SELF-HEALING NETWORKS: REDUNDANCY AND STRUCTURE Guido Caldarelli IMT, CNR-ISC and LIMS, London UK DTRA Grant HDTRA1-11-1-0048
INTRODUCTION The robustness and the shape Baran, P. On distributed Communications (1964) Using a mini-computer, Baran and his team developed simulation programs to test basic connectivity of an array of nodes with varying degrees of linking. That is, a network of n- ary degree of connectivity would have n links per node. The simulation randomly destroyed nodes and subsequently tested the percentage of nodes who remained connected. The result of the simulation revealed that networks where n>=3 had a significant increase in resilience against even as much as 50% node loss. Baran's conviction gained from the simulation was that redundancy was the key.
INTRODUCTION From Arpanet to Internet Baran, P. On distributed Communications (1964)
BASIC OF NETWORKS The difference between models 1 Degree frequency density P(k) = how many times you find a vertex whose degree is k P(k) P( k) e pn ( pn) k! k P ( k) k k 2 Degree Correlation Knn (k) = average degree of a neighbour of a vertex with degree k 3 Clustering Coefficient (k) = the average value of c for a vertex whose degree is k 4
Real world systems show a large number of interdependencies: physical interdependency; cyber interdependency; geographic interdependency; logical interdependency that need financial and political coordination. We need to move beyond topological characterization, and understand how to characterize, observe and control the dynamics of real networks. System A System B During the project Long term Paradigm shift Develop data driven models that go beyond the methodological and disciplinary boundaries of a specific approach. Theoretical, algorithmic, and computational framework that will enable us to evaluate the onset of tipping points, emergent phenomena, cooperative phenomena in multilevel networks. Use the tools of control theory to go beyond network characterization and understand the control of real networks. Breakthrough in the application of complex systems, algorithmic and network theory to the modeling and analysis of the integration of social interactions and technological and communication networks Learn to measure (observe), quantify, predict, and control complex systems.
INTRODUCTION TOPOLOGY AND ROBUSTNESS
INTRODUCTION TOPOLOGY AND ROBUSTNESS The idea is that as Baran noticed, SF is more fragile under attack (RED) But For failures SF is more robust (BLUE)
REDUNDANCY AND BIOLOGY SPECIES PREDATE WITH REDUNDANCY A spanning tree of a connected directed graph is any of its connected directed subtrees with the same number of vertices. In general, the same graph has more spanning trees with different topologies. Since the peculiarity of the system (FOOD WEBS),some are more sensible than the others.
REDUNDANCY AND BIOLOGY Trees key quantities 1 1 1 3 5 1 Out-component size: 0,6 0,5 A X Y nn w X XY A Y 1 Out-component size distribution P(A) : 0,5 P(A) 5 2 11 8 22 10 3 1 33 Sum of the sizes: C X A Y Y X Allometric relations: C C A C C A 35 30 X X C(A) X 33 0,4 0,3 0,2 0,1 0 0,1 0,1 0,1 0,1 0,1 1 2 3 4 5 6 7 8 9 10 A 25 22 20 15 10 11 5 5 3 0 1 0 2 4 6 8 10 12 A
REDUNDANCY AND BIOLOGY Topology to optimise flow on the tree
1 0 1 REDUNDANCY AND BIOLOGY Various Trees 0 C( A) A efficient P( A) stable A1 C( A) A 1 2 P( A) A 0 2 C( A) A inefficient P(A) cost unstable
REDUNDANCY AND OPTIMISATION COST OF FLOW A Capitalist solution, everyone connects, socially (total pipe length) costly Individually (#passages) rewarding B Socialist solution, strict rule to connect socially (total pipe length) rewarding Individually (#passages) costly C Fractal (natural) solution, socially (total pipe length) rewarding Individually (#passages) rewarding
SELF-HEALING DEFINITION Robustness vs Resilience Robustness is important But we want also to check Resilience (how you come back to business) The latter point has been considered less often
We describe as networks the pipelines of some service We model the networks with trees + redundant link We assume that the service follows some sort of Kirchoff s laws Service travels along the links On interruptions some nodes get isolated MODEL OF SELF HEALING SIMPLIFIED ASSUMPTIONS We have an interaction protocol only with the set of neighbouring nodes Unserved nodes try to reconnect to the active tree by waking up (activating) dormant backup links (assuming you can t repair the old)
MODEL OF SELF HEALING A Simple Example Self-healing capabilities are given by distributed communication protocols that exploit redundant links to recover the connectivity of the system
MODEL OF SELF HEALING How to measure success A natural metric to quantify the success of such a procedure is the fraction of served nodes (FoS). Which topology maximizes the FoS? We study the effects of varying the fraction of backup links (redundancy) according to different underlying connectivity patterns with respect to multiple random failures
MODEL OF SELF HEALING Topology used to study model properties
FRACTION OF SERVED NODES REGULAR GRID Regular grid The average fraction <FoS> of nodes that the self-healing protocol is able to restore decreases with the number of faults k with no relevant dependency on the redundancy; results are shown for a 104 nodes network.
FRACTION OF SERVED NODES BARABÁSI-ALBERT Barabási-Albert The average fraction <FoS> of nodes that the self-healing protocol is able to restore decreases with the number of faults k with no relevant dependency on the redundancy; results are shown for a 104 nodes network.
FRACTION OF SERVED NODES REGULAR GRID Small world The average fraction <FoS> of nodes that the self-healing protocol is able to restore decreases with the number of faults k with no relevant dependency on the redundancy; results are shown for a 104 nodes network.
FRACTION OF SERVED NODES COMPARISON OF TOPOLOGIES In fact, the most robust networks as expected are based on the SF topology; however, such a topology is unrealistic for technological networks. Our results on SW topologies hint that a very effective strategy to strengthen realistic networks is to add long range links.
MODEL OF SELF HEALING Algorithm We assume that nodes have only a local knowledge of the networks i.e., only about the state (active, dormant or failed) of their incoming links. To build the maximal connected tree, nodes communicate with their neighbors via a suitable distributed protocol allowing fault nodes to join the active network by activating dormant edges. In other words, nodes are endowed only with the minimal requirements of routing needed to reconstruct a spanning tree
REDUNDANCY HOLDS FOR OTHER NETWORKS AS WELL
Redundancy is key to assess resilience Natural environment already use We can possibly design systems to optimise resilience probability Not in all cases anyhow this is possible CONCLUSIONS Summary
FURTHER INFO Grants DTRA Grant HDTRA1-11-1-0048 www.simpolproject.eu www.multiplexproject.eu