LATEX TikZposter

28: Structure and Dynamics of DirectedScale-free Networks

Christian Monch

Johannes Gutenberg-Universitat Mainz

28: Structure and Dynamics of DirectedScale-free Networks

Christian Monch

Johannes Gutenberg-Universitat Mainz

Applicant identifiers: MO 3831/2-1 Possible connections to projects: 14, 16, 17, 29Requested positions: Own positions: 1

Sparse Scale-free Digraphs

•Background

.Many real world networks are directed

.Key examples: citation networks, follower networks in social media, financial networks, link-structure of WWW

.Directed models mainly in computer science and physics literature

∗Numerical/data driven

∗Analytical heuristics

•Modelling framework

.Random finite digraphs GN = (VN , EN), vertices VN , directed edges / arcs EN

. System size parameter N – study large network asymptotics as N →∞

.Focus on sparse networks with O(|EN |) = O(|VN |)

. Important special case: scale-free graphs 1VN

∑v∈VN 1{v has degree k in GN} = k−τ+o(1)

.Networks often dispay clustering → consider digraphs embedded in space Twitter network of the Victoria and Albert Museum, London, in 2015 [2]

Directed random connection model with power law weights on the 2d-torus

Local vs. Global Structure / Power Laws vs. Geometry

•Sketching possible models

. VN = Zd or unit rate Poisson process restricted to [−N/2, N/2]d

.Generate arcs independently with probability

φ (U→v , U←w )

|v − w|δd , v, w ∈ VN∗ I.i.d. bivariate weight sequence (U←v , U

→v )v∈VN generates local randomness and in-/outdegree correlations

∗Profile parameter δ > 1 modulates effect of geometry

∗Kernel φ produces structural features – power law degrees, preferential attachment-like effects, etc.

.Poisson model yields Directed Random Connection Model with Weights – directed version of [A]

.Lattice model includes Directed Scale-Free Percolation – directed version of [B]

.Combination of scale-free graphs with (long-range) percolation

•Structural properties

.Local behaviour – Large Deviation Theory: strengthen the marked point process-LD approach of [C]

.Global behaviour: percolation & robustness vs. targeted attacks, typical distances – adapt techniques from undirectedsetting e.g [A,B,D]

Network Dynamics: Random Walk, Infection and Information

•Random walk

. . . . on local limits – building on work in [A]

∗Recurrence/transience of supercritical strongly connected clusters in spatial models

∗ Invariance principles? – also unknown for undirected scale-free setting

. . . . on supercritical strongly connected clusters for large N : mixing/cover times

∗Directedness helps to gain independence, cf. [E]

∗“Spreading out” interpolates between spatial and non-spatial (di)graphs

•SI-type dynamics/cascade models

.Multitude of models for spread of infection, rumours, accumulation of systemic risk

.Example: Threshold Contact Process approximation to Boolean networks

∗Threshold dynamics are common in information diffusion and neural network models – transmission rates notproportional to number of infected neighbours

∗Particular interest on critical setting: rigorously verify exponents obtained in [F]

• The normalized community value simply divides the community value by the size of the totalinfected or activated nodes participating in the contagion process (story). This measure gives arough estimate of on average, how many of a voter’s friends have voted on a story, before a voterherself votes on it.

The characteristics of these observed aggregated properties of contagion process occurring on the networkare indicative of how the nature of the underlying network may affect the spread of information over it.

(a) Global cascade size on Digg (b) Largest cascade size on Twitter

(c) Principal cascade size on Digg (d) Principal cascade size on Twitter

Figure 6. Distribution of cascade sizes in Digg and Twitter

Cascade Size Distribution

Given the follower graph and a time sequence of votes, we extract individual cascades generated by allDigg and Twitter stories using using the methodology described in [27]. Figure 6 shows the probabilitydistribution of the cascade sizes. The lognormal or stretched exponential (Weibull) gives a good fit ofthe global cascade size for Digg, Fig. 6(a), while the power law accounts for just a small percentageat the tail of the distribution [27]. The largest cascade size distribution on Twitter also has a similarlong tail distribution. The principal cascade size distribution on Digg takes the log normal form of thepopularity distribution, with the most common size for a Digg cascade being about 200. This observationcan be explained by the fact that Digg activity is dominated by top users [30], who not only submit adisproportionate share of promoted stories, but are also likely to be connected to one another. We believethat the peak in the principal cascade size distribution reflects the influence of the top users.

