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**Published in:** LIPIcs, Volume 257, 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)

Full-bond percolation with parameter p is the process in which, given a graph, for every edge independently, we keep the edge with probability p and delete it with probability 1-p. Bond percolation is studied in parallel computing and network science to understand the resilience of distributed systems to random link failure and the spread of information in networks through unreliable links. Moreover, the full-bond percolation is equivalent to the Reed-Frost process, a network version of SIR epidemic spreading.
We consider one-dimensional power-law small-world graphs with parameter α obtained as the union of a cycle with additional long-range random edges: each pair of nodes {u,v} at distance L on the cycle is connected by a long-range edge {u,v}, with probability proportional to 1/L^α. Our analysis determines three phases for the percolation subgraph G_p of the small-world graph, depending on the value of α.
- If α < 1, there is a p < 1 such that, with high probability, there are Ω(n) nodes that are reachable in G_p from one another in 𝒪(log n) hops;
- If 1 < α < 2, there is a p < 1 such that, with high probability, there are Ω(n) nodes that are reachable in G_p from one another in log^{𝒪(1)}(n) hops;
- If α > 2, for every p < 1, with high probability all connected components of G_p have size 𝒪(log n).

Luca Becchetti, Andrea Clementi, Francesco Pasquale, Luca Trevisan, and Isabella Ziccardi. Bond Percolation in Small-World Graphs with Power-Law Distribution. In 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 257, pp. 3:1-3:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{becchetti_et_al:LIPIcs.SAND.2023.3, author = {Becchetti, Luca and Clementi, Andrea and Pasquale, Francesco and Trevisan, Luca and Ziccardi, Isabella}, title = {{Bond Percolation in Small-World Graphs with Power-Law Distribution}}, booktitle = {2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)}, pages = {3:1--3:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-275-4}, ISSN = {1868-8969}, year = {2023}, volume = {257}, editor = {Doty, David and Spirakis, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2023.3}, URN = {urn:nbn:de:0030-drops-179392}, doi = {10.4230/LIPIcs.SAND.2023.3}, annote = {Keywords: Information spreading, gossiping, epidemics, fault-tolerance, network self-organization and formation, complex systems, social and transportation networks} }

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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Spectral techniques have proved amongst the most effective approaches to graph clustering. However, in general they require explicit computation of the main eigenvectors of a suitable matrix (usually the Laplacian matrix of the graph).
Recent work (e.g., Becchetti et al., SODA 2017) suggests that observing the temporal evolution of the power method applied to an initial random vector may, at least in some cases, provide enough information on the space spanned by the first two eigenvectors, so as to allow recovery of a hidden partition without explicit eigenvector computations. While the results of Becchetti et al. apply to perfectly balanced partitions and/or graphs that exhibit very strong forms of regularity, we extend their approach to graphs containing a hidden k partition and characterized by a milder form of volume-regularity. We show that the class of k-volume regular graphs is the largest class of undirected (possibly weighted) graphs whose transition matrix admits k "stepwise" eigenvectors (i.e., vectors that are constant over each set of the hidden partition). To obtain this result, we highlight a connection between volume regularity and lumpability of Markov chains. Moreover, we prove that if the stepwise eigenvectors are those associated to the first k eigenvalues and the gap between the k-th and the (k+1)-th eigenvalues is sufficiently large, the Averaging dynamics of Becchetti et al. recovers the underlying community structure of the graph in logarithmic time, with high probability.

Luca Becchetti, Emilio Cruciani, Francesco Pasquale, and Sara Rizzo. Step-By-Step Community Detection in Volume-Regular Graphs. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 20:1-20:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{becchetti_et_al:LIPIcs.ISAAC.2019.20, author = {Becchetti, Luca and Cruciani, Emilio and Pasquale, Francesco and Rizzo, Sara}, title = {{Step-By-Step Community Detection in Volume-Regular Graphs}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {20:1--20:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.20}, URN = {urn:nbn:de:0030-drops-115163}, doi = {10.4230/LIPIcs.ISAAC.2019.20}, annote = {Keywords: Community detection, Distributed algorithms, Dynamics, Markov chains, Spectral analysis} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Consider the following asynchronous, opportunistic communication model over a graph G: in each round, one edge is activated uniformly and independently at random and (only) its two endpoints can exchange messages and perform local computations. Under this model, we study the following random process: The first time a vertex is an endpoint of an active edge, it chooses a random number, say +/- 1 with probability 1/2; then, in each round, the two endpoints of the currently active edge update their values to their average.
We provide a rigorous analysis of the above process showing that, if G exhibits a two-community structure (for example, two expanders connected by a sparse cut), the values held by the nodes will collectively reflect the underlying community structure over a suitable phase of the above process. Our analysis requires new concentration bounds on the product of certain random matrices that are technically challenging and possibly of independent interest.
We then exploit our analysis to design the first opportunistic protocols that approximately recover community structure using only logarithmic (or polylogarithmic, depending on the sparsity of the cut) work per node.

