myprojects-hg-reborn

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%\fancyhead[RO,LE]{\bfseries DynaNets 233847}
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\fancyfoot[LO,CE]{V0.0 Complex Network Modeling  WP3 May, 19th}
%\fancyfoot[CO,RE]{To: Dean A. Smith}
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\chapter*{D3.1: Software for generators together with report on performance}
\section*{Introduction}
 \addcontentsline{toc}{section}{Introduction}
Workpackage 3 (WP3) revolves around designing and implementing a
computational framework for  modeling  evolving networks.
The aim is to provide  flexible  tools of analysis for the generation and representation of dynamical networks.
The tools for the representation and analysis of evolving networks
developed within the scope of WP3 have been tested on several
different empirical data sets such as networks of human interactions and the HIV infection network.
% and network still to be provided  by Laszlo ({\bf *time to ping?}).  
The modeling and representation of each of these networks required the development of \emph{ad
hoc} tools, but their  field of applicability is by no means limited to the
datasets/case studies used  for validation and  stress-testing. In
particular these tools can be applied both to empirical networks and
to networks  generated according to a model.

On a general ground, we note  that essentially all
networks are dynamic,  even those studied in a static setting. 
That is because real-world networks are typically mutable and exist in
time; it is only the act of data collection that often results in a
one-time sampling  of these dynamic networks, thus rendering them static. Working with dynamic
networks (either
 collecting longitudinal samples of networks or trying to model the
 evolution of networks in time),
 one also realises that sampling always involves the act of
 aggregation (cumulating interactions that
 happened in a certain time window into one single static network
 instance). This problem is unavoidable,
 but poorly studied. Therefore, we pay special attention to the length
 of the cumulation time window and 
its effect to the aggregated network's properties.



This deliverable mainly focuses on networks of human interactions,
since they are those for which the highest amount of detailed
information is available.
Analysis of large datasets about human activities and interactions
have long been limited by the difficulty of gathering information. 
The ever-increasing use and availability of  digital information are
however widely enabling the collection and analysis of massive amounts
of data about many aspects of human behavior. 
In particular, modern portable sensing devices allow
researchers to collect and datamine massive amounts of information about
human activities, thus changing substantially the approach to the study of
human and social interaction. Bluetooth or wifi technology give access to
proximity patterns \cite{Hui:2005,Eagle:2006,Kostakos,Pentland:2008,persistence}, and even face-to-face interactions
can be resolved \cite{Sociopatterns,Cattuto:2010,alani,percol} both spatially and temporally.
In this context, the use
of a description in terms of complex networks has been a widely 
successful tool \cite{science,reviews,watts-short,Wasserman:1994}, thanks to its versatility.   
% Recent technological steps forward enable even data gathering about
% real world interactions.  In particular, modern portable sensing
% devices allow researchers to collect and datamine massive amounts of
% information about human activities, thus changing substantially the
% approach to the study of human and social interaction.  All these
% important progresses allow researchers to go beyond the study of small
% scale networks, and to gather longitudinal data, up to now difficult
% to study in social network analysis
% \cite{Padgett:1993,Lubbers:2010}. 
% Viewing interaction networks as
% dynamical objects opens many interesting issues, such as for instance
% the consequence of the interaction dynamics on the dynamical processes
% taking place on these networks.
 
We developed software tools for the analysis of the collected data and
the exploration of the network structural and dynamical properties (degree
distribution, assortativity, weight and strength distribution to name
 but  a few).
% As far as networks of human interactions are concerned,  we focused on
% two different case studies, namely
The datasets under study are the following
\bi
\item face-to-face human interaction data collected in two different
  experiments \cite{dublin,ht2009}  carried out within the SocioPatterns
  project \cite{Sociopatterns} and kindly made available to Dynanets;



\item historical data on the HIV infection spread among men
who have sex with men  (from Amsterdam  \cite{amsterdam} and San
Francisco \cite{sanfrancisco} 
cohort studies).
%\item the data held by Laszlo...
% \item an exhibition held at the Science
%   Gallery in Dublin, Ireland, from April $17^{\rm th}$ to July $17^{\rm
%     th}$,
% 2009, hereafter referred to as Dublin and
% \item   a scientific conference (Hypertext 2009, or
% HT2009), hosted by the Institute for Scientific Interchange Foundation
% in Turin, Italy, from June $29^{\rm th}$ to July $1^{\rm st}$,
% 2009.
\ei

    Furthermore, within the scope of WP3, we complemented the tools for network
 analysis with tools for network generation and  basic  tools for network
visualization. In particular
\bi
 \item we developed an 
 agent-based model for human dynamics  \cite{alain-model}. 
The model enforces a simple rule for human dynamics which can be stated as ``the longer an agent interacts with
a group, the less it is likely to leave the group; the more
the agent is isolated the less likely it is to interact with
a group''. 
Despite its simplicity, this agent-based model is able to capture some of the
essential features of human dynamics (e.g. broad contact duration distribution) as described in the following.
 
