Multidimensional Analysis of Complex Networks

Motivation Multidimensional Spatial analysis Growth analysis

Multidimensional analysis of complex networks

Possamai Lino

Alma Mater Studiorum Università di Bologna
Università di Padova

Ph.D. Dissertation Defense
February 21st, 2013

Possamai Lino Università di Bologna - Università di Padova
Multidimensional analysis of complex networks 1/39


Publications and conferences list

Plos One 2012 Thi Hoang, Sun, Possamai, JafariAsbagh, Patil, Menczer
Scholarometer: A Social Framework for Analyzing Impact across Disciplines

IPM 2012 Sun, Kaur, Possamai, Menczer
Ambiguous Author Query Detection using Crowdsourced Digital Library Annotations

SocialCom11 2011 Sun, Kaur, Possamai and Menczer
Detecting Ambiguous Author Names in Crowdsourced Scholarly Data

PSB2010 2010 Biasiolo, Forcato, Possamai, Ferrari, Agnelli, Lionetti, Todoerti, Neri, Marchiori et al.
Critical analysis of transcriptional and post-transcriptional regulatory networks in
Multiple Myeloma

Sunbelt2010 2010 Marchiori, Possamai
Telescopic analysis of complex networks

PRIB2009 2009 Forcato, Possamai, Ferrari, Agnelli, Todoerti, Lambertenghi, Bortoluzzi, Marchiori et al.
Reverse Engineering and Critical Analysis of Gene Regulatory Networks
in Multiple Myeloma

(under submission) 2013 Toward an optimized evolution of social networks

(under submission) 2013 Micro-macro analysis of complex networks



Outline

1 Motivation

2 Multidimensional
Introduction

3 Spatial analysis
Introduction
Algorithm
Datasets
Results

4 Growth analysis
Motivation
Growth dynamics
Simulations
Results



Domain

A complex system is a network of elements that
interacts in a non-linearly way, resulting in an
overall behavior that is difﬁcult to predict.



Domain

The digitalization of every day’s actions allows a
deeper investigation on how persons, computers,
animals, companies etc interact



Domain

Networks are everywhere in Nature: from ecology
to the WWW, to food chain, to social networks, to
ﬁnance



Domain

Networks are everywhere in Nature: from ecology
to the WWW, to food chain, to social networks, to
ﬁnance
This opened up many interdisciplinary research
areas that are very active



History

˝
Started with mathematicians Erdos–Rényi and graph theory
Watts and Strogatz, small world and L , C metrics
Barabási-Albert ﬁrst introduced the scale-free model, identiﬁed hubs and power
law in the degree distribution
Many other works that followed, proposed improvements in the basic statistics and
in the generative models



Motivation

The aim of this Thesis was to study Complex Networks (CN) under the most important
dimensions. Key points are the following:
Currently, many studies on CN underestimate the effect of spatial constraints on
the overall evolution
Many models have been proposed in order to create CNs with the same
properties of the observed networks
However, they are not sufﬁcient to describe precisely how networks evolve
That is why other instincts might be at the root of the growth
No methods have been proposed to increase the commitment in users’ communities
For these reasons, we worked on a new framework that is based on these lacking
features. We call it multidimensional.



Introduction

So what do we mean by multidimensional?
We mean a novel framework that analyzes complex
networks (CN) along the two fundamental
informative axes:



Introduction

informative axes:
Space



Introduction

informative axes:
Space
Time
The study of these dimensions was performed by
freezing one axis and simulating the evolution of
the other



Introduction

T HE SPACE DIMENSION



Introduction

Space dimension

The structure of a CN is not 100% completely deﬁned because it
depends on the level of detail with which the system is observed
For instance, biological networks could be analyzed at different
layers. Nodes could be represented as atoms, proteins, cells,
neurons and so on
Until now, no one has considered to study CN as a function of the
detail levels.
Results, properties, features that are valid in a speciﬁc level might
not hold in other levels.



Algorithm

Spatial Analysis

So what does it means to view a network at a particular
level?
Let us take a spatial network with information about nodes’
positions over a plane.



Algorithm

Spatial Analysis

level?
Viewing a network at different precision levels corresponds
to viewing the network at a difference distance from a point
of view.