Luca Becchetti, Andrea Clementi, Pasin Manurangsi, Emanuele Natale, Francesco Pasquale, Prasad Raghavendra, and Luca Trevisan. Average Whenever You Meet: Opportunistic Protocols for Community Detection. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{becchetti_et_al:LIPIcs.ESA.2018.7, author = {Becchetti, Luca and Clementi, Andrea and Manurangsi, Pasin and Natale, Emanuele and Pasquale, Francesco and Raghavendra, Prasad and Trevisan, Luca}, title = {{Average Whenever You Meet: Opportunistic Protocols for Community Detection}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {7:1--7:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.7}, URN = {urn:nbn:de:0030-drops-94705}, doi = {10.4230/LIPIcs.ESA.2018.7}, annote = {Keywords: Community Detection, Random Processes, Spectral Analysis} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 5031, Algorithms for Optimization with Incomplete Information (2005)

In this paper we introduce the notion of smoothed competitive analysis of online algorithms. Smoothed analysis has been proposed by Spielman and Teng to explain the behaviour of algorithms that work well in practice while performing very poorly from a worst case analysis point of view. We apply this notion to analyze the Multi-Level Feedback (MLF) algorithm to minimize the total flow time on a sequence of jobs released over time when the processing time of a job is only known at time of completion. The initial processing times are integers in the range $[1,2^K]$. We use a partial bit randomization model, i.e., the initial processing times are smoothed by changing the $k$ least significant bits under a quite general class of probability distributions. We show that MLF admits a smoothed competitive ratio of $O((2^k/\psigma)^3 + (2^k/\psigma)^2 2^{K-k}})$, where $\sigma$ denotes the standard deviation of the distribution. In particular, we obtain a competitive ratio of $O(2^{K-k})$ if $\sigma = \Theta(2^k)$. We also prove an $\Omega(2^{K-k})$ lower bound for any deterministic algorithm that is run on processing times smoothed according to the partial bit randomization model. For various other smoothing models, including the additive symmetric smoothing one, which is a variant of the model used by Spielman and Teng, we give a higher lower bound of $\Omega(2^K)$. A direct consequence of our result is also the first average case analysis of MLF. We show a constant expected ratio of the total flow time of MLF to the optimum under several distributions including the uniform one.

Luca Becchetti, Stefano Leonardi, Alberto Marchetti-Spaccamela, Guidouca Schaefer, and Tjark Vredeveld. Average Case and Smoothed Competitive Analysis of the Multi-Level Feedback Algorithm. In Algorithms for Optimization with Incomplete Information. Dagstuhl Seminar Proceedings, Volume 5031, pp. 5-28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{becchetti_et_al:DagSemProc.05031.7, author = {Becchetti, Luca and Leonardi, Stefano and Marchetti-Spaccamela, Alberto and Schaefer, Guidouca and Vredeveld, Tjark}, title = {{Average Case and Smoothed Competitive Analysis of the Multi-Level Feedback Algorithm}}, booktitle = {Algorithms for Optimization with Incomplete Information}, pages = {5--28}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2005}, volume = {5031}, editor = {Susanne Albers and Rolf H. M\"{o}hring and Georg Ch. Pflug and R\"{u}diger Schultz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05031.7}, URN = {urn:nbn:de:0030-drops-752}, doi = {10.4230/DagSemProc.05031.7}, annote = {Keywords: Competitive analysis , average case analysis , smoothed analysis , scheduling} }

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