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The Networked Nature of Society Networked Life CSE 112 Spring 2005 Prof. Michael Kearns 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 structural properties – 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 examine the results of our own last-names exercise • 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 • More recent and interesting: thefacebook • 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 MK’s Friendster NW, 1/19/04 • • • • • • • 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: – 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 – 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 Agenda: Thursday, Jan 20 • Finish up “Networked Nature…” lectures • Detail and due date for first network construction project task • Introduction to Lifester and first assignment 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 – but often dynamics of transmission – what about dynamics involving deliberation, rationality, etc.? 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 Structure, Dynamics, Formation: Two Case Studies Case Study 1: A (Brain-Dead) Model of Economic Exchange • Imagine an(y) undirected, connected network of individuals – no model of network formation • Start each individual off with some amount of currency • At each time step: – – – – each vertex divides their current cash equally among their neighbors (or chooses a random neighbor to give it all to) each vertex thus also receives some cash from its neighbors repeat – – – – vertex i will have fraction deg(i)/D of the wealth; D = sum of deg(i) degree distribution entirely determines outcome! “connectors” are the wealthiest not obvious: consider two degree = 2 vertices… • A transmission model of economic exchange --- no “rationality” • Q: How does network structure influence outcome? • A: As time goes to infinity: Case Study 2: Grandmother Cells, Associative Memory, 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 – closest to a crude “random network” • all connections equally likely • Grandmother cells: • Hebbian learning of correlations: • Problem of associative memory: – localist (vs. “holistic”) representation – single (vs. multi-) cell recognition – cells learn to fire when highly correlated neighboring cells fire – entirely decentralized allocation process – consider the Pelican Brief – or Networked Life A Back-of-the Envelope Analysis • Let’s try assuming: – – – – all connections equally likely independent with probability p grandmother cell representation Hebbian learning of conjunctions • So: – at some point have learned “pelican” and “brief” in separate cells – need to have cells connected to both to learn conjunction – but not too many such cells! • In this model, p ~ 1/sqrt(N) results in any pair of cells having just a few common neighbors! • 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