\item We also  developed computational tools for the HIV infection
network. The investigation of the spreading of HIV and its drug resistance
requires
a holistic approach of various dynamics at multiple spatiotemporal
scales. In addition,
the complex networks that arise from such models will be so large  that
performance
becomes an issue.
Multiscale, multiphysics systems are too complex for accurate  analytical
treatments 
while their numerical simulation is often a  daunting task, yet such systems arise
everywhere from modeling the immune system and proteine interaction
to epidemic spread in a human population. Unfortunately at present,
researchers create their own ad-hoc programs for their particular study.
To address  this problem we present the Simulator for Efficient Evolution
on Complex Networks (SEECN~\cite{simulator}), an expressive simulator of complex
systems  whose salient features are
described in the following.

\item Finally, we  created and studied Elementary Dynamic Network models
(EDNs), the dynamic counterparts of the classic network models for
static networks: 
like Erdos-Renyi (ER), Watts-Strogatz (WS) and Barabasi-Albert (BA)
networks. We investigated in detail the correlation of  the average
betweenness centrality (BW) with the network density (defined as the ratio of
 the actual number of links in the network to the maximum possible number of
 links in the network) while performing a series of operations on
the network \cite{laszlo}.
The importance of the  BW of a node stems from its expressing the fraction of shortest paths
that would be cut by the removal of that node, hence the average
BW is a proxy for the expected number of shortest paths
being affected by the deletion of a random node. Similarly,
maximum BW points to the maximum damage that can occur
by the removal of a single node. As a consequence, the average BW is
an important structural indicator of network robustness.
\ei

% We also developed a computational platform so simulate  disease spreading between
% different cities. To this aim, we adopted a hierarchical network
% approach which allows for the representation of the different time and
% space scales involved, ranging from the  individual to the city level
% and  to the transportation network connecting different cities.
% The platform has been developed bearing in mind the possibility of
% future extensions to include the ever-increasing availability of data
% about human mobility. 

We also developed within the COSMO toolkit an algorithm to simulate disease spreading
        at the individual level, in a multi-scale fashion. To this aim, we adopted a hierarchical network
        approach
        which allows for the representation of the different time and space
        scales involved,
        ranging from the individual to the city level and to the mobility
        network
        connecting different cities. The algorithm has been developed bearing in
        mind the
        possibility of future extensions to include the ever-increasing
        availability of data
        about human activities.


\section*{Network Representation}
 \addcontentsline{toc}{section}{Network Representation}    


The efficient representation of the evolving network is crucial
for the performance of the tools for data analysis.
Due to its simplicity and efficiency, we resorted to a time-dependent
edge list.
Every node is  identified by an
integer number chosen as its ID. %   Due to this identification, in the following we
% will refer to interaction among tags A and B as a shorthand to mean
% interaction among individuals carrying tags A and B.  
% Individuals are then mapped into nodes of the
% human interaction network and the contacts among different individuals
% into network edges.
   The interaction between nodes is then mapped into network 
edges.
The interpretation of the interaction establishing a link between two
nodes depends of course on the system in exam (e.g. a
face-to-face interaction for the human interaction networks  and  a
sexual contact when dealing with the HIV infection network).
% The evolving network is then expressed as  a time dependent
% edge list 
%  where at every (discrete) time $t_{i}$ we report the ID of the
%  interacting tags (which are establishing an edge in the human
%  interaction network). An isolated tag at time $t_{i}$ is expressed as a self-interacting
%  tag like for example tag $1300$ at time $t_{0}$ in Table~\ref{time-edge-list}. 
The evolving network is then expressed as  a time-dependent
edge list 
 where at every (discrete) time $t_{i}=i\delta$ we report the ID of the
 interacting nodes. Here $\delta$ stands for an  time window
 corresponding to the minimum time scale one can resolve in the
 system. Of course this time scale depends on the system under
 scrutiny (it would correspond to 20s for the networks of
 human contacts and to 3 months for the HIV spreading simulations).
 An isolated node at time $t_{i}$ is expressed as a self-interacting
 node like for example node 1300 at time $t_{0}$ in
 Table~\ref{time-edge-list}. 
%Every discrete time $t_{i}$ is in turn a time window. 

