CSCI4190_01

一位local女教授，既Jimmy之后第二位不准手机铃声的教授。。。

第一节课是拿一个英文的科普视频 放一段讲一段。。。太真实了吧？。。。

C1 Strong and Weak ties 17

Basic definition

Path

• A path in a graph is a sequence of nodes with the property that each consecutive pair in the sequence is connected by an edge.
• A path can have repeat nodes;
• Path without repeat nodes is a simple path

Cycle

A cycle is a path with at least three edges, in which the first and last nodes are the same, but otherwise all nodes are distinct.

Bridge

the edge joining two nodes if deleting that edge would cause the two nodes in two different components.

Local Bridge

Bridge is rare

the edge joining nodes A and B, if its endpoints A and B have no friends in common.

Span of a local bridge:

the distance between the end points if that edge were deleted.(original:1)

Embeddedness(edge)

Embeddedness of an edge in a network is the number of common neighbors the two endpoints have.

Embeddedness(Local bridge) =0

Structure hole (几个不同contacts的中间人)

• access information originating in multiple, non-interacting parts of the network.
• have opportunities for innovations arose from the unexpected synthesis of multiple ideas, each of them on their own perhaps well-known, but well-known in distinct and unrelated bodies of expertise.
• have an opportunity for a kind of social “gate-keeping”; a source of power in the organization.

Connected

A graph is connected if for every pair of nodes, there is a path between them

Traffic in a Network

For each pair of nodes A and B: 1 unit of traffic “flow” from A to B if A and B belong to same component. the flow divides itself evenly along all the possible shortest paths from A to B

Betweenness

• Edge: total amount of flow that the edge carries, counting flow between all pairs of nodes using this edge.
• Node: total amount of flow that the node carries, when a unit of flow between each pair of nodes is divided up evenly over shortest paths

Triadic closure

朋友的朋友更容易成为朋友

cluster coefficient(a)

a的朋友们也互相是朋友的概率

Strong Triadic Closure Property

如果a 和bc关系很好，那么bc会成为朋友。

如果不是(no matter strong or weak)，violate the property

Strength of weak ties

neighborhood overlap (edge)

no of nodes who are neighbors of both A and B**/** no of nodes who are neighbors of at least one of A of B.

• Neighborhood overlap increases with the strength
• Local level : tie strength increases with neighborhood
• Global level : weak ties serve to link together different tightly-knit communities ?

Weak ties provide the more crucial(关键) connective structure
• for holding together disparate communities and
• for keeping the global structure of the giant components intact.

Graph patitioning

division

移除最可能连接两大块的（betweenness最大值）

agglomerative(凝聚)

把nodes that are likely to belong to the same region merge起来

Girvan-Newman Method

• Find the edge of highest betweenness and remove these edges from the graph.
• Recalculate all betweennesses, again remove the edge or edges of highest betweenness.

C2 Networks in their Surrounding Contexts

Homophily

The tendency of individuals to associate and bond with similar others

Friendship that forms because two people

• have a common friend (intrinsic network structure, triadic closure)
• attend the same school or work for the same company (contextual factor)情境因素

Measuring Homophily

Homophily Test: If the fraction of cross-gender edges is significantly less than 2pq(p = fraction of males,q = fraction of females), then there is evidence for homophily.

The edge is heterogeneous(异质) if two end nodes do not share the same characteristic

Mechanisms Underlying Homophily

Selection

select friends with similar characteristic. tends to drive the network towards smaller clusters of like-minded individuals (balkanization)

• individual characteristics drive the formation of links (邻居)
• involves immutable(不变) characteristics (determined at birth)
• 物以类聚。

Socialization and social influence

modify their behaviors to bring them more closely into alignment with the behaviors of their friends.

– reverse of selection
– Involves mutable characteristics
– 近朱者赤， 近墨者黑/lies down with dogs, rises with fleas

Social influence

can produce network-wide uniformity, as new behavior spreads across the links.

Affiliation(联系) Networks 17

Include the contextual factors (the set of activities) into the network. expressed as a bipartite(even)
graph.

• node: person or focus(activity)
• edge: (person,focus)

• A graph is bipartite if its nodes can be divided into two sets in such a way that every edge connects a node in one set to a node in the other set

• Two people sharing the same focus → an opportunity to become friends.

