Although there are some techniques for dealing with sparse and concentrated discrete data, standard time-series analyses appear ill-suited to understanding the temporal patterns of terrorist attacks ...due to the sparsity of the events. This article addresses these issues by proposing a novel technique for analysing low-frequency temporal events, such as terrorism, based on their cumulative curve and corresponding gradients. Using an iterative algorithm based on a piecewise linear function, our technique detects trends and shocks observed in the events associated with terrorist groups that would not necessarily be visible using other methods. The analysis leverages disaggregated data on political violence from the Armed Conflict Location & Event Data Project (ACLED) to analyse the intensity of the two most violent terrorist organisations in Africa: Boko Haram (including its splinter group, the Islamic State West Africa Province), and Al-Shabaab. Our method detects moments when terrorist groups change their capabilities to conduct daily attacks and, by taking into account the directionality of attacks, highlights major changes in the government’s strategies. Results suggest that security policies have largely failed to reduce both groups’ forces and restore stability.
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DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, SIK, UILJ, UKNU, UL, UM, UPUK
Both results come from considering the hard-core model from statistical physics: a random independent set I drawn from a graph with probability proportional to \lambda ^{\vert I\vert}, for a fugacity ...parameter \lambda >> We conclude by stating several conjectures on the relationship between the average and maximum size of an independent set in a triangle-free graph and give some consequences of these conjectures in Ramsey theory.>
We give tight upper and lower bounds on the internal energy per particle in the antiferromagnetic $q$-state Potts model on $4$-regular graphs, for $q\ge 5$. This proves the first case of a conjecture ...of the author, Perkins, Jenssen and Roberts, and implies tight bounds on the antiferromagnetic Potts partition function. The zero-temperature limit gives upper and lower bounds on the number of proper $q$-colourings of $4$-regular graphs, which almost proves the case $d=4$ of a conjecture of Galvin and Tetali. For any $q \ge 5$ we prove that the number of proper $q$-colourings of a $4$-regular graph is maximised by a union of $K_{4,4}$'s.
We prove that the ‘Upper Matching Conjecture’ of Friedland, Krop, and Markström and the analogous conjecture of Kahn for independent sets in regular graphs hold for all large enough graphs as a ...function of the degree. That is, for every d and every large enough n divisible by 2d, a union of n/(2d) copies of the complete d-regular bipartite graph maximizes the number of independent sets and matchings of size k for each k over all d-regular graphs on n vertices. To prove this we utilize the cluster expansion for the canonical ensemble of a statistical physics spin model, and we give some further applications of this method to maximizing and minimizing the number of independent sets and matchings of a given size in regular graphs of a given minimum girth.
Independent sets, matchings, and occupancy fractions Davies, Ewan; Jenssen, Matthew; Perkins, Will ...
Journal of the London Mathematical Society,
August 2017, 2017-08-00, Letnik:
96, Številka:
1
Journal Article
Recenzirano
We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d‐regular graphs. For independent sets, this theorem is a strengthening of the results of ...Kahn, Galvin and Tetali, and Zhao showing that a union of copies of Kd,d maximizes the number of independent sets and the independence polynomial of a d‐regular graph.
For matchings, this shows that the matching polynomial and the total number of matchings of a d‐regular graph are maximized by a union of copies of Kd,d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markström.
In probabilistic language, our main theorems state that for all d‐regular graphs and all λ, the occupancy fraction of the hard‐core model and the edge occupancy fraction of the monomer‐dimer model with fugacity λ are maximized by Kd,d. Our method involves constrained optimization problems over distributions of random variables and applies to all d‐regular graphs directly, without a reduction to the bipartite case.
We develop a theory of regularity inheritance in 3-uniform hypergraphs. As a simple consequence we deduce a strengthening of a counting lemma of Frankl and Rödl. We believe that the approach is ...sufficiently flexible and general to permit extensions of our results in the direction of a hypergraph blow-up lemma.
We prove tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the anti‐ferromagnetic Potts model on cubic graphs at every ...temperature and for all q≥2. This immediately implies corresponding tight bounds on the anti‐ferromagnetic Potts partition function. Taking the zero‐temperature limit gives new results in extremal combinatorics: the number of q‐colorings of a 3‐regular graph, for any q≥2, is maximized by a union of K3,3's. This proves the d = 3 case of a conjecture of Galvin and Tetali.
A robust Corrádi–Hajnal theorem Allen, Peter; Böttcher, Julia; Corsten, Jan ...
Random structures & algorithms,
August 2024, Letnik:
65, Številka:
1
Journal Article
Recenzirano
Odprti dostop
For a graph G$$ G $$ and p∈0,1$$ p\in \left0,1\right $$, we denote by Gp$$ {G}_p $$ the random sparsification of G$$ G $$ obtained by keeping each edge of G$$ G $$ independently, with probability p$$ ...p $$. We show that there exists a C>0$$ C>0 $$ such that if p≥C(logn)1/3n−2/3$$ p\ge C{\left(\log n\right)}^{1/3}{n}^{-2/3} $$ and G$$ G $$ is an n$$ n $$‐vertex graph with n∈3ℕ$$ n\in 3\mathbb{N} $$ and δ(G)≥2n3$$ \delta (G)\ge \frac{2n}{3} $$, then with high probability Gp$$ {G}_p $$ contains a triangle factor. Both the minimum degree condition and the probability condition, up to the choice of C$$ C $$, are tight. Our result can be viewed as a common strengthening of the seminal theorems of Corrádi and Hajnal, which deals with the extremal minimum degree condition for containing triangle factors (corresponding to p=1$$ p=1 $$ in our result), and Johansson, Kahn and Vu, which deals with the threshold for the appearance of a triangle factor in G(n,p)$$ G\left(n,p\right) $$ (corresponding to G=Kn$$ G={K}_n $$ in our result). It also implies a lower bound on the number of triangle factors in graphs with minimum degree at least 2n3$$ \frac{2n}{3} $$ which gets close to the truth.
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