The time-dependent edge list representation of a dynamic network
amounts to stitching together several consecutive network snapshots. 
One can easily track e.g. a single
node  (for instance to investigate its contact duration distribution, the
number of different nodes he established a contact with and so
on) by simply looking for the occurrences of its
corresponding ID number.
This representation has proved particularly useful to generate aggregated
networks of human contacts (discussed in detail in the following), but it is a quite general representation
of an evolving network.



\begin{table}
\caption{Representation of a dynamic network as a time-dependent edge list.}\label{time-edge-list}
\begin{center}
    \begin{tabular}{ | l | l | l | p{5cm} |}
    \hline
    Time & ID of node A  & ID of node B  \\
 \hline
    $t_{0}$ & 1100 & 1200 
 \\
 \hline
    $t_{0}$ & 1300 & 1300 
 \\ 
 \hline
    $\cdots$ & $\cdots$ & $\cdots$ 
 \\ 

\hline
    $t_{1}$ & 900 & 1000  \\
    \hline

    $t_{1}$ & 1100 & 1500 
 \\ 
 \hline
    $t_{1}$ & 1100 & 1200 
 \\ 

\hline
    $\cdots$ & $\cdots$ & $\cdots$ 
 \\ 
 \hline
    $t_{i}$ & 1400 & 1800 
 \\ 




\hline
    $\cdots$ & $\cdots$ & $\cdots$ 
\\
\hline


    \end{tabular}
\end{center}
\end{table}



\section*{Data}
 \addcontentsline{toc}{section}{Data}




%\subsection{Representation of Dynamical Networks}

The data about human interactions was collected in two strongly
different contexts. 
The first one is an exhibition held at the Science Gallery
in Dublin, Ireland, from April 17th to July 17th , 2009 (hereafter
referred to as SG).
 The second is a scientific conference (Hypertext 2009, or HT09), hosted by the Institute for
Scientific Interchange Foundation in Turin, Italy, from June 29$^{
  \rm th}$ to July 1$^{\rm st}$, 
2009. Intuitively, interactions among conference participants differ
from interactions
 among museum visitors, and the concerned individuals have very
different goals in both settings.
The deployed infrastructure  to collect data about human interaction uses radio frequency devices (RFID)
embedded into conference badges engaging in ultra-low power
bidirectional packet exchange, as described in \cite{Sociopatterns,Cattuto:2010,alani,percol}.
Conference participants at HT09 and museum visitors volunteer to
carry these wearable RFID devices.
% Exchange of data
% packets between badges is only possible when two persons are at close
% range ($\sim 1$m) and facing each other, as the human body acts as a
% RF shield and blocks the exchange of low-power packets. The devices
% emitting and receiving rates are tuned so that the face-to-face
% proximity of two individuals wearing the RFID tags can be assessed
% with a probability in excess of 99\% over an interval of 20 seconds,
% which corresponds to a typical timescale for social interactions. When
% such proximity (or ``contact'') is detected, it is reported (on a
% different radio channel) to radio receivers, called RFID readers,
% installed in the deployment area. The radio receivers are connected to
% a central computer system by means of a Local Area Network. After a
% contact has been established, it is considered ongoing as long as the
% involved devices continue to exchange at least one such packet for every
% subsequent 20s interval. Conversely, a contact is considered broken if
% a 20s interval elapses with no low-power packets exchanged. As a
% consequence, time is treated as a discrete variable with a granularity
% given by the 20s time window described above.
The deployments at the Science Gallery in Dublin \cite{dublin} and at
HT09 conference \cite{ht2009} in Turin involved a vastly different number of
individuals and stretched along different time scales. The former
lasted about three months and recorded the interactions of more than
14,000 visitors (more than 230,000 recorded face-to-face contacts),
whereas the latter took place during three days involving about 100
conference participants (about 10,000 contacts).%   The contexts are
% also very different: in a museum, visitors spend a restricted amount
% of time (at most few hours, well below the maximum stay permitted by
% the museum opening hours from 12:00 to 20:00), do not return, and
% follow a pre-defined path between the different parts of the
% exhibit. In a conference on the other hand, most attendees remain for
% the whole conference (few days), and move at will between different
% areas (conference room, coffee and lunch break areas, etc...).  In the
% case of the HT09 conference almost all of the participants volunteered in the
% experiment, likewise a high percentage of museum visitors were
% equipped with a RFID tag upon entering the science gallery.


% \subsection*{Network generators}
%  \addcontentsline{toc}{section}{Network generators}

The historical data about the HIV infection describe the evolution of
the number of infected patients at    San
Francisco and  Amsterdam 
cohort studies. The former study   ran 
from 1980 to 1999. During such period  26,176 first-time interviews 
with men having sex with men were conducted.
The latter study started  in 1984 and is still ongoing. In total about 24,000 men
having sex with men  
have participated in the study so far.
%   The data covers a period of $x$ and
% $y$ years and includes an overall $t$ and $z$ individuals in Amsterdam and
% in San Francisco, respectively.   
Unlike the case of the networks of human interactions, the modeling of HIV infection network  has to cope with the incomplete
information about the network itself, hence a different approach is
called for.



\section*{Models}
 \addcontentsline{toc}{section}{Models}


In SEECN, a complex network represents the system where the nodes and
edges have specified properties which dictate the dynamics of the network
over time. The developed application is a detailed model of HIV spread among men
who have sex with men and serves to show the simulator's expressiveness
and to evaluate its performance.

