MATH MODEL EXPLAINS “SIX DEGREES” PHENOMENON

August 25, 2000

SCIENCE & ENGINEERING NEWS

Ithaca, N.Y. — We all know it’s a small world: Any one of us is only about six acquaintances away from anyone else. Even in the vast confusion of the World Wide Web, on the average, one page is only about 16 to 20 clicks away from any other. But how, without being able to see the whole map, can we get a message to a person who is only “six degrees of separation” away?

A Cornell University computer scientist has concluded that the answer lies in personal networking: We use “structural cues” in our local network of friends. “It’s a collective phenomenon. Collectively the network knows how to find people even if no one person does,” says Jon Kleinberg, assistant professor of computer science, who published his explanation in the latest issue (Aug. 24) of the journal Nature.

His research is based on a computer model showing that an “ideal” network structure is one in which connections spread out in an “inverse square”

pattern. In human terms that means that an “ideal” person in the model would have just about as many friends in the rest of the county as in the neighborhood, just as many in the rest of the state as in the county, just as many in the whole nation as in the state, and so on, as you might find in a highly mobile society.

Kleinberg’s answers might have a very practical use in helping to reduce the number of clicks needed when surfing the web, as well as helping to speed up other kinds of networks.

Although Kleinberg has been instrumental in the development of improved search engines for the web, he doesn’t see this work as applying to traditional search engines. They already have the “big picture” of the network, he explains, since they work from indexes of the web. Rather, he sees it being useful in the construction of “agents,” computer programs that will jump around the web looking for specific information.

It could also apply to the distribution of data over the Internet, where computers called routers must send packets of information on their way toward their destinations without knowing what the state of the network is outside of their own immediate neighborhood.

Kleinberg has shown that a computer algorithm (the basic design for a program) can choose the best way to send a message to a faraway place in a network even if it has knowledge only about the characteristics of its immediate neighborhood. “The correlation between local structure and long-range connections provides fundamental cues for finding paths through the network,” he writes in the Nature paper.

Kleinberg’s work is a refinement of an earlier study by two other Cornellians, Steven H. Strogatz, professor of theoretical and applied mechanics, and his graduate student, Duncan Watts, now an assistant professor in Columbia University’s sociology department.

Strogatz and Watts offered a mathematical explanation for the results of a landmark experiment performed in the 1960s at Harvard by social psychologist Stanley Milgram. The researcher gave letters to randomly chosen residents of Omaha, Neb., and asked them to deliver the letters to people in Massachusetts by passing them from one person to another. The average number of steps turned out to be about six, giving rise to the popular notion of “six degrees of separation,” and eventually the “six degrees of Kevin Bacon” game in which actors are connected by their movie appearances with other actors.

Strogatz and Watts created a mathematical model of a network in which each point, or node, is closely connected to many other nodes nearby. When they added just a few random connections between a few widely separated nodes, messages could travel from one node to any other much faster than the size of the network would suggest. The six degrees of separation idea works, they said, because in every small group of friends there are a few people who have wider connections, either geographically or across social divisions. They also showed that such cross-connected networks exist not only between human beings but also in such diverse places as computer networks, power grids and the human brain.

But Kleinberg has found mathematically that the model proposed by Strogatz and Watts doesn’t explain how messages can travel so quickly through real human networks. “The Strogatz-Watts model had random connections between nodes. Completely random connections bring everyone closer together,”

Kleinberg explains, “but a computer algorithm would have only local information. The long-range connections are so random that it [the algorithm] gets lost.”

So Kleinberg designed a model in which nodes are arranged in a square grid and each node is connected randomly to others but with “a bias based on geography.” As a result each node is connected to many nearby, a few at a longer distance and even fewer at a great distance – the “inverse square”

structure. “This bias builds in the structural cues in my long-range connections, and it’s the bias that is implicitly guiding you to the target,”

Kleinberg explains. “In the Strogatz-Watts model, there is no bias at all and, hence, no cues – the structure of the long-range connections gives you no information at all about the underlying network structure.”

The sender of a message in this system doesn’t know where all the connections are but does know the general geographic direction of the destination, and if messages are sent in that direction, Kleinberg says, they arrive much faster than they would by completely random travel.

Kleinberg explains, “The Watts and Strogatz model is sort of like a large group of people who know each other purely through electronic chat on the Internet. If you are given the user ID of someone you don’t know, it’s really hard to guess which of your friends is liable to help you forward a message to them.

“The inverse square model is more like the geographic view of Milgram’s experiment – if you live on the West Coast and need to forward a message to someone in Ithaca, you can guess that a resident of New York state is a good first step in the chain. They are more likely to know someone in the Finger Lakes region, who in turn is more likely to know someone in Ithaca and so forth. Knowing that our distribution of friends is correlated with the geography lets you form guesses about how to forward the message quickly.”

The geographic information on the grid, he adds, is an analogue of the social connections between people. Just as nodes on his simulated network choose the correct geographical direction to send a message, so humans may choose a social direction: A mathematician who wants to send a message to a politician might start by handing it to a lawyer.

