Titles and Abstracts

Elisabetta Candellero: Oil and water model on vertex transitive graphs

The oil and water model is an interacting particle system with two types of particles and a dynamics that conserves the number of particles, which belongs to the so-called class of Abelian networks. Widely studied processes in this class are sandpiles models and activated random walks, which are known (at least for some choice of the underlying graph) to undergo an absorbing-state phase transition. In this work we show that the oil and water model is substantially different from such models, as it does not undergo an absorbing-state phase transition and is in the regime of fixation at all densities. Joint work with A. Stauffer and L. Taggi.

Giacomo Como: On network centrality, influence, and games

Several social, economic, and financial systems can often be fruitfully modelled as networks of interacting agents whereby different and possibly complex behaviours may emerge as a result of the graph topology and of the local interaction rules. In particular, graph-theoretic notions of centrality, connectivity, and conductance often recur in characterising such emerging behaviours. In this talk, we first review some models and results in opinion dynamics, discussing conditions for the emergence of consensus vs the persistence of disagreement and polarization, highlighting the role of network centralities as measures of social influence, and formulating optimal intervention and targeting problems. Then, we present a network formation game where the nodes choose strategically whom to link to and discuss the structure of Nash equilibria and the behaviour of asynchronous best response dynamics in such game.

Umberto De Ambroggio: On dynamic random graphs with degree homogenization via anti-preferential attachment probabilities

We analyze a dynamic random undirected graph in which newly added vertices are connected to those already present in the graph either using, with probability p, an anti-preferential attachment mechanism or, with probability 1−p, a preferential attachment mechanism. We derive the asymptotic degree distribution in the general case and study the asymptotic behaviour of the expected degree process in the general and that of the degree process in the pure anti-preferential attachment case. Degree homogenization mainly affects convergence rates for the former case and also the limiting degree distribution in the latter.

Nicola Del Giudice: Vanishing of cohomology groups of random simplicial complexes

Inspired by the existence of a threshold for the connectedness of the binomial random graph, it is interesting to investigate higher-dimensional analogues of this phenomenon. In particular, random hypergraphs and random simplicial complexes have received considerable attention in this setting. We study a model which represents a bridge between these two approaches: random simplicial k-complexes that arise as the downward-closure of random (k+1)-uniform hypergraphs, for each integer k ≥ 2. As connectedness counterpart, we consider 𝔽₂-cohomological j-connectedness for each j ∈ [k-1], that is the vanishing of the cohomology groups with coefficients in the two-element field 𝔽₂, of dimension up to j. As it turns out, 𝔽₂-cohomological j-connectedness for our model is not a monotone property, so the existence of a single threshold is not guaranteed. Nevertheless we determine such threshold, relating the vanishing of the cohomology groups to the disappearance of the last minimal obstruction. This is joint work with O. Cooley, M. Kang and P. Sprüssel.

Thomas M. Michelitsch: Non-Markovian continuous time random walks based on generalized fractional Poisson process (joint work with Alejandro Pérez Riascos)

We analyze a non-Markovian generalization of Laskin’s fractional Poisson process, the "generalized fractional Poisson process" (GFPP), first introduced by Cahoy and Polito [1]. This process contains two index parameters 0 < β ≤ 1, α > 0 and a time scale parameter. We discuss the fat-tailed asymptotic power-law features in the GFPP, the non-Markovian memory effect and the probabilities of n arrivals ("generalized fractional Poisson distribution"). With these results for the GFPP renewal process, we construct Montroll-Weiss continuous time random walks (CTRWs) on undirected networks. We develop CTRWs subordinated to the GFPP [2]. We derive the "generalized fractional Kolmogorov-Feller (K-F) equation" for this walk and demonstrate that the classical cases are contained (i.e. fractional K-F equation for 0 < β < 1 with α = 1 corresponding to Laskin’s fractional Poisson, and the standard K-F equation for β = 1, α = 1 of standard Poisson). Then we apply these results to the integer lattice ℤ^d. For this stochastic motion, we analyze the "well-scaled diffusion limit" and obtain the same type of fractional diffusion equation as for the fractional Poisson process. This fractional diffusion equation exhibits a subdiffusive t^β-power law (0 < β < 1) for the mean-square displacement. The occurrence of fractional diffusive features reflects the asymptotic universality of the Laskin fractional Poisson process and the corresponding Mittag-Leffler waiting time PDF as generally demonstrated by Gorenflo and Mainardi [3]. The remarkably rich dynamics introduced by the GFPP and further generalizations open a wide field of applications in anomalous transport [4] and the dynamics of complex systems.

