involving an elliptic integral . This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless C 0 = 2 {\displaystyle C_{0}=2} ). Another way to approach

3560-495: Is unfeasible due to a high risk of message distortion or delivery failure. Centrality : Centrality refers to a group of metrics that aim to quantify the "importance" or "influence" (in a variety of senses) of a particular node (or group) within a network. Examples of common methods of measuring "centrality" include betweenness centrality , closeness centrality , eigenvector centrality , alpha centrality , and degree centrality . Density : The proportion of direct ties in

3649-414: Is used in fields such as engineering , physics , economics , and ecology . Linearizations of a function are lines —usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function y = f ( x ) {\displaystyle y=f(x)} at any x = a {\displaystyle x=a} based on

Nonlinear system - Misplaced Pages Continue

3738-511: Is very general in that x {\displaystyle x} can be any sensible mathematical object (number, vector, function, etc.), and the function f ( x ) {\displaystyle f(x)} can literally be any mapping , including integration or differentiation with associated constraints (such as boundary values ). If f ( x ) {\displaystyle f(x)} contains differentiation with respect to x {\displaystyle x} ,

3827-515: Is very likely to morph into a balanced cycle, such as one where B only has a good relationship with A, and both A and B have a negative relationship with C. By using the concept of balanced and unbalanced cycles, the evolution of signed social network graphs can be predicted. Especially when using social network analysis as a tool for facilitating change, different approaches of participatory network mapping have proven useful. Here participants / interviewers provide network data by actually mapping out

3916-397: Is very nearly the value of the tangent line at the point ( x + h , L ( x + h ) ) {\displaystyle (x+h,L(x+h))} . The final equation for the linearization of a function at x = a {\displaystyle x=a} is: y = ( f ( a ) + f ′ ( a ) ( x −

4005-529: The Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology. One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle . A good example of this

4094-464: The September 11 attacks . Large textual corpora can be turned into networks and then analysed with the method of social network analysis. In these networks, the nodes are Social Actors, and the links are Actions. The extraction of these networks can be automated by using parsers. The resulting networks, which can contain thousands of nodes, are then analysed by using tools from network theory to identify

4183-518: The eigenvalues of the Jacobian matrix evaluated at a hyperbolic equilibrium point to determine the nature of that equilibrium. This is the content of the linearization theorem . For time-varying systems, the linearization requires additional justification. In microeconomics , decision rules may be approximated under the state-space approach to linearization. Under this approach, the Euler equations of

4272-467: The utility maximization problem are linearized around the stationary steady state. A unique solution to the resulting system of dynamic equations then is found. In mathematical optimization , cost functions and non-linear components within can be linearized in order to apply a linear solving method such as the Simplex algorithm . The optimized result is reached much more efficiently and is deterministic as

4361-585: The 1930s Jacob Moreno and Helen Jennings introduced basic analytical methods. In 1954, John Arundel Barnes started using the term systematically to denote patterns of ties, encompassing concepts traditionally used by the public and those used by social scientists: bounded groups (e.g., tribes, families) and social categories (e.g., gender, ethnicity). Starting in the 1970s, scholars such as Ronald Burt , Kathleen Carley , Mark Granovetter , David Krackhardt , Edward Laumann , Anatol Rapoport , Barry Wellman , Douglas R. White , and Harrison White expanded

4450-520: The R package SIENA (Simulation Investigation for Empirical Network Analyses), developed by Tom Snijders and colleagues. Longitudinal social network analysis became mainstream after the publication of a special issue of the Journal of Research on Adolescence in 2013, edited by René Veenstra and containing 15 empirical papers. Social network analysis is also used in intelligence, counter-intelligence and law enforcement activities. This technique allows

4539-704: The SNP of respondents and by targeting High SNP respondents, the strength and relevance of quantitative marketing research used to drive viral marketing strategies is enhanced. Variables used to calculate an individual's SNP include but are not limited to: participation in Social Networking activities, group memberships, leadership roles, recognition, publication/editing/contributing to non-electronic media, publication/editing/contributing to electronic media (websites, blogs), and frequency of past distribution of information within their network. The acronym "SNP" and some of

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4628-405: The accompanying diagram shows the tangent line of f ( x ) {\displaystyle f(x)} at x {\displaystyle x} . At f ( x + h ) {\displaystyle f(x+h)} , where h {\displaystyle h} is any small positive or negative value, f ( x + h ) {\displaystyle f(x+h)}

4717-476: The analysis is on the "connections" made among the participants – how they interact and communicate – as opposed to how each participant behaved on his or her own. There are several key terms associated with social network analysis research in computer-supported collaborative learning such as: density , centrality , indegree , outdegree , and sociogram . In-degree and out-degree variables are related to centrality. Researchers employ social network analysis in

4806-405: The analysts to map covert organizations such as an espionage ring, an organized crime family or a street gang. The National Security Agency (NSA) uses its electronic surveillance programs to generate the data needed to perform this type of analysis on terrorist cells and other networks deemed relevant to national security. The NSA looks up to three nodes deep during this network analysis. After

