PERCOLATION MODELBabak BagheriMohammad Reza Dehghani Tafti

Shahid Beheshti UniversityNovember 2016

A simple example

Basic properties: Bond percolation Site percolation Each site can be occupied with probability p Spanning cluster In the limit of an infinite lattice there exists a threshold probability above which there suddenly emerges a large cluster that spans the system.

Order parameter

On a given lattice percolation threshold for bonds is smaller than percolation threshold foe site. site-bond percolation: Sites are occupied with probability (p) and bonds are occupied with probability q. Can be relevant for gelation, epidemic process and Ising model in dilute media.

Cluster size distribution The probability that an arbitrary site belongs to a finite cluster of size s is then given by ) so:

the probability an occupied site belongs to a cluster of finite mass s :

Mean cluster size is

And we can state for the number of finite cluster that: and near :

these critical exponent are not independent, and they are universal which means they only depends on dimension of lattice.

And for : and for the percolation process behaves in the same manner as percolation on an infinite regular tree, and their critical exponents take on the corresponding values given by mean-field theory

So the upper critical dimension

Percolation on : There is no infinite open cluster at for and For other dimension it seems to be held but its proof is an open problem.

Critical threshold for bond percolation and site percolation satisfy:

It can be showed: so What about number of infinite cluster?It was shown by Newman and Schulman that number of infinite cluster can be 0 or 1 or is impossible on there are some graph like trees infinite clusters can coexist.

Directed percolation: the water propagation is not isotropic but directed. For this model, the percolation can only occur along a given spatial direction

Nonequilibrium phase transitions, a continuous phase transition It turned out to describe a wide range of spreading models , and forest re models

isotropic percolation: Notice that different origins may generate the same cluster. Consequently the whole lattice decomposes into a set of disjoint clusters.

DP: each site generates an individual cluster, a decomposition of the lattice into disjoint clusters is no longer possible.

In the supercritical phase p > pc the medium is permeable (P∞ > 0) while in the subcritical phase p < pc the medium becomes impermeable (P∞ = 0).

directed percolation as a dynamic process: directed percolation may be interpreted as a d+1 dimensional dynamic process that describes the spreading of some non-conserved agent.

percolation on a dynamic network can be mapped onto the problem of directed percolation in infinite dimensions.

1+1:

The set s = {si} at a given time t specifies the configuration of the system.

1.fully occupied lattice Initial state: 2. random initial conditions 3.single particle at the origin (also called ‘active seed’).

The Importance of Robustness in Complex Networks:

Medicine Ecology Engineering Biology Politics

Factors impacting Robustness of Complex Networks

Distribution Function of Complex Networks

Failure vs Attack

Cascading Failure

Number of Hubs

Assortativity

FUTURE WORK Robustness in Small World Complex Network

Robustness in Human Social Networks

PROPAGATION OF FIRE

FOREST FIRE Each square is occupied by a tree(green) with probability p or is empty with 1-p

Green red black (?) Does fire on one side of forest penetrate through the whole forest? near Pc : critical slowing down fire needs more time to find out that it can not penetrate the forest

OIL & GAS IN RESERVOIRS

OIL FIELD Unoccupied site (1-p) regions filled with hard rocks Occupied site (p) regions filled with oil M = number of points within the frame that belong to the same cluster

for at (fractal) average density is not constant!

DIFFUSION IN DISORDERED MEDIA

Particles can move only from one occupied site to a nearest neighbor which is also occupied. so the motion is restricted to the cluster of percolation theory. This problem is called ‘’ant in the labyrinth” 1. An ant sits on an occupied site2. An every time selects one of its neighbor randomly.3. Moves to selected site of it is occupied and stays at its old site if the

selected site is empty.

Root mean square(?) For regular lattice (p=1) At At : there are finite clusters the motion restricted over finite distances R approaches to a constant as

At :there are some holes in the network “for distances larger than hole size disorder acts as friction”

Ant feels fractal dimension!

AGGREGATION

Sites are occupied by monomers with probability p Each monomer has a functionality f (how many of its bond can be active f=1,2,3,4)

1. Initiator placed randomly on the monomers activate available bonds by a random walk process

2. Initiators jump randomly to a neighbor site 3. If the site is monomer with , f is reduced by 1 and the bond is activated.4. If or the new site is a solvent molecule the jump is rejected.5. Time is increased by 6. If tow initiator meet at the same monomer they annihilate.The concentration of occupied bonds increases with tAt a critical concentration a spanning cluster exists for first time. critical behavior is described by and as in normal percolation

SPREADING

AN IMPROVED MODEL:

An improved model: An infested site infects it neighbor only with probability q and within a time interval

Absence of immune sites p=1 bond percolation For p<1 site-bond percolation

CONTROLLABILITY OF COMPLEX NETWORKS According to control theory, a dynamical system is controllable if, with a suitable choice of inputs, it can be driven from any initial state to any desired final state within finite time.

Although control theory is a mathematically highly developed branch of engineering with applications to electric circuits, manufacturing processes, communication systems, aircraft, spacecraft and robots, fundamental questions pertaining to the controllability of complex systems emerging in nature and engineering have resisted advances

The difficulty is rooted in the fact that two independent factors contribute to controllability, each with its own layer of unknown: (1) the system’s architecture, represented by the network encapsulating which components interact with each other; and (2) the dynamical rules that capture the time-dependent interactions between the components.

CONTROLLABILITY OF COMPLEX NETWORKS If we wish to control a system, we first need to identify the set of nodes that, if driven by different signals, can offer full control over the network.

We will call these ‘driver nodes’. We are particularly interested in identifying the minimum number of driver nodes, denoted by Nd, whose control is sufficient to fully control the system’s dynamics

Kalman’s controllability rank condition: Identify the minimum number of driver nodes need to know the weight of each link

structurally controllable: Thus, structural controllability helps us to overcome our inherently incomplete knowledge of the link weights

CONTROLLABILITY OF COMPLEX NETWORKS Albert Barabasi: as a group, gene regulatory networks display high Nd (0.8), indicating that it is necessary to independently control about 80% of nodes to control them fully. In contrast, several social networks are characterized by some of the smallest Nd values, suggesting that a few individuals could in principle control the whole system.

Robustness of control: ability to control a network under unavoidable link failure, we classify each link into one of three categories: ‘critical’ if in its absence we need to increase the number of driver nodes to maintain full control; ‘redundant’ if it can be removed without affecting the current set of driver nodes; or ‘ordinary’ if it is neither critical nor redundant

FUTURE WORK Controllability Metrics, Limitations and Algorithms for Complex Networks

The Role of Network Centrality in the Controllability of Complex Networks

The Role of Diameter in the Controllability of Complex Networks