Extension:

• Node: person/focus
• edge: (person,focus) or (person person)

3 Closure Processes

• Triadic closure: 朋友的朋友更容易成为朋友
• Membership closure(social influence): 朋友参加的活动更容易参加
• Focal closure (selction): 一个活动中的两人更容易成为朋友

Tracking Link Formation

1. – For each k, identify all pairs of nodes who have exactly k friends in common, but who are not directly connected by an edge.
2. (Difference time with step1 ) T (k) = the fraction of these pairs that have formed an edge.

• Triadic Closure： HIGH comon friends -> higher fraction
• Focal closure: high -> not so high

The Schelling Model

How global patterns of spatial segregation can arise from the effect of homophily operating at a local level (Thomas Schelling)

• Each agent(population) wants to have at least t agents of its own type as neighbors
• Unsatisfied agents: – fewer than t neighbors of the same type as itself and move to a new cell

homophily draws people together along immutable characteristics

The Schelling model creates a natural tendency for mutable characteristics

C3 Positive and Negative Relationships

clique/complete graph: graph which an edge connecting each pair of nodes

• Balanced Structure(triangle)
  • Three mutual(相互的) friends
  • A third mutual enemy
• Unbalanced Structure(triangle)
  • A,和BC 玩的好。BC仇敌。
  • All are enermies

A complete graph is structurally balanced if its everyone triangle is balanced.

Balance Theorem

If a labeled complete graph is balanced, then

• either all pairs of nodes are friends,
• or else the nodes can be divided into two groups, X and Y ,

Weak Structural Balance Property:

There is no set of three nodes such that the edges among them consist of exactly two positive edges and one negative edge . ( all are enermy are possible)

Structural Balance in Arbitrary (Non-Complete) Networks

A signed graph is balanced if and only if it contains no cycle with an odd number of negative edges.

Searching for balanced division

• find the connected components using positive edges. Declare each component to be a supernode.
• the supernodes form the reduced graph (negative edges only)
• BFS: edges connect two nodes in the same layer: a cycle with odd number of nodes

Generalizing the Definition of Structural Balance

Approximately Balance Theorem

If at least 99.9% (1-$\epsilon$)of all triangles in a labeled complete graph are balanced, then

• either there is a set consisting of at least 90%(1-$\epsilon^{1/3}$)of the nodes in which at least 90% of all pairs are friends
• or else the nodes can be divided into two groups, X and Y ,
  • at least 90% of the pairs in X like each other
  • at least 90% of the pairs in Y like each other, and
  • at least 90% of the pairs with one end in X and the other end in Y are enemies.

C4 Information cascade 17

Reasons why individual might imitate the behavior of others

• – Informational effect :

  • The behavior of others conveys information about what they know
  • Peer pressure
  • Imitate what others (e.g. idols) are doing

• Network effect (direct benefit effect) :(C5)

  you incur an explicit benefit when you align behavior with the behavior of others.

Bayes Rule

$P(A|B) = \frac{A\cap B}{B}=\frac{P(A)P(B|A)}{P(B)}$, 一般 ，pa, p(b|a)题目都给了。问题是求pb：

若A好算:$p(b) = P(A)P(B|A)+P(A^c)P(B|A^C)$

A Simple General Cascade Model 17

• States: G(Good) with probability p, and B with 1-p

• Accepting select which state

• Private information

• Signal H(High) before making decision: accepting is a good idea, L(Low) : bad idea

  P(H|G) = q > 0.5, P(L|G) = 1-q < 0.5

• Payoff: 0 reject. Vg: accepting a good option Vb: accept a bad option

Suppose one receives a H signal

• Expected payoff = Vg * P(G|H) + Vb *P(B|H)
• P(G|H) = $\frac{p(G)p(H|G)}{p(G)p(H|G)+p(B)p(H|B)}= \frac{pq}{pq+(1-p)(1-q)} >p$

Multiple signals

S = a high signals + b low signals

• posterior probability Pr [G | S] is greater than the prior Pr [G] (p) when a > b;

C5 Network Effects

Externalities（和外部坏境联系的场景 如 交通事故？party）

situation that the welfare of an individual is affected by the actions of other individuals

• Positive: welfare increases when others are joining
• Negative: 相反。

Without network effect, only consider Reservation price

the max amount one is willing to pay for one unit of the good. If consumer x (0 ≤ x ≤ 1) has a higher reservation price r(x) than consumer y, then x < y.

