Fat Tail Distributions and Efficiency of Flow Processing on Complex Networks

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Fat Tail Distributions and Efficiency of Flow
Processing on Complex Networks
Zoltán Toroczkai
Center for Nonlinear Studies, and Complex
Systems Group, Theoretical Division, Los Alamos
National Laboratory
LA-UR-03-5542
LANL LDRD-DR S.P.I.N. Project, 2003-06
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What are Networks?
Interacting many particle systems where the
interactions are propagated through a discrete
structure, a graph (not a continuum).
Node (the particle)
Link (edge)
The links edges represent interactions or
associations between the nodes.
Graph

• - undirected

• - directed

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Where are Networks?

• Infrastructures transportation nw-s (airports,
  highways, roads, rail, water) energy transport
  nw-s (electric power, petroleum, natural gas)
• Communications telephone, microwave backbone,
  internet, email, www, etc.
• Biology protein-gene interactions,
  protein-protein interactions, metabolic nw-s,
  cell-signaling nw-s, the food web, etc.
• Social Systems acquaintance (friendship) nw-s,
  terrorist nw-s, collaboration networks, epidemic
  networks, the sex-web
• Geology river networks

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Communication Networks
Skitter data depicting a macroscopic snapshot of
Internet connectivity, with selected backbone
ISPs (Internet Service Provider) colored
separately by K. C. Claffy email kc_at_caida.org
http//www.caida.org/Papers/Nae/
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Networks in Biology
The metabolic pathway
Chemicals
Bio-Chemical reactions
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Networks in Biology
The metabolic pathway
Chemicals
Bio-Chemical reactions
Biochemical Pathways - Metabolic Pathways,
Source ExPASy
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The protein network
proteins
Binding
H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N.
Oltvai, Nature 411, 41-42 (2001)
P. Uetz, et al. Nature 403, 623-7 (2000).
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Structural properties degree distributions and
the scale-free character
Node degree number of neighbors
ki5
i
Degree distribution, P(k) fraction of nodes
whose degree is k (a histogram over the ki s.)
Observation networks found in Nature and human
made, are in many cases scale-free (power-law)
networks
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For the sake of definitions
The Erdos-Rényi Random Graph (also called the
binomial random graph)

• Consider N nodes (dots).

• Take every pair (i,j) of nodes and connect them
  with an edge with probability p.

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What is scale-free?
Poisson distribution
Power-law distribution
?ltkgt
Capacity achieving degree distribution of Tornado
code. The decay exponent -2.02.
Erdos-Rényi Graph
Non-Scale-free Network
M. Luby, M. Mitzenmacher, M.A. Shokrollahi, D.
Spielman and V. Stemann, in Proc. 29th ACM Symp.
Theor. Comp. pg. 150 (1997).
Scale-free Network
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Science citations
www, out- and in- link distributions
Internet, router level
Archaea
Bacteria
Eukaryotes
Bacteria
Eukaryotes
Metabolic network
Sex-web
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Scale-free Networks Coincidence or Universality?

• No obvious universal mechanism identified
• As a matter of fact we claim that there is none
  (universal that is).
• Instead, our statement is that at least for a
  large class of networks (to be specified) network
  structural evolution is governed by a selection
  principle which is closely tied to the global
  efficiency of transport and flow processing by
  these structures, and
• Whatever the specific mechanism, it is such as
  to obey this selection principle.

