### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# 589 Class Note for PHYS 597A with Professor Albert at PSU

### View Full Document

## 20

## 0

## Popular in Course

## Popular in Department

This 17 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 20 views.

## Similar to Course at Penn State

## Reviews for 589 Class Note for PHYS 597A with Professor Albert at PSU

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 02/06/15

Idea generate random graphs with a powerlaw degree distribution Fixed N Pkk397 Configuration model choose a degree sequence NkN Pk give the nodes k stubs according to Nk connect stubs randomly Ex 1 Construct a graph with 10 nodes and degree sequence N14 N23 N32 N41 What is a necessary condition for the graph construction Scalefree random graphs use the degree distribution as an input Fixed N Pk Akquot k lt K Network assembly random edges but enforcing the right Pk The properties of the graph depend on 7 1 Kk y1 A lt k gt 21 wa 2f kquot The number of edges increases as 7 decreases Theory of scalefree random graphs The theory of scalefree random graphs is in many ways similar to that of ErdosR nyi random graphs Graph properties depend on 7 giant cluster 7 S 347 connected 7 g 2 W Aiello F Chung L Lu Proc 32th ACM Theor Comp 171 2000 M E J Newman S H Strogatz D J Watts Phys Rev E 64 026118 2001 Average path length of scalefree random graphs Network N Pkzk7 for kSK 1nNB A M E J Newman S H Strogatz D J Watts Phys Rev E 64 026118 2001 Prediction 1 1 ABf7K I I I Vfood webs ltgt Elmetabolic networks 15 Olnternet Acollaboration networks 0 OWN qualitative agreement co io A worse than a random N A 2 A0 graph 5 D o 9 10 11392 10 16 10 10 Clustering coefficient of scalefree random graphs Cltkgtltk2gt ltkgt22 N ltkgt2 The second term depends on the variance of the degree distribution Pk z kquot C z N sy oy l For 7 lt 73 C increases with N M E J Newman SIAM Review 45 167 2003 We need to uncover the mechanisms responsible for the scalefree Pk random graphs smallworld networks Stat39c scalefree random graphs Real networks continuously change random graphs Homogeneous smallworld networks Scalefree degree distribution the nodes are not equivalent We need to model the evolution of networks not just their topology Model the network assembly and evolution Barabasi Albert model Barabasi Albert Science 286 509 2000 Start with a small seed of mo nodes and m0m012 edges growth a node and m edges added at every step ki 23 k J J preferential attachment Hkl General properties of the network nr of nodes N m0 t m m 1 onrofedges nmt 10392 2n g average degree ltkgt W gt 2111 l 10 degree distribution 1039 PM Ak393 Lgrvs I Although the network grows the degree distribution becomes stationary Analytical determination of Pk The degree of old nodes increases by acquiring new edges The probability of this process is mHklm ki z k 21 J J 1 with prob Degree increase Aki 0 otherwise Choices follow the increase in the number of nodes with degree k rate equation approach follow the increase in time in kcontinuum theory Rate equation approach P Krapivsky S Redner F Leyvraz Phys Rev Lett 85 4629 2000 Change in average number of nodes with degree k M k 1Nk1t kat m dt 2ka t Asymptotically PkiEgNktt 6km Pk l f0rkgtm Pm 2 HM 2mm1 Nk3 f0rkm kk1k2 m2 Stationary power law with an exponent 7 3 Continuum theory 139 1 with prob Degree increase M 21 l 0 otherwise Assume that there is a continuous rate of change of k dkl N k 1 dt z t Initial condition kltl m t time when node iwas added to the network Solve for k kittl mg Continuum theory kitgtti n1tZ Calculate the cumulative probability that klttl lt k 2 m k2 mzt PkitltkPt gt 1 It lt k2 I All t values are equiprobable pa mg t mzt mzt k2 k2m0t Ptl lt 6Pkltltk 2m2t 1 N k3 6k m0 t k3 Degree distribution Pk Stationary power law with an exponent 7 3 Simulations and theory agree u u u 10 my x ocuuu 0 N if XA imam Eu 1 3 Mu mmmuuu u cfwuu vac m m I 1 a a a D a x Ex 1 Start from a seed of two nodes connected by an edge At each step add a new node and connect it by a single edge to a preexisting node Let the probability of selection be directly proportional with the degree of the old node Continue growing the graph until you reach 15 nodes Describe the graph average degree degree distribution clustering coefficient maximum distance Ex 2 How will the properties of the graph change if at each step a new node and two new edges are added Model A grovv l re 39k uniform AnltkL at m0t 1 1039 kiltrm1nltm1 mQ 1 Pltk3explt 5equot m m a Bmle mm O D 3 laid 3 9a quot1 o a a 5 a 00 E 0 A Q 0 M O to 50 100 Model B M preferential attachment aki I N 1 10 A139Iki t at N N 12t N ftt N 101 2N1 tCt2N 1i NN 2 t 10393 a Pk power law initially gt Gaussian 10 a 1oa Scalefree algorithm with directed edges New edges are directed from the new to the old nodes kl t 2 for i gt m0 t0 t2 t3 4 kl varies k m akr at t mt mt 1 Pk t ltk 1 P k k392 km0t S m m tk2 The degree exponent ofthe directed scalefree network is 2

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "Selling my MCAT study guides and notes has been a great source of side revenue while I'm in school. Some months I'm making over $500! Plus, it makes me happy knowing that I'm helping future med students with their MCAT."

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.