8

Figure 2: | Symmetric average avalanche shapes. (Left column: panels (a), (c) and (e)):Average avalanche shapes from equation (4), for Poisson offspring distribution qk, with mean ξ.Each panel in the right column (panels (b), (d) and (e)) shows the same functions as in the panelto its left, but plotted (as, for example, in [7]) in rescaled time t/T and rescaled vertically to havea maximum value of 1. Panels (a) and (b): critical case with ξ = 1; panels (c) and (d): subcriticalcase, ξ = 0.8; panels (e) and (f): supercritical case, ξ = 1.2. The dashed green curve in panels (b),(d), and (f) is the parabola 4 t

T

(1− t

T

).

6

Left: Stylised representation of competing SI-type dynamcis on a directed graph.Right, top: Empirical distribution of information cascade total sizes (‘avalanches’)in social networks Digg (blue) and Twitter (red) [3].Right, bottom: Temporal avalanche shapes in idealised ODE model [4]. Top 2images: critical dynamics – universality conjectured. Bottom 2 images:supercritical dynamics. RHS: profiles rescaled by total avalanche duration.

References

[A ] P. Gracar, M. Heydenreich, C. Monch and P. Morters, Recurrence versus Transience for Weight-Dependent Random Connection Models, arXiv preprint arXiv:1911.04350 (2020+).

[B ] M. Deijfen, R. van der Hofstad and G. Hooghiemstra, Scale-free percolation, Annales de l’IHP Probabilites et statistiques 49(3), pp. 817-838 (2013).

[C ] C. Hirsch and C. Monch, Distances and large deviations in the spatial preferential attachment model, Bernoulli, 26(2), pp. 927-947 (2020).

[D ] M. Heydenreich, T. Hulshof and J. Jorritsma, Structures in supercritical scale-free percolation, The Annals of Applied Probability 27(4), pp. 2569-2604.

[E ] C. Bordenave, P. Caputo and J. Salez, Random walk on sparse random digraphs, Probability Theory and Related Fields, 170(3-4), pp. 933-960 (2018).

[F ] M. Moller and B. Drossel, Scaling laws in critical random Boolean networks with general in-and out-degree distributions, Physical Review E 87(5), 052106 (2013).

Image credits

[1 ] A. Broder et al., Graph structure in the web, Computer networks, 33(1-6), 309-320 (2000). The drawing is a reimaging of Fig. 9, p. 318.

[2 ] A. Espinos, Museums on Twitter: Three case studies of the relationship between a museum and its environment, MW2015: Museums and the Web 2015 (2015). Available via https://mw2015.museumsandtheweb.com/paper/museums-on-twitter-three-case-studies-of-the-relationship-between-a-museum-and-its-environment-museum-professionals-on-twitter/.

[3 ] K. Lerman and R. Ghosh, Information contagion: An empirical study of the spread of news on Digg and Twitter social networks, 4th International AAAI Conference on Weblogs and Social Media (2010). Image reproduced from preprint arXiv:1202.3162, Fig. 6.

[4 ] J. P. Gleeson and R. Durrett, Temporal profiles of avalanches on networks, Nature Communications 8:1227, pp.1-13 (2017). The image shown represents the Poisson setting, cf. Fig. 2, p.4.

Rigorous mathematicalresults are scarce.

‘Bowtie’ structure of the WWW [1]

Spatial random digraphs arenot locally treelike – theydisplay realistic clustering.

Whichproperties arealmost local?

Focus onmodels

requiringdigraphs /display new

features