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### Efficiently Constructible Huge Graphs That Preserve First Order Properties of Random Graphs [chapter]

Moni Naor, Asaf Nussboim, Eran Tromer
2005 Lecture Notes in Computer Science
We also demonstrate that 0/1 laws do not hold for random graphs w.r.t. properties of significantly larger quantifier depth.  ...  Random graphs are known to have remarkable structure w.r.t. first order properties, as indicated by the following 0/1 law: for a variety of choices of p(n), any fixed first-order property φ holds for G  ...  Although we can provide graphs satisfying D(n)-0/1 laws without achieving quasi-randomness, it is not clear whether D(n)-0/1 laws combined with D(n)equivalence to random graphs implies quasi-randomness  ...

### On failure of 0-1 laws [article]

Saharon Shelah
2021 arXiv   pre-print
Let α∈ (0,1)_ℝ be irrational and G_n = G_n,1/n^α be the random graph with edge probability 1/n^α; we know that it satisfies the 0-1 law for first order logic.  ...  We deal with the failure of the 0-1 law for stronger logics: 𝕃_∞,k,k large enough and the inductive logic.  ...  The 0-1 law holds if α is irrational, but we have elimination of quantifiers only up to (Boolean combination of) existential formulas. Do we have 0-1 law also for those stronger logics?  ...

### Zero-One Laws [chapter]

Leonid Libkin
2004 Elements of Finite Model Theory
do not.  ...  As one step in doing this, we will construct a model for EA called the random graph.  ...

### A Zero-One Law for First-Order Logic on Random Images [chapter]

David Coupier, Agnès Desolneux, Bernard Ycart
2004 Mathematics and Computer Science III
For an n×n random image with independent pixels, black with probability p(n) and white with probability 1−p(n), the probability of satisfying any given firstorder sentence tends to 0 or 1, provided both  ...  p(n)n 2 k and (1 − p(n))n 2 k tend to 0 or +∞, for any integer k.  ...  - Fagin theorem, the zero-one law holds for first-order propositions on random digraphs.  ...

### Pursuit-Evasion in Models of Complex Networks

Anthony Bonato, Paweł Prałat, Changping Wang
2007 Internet Mathematics
The cop number of the core of random power law graphs is investigated, and is proved to be of smaller order than the order of the core. 1991 Mathematics Subject Classification. 05C80, 68R10, 94C15.  ...  We find bounds for the cop number of G(n, p) for a large range of p as a function of n. We prove that the cop number of random power law graphs with n vertices is asymptotically almost surely Θ(n).  ...  For a random power law graph G ∈ G(w) with power law exponent β ∈ (2, 3) a.a.s. the cop number of the core G of G satisfies N (1+o(1))(3−β)/ log log N ≤ c( G) ≤ N 1−(1+o(1))(β−1)(3−β)/(β−2) log log n .  ...

### Logical properties of random graphs from small addable classes [article]

Anuj Dawar, Eryk Kopczyński
2019 arXiv   pre-print
We establish zero-one laws and convergence laws for monadic second-order logic (MSO) (and, a fortiori, first-order logic) on a number of interesting graph classes.  ...  In particular, we show that MSO obeys a zero-one law on the class of connected planar graphs, the class of connected graphs of tree-width at most k and the class of connected graphs excluding the k-clique  ...  Moreover, a limit law holds for random graphs of such classes which do not have to be connected.  ...

### Locally interacting diffusions as space-time Markov random fields [article]

Daniel Lacker, Kavita Ramanan, Ruoyu Wu
2020 arXiv   pre-print
of histories of the processes at different vertices forms a second-order Markov random field on path space.  ...  Under general conditions on the coefficients, we show that if the initial conditions form a second-order Markov random field on d-dimensional Euclidean space, then at any positive time, the collection  ...  Precisely, even on a finite graph with gradient drift, in general the collection of histories (X v [t]) v∈V do not form a first-order Markov random field, nor do the time-t marginals (X v (t)) v∈V exhibit  ...

### Convergence law for hyper-graphs with prescribed degree sequences [article]

Nans Lefebvre
2015 arXiv   pre-print
We develop the logical analysis of this framework and first prove a convergence law for first-order logic, then characterise the limit first-order theories defined by a wide class of degree distributions  ...  Convergence laws of other models follow, and in particular for the classical Erdős-Rényi graphs and k-uniform hyper-graphs.  ...  The power laws with exponent smaller than 3 do not have a sublinear fourth moment so do not satisfy condition (iii).  ...

