NETWORK MODELING IN INTERNATIONAL COUNTERACTING GLOBAL THREATS A. Tikhomirov ,
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Transcript of NETWORK MODELING IN INTERNATIONAL COUNTERACTING GLOBAL THREATS A. Tikhomirov ,
ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011
NETWORK MODELING IN INTERNATIONAL COUNTERACTING GLOBAL THREATS
A. Tikhomirov , International Informatization Academy, Moscow , RF
A.Trufanov , Irkutsk State Technical University, Irkutsk, RF,e-mail: troufan.istu.edu
A.Caruso, Court of Auditors, Regional Chamber of Control , Milan, Italy
A.Rossodivita , San Raffaele Hospital Scientific Foundation, Milan, Italy
E. Shubnikov,Institute of Internal Medicine, Novosibirsk, RF
R.Umerov, Crimean Engineering and Pedagogical University, Simferopol, Ukraine
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Networking in counteracting global threats: policy, research, education and practice
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Why do we care networking and networks ?
Because networks are of great value for Disasters and Emergencies
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Graph Theory originated in the moment when Leonhard Euler, Swiss, German and Russian mathematician, decided to prove that a passerby can not get around Konigsberg (modern Kaliningrad), using only one each of the seven city bridges.
Its key conclusion is: structural characteristics of graphs (networks) define a potential for their use.
The first example of using the methods of modern algebra in graph theory accounts for the work of the physicist Gustav Robert Kirchhoff, in 1845 he formulated so called Kirchhoff's laws to calculate voltages and currents in electrical circuits.
Advances in theory and practice of networks
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Introduction of probabilistic methods in graph theory, especially in research of Paul Erdős and Alfréd Rényi
on asymptotic probabilities of graphs created another branch known as theory of random graphs
Mathematician Dénes Kőnig published in 1936 a book titled "Theory of finite and infinite graphs” - the first textbook in the field of graph theory
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Four structural models for networks
• Regular networks ( e.g. crystal lattice)
• Random networks
• Small-world networks
• Scale-free networks
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Efficiency of a Network
The network efficiency E (G) is a measure to quantify how efficiently the nodes of the network exchange information.To define efficiency of G first we calculate the shortest path lengths {dij} between two nodes i and j. Suppose that every node sends information along the network, through its edges. The efficiency ij in the communication between vertex i and j is inversely proportional to the shortest distance dij: ij = 1/dij
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Current classification of complex networks
Three important complex network models:
• random graph model
• small-world network
• scale-free network model
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Erdös-Renyi Random graphs
Paul Erdös (1913-1996)
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Erdös-Renyi Random/Exponential / Homogeneous Graphs
• The Gn,p model
– n : the number of vertices– 0 ≤ p ≤ 1– for each pair (i,j),
generate the edge (i,j)
independently
with probability p
Exponential Graphs (Networks)
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Small world phenomena• Small worlds: networks
with short paths
Psychologist Stanley Milgram (1933-1984): “The man who shocked the world”
Measuring the small world phenomenon
dij = shortest path between i and j nodes
ijji,
dmaxd
ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011Watts and Strogatz model – WS that analyses Milgram’s theory
• Start with a ring, where every node is connected to the next k nodes (regular network)
• With probability p, rewire every edge (or, add a shortcut) to a uniformly chosen destination.