**r: strictly decreasing **

p’: production cost of one unit of the good (the minimum price a producer is willing to accept to sell a good)

x’: equilibrium quantity of the good so that r(x’) = p’

The Economy with Network Effects

• intrinsic interest. r(x). eg. r(1 ) = 0

• the no. of other people using the good

  • ( z: fraction of people in use).
  • f(z) measures the benefit to each consumer from those who use the good.f(z) is increasing in z, f(0) = 0

• reservation price of consumer x = r(x)f(z) . here z is the fraction expected by x

• self-fulfilling expectations equilibrium p’ = r(z)f(z)

Stability, Instability, and Tipping Points

• 0 < z < z’ 买家不想买。“downward pressure” on the consumption of the good : this would push demand downward.
• z’ < z < z” 买家想买。 “upward pressure” on the consumption of the
  good: this would drive demand upward
• z” < z < 1 “downward pressure”
• z’ : critical point/tipping point

A Dynamic View of the Market

z^: fraction of population who buy the product (actual buyer)

r(z^)f(z) = p = >

z^ = $r^{-1}(p/f(z))$ = g(z)

For goods with network effects, the equilibria are typically not social optimal.(社会所有人的总利益)

Being the first to reach this tipping point is more important than being “best”.

Mixing Individual Effects with Population-Level Effects

f(Z) = 1 + az^2

r(X) = 1-x

C6 Power Laws and Rich-Get-Richer Models 17

Central Limit Theorem says that if we take any sequence of small independent random quantities, then in the limit their sum (or average) will be distributed according to the normal distribution.

A function that decreases as k to some fixed power, such as 1/k 2(fraction) is called a power law

Rich-Get-Richer Models

Model the growth of the popularity.

• Created pages in order : 1, 2, 3, . . . , j.
• When j is created, With probability p, page j, chooses a page i uniformly at random from among all earlier pages, and creates a link to this page i.
• With probability 1-p, page j creates a link to the page that i links to. (i also uniformly random distributed)
  • new version:with probability 1 − p, page j chooses a page l with probability
    proportional to l ’s current number of in-links, and creates a link to l.

preferential attachment: : “preferentially” to pages that already have high popularity. (人们更愿意打开已经火了的页面,copy别人的)

No. of in-links of node j = $x_j = p/q[(t/j)^q-1]$ . t: simulated steps. q = 1-p

At time t, we have nodes x1, x2, … xt. The fraction of nodes with at least k links is

$[\frac{q}{p}k +1]^{-1/q}$

f(k) = the fraction of nodes with exactly k links = $1/p [1+\frac{1-p}{p}k]^{-(1+\frac{1}{1-p})}$. when p -0, power law exponent is 2.

The long tail

a small set of items that are enormously popular

Zipf’s Law : the frequency of the j th most common word in English (or most other widespread human languages) is proportional to 1/j.

C6 Cascading behavior in networks 17

Structure of the network and how individuals are influenced by their particular network neighbors (instead of everyone else in C5)

Why an innovation can fail to spread through a population ?

• tightly-knit(严密) social community;
• complexity
• observability observe别人也在用
• trialability 可试验性。（一步一步慢慢来）
• compatibility with social system

the set of initial adopters causes a complete cascade at threshold q(大于百分之多少才换）.

cluster of density p

cluster of density p = a set of nodes such that each node in the set has at least a p fraction of its network neighbors in the set.

cluster of density 1:

The Relationship between Clusters and Cascades

• If the remaining network contains a cluster of density greater than 1 − q, then the set of initial adopters will not cause a complete cascade.(cluster 是cascade的阻碍 必要条件)
• Moreover, whenever a set of initial adopters does not cause a complete cascade with threshold q, the remaining network must contain a cluster of density greater than 1 − q（充分条件）

Extensions of the Basic Cascade Model 17

Heterogeneous Thresholds： each node has a specific payoff and hence threshold

A blocking cluster in the network is a set of nodes for which each node v has more than a 1 − qv fraction of its friends also in the set.

cascade capacity of the network: 有限初始集能造成complete cascade的最大threshold

最大值：1/2

proof number of AB edges is decreasing when switching w from B to A

payoff from choosing A = a’（两边都是A/AB) 则double a
payoff from choosing B = b’
payoff from choosing AB= a’+b’-c

C7 Small World Phenomenon

Watts-Strogatz model

Two type of links:

Homophily(和其他r个小格内点相连的link)and

Weak ties(剩余的k 个远link).

