Complex Networks: Connectivity and Functionality. Milena Mihail Georgia Tech. Search and routing networks, like the WWW , the internet , P2P networks, ad-hoc ( mobile, wireless, sensor ) networks are pervasive and scale at an unprecedented rate.
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Complex Networks:Connectivity and Functionality
Search and routing networks,
like the WWW, the internet, P2P networks,
ad-hoc (mobile, wireless, sensor) networks
are pervasive and scale at an unprecedented rate.
Performance analysis/evaluation in networking:
measure parameters hopefully predictive of performance.
Which are criticalnetwork parameters/metrics
that determinealgorithmic performance?
Important in network simulation and design.
Predictive of routing and searching performance
is conductance, expansion, spectral gap.
This talk: The case of
internet routing topology
How can network models capture the parameters/metrics
that are critical in network performance?
This talk: The case of
Can we design network algorithms/protocols
that optimize these critical network parameters?
Current Models for Internet Routing Topologies
focus on large degree-variance
Erdos-Renyi-like, Configurational :
A random graph with given degrees
macroscopic and microscopic :
The graph grows one vertex at a time
and attaches preferentially to degrees
or according to some optimization criterion
The case of modeling the internet routing topology
Nodes are routers or Autonomous Systems
Two nodes connected by a link if they
are involved in direct exchange of traffic
Sparse small-world graphs
with large degree-variance
But are degrees the “right” parameter to measure?
An important metric:
Conductance and the second eigenvalue
of the stochastic normalization of the adjacency matrix
characterize routing congestion under link capacities,
mixing rate, cover time.
How does the second eigenvalue
(more generally the principal eigenvalues)
scale as the size of the network increases?
This is also another point of view
of the small-world phenomenon
This also says that congestion
under link capacities scales smoothly
Second eigenvalue of internet is larger than that of random graphs
but spectral gap remains constant as number of nodes increases.
Open problem: Erdos-Renyi like, configurational models
which include spectral gap parameter?
Some evolutionary random graph models
may capture clustering
Growth & Preferential Attachment
One vertex at a time
New vertex attaches to existing vertices
Open Problem: characterize clustering as
a function of model parameter
number of nodes
ie, indicate which parameter ranges
are important in simulations
Other discrepancies of random graph models from
real internet topologies:
Li, Alderson, Willinger, Doyle
high degree nodes away from “network core”
high degrees mostly connected to low degrees
“core” of low degrees
what do internet topologies “optimize” ?
Algorithms improving congestion
conductance and spectral gap
Given total length l and n random points in a metric space
construct a graph with total link length l that
- maximizes spectral gap, conductance
- minimizes congestion under node capacities
Algorithms optimizing connectivity
How do you maintain a
P2P network with good
search performance ?
The case of Peer-to-Peer Networks
each node has resources
and knows a
constant size neighborhood
Search for content, e.g. by flooding or random walk
Must maintain well connected topology,
e.g. a random graph, an expander
P2P Network Topology Maintenance by Constant Randomization
Gnutella: constantly drops existing connections
and replaces them with new connections
random graph, expander
Theorem[Cooper, Frieze & Greenhill 04]:
The Markov chain corresponding to a 2-link switch on d-regular graphs is rapidly mixing.
Theorem[Feder, Guetz, M, Saberi 06]:
The Markov chain on d-regular graphs is rapidly mixing,
even under local 2-link switches or flips.
Question: How does the network “pick” a random 2-link switch?
In reality, the links involved in a switch are within constant distance.
The proof is a Markov chain comparison argument
Space of connected d-regular graphs
local Flip Markov chain
Space of d-regular graphs
general 2-link switch Markov chain
Define a mapping from to such that
(b) each edge in maps to a path of constant length in
Question: How do we add new nodes with low network overhead?
Question: How do we delete nodes with low network overhead?
Algorithms developing topology awareness
A node has
a query and a budget
the remaining budget
the parts to the neighbors
Let be a graph.
Assign symmetric transition probabilities to links in (and self loops)
so that the resulting matrix is stochastic
and the second in absolute value largest eigenvalue is minimized.
Fastest Mixing Markov Chain
Fastest Mixing Markov Chain Subgradient Algorithm
is some vector on of initial transition probabilities
is the eigenvector corresponding to
second in absolute value largest eigenvalue
is a vector on with
projection to feasible subspace
Is there a decentralized implementation or algorithm?
The Case of Ad-Hoc Wireless Networks
How does Capacity/Throughput/Delay Scale?
Capacity of Wireless Networks, Gupta & Kumar, 2000
Is there a connection with
Lipton & Tarjan’s separators for planar graphs?
Mobility Increases Capacity, Grossglauser & Tse, 2001
Capacity, Delay and Mobility in Wireless Networks, Bansal & Liu 2003
Throughput-delay Trade-off in Wireless Networks, El Gamal, Mammen, Prabhakar & Shah 2004