NETWORK MODELING IN INTERNATIONAL COUNTERACTING GLOBAL THREATS A. Tikhomirov ,

1. 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

2. Networking in counteracting global threats: policy, research, education and practice

3. Because networks are of great value for Disasters and Emergencies Why do we care networking and networks ?

4. Advances in theory and practice of networks • 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.

5. Mathematician Dénes Kőnig published in 1936 a book titled "Theory of finite and infinite graphs” - the first textbook in the field of graph theory 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

6. Regular networks ( e.g. crystal lattice) Random networks Small-world networks Scale-free networks Four structural models for networks

7. 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

8. Three important complex network models: random graph model small-world network scale-free network model Current classification of complex networks

9. Erdös-Renyi Random graphs Paul Erdös (1913-1996)

10. 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) Erdös-Renyi Random/Exponential / Homogeneous Graphs

11. 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

12. 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. Watts and Strogatz model – WS that analyses Milgram’s theory Small World order randomness p = 0 0 < p < 1 p = 1

13. 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) Properties of Small-world Graphs

14. Barabasi-Albert (BA) model ( Barabasi model/ Scale Free model) The next step

15. Power – law distributionfor evolving self-organized networks was proposed by Barabasi, Albert and collaborators Typical range for  These networks have no natural average number of edges and are called scale-free

16. Two types of network connectivity • 1. Homogeneous network connectivity • 2. Inhomogeneous network connectivity • Red nodes are most connected nodes ( cluster centers )

17. Two types of network connectivity • 2. ASI Slavonski Brod structure • 1. Bad Workshops

18. Power law: Exponential: HUBS random graphs (Erdös-Réyni) model Power law tail Exponential tail

19. 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

21. Two types of network connectivity • 2. Inhomogeneous network connectivity • 1. Homogeneous network connectivity

22. Two types of network connectivity, but… In real life we may encounter Variations on the Barabasi-Albert Model

23. 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

24. Aging and Costexplain the “exponential cut-off” in power law networks One can consider that, in real networks Link cost The cost of hosting new link increases with the number of links E.g., for a Web site this implies adding more computational power, for a router this means buying a new powerful router Node Aging The possibility of hosting new links decreased with the “age” of the node E.g. nodes get tired or out-of date

25. 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 BA model advances

26. Power law distribution with Scale –Free properties ( that means that these networks have no specific scale contrary to random (exponential) ones BA model advances 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.

27. Preferential Attachment and Growth of a Network ( dynamics ) BA model advances • At every time step t, • A new node is connected to node i • depends on the connectivity kiof node i • The probability • ∏i =ki / ∑j kj

29. Example: In practice sophisticated terms of Theory of Graphs are similar to Chinese ABC BA model advances Simple and clear terminology for all interested societies: nodes and links instead vertexes and edges of Theory of Graphs )

30. Vulnerability of Networks: these are not left alone

31. 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. The error tolerance/ robustness of the network

32. 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. Attack 2 1

33. Fragmentation When nodes are removed from a network, Clusters of nodes may be cut off (fragmented) from the main cluster. Fragmentation

34. Attack Evaluation – one more metrics S; The size of the largest cluster Divided by the initial total system size to normalize. Evaluation

35. Regular networks Random networks Small-world networks Scale-free networks So there are two important complex network models to explore

36. Exponential model attack

37. The size of the largest cluster - exponential model • Failure/ unintentional attack; Attack/ intentional attack

38. Scale free model attack

39. The size of the largest cluster- scale free model • Failure/ unintentional attack; Attack/ intentional attack

40. 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 resolvingcommunity structure in complex networks. PNAS . May 1, 2007, vol. 104 , N 18 , 7327–7331

41. Diversity of attacks The impact of failures and attacks on the network structure BA model in not enough to explore all aspect of attacks Complexity of a real problem

42. Regular Model Erdös-Renyi (ER) Watts and Strogatz (WS) Barabasi-Albert (BA) Levitin G, Hausken K. Rossodivita- Trufanov (RT) Xiao, Xiao and Cheng Caruso- Rossodivita- Shubnikov-Tikhomirov-Trufanov -Umerov Aminova - Rossodivita- Tikhomirov-Trufanov

43. S. Xiao, G. Xiao ,T. H. Cheng Tolerance of local information-based intentional attacks in complex networks. J. Phys. A: Math. Theor. 43 (2010) 335101 Distributed 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 the local network-topology information. Xiao models Division of Communication Engineering, School of Electrical and Electronic Engineering,Nanyang, Singapore

44. S. Xiao, G. Xiao On Degree-Based Decentralized Search in Complex Networks arXiv 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

45. Caruso- Rossodivita- Shubnikov-Tikhomirov-Trufanov -Umerovmodels • Real life attacks : • mixture of Failures and Attacks • Combined attacks model : • Sequence ofFailures and Sequence of Attacks • Sequence of Attacks and Sequence of Failures • Failure/ unintentional attack; Attack/ intentional attack

46. Failures+Attacks / Attacks + Failures

47. Network connectivity - Inhomogeneous network connectivity

48. Caruso- Rossodivita- Shubnikov-Tikhomirov-Trufanov -Umerovmodels • 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

49. Caruso- Rossodivita- Shubnikov-Tikhomirov-Trufanov -Umerovmodels • Real life : • protection of nodes and links • Node Protection model : • Metrics of d: Investments (Money) • d ~ Funding • Failure/ unintentional attack; Attack/ intentional attack

50. d2 d1 d0<d1<d2 d0 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