### The Small World Phenomenonin Hybrid Power Law Graphs [chapter]

Fan Chung, Linyuan Lu
2004 Lecture Notes in Physics
The global graph is modeled by a random graph with a power law degree distribution, while the local graph has specified local connectivity.  ...  We consider a hybrid graph model that incorporates both properties by combining a global graph and a local graph.  ...  In Kleinberg's model and the model of Watts and Strogatz, the graphs have the same expected degree at every node and do not have a power law degree distribution.  ...

### Conditions and Assumptions for Constraint-based Causal Structure Learning [article]

2022 arXiv   pre-print
We also provide a set of assumptions, under which any natural structure-learning algorithm outputs Markov equivalent graphs to the causal graph.  ...  We provide conditions for a "natural" family of constraint-based structure-learning algorithms that output graphs that are Markov equivalent to the causal graph.  ...  to Steffen Lauritzen for providing Example 18, to Joris Mooij for stimulating discussions with the first author especially on the examples involving modular arithmetic, and to Jonas Peters for hosting a  ...

### Asymptotic Theories of Classes Defined by Forbidden Homomorphisms [article]

Manuel Bodirsky, Colin Jahel
2022 arXiv   pre-print
We study the first-order almost-sure theories for classes of finite structures that are specified by homomorphically forbidding a set ℱ of finite structures.  ...  In our proof, we establish a result of independent interest, namely that every constraint satisfaction problem for a finite digraph has first-order convergence, and that the corresponding asymptotic theory  ...  As demonstrated in Example 4, these classes do not always have a first-order 0-1 law.  ...

### Spectra of random graphs with given expected degrees

F. Chung, L. Lu, V. Vu
2003 Proceedings of the National Academy of Sciences of the United States of America
We will prove that (under certain mild conditions) the eigenvalues of the (normalized) Laplacian of a random power law graph follow the semi-circle law while the spectrum of the adjacency matrix of a power  ...  law graph obeys the power law.  ...  For a random power law graph with exponent β > 2.5, the largest eigenvalue of a random power law graph is almost surely (1 + o(1)) √ m where m is the maximum degree.  ...

### The Spectra of Random Graphs with Given Expected Degrees

Fan Chung, Linyuan Lu, Van Vu
2004 Internet Mathematics
We will prove that (under certain mild conditions) the eigenvalues of the (normalized) Laplacian of a random power law graph follow the semi-circle law while the spectrum of the adjacency matrix of a power  ...  law graph obeys the power law.  ...  For a random power law graph with exponent β > 2.5, the largest eigenvalue of a random power law graph is almost surely (1 + o(1)) √ m where m is the maximum degree.  ...

### The logic of random regular graphs

Simi Haber, Michael Krivelevich
2010 Journal of Combinatorics
On the contrary, if d = n α for rational 0 < α < 1, then there is a (theoretically explicit) first order property with no limiting probability in G n,d .  ...  In particular we prove that if the degree d is linear in the number of vertices n, or if d = n α for 0 < α < 1 irrational, then the random d-regular graph G n,d obeys the Zero-One law.  ...  If AST(G(n)) is complete we say that G satisfies the Zero-One law. In other words, if for every first order property A one has lim n→∞ Pr[G(n) |= A] ∈ {0, 1} then G satisfies the Zero-One law.  ...

### Greed is Good for Deterministic Scale-Free Networks

Ankit Chauhan, Tobias Friedrich, Ralf Rothenberger
2020 Algorithmica
Chung-Lu Random Graphs satisfy PLB-(U,N): Chung-Lu Random Graphs (CLRGs) [18] assume a sequence of expected degrees w 1 , w 2 , ... , w n and each edge (i, j) exists independently at random with probability  ...  Theorem 4.14 Let G be a HRG with H > 1 2 . Then, G almost surely fulfills PLB-U and PLB-N with = 2 H + 1 − , t = 0 , any constant > 0 , and some constants c 1 and c 3 .  ...  Therefore, ( , )-Power Law Graphs are too constrained and do not capture most real networks. To allow for those deviations in the degree distribution Brach et al.  ...
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