Algorithm

Spatial Analysis

level?
Viewing a network at different precision levels corresponds
to viewing the network at a difference distance from a point
of view.
This process is modeled utilizing a concept that comes
from the human eyes ability to distinguish two points at
some distance from the observer.
The points are nodes of the network with x, y coordinates
over a plane.



Algorithm

Spatial Analysis

Generally, the telescopic algorithm is a function t : (G × f ) → G′ that takes as
input:
a graph G
fuzziness f (distance)
and produces a resulting graph G′



Algorithm

Spatial Analysis

Generally, the telescopic algorithm is a function t : (G × f ) → G′ that takes as
input:
a graph G
fuzziness f (distance)
and produces a resulting graph G′
In order to emulate the network abstraction capability, we placed a virtual grid on
top of the input graph.
Cell’s dimensions depend on the fuzziness value.



Algorithm

Spatial Analysis

All the nodes belonging to the same cell are collapsed and
represented by a unique node in the new graph.
If there is an edge from at least one node of the i cell to at
least one of the j cell then the (i, j) edge exists in the new
graph G′ .
With these rules, the long range edges are preserved.



Algorithm

Spatial Analysis

By repeatedly applying this function we create a
fuzziness-varying family of graphs T = {G0 , G1 , . . . Gp }
where p is the number of precision levels.
G0 is the micro view and Gp is the macro view.
This novel analysis then allows creating the telescopic
spectrum of a network, and study, wrt each property of
interest, what changes in the micro-macro shift (in
[Sunbelt2010]).



Datasets

Tracking properties

To characterize the structural properties during the abstraction process, we
consider several features widely used in network literature
Number of nodes, edges, kmax , kmean , standard deviation of k
Physical, topological and metrical diameter
Topological and metrical efﬁciency:

t 1 1 m 1 1
Eglob = Eglob =
n(n − 1) i=j
hij n(n − 1) i=j
δij

Topological and metrical local efﬁciency
Topological and metrical costs:

|E| i=j aij lij
Ct = Cm =
n(n − 1)/2 i=j lij

Homophily (degree correlation)



Datasets

Network datasets

Two different classes of networks are considered:



Datasets

Network datasets

Four subway networks are considered: two from
the U.S., Boston and New York and two from
Europe, Paris and Milan



Datasets

Network datasets

The US airline network



Datasets

Network datasets

The US airline network
The VirtualTourist online social network (*)
They all are undirected networks.



Results

Global Efficiency

1 1
Topological Eglob

0.8 0.8

Metrical Eglob
0.6 0.6
0.4 Bos 0.4 Bos
NYC NYC
0.2 Par 0.2 Par
Mil Mil
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Fuzziness Fuzziness

We found different results by considering topological and metrical efficiency
Topological: networks with high efficiency at macro level might have low Eglob at
micro
Metrical: stable under detail levels variation.



Results

Global Efﬁciency

1 1
Topological Eglob

0.8 0.8

Metrical Eglob
0.6 IT 0.6 IT
UK UK
0.4 NL 0.4 NL
AU AU
0.2 IN 0.2 IN
Air Air
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Fuzziness Fuzziness

All the curves start at higher values because of the better structure of SM-SF
networks
Both subways and SM-SF networks will be simpler as f increases, more efﬁcient,
but indistinguishable



Results

Local Efﬁciency

1 1
Bos Bos
Topological Eloc

0.8 Par 0.8 Par

Metrical Eloc
Mil Mil
0.6 NYC 0.6 NYC

0.4 0.4
0.2 0.2
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Fuzziness Fuzziness

1 1
IT
Topological Eloc

0.8 0.8 UK

Metrical Eloc
NL
0.6 IT 0.6 AU
UK IN
0.4 NL 0.4 Air
AU
0.2 IN 0.2
Air
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Fuzziness Fuzziness

Eloc is stable under our telescopic framework. Low values of local clustering
maintained throughout the spectrum
Results strongly differ from subways. This clearly means that the abstraction
process is able to distinguish the two different principles that guided the evolution