% The model of HIV that has been simulated with SEECN is as follows. A million MSM
% can be either of Healthy, Acute HIV-infected (3 months), Asymptomatic
% HIV-infected without treatment, Asymptomatic HIV-infected with treatment,
% and AIDS.
The epidemic model simulated with SEECN is fairly complex and
represents the HIV infection network in
a population of homosexual men, therefore each node has the same type (male). This model assumes a hierarchical network
with power-law exponent 1.6 (i.e. the probability that a homosexual 
has $n$ sexual partners in a 3-month time window decays
like 
$P(n)\propto n^{-1.6}$), and classifies nodes as healthy, acute, (un)treated
asymptomatic, or (un)treated AIDS.


We assume that an untreated HIV patient has an expected progression time
into AIDS of thirteen years, and a slightly lower median. The expected progression
time of a treated HIV patient into AIDS is taken to be about twenty-two years. Acute
HIV lasts for approximately three months, which we take as the length of one time
step.
Treatment uptake is typically modeled as a percentage of untreated patients per
quarter. We calculate the uptake to ensure that about   70\%
of a node's time of
infection is under treatment, and assume that    30\%
of the treatments fail. % In the first
% simulation step the uptake is set to zero.
Various drug treatments may
be introduced
at different times, but in our simulations we model only one treatment type that is
available at time step zero, and the use of which gradually increases.
We further assume that about  60\%
of untreated infected nodes is actually diagnosed and aware of their
status, which results in  25\% 
less risky behavior. 
    Nodes in
the acute HIV stage are taken to be much more infectious due to a peak of viral body count.
%  each of which have distinct infectiousness,
% life expectancy and duration of relationships.
  Dynamics and parameters also   include condom use, % diagnosis rate,
% treatment uptake, infection probability reduction due to treatment, AIDS onset,
scale-free distribution of number of partnerships including community
structure, partnership duration, and
distinction of steady and casual partnerships. For a detailed list of the parametrization of node
and edge dynamics in the HIV infection network, see  appendix A of
Ref.~\cite{rick-master}.
  We have found
remarkable resemblance
to historical data (from Amsterdam Cohort Study and San Francisco
cohort studies). 
% Our domain is the homosexual community, so each node has the same type (male).
% % The infection status can be one of healthy, acute HIV, untreated HIV, treated HIV and
% % AIDS.

 



The model for human dynamics introduced in Ref. \cite{alain-model},
instead, characterizes every agent with only two parameters, namely
the number $p_{i}$ 
of other agents with which it is in
contact (i.e. its degree in the network) and the time $t_{i}$  at
which $p_{i}$ last evolved.  
Each agent
can either be isolated or belong to a group with other
agents, and the groups define an instantaneous contact
network. During the dynamics, agents can join other
agents or on the contrary leave the group they belong to.
One can calculate numerically several quantities of 
  interest such as for instance time spent by agents in
each state, the duration of contacts between two agents,
and the time intervals between successive contacts of
an agent.
When we impose the  self-reinforcement mechanism according to which
the more an agent is in a state,
the less likely it is to change its state, then in all cases, 
broad distributions are obtained for all the aforementioned quantities
of interest, in agreement with what
observed empirically \cite{alain-preprint}.
To fix the ideas, we show some quantities of interest in
Fig.\ref{tau-alain} where we plot the distribution of contact
durations between two agents (a),  time intervals between the beginnings of successive
contacts of an agent $A$ with two different agents $B$ and $C$  (a
measure of the ability of an agent to efficiently spread information)
(b) 
and the triangle duration probability distribution (c) for two
different values of the parameter $b_{1}$, which parametrizes the
tendency of an agent to leave an existing group \cite{alain-preprint}.

\begin{figure}  
\begin{center}
\includegraphics[width=0.7\columnwidth, height=5.5cm]{tau-tABAC-triangles.pdf}
\caption{ Distributions of (a) duration of a contact between two
agents; (b) time intervals between the beginnings of successive
contacts of an agent $A$ with two different agents $B$ and $C$; (c)
duration of a triangle.  }\label{tau-alain}
\end{center}
\end{figure}


Finally, we describe briefly the EDNs used to carry out the numerical experiments

