What is discrete math? • The real numbers are continuous in the senses that: * between any two real numbers there is a real number • The integers do not share this property. In this sense the integers are lumpy, or " discrete " So discrete math is the study of mathematical objects that are discrete. " It's all the math that counts " Some discrete mathematical concepts: • Integers: Between two integers there is not another integer. • Propositions: Either true or false, there are no 1/2 truths (in math) • Sets: An item is either in a set or not in a set, never partly in and partly out. • Relations: A pair of items are related or not. • Networks (graphs): Between two terminals of a network connection there are no terminals. Propositional Logic Propositions are statements that are either true or false. Principles: Substituting an equivalent statement. Replacing a logic variable in a tautology. Defn algebraic proof

Discrete math = 離散数学

The manual identification and count of laminae in layered textures is a common practice in the study of geological records, which can be time consuming and carry large uncertainty for dense or disturbed lamina textures. We present here a novel image analysis approach to detect and count laminae in geoscientific imagery, called WlCount. Based on Dynamic Time Warping and Wavelet analysis, WlCount firstly aligns persistent vertical elements to increase the continuity of the lamina structure. Then, using a graphical interface, the user extracts the most significant signal frequencies and allows the automatic count of the laminae. The software, tested on a series of stalagmite cut images showing different types of laminations and a tree-ring image, provides an estimation of the laminae detection and count comparable to the manual one. WlCount presents as a useful open-source tool to help geoscientists, sensibly speeding up the lamination count process.

Computers and Geosciences

Theoretical Computer Science Volume 213-214

Hierarchical networks as fundamental models to describe the complex networks, have many applications in networks science, engineering technology and so on. In this paper, we first propose a new class of hierarchical networks with fractal structure, which are the networks with triangles compared to traditional hierarchical networks. Second, we study the precise results of some structural properties to derive small-world effect and scale-free feature. Third, it is found that the constructed network is sparse through the average degree and density. Fourth, it is also demonstrated that the degree distributions of hub nodes and the bottom nodes are the power law and exponential, respectively. Finally, we prove that clustering coefficient with a definite value [Formula: see text] tends to stabilize at a lower bound as [Formula: see text] iterates to a certain number, and the average distance of [Formula: see text] has an increasing relationship along with the value of [Formula: see text]. 

/pdf/analyses-of-some-structural-properties-on-a-class-of-1v1yp4pw.pdf

Analyses of some structural properties on a class of hierarchical scale-free networks

Covid-19 is a disease of the virus that is shaking the world and has been designated by WHO as a pandemic. This case of Covid-19 can be a place of dissemination of disinformation that can be utilized by some parties. The dissemination of information in this day and age has turned to the internet, namely social media, Twitter is one of the social media that is often used by Indonesians and the data can be analyzed. This study uses the social network analysis method, conducted to be able to find nodes that affect the ongoing interaction in the interaction network of information dissemination related to Covid-19 in Indonesia and see if the node is directly proportional to the value of its popularity. As well as to know in identifying the source of Covid-19 information, whether dominated by competent Twitter accounts in their fields. The data examined 19,939 nodes and 12,304 edges were taken from data provided by the web academic.droneemprit.id on the project "Analisis Opini Persebaran Virus Corona di Media Sosial", using the period of December 2019 to December 2020 on social media Twitter. The results showed that the @do_ra_dong account is an influential actor with the highest degree centrality of 860 and the @detikcom account is the actor with the highest popularity value of follower rank of 0.994741605. Thus actors who have a high degree of centrality value do not necessarily have a high follower rank value anyway. The study ignores if there are buzzer accounts on Twitter.

/pdf/analisis-akun-twitter-berpengaruh-terkait-covid-19-1onr9072h4.pdf

Analisis Akun Twitter Berpengaruh terkait Covid-19 menggunakan Social Network Analysis

The problem of how to clearer account for the significant evolutional mechanism gradually driving seemingly random complex networks into ones that have a few typical characters is of great interest in various science communities. Such discussed characters include small-world property, scale-free feature, and self-similarity, etc. However, most of previously generated models have no time memory, that is, the probability for each old vertex i of degree k i to obtain a new link from young vertex does only depend on its current degree. To address this issue, in this paper, we propose a class of prototype of evolving complex network models based on the additional feature, i.e., time memory. In contrast with presented models lacking of time memory, we can find that our model still follows power-law form of vertex-degree distribution and has almost the smallest diameter. More interestingly, this diameter of our model does not increase continuously with the development of model over time, but has discontinuity growth. From the generalization point of view, this thought behind the construction of our model can be thought of as a more general fashion in which a large number of published models can be rebuilt. With the validity of our model, it is worth to note that in the growth process of deterministic network models the number m of each existing vertex i obtaining new edges may be proportional to its degree k i , not always an integer multiple of k i . So as to describe and generate more available deterministic models as complex networks, we here introduce another parameter, vertex-velocity, corresponding to dynamic function on networks, which will be carefully discussed in this paper.