References

[1] D. O. Cahoy, F. Polito, Renewal processes based on generalized Mittag-Leffler waiting times, Commun Nonlinear Sci Numer Simul, Vol. 18 (3), 639-650, 2013.

[2] T. Michelitsch, A.P. Riascos, Continuous time random walk and diffusion with generalized fractional Poisson process, In Press, Physica A, https://doi.org/10.1016/j.physa.2019.123294 , (Revised part on diffusion limit: arXiv:1907.03830v2 [cond-mat.stat-mech]).

[3] R. Gorenflo, F. Mainardi, The asymptotic universality of the Mittag-Leffler waiting time law in continuous time random walks, Invited lecture at the 373. WE-Heraeus-Seminar on Anomalous Transport: Experimental Results and Theoretical Challenges, Physikzentrum Bad-Honnef (Germany), 12-16 July 2006.

[4] T. Michelitsch, A.P. Riascos, B. Collet, A. Nowakowski, F. Nicolleau, Generalized space-time fractional dynamics in networks and lattices, Preprint: arXiv:1910.05949 [cond-mat.stat-mech].

Angelica Pachon: Preferential attachment models with edge removal

Although the Barabási-Albert model has a simple description and it has an important property observed in real networks, which is an asymptotic power-law degree distribution, a more realistic model should also take into account features like node or edge removal. Motivated by some models which have been proposed in the study of the evolution of growing populations and the theory of evolutionary dynamics, we describe in this work two preferential attachment models with detachment edges. Some numerical results related to these models are presented in [1] and [2]. Here we demonstrate that the introduction of detachment of edges affect considerably the in-degree distribution of an uniformly random chosen vertex, as well as the empirical in-degree distribution, leading to a topological phase transition in both models: from a power-law in-degree distribution in the super-critical case, to a faster than a power law tail decays in the critical case, until an exponentially fast tail decays in the sub-critical case. Furthermore, we can show that the the critical point will depend on the probabilities of adding and removing edges. We relate the random graph processes associated with the preferential attachment models with a generalised Yule model studied in [3], which is in turn related to a birth-death process.

References

[1] S. Manrubia and D. Zanette. At the boundary between biological and cultural evolution:the origin of surname distributions. J. theor. Biol., (216):461–477, 2002.

[2] Y. E. Maruvka, D. A. Kessler, and N. M. Shnerb. The birth-death-mutation process: A new paradigm for fat tailed distributions. PLOS ONE, 6(11):1–7, 11 2011.

[3] P. Lansky, F. Polito, and L. Sacerdote. The role of detachment of in-links in scale-free networks. Journal of Physics A: Mathematical and Theoretical, 47(34):345002, 2014.

Guillem Perarnau: Diameter of the directed configuration mode

We asymptotically determine the diameter of the directed configuration model for degree sequences with finite variance. While the typical behaviour of in- and out-neighbourhood is the same, we show that the their critical behaviour is different. Thus, the diameter depends on the growth rate of the branching processes with the in- and out- size-biased distributions conditional on extinction. This is joint work with Xing Shi Cai.

Matt Roberts: Exceptional times of the critical Erdős-Rényi graph

It is well known that the largest components in the critical Erdős-Rényi graph have size of order n^2/3. We introduce a dynamic Erdős-Rényi graph by rerandomising each edge at rate 1, and ask whether there exist times in [0,1] at which the largest component is significantly larger than n^2/3.