\bi
\item ER1: Starting from an arbitraty network, in each time step, any
  non-existent edge is added with 
probability $p$ and each existing edge is deleted with probability $q$.
\item ER2: Starting from an arbitrary network, in each time step,
  $k_{1}$ existing edges are deleted (selected uniformly random from 
 existing edges) and $k_{2}$ non-existing edges are added (selected uniformly random from non-existing edges).
\item ER3
Starting from an arbitrary network, in each time step, $k$
 existing edges are rewired by moving one end of the link to another
 node.
\ei
%  Since many network properties are known to be sensitive to network
%  density, 
% we concentrated on density preserving EDNs i.e. we implemented dynamic network models where the density of the network stays
%  about constant over time \cite{laszlo} (network density is defined as the ratio of
%  the actual number of links in the network to the maximum possible number of
%  links in the network).  
In order to understand the impact of one-time sampling an evolving
network,
  we experimented with four EDNs, ER1, ER2 and ER3, as described
  above, 
 plus a dynamic version of the classic Watts-Strogatz model.
%  statistics (density, degree distribution, 
% clustering, number of components, size of the largest component,
% average path length, etc.) about these networks in
%  each point in time, studying time-local and cumulative networks in
%  parallel \cite{laszlo}.
% Our initial results imply the sensitivity
%  of some of these measures on both the length of the cumulation time
%  window and on the particular
%  model version applied.
% In addition to the basic studies described above% ,
Here we only report the results of the experiments
  we carried out
  with decreasing densities,  in part, 
 due to their immediate connections to existing works on robustness. In
 network theory, robustness is understood as the resilience
  of networks in terms of connectivity and average path length, subject
  to the repeated removal of nodes (and links connected to them).
  We first successfully replicated the results in
  Ref. \cite{barabasi-resilience}
  regarding the random and degree-based removal of nodes.
  % We then extended these results by exploring a larger region of initial
  % densities and by analyzing the change in betweenness
  % centrality.  
  Then we introduced another removal scheme (a
  second ``attack scenario''), i.e., the removal of high
  betweenness nodes. Our results show an initial decline in average
  betweennes in both Erdos-Renyi and Barabasi-Albert networks, 
 as expected. However, after a critical loss, this is followed by a
 new characteristic peak, as shown in Fig. \ref{laszlo-networks},
 where each line is obtained by averaging the results on one hundred EDNs.
% {\bf Ask laszlo to provide a figure? Should this last part be made a
%   bit clearer?}.



\begin{figure}  
\includegraphics[width=6cm, height=6cm]{las3b.pdf} 
\includegraphics[width=6cm,height=6cm]{las1b.pdf} \\
\includegraphics[width=6cm, height=6cm]{las4b.pdf} 
\includegraphics[width=6cm,height=6cm]{las2b.pdf} 
\caption{Average BW dynamics of ER and BA networks 
in time using various density lowering schemes. ER networks are on the left, BAs are on the
right. In the first row the initial density is $d = 0.004$ (average
degree is $k=4$), in the second $d = 0.008$ ($k = 8$).
%  (These correspond to link probabilities
% $p = 0.004, 0.008$ in the ER model and $e = 2, 4 link per newly arriving node in the BA model.)
Blue lines depict random node removals, reds are
maximum degree-based, while greens maximum-BW based removals.
   }\label{laszlo-networks}
\end{figure}

% Always within the scope of WP3, we began the development of a computational platform
% aimed at simulating the spread of a disease (e.g. influenza) among different cities.
% To this aim, we resort to a hierchical network comprising
% different spatial and temporal scales, hereafter referred to as levels.
% So far we have implemented human
% dynamics in a subpopulation   representing a city (individual level) and the migration flow
% of individual between cities (city level).
% Human dynamics  at the individual level is simulated by resorting to an
% agent-based model where every individual is a node  characterized by an
% infection status (according to an SIR model), a viral load, its mobility
% properties and its relations with other individuals.
% At the city level, instead, every node stands for a city characterized by properties
% such as epidemic seasonality, natality/mortality rates and a migration
% rate establishing a link between different cities.
% The computational platform allows for the future integration of real
% mobility data which will lead to the inclusion of a new level in the model,
% thus further enriching the dynamics of disease spreading.
 

Always within the scope of WP3, we developed within the COSMO toolkit an algorithm
        aimed at simulating the spread of a disease (e.g. influenza) in a multi-scale fashion.
        To this aim, we resort to a hierarchical network comprising
        different spatial and temporal scales, hereafter referred to as levels.
        So far we have implemented human
        dynamics at the individual level within a subpopulation representing a city, and
        the migration flow
        of individual between cities (city level).
        Human dynamics  at the individual level is simulated by resorting to an
        agent-based model where every individual stands for a node
        characterized by an
        infection status (modeled according to an SIR model), a viral load, its
        mobility properties and its relations with other individuals.
        In the network at the city level, instead, every node stands for a city
        characterized by properties
        such as seasonality, natality/mortality rates and a migration
        rate establishing a link between different cities.
        The algorithm allows for the future integration of real data for mobility and node features
        for specific applications to the dynamics of disease spreading.
        



\section*{Aggregated networks and temporal analysis}
 \addcontentsline{toc}{section}{Aggregated networks and structural analysis}
% The time-dependent edge list representation shown in
% Table~\ref{time-edge-list} is
%  an extremely convenient representation of a
% dynamical network. 