• d(v, w) :the number of grid steps between nodes v and w
• Probability of an edge is proportional to d(v, w)^-q (Small q, more long distances edges)

time: at most a2(log n)^2

A Rough Calculation Motivating the Inverse-Square Network

The probability that a random edge from center point v links into any node in this group is approximately independent of the value of d. ( in d - 2d circles)

Rank-Based Friendship

rank(w) = the number of other nodes that are closer to v than w is. (比w 还离v近一点的)

social distance

dist(v, w) = the social distance between nodes v and w = the size of the smallest focus that contains both v and w

A link between each pair of nodes v and w with probability proportional to dist(v,w)^-p

Efficient decentralized search when p = 1. 17

C8 Epidemics

•The basic reproductive number ( R0) = the expected number of new cases of the disease caused by a single individual = p(probability)k(new meet num).

Percolation : static view of the model. edge v-w is open is w is infected by v

SIR . susceptible infectious removed

SIS susceptible infectious back_TO_susceptile

SIRS susceptible infectious 免疫期 back_TO_susceptile

transient contacts — contact networks in which each edge is annotated with the period of time during which it existed

Analysis of Branching Processes

qn = the probability that the epidemic survives for at least n waves

q∗ = qn when n is infinite = the probability that the epidemic persists indefinitely.

Wright-Fisher model

1. 每一代 N个人
2. 每个孩子都只有一个家长。一个家长可以有多个孩子也可能没有。

consider the time that k个个体是一个祖先

C9 Link Analysis and Web Search

• Ranking of web pages

Information Retrieval

1960s: Search repositories of newspaper articles, scientific papers, patents, legal abstracts, and other document collections in response to keyword queries.

vote by link 缺点：offtopic, criticism, advertisement

each page有两种score

authorities 权威网站 更新： 所有指向他的page的hub score和

hub 首页 更新 类似

initial are score = 1
select k steps
for i in k:
	update auth
	update hub
normalize hub and auth

(A-\lamda I)x = 0

\lamda = eigenvalue

x = eigenvector

pagerank

每个node指出k个，其value平均分k，到新的node。

最后把收到的value sum。

Scaling Factor

sum之后乘以一个factor 。 然后把factor多出来的那一部分均匀分给每个人。

Random walks

随便选一个。然后继续选out-going的。和pagerank思路一样。停在哪个page的概率和pagerank概率一样。

• 有相关text的超链接权重更高。
• 用户点击数据 更能反映是否相关。

C10 Sponsored Search Market

SPONSORED SEARCH AD:关键词

branding ad ：直接投放在网站上

contextual ad： 根据用户特征投放广告

拍卖

• The Second Price Sealed Auction 由出价最高的买家获得物品，但他只需要支付所有投标者中的第二高价。

click-through rates：（r_i) The number of clicks per hour it will receive

revenue(收入) v_j per click of advertiser j,

ri*vj = benifit

Matching Market

if the search engine knew all the advertisers’ valuations for clicks

For buyer:

$v_{ij}$ = valuation for item i by buyer j

$p_i$ = price announced by the seller

v- p 就是买家收益。

Market clearing prices 所有买家选到最赚的而且不冲突

Constructing the set of market clearing price

price先重置为0

将最多人要买的slot price 加一

重新计算

重新加一

直到可以market clearing

Vickrey-Clarke-Groves (VCG) mechanism

the advertisers’ valuations are not known

算没有 A 后 BC 收益。比如600

有A后BC收益 比如 200

那么A 需要支付 600-200

GSP

bi bids per click

GSP charges a cumulative price of ri bi+1 for slot i.

Nash equilibrium

The pair of strategies (S; T) is a Nash equilibrium if S is a best response to T, and T is a best response to S.

Best response: $P_1(S,T)>= P_1(S’,T)$ T: stratage of 2

STrict: 没有等于。

Dominant strategy： a strategy that is a best response to every strategy of Player 2.

A strictly dominant strategy .