1. Percolation and diffusion in network models Shai Carmi, Department of Physics, Bar-Ilan University Networks Percolation Diffusion Background picture: The Internet at the Autonomous Systems level. Nodes are colored by their proximity to the network’s core. Node size is proportional to its number of links (degree). S. Carmi, S. Havlin, S. Kirkpatrick, Y. Shavitt, and E. Shir. PNAS, 104, 11150 (2007). Networks are everywhere: technology, society, economy, transportation, biology, and others. How to model networks? Random network model (Erdos-Renyi, ER) Connect each pair of nodes with equal small probability p. Studied since the 1960s. Nodes have typical degree (number of connections). Degree distribution is Poissonian. Easy to solve for various properties. Small world. But cannot account for properties of real networks. Scale free (SF) network model The distribution of node degrees is broad, usually a power-law. Some nodes have extremely high degree (hubs). Nodes do not have typical degree; degrees span several order of magnitudes. Asymmetry (or heterogeneity) in network structure. Hubs can be attracted or repelled from other hubs. SF topology naturally results from growth with preferential attachment. Almost all natural networks are scale-free. Technology Internet Wireless networks Electric circuits Biology Protein interaction Genetic regulation Biochemical pathways Society Social networks Online networks Business relations Others Transportation networks Linguistic networks WWW Energy landscapes Questions How and why did so many networks evolve to be scale-free? Which are the most important nodes and links? Do networks have hierarchical organization? What is the typical distance between nodes? Can scale-free networks be embedded in a lattice/fractal? How to find communities in a network? Is scale-free topology optimal and in what sense? How does scale-free topology affect network function and dynamics (random walk, routing, synchronization, conductance)? How does the network structure affect its stability? Are networks stable? For scale-free networks, depends on how nodes are removed: Failure Nodes are randomly removed Attack Nodes of highest degree are removed first Almost all remaining nodes are connected. Scale-free networks are resilient to random failure! The percolation threshold is: where k is the degree. For scale-free networks, κ→∞, and thus p c =0, meaning all nodes need to be removed before the network disintegrates. E. Lopez, R. Parshani, R. Cohen, S. Carmi, and S. Havlin. Phys. Rev. Lett. 99, 188701 (2007). M. Maragakis, S. Carmi, D. ben-Avraham, S. Havlin, and P. Argyrakis. Phys. Rev. E (RC) 77, 020103 (2008). Attacked network is fragmented. Scale-free networks are highly sensitive to targeted attack on their hubs! The critical fraction of nodes that need to remain before the network breaks down is called the percolation threshold and is denoted by p c. For attack on scale-free networks, p c >0. Removal of nodes might leave the network connected, but eliminates many shortcuts. In many real systems, nodes connected only through very long paths are practically disconnected. Communication: Packets accumulate errors or are discarded after many transfers. Society: Information is irrelevant if it arrives too late. Transportation: Commuting is pointless if commute time is too long Percolation theory of networks addresses the question: Epidemics: Individuals very distant from the origin of the epidemics might not be infected due to mutation or vaccination. New percolation model is needed! A fraction p of the network nodes are removed. However, nodes are considered disconnected if the new distance is more than a times the original distance. We find new results for the percolation threshold and for the size of the largest connected component S a. SF networks are assumed to have degree distribution. Consider a simple random walk, or diffusion. In a lattice, equilibrium particle density is homogeneous. Networks are heterogeneous substrates: particle density is proportional to degree. If the total particle density is , the density at a node of degree k is:  k =  k/. → hubs are loaded. What is the relation between network topology and dynamics? p=1/5 In communication networks, packets often have inherent priorities. Low priority packets are handled only after high priority packets are handled. How is the diffusion affected? Model: Particles of two types: A (high priority) and B (low priority), are randomly traversing a network. At each time step, a B particle is allowed to move only if its current node is clear of A particles. Selection of particles is random, by uniformly choosing either a site (the site protocol) or a particle (the particle protocol). What are the diffusion coefficients when priority is applied? Analytically solvable for lattices! Find first the fraction f 0 of sites empty of the high priority particles. Proceed to find the diffusion coefficients (site protocol): For the particle protocol, calculate first the probability of a low priority particle B to be free from which the diffusion coefficients are easily derived. In networks, the hubs are almost always occupied: The waiting times of the B particles increase exponentially with node degree: Real Internet In scale-free networks, freely moving A’s aggregate at the hubs. B’s can’t escape from the hubs and aggregate there as well, leading to total inhibition of their motion. The waiting times of low priority particles in scale-free networks is compatible with sub-diffusion. Conclusion: network topology should be taken into account when designing routing protocols. Advisor: Prof. Shlomo Havlin Limited Path Percolation (LPP):