The time-dependent edge list Table~\ref{time-edge-list} allows one
easily obtain the  aggregated network along a given time interval.
Indeed aggregating the network in the time interval $[t_{\rm ini}, t_{\rm
  fin}]$ amounts simply to selecting the corresponding time window in the
time-dependent edge list.
In the aggregated network   an edge is drawn between node $A$ and node
$B$ if at least one contact was detected between those nodes during the aggregation time
interval. 
The duration of an interaction between node
 $A$ and node $B$ can then be
easily calculated by simply counting the occurrences of $A\leftrightarrow
B$ edges multiplied by the time window $\delta$. 
We stress that the strategy described above is by no means limited to
the datasets  analyzed in this deliverable but it can be applied to  any dynamic network whose time-dependent edge list is known
and for which it makes sense to introduce 
 the concept of link duration. 
Furthermore,  finding the neighbors of  node $A$ (and therefore the degree
distribution of the aggregated network when we iterate the procedure on
all the nodes) simply amounts to identifying
the number of unique $A\leftrightarrow B$ links, with $B\neq A$,   in the aggregated
network.
Finally, the calculation the lifetime of a node in the aggregated network
(which would be e.g. the time spent by a visitor in the SG in the case
of networks of human interactions) is reduced to identifying the first
and last occurrence of that node in the aggregated network. 
  
These methodologies have been implemented and extensively used in the
case of networks of human interactions.
In the case of the network of face-to-face interactions we will refer
to interaction
 among tags $A$ and $B$ as a shorthand to mean
 interaction among individuals carrying tags $A$ and $B$.  
% In the case of the network of face-to-face interactions we will refer
% to interaction
%  among tags $A$ and $B$ as a shorthand to mean
%  interaction among individuals carrying tags $A$ and $B$.  
For the aggregated networks of human contacts,  each edge is thus naturally weighted by the total duration of the
various contact events occurred between these tags, i.e., by the total time
during which the individuals have been in contact.
    The choice of daily time windows seems quite natural as it 
would represent for instance a typical time scale for a representation of social
networks based on surveys of the participants, in which each participant
would (ideally) record who s/he has encountered during the day. Such a
choice of time-window, albeit natural, is by no means compulsory. For
instance, we have also studied the museum data along longer periods (weeks,
months) to investigate the stationarity of some properties of the collected
data. Shorter aggregation times of the order of a few minutes are also
useful for instance in order to investigate the variation of the number of
museum visitors within a single day.


Examples of aggregated networks from both a day at HT09 and a few
representative days at the SG are  given in
Fig. \ref{aggregated-networks}, where we highlight the network
diameter.
\begin{figure}  
\includegraphics[width=0.5\columnwidth, height=5.5cm]{original_network.pdf}
\includegraphics[width=0.5\columnwidth,height=5.5cm]{original_network-14-7.pdf}
\includegraphics[width=0.5\columnwidth, height=5.5cm]{original_network-19-5.pdf}
\includegraphics[width=0.5\columnwidth,height=5.5cm]{original_network-20-5.pdf}
\caption{Aggregated network along one day (with highlighted network
  diameter). Clockwise from top: HT09 conference  and three
  representative days 
at the Science Gallery Museum in Dublin.   }\label{aggregated-networks}
\end{figure}
The aggregated network at the HT09 exhibits a short diameter and a high connectivity, whereas the
aggregated networks at the SG  have a much longer diameter and are
often split into two connected components (CC).
After aggregating the network on the desired time interval, the tools
developed for WP3 can be used to perform a number of statistical
analysis on the network. 
As an example, we calculate 
the degree distribution $P(k)$, i.e. the probability that a randomly
chosen node has $k$ neighbors, which is one of the most important
quantities  
 to characterize a network topology. In Fig.~\ref{P-k} we show the degree
distributions obtained by gathering together the data from the
aggregated networks on a $24$-hour basis for the whole duration of the
deployments in the museum (right) and at the HT09 conference
(left). For the museum data, we left out the (few) isolated nodes
which obviously contribute only with $k=0$ to the degree
distribution.
\begin{figure}  
\includegraphics[width=0.5\columnwidth, height=0.5\columnwidth]{N_P_k_ht2009_plot_tik.pdf}
\includegraphics[width=0.5\columnwidth,height=0.5\columnwidth]{N_P_k_dublin_plot_tik.pdf}
\caption{Left: degree distribution $P(k)$ 
at HT09. Right: as in left  for the aggregated 
networks from the SG. }\label{P-k} 
\end{figure}   





The tools for structural network analysis include the generation  a
randomized version of the original
  network by resorting to the rewiring
procedure described in Ref.~\cite{rewiring}, which preserves the
degree distribution $P(k)$ while destroying correlations. % Care is
% taken to prevent the rewiring from merging initially distinct CC. The
% rewiring procedure cannot be extended to the rare CC consisting of
% fewer than four nodes. Such small CC are thus removed from the
% aggregated networks before rewiring.
\begin{figure}  
\includegraphics[width=0.5\columnwidth, height=5cm]{ht2009-rewired.pdf}
\includegraphics[width=0.5\columnwidth,height=5cm]{dublin-rewired.pdf}
\caption{Left: randomized version of the aggregated networks in the
  top row of Fig.~\ref{aggregated-networks}. Left: HT09,
June, 30$^{\rm th}$. Right:  SG, July, 14$^{\rm th}$.}\label{rewired-networks}
\end{figure}
A randomized network is often
needed as a null model to carry out statistical analysis.
  In Fig.~\ref{rewired-networks} we
plot a single realization of the null model for the networks in the
top row of Fig.~\ref{aggregated-networks}. One notices that the
rewired version of the aggregated network at the HT09 is very
similar to the original aggregated network, whereas the null model for
the aggregated network in the museum on July, 14$^{\rm th}$ is more compact
than the original network and exhibits a much shorter
diameter. Similar considerations hold for the other aggregated
networks of the museum setting.
Once again, we stress that the tools for network analysis are not
limited to the networks of human interaction used as one of the case studies in this deliverable.