Generating Fibonacci-model as evolution of networks with vertex-velocity and time-memory

More attention has been paid to research of various kinds of fractals due to a great number of applications in different fields over the past years. In this paper, we present stochastic generalized Vicsek fractal networks to model underlying structures on many hyperbranched polymers. Then, we consider random walks, the widely-studied dynamical behavior, on stochastic models built in order to better understand structural properties. Specifically, we analytically derive the closed-form solution to mean first-passage time for random walks, which is a quantity that allows one to better understand the underlying structure of model in question, using a more effective approach compared with the typical computational methods including spectral technique. At the same time, some other fundamental structural parameters including Kirchhoff index and fractal dimension are obtained analytically as well. The results suggest that the fractal characteristic makes the underlying structure of network more loose, and thus leads the efficiency of delivering information in a random-walk-based manner to become lower. Finally, we conduct extensive simulations to demonstrate that theoretical analysis and numerical simulations are in perfect agreement with each other.

Random Walks on Stochastic Generalized Vicsek Fractal Networks: Analytic Solution and Simulations

Degree distribution, or equivalently called degree sequence, has been commonly used to be one of most significant measures for studying a large number of complex networks with which some well-known results have been obtained. By contrast, in this paper, we report a fact that two arbitrarily chosen networks with identical degree distribution can have completely different other topological structure, such as diameter, spanning trees number, pearson correlation coefficient, and so forth. Besides that, for a given degree distribution (as power-law distribution with exponent $\gamma=3$ discussed here), it is reasonable to ask how many network models with such a constraint we can have. To this end, we generate an ensemble of this kind of random graphs with $P(k)\sim k^{-\gamma}$ ($\gamma=3$), denoted as graph space $\mathcal{N}(p,q,t)$ where probability parameters $p$ and $q$ hold on $p+q=1$, and indirectly show the cardinality of $\mathcal{N}(p,q,t)$ seems to be large enough in the thermodynamics limit, i.e., $N\rightarrow\infty$, by varying values of $p$ and $q$. From the theoretical point of view, given an ultrasmall constant $p_{c}$, perhaps only graph model $N(1,0,t)$ is small-world and other are not in terms of diameter. And then, we study spanning trees number on two deterministic graph models and obtain both upper bound and lower bound for other members. Meanwhile, for arbitrary $p(\neq1)$, we prove that graph model $N(p,q,t)$ does go through two phase transitions over time, i.e., starting by non-assortative pattern and then suddenly going into disassortative region, and gradually converging to initial place (non-assortative point). Among of them, one "null" graph model is built.

An ensemble of random graphs with identical degree distribution

Here, we propose a simple algorithmic framework for creating power-law graphs with small diameters and then study structural properties, for instance, average degree, on graphs built. The results show that our graphs have not only some commonly seen properties including scale-free feature, small-world property, and disassortative structure, but also many rarely found characteristics, such as the density feature due to power-law exponents equal to 2 and the diameter equivalent to 2, compared to most previous scale-free models. In addition, we also consider the trapping problem on the proposed graphs and then find that they have more optimal trapping efficiency by means of their own average trapping times achieving the theoretical lower bound, a phenomenon that is seldom observed in existing scale-free models. We conduct extensive simulations, and the results show that empirical simulations are consistent with theoretical analysis.

Power-law graphs with small diameter: Framework, structural properties, and average trapping time.

Geodesic distance, sometimes called shortest path length, has proven useful in a great variety of applications, such as information retrieval on networks including treelike networked models. Here, our goal is to analytically determine the exact solutions to geodesic distances on two different families of growth trees which are recursively created upon an arbitrary tree $\mathcal{T}$ using two types of well-known operations, first-order subdivision and ($1,m$)-star-fractal operation. Different from commonly-used methods, for instance, spectral techniques, for addressing such a problem on growth trees using a single edge as seed in the literature, we propose a novel method for deriving closed-form solutions on the presented trees completely. Meanwhile, our technique is more general and convenient to implement compared to those previous methods mainly because there are not complicated calculations needed. In addition, the closed-form expression of mean first-passage time ($MFPT$) for random walk on each member in tree families is also readily obtained according to connection of our obtained results to effective resistance of corresponding electric networks. The results suggest that the two topological operations above are sharply different from each other due to $MFPT$ for random walks, and, however, have likely to show the similar performance, at least, on geodesic distance.

pdf/a-method-for-geodesic-distance-on-subdivision-of-trees-with-1x8ro2bjhz.pdf

A Method for Geodesic Distance on Subdivision of Trees with Arbitrary Orders and Their Applications

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