order randomness
p = 0 p = 10 < p < 1
Small World
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Properties of Small-world Graphs
– Large networks (n >> 1)– Sparse connectivity (avg degree k << n)
– No central node (kmax << n)
– Large clustering coefficient (larger than in random graphs of same size)
– Short average paths (~log n, close to those of random graphs of the same size)
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The next step
Barabasi-Albert (BA) model
( Barabasi model/ Scale Free model)
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P(k) k
32 These networks have no natural average number of edges and are called scale-free
Typical range for
Power – law distributionfor evolving self-organized networks was
proposed by Barabasi, Albert and collaborators
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Two types of network connectivity
1. Homogeneous network connectivity 2. Inhomogeneous network connectivity
Red nodes are most connected nodes ( cluster centers )
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Two types of network connectivity
1. Bad Workshops 2. ASI Slavonski Brod structure
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Power law tail kkP )(
kekP )( Exponential tail
random graphs (Erdös-Réyni) model
Exponential: Power law:
HUBS
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Scale free model :• The degree distributions of most real-life networks follow
a power law• there is a non-negligible fraction of nodes that has very
high degree (hubs)• scale-free: no characteristic scale, average is not
informative
Contrary random model :• highly concentrated around the mean• the probability of very high degree nodes is exponentially
small
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ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011
Two types of network connectivity
1. Homogeneous network connectivity 2. Inhomogeneous network connectivity
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Two types of network connectivity, but…
In real life we may encounter Variations on the Barabasi-Albert Model
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Truncated power-law
is the cut-off, s. t. the number of connections is less than expected for pure scale-free networks for
and the behaviour is approximately scale-
free within the range
0 1 2 3 4
02
46
8
log(1:l)
log(tabula
te(lin
ks1))
)/exp(~)( ckkkkp
ck
ckk
ckk 1
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One can consider that, in real networks
Link costThe cost of hosting new link increases with the number of linksE.g., for a Web site this implies adding more computational power, for a router this means buying a new powerful router
Node AgingThe possibility of hosting new links decreased with the “age” of the nodeE.g. nodes get tired or out-of date
Aging and Cost explain the “exponential cut-off” in power law
networks
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BA model advances
Power law distribution with Scale –Free properties ( that means that these networks have no specific scale contrary to random /exponential/ ones )
Preferential Attachment and Growth of a Network / Dynamic
Simple and clear terminology for all interested societies
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Power law distribution with Scale –Free properties ( that means that these networks have no specific scale contrary to random (exponential) ones
This implies that scale-free networks are self-similar, i.e. any part of the network is statistically similar to the whole network and parameters are assumed to be independent of the system size.
BA model advances
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Preferential Attachment and Growth of a Network ( dynamics )
• At every time step t, – A new node is connected to node i
• depends on the connectivity ki of node i
– The probability • ∏i = ki / ∑j kj
BA model advances
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ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011
Example: In practice sophisticated terms of Theory of Graphs are similar to Chinese ABC
Simple and clear terminology for all interested societies: nodes and links instead vertexes and edges of Theory of Graphs )
BA model advances
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Vulnerability of Networks: these are not left alone
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The error tolerance/ robustness of the network
Threat-Attack-Damage• Evaluation
– The changes in diameter when a small fraction f of the nodes is removed.
– The absence of any node in general increases the distances between the remaining nodes.
• How to remove nodes– Failure/ unintentional attack;
• Any node is removed with the equivalent probability– Attack/ intentional attack;
• The node which has the most connectivity is removed first.
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Attack• Attack / intentional attack (initiated by human beings)
– Will be on the most connected node rather than randomly
• Attack model– Remove the most connected node, – Continue selecting and removing nodes in decreasing order of
their connectivity k.
1 2
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Fragmentation
• Fragmentation– When nodes are removed from a network,
• Clusters of nodes may be cut off (fragmented) from the main cluster.
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Evaluation
• Attack Evaluation – one more metrics– S; The size of the largest cluster
• Divided by the initial total system size to normalize.
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So there are two important complex network models to explore
• Regular networks
• Random networks
• Small-world networks
• Scale-free networks
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• Exponential model attack
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The size of the largest cluster - exponential modelFailure/ unintentional attack; Attack/ intentional attack
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• Scale free model attack
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The size of the largest cluster- scale free modelFailure/ unintentional attack; Attack/ intentional attack
ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011
Complex Network tools have been successfully applied to understanding and counteracting such threats as infection diseases spread and terrorist activity.
Martin Rosvall† and Carl T. Bergstrom . An information-theoretic framework for resolving community structure in complex networks. PNAS . May 1, 2007, vol. 104 , N 18 , 7327–7331
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Complexity of a real problem
• Diversity of attacks–The impact of failures and attacks on the
network structure
BA model in not enough to explore all aspect of attacks
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Regular Model
Watts and Strogatz (WS)
Erdös-Renyi (ER)
Barabasi-Albert (BA)
Rossodivita- Trufanov (RT)
Caruso- Rossodivita- Shubnikov-Tikhomirov-Trufanov -Umerov
Levitin G, Hausken K.