The tools developed for WP3 allow one to represent the network
longitudinal dimension by studying  several temporal properties of the
nodes and/or the links.
For instance, in  Fig. \ref{P-visits} we show the distribution of visit durations
obtained collecting the data for the whole duration of the
experiment at the SG and a fit to a lognormal distribution (red line)  
with geometric mean
around $\mu\simeq 35$ minutes.
\begin{figure}  
\begin{center}
\includegraphics[width=0.5\columnwidth,height=0.5\columnwidth]{visit_histogram_tik.pdf}
\end{center}
\caption{Visit duration distribution at the SG (histogram) and fit to a
  lognormal distribution (red line). }\label{P-visits}
\end{figure}  
 This shows that, unlike the case of
the conference, here one can meaningfully introduce the concept of a
characteristic visit duration which turns out to be well below the
cutoff imposed by museum opening hours.
The existence of a characteristic visit duration sheds light on the
elongated aspect of the aggregated networks of visitor interactions
(see Fig.~\ref{aggregated-networks}). Indeed museum visitors are
unlikely to interact directly with other visitors entering the museum
more than an hour after them, thus preventing the aggregated network
from exhibiting small-world properties.
In Fig \ref{node-color-code} we show the aggregated networks for two
different days at the SG. Each node is colored according to the
invidividual's entry time slot. Again, the network diameter is
highlighted. The network diameter connects early visitors with late
visitors, thus exhibiting a temporal beside a topological meaning.
This also shows that network topology and network longitudinal
dimension are deeply interwoven.
\begin{figure}  
\includegraphics[width=\columnwidth, height=0.5\columnwidth]{dublin_color_panel.pdf}
%\includegraphics[width=0.5\columnwidth,height=5cm]{visit_histogram.pdf}
\caption{Aggregated networks at two different days in the museum. Nodes
  are colored according to the visitor's entry time slot. The network  
  diameter is highlighted.  }\label{node-color-code}
\end{figure}  

Many other temporal quantities of interest can be investigated to shed
light on the human dynamics.
For instance, we can calculate the contact  duration and cumulative
contact duration (weight $w_{ij}$) distribution, as shown in
Fig. \ref{P-contact-and-weight}.
We notice the broadness of both distributions which decay only
slightly faster than a power law.
Finally, we mention the strength distribution $P(s)$, where $s$ (node
strength) is the total time an individual spends interacting with
other individuals. As shown in Fig. \ref{P-s} (right), $s$ spans several
orders of magnitude and $P(s)$ is a monotonically decreasing quantity
for the SG, with a plateau in the range from 2 to about 20
minutes.
On the other hand (see the left diagram in Fig. \ref{P-s}) for the
HT09 $P(s)$ exhibits a peak for $s\simeq 5$ minutes.

The network visualizations  in this section were produced  
using the igraph library \cite{igraph}. Beside being visually appealing, they help
researchers to intuitively grasp some of the most relevant (non
necessarily only topological) 
features of the aggregated networks, which can be then be investigated
 quantitatively with the tools provided by WP3.


To conclude this section, we remark that the tools developed within
WP3 allow for the investigation of many structural and longitudinal
network properties, thus enabling the topological and dynamical
characterization of the evolving network.



\begin{figure}  
%\includegraphics[width=0.5\columnwidth,height=5.5cm]{Comparison-contact.pdf}
\includegraphics[width=0.5\columnwidth,height=0.5\columnwidth]{Comparison-contact_tik.pdf}
\includegraphics[width=0.5\columnwidth, height=0.5\columnwidth]{P_w_comparison_tik.pdf}
\caption{Left: contact duration distributions  at
  HT09 (triangles) and  at the SG (circles). Right: distributions of the daily aggregated networks
  HT09 (triangles) and  at the SG (circles).}\label{P-contact-and-weight}  
\end{figure}  
 

 
% \begin{figure}  

% \caption{Weight distributions of the daily aggregated networks
%   HT2009 (triangles) and  in the museum (circles).}\label{P-w}
% \end{figure}  


\begin{figure}  
\includegraphics[width=0.5\columnwidth, height=0.5\columnwidth]{P_s_ht2009_plot_tik.pdf}
\includegraphics[width=0.5\columnwidth,height=0.5\columnwidth]{P_s_dublin_plot_tik.pdf}
\caption{Left: Strength distribution $P(s)$ at HT09. Right: as in left  for the aggregated
  networks at the SG. }\label{P-s}
\end{figure}  



\section*{Performance Report}
 \addcontentsline{toc}{section}{Performance Report}
All the software tools for the analysis of networks of human
interactions were developed either using Python or the R language for
statistical analysis. 
Both scripting languages are widely spread in the scientific community
and provide a wide choice of high quality packages for network
analysis which are typically  a thin layer over modules written in C/C++ or
Fortran for performance reasons.

