Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne [email protected] :: @LHDnets Joint work with Antoine Allard Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Percolation on complex networks 1 Take a network. 2 Randomly keep a fraction p of edges (“occupied” edges). 3 Measure the number and size of components. Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Percolation on complex networks 1 Take a network. 2 Randomly keep a fraction p of edges (“occupied” edges). 3 Measure the number and size of components. • Occupied edges can be “good” (active connections on the Internet) or “bad” (transmissions of diseases). • Models resilience to failures, or the outcome of an epidemic. • It is also a benchmark for the quality of our network models. Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Percolation on complex networks 0.0 0.2 0.4 0.6 0.8 1.0 Probability of existence of edges (p) 0.0 0.2 0.4 0.6 0.8 1.0 Fraction of vertices Power grid LCC Power grid from Watts & Strogatz, Nature (2000) and others. Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Phase transitions in percolation 0.0 0.2 0.4 0.6 0.8 1.0 Probability of existence of edges (p) 0.0 0.2 0.4 0.6 0.8 1.0 Fraction of vertices Power grid LCC How to detect the phase transition of a real system? Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Phase transitions in percolation 0.0 0.2 0.4 0.6 0.8 1.0 Probability of existence of edges (p) 0.0 0.2 0.4 0.6 0.8 1.0 Fraction of vertices LCC 25 × s 0 10 20 30 40 50 60 70 80 90 Number of vertices Power grid Allard, Althouse, Scarpino & H´ ebert-Dufresne, PNAS ’17 Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Phase transitions in percolation 0.0 0.2 0.4 0.6 0.8 1.0 Probability of existence of edges (p) 0.0 0.2 0.4 0.6 0.8 1.0 Fraction of vertices LCC 25 × s χ1 0 50 100 150 200 Number of vertices Power grid Radicchi, Predicting percolation thresholds. . . , PRE ’15 Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Phase transitions in percolation 0.0 0.2 0.4 0.6 0.8 1.0 Probability of existence of edges (p) 0.0 0.2 0.4 0.6 0.8 1.0 Fraction of vertices LCC 25 × s χ1 S2 0 100 200 300 400 500 600 Number of vertices Power grid Zhang, Spectral estimation of the percolation transition, PRE ’17 Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Results on real networks 0.0 0.2 0.4 0.6 0.8 1.0 Probability of existence of edges (p) 0.0 0.2 0.4 0.6 0.8 1.0 Fraction of vertices 0 10 20 30 40 50 60 Number of vertices PGP Pretty-Good-Privacy network Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Phase transitions in percolation 0.0 0.2 0.4 0.6 0.8 1.0 Probability of existence of edges (p) 0.0 0.2 0.4 0.6 0.8 1.0 Fraction of vertices LCC 25 × s χ1 S2 0 100 200 300 400 500 600 Number of vertices Power grid Is this what we want? Plus some finite-size effects? Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Phase transitions in percolation 0.0 0.2 0.4 0.6 0.8 1.0 Probability of existence of edges (p) 0.0 0.2 0.4 0.6 0.8 1.0 Fraction of vertices LCC 25 × s χ1 S2 0 100 200 300 400 500 600 Number of vertices Power grid For comparison, message passing predicts pc = 0.16. . . Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Smeared phase transitions 0.0 0.2 0.4 0.6 0.8 1.0 Bond occupation probability (p) 0.0 0.2 0.4 0.6 0.8 1.0 Percolation (power grid) S1 /N χ/χmax 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 Temperature (T) 0.0 0.2 0.4 0.6 0.8 1.0 Ising M χ/χmax Sknepnek & Volta, Smeared phase transition. . . , PRB (2004) Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Smeared phase transitions Phase transitions with non-vanishing variance in the distribution of the local order parameter. Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Smeared phase transitions Phase transitions with non-vanishing variance in the distribution of the local order parameter. In percolation: • Order parameter: Fractional size S of the largest component (LCC). • Local order parameter: Probability Pi that node i is in the LCC. • Related through S = Pi /N. Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Smeared phase transitions Distribution of local order parameter in small Erd˝ os-R´ enyi graphs. Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Smeared phase transitions Distribution of local order parameter in complex networks. Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Smeared Sequential phase transitions Nodes tagged by degree. Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Smeared Sequential phase transitions Nodes tagged by coreness (k-core centrality). Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Smeared Sequential phase transitions Nodes tagged by layer [onion decomp.: LHD et al, Sci Rep (2015)]. Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Exploration in toy-models Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Exploration in toy-models Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Exploration in toy-models Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Exploration in toy-models Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Exploration in toy-models Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Exploration in toy-models Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Finite-size effects? Hard to tell. If we double the power grid, do we expect 1 the same core, 2 twice as many cores. 3 or one core twice as large? Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Finite-size effects? Hard to tell. If we double the power grid, do we expect 1 the same core, 2 twice as many cores. 3 or one core twice as large? The choice is best left to experts. But our mathematical models have to choose, and they pick option # 3. Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

A problem with message passing? MPA predicts a transition at pc = 0.184. Is that what we want? Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

How do we detect smeared transitions? Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

How do we detect smeared transitions? Option #1: Look at set of Pi(p) curves, or aggregated over some centrality metrics (onion layer). Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

How do we detect smeared transitions? Option #2: Measure of local susceptibility: χlocal = d2σ(Pi(p))/dp2 0.0 0.2 0.4 0.6 0.8 1.0 Transmissibility (T) 0.0 0.2 0.4 0.6 0.8 1.0 Fraction of vertices S1 /N χlocal χ1 0 20 40 60 80 100 Number of vertices PolishGrid Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

How do we detect smeared transitions? Option #2: Measure of local susceptibility: χlocal = d2σ(Pi(p))/dp2 Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

Take-home messages 1 Complex networks do not have a clean phase transition. 2 The transition is smeared by sequential subgraph activation. 3 Measures of local susceptibility can identify these transitions. χlocal = d2 dp2 σ(Pi(p)) Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne

If you have any questions The smeared phase transition of percolation on real complex networks Preprint by H´ ebert-Dufresne & Allard available soon. Until then: [email protected] or @LHDnets Sequential phase transitions in percolation on complex networks Laurent H´ ebert-Dufresne