4895-424: The behavior of a nonlinear system is described in mathematics by a nonlinear system of equations , which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations ) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations,

4984-404: The case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation. Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations. A classic, extensively studied nonlinear problem is the dynamics of

5073-474: The context of social marketing intelligence was Communities Dominate Brands by Ahonen & Moore in 2005. In 2012, Nicola Greco ( UCL ) presents at TEDx the Social Networking Potential as a parallelism to the potential energy that users generate and companies should use, stating that "SNP is the new asset that every company should aim to have". Social network analysis is used extensively in

5162-553: The data is done through displaying nodes and ties in various layouts, and attributing colors, size and other advanced properties to nodes. Visual representations of networks may be a powerful method for conveying complex information, but care should be taken in interpreting node and graph properties from visual displays alone, as they may misrepresent structural properties better captured through quantitative analyses. Signed graphs can be used to illustrate good and bad relationships between humans. A positive edge between two nodes denotes

5251-582: The emergence of new data available about online social networks as well as "digital traces" regarding face-to-face networks. Computational SNA has been extensively used in research on study-abroad second language acquisition. Even in the study of literature, network analysis has been applied by Anheier, Gerhards and Romo, Wouter De Nooy, and Burgert Senekal. Indeed, social network analysis has found applications in various academic disciplines as well as practical contexts such as countering money laundering and terrorism . Size: The number of network members in

5340-641: The equation the linearized system can be written as where x 0 {\displaystyle \mathbf {x_{0}} } is the point of interest and D F ( x 0 , t ) {\displaystyle D\mathbf {F} (\mathbf {x_{0}} ,t)} is the x {\displaystyle \mathbf {x} } - Jacobian of F ( x , t ) {\displaystyle \mathbf {F} (\mathbf {x} ,t)} evaluated at x 0 {\displaystyle \mathbf {x_{0}} } . In stability analysis of autonomous systems , one can use

5429-580: The equation is not a linear function of u {\displaystyle u} and its derivatives. Note that if the u 2 {\displaystyle u^{2}} term were replaced with u {\displaystyle u} , the problem would be linear (the exponential decay problem). Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered. Common methods for

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5518-627: The equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it. As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations ( linearization ). This works well up to some accuracy and some range for

5607-402: The first algorithms developed to quantify an individual's social networking potential were described in the white paper "Advertising Research is Changing" (Gerstley, 2003) See Viral Marketing . The first book to discuss the commercial use of Alpha Users among mobile telecoms audiences was 3G Marketing by Ahonen, Kasper and Melkko in 2004. The first book to discuss Alpha Users more generally in

5696-525: The function f ( x ) = x {\displaystyle f(x)={\sqrt {x}}} at x {\displaystyle x} . Substituting in a = 4 {\displaystyle a=4} , the linearization at 4 is y = 2 + x − 4 4 {\displaystyle y=2+{\frac {x-4}{4}}} . In this case x = 4.001 {\displaystyle x=4.001} , so 4.001 {\displaystyle {\sqrt {4.001}}}

5785-501: The initial mapping of the social network is complete, analysis is performed to determine the structure of the network and determine, for example, the leaders within the network. This allows military or law enforcement assets to launch capture-or-kill decapitation attacks on the high-value targets in leadership positions to disrupt the functioning of the network. The NSA has been performing social network analysis on call detail records (CDRs), also known as metadata , since shortly after

5874-457: The input values, but some interesting phenomena such as solitons , chaos , and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of

5963-414: The key actors, the key communities or parties, and general properties such as robustness or structural stability of the overall network, or centrality of certain nodes. This automates the approach introduced by Quantitative Narrative Analysis, whereby subject-verb-object triplets are identified with pairs of actors linked by an action, or pairs formed by actor-object. In other approaches, textual analysis

6052-406: The minimum number of members who, if removed from a group, would disconnect the group. Visual representation of social networks is important to understand the network data and convey the result of the analysis. Numerous methods of visualization for data produced by social network analysis have been presented. Many of the analytic software have modules for network visualization. Exploration of

6141-406: The network (with pen and paper or digitally) during the data collection session. An example of a pen-and-paper network mapping approach, which also includes the collection of some actor attributes (perceived influence and goals of actors) is the * Net-map toolbox . One benefit of this approach is that it allows researchers to collect qualitative data and ask clarifying questions while the network data

6230-408: The nonlinear equation has u = 1 x + C {\displaystyle u={\frac {1}{x+C}}} as a general solution (and also the special solution u = 0 , {\displaystyle u=0,} corresponding to the limit of the general solution when C tends to infinity). The equation is nonlinear because it may be written as and the left-hand side of

6319-511: The number of solutions. A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences . Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and

6408-540: The output of a function near x = a {\displaystyle x=a} . For example, 4 = 2 {\displaystyle {\sqrt {4}}=2} . However, what would be a good approximation of 4.001 = 4 + .001 {\displaystyle {\sqrt {4.001}}={\sqrt {4+.001}}} ? For any given function y = f ( x ) {\displaystyle y=f(x)} , f ( x ) {\displaystyle f(x)} can be approximated if it

