Case Studies in Complex Networks

Transcription

1 Case Studies in Complex Networks Introduction to Scientific Modeling CS 365 George Bezerra 08/27/2012

2 The origin of graph theory Königsberg bridge problem Leonard Euler ( )

3 The Königsberg Bridge Problem Graph representation

4 The Königsberg Bridge Problem Graph representation The answer is no. Every vertex must have an even number of edges connected to it.

5 Theory of Random Graphs Erdös and Rényi (1960) Studied the evolution of random graphs as the mean degree is increased. Properties in random graphs emerge not gradually, but suddenly (phase transitions). E.g., giant component Paul Erdös ( )

6 Small-World Experiment Travers and Milgram (1969) Send a letter to individuals and ask them to forward it to someone that might know the target person. 296 individuals from Omaha, Nebraska, and Boston were recruited. Target person lives in Sharon, Massachussets.

7 HOW TO TAKE PART IN THIS STUDY: 1) ADD YOUR NAME TO THE ROSTER AT THE BOTTOM OF THIS SHEET, so that the next person who receives this letter will know who it came from. 2) DETATCH ONE POSTCARD. FILL IT OUT AND RETURN IT TO THE HARVARD UNIVERSITY. No stamp is needed. The postcard is very important. It allows us to keep track of the progress of the folder as it moves toward the target person. 3) IF YOU KNOW THE TARGET PERSON ON A PERSONAL BASIS, MAIL THIS FOLDER DIRECTLY TO HIM (HER). Do this only if you have previously met the target person and know each other on a first name basis. 4) IF YOU DO NOT KNOW THE TARGET PERSON ON A PERSONAL BASIS, DO NOT TRY TO CONTACT HIM DIRECLTY. INSTEAD MAIL THIS FOLDER TO A PERSONAL ACQUAINTANCE WHO IS MORE LIKELY THAN YOU TO KNOW THE TARGET PERSON.

8 Small-World Experiment 29% of the letters reached the target The number of intermediate acquaintances varied from 1 to 11. The median being 5.2. (Six degrees of separation!) Criticisms Most letters didn't reach the target. There is no guarantee the letters followed the shortest path.

9 The Erdös Number Co-authorship network of scientific papers Erdös published more than 1500 articles with 500 co-authors. Erdös has Erdös number 0.

10 Erdös number person Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people Erdös number people

11 Small-world networks What is a small world network? Mean geodesic distance (diameter) Erdös is a hub in the scientific world.

12 Scale-free networks The degree of nodes in real-world networks follows a power-law distribution What is a power-law?

13 Scale-free networks The degree of nodes in real-world networks follows a power-law distribution What is a power-law?

14 Power laws a>1 0<a<1 a<0

15 Log-log plots What happens when we plot a power-law in log-log scale?

16 Population in cities Histogram of the populations of all US cities US with population or more. Data from the 200 US Census.

17 Zipf's law Number of occurrences of words in the book Moby Dick.

18 Earthquakes Earthquake magnitude distribution over 6 decades. (K. Christensen, L. Danon, T. Scanlon, and Per Bak Unified scaling law for earthquakes, PNAS : )

19 The World Wide Web Albert, Jeong, Barabasi (1999)

20 The physical structure of the Internet

21 Protein Interaction Networks

22 Examples Social networks Semantic networks Airport networks Neuronal networks Scientific collaboration Gene regulatory networks network Terrorist networks Metabolic network Software networks Actors network Food webs

23 The rich get richer The Preferential Attachment model (BarabasiAlbert) The network grows Probability of connecting to a node is porportional to its degree

24 Fractal networks

25 Trees and vascular systems

26 Hierarchical modularity

27 Brain

28 Metabolism Brain

29 Fractality in computer design Rent's rule (1963) CPU

30 Rent s rule Rent s rule is a scaling relationship observed in the interconnection structure of VLSI circuits. C = communication N = circuit size p = Rent s exponent (0 p 1)

31 Hierarchical partitioning

32 Communication (C) Log-log plot of C vs N Size (N) Rent's rule for benchmark circuit c3540.

33 Communication (C) Log-log plot of C vs N Size (N) Rent's rule for benchmark circuit c3540.

34 Communication (C) Log-log plot of C vs N Region I Region II Size (N) Rent's rule for benchmark circuit c3540.

35 Case studies 1) Modeling the scaling of metabolic rate in biological organisms 2) Modeling the scaling of power consumption and performance in computers

36 Keiber's law

37 Keiber's law

38 Modeling approach Hypothesis: the design of vascular networks determines the rate at which nutrients are delivered to cells. Approach: determine how the flow of blood scales with size. Tools: fractal geometry.

39 The pipe model WBE model i=3 i=2 i=1 i=0

40 Modeling metabolic rate Metabolic rate (B) is proportional to the number of capillaries (N) Body mass (M) is proportional to the volume of blood (volume of the network) In oder to compute metabolic rate we must compute how the volume of the network scales.

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42 Back to metabolism

43 Power consumption in computers There is a large interest in modeling the scaling of power. Power consumption in chips is analogous to metabolic rate in organisms The number of transistors in a computer microprocessor has scaled by several orders of magnitude (2,000 to 2 billion). Can we use a similar approach to model power in computers?

44 Modeling approach Hypothesis: the design of computer interconnects determines the rate at which information is delivered to transistors. Approach: determine how the flow of information scales with transistor count. Tools: fractal geometry (Rent's rule).

45 A network of transistors and wires

46 A network of transistors and wires

47 Scaling of communication There are multiple wires per module The scaling of communication (number of wires) with the hierarchical level is given by Rent's rule as:

48 Network geometry

49 Physical properties Resistance: Capacitance: Latency:

50 Power and performance scaling For Dl = 2, Dr = 2, and Dw = 2: Power increases as N1/2 Throughput increases as N

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