Results

Cost

1 1
Bos Bos
0.8 NYC 0.8 NYC
Par Par
0.6 Mil 0.6 Mil

Cm
Ct

0.4 0.4
0.2 0.2
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Fuzziness Fuzziness

It might be counterintuitive that simple (abstracted) networks are expensive
The cost is directly connected to the efﬁciency of a network



Results

Cost

1 1
0.8 0.8
0.6 IT 0.6 IT

Cm
Ct

UK UK
0.4 NL 0.4 NL
AU AU
0.2 IN 0.2 IN
Air Air
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Fuzziness Fuzziness

However, when compared to SM-SF networks turn out that the inborn economic
principles that characterize subways are maintained



Results

Randomized fuzziness-varying graphs

In order to understand how the topological and metrical structure of CNs is
affected by the spatial analysis, we used also null models in our simulations



Results

Randomized fuzziness-varying graphs

In order to understand how the topological and metrical structure of CNs is
affected by the spatial analysis, we used also null models in our simulations
In particular, we provided four models that account for different perturbations
+n, shufﬂing nodes’ positions
+a, rewiring edges
+r, that is the union of +n and +a
+s, scale-free structure (using BA model)



Results

Evolution on randomized networks

1 1
Boston Boston
Topological Eglob

0.8 0.8

Metrical Eglob
0.6 0.6
Norm Norm
0.4 +r 0.4 +r
+a +a
0.2 +n 0.2 +n
+s +s
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Fuzziness Fuzziness
t
In Eglob , randomizations increase the efﬁciency because they create the right
shortcuts that drop L
Conversely, randomness in a spatial context destroys the global efﬁciency. Indeed,
when f > 0.3 all the networks will be indistinguishable.



Results

Evolution on randomized networks

1 1
Aus Aus
Topological Eglob

0.8 0.8

Metrical Eglob
0.6 0.6
Norm Norm
0.4 +r 0.4 +r
+a +a
0.2 +n 0.2 +n
+s +s
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Fuzziness Fuzziness

Random perturbations do not alter Eglob because random networks are by
definition very efficient
The destroying effect found in subways is also present but constrained to small
values of f in metrical efficiency
SM-SF are robust because the randomizations do not alter considerably the
networks on the spectrum



Motivation

T HE TIME DIMENSION



Motivation

Time analysis

Many researches in the literature have dealt with proposing generative models
that uncover the key ingredients of network evolution
These are based on simple and advanced local rules that produce a global
behavior that is similar to the steady-state target’s network
Since many of them are based on social systems, we also concentrate on these
types of CNs



Growth dynamics

Growth rule I

The random rule assumes that:

Deﬁnition
Nodes of the networks randomly connect each other with
uniform probability

pij = k

Empirical tests discovered that real world networks are far from
being random



Growth dynamics

Growth rule II

The rule of Preferential attachment assumes that:

Deﬁnition
Older nodes are more likely to acquire new links
compared to new ones.

ki
Π(ki ) =
j kj



Growth dynamics

Growth rule III

The Social rule assumes that:

Deﬁnition
if two people have a friend in common then there is an increased
likelihood that they will become friend in the future

This rule is at the root of the local clustering property (found in
many networks)
Clearly, these rules are not sufﬁcient to completely describe the
evolution of social networks.
There must be some other instincts that trigger the network
evolution



Growth dynamics

Settings with special nodes

The contribution of this Thesis is to understand
whether new instincts on top of the previous growth
models can leverage the users’ commitment in
networks
Insight on network evolution with special nodes



Growth dynamics


The contribution of this Thesis is to understand mad
networks
m = number of sirens (6,12)
a = attractiveness
d = activation time span



Growth dynamics


The contribution of this Thesis is to understand mad
networks
m = number of sirens (6,12)
a = attractiveness
d = activation time span
conﬁgurations ci = (m, a, d)
conﬁgurations cost Cs = m · a · d



Simulations

Simulations

Both sequential and simultaneous simulations are considered
The network evolves according to one of the following rules random, aristocratic or
social both at the users and sirens levels
The entire system dynamics is accounted by two almost independent user and
siren subprocesses that evolve according to the previous local rules
In both cases, the future evolution Gt+1 will depend on Gt