Need to define first a flow process on these
networks.
Z. Toroczkai and K.E. Bassler, Jamming is
Limited in Scale-free Networks, Nature, 428, 716
(2004)
Z. Toroczkai, B. Kozma, K.E. Bassler, N.W.
Hengartner and G. Korniss Gradient Networks,
http//www.arxiv.org/cond-mat/0408262
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Gradient Flow Networks
Gradients of a scalar (temperature,
concentration, potential, etc.) induce flows
(heat, particles, currents, etc.).
Naturally, gradients will induce flows on
networks as well.
Ex.
Load balancing in parallel computation and packet
routing on the internet
Y. Rabani, A. Sinclair and R. Wanka, Proc. 39th
Symp. On Foundations of Computer Science (FOCS),
1998 Local Divergence of Markov Chains and the
Analysis of Iterative Load-balancing Schemes
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Setup
Let GG(V,E) be an undirected graph, which we
call the substrate network.
The vertex set
The edge set
A simple representation of E is via the Nx N
adjacency (or incidence) matrix
A
(1)
Let us consider a scalar field
Set of nearest neighbor nodes on G of i
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Definition 1
The gradient ?h(i) of the field h in node i is
a directed edge
(2)
Which points from i to that nearest neighbor
for G for which the increase in the
scalar is the largest, i.e.,
(3)
The weight associated with edge (i,?) is given by
The self-loop
.
.
is a loop through i
with zero weight.
Definition 2
The set F of directed gradient edges on G
together with the vertex set V forms the gradient
network
If (3) admits more than one solution, than the
gradient in i is degenerate.
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In the following we will only consider scalar
fields with non-degenerate gradients. This means
Theorem 1
Non-degenerate gradient networks form forests.
Proof
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Theorem 2
The number of trees in this forest number of
local maxima of h on G.
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In-degree distribution of the Gradient Network
when GGN,p . A combinatorial derivation
Version Balazs Kozma (RPI)
Assume that the scalar values at the nodes are
i.i.d, according to some distribution ?(h).
First, distribute the scalars on the node set V,
then find those link configurations which
contribute to R(l) when building the GN,p graph.
Without restricting the generality, calculate
R(l) for node 0.
Consider the set of nodes with the property
Let the number of elements in this set be n, and
the set be denoted by ?n.
The complementary set of ?n in V\0 is
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In order to have exactly l nodes pointing their
gradient edges into 0

• they have to be connected to node 0 on the
  substrate
• they must NOT be connected to the set ?n

For l nodes
Also need to require that no other nodes will be
pointing their gradient directions into node 0
(Obviously none of the ?n will.)
So, for a fixed h0 and a specific set ?n
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The probability Qn for such an event for a given
n while letting h-s vary according to their
distribution
For one node to have its scalar larger than h0
For exactly n nodes
Thus
Combining
Finally
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What happens when the substrate is a scale-free
network?
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Gradient Networks and Transport Efficiency

• every node has exactly one out-link (one
  gradient direction) but it can have more than one
  in-link (the followers)
• the gradient network has N-nodes and N
  out-links. So the number of out-streams is
  Nsend N
• the number of RECEIVERS is

• J is a congestion (pressure) characteristic.
• 0 ? J ? 1. J0 minimum congestion, J1
  maximum congestion

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In the scaling limit
- for large networks we get maximal congestion!
In the scaling limit
- becomes congested for large average degree.
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- For scale-free structures, the congestion
factor becomes independent on the system
(network) size!!
For LARGE and growing networks, where the
conductance of edges is the same, and the flow is
generated by gradients, scale-free networks are
more likely to be selected during network
evolution than scaled structures.
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The Configuration model
A. Clauset, C. Moore, Z.T., E. Lopez, to be
published.
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Generating functions
K-th Power of a Ring
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Degree distribution of the gradient network for
the K-th power of a ring
So
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where
is an n-subset of the set 1,2,,N-1.
denotes the set of all possible n-subsets of
1,2,N-1.
is always zero, if there is a node from the
n-subset connected to i, or i belongs to the
n-subset.
Let
which is the union of the disks of all nodes from
the n-subset.
Thus, one needs to find the number of coverings
of the ring with n disks, each of radius K, that
misses exactly l nearest neighbors of the origin.
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Power law with exponent - 3
2Kl
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Competition Games on Networks
Collaboration with

• Marian Anghel (CCS-3)
• Kevin E. Bassler (U. Houston)
• György Korniss (Rensselaer)