6497-471: The private sector, businesses use social network analysis to support activities such as customer interaction and analysis, information system development analysis, marketing, and business intelligence needs (see social media analytics ). Some public sector uses include development of leader engagement strategies, analysis of individual and group engagement and media use , and community-based problem solving . Large numbers of researchers worldwide examine

6586-535: The problem is to linearize any nonlinearity (the sine function term in this case) at the various points of interest through Taylor expansions . For example, the linearization at θ = 0 {\displaystyle \theta =0} , called the small angle approximation, is since sin ⁡ ( θ ) ≈ θ {\displaystyle \sin(\theta )\approx \theta } for θ ≈ 0 {\displaystyle \theta \approx 0} . This

6675-429: The qualitative analysis of nonlinear ordinary differential equations include: The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly linear). Sometimes, the equation may be transformed into one or more ordinary differential equations , as seen in separation of variables , which

6764-426: The quality, topic, and strategies of communication. Additionally, SNA can focus on specific aspects of the network connection, or the entire network as a whole. It uses graphical representations, written representations, and data representations to help examine the connections within a CSCL network. When applying SNA to a CSCL environment the interactions of the participants are treated as a social network. The focus of

6853-498: The related nonlinear system identification and analysis procedures. These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains. A system of differential equations is said to be nonlinear if it is not a system of linear equations . Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are

6942-420: The result will be a differential equation . A nonlinear system of equations consists of a set of equations in several variables such that at least one of them is not a linear equation . For a single equation of the form f ( x ) = 0 , {\displaystyle f(x)=0,} many methods have been designed; see Root-finding algorithm . In the case where f is a polynomial , one has

7031-645: The small angle approximation, this approximation is unstable, meaning that | θ | {\displaystyle |\theta |} will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state. One more interesting linearization is possible around θ = π / 2 {\displaystyle \theta =\pi /2} , around which sin ⁡ ( θ ) ≈ 1 {\displaystyle \sin(\theta )\approx 1} : This corresponds to

7120-399: The social networks of children and adolescents. In questionnaires, they list all classmates, students in the same grade, or schoolmates, asking: "who are your best friends?". Students may sometimes nominate as many peers as they wish; other times, the number of nominations is limited. Social network researchers have investigated similarities in friendship networks. The similarity between friends

7209-474: The strategies used to communicate within the group. Some authors also suggest that SNA provides a method of easily analyzing changes in participatory patterns of members over time. A number of research studies have applied SNA to CSCL across a variety of contexts. The findings include the correlation between a network's density and the teacher's presence, a greater regard for the recommendations of "central" participants, infrequency of cross-gender interaction in

7298-417: The study of computer-supported collaborative learning in part due to the unique capabilities it offers. This particular method allows the study of interaction patterns within a networked learning community and can help illustrate the extent of the participants' interactions with the other members of the group. The graphics created using SNA tools provide visualizations of the connections among participants and

7387-482: The superposition principle An equation written as is called linear if f ( x ) {\displaystyle f(x)} is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if C = 0 {\displaystyle C=0} and f ( x ) {\displaystyle f(x)} is a homogeneous function . The definition f ( x ) = C {\displaystyle f(x)=C}

7476-412: The system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology. Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others: Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals. In mathematics ,

7565-478: The use of systematic social network analysis. Beginning in the late 1990s, social network analysis experienced a further resurgence with work by sociologists, political scientists, economists, computer scientists, and physicists such as Duncan J. Watts , Albert-László Barabási , Peter Bearman , Nicholas A. Christakis , James H. Fowler , Mark Newman , Matthew Jackson , Jon Kleinberg , and others, developing and applying new models and methods, prompted in part by

7654-491: The value and slope of the function at x = b {\displaystyle x=b} , given that f ( x ) {\displaystyle f(x)} is differentiable on [ a , b ] {\displaystyle [a,b]} (or [ b , a ] {\displaystyle [b,a]} ) and that a {\displaystyle a} is close to b {\displaystyle b} . In short, linearization approximates

7743-545: The web. The connections between organizations has been analyzed via hyperlink analysis to examine which organizations within an issue community. Another concept that has emerged from this connection between social network theory and the Internet is the concept of netocracy , where several authors have emerged studying the correlation between the extended use of online social networks, and changes in social power dynamics. Social network analysis has been applied to social media as

7832-419: Was developed by sociologist Ronald Burt , and is sometimes referred to as an alternate conception of social capital. Tie Strength: Defined by the linear combination of time, emotional intensity, intimacy and reciprocity (i.e. mutuality). Strong ties are associated with homophily, propinquity and transitivity, while weak ties are associated with bridges. Groups are identified as ' cliques ' if every individual

7921-691: Was established as far back as classical antiquity. Resemblance is an important basis for the survival of friendships. Similarity in characteristics, attitudes, or behaviors means that friends understand each other more quickly, have common interests to talk about, know better where they stand with each other, and have more trust in each other. As a result, such relationships are more stable and valuable. Moreover, looking more alike makes young people more confident and strengthens them in developing their identity. Similarity in behavior can result from two processes: selection and influence. These two processes can be distinguished using longitudinal social network analysis in

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