Simulations

Simulations

Sirens are used for a limited time span (d) after that the system will evolve by itself
Sirens acquire new links constantly over time as

es = |V s | · |V | · a

a is the attractiveness of the sirens
q(s)
a(s) = q(u) = 10 ∀u ∈ V s q(u) = 1 ∀u ∈ V
u∈V ∪V s q(u)

In simultaneous simulations, many edges can be created and this number varies
as a function of Eglob

E(Gt−1 )
et = 1 + C · · (nart−1 − 1)
E(Gideal )



Results

Results and Datasets

At this point, based on the framework we provided, we are now able to answer the
following set of fundamental questions:
Are the sirens effective in leveraging users’ commitment in new on line social networks?



Results


What are the best parameters for the same cost conﬁgurations?



Results


Is the beneﬁt of sirens proportional to the amount of money involved?



Results


Is the beneﬁt of sirens proportional to the amount of money involved?
We were particularly interested in on line social networks like VirtualTourist and
Communities



Results

Q1: Effectiveness

In order to understand whether sirens are effective we compare the simulations
with and without sirens
0.25 0.25
rnd rnd
ari ari
0.2 soc 0.2 soc

0.15 0.15
Eglob

Eglob
0.1 0.1

0.05 CM 0.05 CM + Sir

0 0
0 600 1200 1800 2400 3000 0 20 40 60 80 100 120 140
Step Step



Results

Q2: Best parameter

What are the best parameters in the siren conﬁgurations ci = (m, a, d)?
The conﬁgurations that have the higher value of attractiveness are the ones that
perform best
Results are valid for all the rules and networks considered

0.25 aristocratic 0.25 aristocratic
Cs = 1200 Cs = 2400
0.2 0.2

0.15 0.15
Eglob

Eglob
0.1 0.1

0.05 (12,10,10) 0.05 (12,10,20)
(6,10,20) (12,20,10)
(6,20,10) (6,20,20)
0 0
0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70
Step Step



Results

Q3: Benefit

We set the number of sirens and see how the other configuration parameters
influence the growth behavior
We clearly see that the benefit increases, as the cost gets higher. In fact, it is not
proportional to Cs .

0.25 aristocratic 4000
rnd pref
ari pref
0.2 CM+Sir 3000 soc pref

0.15
Eglob

Cs
2000
0.1
(6,10,10) 1000
0.05 (6,10,20)
(6,20,10)
(6,20,20) 0
0 40 60 80 100 120 140
0 40 80 120 160 200 Tmin
Step



Results

Recap of contributions

We introduced a new framework in which we consider the two most important
informative axes along with a CN evolves



Results


The ﬁrst, spatial analysis, deals with analyzing a network under different detail
levels
Subway networks indexes tend to be more stable under the telescopic variations
Network properties change in the telescopic spectrum: their micro and macro behavior
are different



Results


The first, spatial analysis, deals with analyzing a network under different detail
levels
Subway networks indexes tend to be more stable under the telescopic variations
Network properties change in the telescopic spectrum: their micro and macro behavior
are different
The second, time analysis, models the growth of social networks by using a set of
privileged nodes that promote network evolution
These special nodes are an effective way to increase network efficiency
The benefit increases as cost increases, however it is not proportional
Invest on attractiveness



Results

Referees reports

From leading expert in the Complex System area
Jesús Gómez Gardeñes (University of Zaragoza)
Overall positive feedback
Acknowledged contributions to state-of-the-art



Results

Closing remarks and ongoing activities

Consider more spatial networks in order to have a broader coverage and test
whether our ﬁndings are still valid
Study force-based network permutations such as Kamada-Kawai and
Fruchterman-Reingold



Results

Closing remarks and ongoing activities

Consider more spatial networks in order to have a broader coverage and test
whether our ﬁndings are still valid
Study force-based network permutations such as Kamada-Kawai and
Fruchterman-Reingold
Deﬁne network growth that consider mixed rules instead of independent ones
Study the evolution by simultaneously varying the two axes
Continue the work done at Indiana University and in particular verify whether the
idea of “duplex” networked systems can be extended to digital libraries



Results

Thank you


Multidimensional Analysis of Complex Networks

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