We took advantage of time-dependent edge list representation allowing one to split
different tasks. 
For instance, the calculation of the degree distribution $P(k)$ or 
network rewiring require necessarily the previous generation and
storage in memory of the corresponding
aggregated network.

On the other hand, the calculation of  several other quantities of interest like e.g. 
the contact duration distribution $P(w_{ij})$ essentially amounts to
looking for the occurrences of node IDs in the time-dependent edge
list. As a consequence, we are able to separate to some extent the
analysis of network topology from the analysis of some network
temporal properties, thus saving computational time since both tasks
can be carried out independently.
% As a consequence, the calculation of $P(w_{ij})$, $P(\Delta)$, and the
% visit duration distribution can be carried out without constructing the
% corresponding aggregated network. This means that, if needed, some 
% node temporal properties can be accessed without the need of generating the
% aggregated network.
    




 % The analysis of networks of human contacts  described in this deliverable are all within the
 % computational power of a modern up-to-date desktop PC and can be
 % performed on a time scale of a few hours at most.

As a benchmark, we report that all the analysis of networks of human
contacts presented in  this study can be carried out within 1-2 days
using a single up-to-date desktop PC. 
% The crucial factor for performance is not the length of the collected
% dataset, but rather the size of each aggregated network.
Indeed even   the large dataset collected at the SG recording the
interactions of tens of thousands visitors ends up broken
into   about 80 daily
aggregated networks each consisting of a few hundred nodes.
% This eases the numerical work, but we stress that we were able to run
% the same kind of 
% analysis on longer aggregation periods (weeks or months) always
% within the time scale of one or two days  on a single
% multi-core desktop PC.
% As a consequence, we validated and tested the software tools on
% networks ranging from a few tens to several thousands of nodes.
Finally, we mention that the software tools for network analysis make
extensive use of the igraph library \cite{igraph} which has been
reported to be able to handle network sizes of several hundred thousands of
nodes or more. 
As a consequence, we predict the scalability of the software tools
developed for the analysis of networks of human contacts up to
networks consisting of $\sim 10^{5}$ nodes at least.
Furthermore, we remark that we used the same hardware to generate all
the network visualizations in this report. The network plots are
generated once again resorting to the igraph library \cite{igraph}
within a few tens of seconds at most.

The complex network simulator deployed for the HIV, SEECN, is  optimized for both
single-core and multi-core performance. It features the capability of
integrating the information from different time scales while confirming the  computational
feasibility of agent-based modeling
combined with complex networks.
The required computational time depends of course on the size and
complexity of the desired agent-based simulation, but as a benchmark
we report that a detailed simulation of HIV among one
million persons
in a hierarchical and scale-free network over 25 years (100 time steps) takes
two minutes using 16 processes.

As to the EDNs experiments, we point out the that the numerical
experiments we ran were not 
computationally expensive, 
as the basic models that we experimented with had a size $N\simeq 100$. 
However, considering the size of real networks (where 100 nodes count 
as extremely small) and the possible
 number of connections that increases quadratically with the number of
 nodes, simulations of EDNs may require a 
considerable computational effort. Especially, as for a full
experimental treatment, the same simulation has to be
 run several times with the same parameter combinations (but with
 different random number seeds) in order to assess 
the robustness of our findings. Moreover, we also intend to do
sensitivity analysis (i.e., the replication of our
 measurements with different parameter combinations). All these
 together will amount to a significant computational
 effort, but all within the controllable range. A single simulation
 run with 1000 nodes, performing all the intended 
network statistics in each time step takes us about one week on a single core PC. 
 
% \paragraph{paragraph title}
% \subparagraph{subparagraph title}

\section*{Conclusion}
 \addcontentsline{toc}{section}{Conclusion}
Within the scope of WP3 we developed computational tools for the
generation and temporal modeling of dynamical networks. Such tools
allow for 
the generation both of an aggregated network from time-ordered network snapshots 
 and  of a   network    parameterized by a set of
statistical parameters. We tested their scalability to network
consisting of tens to hundred of thousands of nodes.  Such tools have been developed bearing 
in mind their future extension to incorporate the dynamics of
processes both of the network and on the network, which will be the
object of the next deliverable and where the attention of the WP3 team
is already focusing.
  


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\end{thebibliography}
  
\end{document}