D.Yu. Ignatov, A.N. Filippov, A.D. Ignatov, X. Zhang 5 October 2015 Automatic Analysis, Decomposition and Parallel Optimization of Large Homogeneous Networks D.Yu. Ignatov, A.N. Filippov, A.D. Ignatov, X. Zhang ISPRAS Open 2016

Homogeneous Networks Communication Networks Road network Wired 5 October 2015 Communication Networks Road network Wired Wireless Signals of sector antennas A0 – A7 S1 A3 A6 A0 A2 A5 A1 A7 A4 Element Crossroad Switch Antenna Optimized integral index Average speed of traffic Average power of prevalent radio signal Correlation formula for 2 elements 1 / (1 + [elements quantity on the shortest path]) 1 / (distance between antennas)

Background: Sector Planning with full optimization 5 October 2015 Graph-based representation Homogeneous network is represented as a weighted complete graph, where each vertex corresponds to network element each edge has weight equals to correlation between correspondent elements Optimization loop: Decomposition of network into subnets by the rule of minimal sum of crossing edges weights Parallel optimization of subnets Update of network parameters While stopping criterion isn’t met Full network DECOMPOSITION 1 1 1 1 1 2 2 UPDATE 2 2 2 2 Main drawback: Crossing edges are ignored => poor accuracy 1 2 Agenda Optimization of all parameters Thread Thread - Element 1 2 - Subnets

Idea 1: Alternative splitting 5 October 2015 Full network Discard light edges under threshold Reduced network Alternative splits 1 2 1 1 2 4 2 3 4 4 3 3 1 While stopping criterion is not met 2 UPDATE NETWORK 3 4 Thread Optimization Split by split Thread Thread Thread Agenda - Element 1 2 - Subnets 3 4 Advantage: All crossing edges are taken into account

Idea 2: Independent optimization 5 October 2015 FOR EVERY OPTIMIZED UNIT FIND SUBNET Reduced network COMBINE NON-OVERLAPPING SUBNETS 1 2 Agenda - Optimized unit 2 1 - Border element 1 2 - Subnets 1 2 Advantage: Parallel optimization of the fully independent elements

Idea 3: Regulation of threshold Optimized unit = 2 elements 5 October 2015 Optimized unit = 2 elements O p t i m i z a t i o n 1 subnet Threshold increasing Splitting with threshold 2 subnets 3 subnets Advantage: Optimization process is regulated by threshold

… Full cycle of alternative splitting with regulated threshold 5 October 2015 Full network DISCARD EDGES UNDER THRESHOLD Reduced network DECOMPOSITION Alternative splits of network into non-overlapping subnets for optimized units: 1 2 If threshold under limit increase its value and continue 1 1 1 1 1 1 1 2 2 … 2 2 2 2 UPDATE NETWORK If stop-ping criterion is not met 1 2 Optimization of independent elements Thread Thread Split by split Agenda - Optimized unit - Border element 1 2 - Subnets

Optimization process regulation 5 October 2015 Usage of all computational resources on computer / cluster Quantity Quantity of alternative calculations Subnets Quantity of processes = available cores Strength of distant interactions Precision of optimization Com-plexity Precision of optimizing procedure Rough search of global optimum in small number of complex subnets (avoiding of stuck in local optimum) Precise search of optimums in big number of simple subnets (maximal precision at the end) Start End Colored arrows ( ) – increase (up) or decrease (down) of parameter value Empty arrows – impact on optimization process

Optimization of mobile network coverage and quality 16 October 2015 Optimization of mobile network coverage and quality 2014/9/8 The optimized mobile network consists of sector antennas which transmit radio signal (green and yellow colors). And we can see the red problem areas with low level of radio signal and/or high level of noise. Coverage and Quality of network are regulated by power, height, tilt and azimuth of Antennas Optimized model includes non-differentiable functions: Radio Propagation models which depend on the type of area, signal-to-interference-plus-noise ratio - integral index which consists of ratio between prevalent signal and noise. This function has large optimization complexity (number of states): For example in optimized network with 300 sector antennas and 4 parameters, if every parameter has n states, then total number of states is n1200 So, we optimize multivariable non-differentiable function with large optimization complexity. Initial subnet Optimized subnet Regulated parameters of sector antenna: power, height, tilt, azimuth High optimization complexity: 300 antennas * 4 parameters, n states of parameter => n1200 states

Alternative splitting vs. Sector planning 5 October 2015 Optimized integral index – average Signal to Interference plus Noise Ratio: n – number of optimized areas in network Pi – the power of the prevalent signal in i-th area Ii – the power of the other (interfering) signals in i-th area Ni – other noise in i-th area Optimizing procedure – modified Simulated Annealing: procedure optimize(S0, precision) { Snew := S0 step := maxStep ∙ (1 – precision) Sgen := random neighbor of S0 within step t := T(1 – precision) if A(E(S0), E(Sgen), t) ≥ random(0, 1) then Snew := Sgen Output: state Snew } S0, Sgen, Snew – current, generated, new states of subnet maxStep – maximal value of step T, E, A – temperature, energy, acceptance functions 9 times faster 1 hour – SINR 15 % higher (p < 0.01)

Benefits of Independent optimization of alternatives 5 October 2015 Faster optimization due to reduction of optimization complexity and efficient usage of all computational resources Better quality of optimization due to progressive shift of optimization strategy from rough search of global optimum at the beginning of optimization process to precise search of optimum at the end of optimization

D.Yu. Ignatov, A.N. Filippov, A.D. Ignatov, X. Zhang 5 October 2015 Automatic Analysis, Decomposition and Parallel Optimization of Large Homogeneous Networks D.Yu. Ignatov, A.N. Filippov, A.D. Ignatov, X. Zhang ISPRAS Open 2016