scholarly journals A Note on the Warmth of Random Graphs with Given Expected Degrees

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Statistical Mechanics ◽  
Chromatic Number ◽  
Random Graph ◽  
Lower Bounds ◽  
Random Graphs ◽  
Upper Bound ◽  
Degree Sequence ◽  
Graph Model ◽  
Upper And Lower Bounds ◽  
Random Graph Model

We consider the random graph modelG(w)for a given expected degree sequencew=(w1,w2,…,wn). Warmth, introduced by Brightwell and Winkler in the context of combinatorial statistical mechanics, is a graph parameter related to lower bounds of chromatic number. We present new upper and lower bounds on warmth ofG(w). In particular, the minimum expected degree turns out to be an upper bound of warmth when it tends to infinity and the maximum expected degreem=O(nα)with0<α<1/2.

Random graphs

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Large Scale ◽  
Random Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Random Graph Model ◽  
Definition Of ◽  
Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

The Impact of Edge Correlations in Random Networks

2021 ◽  
Vol 30 (4) ◽  
pp. 525-537
Author(s):  
András Faragó ◽  
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Real Life ◽  
Graph Model ◽  
Random Networks ◽  
Random Graph Model ◽  
Special Cases ◽  
Graph Parameters ◽  
The Impact ◽  
Good Number

Random graphs are frequently used models of real-life random networks. The classical Erdös–Rényi random graph model is very well explored and has numerous nontrivial properties. In particular, a good number of important graph parameters that are hard to compute in the deterministic case often become much easier in random graphs. However, a fundamental restriction in the Erdös–Rényi random graph is that the edges are required to be probabilistically independent. This is a severe restriction, which does not hold in most real-life networks. We consider more general random graphs in which the edges may be dependent. Specifically, two models are analyzed. The first one is called a p-robust random graph. It is defined by the requirement that each edge exist with probability at least p, no matter how we condition on the presence/absence of other edges. It is significantly more general than assuming independent edges existing with probability p, as exemplified via several special cases. The second model considers the case when the edges are positively correlated, which means that the edge probability is at least p for each edge, no matter how we condition on the presence of other edges (but absence is not considered). We prove some interesting, nontrivial properties about both models.

scholarly journals On Komlós’ tiling theorem in random graphs

2019 ◽  
Vol 29 (1) ◽  
pp. 113-127
Author(s):  
Rajko Nenadov ◽  
Nemanja Škorić
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Random Graphs ◽  
Upper Bound ◽  
Minimum Degree ◽  
Arbitrary Graph ◽  
Spanning Subgraph ◽  
Delta G ◽  
Image Position ◽  
Vertex Disjoint

AbstractGiven graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$ contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most $\gamma {(C/p)^{{m_2}(H)}}$ vertices, which is strictly smaller when $p \ge C{n^{ - 1/{m_2}(H)}}$ . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.

The Largest Component in Subcritical Inhomogeneous Random Graphs

2010 ◽  
Vol 20 (1) ◽  
pp. 131-154 ◽  
Author(s):  
TATYANA S. TUROVA
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Graph Model ◽  
Exact Formula ◽  
Connected Component ◽  
Subcritical Case ◽  
Random Graph Model ◽  
Asymptotic Size ◽  
Inhomogeneous Random Graphs ◽  
Inhomogeneous Random Graph

We study the ‘rank 1 case’ of the inhomogeneous random graph model. In the subcritical case we derive an exact formula for the asymptotic size of the largest connected component scaled to log n. This result complements the corresponding known result in the supercritical case. We provide some examples of applications of the derived formula.

scholarly journals Analysis of E-Commerce Product Graphs

2020 ◽  
Author(s):  
Shalin Shah
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
Graph Model ◽  
Graph Analysis ◽  
Random Graph Model ◽  
Product Graphs ◽  
Clustering Coefficients

<p>Consumer behavior in retail stores gives rise to product graphs based on copurchasing</p><p>or co-viewing behavior. These product graphs can be analyzed using</p><p>the known methods of graph analysis. In this paper, we analyze the product graph</p><p>at Target Corporation based on the Erd˝os-Renyi random graph model. In particular,</p><p>we compute clustering coefficients of actual and random graphs, and we find that</p><p>the clustering coefficients of actual graphs are much higher than random graphs.</p><p>We conduct the analysis on the entire set of products and also on a per category</p><p>basis and find interesting results. We also compute the degree distribution and</p><p>we find that the degree distribution is a power law as expected from real world</p><p>networks, contrasting with the ER random graph.</p>

scholarly journals Analysis of E-Commerce Product Graphs

2020 ◽  
Author(s):  
Shalin Shah
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
Graph Model ◽  
Graph Analysis ◽  
Random Graph Model ◽  
Product Graphs ◽  
Clustering Coefficients

<p>Consumer behavior in retail stores gives rise to product graphs based on copurchasing</p><p>or co-viewing behavior. These product graphs can be analyzed using</p><p>the known methods of graph analysis. In this paper, we analyze the product graph</p><p>at Target Corporation based on the Erd˝os-Renyi random graph model. In particular,</p><p>we compute clustering coefficients of actual and random graphs, and we find that</p><p>the clustering coefficients of actual graphs are much higher than random graphs.</p><p>We conduct the analysis on the entire set of products and also on a per category</p><p>basis and find interesting results. We also compute the degree distribution and</p><p>we find that the degree distribution is a power law as expected from real world</p><p>networks, contrasting with the ER random graph.</p>

scholarly journals On the Number of Non-Zero Elements of Joint Degree Vectors

10.37236/6385 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Éva Czabarka ◽  
Johannes Rauh ◽  
Kayvan Sadeghi ◽  
Taylor Short ◽  
László Székely
Keyword(s):  
Random Graph ◽  
Upper Bound ◽  
Graph Model ◽  
Upper Bounds ◽  
Exponential Random Graph Model ◽  
Lower And Upper Bounds ◽  
Sufficient Statistics ◽  
Random Graph Model ◽  
Exponential Random Graph ◽  
Vertex Graph

Joint degree vectors give the number of edges between vertices of degree $i$ and degree $j$ for $1\le i\le j\le n-1$ in an $n$-vertex graph. We find lower and upper bounds for the maximum number of nonzero elements in a joint degree vector as a function of $n$. This provides an upper bound on the number of estimable parameters in the exponential random graph model with bidegree-distribution as its sufficient statistics.

scholarly journals The natural connectivity of colored random graphs

2011 ◽  
Vol 20 (2) ◽  
pp. 197-202
Author(s):  
YILUN SHANG ◽  
Keyword(s):  
Complex Network ◽  
Random Graph ◽  
Random Graphs ◽  
Graph Model ◽  
Graph Spectrum ◽  
Random Graph Model ◽  
Robustness Measure ◽  
Theoretical Results ◽  
Inhomogeneous Random Graph

The natural connectivity as a robustness measure of complex network has been proposed recently. It can be regarded as the average eigenvalue obtained from the graph spectrum. In this paper, we introduce an inhomogeneous random graph model, G(n, {ci}, {pi}), and investigate its natural connectivity. Binomial random graph ... . Simulations are performed to validate our theoretical results.

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