复旦大学：《网络科学导论 Introduction to Network Science》教学成果_学生翻转课堂展示1 INTRODUCTION TO COMPLEX NETWORKS

INTRODUCTION TO COMPLEX NETWORKS NN-CLASS PRESENTATION GROUP1 How social networks predict epidemics' by Nicholas Christakis

INTRODUCTION TO COMPLEX NETWORKS

Our mission 308"&6’30"the“S”- shape curve; Explain the difference between the two curves (yellow line and red dot); why the yellow curve shifts to left side?

Our mission 3'08'' & 6'30'' the “S”-shape curve; Explain the difference between the two curves (yellow line and red dot) ; why the yellow curve shifts to left side?

CONTENTS PART ONE PART THREE Why the yellow curve shifts to what is an S"-shape curve? left side? PART TWO Differences of two curve

CONTENTS PART ONE PART TWO what is an “S”-shape curve? Differences of two curve. PART THREE Why the yellow curve shifts to left side?

01 what is an“S”- shape curve?

what is an “S”-shape curve?

what is an"S-shape"curve? A sigmoid function is a mathematical function having a characteristic s-shaped curve or sigmoid curve

what is an “S-shape”curve? A sigmoid function is a mathematical function having a characteristic "S"-shaped curve or sigmoid curve

s Curves ErDqrersivesuasesU “Ss- shape" curve A wide variety of sigmoid functions have been used as the activation function of artificial neurons. Sigmoid curves are also common in statistics as cumulative distribution functions(which go from 0 to 1), such as the Virtuous cycle what you kno integrals of the logistic field of the job, how distribution. the norma things can be growth distribution and student's t Vicious cycle distribution probability density functions o2012 Juan C, Mendez and whitney Johnson, all rights reserved

“S-shape”curve A wide variety of sigmoid functions have been used as the activation function of artificial neurons. Sigmoid curves are also common in statistics as cumulative distribution functions (which go from 0 to 1), such as the integrals of the logistic distribution, the normal distribution, and Student's t￾distribution probability density functions

“Ss- shape" curve Logistic function Error function 1 f(a) f(r)=erf(a)=2 1+e-x / hyperbolic tangent Generalised logistic function f(a)=tanh a f(x)=(1+e2)a,a>0 Smoothstep function arctangent function -1 c≤1 f(a)=arctan a (1-2)dn) ∫(1-u2) N≥ sgn(a) c≥1 Gudermannian function Specific algebraic functions f(a)=gd(e) =/ cosh t f(x)= Error function f(a)=erf(c 2 /

“S-shape”curve

S-shape"curve Often sigmoid function refers to the special case of the logistic function shown in the first figure and √1+x defined by the formula tanh(r) arctan(2) S(a) 1+e-x 一gd(票x) 0.5 1+|x Other examples of similar shapes include the gompertz curve (used in modeling systems that 0.5 saturate at large values of x) and 0.5 the ogee curve(used in the of some dams). Sigmoid functions have have domain of all real numbers. with return value monotonically increasing most often from o to 1 or alternative 1 to 1

“ S -shape ”curve Often, sigmoid function refers to the special case of the logistic function shown in the first figure and defined by the formula Other examples of similar shapes include the Gompertz curve (used in modeling systems that saturate at large values of x) and the ogee curve (used in the spillway of some dams). Sigmoid functions have have domain of all real numbers, with return value monotonically increasing most often from 0 to 1 or alternatively from −1 to 1

“Ss- shape" curve a logistic function or logistic curve is a common"S" shape(sigmoid curve), with equation L where 1+e-k(x-20) 0.5 e=the natural logarithm base(also known as Euler's number) x0= the x-value of the sigmoid's midpoint, L= the curve's maximum value. and k= the steepness of the curve The logistic function finds applications in a range of fields, including artificial neural networks, biology (especially ecology ), biomathematics, chemistry, demography, economics, geoscience, mathematical psychology, probability, sociology, political science, linguistics, and statistics

A logistic function or logistic curve is a common "S" shape (sigmoid curve), with equation: where e = the natural logarithm base (also known as Euler's number), x0 = the x-value of the sigmoid's midpoint, L = the curve's maximum value, and k = the steepness of the curve. The logistic function finds applications in a range of fields, including artificial neural networks, biology (especially ecology), biomathematics, chemistry, demography, economics, geoscience, mathematical psychology, probability, sociology, political science, linguistics, and statistics. “S-shape”curve

“Ss- shape" curve Diffusion of innovations is a theory that 100 seeks to explain how, why, and at what rate new ideas and technology spread Rogers proposes that four main elements influence the spread of a new idea: the innovation itself communication channels, time, and a social system this process relies heavily on human capital 0 Innovators Early Early Laggards The innovation must be widely adopted 2.5 Adopters Majority Majority 13.5%34% 34% in order to self-sustain Within the rate of adoption there is a point at which an innovation reaches critical mass

Diffusion of innovations is a theory that seeks to explain how, why, and at what rate new ideas and technology spread. Rogers proposes that four main elements influence the spread of a new idea: the innovation itself, communication channels, time, and a social system. This process relies heavily on human capital. The innovation must be widely adopted in order to self-sustain. Within the rate of adoption, there is a point at which an innovation reaches critical mass. “S-shape”curve