Aminova - Rossodivita- Tikhomirov-Trufanov
Xiao, Xiao and Cheng
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S. Xiao, G. Xiao ,T. H. Cheng Tolerance of local information-based intentional attacks in complex networks.J. Phys. A: Math. Theor. 43 (2010) 335101Distributed attacks basically target on some or all of the live nodes adjacent to the
crashed nodes in each step, and the selections of the targets depend on only thelocal network-topology information.
Xiao models
Division of Communication Engineering, School of Electrical and Electronic Engineering,Nanyang, Singapore
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S. Xiao, G. Xiao On Degree-Based Decentralized Search in Complex NetworksarXiv e-print (arXiv:cs/0610173)
Decentralized search aims to find the target node in a large network by using only local information
Xiao, Xiao and Cheng models
Division of Communication Engineering, School of Electrical and Electronic Engineering,Nanyang, Singapore
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Caruso- Rossodivita- Shubnikov-Tikhomirov-Trufanov -Umerov models
Real life attacks : mixture of Failures and Attacks
Combined attacks model :
• Sequence of Failures and Sequence of Attacks
• Sequence of Attacks and Sequence of Failures
Failure/ unintentional attack; Attack/ intentional attack
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Failures+Attacks / Attacks + Failures
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Network connectivity
- Inhomogeneous network connectivity
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Real life : protection of nodes and links
Node Protection model :
•Protection barriers are constructed for network nodes with “thickness” d.d is sum of traditional protection measures:Ethical; Legal; Organizational; Technological; Physical; Math• Attenuation of any attack is proportional to exp(-µd) , where µ is a coefficient;µd=(µd)E+ (µd)L +(µd)O + (µd)T+ (µd)P+ (µd)M
Failure/ unintentional attack; Attack/ intentional attack
Caruso- Rossodivita- Shubnikov-Tikhomirov-Trufanov -Umerov models
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Real life : protection of nodes and links
Node Protection model :
•Metrics of d: Investments (Money)
d ~ Funding
Failure/ unintentional attack; Attack/ intentional attack
Caruso- Rossodivita- Shubnikov-Tikhomirov-Trufanov -Umerov models
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d0
d1
d2
d0<d1<d2
F. Galindo, N.V.Dmitrienko, A.Caruso, A. Rossodivita, A.A.Tikhomirov, A. I.Trufanov, E. V. Shubnikov, Modeling of Aggregate Attacks on Complex Networks. Information Security Technologies , Moscow – 2010, N3, P.115-121
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Real life : protection strategy of nodes and links
Node Protection Strategy model :
• how Investments should be distributed among different nodes
1.d ~ Const Funding 2.d ~ Funding (k)3.d ~ Funding (k2)
Failure/ unintentional attack; Attack/ intentional attack
Caruso- Rossodivita- Shubnikov-Tikhomirov-Trufanov -Umerov models
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d0
dC
dk
dk2
F. Galindo, N.V.Dmitrienko, A.Caruso, A. Rossodivita, A.A.Tikhomirov, A. I.Trufanov, E. V. Shubnikov, Modeling of Aggregate Attacks on Complex Networks. Information Security Technologies , Moscow – 2010, N3, P.115-121
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ConclusionER, BA & XXC, RT, СRSTTU - ( protected network,
aggregated attacks)
These models show how to built a Robust Network
RT model is complex and close to real secure Networking
CRSTTU - model is under developing…
ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011
Authors express their gratitude and sincere respect to NATO SPS Program
(with its ASI and ARW Institutions had led by Dr. F.Linkov, 2005;
Dr. P.Rumm and Prof. E.Stikova,2006; Prof. J.-G.Fontaine, 2010;
Dr. E.Gursky and Dr. B.Hreckovski, 2011 )