References
M. Anghel, Z. Toroczkai, K.E. Bassler and G.
Korniss, Competition-driven Network Dynamics
Emergence of a Scale-free Leadership Structure
and Collective Efficiency, Phys.Rev.Lett. 92,
058701 (2004)
Z. Toroczkai, M. Anghel, G. Korniss and K.W.
Bassler, Effects of Inter-agent Communications on
the Collective, in Collectives and the Design of
Complex Systems, eds. K. Tumer and D.H. Wolpert,
Springer, 2004.
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Resource limitations lead in human, and most
biological populations to competitive dynamics.
The more severe the limitations, the more fierce
the competition.
Amid competitive conditions certain agents may
have better venues or strategies to reach the
resources, which puts them into a distinguished
class of the few, or elites. Elites form a
minority group.
In spite of the minority character, the elites
can considerably shape the structure of the whole
society
since they are the most successful (in the given
situation), the rest of the agents will tend to
follow (imitate, interact with) the elites
creating a social structure of leadership in the
agent society.
Definition a leader is an agent that has at
least one follower at that moment. The influence
of a leader is measured by the number of
followers it has. Leaders can be following other
leaders or themselves.
The non-leaders are coined followers.
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The El Farol bar problem
W. B Arthur(1994)
A
B

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A binary (computer friendly) version of the El
Farol bar problem
The Minority Game (MG)
Challet and Zhang (1997)
A 0 (bar ok, go to the bar) B 1 (bar
crowded, stay home)
latest bit
? l ? 0,1,..,2m-1
World utility(history)
(011..101)
m bits
S(i)1(l)
S(i)2(l)
(Strategies)(i)
(Scores)(i) C (i)(k), k 1,2,..,S.
?
S(i)S(l)
(Prediction) (i)
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A(t)
t
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Attendance time-series for the MG
World Utility Function
?
Agents cooperate if they manage to produce
fluctuations below (N1/2)/2 (RCG).
Scaling variable
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The El Farol bar game on a social network
A
B

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The Minority Game on Networks (MGoN)
Agents communicate among themselves.
Social network
2 components
1) Aquintance (substrate) network G
(non-directed, less dynamic)
2) Action network A (directed and dynamic)
G
A ? G
A
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Communication types (more bounded rationality)
Minority rule
Majority rule
(not rational)
(not rational)
Critics rule an agent listens to the
OPINION/PREDICTION of all neighboring agents on
G, scores them (self included) based on their
past predictions, and ACTS on the best score.
(more rational, uses reinforcement learning)
(Links)(i)
(Scores)(i) F (i)(j), j 1,2,..,K.
?
i
(Prediction) (i)
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Emergence of scale-free leadership structure
m6

• Robust leadership hierarchy

• RCG on the ER network produces the scale-free
• backbone of the leadership structure

• The influence is evenly distributed
• among all levels of the leadership
• hierarchy.

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• The followers (sheep) make up most of the
  population (over 90) and their number scales
  linearly with the total number of agents.

• Structural un-evenness appears in the leadership
  structure for low trait diversity.

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• Leadership position Symmetric-Asymmetric phase
  transition

• In low m regime, where trait diversity is low
  (as in a dictatorship) leaders leave longer!

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Network Effects Improved Market Efficiency

• A networked, low trait diversity system
• is more effective as a collective
• than a sophisticated group!

• Can we find/evolve networks/strategies
• that achieve almost perfect volatility
• given a group and their strategies
• (or the social network on the group)?

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Conclusions

• We defined Gradient Networks as directed
  sub-graphs formed by local gradients of a scalar
  distributed on a substrate graph G.
• When the gradient direction is unique these
  Gradient Networks form forests.
• Gradient Networks typically arise when there is
  a local extremizing dynamics at the node level
  (Agent-based Systems such as markets, routers,
  parallel computers, etc..)).
• Gradient Networks can be scale-free graphs even
  on substrate networks that are NOT scale-free
  networks (such as E-R graphs)!!
• Gradient Networks can be highly dynamic, their
  evolution driven by the dynamics of the scalar
  field on G and they are not solely defined
  through the topological properties of G!! (such
  as in the case of preferential attachment).
• G. N.-s give a natural explanation for why
  scale-free large networks might emerge if the
  edges have the same conductance and the flows are
  generated by gradients.