On the other hand, he says, “When long-range connections are generated uniformly at random, our model describes a world in which short chains exist but individuals, faced with a disorienting array of social contacts, are unable to find them.” The paper in Nature is titled “Navigation in a Small World.”

============================================================

Subscribe to HPCwire's Weekly Update!

Be the most informed person in the room! Stay ahead of the tech trends with industy updates delivered to you every week!

Mira Supercomputer Enables Cancer Research Breakthrough

November 11, 2019

Dynamic partial-wave spectroscopic (PWS) microscopy allows researchers to observe intracellular structures as small as 20 nanometers – smaller than those visible by optical microscopes – in three dimensions at a mill Read more…

By Staff report

IBM Adds Support for Ion Trap Quantum Technology to Qiskit

November 11, 2019

After years of percolating in the shadow of quantum computing research based on superconducting semiconductors – think IBM, Rigetti, Google, and D-Wave (quantum annealing) – ion trap technology is edging into the QC Read more…

By John Russell

Tackling HPC’s Memory and I/O Bottlenecks with On-Node, Non-Volatile RAM

November 8, 2019

On-node, non-volatile memory (NVRAM) is a game-changing technology that can remove many I/O and memory bottlenecks and provide a key enabler for exascale. That’s the conclusion drawn by the scientists and researcher Read more…

By Jan Rowell

What’s New in HPC Research: Cosmic Magnetism, Cryptanalysis, Car Navigation & More

November 8, 2019

In this bimonthly feature, HPCwire highlights newly published research in the high-performance computing community and related domains. From parallel programming to exascale to quantum computing, the details are here. Read more…

By Oliver Peckham

Machine Learning Fuels a Booming HPC Market

November 7, 2019

Enterprise infrastructure investments for training machine learning models have grown more than 50 percent annually over the past two years, and are expected to shortly surpass $10 billion, according to a new market fore Read more…

By George Leopold

AWS Solution Channel

Making High Performance Computing Affordable and Accessible for Small and Medium Businesses with HPC on AWS

High performance computing (HPC) brings a powerful set of tools to a broad range of industries, helping to drive innovation and boost revenue in finance, genomics, oil and gas extraction, and other fields. Read more…

IBM Accelerated Insights

Atom by Atom, Supercomputers Shed Light on Alloys

November 7, 2019

Alloys are at the heart of human civilization, but developing alloys in the Information Age is much different than it was in the Bronze Age. Trial-by-error smelting has given way to the use of high-performance computing Read more…

By Oliver Peckham

IBM Adds Support for Ion Trap Quantum Technology to Qiskit

November 11, 2019

After years of percolating in the shadow of quantum computing research based on superconducting semiconductors – think IBM, Rigetti, Google, and D-Wave (quant Read more…

By John Russell

Tackling HPC’s Memory and I/O Bottlenecks with On-Node, Non-Volatile RAM

November 8, 2019

On-node, non-volatile memory (NVRAM) is a game-changing technology that can remove many I/O and memory bottlenecks and provide a key enabler for exascale. Th Read more…

By Jan Rowell

MLPerf Releases First Inference Benchmark Results; Nvidia Touts its Showing

November 6, 2019

MLPerf.org, the young AI-benchmarking consortium, today issued the first round of results for its inference test suite. Among organizations with submissions wer Read more…

By John Russell

Azure Cloud First with AMD Epyc Rome Processors

November 6, 2019

At Ignite 2019 this week, Microsoft's Azure cloud team and AMD announced an expansion of their partnership that began in 2017 when Azure debuted Epyc-backed ins Read more…

By Tiffany Trader

Nvidia Launches Credit Card-Sized 21 TOPS Jetson System for Edge Devices

November 6, 2019

Nvidia has launched a new addition to its Jetson product line: a credit card-sized (70x45mm) form factor delivering up to 21 trillion operations/second (TOPS) o Read more…

By Doug Black

In Memoriam: Steve Tuecke, Globus Co-founder

November 4, 2019

HPCwire is deeply saddened to report that Steve Tuecke, longtime scientist at Argonne National Lab and University of Chicago, has passed away at age 52. Tuecke Read more…

By Tiffany Trader

Spending Spree: Hyperscalers Bought $57B of IT in 2018, $10B+ by Google – But Is Cloud on Horizon?

