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The Networked Nature of Society Networked Life CSE 112 Spring 2004 Prof. Michael Kearns Course News and Notes 1/20 • Course mailing list: [email protected] • Updates to course web site • Recitation sections: – Tuesday (today) 6-7 PM; room announced later today – Wednesday 5-6 in Towne 313 Course News and Notes 1/15 • Please give last-names exercise to Nick now • If you missed the first lecture, please get the handouts • Extra credit for identifying papers, web sites, demos or other material related to class, if it gets used • Prof. Kearns office hours today 10:30-11:30 • Tuesday recitation may be moved to 6 PM; stand by • Please start reading “The Tipping Point” – required chapters: 1,2,4,5 What is a Network? • • • • • • • A collection of individual or atomic entities Referred to as nodes or vertices Collection of links or edges between vertices Links represent pairwise relationships Links can be directed or undirected Network: entire collection of nodes and links Extremely general, but not everything: – actors appearing in the same film – lose information by pairwise representation • We will be interested in properties of networks – often statistical properties of families of networks Some Definitions • Network size: total number of vertices (denoted N) • Maximum number of edges: N(N-1)/2 ~ N^2/2 • Distance between vertices u and v: – number of edges on the shortest path from u to v – can consider directed or undirected cases – infinite if there is no path from u to v • Diameter of a network: – worst-case diameter: largest distance between a pair – average-case diameter: average distance • If the distance between all pairs is finite, we say the network is connected; else it has multiple components • Degree of vertex u: number of edges Examples of Networks “Real World” Social Networks • Example: Acquaintanceship networks – – – – vertices: people in the world links: have met in person and know last names hard to measure let’s do our own Gladwell estimate • Example: scientific collaboration – – – – – – vertices: math and computer science researchers links: between coauthors on a published paper Erdos numbers : distance to Paul Erdos Erdos was definitely a hub or connector; had 507 coauthors MK’s Erdos number is 3, via Mansour Alon Erdos how do we navigate in such networks? Online Social Networks • A very recent example: Friendster – vertices: subscribers to www.friendster.com – links: created via deliberate invitation – Here’s an interesting visualization by one user • Older example: social interaction in LambdaMOO – – – – LambdaMOO: chat environment with “emotes” or verbs vertices: LambdaMOO users links: defined by chat and verb exchange could also examine “friend” and “foe” sub-networks Update: MK’s Friendster NW, 1/19/03 • If you didn’t get my email invite, let me know – send mail to [email protected] • • • • • • • Number of friends (direct links): 8 NW size (<= 4 hops): 29,901 13^4 ~ 29,000 But let’s look at the degree distribution So a random connectivity pattern is not a good fit What is??? Another interesting online social NW: [thanks Albert Ip!] – AOL IM Buddyzoo Content Networks • Example: document similarity – – – – vertices: documents on the web links: defined by document similarity (e.g. Google) here’s a very nice visualization not the web graph, but an overlay content network • Of course, every good scandal needs a network – vertices: CEOs, spies, stock brokers, other shifty characters – links: co-occurrence in the same article • Then there are conceptual networks – vertices: concepts to be discussed in NW Life – links: arbitrarily determined by Prof. Kearns • Update: here are two more examples [thanks Hanna Wallach!] – a thesaurus defines a network – so do the interactions in a mailing list Business and Economic Networks • Example: eBay bidding – vertices: eBay users – links: represent bidder-seller or buyer-seller – fraud detection: bidding rings • Example: corporate boards – vertices: corporations – links: between companies that share a board member • Example: corporate partnerships – vertices: corporations – links: represent formal joint ventures • Example: goods exchange networks – vertices: buyers and sellers of commodities – links: represent “permissible” transactions Physical Networks • Example: the Internet – – – – – vertices: Internet routers links: physical connections vertices: Autonomous Systems (e.g. ISPs) links: represent peering agreements latter example is both physical and business network • Compare to more traditional data networks • Example: the U.S. power grid – vertices: control stations on the power grid – links: high-voltage transmission lines – August 2003 blackout: classic example of interdependence Biological Networks • Example: the human brain – – – – – – – vertices: neuronal cells links: axons connecting cells links carry action potentials computation: threshold behavior N ~ 100 billion typical degree ~ sqrt(N) we’ll return to this in a moment… Network Statics • Emphasize purely structural properties – size, diameter, connectivity, degree distribution, etc. – may examine statistics across many networks – will also use the term topology to refer to structure • Structure can reveal: – – – – community “important” vertices, centrality, etc. robustness and vulnerabilities can also impose constraints on dynamics • Less emphasis on what actually occurs on network – web pages are linked, but people surf the web – buyers and sellers exchange goods and cash – friends are connected, but have specific interactions Network Dynamics • Emphasis on what happens on networks • Examples: – mapping spread of disease in a social network – mapping spread of a fad – computation in the brain • Statics and dynamics often closely linked – rate of disease spread (dynamic) depends critically on network connectivity (static) – distribution of wealth depends on network topology • Gladwell emphasizes dynamics Network Formation • Why does a particular structure emerge? • Plausible processes for network formation? • Generally interested in processes that are – – – – – decentralized distributed limited to local communication and interaction “organic” and growing consistent with measurement • The Internet versus traditional telephony Course News and Notes 1/22 • Prof. Kearns’ Friendster NW: please send mail to [email protected] if you want to join • Prof. Kearns’ office hours today 10:30-12 • Two new articles added to required readings • Homework 1 distributed today, due in class Feb 5 • NO CLASS on Feb 3 • Today’s agenda: – – – – recap of “brain analysis” from NW Nature of Society lectures further examples Contagion, Tipping and NW material cont’d distribution and discussion of HW 1 Brief Case Study: Associative Memory, Grandmother Cells, and Random Networks • A little more on the human brain: – (neo)cortex most recently evolved – memory and higher brain function – long distance connections: • pyramidal cells, majority of cortex • white matter as a box of cables – closet to a crude “random network” • Associative memory: – consider the Pelican Brief – or Networked Life • Grandmother cells: – localist vs. “holistic” representation – single vs. multi-cell recognition A Back-of-the Envelope Analysis • Let’s try assuming: – – – – – all connections equally likely independent with probability p grandmother cell representation decentralized learning correlated learning of conjunctions • So: – at some point have learned “pelican” and “brief” in separate cells – need to have cells connected to both – but not too many such cells! • In this model, p ~ 1/sqrt(N) balances near-certain upper and lower bounds on common neighbors for random pairs • Broadly consistent with biology Remarks • Network formation: – random long-distance – is this how the brain grows? • Network structure: – common neighbors for arbitrary cell pairs – implications for degree distribution • Network dynamics: – distributed, correlation-based learning • There is much that is broken with this story • But it shows how a set of plausible assumptions can lead to nontrivial constraints Recap • We chose a particular, statistical model of network generation – each edge appears independently and with probability p – why? broadly consistent with long-distance cortex connectivity – a statistical model allows us to study variation within certain constraints • We were interested in the NW having a certain global property – any pair of vertices should have a small number of common neighbors – why? corresponds to controlled growth of learned conjunctions, in a model assuming distributed, correlated learning and “grandmother cells” • We asked whether our NW model and this property were consistent – yes, assuming that p ~ 1/sqrt(N) – this implies each neuron (vertex) will have about p*N ~ sqrt(N) neighbors – and this is roughly what one finds biologically • Note: this statement is not easy to prove Another Example • We choose a particular, statistical model of network generation – each edge appears independently and with probability p – why? might propose it as a first guess as to how social networks form • We are interested in the NW having a certain global property – that the network be connected, yet not have large (say, log(N)) cliques – why? want a connected society without overly powerful subgroups • We ask whether our NW model and this property are consistent – I don’t know the answer to this one yet • We’ll follow this kind of questioning and analysis frequently A Non-Statistical Example • Often interested in the mere existence of certain NWs – – – – – let D be the largest degree allowed why? e.g. because there is a limit to how many friends you can have let D be a bound on the diameter of the network why? because many have claimed that D is often small let N(D,D) = size of the largest possible NW obeying D and D • Exact form of N(D,D) is notoriously elusive – but known that it is between (D/2)^D and 2D^D • So, for example, if we want N ~ 300M (U.S. population): – if D = 150 (e.g. see Gladwell) and D = 6 (6 degrees): NW exists – D = 6, N = 300M, solve 2D^D > N: get D > 23