26,878 research outputs found
Motif Counting in Preferential Attachment Graphs
No full textNetwork motifs are small patterns that occur in a network significantly more often than expected. They have gathered a lot of interest, as they may describe functional dependencies of complex networks and yield insights into their basic structure [Milo et al., 2002]. Therefore, a large amount of work went into the development of methods for network motif detection in complex networks [Kashtan et al., 2004; Schreiber and Schwöbbermeyer, 2005; Chen et al., 2006; Wernicke, 2006; Grochow and Kellis, 2007; Alon et al., 2008; Omidi et al., 2009]. The underlying problem of motif detection is to count how often a copy of a pattern graph H occurs in a target graph G. This problem is #W[1]-hard when parameterized by the size of H [Flum and Grohe, 2004] and cannot be solved in time f(|H|)n^o(|H|) under #ETH [Chen et al., 2005].
Preferential attachment graphs [Barabási and Albert, 1999] are a very popular random graph model designed to mimic complex networks. They are constructed by a random process that iteratively adds vertices and attaches them preferentially to vertices that already have high degree. Preferential attachment has been empirically observed in real growing networks [Newman, 2001; Jeong et al., 2003].
We show that one can count subgraph copies of a graph H in the preferential attachment graph G^n_m (with n vertices and nm edges, where m is usually a small constant) in expected time f(|H|) m^O(|H|^6) log(n)^O(|H|^12) n. This means the motif counting problem can be solved in expected quasilinear FPT time on preferential attachment graphs with respect to the parameters |H| and m. In particular, for fixed H and m the expected run time is O(n^(1+epsilon)) for every epsilon>0.
Our results are obtained using new concentration bounds for degrees in preferential attachment graphs. Assume the (total) degree of a set of vertices at a time t of the random process is d. We show that if d is sufficiently large then the degree of the same set at a later time n is likely to be in the interval (1 +/- epsilon)d sqrt(n/t) (for epsilon > 0) for all n >= t. More specifically, the probability that this interval is left is exponentially small in d
Author Peter FitzSimons speaking at the National Library of Australia, Canberra, 13 November 2012 /
No full textTitle from acquisitions documentation.; Part of the collection: Portraits of author Peter FitzSimons speaking at the National Library of Australia, Canberra, 13 November 2012.; Acquired in digital format; access copy available online.; Mode of access: Online.; Photographed by a staff member of the National Library of Australia
Hardness of FO Model-Checking on Random Graphs
No full textIt is known that FO model-checking is fixed-parameter tractable on Erdős - Rényi graphs G(n,p(n)) if the edge-probability p(n) is sufficiently small [Grohe, 2001] (p(n)=O(n^epsilon/n) for every epsilon>0). A natural question to ask is whether this result can be extended to bigger probabilities. We show that for Erdős - Rényi graphs with vertex colors the above stated upper bound by Grohe is the best possible.
More specifically, we show that there is no FO model-checking algorithm with average FPT run time on vertex-colored Erdős - Rényi graphs G(n,n^delta/n) (0 < delta < 1) unless AW[*]subseteq FPT/poly. This might be the first result where parameterized average-case intractability of a natural problem with a natural probability distribution is linked to worst-case complexity assumptions.
We further provide hardness results for FO model-checking on other random graph models, including G(n,1/2) and Chung-Lu graphs, where our intractability results tightly match known tractability results [E. D. Demaine et al., 2014]. We also provide lower bounds on the size of shallow clique minors in certain Erdős - Rényi and Chung - Lu graphs
Moral Good, the Beatific Vision, and God’s Kingdom Writings by Germain Grisez and Peter Ryan, S.J.. Edited by Peter J. Weigel
No full textFor close to half a century, the work of Germain Grisez has been highly influential, and his writings continue to receive considerable attention from philosophers and theologians of diverse viewpoints. His co-author for this work is the professor and noted moral theologian Fr. Peter Ryan, S.J., currently the executive director of the Secretariat of Doctrine and Canonical Affairs of the United States Conference of Catholic Bishops (USCCB). These two eminent scholars explore fundamental questions about Christian eschatology, moral theory, the purpose of human life, and the promise of human fulfilment. The authors examine Christian teaching on the final destiny of persons, investigating the meaning of God's kingdom, the hope of the beatific vision, and the centrality of moral goodness and divine grace in one's final end. This work is an ideal source for students, scholars, ministers and lay persons interested in basic questions of Christian theology, the philosophy of religion, ethical theory, and Catholic doctrin
Murder on the mountain: author talk with Peter J. Wosh
No full textAuthor talk by Peter J. Wosh on May 5th, 2022, on his book, "Murder on the Mountain: crime, passion, and punishment in gilded age New Jersey.
Maximum Shallow Clique Minors in Preferential Attachment Graphs Have Polylogarithmic Size
No full textPreferential attachment graphs are random graphs designed to mimic properties of real word networks. They are constructed by a random process that iteratively adds vertices and attaches them preferentially to vertices that already have high degree. We prove various structural asymptotic properties of this graph model. In particular, we show that the size of the largest r-shallow clique minor in Gⁿ_m is at most log(n)^{O(r²)}m^{O(r)}. Furthermore, there exists a one-subdivided clique of size log(n)^{1/4}. Therefore, preferential attachment graphs are asymptotically almost surely somewhere dense and algorithmic techniques developed for structurally sparse graph classes are not directly applicable. However, they are just barely somewhere dense. The removal of just slightly more than a polylogarithmic number of vertices asymptotically almost surely yields a graph with locally bounded treewidth
Lunchtime Talk with Author and Attorney Peter Godwin
No full textAuthor and attorney Peter Godwin gave a lunchtime talk about the topics discussed in his book, The Fear, which focuses on the human rights situation in Zimbabwe under the rule of Robert Mugabe
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