October 31, 2019

Hyperscalers are the masters of the IT universe, gravitational centers of increasing pull in the emerging age of data-driven compute and AI.  In the high-stake Read more…

By Doug Black

Cray Debuts ClusterStor E1000 Finishing Remake of Portfolio for ‘Exascale Era’

October 30, 2019

Cray, now owned by HPE, today introduced the ClusterStor E1000 storage platform, which leverages Cray software and mixes hard disk drives (HDD) and flash memory Read more…

By John Russell

Supercomputer-Powered AI Tackles a Key Fusion Energy Challenge

August 7, 2019

Fusion energy is the Holy Grail of the energy world: low-radioactivity, low-waste, zero-carbon, high-output nuclear power that can run on hydrogen or lithium. T Read more…

By Oliver Peckham

Using AI to Solve One of the Most Prevailing Problems in CFD

October 17, 2019

How can artificial intelligence (AI) and high-performance computing (HPC) solve mesh generation, one of the most commonly referenced problems in computational engineering? A new study has set out to answer this question and create an industry-first AI-mesh application... Read more…

By James Sharpe

Cray Wins NNSA-Livermore ‘El Capitan’ Exascale Contract

August 13, 2019

Cray has won the bid to build the first exascale supercomputer for the National Nuclear Security Administration (NNSA) and Lawrence Livermore National Laborator Read more…

By Tiffany Trader

DARPA Looks to Propel Parallelism

September 4, 2019

As Moore’s law runs out of steam, new programming approaches are being pursued with the goal of greater hardware performance with less coding. The Defense Advanced Projects Research Agency is launching a new programming effort aimed at leveraging the benefits of massive distributed parallelism with less sweat. Read more…

By George Leopold

AMD Launches Epyc Rome, First 7nm CPU

August 8, 2019

From a gala event at the Palace of Fine Arts in San Francisco yesterday (Aug. 7), AMD launched its second-generation Epyc Rome x86 chips, based on its 7nm proce Read more…

By Tiffany Trader

D-Wave’s Path to 5000 Qubits; Google’s Quantum Supremacy Claim

September 24, 2019

On the heels of IBM’s quantum news last week come two more quantum items. D-Wave Systems today announced the name of its forthcoming 5000-qubit system, Advantage (yes the name choice isn’t serendipity), at its user conference being held this week in Newport, RI. Read more…

By John Russell

Ayar Labs to Demo Photonics Chiplet in FPGA Package at Hot Chips

August 19, 2019

Silicon startup Ayar Labs continues to gain momentum with its DARPA-backed optical chiplet technology that puts advanced electronics and optics on the same chip Read more…

By Tiffany Trader

Crystal Ball Gazing: IBM’s Vision for the Future of Computing

October 14, 2019

Dario Gil, IBM’s relatively new director of research, painted a intriguing portrait of the future of computing along with a rough idea of how IBM thinks we’ Read more…

By John Russell

Leading Solution Providers

ISC 2019 Virtual Booth Video Tour

CRAY
CRAY
DDN
DDN
DELL EMC
DELL EMC
GOOGLE
GOOGLE
ONE STOP SYSTEMS
ONE STOP SYSTEMS
PANASAS
PANASAS
VERNE GLOBAL
VERNE GLOBAL

Intel Confirms Retreat on Omni-Path

August 1, 2019

Intel Corp.’s plans to make a big splash in the network fabric market for linking HPC and other workloads has apparently belly-flopped. The chipmaker confirmed to us the outlines of an earlier report by the website CRN that it has jettisoned plans for a second-generation version of its Omni-Path interconnect... Read more…

By Staff report

Kubernetes, Containers and HPC

September 19, 2019

Software containers and Kubernetes are important tools for building, deploying, running and managing modern enterprise applications at scale and delivering enterprise software faster and more reliably to the end user — while using resources more efficiently and reducing costs. Read more…

By Daniel Gruber, Burak Yenier and Wolfgang Gentzsch, UberCloud

Dell Ramps Up HPC Testing of AMD Rome Processors

October 21, 2019

Dell Technologies is wading deeper into the AMD-based systems market with a growing evaluation program for the latest Epyc (Rome) microprocessors from AMD. In a Read more…

By John Russell

Intel Debuts Pohoiki Beach, Its 8M Neuron Neuromorphic Development System

July 17, 2019

Neuromorphic computing has received less fanfare of late than quantum computing whose mystery has captured public attention and which seems to have generated mo Read more…

By John Russell

Rise of NIH’s Biowulf Mirrors the Rise of Computational Biology

July 29, 2019

The story of NIH’s supercomputer Biowulf is fascinating, important, and in many ways representative of the transformation of life sciences and biomedical res Read more…

By John Russell

Xilinx vs. Intel: FPGA Market Leaders Launch Server Accelerator Cards

August 6, 2019

The two FPGA market leaders, Intel and Xilinx, both announced new accelerator cards this week designed to handle specialized, compute-intensive workloads and un Read more…

By Doug Black

When Dense Matrix Representations Beat Sparse

September 9, 2019

In our world filled with unintended consequences, it turns out that saving memory space to help deal with GPU limitations, knowing it introduces performance pen Read more…

By James Reinders

With the Help of HPC, Astronomers Prepare to Deflect a Real Asteroid

September 26, 2019

For years, NASA has been running simulations of asteroid impacts to understand the risks (and likelihoods) of asteroids colliding with Earth. Now, NASA and the European Space Agency (ESA) are preparing for the next, crucial step in planetary defense against asteroid impacts: physically deflecting a real asteroid. Read more…

By Oliver Peckham

  • arrow
  • Click Here for More Headlines
  • arrow
Do NOT follow this link or you will be banned from the site!
Share This