Dsouza Supernova 2008

Layers of Networks
(Towards a Science of Networks)

Raissa D’Souza UC Davis
Dept of Mechanical and Aeronautical Eng.
Complexity Sciences Center
Santa Fe Institute

Transportation
Networks/
Networks: Power grid
(distribution/
collection networks)

Computer
networks

Biological networks
- protein interaction Social networks
- genetic regulation - Immunology
- drug design - Information
22 January 2007 CSE Advance
- Commerce 2

Networks: Physical, Biological, Social

• Geometric versus virtual (Internet versus WWW).

• Natural / spontaneously arising versus engineered / built.

• Each network optimizes something unique.

• Identifying similarities and fundamental differences can
guide future design/understanding.
1. How do we build a coherent distributed energy system integrating solar,
wind, hydropower, bio-diesel, hydrogen, etc.
2. Is old infrastructure introducing vulnerabilities in telecom?

• Deﬁnition of node can depend on level of representation.

Studying each network individually
(Though we know they interact)

• Topology (Statistical properties of node and edges)
– degree and degree distribution (extremely varied)
– diameter (“small-world”)
– clustering coefﬁcients
– assortative mixing
– betweenness, communities/partitioning, etc.

• Activity (Information ﬂows)
– epidemiology (humans and computers)
– Web search (ranking the web map)
– consensus formation / tipping points / phase transitions

Interactions between structure and function.

Software call graphs and
OSS Developer networks

• Highly evolveable, modular, robust to mutation, exhibit punctuated eqm
• Open-source software as a “systems” / organization paradigm.

D’Souza, Filkov, Devanbu, Swaminathan, Hsu

NETWORK TOPOLOGY

Connectivity matrix, M :

1 if edge exists between i and j
Mij =
0 otherwise.

 
1 1 1 1 0

 1 1 0 1 0 

1 0 1 0 0 =M
 

1 1 0 1 1
 
 
0 0 0 1 1

Node degree is number of links.

Broad Heterogeneity in node degree
e.g., The “Who-is-Who” network in Budapest
´ ¨ ´ ´
(Balazs Szendroi and Gabor Csanyi)

Bayesian curve ﬁtting → p(k) = ck −γ e−αk

istribution peaked at k and decaying 20

10
Random Power Law Graphs:
Attack
(e.g., “Preferential Attachment”, Barabasi and Albert, Science 1999)
15
Hubs and leaves
5

b Failure
0 10
Albert, Jeong and Barabasi, Nature, 406 (27) 2000.
0.00 0.01 0.02 0.00
N=130, E=215
f
Red five highest degree nodes;
Figure 2 Changes in theGreen theirthe network as a func
diameter d of neighbors.
removed nodes. a, Comparison between the exponential (E) a
models, each containing N ¼ 10;000 nodes and 20,000 links
“Robust” to random failure,
symbols correspond to the diameter of to targeted. (triang
fragile the exponential
(squares) networks when a fraction f of the nodes are removed
Red symbols show the response of the exponential (diamonds)
Is connectivity a good thing?
networks to attacks, when the most connected nodes are rem
Scale-free
dependence of the diameter for different system sizes (N ¼ 1
rence between an exponential and a scale-free found that the obtained curves, apart from a logarithmic size
s homogeneous: most nodes have approximately those shown in a, indicating that the results are independent o
Engineered networks (e.g., the Internet) are not random!
e-free network is inhomogeneous: the majority of note that the diameter of the unperturbed (f ¼ 0) scale-free n
few nodes have a large number of links, of the exponential network, indicating that scale-free network
connected. Red, the five nodes with the highest them more efficiently, generating a more interconnected web

Optimization in network growth
(D’Souza, Borgs, Chayes, Berger, Kleinberg, PNAS 2007)

o oo
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oo o o
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o o oo o o
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oo o oo o oo o o
o
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(Competing objectives)

Network Activity: FLOWS on NETWORKS
(Spread of disease, routing data, materials transport/ﬂow,
gossip spread/marketing)

Random walk on the network has state transition matrix, P :

 
1/4 1/3 1/2 1/4 0

 1/4 1/3 0 1/4 0 

1/4 0 1/2 0 0 =P
 

1/4 1/3 0 1/4 1/2
 
 
0 0 0 1/4 1/2

The eigenvalues and eigenvectors convey much information.
Markov Chains, Spectral Gap.

Feedback and network growth
of Hierarchical organizations

• Functional = efficient information flow throughout organization.

• More functional → grow faster
(but each new attachment less optimal)

• Less functional → grow slower but more balanced
(each new attachment
more considered)

(more balanced, efficient structures:

respond to changing circumstances)

Building a “science of networks”

• Last ten years, since 1999.
• Understanding activity and topology of individual networks.
• “Nodes”, “Robustness” (e.g., connectivity) context dependent.

“all our modern critical infrastructure
relies on networks”

Our modern infrastructure
Layered, interacting networks

• MATHEMATICS NEEDED:
Multiple info streams; Layered interactions; PDEs (calculus)

Dsouza Supernova 2008

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